FLUID DYNAMICS IN HORIZONTAL CYLINDRICAL

FLUID DYNAMICS IN HORIZONTAL CYLINDRICAL

CONTAINERS AND

LIQUID CARGO VEHICLE DYNAMICS

A Dissertation

Submitted to the Faculty of Graduate Studies and Research

In Partial Fulfillment of the Requirements

for the Degree of

Doctor of Philosophy

In Engineering

University of Regina

By

Liang Xu

Regina, Saskatchewan

December 2005

Copyright 2005: Liang Xu

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

FLUID DYNAMICS IN HORIZONTAL CYLINDRICAL

CONTAINERS AND

LIQUID CARGO VEHICLE DYNAMICS

A Dissertation

Submitted to the Faculty of Graduate Studies and Research

In Partial Fulfillment of the Requirements

for the Degree of

Doctor of Philosophy

In Engineering

University of Regina

By

Liang Xu

Regina, Saskatchewan

December 2005

Copyright 2005: Liang Xu


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Bibliotheque et Archives Canada

Published Heritage Branch

395 Wellington Street Ottawa ON K1A 0N4 Canada

Your file Votre reference ISBN: 978-0-494-18871-2 Our file Notre reference ISBN: 978-0-494-18871-2

Direction du Patrimoine de I'edition

395, rue Wellington Ottawa ON K1A 0N4 Canada

NOTICE:The author has granted a nonexclusive license allowing Library and Archives Canada to reproduce, publish, archive, preserve, conserve, communicate to the public by telecommunication or on the Internet, loan, distribute and sell theses worldwide, for commercial or noncommercial purposes, in microform, paper, electronic and/or any other formats.

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The author retains copyright ownership and moral rights in this thesis. Neither the thesis nor substantial extracts from it may be printed or otherwise reproduced without the author's permission.

L'auteur conserve la propriete du droit d'auteur et des droits moraux qui protege cette these.Ni la these ni des extraits substantiels de celle-ci ne doivent etre imprimes ou autrement reproduits sans son autorisation.

In compliance with the Canadian Privacy Act some supporting forms may have been removed from this thesis.

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CanadaReproduced with permission of the copyright owner. Further reproduction prohibited without permission.

UNIVERSITY OF REGINA

FACULTY OF GRADUATE STUDIES AND RESEARCH

SUPERVISORY AND EXAMINING COMMITTEE

Liang Xu, candidate for the degree of Doctor of Philosophy, has presented a thesis titled,

Fluid Dynamics in Horizontal Cylindrical Containers and Liquid Cargo Vehicle Dynamics,

in an oral examination held on November 25, 2005. The following committee members have

found the thesis acceptable in form and content, and that the candidate demonstrated

satisfactory knowledge of the subject material.

External Examiner: Dr. Pei Yu, University of Western Ontario

Supervisor: Dr. Liming Dai, Faculty of Engineering

Committee Member: Dr. Adisorn Aroonwilas, Faculty of Engineering

Committee Member: Dr. Nader Mobed, Department of Physics

Committee Member: Dr. Jing Tao Yao, Department of Computer Science

Chair of Defense: Dr. David Malloy, Faculty of Graduate Studies and Research


UNIVERSITY OF REGINA

FACULTY OF GRADUATE STUDIES AND RESEARCH

SUPERVISORY AND EXAMINING COMMITTEE

Liang Xu, candidate for the degree o f Doctor o f Philosophy, has presented a thesis titled,

Fluid Dynamics in Horizontal Cylindrical Containers and Liquid Cargo Vehicle Dynamics,

in an oral examination held on November 25, 2005. The following committee members have

found the thesis acceptable in form and content, and that the candidate demonstrated

satisfactory knowledge o f the subject material.

External Examiner: Dr. Pei Yu, University of Western Ontario

Supervisor: Dr. Liming Dai, Faculty of Engineering

Committee Member: Dr. Adisom Aroonwilas, Faculty of Engineering

Committee Member: Dr. Nader Mobed, Department of Physics

Committee Member: Dr. Jing Tao Yao, Department of Computer Science

Chair o f Defense: Dr. David Malloy, Faculty of Graduate Studies and Research


ABSTRACT

A new mathematical method especially for liquid motion in horizontal cylindrical

tanks has been developed to investigate the fluid dynamics inside liquid cargo tank

vehicles. The governing equations based on potential flow theory are rearranged by

continuous coordinate mappings in such a way that the difficulties of direct discretization

for numerical calculation are avoided. Corresponding numerical procedures have been

established for sloshing problems in 2D partially filled road tanks to study transient

lateral liquid responses under turning, lane change and double lane change manoeuvres.

The newly developed method has been extended to solve dynamic liquid

behaviour in partially filled 3D horizontal cylindrical tanks in a completely 3D manner.

The transient longitudinal liquid motion and corresponding liquid forces and moments

have been calculated for the tanks subjected to longitudinal acceleration input during the

accelerating/braking operations. The influence of different accelerations, fill levels,

hemispherical heads, the configuration of compartmented tanks and liquid distribution

has been analyzed in detail in different situations. This methodology can be used for road

tanks of arbitrarily shaped walls. It can also be easily integrated into coupled liquid-

structure systems to systematically study the dynamics of vehicle systems subjected to

liquid sloshing and other loadings.

Longitudinal liquid cargo vehicle dynamics has been investigated by equivalent

mechanical models for two cases. The ride performance of partially filled compartmented

tank vehicles has been investigated by using a linearized multi-degree-of-freedom

dynamic model. The liquid motion in the partially filled tank is described as a linear

I


ABSTRACT

A new mathematical method especially for liquid motion in horizontal cylindrical

tanks has been developed to investigate the fluid dynamics inside liquid cargo tank

vehicles. The governing equations based on potential flow theory are rearranged by

continuous coordinate mappings in such a way that the difficulties o f direct discretization

for numerical calculation are avoided. Corresponding numerical procedures have been

established for sloshing problems in 2D partially filled road tanks to study transient

lateral liquid responses under turning, lane change and double lane change manoeuvres.

The newly developed method has been extended to solve dynamic liquid

behaviour in partially filled 3D horizontal cylindrical tanks in a completely 3D manner.

The transient longitudinal liquid motion and corresponding liquid forces and moments

have been calculated for the tanks subjected to longitudinal acceleration input during the

accelerating/braking operations. The influence of different accelerations, fill levels,

hemispherical heads, the configuration of compartmented tanks and liquid distribution

has been analyzed in detail in different situations. This methodology can be used for road

tanks o f arbitrarily shaped walls. It can also be easily integrated into coupled liquid-

structure systems to systematically study the dynamics o f vehicle systems subjected to

liquid sloshing and other loadings.

Longitudinal liquid cargo vehicle dynamics has been investigated by equivalent

mechanical models for two cases. The ride performance of partially filled compartmented

tank vehicles has been investigated by using a linearized multi-degree-of-ffeedom

dynamic model. The liquid motion in the partially filled tank is described as a linear

I


spring-mass model. The power spectral density of seat accelerations has been utilized to

study the influence of liquid motion on the ride quality under different conditions,

including fill levels, vehicle speeds, road conditions, and types of liquid being carried. A

nonlinear impact mechanical system that describes the liquid motion as a linear spring-

mass system with an impact subsystem has been developed to investigate the longitudinal

dynamic behaviour of partially filled tank vehicles under rough road conditions.

The established methodology will provide a useful tool for researchers, in

performing investigations on liquid behaviour and dynamics of liquid-vehicle systems

with horizontal cylindrical tanks. The research results will also benefit engineers in

vehicle structure designing and manufacturing.

II


spring-mass model. The power spectral density of seat accelerations has been utilized to

study the influence o f liquid motion on the ride quality under different conditions,

including fill levels, vehicle speeds, road conditions, and types o f liquid being carried. A

nonlinear impact mechanical system that describes the liquid motion as a linear spring-

mass system with an impact subsystem has been developed to investigate the longitudinal


The established methodology will provide a useful tool for researchers, in

performing investigations on liquid behaviour and dynamics o f liquid-vehicle systems

with horizontal cylindrical tanks. The research results will also benefit engineers in

vehicle structure designing and manufacturing.

II


ACKNOWLEDGEMENTS

The author wishes to express his sincere appreciation to his Ph.D. supervisor, Dr.

Liming Dai, for his guidance throughout the course of this investigation. Dr. Dai's

patience, encouragement and financial support are crucial to the successful completion of

this research endeavour.

The author is grateful to Dr. Mehran Mehrandezh, Dr. Mingzhe Dong, Dr. Andy

Aroonwilas, Dr. Jing Tao Yao and Dr. Nader Mobed for their guidance and helps during

his study and thesis work. The helps provided by Mr. Robert D. Jones are significant for

the numerical computation of the author's research. Thanks are also due to faculty, staff

and colleagues for their contributions to this research.

The author also wishes to acknowledge the Faculty of Graduate Studies and

Research for the financial support provided in the form of Graduate Scholarships, the

Sampson J. Goodfellow Scholarship, the John Spencer Middleton & Jack Spencer

Gordon Scholarship, and the Teaching Fellowship. The work opportunities as a sessional

lecturer and teaching assistant provided by the Faculty of Engineering at the University

of Regina are also highly appreciated.

Finally, the author would like to express his special thanks to his parents, his

wife's parents, his wife and children for their continuous encouragement and support.

III


ACKNOWLEDGEMENTS

The author wishes to express his sincere appreciation to his Ph.D. supervisor, Dr.

Liming Dai, for his guidance throughout the course o f this investigation. Dr. Dai’s

patience, encouragement and financial support are crucial to the successful completion of

this research endeavour.

The author is grateful to Dr. Mehran Mehrandezh, Dr. Mingzhe Dong, Dr. Andy

Aroonwilas, Dr. Jing Tao Yao and Dr. Nader Mobed for their guidance and helps during

his study and thesis work. The helps provided by Mr. Robert D. Jones are significant for

the numerical computation o f the author’s research. Thanks are also due to faculty, staff

and colleagues for their contributions to this research.

The author also wishes to acknowledge the Faculty o f Graduate Studies and

Research for the financial support provided in the form of Graduate Scholarships, the

Sampson J. Goodfellow Scholarship, the John Spencer Middleton & Jack Spencer

Gordon Scholarship, and the Teaching Fellowship. The work opportunities as a sessional

lecturer and teaching assistant provided by the Faculty o f Engineering at the University

o f Regina are also highly appreciated.

Finally, the author would like to express his special thanks to his parents, his

w ife’s parents, his wife and children for their continuous encouragement and support.

Ill


TABLE OF CONTENTS

ABSTRACT I

ACKNOWLEDGEMENTS III

TABLE OF CONTENTS IV

LIST OF TABLES VIII

LIST OF FIGURES IX

NOMENCLATURE XII

CHAPTER 1 INTRODUCTION 1

1.1 Background 1

1.2 Research objective 2

1.3 Outline of the dissertation 3

CHAPTER 2 LITERATURE REVIEW 6

2.1 General sloshing problems 6

2.2 Liquid-structure systems 14

2.3 Sloshing in horizontal cylindrical tanks 18

2.4 Dynamics of liquid cargo vehicles 22

2.5 Summary 35

CHAPTER 3 DYNAMIC LIQUID MOTION IN 2D HORIZONTAL TANKS 38

3.1 Introduction 38

3.2 Mathematical model using potential flow theory 39

3.3 Mathematical method 41

3.3.1 First transformation 44

3.3.2 Second transformation 47

IV


TABLE OF CONTENTS

ABSTRACT.................................................................................................................................... I

ACKNOWLEDGEMENTS....................................................................................................... Ill

TABLE OF CONTENTS........................................................................................................... IV

LIST OF TABLES................................................................................................................... VIII

LIST OF FIGURES.................................................................................................................... IX

NOMENCLATURE................................................................................................................. XII

CHAPTER 1 INTRODUCTION.................................................................................................1

1.1 Background.......................................................................................................................... 1

1.2 Research objective..............................................................................................................2

1.3 Outline o f the dissertation..................................................................................................3

CHAPTER 2 LITERATURE REVIEW .....................................................................................6

2.1 General sloshing problems................................................................................................ 6

2.2 Liquid-structure systems..................................................................................................14

2.3 Sloshing in horizontal cylindrical tanks.........................................................................18

2.4 Dynamics o f liquid cargo vehicles................................................................................. 22

2.5 Summary............................................................................................................................35

CHAPTER 3 DYNAMIC LIQUID MOTION IN 2D HORIZONTAL TANKS...............38

3.1 Introduction........................................................................................................................38

3.2 Mathematical model using potential flow theory........................................................ 39

3.3 Mathematical method....................................................................................................... 41

3.3.1 First transformation...................................................................................................44

3.3.2 Second transformation............................................................................................. 47

IV


3.3.3 Third transformation 51

3.4 Numerical method 54

3.5 Results and discussion 60

3.5.1 Sloshing in circular tanks under harmonic excitations with small amplitudes 60

3.5.2 Sloshing in circular tanks under harmonic excitations with finite amplitude

near resonance 61

3.5.3 Transient liquid oscillations in circular tanks 67

3.6 Liquid motion in 2D elliptical tanks 74

3.6.1 Statement of liquid motion in 2D elliptical tanks 74

3.6.2 Natural frequencies 76

3.7 Summary 83

CHAPTER 4 LIQUID MOTION IN 3D HORIZONTAL CYLINDRICAL TANKS 86

4.1 Introduction 86

4.2 Statement of the problem 88

4.3 Mathematical approach 90

4.3.1 Continuous coordinate mappings 90

4.3.2 Formulae derivation 93

4.3.3 Numerical method 97

4.3.4 Calculation procedures 101


4.4.1 Natural frequencies 103

4.4.2 Transient liquid dynamics 105

4.5 Summary 118

V


3.3.3 Third transformation

3.4 Numerical method...........

3.5 Results and discussion....

51

54

60

3.5.1 Sloshing in circular tanks under harmonic excitations with small amplitudes 60


near resonance..................................................................................................................... 61

3.5.3 Transient liquid oscillations in circular tanks........................................................ 67

3.6 Liquid motion in 2D elliptical tanks..............................................................................74

3.6.1 Statement o f liquid motion in 2D elliptical tanks.................................................74

3.6.2 Natural frequencies...................................................................................................76

3.7 Summary............................................................................................................................83

CHAPTER 4 LIQUID MOTION IN 3D HORIZONTAL CYLINDRICAL TANKS 8 6

4.1 Introduction........................................................................................................................8 6

4.2 Statement o f the problem.................................................................................................8 8

4.3 Mathematical approach....................................................................................................90

4.3.1 Continuous coordinate mappings............................................................................90

4.3.2 Formulae derivation..................................................................................................93

4.3.3 Numerical method.....................................................................................................97

4.3.4 Calculation procedures.......................................................................................... 101

4.4 Results and discussion................................................................................................... 103

4.4.1 Natural frequencies.................................................................................................103

4.4.2 Transient liquid dynamics...................................................................................... 105

4.5 Summary.......................................................................................................................... 118

V


CHAPTER 5 INFLUENCE OF LIQUID MOTION ON RIDE QUALITY OF LIQUID

CARGO TANK VEHICLES 120

5.1 Introduction 120

5.2 Vehicle model 121

5.3 Analysis procedure 128


5.4.1 Frequency characteristics of partially filled liquid cargo vehicles 133

5.4.2 Ride performance under variable fill conditions 134

5.4.3 Ride performance under variable liquid types 138

5.4.4 Ride performance under variable vehicle speeds 139

5.4.5 Ride performance under variable road conditions 140

5.4.6 Ride performance of different seat suspensions 141

5.5 Summary 143

CHAPTER 6 INFLUENCE OF NONLINEAR IMPACT ON LIQUID CARGO TANK

VEHICLES 144

6.1 Introduction 144

6.2 Nonlinear impact model of liquid sloshing 147

6.3 Tank vehicle model in the pitch plane 150

6.3.1 Horizontal accelerations of the tractor and the tank on rough roads 150

6.3.2 Equations of the semi-trailer 152

6.3.3 Equations of the tractor 156


VI


CHAPTER 5 INFLUENCE OF LIQUID MOTION ON RIDE QUALITY OF LIQUID

CARGO TANK VEHICLES...................................................................................................120

5.1 Introduction......................................................................................................................120

5.2 Vehicle m odel................................................................................................................. 121

5.3 Analysis procedure..........................................................................................................128

5.4 Results and discussion...................................................................................................132

5.4.1 Frequency characteristics o f partially filled liquid cargo vehicles.................. 133

5.4.2 Ride performance under variable fill conditions................................................134

5.4.3 Ride performance under variable liquid types.................................................... 138

5.4.4 Ride performance under variable vehicle speeds................................................139

5.4.5 Ride performance under variable road conditions..............................................140

5.4.6 Ride performance o f different seat suspensions.................................................141

5.5 Summary.......................................................................................................................... 143

CHAPTER 6 INFLUENCE OF NONLINEAR IMPACT ON LIQUID CARGO TANK

VEHICLES.................................................................................................................................144

6.1 Introduction......................................................................................................................144

6.2 Nonlinear impact model o f liquid sloshing................................................................. 147

6.3 Tank vehicle model in the pitch plane.........................................................................150

6.3.1 Horizontal accelerations of the tractor and the tank on rough roads...............150

6.3.2 Equations o f the semi-trailer..................................................................................152

6.3.3 Equations o f the tractor.......................................................................................... 156

6.4 Results and discussion................................................................................................... 158

VI


6.4.1 Horizontal accelerations of the tractor and tank under rough road conditions

158

6.4.2 Comparison between linear model and nonlinear impact model 160

6.4.3 Dynamic fifth wheel loads 161

6.4.4 Dynamic normal axle loads 162

6.5 Summary 164

CHAPTER 7 CONCLUSIONS AND RECOMMENDATIONS 166

7.1 Conclusions 166

7.2 Recommendations for future work 172

REFERENCES 176

APPENDIX A: STRUCTURAL ANALYSIS OF A B-TRAIN TANK TRUCK

SUBFRAME SUBJECTED TO BRAKING/ACCELERATING 193

A.1 Introduction 193

A.2 B-train tank truck model 195

A.3 Finite element model of the subframe 197

A.4 Results and discussion 199

A.5 Summary 209

VII



.............................................................................................................................................158

6.4.2 Comparison between linear model and nonlinear impact model..................... 160

6.4.3 Dynamic fifth wheel loads.....................................................................................161

6.4.4 Dynamic normal axle loads...................................................................................162

6.5 Summary........................................................................................................................ 164

CHAPTER 7 CONCLUSIONS AND RECOMMENDATIONS....................................... 166

7.1 Conclusions......................................................................................................................166

7.2 Recommendations for future w ork .............................................................................. 172

REFERENCES.......................................................................................................................... 176

APPENDIX A: STRUCTURAL ANALYSIS OF A B-TRAIN TANK TRUCK

SUBFRAME SUBJECTED TO BRAKING/ACCELERATING......................................193

A .l Introduction.....................................................................................................................193

A.2 B-train tank truck m odel...............................................................................................195

A.3 Finite element model of the subframe.........................................................................197

A.4 Results and discussion.................................................................................................. 199

A.5 Summary.........................................................................................................................209

VII


LIST OF TABLES

Table 3.1 First eigenvalue of liquid motion in an elliptical tank 79

Table 3.2 Second eigenvalue of liquid motion in an elliptical tank 80

Table 3.3 Third eigenvalue of liquid motion in an elliptical tank 80

Table 3.4 Fourth eigenvalue of liquid motion in an elliptical tank 80

Table 3.4 Fifth eigenvalue of liquid motion in an elliptical tank 81

Table 5.2 Values of C sp and N, for the power spectral density function for various road

surfaces 130

Table 5.3 Masses and moments of inertia of tractor semi-trailer components 131

Table 5.4 Dimensions of tractor semi-trailer (m) 131

Table 5.5 Spring and damping coefficients of tractor semi-trailer components 132

Table 5.6 Natural frequencies (Hz) of tractor semi-trailer 134

Table 6.1 Simulation parameters 158

VIII


LIST OF TABLES

Table 3.1 First eigenvalue o f liquid motion in an elliptical tank.......................................... 79

Table 3.2 Second eigenvalue o f liquid motion in an elliptical tank......................................80

Table 3.3 Third eigenvalue o f liquid motion in an elliptical tan k ........................................80

Table 3.4 Fourth eigenvalue of liquid motion in an elliptical tank.......................................80

Table 3.4 Fifth eigenvalue of liquid motion in an elliptical tank.......................................... 81

Table 5.2 Values o f Csp and Nr for the power spectral density function for various road

surfaces.............................................................................................................................. 130

Table 5.3 Masses and moments o f inertia o f tractor semi-trailer components................. 131

Table 5.4 Dimensions of tractor semi-trailer (m )................................................................. 131

Table 5.5 Spring and damping coefficients of tractor semi-trailer components...............132

Table 5.6 Natural frequencies (Hz) of tractor semi-trailer.................................................. 134

Table 6.1 Simulation parameters.............................................................................................158

VIII


LIST OF FIGURES

Figure 3.1 Sketch of liquid sloshing in a circular tank 39

Figure 3.2 First coordinate transformation 45

Figure 3.3 Second coordinate transformation 48

Figure 3.4 Numerical procedures 56

Figure 3.5 Nondimensional liquid height subjected to harmonic excitations with small

amplitudes 63

Figure 3.6 Volume error of calculating liquid motion subjected to harmonic excitations

with small amplitudes 65

Figure 3.7 Free surface development in a circular tank subjected to a harmonic excitation

near resonance with finite amplitude 66

Figure 3.8 Acceleration input 68

Figure 3.9 Nondimensional liquid height in a circular tank during turning under different

final accelerations 69

Figure 3.10 Nondimensional liquid height in a horizontal circular tank during turning

under different input time 70

Figure 3.11 Wave profile in a horizontal circular tank 72

Figure 3.12 Sketch of liquid motion in an elliptical tank 75

Figure 3.13 Eigenvalue in a half-full circular tank 78

Figure 3.14 Eigenvalue in a circular tank 78

Figure 3.15 First eigenvalue of liquid motion in an elliptical tank 81

Figure 3.16 Second eigenvalue of liquid motion in an elliptical tank 82

Figure 3.17 Third eigenvalue of liquid motion in an elliptical tank 82

IX


LIST OF FIGURES

Figure 3.1 Sketch of liquid sloshing in a circular tank .......................................................... 39

Figure 3.2 First coordinate transformation..............................................................................45

Figure 3.3 Second coordinate transformation..........................................................................48

Figure 3.4 Numerical procedures.............................................................................................. 56


amplitudes............................................................................................................................63


with small amplitudes........................................................................................................ 65


near resonance with finite amplitude............................................................................... 6 6

Figure 3.8 Acceleration input....................................................................................................6 8


final accelerations...............................................................................................................69


under different input tim e ..................................................................................................70

Figure 3.11 Wave profile in a horizontal circular tank.......................................................... 72

Figure 3.12 Sketch of liquid motion in an elliptical tan k ......................................................75

Figure 3.13 Eigenvalue in a half-full circular tank.................................................................78

Figure 3.14 Eigenvalue in a circular tank........................................................................................ 78

Figure 3.15 First eigenvalue o f liquid motion in an elliptical tank.......................................81

Figure 3.16 Second eigenvalue of liquid motion in an elliptical tank ................................. 82

Figure 3.17 Third eigenvalue o f liquid motion in an elliptical tank .....................................82

IX


Figure 3.18 Fourth eigenvalue of liquid motion in an elliptical tank 83

Figure 3.19 Fifth eigenvalue of liquid motion in an elliptical tank 83

Figure 4.1 Sketch of horizontal cylindrical tanks 89

Figure 4.2 First eigenvalue in the longitudinal direction for liquid motion in a cylindrical

tank 106

Figure 4.3 Force and moment calculation by fluid dynamics 107

Figure 4.4 Force and moment calculation by mass centre 107

Figure 4.5 Force and moment under different accelerations 109

Figure 4.6 Free surface development under different accelerations 110

Figure 4.7 Force and moment under different fill levels 112

Figure 4.8 Force and moment for different tank shapes 113

Figure 4.9 Free surface development in a tank with hemispherical heads 113

Figure 4.10 Axial forces in compartmented tanks 117

Figure 4.11 Influence of input time 115

Figure 5.1 Pitch plane model of the tractor semi-trailer 124

Figure 5.2 Influence of variable fill levels (20%) 135



Figure 5.5 Influence of liquid densities 139

Figure 5.6 Influence of vehicle speed 140

Figure 5.7 Influence of road condition 141

Figure 5.8 Influence of seat suspension 142

Figure 6.1 Tractor semi-trailer model and motion profile 150

X


Figure 3.18 Fourth eigenvalue of liquid motion in an elliptical tan k .................................. 83

Figure 3.19 Fifth eigenvalue of liquid motion in an elliptical tan k ......................................83

Figure 4.1 Sketch of horizontal cylindrical tanks................................................................... 89


tank...................................................................................................................................... 106

Figure 4.3 Force and moment calculation by fluid dynamics.............................................. 107

Figure 4.4 Force and moment calculation by mass centre................................................. 107

Figure 4.5 Force and moment under different accelerations...............................................109

Figure 4.6 Free surface development under different accelerations...................................110

Figure 4.7 Force and moment under different fill levels..................................................... 112

Figure 4.8 Force and moment for different tank shapes...................................................... 113

Figure 4.9 Free surface development in a tank with hemispherical heads........................113

Figure 4.10 Axial forces in compartmented tanks................................................................ 117

Figure 4.11 Influence of input tim e........................................................................................ 115

Figure 5.1 Pitch plane model of the tractor semi-trailer...................................................... 124

Figure 5.2 Influence o f variable fill levels (20%)................................................................. 135

Figure 5.3 Influence of variable fill levels (50%)................................................................. 136

Figure 5.4 Influence o f variable fill levels (80%)................................................................. 137

Figure 5.5 Influence o f liquid densities..................................................................................139

Figure 5.6 Influence o f vehicle speed.....................................................................................140

Figure 5.7 Influence o f road condition...................................................................................141

Figure 5.8 Influence o f seat suspension..................................................................................142

Figure 6.1 Tractor semi-trailer model and motion profile................................................... 150

X


Figure 6.2 Loading configuration of the semi-trailer 154

Figure 6.3 Loading configuration of the tractor 156

Figure 6.4 Horizontal accelerations of the tractor and tank 159

Figure 6.5 Nondimensional sloshing mass displacement (wavelength: 75m) 160

Figure 6.6 Nondimensional sloshing mass displacement (wavelength: 100m) 161

Figure 6.7 Fifth wheel loads 163

Figure 6.8 Normal axle loads 164

Figure A.1 Schematic of a B-train tank truck 196

Figure A.2 Subframe model 198

Figure A.3 Load shift during acceleration 200

Figure A.4 Load shift during braking 201

Figure A.5 Forces at hitch point during acceleration 202

Figure A.6 Forces at hitch point during braking 203

Figure A.7 Force difference between the liquid cargo and equivalent rigid cargo 204

Figure A.8 Stress distributions of the subframe 206

Figure A.9 Von Mises stress at critical node 208

XI


Figure 6.2 Loading configuration of the semi-trailer............................................................154

Figure 6.3 Loading configuration o f the tractor.....................................................................156

Figure 6.4 Horizontal accelerations o f the tractor and tank.................................................159

Figure 6.5 Nondimensional sloshing mass displacement (wavelength: 75m ).................. 160

Figure 6 .6 Nondimensional sloshing mass displacement (wavelength: 100m)................161

Figure 6.7 Fifth wheel loads.................................................................................................... 163

Figure 6 .8 Normal axle loads................................................................................................... 164

Figure A. 1 Schematic o f a B-train tank truck.......................................................................196

Figure A.2 Subframe model..................................................................................................... 198

Figure A.3 Load shift during acceleration.............................................................................200

Figure A.4 Load shift during braking.....................................................................................201

Figure A.5 Forces at hitch point during acceleration........................................................... 202

Figure A .6 Forces at hitch point during braking...................................................................203

Figure A.7 Force difference between the liquid cargo and equivalent rigid cargo 204

Figure A .8 Stress distributions of the subframe.................................................................... 206

Figure A.9 Von Mises stress at critical node.........................................................................208

XI


NOMENCLATURE

a: half of the major axis length of elliptical tanks

al , a2 , a3, a30 , a4 , ael , ae2 , a , 3 , ae4 , as : geometric parameters of tractor semi-trailer

aft: horizontal acceleration of the tractor in the local coordinate system

an : amplitude of the road contour

art: horizontal acceleration of the tank in the local coordinate system

at: applied acceleration on the equivalent mass

A,B,C,D,E,F,G: parameters used in calculation

Ao: final acceleration value when applied by a ramp function

Ag, Bg: parameter for adjusting the grid clustering

At, Az: lateral and longitudinal accelerations applied on the liquid

b: half of the minor axis length of elliptical tanks

,b5 : geometric parameters of tractor semi-trailer

bn : positive constant impact parameter

B: parameter for adjusting the grid clustering

Bir, B2r: coefficient matrices

B, G,, Ei, Di, Ii, Hi: coefficients used in the transformations

C:

CO:

elf :

e l2 :

half length of horizontal cylindrical tanks with flat heads

and half length of cylindrical section of tanks with hemispherical heads

distance between the still liquid surface and the coordinate system origin

damping coefficient of the tractor front suspension

damping coefficient of the tractor rear suspension

XII


NOMENCLATURE

a: half of the major axis length of elliptical tanks

ax,a 2,a 2, ai0, a4, aei, ae2, aei, ae4, as : geometric parameters o f tractor semi-trailer

a/x: horizontal acceleration o f the tractor in the local coordinate system

a„ : amplitude of the road contour

arx'. horizontal acceleration of the tank in the local coordinate system

ax: applied acceleration on the equivalent mass

A ,B ,C ,D ,E ,F ,G : parameters used in calculation

Ao: final acceleration value when applied by a ramp function

Ag, Bg. parameter for adjusting the grid clustering

AXf Az: lateral and longitudinal accelerations applied on the liquid

b : half o f the minor axis length of elliptical tanks

b{, b20, b4, bs : geometric parameters of tractor semi-trailer

bn : positive constant impact parameter

B\ parameter for adjusting the grid clustering

B\r, B2/. coefficient matrices

Bj Gj Ejf Df Ii Hf. coefficients used in the transformations

c : half length of horizontal cylindrical tanks with flat heads

and half length of cylindrical section of tanks with hemispherical heads

co: distance between the still liquid surface and the coordinate system origin

cn : damping coefficient o f the tractor front suspension

c, 2 : damping coefficient of the tractor rear suspension

XII


C21 :

Cf.

Cni:

C,:

C,11

C,12

Cal

damping coefficient of the semi-trailer suspension

artificial damping coefficient

equivalent damping coefficient of the equivalent mass-spring system

damping coefficient of the seat

damping coefficient of the tractor front axle

damping coefficient of the tractor rear axle

damping coefficient of the semi-trailer axle

Ci coefficients used in the transformations

Cr:

d:

D:

Do:

f f :

f.:

f

Fd:

F m:

F1• •

Fim:

damping matrix

constant for the road surface

still liquid height

tank diameter

amplitude of the tank displacement

tank displacement

frequency in the unit of Hz

tractor front axle rolling resistance coefficient

tractor rear axle rolling resistance coefficient

semi-trailer axle rolling resistance coefficient

driving force

tractor rear axle normal force

tractor front axle normal force

impact force

XIII


c21: damping coefficient of the semi-trailer suspension

cf. artificial damping coefficient

cn\: equivalent damping coefficient o f the equivalent mass-spring system

cs : damping coefficient of the seat

clU : damping coefficient o f the tractor front axle

c, 12 : damping coefficient o f the tractor rear axle

cm : damping coefficient of the semi-trailer axle

Ci K if J if Mi, Liw Nf. coefficients used in the transformations

C,: damping matrix

Cv - constant for the road surface

d: still liquid height

D: tank diameter

Do: amplitude o f the tank displacement

Dx: tank displacement

f i frequency in the unit of Hz

fr tractor front axle rolling resistance coefficient

fm- tractor rear axle rolling resistance coefficient

fr - semi-trailer axle rolling resistance coefficient

Fd: driving force

Fm: tractor rear axle normal force

Fr tractor front axle normal force

F ■1 im• impact force

XIII


FL:

Fr :

Frp

Fx :

Fy :

FZ :

g:

Gy:

h:

H:

H:

H*:

He:

force vector caused by liquid motion

liquid fill level

semi-trailer axle normal force

Laplace transform vector

interaction force between the liquid and the tank walls

horizontal force in the global coordinate system on the fifth wheel

vertical force in the global coordinate system on the fifth wheel

longitudinal liquid force

acceleration due to gravitation

frequency response of a given output yr in response to the road input

dynamic liquid height in the transformed coordinate system


smoothed dynamic liquid height in the transformed coordinate system


equivalent still liquid height

hoi,h1,, h2,14, H1,H2: geometric parameters of the tractor semi-trailer

j, k:

/ 20

I rl •

k11 :

k12 :

indices

moment of inertia of the empty semi-trailer

moment of inertia of the tractor

combined moment of inertia of semi-trailer and all fixed parts of the liquid

spring coefficient of the tractor front suspension

spring coefficient of the tractor rear suspension

XIV


F[ : force vector caused by liquid motion

FL: liquid fill level

Fr : semi-trailer axle normal force

Frp: Laplace transform vector

Fs\ interaction force between the liquid and the tank walls

Fx : horizontal force in the global coordinate system on the fifth wheel

Fy : vertical force in the global coordinate system on the fifth wheel

F : longitudinal liquid force

g: acceleration due to gravitation

Gy. frequency response o f a given output jv in response to the road input

h: dynamic liquid height in the transformed coordinate system

H: dynamic liquid height in the transformed coordinate system

H : smoothed dynamic liquid height in the transformed coordinate system

F t : dynamic liquid height in the transformed coordinate system

He\ equivalent still liquid height

Aoy, hy, li2,hw, H\,H 2 '. geometric parameters o f the tractor semi-trailer

i,j, k: indices

I 20: moment o f inertia o f the empty semi-trailer

I rt: moment o f inertia o f the tractor

I r2: combined moment o f inertia o f semi-trailer and all fixed parts o f the liquid

ki,: spring coefficient of the tractor front suspension

kn : spring coefficient o f the tractor rear suspension

XIV


k 21

kf

kg:

kil:

k :

k t 1 1 :

k t 12 :

k t 21 :

1, :

L:

Le:

LI :

m20 • '

ml:

Mnl

mri

M r2

ms :

spring coefficient of the semi-trailer suspension

artificial soft spring coefficient

parameter for adjusting the grid clustering

equivalent stiffness coefficient of the equivalent mass-spring system

spring coefficient of the seat

spring coefficient of the tractor front axle

spring coefficient of the tractor rear axle

spring coefficient of the semi-trailer axle

stiffness matrices

L,„: geometric parameters of the tractor semi-trailer

distance from the still free surface to the centre of fixed mass

distance from the still free surface to the centre of sloshing mass

distance between the tractor rear axle and semi-trailer axle

Lagrangian

equivalent tank length

distance between the tractor front axle and tractor rear axle

mass of the empty semi-trailer

total liquid mass inside the tank

equivalent mass of the equivalent mass-spring system

mass of the tractor

combined mass of the empty semi-trailer and all fixed parts of the liquid

mass of the seat and the driver

XV


k2l: spring coefficient o f the semi-trailer suspension

kf. artificial soft spring coefficient

kg. parameter for adjusting the grid clustering

kn\. equivalent stiffness coefficient o f the equivalent mass-spring system

ks - spring coefficient o f the seat

k, 11 • spring coefficient o f the tractor front axle

kt 12 • spring coefficient of the tractor rear axle

kt 21. spring coefficient of the semi-trailer axle

Kr. stiffness matrices

Ij, kjt lg> Av, k w. geometric parameters of the tractor semi-trailer

1/nO '• distance from the still free surface to the centre of fixed mass

AbI '• distance from the still free surface to the centre o f sloshing mass

k- distance between the tractor rear axle and semi-trailer axle

L : Lagrangian

Le: equivalent tank length

V distance between the tractor front axle and tractor rear axle

m2o : mass o f the empty semi-trailer

mi: total liquid mass inside the tank

m„ i : equivalent mass of the equivalent mass-spring system

mrX\ mass of the tractor

mr2: combined mass o f the empty semi-trailer and all fixed parts o f the liq

ms : mass o f the seat and the driver

XV


mill mass of the tractor front axle

Mt12 : mass of the tractor rear axle

Mt21 mass of the semi-trailer axle

MI : moment caused by liquid motion

/1-;/, : moment vector caused by liquid motion

Mr: mass matrix

n: normal direction on the curved walls

nn : positive impact integer

N, M, L: total numbers of cells in x*,Y* and Z* directions

Nn: number of compartments

Nr : constant for the road surface

P: liquid pressure

Q,: jth generalized force

R: radius of the cylindrical tank

Rf •• tractor front axle rolling resistance

R„, : tractor rear axle rolling resistance

Rr : semi-trailer axle rolling resistance

Re: Reynolds number

S: cross-section area of the liquid in the cylindrical section

Sy : power spectral density of a given output variable

S zii : power spectral density function of the elevation of the road surface profile

t: time

XVI


m ,u : mass o f the tractor front axle

mt\2 • mass o f the tractor rear axle

m, 21: mass o f the semi-trailer axle

M r. moment caused by liquid motion

Mr- moment vector caused by liquid motion

Mr\ mass matrix

n: normal direction on the curved walls

n„ : positive impact integer

N, M, L: total numbers o f cells in X*, Y* and Z* directions

Nn: number of compartments

Nr : constant for the road surface

P- liquid pressure

Qrr yth generalized force

R : radius of the cylindrical tank

Rf : tractor front axle rolling resistance

Rm- tractor rear axle rolling resistance

K - semi-trailer axle rolling resistance

Re: Reynolds number

S: cross-section area of the liquid in the cylindrical section

Sy : power spectral density o f a given output variable

•• power spectral density function o f the elevation o f the road surface profile

t: time

XVI


to: acceleration input time when applied by a ramp function

: delay time for the tractor rear axle

12 : delay time for the semi-trailer axle

ur: horizontal velocity of the tank in the local coordinate system

Uf : horizontal velocity of the tractor in the global coordinate system

horizontal velocity in the global coordinate system

Ur: horizontal velocity of the tank in the global coordinate system

Urp: vector of instantaneous values of vertical displacements of the road profile

at each axle location

v: speed of the vehicle

of vertical velocity of the tractor in the local coordinate system

vr: vertical velocity of the tank in the local coordinate system

V: liquid volume in the hemispherical head

Vj vertical velocity of the tractor in the global coordinate system

Vn: vertical velocity in the global coordinate system

Vr: vertical velocity of the tank in the global coordinate system

w: longitudinal length of the tank compartment

weight of the jth fixed mass

Wif • weight of the jth sloshing mass •

coefficients used in the transformations

W k : weight of the tractor

WL: wavelength of the road contour

XVII


t0: acceleration input time when applied by a ramp function

t{: delay time for the tractor rear axle

t2: delay time for the semi-trailer axle

ur'. horizontal velocity o f the tank in the local coordinate system

U f: horizontal velocity o f the tractor in the global coordinate system

U„: horizontal velocity in the global coordinate system

Ur: horizontal velocity o f the tank in the global coordinate system

Urp. vector of instantaneous values of vertical displacements o f the road profile

at each axle location

v: speed of the vehicle

v/. vertical velocity of the tractor in the local coordinate system

vr: vertical velocity of the tank in the local coordinate system

V: liquid volume in the hemispherical head

V/. vertical velocity of the tractor in the global coordinate system

Vn: vertical velocity in the global coordinate system

Vr\ vertical velocity o f the tank in the global coordinate system

w: longitudinal length of the tank compartment

W0 j : weight of they'th fixed mass

Wyj : weight o f the yth sloshing mass

Wi Pi, Oi Si, Qi, Ri. coefficients used in the transformations

Wk : weight o f the tractor

WL: wavelength of the road contour

XVII


W:

x, y, z:

x1,y1,z1:

xc :

xej:

xn, yn:

Xn0

Xr 1 :

Xr2:

Xs:

Xill:

Xt 1 2:

Xt21:

X, Y, Z:

Y Z :

Xfi

X., Y.:

Xn, Yn:

Xr, Yr:

Yr;

Yrp:

weight of the empty semi-trailer

coordinate system for coordinate transformation

coordinates for liquid sloshing inside tanks

nondimensional displacement

horizontal displacement of jth sloshing mass

local coordinate system on the tractor and the tank

displacement when the equivalent mass reaches the compartment walls

horizontal displacement of the tractor mass centre

Horizontal displacement of the semi-trailer mass centre

horizontal displacement of the seat

horizontal displacement of the tractor front axle

horizontal displacement of the tractor rear axle

horizontal displacement of the semi-trailer axle



global coordinates of tractor front tire-ground contact point

global coordinates of tractor rear tire-ground contact points

global coordinates of the tractor semi-trailer

global coordinates of semi-trailer tire-ground contact points

jth generalized coordinate of the vehicle system

vertical and longitudinal distances between the mass centre and

the selected axis for moment calculation

independent generalized coordinate vector

XVIII


Wt : weight of the empty semi-trailer

x, y, z: coordinate system for coordinate transformation

jcijyz,: coordinates for liquid sloshing inside tanks

x c : nondimensional displacement

xej\ horizontal displacement ofy'th sloshing mass

x„, y„: local coordinate system on the tractor and the tank

x„o: displacement when the equivalent mass reaches the compartment walls

xr\: horizontal displacement of the tractor mass centre

xry. Horizontal displacement of the semi-trailer mass centre

Xyi horizontal displacement of the seat

xt\\\ horizontal displacement of the tractor front axle

xtn- horizontal displacement of the tractor rear axle

xa\. horizontal displacement of the semi-trailer axle

X, Y, Z: coordinate system for coordinate transformation

X*, Y*, Z*\ coordinate system for coordinate transformation

Xf, Yf. global coordinates of tractor front tire-ground contact point

X m, Ym: global coordinates o f tractor rear tire-ground contact points

X„, Yn: global coordinates o f the tractor semi-trailer

X r, Yr: global coordinates of semi-trailer tire-ground contact points

y rJ : yth generalized coordinate o f the vehicle system

y x, Zj: vertical and longitudinal distances between the mass centre and

the selected axis for moment calculation

Yrp\ independent generalized coordinate vector

XVIII


Z11:

Z12:

Z21:

Zej:

Zrl:

Zr2:

Zs:

Ztll:

Zt12:

Z121:

a,13, y:

an, fin:

x:

8:

(1):

17:

17n:

f:

corn:

cor:

K:

road profile at the tractor front axle

road profile at the tractor rear axle

road profile at the semi-trailer axle

vertical displacement of jth sloshing mass

vertical displacement of the tractor mass centre

vertical displacement of the semi-trailer mass centre

vertical displacement of the seat

vertical displacement of the tractor front axle

vertical displacement of the tractor rear axle

vertical displacement of the semi-trailer axle


angle of the tank with respect to the X„ and Y, coordinates

eigenvector

parameter for different tank head types

velocity potential in the transformed coordinate system

free surface elevation in the transformed coordinate system

positive impact integer

velocity potential in the original coordinate system

road profile tangent angle at tire-ground contact point of tractor front axle

road profile tangent angle at tire-ground contact point of tractor rear axle

road profile tangent angle at tire-ground contact point of semi-trailer axle

eigenvalue

XIX


z 1 1 : road profile at the tractor front axle

z\2 '. road profile at the tractor rear axle

Z2 \: road profile at the semi-trailer axle

zef. vertical displacement ofy'th sloshing mass

zr\: vertical displacement o f the tractor mass centre

z r2 '■ vertical displacement of the semi-trailer mass centre

zv: vertical displacement o f the seat

zt11: vertical displacement of the tractor front axle

zt\2- vertical displacement of the tractor rear axle

za\. vertical displacement of the semi-trailer axle

a, p, y: coordinate system for coordinate transformation

an p n\ angle o f the tank with respect to the X„ and Yn coordinates

X '■ eigenvector

8 : parameter for different tank head types

<p. velocity potential in the transformed coordinate system

77: free surface elevation in the transformed coordinate system

T]n: positive impact integer

qr. velocity potential in the original coordinate system

<pf : road profile tangent angle at tire-ground contact point o f tractor front axle

(pm: road profile tangent angle at tire-ground contact point o f tractor rear axle

cpr : road profile tangent angle at tire-ground contact point o f semi-trailer axle

k : eigenvalue

XIX


ith eigenvalue

p: damping coefficient in the modified Rayleigh term

7r 3.1415926

A free surface angle when liquid motion modeled by mass centre models

01: angular displacement of the tractor

02: angular displacement of the semi-trailer

0„ : pendulum angular displacement

0„n : angular displacement when the pendulum reached the container walls

P: liquid density

co: excitation frequency

co,: natural frequency of liquid motion

con: first liquid sloshing frequency

free surface elevation above still liquid level in xiy,z, coordinate system

nondimensional damping coefficient

0: velocity potential in the transformed coordinate system

velocity potential in the transformed coordinate system

II : impact potential energy function

O : coefficient matrix in the eigenvalue problem

: spatial frequency

: coefficient matrix in the eigenvalue problem

XX


X{. ith eigenvalue

ji\ damping coefficient in the modified Rayleigh term

tv. 3.1415926

&. free surface angle when liquid motion modeled by mass centre models

9\ : angular displacement o f the tractor

6 2 ' angular displacement o f the semi-trailer

dn: pendulum angular displacement

0 nQ: angular displacement when the pendulum reached the container walls

p\ liquid density

co\ excitation frequency

C0i\ natural frequency o f liquid motion

con: first liquid sloshing frequency

free surface elevation above still liquid level in x xy xz x coordinate system

Q. nondimensional damping coefficient

d>. velocity potential in the transformed coordinate system

0 *: velocity potential in the transformed coordinate system

1 1 : impact potential energy function

0 : coefficient matrix in the eigenvalue problem

Q : spatial frequency

W : coefficient matrix in the eigenvalue problem

XX


CHAPTER 1 INTRODUCTION

1.1 Background

Horizontal cylindrical tanks are widely used in road transportation and civil

engineering for carrying and storing liquids. Partial fill conditions are quite common

during the service time of these tanks. When the tanks are subjected to translatory, roll,

yaw and pitch perturbations on a tank vehicle caused by driving operations and road

surface irregularities, liquid inside the tanks will be excited to undergo oscillatory

motion. At the same time, dynamic liquid behaviour will affect the motion of the tanks

and supporting structures, which causes a complicated coupled liquid-structure problem.

Liquid motion inside the tanks changes the pressure distribution on tank walls,

which often generates adverse forces and moments of considerable magnitudes.

Generally, the influence of the dynamic liquid motion inside the tanks can be found in

three different areas. First, the dynamic interaction between the vehicle and the liquid

cargo can cause problems in vehicle controllability and stability. For example, the

rollover immunity levels of liquid cargo vehicles are lower than those of rigid cargo

vehicles. Second, the structural integrity problem of the tanks and supporting structures is

a big concern for vehicle structure design. As a matter of fact, cracks and fatigue failure

are the major failure modes of the liquid cargo vehicle structures. These failures reduce

the profits for the vehicle users. Third, the ride quality of liquid cargo vehicles can be

deteriorated by the liquid motion due to the coupling effect of the liquid sloshing and

various vehicle vibration modes. This negative influence of the liquid motion on the

driving comfort will cause driver fatigue. All of the above problems are directly or

1


CHAPTER 1 INTRODUCTION

1.1 Background

Horizontal cylindrical tanks are widely used in road transportation and civil

engineering for carrying and storing liquids. Partial fill conditions are quite common

during the service time o f these tanks. When the tanks are subjected to translatory, roll,

yaw and pitch perturbations on a tank vehicle caused by driving operations and road

surface irregularities, liquid inside the tanks will be excited to undergo oscillatory

motion. At the same time, dynamic liquid behaviour will affect the motion o f the tanks

and supporting structures, which causes a complicated coupled liquid-structure problem.

Liquid motion inside the tanks changes the pressure distribution on tank walls,

which often generates adverse forces and moments of considerable magnitudes.

Generally, the influence of the dynamic liquid motion inside the tanks can be found in

three different areas. First, the dynamic interaction between the vehicle and the liquid

cargo can cause problems in vehicle controllability and stability. For example, the

rollover immunity levels o f liquid cargo vehicles are lower than those o f rigid cargo

vehicles. Second, the structural integrity problem of the tanks and supporting structures is

a big concern for vehicle structure design. As a matter o f fact, cracks and fatigue failure

are the major failure modes o f the liquid cargo vehicle structures. These failures reduce

the profits for the vehicle users. Third, the ride quality o f liquid cargo vehicles can be

deteriorated by the liquid motion due to the coupling effect o f the liquid sloshing and

various vehicle vibration modes. This negative influence o f the liquid motion on the

driving comfort will cause driver fatigue. All o f the above problems are directly or

1


indirectly related to public safety concerns if the tank vehicles are involved in accidents,

considering the flammable, explosive and toxic features of most liquids being carried by

the tank vehicles.

The research topics in this study are taken directly from a research project

collaborated between the University of Regina and MaXfield, Inc., a Canadian

manufacturer in Calgary, which produces B-train tank trucks. In order to improve the

existing vehicle structure design by increasing the vehicle structure strength and

extending the vehicle service life, a comprehensive understanding of the coupled liquid-

structure-vehicle system under different operation conditions is necessary. An effective

way to describe the liquid motion in partially filled horizontal cylindrical tanks is actually

one of the key factors in helping to reveal the relationship between the liquid motion and

the vehicle dynamics and structure strength. Research results will benefit the road

transportation industry and vehicle manufacturers, as well as improving public safety.

1.2 Research objective

Based on the state of arts of the studies on liquid motion in horizontal cylindrical

tanks and liquid cargo tank vehicle dynamics, as well as the current needs in industry, the

main objectives of the research are as follows.

1. Develop an effective mathematical method to solve the liquid motion problem in the

partially filled 2D horizontal tanks under normal operation conditions. This new

mathematical method should be able to overcome some of the difficulties of the

conventional methods in solving the liquid motion problem in 2D circular tanks. The

method would be used to study the lateral dynamics of the liquid motion. It should be

2


indirectly related to public safety concerns if the tank vehicles are involved in accidents,

considering the flammable, explosive and toxic features of most liquids being carried by

the tank vehicles.

The research topics in this study are taken directly from a research project

collaborated between the University o f Regina and MaXfield, Inc., a Canadian

manufacturer in Calgary, which produces B-train tank trucks. In order to improve the

existing vehicle structure design by increasing the vehicle structure strength and

extending the vehicle service life, a comprehensive understanding of the coupled liquid-

structure-vehicle system under different operation conditions is necessary. An effective

way to describe the liquid motion in partially filled horizontal cylindrical tanks is actually

one of the key factors in helping to reveal the relationship between the liquid motion and

the vehicle dynamics and structure strength. Research results will benefit the road

transportation industry and vehicle manufacturers, as well as improving public safety.

1.2 Research objective

Based on the state o f arts o f the studies on liquid motion in horizontal cylindrical

tanks and liquid cargo tank vehicle dynamics, as well as the current needs in industry, the

main objectives o f the research are as follows.

1. Develop an effective mathematical method to solve the liquid motion problem in the

partially filled 2D horizontal tanks under normal operation conditions. This new

mathematical method should be able to overcome some o f the difficulties of the

conventional methods in solving the liquid motion problem in 2D circular tanks. The

method would be used to study the lateral dynamics o f the liquid motion. It should be

2


easy to apply to tanks of other shapes, such as elliptical tanks. It should also be easy

to extend the method to solve liquid motion problems in 3D horizontal cylindrical

tanks.

2. Develop an effective mathematical method to solve the liquid motion in the partially

filled 3D horizontal cylindrical tanks under normal operation conditions. This

situation had seldom been studied in the liquid cargo vehicle dynamics due to the lack

of an effective algorithm to describe the liquid motion in horizontal cylindrical tanks

in a completely 3D manner. The method should be able to be used to study the

longitudinal liquid dynamics, as well as the combined longitudinal and lateral

dynamics of the liquid motion.

3. Establish the numerical method and procedures needed to solve the liquid motion

inside the 2D and 3D tanks based on the methodology developed above, and study the

lateral liquid dynamics under transversal excitation for 2D tanks and longitudinal

liquid dynamics for 3D tanks under typical operations such as braking/accelerating.

4. Investigate the liquid cargo vehicle dynamics in the longitudinal direction by using

equivalent mechanical models for situations where the newly developed methodology

cannot be used, such as the ride comfort problem in the frequency domain and the

nonlinear impact problem in the pitch plane.

1.3 Outline of the dissertation

The content of the different chapters of this dissertation is briefly described below.

In this chapter, an introduction, including the background of the research, the objectives

of the research and the organization of the dissertation, is presented. In Chapter 2, a

3


easy to apply to tanks of other shapes, such as elliptical tanks. It should also be easy

to extend the method to solve liquid motion problems in 3D horizontal cylindrical

tanks.

2. Develop an effective mathematical method to solve the liquid motion in the partially

filled 3D horizontal cylindrical tanks under normal operation conditions. This

situation had seldom been studied in the liquid cargo vehicle dynamics due to the lack

of an effective algorithm to describe the liquid motion in horizontal cylindrical tanks

in a completely 3D manner. The method should be able to be used to study the

longitudinal liquid dynamics, as well as the combined longitudinal and lateral

dynamics o f the liquid motion.

3. Establish the numerical method and procedures needed to solve the liquid motion

inside the 2D and 3D tanks based on the methodology developed above, and study the

lateral liquid dynamics under transversal excitation for 2D tanks and longitudinal

liquid dynamics for 3D tanks under typical operations such as braking/accelerating.

4. Investigate the liquid cargo vehicle dynamics in the longitudinal direction by using

equivalent mechanical models for situations where the newly developed methodology

cannot be used, such as the ride comfort problem in the frequency domain and the

nonlinear impact problem in the pitch plane.

1.3 Outline of the dissertation

The content o f the different chapters o f this dissertation is briefly described below.

In this chapter, an introduction, including the background of the research, the objectives

of the research and the organization of the dissertation, is presented. In Chapter 2, a

3


comprehensive literature review is presented. The review has been carried out under

various subject headings in different sections, i.e., general sloshing problem, liquid-

structure systems, sloshing in horizontal cylindrical tanks, as well as the dynamics of

liquid cargo vehicles. More attention has been paid to horizontal cylindrical tanks due to

the wide application of this special configuration in liquid cargo vehicles in road

transportation industry, for which the liquid sloshing problem has only been investigated

by limited studies in comparison with the liquid motion analyses in the other fields.

In Chapter 3, a new mathematical method used to study the dynamic liquid

behaviour in partially filled horizontal circular tanks has been developed. Sloshing

problems in 2D circular tanks subjected to harmonic motions with small and finite

amplitudes are simulated to show the efficiency of the new method. Transient responses

of the liquid in the road tanks have been studied in detail under turning, lane change and

double lane change manoeuvres. The natural frequencies of liquid motion in 2D elliptical

tanks with different aspect ratios and under different liquid fill levels have been solved by

the current method for the first five liquid modes.

In Chapter 4, the mathematical method is further developed to study the liquid

dynamics in partially filled 3D horizontal cylindrical tanks, based on the method

developed for the 2D circular and elliptical tanks in Chapter 3. The transient liquid

motion and corresponding liquid forces and moments acting on the tank walls have been

calculated for tanks subjected to longitudinal acceleration input. The influence of the tank

shapes, tank configurations and liquid fill levels on the transient liquid motion has been

studied in detail.

In Chapter 5, the ride performance of a partially filled compartmented tank

4


comprehensive literature review is presented. The review has been carried out under

various subject headings in different sections, i.e., general sloshing problem, liquid-

structure systems, sloshing in horizontal cylindrical tanks, as well as the dynamics of

liquid cargo vehicles. More attention has been paid to horizontal cylindrical tanks due to

the wide application of this special configuration in liquid cargo vehicles in road

transportation industry, for which the liquid sloshing problem has only been investigated

by limited studies in comparison with the liquid motion analyses in the other fields.

In Chapter 3, a new mathematical method used to study the dynamic liquid

behaviour in partially filled horizontal circular tanks has been developed. Sloshing

problems in 2D circular tanks subjected to harmonic motions with small and finite

amplitudes are simulated to show the efficiency o f the new method. Transient responses

o f the liquid in the road tanks have been studied in detail under turning, lane change and

double lane change manoeuvres. The natural frequencies of liquid motion in 2D elliptical

tanks with different aspect ratios and under different liquid fill levels have been solved by

the current method for the first five liquid modes.

In Chapter 4, the mathematical method is further developed to study the liquid

dynamics in partially filled 3D horizontal cylindrical tanks, based on the method

developed for the 2D circular and elliptical tanks in Chapter 3. The transient liquid

motion and corresponding liquid forces and moments acting on the tank walls have been

calculated for tanks subjected to longitudinal acceleration input. The influence of the tank

shapes, tank configurations and liquid fill levels on the transient liquid motion has been

studied in detail.

In Chapter 5, the ride performance o f a partially filled compartmented tank

4


vehicle by a linearized multi-degree-of-freedom dynamic model has been investigated.

The liquid motion in the partially filled tanks has been described by a linear spring-mass

mechanical model. The power spectral density of the vertical and horizontal seat

accelerations has been utilized to study the influence of liquid motion on the ride quality.

In Chapter 6, a nonlinear impact mechanical model has been integrated into a tractor

semi-trailer vehicle model in order to study the influence of the liquid motion on the

liquid cargo vehicles subjected to rough road excitation.

The major conclusions of the research and recommendations for future work are

presented in Chapter 7. In Appendix A, a finite element analysis for the subframe of a B-

train tank truck has been performed based on the loading conditions obtained from the

vehicle model and the liquid load shift model.

5


vehicle by a linearized multi-degree-of-freedom dynamic model has been investigated.

The liquid motion in the partially filled tanks has been described by a linear spring-mass

mechanical model. The power spectral density of the vertical and horizontal seat

accelerations has been utilized to study the influence of liquid motion on the ride quality.

In Chapter 6 , a nonlinear impact mechanical model has been integrated into a tractor

semi-trailer vehicle model in order to study the influence of the liquid motion on the

liquid cargo vehicles subjected to rough road excitation.

The major conclusions of the research and recommendations for future work are

presented in Chapter 7. In Appendix A, a finite element analysis for the subframe of a B-

train tank truck has been performed based on the loading conditions obtained from the

vehicle model and the liquid load shift model.

5


CHAPTER 2 LITERATURE REVIEW

2.1 General sloshing problems

Liquid sloshing in moving tanks is the oscillatory phenomenon of liquid caused

by the motion of the tanks. It is an important topic in areas where liquid is being carried

or stored. Studies on liquid sloshing have been conducted in aerospace applications, road

transportation, ocean engineering, civil engineering, etc. Since the early 1960s, the

sloshing problem has been of major concern to aerospace researchers and engineers in

studying the influence of liquid propellant sloshing on the flight performance of jet

vehicles. In civil engineering, sloshing problems have been studied to find the effects of

earthquakes on large dams, liquid storage tanks, water reservoirs, and nuclear vessels. In

the road transportation area, the vehicle dynamics of liquid cargo tank trucks and trains

have been investigated by including the influence of liquid motion in the vehicle systems.

Sloshing problems of liquid cargo in ocean-going vessels have also been extensively

studied in the ocean engineering.

The monograph edited by Abramson and Silverman (1966) provided a thorough

review of the publications on sloshing problems prior to 1966, most of which were

related to applications in aerospace engineering. Ibrahim et al (2001) conducted a

detailed review of the research work developed on liquid sloshing dynamics in recent

years. This review contains 1319 references, most of which were published after 1966.

This review covers liquid sloshing in almost all areas until 2001.

Generally, sloshing problems can be mathematically modeled by two sets of

governing equations. The first set of equations is the Navier-Stokes equations (Harlow

6


CHAPTER 2 LITERATURE REVIEW

2.1 General sloshing problems

Liquid sloshing in moving tanks is the oscillatory phenomenon o f liquid caused

by the motion o f the tanks. It is an important topic in areas where liquid is being carried

or stored. Studies on liquid sloshing have been conducted in aerospace applications, road

transportation, ocean engineering, civil engineering, etc. Since the early 1960s, the

sloshing problem has been of major concern to aerospace researchers and engineers in

studying the influence o f liquid propellant sloshing on the flight performance o f jet

vehicles. In civil engineering, sloshing problems have been studied to find the effects of

earthquakes on large dams, liquid storage tanks, water reservoirs, and nuclear vessels. In

the road transportation area, the vehicle dynamics o f liquid cargo tank trucks and trains

have been investigated by including the influence o f liquid motion in the vehicle systems.

Sloshing problems o f liquid cargo in ocean-going vessels have also been extensively

studied in the ocean engineering.

The monograph edited by Abramson and Silverman (1966) provided a thorough

review o f the publications on sloshing problems prior to 1966, most o f which were

related to applications in aerospace engineering. Ibrahim et al (2001) conducted a

detailed review of the research work developed on liquid sloshing dynamics in recent

years. This review contains 1319 references, most o f which were published after 1966.

This review covers liquid sloshing in almost all areas until 2001.

Generally, sloshing problems can be mathematically modeled by two sets of

governing equations. The first set of equations is the Navier-Stokes equations (Harlow

6


and Welch, 1965). For liquid in tanks of arbitrary shapes subjected to different kinds of

motion, the liquid behaviour is usually described by the continuity equation, momentum

equations, kinematic and dynamic conditions on the free surface, and the velocity

conditions on the rigid walls. Based on some simplifying assumptions, the sloshing

problem could be modeled by the second set of equations based on the potential flow

theory. The adopted assumptions include: rigid tank, irrotational flow field, nonviscous

fluid, homogeneous fluid, incompressible fluid, no sinks or sources, and single-valued

velocity potential in any simply connected region (Abramson and Silverman, 1966).

Under these assumptions, the liquid behaviour can be established using the velocity

potential. The governing equation of the liquid motion should satisfy the Laplace

equation inside the liquid domain. The kinematic and dynamic conditions are to be

satisfied on the free surface. Relative normal velocities on the rigid walls should be zero.

(1) Analytical and semi-analytical solution

For many years, mathematical studies of liquid sloshing problems were basically

carried out on how to solve either the Navier-Stokes equations or the equations based on

the potential theory. Analytical or semi-analytical solutions of the Navier-Stokes

equations are not available without the adoption of further assumptions. Numerical

schemes can usually be employed to discretize the Navier-Stokes equations in such a way

that nonlinearities and viscous effect can be retained without any simplification. On the

contrary, analytical or semi-analytical solutions were tried extensively by many

researchers for simple tank configurations on the governing equations based on the

7


and Welch, 1965). For liquid in tanks of arbitrary shapes subjected to different kinds of

motion, the liquid behaviour is usually described by the continuity equation, momentum

equations, kinematic and dynamic conditions on the free surface, and the velocity

conditions on the rigid walls. Based on some simplifying assumptions, the sloshing

problem could be modeled by the second set o f equations based on the potential flow

theory. The adopted assumptions include: rigid tank, irrotational flow field, nonviscous

fluid, homogeneous fluid, incompressible fluid, no sinks or sources, and single-valued

velocity potential in any simply connected region (Abramson and Silverman, 1966).

Under these assumptions, the liquid behaviour can be established using the velocity

potential. The governing equation o f the liquid motion should satisfy the Laplace

equation inside the liquid domain. The kinematic and dynamic conditions are to be

satisfied on the free surface. Relative normal velocities on the rigid walls should be zero.

(1) Analytical and semi-analytical solution

For many years, mathematical studies o f liquid sloshing problems were basically

carried out on how to solve either the Navier-Stokes equations or the equations based on

the potential theory. Analytical or semi-analytical solutions o f the Navier-Stokes

equations are not available without the adoption o f further assumptions. Numerical

schemes can usually be employed to discretize the Navier-Stokes equations in such a way

that nonlinearities and viscous effect can be retained without any simplification. On the

contrary, analytical or semi-analytical solutions were tried extensively by many

researchers for simple tank configurations on the governing equations based on the

7


potential flow theory. These solutions can be divided into two classes, i.e., linear

solutions and non-linear solutions.

When the sloshing problems are further simplified with the assumptions that tanks

are subjected to small displacements or velocities and free surface slopes are small, the

boundary conditions at the free surface can be completely linearized, which makes some

of the sloshing problems solvable by the method of separation of variables. Linear

sloshing theory has been well established for certain kinds of tanks (Abramson and

Silverman, 1966). Liquid behaviour, such as the liquid height inside the tanks, liquid

forces and moments on the tank walls under single harmonic sway, pitch or yaw tank

motion, can be explicitly expressed after the velocity potential is obtained in the liquid

domain. At the same time, the eigenvalue problem, which represents the oscillation of the

free surface of the liquid inside a stationary tank, can be solved by the linear sloshing

theory to obtain the natural frequencies of fluid oscillation and the corresponding

sloshing modes (Moiseev and Petrov, 1966).

The situations where the linear sloshing theory could be applicable are rather

limited. It is apparent that nonlinear effects would be present and would sometimes

govern the character of the liquid motions. Such nonlinear effects might be described in

terms of three classes (Abramson and Silverman, 1966): (a) nonlinear effects arising as a

consequence of the container geometry, (b) nonlinear effects arising as a consequence of

large amplitude excitation and response, and (c) nonlinear effects arising as a

consequence of coupling or instabilities of different sloshing modes.

Ibrahim et al (2001) summarized three of the main theories that were developed

for treating the nonlinear liquid free surface in rectangular and upright cylindrical

8


potential flow theory. These solutions can be divided into two classes, i.e., linear

solutions and non-linear solutions.

When the sloshing problems are further simplified with the assumptions that tanks

are subjected to small displacements or velocities and free surface slopes are small, the

boundary conditions at the free surface can be completely linearized, which makes some

o f the sloshing problems solvable by the method of separation o f variables. Linear

sloshing theory has been well established for certain kinds o f tanks (Abramson and

Silverman, 1966). Liquid behaviour, such as the liquid height inside the tanks, liquid

forces and moments on the tank walls under single harmonic sway, pitch or yaw tank

motion, can be explicitly expressed after the velocity potential is obtained in the liquid

domain. At the same time, the eigenvalue problem, which represents the oscillation of the

free surface of the liquid inside a stationary tank, can be solved by the linear sloshing

theory to obtain the natural frequencies o f fluid oscillation and the corresponding

sloshing modes (Moiseev and Petrov, 1966).

The situations where the linear sloshing theory could be applicable are rather

limited. It is apparent that nonlinear effects would be present and would sometimes

govern the character of the liquid motions. Such nonlinear effects might be described in

terms of three classes (Abramson and Silverman, 1966): (a) nonlinear effects arising as a

consequence o f the container geometry, (b) nonlinear effects arising as a consequence of

large amplitude excitation and response, and (c) nonlinear effects arising as a

consequence o f coupling or instabilities of different sloshing modes.

Ibrahim et al (2001) summarized three o f the main theories that were developed

for treating the nonlinear liquid free surface in rectangular and upright cylindrical

8


containers. The first of these theories is Moiseev's theory (1958), which constructed

normal mode functions and characteristic numbers by integral equations in terms of

Green's function of the second kind. The second of these theories is Penny and Price's

theory (1952). Their method carried out a successive approximate approach where the

potential function was expressed as a Fourier series in space with coefficients that were

functions of time. These coefficients were again approximated by a Fourier time series

using the method of perturbation. The resulting solution was given as a double Fourier

series in space and time. The third of these theories is Hutton's theory (1963). Hutton

investigated the motion of fluid in an upright cylindrical tank subjected to lateral

harmonic vibration at a frequency in the neighbourhood of the lowest resonant frequency

of fluid. The Laplace equation and boundary conditions at the walls and free surface were

solved by the separation of variables and the Rayleigh-Ritz procedure. The investigation

indicated that nonplanar fluid motion was due to a nonlinear coupling between fluid

motions parallel and perpendicular to the plane excitation, and that this coupling took

place through the free surface waves.

All of the above three theories were further developed or applied in later

investigations on nonlinear liquid sloshing or liquid-structure coupling dynamics. A

detailed review of the development of the theoretical studies can be found in Ibrahim et al

(2001).

Analytical and semi-analytical solutions for rectangular tanks have also been

further advanced in the past several years. For example, Faltinsen and co-authors further

studied the nonlinear sloshing problems in 2D and 3D rectangular tanks using the modal

theory (Faltinsen et al., 2000, Faltinsen and Timokha, 2001, Faltinsen and Timokha, 2002,

9


containers. The first o f these theories is Moiseev’s theory (1958), which constructed

normal mode functions and characteristic numbers by integral equations in terms of

Green’s function o f the second kind. The second of these theories is Penny and Price’s

theory (1952). Their method carried out a successive approximate approach where the

potential function was expressed as a Fourier series in space with coefficients that were

functions o f time. These coefficients were again approximated by a Fourier time series

using the method o f perturbation. The resulting solution was given as a double Fourier

series in space and time. The third of these theories is Hutton’s theory (1963). Hutton

investigated the motion o f fluid in an upright cylindrical tank subjected to lateral

harmonic vibration at a frequency in the neighbourhood of the lowest resonant frequency

of fluid. The Laplace equation and boundary conditions at the walls and free surface were

solved by the separation of variables and the Rayleigh-Ritz procedure. The investigation

indicated that nonplanar fluid motion was due to a nonlinear coupling between fluid

motions parallel and perpendicular to the plane excitation, and that this coupling took

place through the free surface waves.

All o f the above three theories were further developed or applied in later

investigations on nonlinear liquid sloshing or liquid-structure coupling dynamics. A

detailed review of the development of the theoretical studies can be found in Ibrahim et al

(2001).

Analytical and semi-analytical solutions for rectangular tanks have also been

further advanced in the past several years. For example, Faltinsen and co-authors further

studied the nonlinear sloshing problems in 2D and 3D rectangular tanks using the modal

theory (Faltinsen et al., 2000, Faltinsen and Timokha, 2001, Faltinsen and Timokha, 2002,

9


Faltinsen et al., 2003, Solaas, 1995). The infinite-dimensional nonlinear modal theory

was first proposed by Miles (1976). Faltinsen et al. (2000) later generalized this method.

In the modal representation, Fourier series with time dependent coefficients (modal

functions) were used to describe the free surface evolution. Instead of prescribing the

acceleration of the tank, which was often adopted in the pure sloshing studies, the motion

of the tank was described by a pair of time-dependent vectors denoting instantaneous

translatory and angular velocities of a mobile Cartesian coordinate system rigidly framed

with the body relative to an absolute coordinate system. This made it possible to include

the instantaneous tank motion by solving the governing equations of the whole structures.

This facilitated the coupled liquid-structure dynamics studies.

Sloshing in upright cylindrical tanks is another area where analytical and semi-

analytical solutions have obtained great progress under both linear and nonlinear

conditions. In recent years, the investigations on this problem have been extended to

different cases, such as frictionless free surface liquid in upright cylindrical tanks

partially covered by an annular rigid surface (Bauer and Eidel, 1999a), viscous liquid

sloshing problems in cylindrical containers (Bauer and Eidel, 1999b), viscous liquid in

cylindrical containers with membrane and plate covers (Bauer and Chiba, 2000),

nonlinear oscillations in circular cylindrical containers separated by diametrical barriers

(Solodun, 2002), and the analytical investigation of vibration characteristics of the

sloshing and bulging modes for a liquid-filled rigid circular cylindrical storage tank with

an elastic annular plate in contact with sloshing surface of liquid (Kim and Lee, 2004).

Lukovsky et al (2002) studied the nonlinear fluid sloshing in tanks with non-vertical

walls using the modal modeling method. The fluid sloshing in circular conic tanks was

10


Faltinsen et al., 2003, Solaas, 1995). The infinite-dimensional nonlinear modal theory

was first proposed by Miles (1976). Faltinsen et al. (2000) later generalized this method.

In the modal representation, Fourier series with time dependent coefficients (modal

functions) were used to describe the free surface evolution. Instead o f prescribing the

acceleration of the tank, which was often adopted in the pure sloshing studies, the motion

o f the tank was described by a pair o f time-dependent vectors denoting instantaneous

translatory and angular velocities o f a mobile Cartesian coordinate system rigidly framed

with the body relative to an absolute coordinate system. This made it possible to include

the instantaneous tank motion by solving the governing equations o f the whole structures.

This facilitated the coupled liquid-structure dynamics studies.

Sloshing in upright cylindrical tanks is another area where analytical and semi-

analytical solutions have obtained great progress under both linear and nonlinear

conditions. In recent years, the investigations on this problem have been extended to

different cases, such as frictionless free surface liquid in upright cylindrical tanks

partially covered by an annular rigid surface (Bauer and Eidel, 1999a), viscous liquid

sloshing problems in cylindrical containers (Bauer and Eidel, 1999b), viscous liquid in

cylindrical containers with membrane and plate covers (Bauer and Chiba, 2000),

nonlinear oscillations in circular cylindrical containers separated by diametrical barriers

(Solodun, 2002), and the analytical investigation o f vibration characteristics o f the

sloshing and bulging modes for a liquid-filled rigid circular cylindrical storage tank with

an elastic annular plate in contact with sloshing surface of liquid (Kim and Lee, 2004).

Lukovsky et al (2002) studied the nonlinear fluid sloshing in tanks with non-vertical

walls using the modal modeling method. The fluid sloshing in circular conic tanks was

10


solved by spectral and variational theorems and a non-conformal mapping technique was

developed to transform the fluid domain to an artificial cylindrical domain, in which five

modes are used to approximate the free surface and the velocity potential.

(2) Numerical solutions

Numerical studies on the liquid sloshing problems were advanced by new

developments in computers and computational technologies in recent years. Different

discretization schemes were applied in two manners. First, nonlinear liquid sloshing

problems under large liquid wave heights were widely investigated for simple containers

such as rectangular and upright cylindrical containers, for which analytical solutions are

only available for linear situations, with semi-analytical solutions being available for

limited nonlinear cases. Second, researchers used the numerical methods to handle

special types of tank geometry, such as containers with non-vertical and non-straight

walls.

Five basic discretization methods are in common use. These are the finite

difference method, finite volume method, finite element method, boundary element

method, and pseudo-spectral method. Of these methods, the boundary element method

was traditionally considered the best choice for studying the liquid sloshing problems

because this discretizes the governing equations only on the rigid walls and the free

surface of liquid, while the others do the same thing in the entire liquid domain, and thus

require longer computational time and larger storage space in memory.

The most difficult thing in simulating free surface flow is the boundary conditions

on the time-varying free surface that needs to be located in the period of calculation.

11


solved by spectral and variational theorems and a non-conformal mapping technique was

developed to transform the fluid domain to an artificial cylindrical domain, in which five

modes are used to approximate the free surface and the velocity potential.

(2) Numerical solutions

Numerical studies on the liquid sloshing problems were advanced by new

developments in computers and computational technologies in recent years. Different

discretization schemes were applied in two manners. First, nonlinear liquid sloshing

problems under large liquid wave heights were widely investigated for simple containers

such as rectangular and upright cylindrical containers, for which analytical solutions are

only available for linear situations, with semi-analytical solutions being available for

limited nonlinear cases. Second, researchers used the numerical methods to handle

special types o f tank geometry, such as containers with non-vertical and non-straight

walls.

Five basic discretization methods are in common use. These are the finite

difference method, finite volume method, finite element method, boundary element

method, and pseudo-spectral method. O f these methods, the boundary element method

was traditionally considered the best choice for studying the liquid sloshing problems

because this discretizes the governing equations only on the rigid walls and the free

surface o f liquid, while the others do the same thing in the entire liquid domain, and thus

require longer computational time and larger storage space in memory.

The most difficult thing in simulating free surface flow is the boundary conditions

on the time-varying free surface that needs to be located in the period o f calculation.

11


Generally, there are three approaches that could be adopted to compute the free surface:

(1) Lagrangian mesh methods, (2) Eulerian mesh methods, and (3) hybrid Eulerian-

Lagrangian mesh methods. In Lagrangian mesh methods, the mesh moves with the fluids

and the edges of the mesh construct the fluid domain. Therefore, the basic task in these

methods is to re-establish the meshes all the time according to the liquid distribution and

liquid motion. In Eulerian methods, the mesh in the calculation domain is fixed, and the

fluid occupies the mesh. The main problem is to determine which cells of the mesh are

occupied by fluid. An indicator function is usually used to mark the fluids on both sides

of the free surface. In the hybrid Eulerian-Lagrangian mesh methods, the free surface is

represented and tracked explicitly by special marker points. However, the grid also

remains fixed.

Among many algorithms of updating the free surface in the free surface flow, the

Volume of Fluid (VOF) method (Hirt and Nichols, 1981) was found to be a very effective

method that could handle relatively large fluid motions, discontinuous surface segments,

multi-values surfaces, and the surface collapsing upon itself. This made the VOF method

capable of simulating some extreme situations such as overturning waves, breaking

waves, impacts on tank top covers, as well as large wave heights. In recent years, the

Volume of Fluid method has been further developed by many researchers (Rudman,

1997, Rider and Kothe, 1998, Harvie and Fletcher, 2000).

Although the sloshing problems are different in some respects from free surface

flow, especially the steady state free surface flow, the Volume of Fluid method has also

been applied in simulating sloshing problems due to the existence of the free surface. For

example, Kim and Lee (2003) and Kim et al (2003) proposed a free surface tracking

12


Generally, there are three approaches that could be adopted to compute the free surface:

(1) Lagrangian mesh methods, (2) Eulerian mesh methods, and (3) hybrid Eulerian-

Lagrangian mesh methods. In Lagrangian mesh methods, the mesh moves with the fluids

and the edges o f the mesh construct the fluid domain. Therefore, the basic task in these

methods is to re-establish the meshes all the time according to the liquid distribution and

liquid motion. In Eulerian methods, the mesh in the calculation domain is fixed, and the

fluid occupies the mesh. The main problem is to determine which cells o f the mesh are

occupied by fluid. An indicator function is usually used to mark the fluids on both sides

o f the free surface. In the hybrid Eulerian-Lagrangian mesh methods, the free surface is

represented and tracked explicitly by special marker points. However, the grid also

remains fixed.

Among many algorithms o f updating the free surface in the free surface flow, the

Volume of Fluid (VOF) method (Hirt and Nichols, 1981) was found to be a very effective

method that could handle relatively large fluid motions, discontinuous surface segments,

multi-values surfaces, and the surface collapsing upon itself. This made the VOF method

capable o f simulating some extreme situations such as overturning waves, breaking

waves, impacts on tank top covers, as well as large wave heights. In recent years, the

Volume o f Fluid method has been further developed by many researchers (Rudman,

1997, Rider and Kothe, 1998, Harvie and Fletcher, 2000).

Although the sloshing problems are different in some respects from free surface

flow, especially the steady state free surface flow, the Volume o f Fluid method has also

been applied in simulating sloshing problems due to the existence of the free surface. For

example, Kim and Lee (2003) and Kim et al (2003) proposed a free surface tracking

12


scheme which is based on the orientation vector to represent the free surface orientation

in each cell, and the baby-cell to determine the fluid volume flux at each cell boundary.

Sloshing simulation for large free surface motion in 3D rectangular oil tanks subjected to

harmonic motions in all six directions were conducted using this scheme in combination

with the finite element method.

For simulations of a free-surface flow problem, it is often very efficient to map

the liquid area into a rectangular computational domain on which the transformed

equations are discretized and solved (Tsai and Yue, 1996, Thompson et al, 1982). In

cases of non-breaking and non-overturning waves with single-valued liquid heights, the

sigma-transformation, which was originally proposed for meteorological forecasting by

Phillips (1957), has recently been applied in investigating the hydrodynamic problem,

non-sloshing free surface flow and free surface sloshing problems in several studies with

different numerical methods. These problems include pressures on dams by Navier-

Stokes equations and the finite difference method (Chen, 1994), 2D sloshing in

rectangular containers by the potential theory and the pseudospectral method (Chem et al,

1999), 3D standing and impulse waves in an upright cylindrical container with a central

cylindrical inclusion by potential theory and the spectral method (Chem et al, 2003), 2D

sloshing in rectangular containers by the potential theory and the finite element method

(Turnbull et al, 2003), 2D sloshing in rectangular containers by the potential theory and

the finite difference method (Frandsen, 2003, Frandsen and Borthwick 2003), and 2D

sloshing in rectangular containers by the potential theory and the Finite volume method

(Bucchignani, 2004). This has also been applied in sloshing problems in horizontal

circular and cylindrical tanks by using continuous coordinate transformations (Dai and

13


scheme which is based on the orientation vector to represent the free surface orientation

in each cell, and the baby-cell to determine the fluid volume flux at each cell boundary.

Sloshing simulation for large free surface motion in 3D rectangular oil tanks subjected to

harmonic motions in all six directions were conducted using this scheme in combination

with the finite element method.

For simulations o f a free-surface flow problem, it is often very efficient to map

the liquid area into a rectangular computational domain on which the transformed

equations are discretized and solved (Tsai and Yue, 1996, Thompson et al, 1982). In

cases o f non-breaking and non-overturning waves with single-valued liquid heights, the

sigma-transformation, which was originally proposed for meteorological forecasting by

Phillips (1957), has recently been applied in investigating the hydrodynamic problem,

non-sloshing free surface flow and free surface sloshing problems in several studies with

different numerical methods. These problems include pressures on dams by Navier-

Stokes equations and the finite difference method (Chen, 1994), 2D sloshing in


1999), 3D standing and impulse waves in an upright cylindrical container with a central

cylindrical inclusion by potential theory and the spectral method (Chem et al, 2003), 2D

sloshing in rectangular containers by the potential theory and the finite element method

(Turnbull et al, 2003), 2D sloshing in rectangular containers by the potential theory and

the finite difference method (Frandsen, 2003, Frandsen and Borthwick 2003), and 2D

sloshing in rectangular containers by the potential theory and the Finite volume method

(Bucchignani, 2004). This has also been applied in sloshing problems in horizontal

circular and cylindrical tanks by using continuous coordinate transformations (Dai and

13


Xu, 2004, Xu and Dai, 2005a, Xu and Dai, 2005b, Xu and Dai, 2005c). By transforming

the physical liquid domain in a rectangular container onto a rectangular region bounded

by horizontal and vertical sides, the solution in the transformed computational domain

exactly fits the free surface boundary.

Liquid motion inside a tank depends largely on the tank geometry, liquid fill

levels, different excitation amplitudes and frequencies. There exists no universal

computational procedure that can be used for general nonlinear sloshing problems.

Although numerous publications (Wu et al, 1998, Liu and Huang, 1994) in numerical

simulations of liquid sloshing could be found in recent years, most of them focused on

the 2D and 3D rectangular and upright cylindrical containers. For tanks of other shapes

and configurations, they cannot be used directly. Therefore, there is still much work that

should be done to establish effective and efficient numerical schemes with good

convergence, high accuracy and a low computational cost for more tank configurations.

One disadvantage of numerical analysis is that it is difficult in the parameter studies.

Since the simulations are all carried out in the time domain, calculation should be

repeated if any one of the system parameters is changed. This sometimes can be time-

consuming.

2.2 Liquid-structure systems

Tanks for carrying or storing liquid are always connected with other structural

components. Depending on different applications, the structure system can be as simple

as the tank itself or as complicated as an aerospace craft or a B-train tank truck. Generally,

there are two types of interaction in the liquid-structure system. In the first type, the

14


Xu, 2004, Xu and Dai, 2005a, Xu and Dai, 2005b, Xu and Dai, 2005c). By transforming

the physical liquid domain in a rectangular container onto a rectangular region bounded

by horizontal and vertical sides, the solution in the transformed computational domain

exactly fits the free surface boundary.

Liquid motion inside a tank depends largely on the tank geometry, liquid fill

levels, different excitation amplitudes and frequencies. There exists no universal

computational procedure that can be used for general nonlinear sloshing problems.

Although numerous publications (Wu et al, 1998, Liu and Huang, 1994) in numerical

simulations o f liquid sloshing could be found in recent years, most o f them focused on

the 2D and 3D rectangular and upright cylindrical containers. For tanks o f other shapes

and configurations, they cannot be used directly. Therefore, there is still much work that

should be done to establish effective and efficient numerical schemes with good

convergence, high accuracy and a low computational cost for more tank configurations.

One disadvantage of numerical analysis is that it is difficult in the parameter studies.

Since the simulations are all carried out in the time domain, calculation should be

repeated if any one o f the system parameters is changed. This sometimes can be time-

consuming.

2.2 Liquid-structure systems

Tanks for carrying or storing liquid are always connected with other structural

components. Depending on different applications, the structure system can be as simple

as the tank itself or as complicated as an aerospace craft or a B-train tank truck. Generally,

there are two types o f interaction in the liquid-structure system. In the first type, the

14


components in the structure system, especially the tanks themselves, are considered as

flexible elements, for which the deformation theory should be adopted to solve both fluid

and solid simultaneously by either analytical or numerical methods. This is important for

situations where sloshing loads contribute considerably to the damage of tank structures,

for example, storage tanks in civil engineering. In the second type, the structure system is

described by the rigid body dynamics. For the complicated structure system such as

liquid cargo tank vehicles, structural components experience large rigid body motions

with small elastic deformations. The flexibility of one individual component has no

significant effect on the overall dynamic behaviour of the system. A rigid body model for

tanks can be employed.

Dynamic behaviour of liquid in the liquid-structure systems is quite different from

that of the pure fluid mechanics problem, in which the tank motions are usually

prescribed to be harmonic oscillations. In coupled liquid-structure systems, the tank

motions should be worked out from the governing equations of the structure system. Tank

motions in three translatory and three rotational directions are quite common. Transient

tank dynamics can sometimes be much more important than the steady state harmonic

responses. At the same time, the forces and moments caused by the pressure distribution

on the tank walls will have significant influence on the tank and whole structure system,

which in turn will change the liquid motion in the tank.

Lui (1990) studied the dynamic coupling of a fluid-tank system under transient

excitations. The tank system was modeled as a spring-damping-mass system and

described by Newton's Second Law of motion. The fluid in a 2D rectangular tank was

solved by the stream function. The forces and moments were calculated to link the two

15


components in the structure system, especially the tanks themselves, are considered as

flexible elements, for which the deformation theory should be adopted to solve both fluid

and solid simultaneously by either analytical or numerical methods. This is important for

situations where sloshing loads contribute considerably to the damage o f tank structures,

for example, storage tanks in civil engineering. In the second type, the structure system is

described by the rigid body dynamics. For the complicated structure system such as

liquid cargo tank vehicles, structural components experience large rigid body motions

with small elastic deformations. The flexibility o f one individual component has no

significant effect on the overall dynamic behaviour o f the system. A rigid body model for

tanks can be employed.

Dynamic behaviour o f liquid in the liquid-structure systems is quite different from

that of the pure fluid mechanics problem, in which the tank motions are usually

prescribed to be harmonic oscillations. In coupled liquid-structure systems, the tank

motions should be worked out from the governing equations o f the structure system. Tank

motions in three translatory and three rotational directions are quite common. Transient

tank dynamics can sometimes be much more important than the steady state harmonic

responses. At the same time, the forces and moments caused by the pressure distribution

on the tank walls will have significant influence on the tank and whole structure system,

which in turn will change the liquid motion in the tank.

Lui (1990) studied the dynamic coupling o f a fluid-tank system under transient

excitations. The tank system was modeled as a spring-damping-mass system and

described by Newton’s Second Law of motion. The fluid in a 2D rectangular tank was

solved by the stream function. The forces and moments were calculated to link the two

15


subsystems. The linearized system was transformed into the frequency domain by the

Laplace transformation. The influence of the fluid on the system was investigated for

rectilinear motion and rolling motion of the tank and compared to the equivalent non-

shifting cargo system. Peterson (1989) investigated the dynamics of a spacecraft model

coupled with the nonlinear low-gravity slosh of a fluid in an upright cylindrical tank. The

coupled nonlinear equations of motion for the fluid-spacecraft system were derived

through the Lagrangian method with two fundamental slosh modes and three secondary

slosh modes. An approximate perturbation solution of the equations of motion was

obtained.

Ibrahim and co-authors (Ibrahim and Barr, 1975a, Ibrahim and Barr, 1975b,

Ibrahim et al 1988b, Ibrahim and Li, 1988, Soundararajan and Ibrahim, 1988) studied the

parametric and autoparametric vibrations of an elevated water tower, in which the

container was an upright cylinder. The non-linear dynamic response of the system

subjected to the prescribed harmonic base excitations was examined by the interaction of

the lowest liquid sloshing mode with structural modes to show the characteristics of the

internal resonance and combination resonance.

Ikeda and Nakagawa (1997) studied the nonlinear vibrations of a structure

coupled with water sloshing in a 2D rectangular tank subjected to horizontal harmonic

excitations theoretically and experimentally. The structure was modeled by a mass-

spring-damping system. The liquid sloshing was solved by the modal method, in which

the velocity potential and the free surface were expressed by the natural linear liquid

modes with dynamic coefficients. The coupled nonlinear governing equations were

solved by numerical integration, and the theoretical resonance curves versus the

16


subsystems. The linearized system was transformed into the frequency domain by the

Laplace transformation. The influence of the fluid on the system was investigated for

rectilinear motion and rolling motion of the tank and compared to the equivalent non

shifting cargo system. Peterson (1989) investigated the dynamics o f a spacecraft model

coupled with the nonlinear low-gravity slosh of a fluid in an upright cylindrical tank. The

coupled nonlinear equations of motion for the fluid-spacecraft system were derived

through the Lagrangian method with two fundamental slosh modes and three secondary

slosh modes. An approximate perturbation solution o f the equations o f motion was

obtained.

Ibrahim and co-authors (Ibrahim and Barr, 1975a, Ibrahim and Barr, 1975b,

Ibrahim et al 1988b, Ibrahim and Li, 1988, Soundararajan and Ibrahim, 1988) studied the

parametric and autoparametric vibrations o f an elevated water tower, in which the

container was an upright cylinder. The non-linear dynamic response of the system

subjected to the prescribed harmonic base excitations was examined by the interaction of

the lowest liquid sloshing mode with structural modes to show the characteristics of the

internal resonance and combination resonance.

Ikeda and Nakagawa (1997) studied the nonlinear vibrations o f a structure

coupled with water sloshing in a 2D rectangular tank subjected to horizontal harmonic

excitations theoretically and experimentally. The structure was modeled by a mass-

spring-damping system. The liquid sloshing was solved by the modal method, in which

the velocity potential and the free surface were expressed by the natural linear liquid

modes with dynamic coefficients. The coupled nonlinear governing equations were

solved by numerical integration, and the theoretical resonance curves versus the

16


excitation frequency were obtained by the harmonic balance method. Ikeda (2003)

studied a similar coupled system for a 2D rectangular container subjected to vertical

harmonic excitations.

The review of the coupled liquid-structure system shows that for 2D and 3D

rectangular containers and upright cylindrical containers, the governing equations based

on the potential theory can be approximately solved by the modal method, in which the

liquid height and the velocity potential are expressed by the sums of the linearized liquid

modes.

The numerical methods for the liquid sloshing problems developed quickly due to

the application of high performance computers, and considerable researches were

conducted using the numerical methods in recent years. The application of these

numerical methods facilitates the studies on dynamics of the coupled liquid-structure

systems. The basic procedure of numerical methods for coupled liquid-structure systems

is quite clear. The liquid motion is solved directly from the Navier-Stokes equations or

the equations based on the potential theory with the assumption of incompressible,

irrotational, and inviscid liquid. Applications can be found by almost all existing

discretization schemes, such as the finite element method, finite difference method,

boundary element method, finite volume method, and spectral method. The forces and

moments caused by liquid motion are then obtained by integrating liquid pressures on the

containers and used as the excitations for the structures. The motion of the tanks can then

be established by the common methods for multi-DOF vibration, such as Newton's law

of motion, Lagrange's method or the Hamilton Principle. Due to the coupling of the

liquid motion and the structure motion, numerical methods, such as the Runge-Kutta

17


excitation frequency were obtained by the harmonic balance method. Ikeda (2003)

studied a similar coupled system for a 2D rectangular container subjected to vertical

harmonic excitations.

The review of the coupled liquid-structure system shows that for 2D and 3D

rectangular containers and upright cylindrical containers, the governing equations based

on the potential theory can be approximately solved by the modal method, in which the

liquid height and the velocity potential are expressed by the sums of the linearized liquid

modes.

The numerical methods for the liquid sloshing problems developed quickly due to

the application o f high performance computers, and considerable researches were

conducted using the numerical methods in recent years. The application o f these

numerical methods facilitates the studies on dynamics of the coupled liquid-structure

systems. The basic procedure of numerical methods for coupled liquid-structure systems

is quite clear. The liquid motion is solved directly from the Navier-Stokes equations or

the equations based on the potential theory with the assumption o f incompressible,

irrotational, and inviscid liquid. Applications can be found by almost all existing

discretization schemes, such as the finite element method, finite difference method,

boundary element method, finite volume method, and spectral method. The forces and

moments caused by liquid motion are then obtained by integrating liquid pressures on the

containers and used as the excitations for the structures. The motion o f the tanks can then

be established by the common methods for multi-DOF vibration, such as Newton’s law

of motion, Lagrange’s method or the Hamilton Principle. Due to the coupling of the

liquid motion and the structure motion, numerical methods, such as the Runge-Kutta

17


method, need to be used to get the dynamic response of the containers that will be

imported as the input excitation for the liquid for the next time step.

2.3 Sloshing in horizontal cylindrical tanks

Horizontal cylindrical tanks with circular and elliptical cross sections are widely

used in road transportation and civil engineering. Liquid sloshing could happen, and

could even become a big problem when the tanks are partially filled. However, unlike the

sloshing problems in rectangular and upright cylindrical tanks, liquid motions in the

horizontal cylindrical tanks and the resultant structure-liquid interactions have only been

studied in limited investigations, due to difficulties in dealing with the boundary

conditions on the curved walls and the free surface in the time-varying domain. A special

review is conducted as follows for the analytical, experimental and numerical

investigations on the liquid sloshing problems in horizontal cylindrical tanks.

Early analytical results on sloshing frequency for a half-full circular container

were given by Lamb (1945). Numerical techniques were later used by several researchers

to obtain the sloshing frequencies for a circular container with different fill levels.

Budiansky (1960) developed an integral-equation approach to analyze the natural

frequencies and natural modes in a 2D circular canal for arbitrary depth of liquid.

Sloshing forces were also obtained. Moiseev and Petrov (1966) described the application

of the Ritz variational method for the numerical calculation of sloshing frequencies in

vessels of various geometries, including the case of a horizontal cylindrical container.

McIver (1989) studied the 2D sloshing frequencies of fluid in a horizontal circular

cylindrical container and the 3D sloshing frequencies of fluid in a spherical container.

18


method, need to be used to get the dynamic response of the containers that will be

imported as the input excitation for the liquid for the next time step.

2.3 Sloshing in horizontal cylindrical tanks

Horizontal cylindrical tanks with circular and elliptical cross sections are widely

used in road transportation and civil engineering. Liquid sloshing could happen, and

could even become a big problem when the tanks are partially filled. However, unlike the

sloshing problems in rectangular and upright cylindrical tanks, liquid motions in the

horizontal cylindrical tanks and the resultant structure-liquid interactions have only been

studied in limited investigations, due to difficulties in dealing with the boundary

conditions on the curved walls and the free surface in the time-varying domain. A special

review is conducted as follows for the analytical, experimental and numerical

investigations on the liquid sloshing problems in horizontal cylindrical tanks.

Early analytical results on sloshing frequency for a half-full circular container

were given by Lamb (1945). Numerical techniques were later used by several researchers

to obtain the sloshing frequencies for a circular container with different fill levels.

Budiansky (1960) developed an integral-equation approach to analyze the natural

frequencies and natural modes in a 2D circular canal for arbitrary depth o f liquid.

Sloshing forces were also obtained. Moiseev and Petrov (1966) described the application

o f the Ritz variational method for the numerical calculation o f sloshing frequencies in

vessels o f various geometries, including the case of a horizontal cylindrical container.

Mclver (1989) studied the 2D sloshing frequencies o f fluid in a horizontal circular

cylindrical container and the 3D sloshing frequencies o f fluid in a spherical container.

18


The linearized wave theory was used to determine the frequencies of free oscillations

under gravity of an arbitrary amount of fluid by bipolar coordinates for circular

containers and toroidal coordinates for spherical containers. The eigenvalues were solved

numerically using integral equations. McIver and McIver (1993) presented analytical

methods to obtain upper and lower bounds of sloshing frequencies in horizontal

cylinders.

Analytical solutions for dynamic liquid behaviour in horizontal cylindrical

containers under forced oscillations are not available even for linearized problems. In

recent years, numerical investigations on sloshing problems in 2D horizontal circular

containers have been carried out by several studies using different discretization

approaches.

Tosaka et al (1989) developed a numerical approximation procedure to compute

the unsteady irrotational motion of an inviscid and incompressible fluid with a free

surface by the boundary element approach. The constant boundary element was used, and

the fluid motion in a half-filled circular container subjected to a forced horizontal

acceleration was simulated. Tosaka and Rugino (1990) further simulated the sloshing in a

partially filled circular container subjected to forced horizontal accelerations with large

amplitudes. The authors also simulated the sloshing in 2D elliptical containers with the

major axis in the gravitational direction and the minor axis in the gravitational direction.

Ortiz et al (1998) applied the boundary element method in simulating the 2D

large-displacement non-linear sloshing in circular containers subjected to prescribed

motions. The free surface position was updated by an adaptive technique and smoothing

and volume correction approaches were used in a polar coordinate system. The fluid

19


The linearized wave theory was used to determine the frequencies o f free oscillations

under gravity of an arbitrary amount o f fluid by bipolar coordinates for circular

containers and toroidal coordinates for spherical containers. The eigenvalues were solved

numerically using integral equations. Mclver and Mclver (1993) presented analytical

methods to obtain upper and lower bounds of sloshing frequencies in horizontal

cylinders.

Analytical solutions for dynamic liquid behaviour in horizontal cylindrical

containers under forced oscillations are not available even for linearized problems. In

recent years, numerical investigations on sloshing problems in 2D horizontal circular

containers have been carried out by several studies using different discretization

approaches.

Tosaka et al (1989) developed a numerical approximation procedure to compute

the unsteady irrotational motion o f an inviscid and incompressible fluid with a free

surface by the boundary element approach. The constant boundary element was used, and

the fluid motion in a half-filled circular container subjected to a forced horizontal

acceleration was simulated. Tosaka and Rugino (1990) further simulated the sloshing in a

partially filled circular container subjected to forced horizontal accelerations with large

amplitudes. The authors also simulated the sloshing in 2D elliptical containers with the

major axis in the gravitational direction and the minor axis in the gravitational direction.

Ortiz et al (1998) applied the boundary element method in simulating the 2D

large-displacement non-linear sloshing in circular containers subjected to prescribed

motions. The free surface position was updated by an adaptive technique and smoothing

and volume correction approaches were used in a polar coordinate system. The fluid

19


motion was modeled by the potential flow theory with the Rayleigh damping. A simple

road container model with horizontal force and vertical road profile input was

investigated. However, this simple model cannot be adopted and extended to simulate the

real road containers because the normal direction of the 2D circular plane was set to be

perpendicular to the driving direction, which was not the case for the horizontal

cylindrical tanks used in tank vehicles, for which the normal direction of the cross section

is the same as the driving direction.

Behr (2004) numerically studied the application of a slip boundary condition on

curved boundaries. Finite element analysis was used on a 2D half-filled circular tank. It

was concluded that a BC-free boundary condition was effective in maintaining both the

stationary hydrostatic solution and the expected non-recirculating transient solution.

Sloshing in a half-full circular container was also simulated using the finite volume

method by Sames et al (2002).

It was found that the existing numerical methods and algorithms that deal with

sloshing problems in 2D circular tanks can hardly be easily extended to corresponding

3D problems, i.e., sloshing in horizontal cylindrical tanks. Partom (1987) extended the

2D Volume of Fluid method to 3D sloshing in horizontal cylindrical containers. The

author gave a feeling for the capabilities and drawbacks of the Volume of Fluid method in

3D sloshing. When the procedure of free surface updating was extended to 3D geometry,

it could be extremely intricate and the algorithm called for considerable programming

sophistication. The stability of the program could sometimes hardly be guaranteed.

For the particular case of an exactly half-full horizontal cylinder and sphere,

Evans and Linton (1993) presented a series-type (semi-analytical) solution to the

20


motion was modeled by the potential flow theory with the Rayleigh damping. A simple

road container model with horizontal force and vertical road profile input was

investigated. However, this simple model cannot be adopted and extended to simulate the

real road containers because the normal direction o f the 2D circular plane was set to be

perpendicular to the driving direction, which was not the case for the horizontal

cylindrical tanks used in tank vehicles, for which the normal direction of the cross section

is the same as the driving direction.

Behr (2004) numerically studied the application of a slip boundary condition on

curved boundaries. Finite element analysis was used on a 2D half-filled circular tank. It

was concluded that a BC-free boundary condition was effective in maintaining both the

stationary hydrostatic solution and the expected non-recirculating transient solution.

Sloshing in a half-full circular container was also simulated using the finite volume

method by Sames et al (2002).

It was found that the existing numerical methods and algorithms that deal with

sloshing problems in 2D circular tanks can hardly be easily extended to corresponding

3D problems, i.e., sloshing in horizontal cylindrical tanks. Partom (1987) extended the

2D Volume of Fluid method to 3D sloshing in horizontal cylindrical containers. The

author gave a feeling for the capabilities and drawbacks of the Volume o f Fluid method in

3D sloshing. When the procedure of free surface updating was extended to 3D geometry,

it could be extremely intricate and the algorithm called for considerable programming

sophistication. The stability of the program could sometimes hardly be guaranteed.

For the particular case of an exactly half-full horizontal cylinder and sphere,

Evans and Linton (1993) presented a series-type (semi-analytical) solution to the

20


eigenvalue problem. The velocity potential was expanded in terms of non-orthogonal

bounded harmonic spatial functions. This method has recently been applied by

Papaspyrou et al (2004) in investigating the response of half-full horizontal cylindrical

vessels under external excitation in the transverse direction. A 2D mathematical model

was developed to describe sloshing effects in rigid vessels. The velocity potential was

expressed in a series form, where each term was the product of a time function and the

associated spatial function. In this geometrical configuration the spatial functions were

not orthogonal, and the problem was not separable. Application of the boundary

conditions resulted in a system of ordinary linear differential equations, which were

solved numerically. Unfortunately, because the solution was based on the linearized

kinematic and dynamic free surface conditions, it cannot be used to solve the nonlinear

problem for exactly half-full containers. It cannot be applied to containers at other fill

levels, either, even for a linearized free surface.

There are several experimental studies on the sloshing problems in horizontal

cylindrical tanks. McCarty and Stephens (1960) experimentally investigated the natural

frequencies of fluids in spherical and cylindrical tanks of different sizes, fullness, and

orientation with respect to the direction of oscillation in non-dimensional forms. The

experimental results were compared with other analytical results available. The results of

Budiansky (1960) were verified by the experimental results for the transverse sloshing in

the horizontal cylindrical container. In case of longitudinal oscillations of fluids in

horizontal circular cylinders, the experimental results indicated that the frequency

parameters for the longitudinal models were essentially independent of tank geometry.

There was no theoretical verification for this case.

21


eigenvalue problem. The velocity potential was expanded in terms of non-orthogonal

bounded harmonic spatial functions. This method has recently been applied by

Papaspyrou et al (2004) in investigating the response o f half-full horizontal cylindrical

vessels under external excitation in the transverse direction. A 2D mathematical model

was developed to describe sloshing effects in rigid vessels. The velocity potential was

expressed in a series form, where each term was the product o f a time function and the

associated spatial function. In this geometrical configuration the spatial functions were

not orthogonal, and the problem was not separable. Application o f the boundary

conditions resulted in a system of ordinary linear differential equations, which were

solved numerically. Unfortunately, because the solution was based on the linearized

kinematic and dynamic free surface conditions, it cannot be used to solve the nonlinear

problem for exactly half-full containers. It cannot be applied to containers at other fill

levels, either, even for a linearized free surface.

There are several experimental studies on the sloshing problems in horizontal

cylindrical tanks. McCarty and Stephens (1960) experimentally investigated the natural

frequencies o f fluids in spherical and cylindrical tanks of different sizes, fullness, and

orientation with respect to the direction of oscillation in non-dimensional forms. The

experimental results were compared with other analytical results available. The results of

Budiansky (1960) were verified by the experimental results for the transverse sloshing in

the horizontal cylindrical container. In case of longitudinal oscillations o f fluids in

horizontal circular cylinders, the experimental results indicated that the frequency

parameters for the longitudinal models were essentially independent o f tank geometry.

There was no theoretical verification for this case.

21


Kobayashi et al (1989) experimentally investigated the liquid natural frequencies

and the resultant slosh forces in horizontal cylindrical tanks. The liquid slosh response of

small and large slosh wave heights was studied in both longitudinal and transverse

directions. Under small slosh waves, the authors developed a calculation method of the

longitudinal slosh by substituting an equivalent rectangular tank for a horizontal

cylindrical tank. The calculated natural frequencies, slosh wave heights and slosh forces

were in good agreement with the experimental ones. Under large slosh waves, impulsive

slosh forces were observed for longitudinal excitation when the slosh liquid hit the top of

the tank. Also, the measured slosh forces, including the impulsive forces, were larger

than the calculated ones. The experiments were parametrically conducted with several

tank aspect ratios, liquid levels and excitation amplitudes. Ye (1990) and Ye and Birk

(1994) measured the fluid pressure in horizontal partially filled cylindrical tanks when

suddenly accelerated by impact along the longitudinal axis. Test results showed that the

peak pressure on the end of the tank was strongly affected by the fill level and the tank

length-diameter ratio. The maximum pressure observed was on the top of the tank.

2.4 Dynamics of liquid cargo vehicles

Numerous analytical vehicle models have been developed for vehicle dynamics

studies in the past. The kind of vehicle model that should be adopted usually depends on

the research objective. Kang (2000) gave a detailed review on the publications relevant to

heavy vehicle dynamics and stability for rigid cargo vehicles. According to this review,

degrees of freedom of vehicle models vary considerably, depending upon the number of

axles and units of the vehicle combinations, the analysis objectives and simplifying

22


Kobayashi et al (1989) experimentally investigated the liquid natural frequencies

and the resultant slosh forces in horizontal cylindrical tanks. The liquid slosh response of

small and large slosh wave heights was studied in both longitudinal and transverse

directions. Under small slosh waves, the authors developed a calculation method of the

longitudinal slosh by substituting an equivalent rectangular tank for a horizontal

cylindrical tank. The calculated natural frequencies, slosh wave heights and slosh forces

were in good agreement with the experimental ones. Under large slosh waves, impulsive

slosh forces were observed for longitudinal excitation when the slosh liquid hit the top of

the tank. Also, the measured slosh forces, including the impulsive forces, were larger

than the calculated ones. The experiments were parametrically conducted with several

tank aspect ratios, liquid levels and excitation amplitudes. Ye (1990) and Ye and Birk

(1994) measured the fluid pressure in horizontal partially filled cylindrical tanks when

suddenly accelerated by impact along the longitudinal axis. Test results showed that the

peak pressure on the end of the tank was strongly affected by the fill level and the tank

length-diameter ratio. The maximum pressure observed was on the top o f the tank.

2.4 Dynamics of liquid cargo vehicles

Numerous analytical vehicle models have been developed for vehicle dynamics

studies in the past. The kind of vehicle model that should be adopted usually depends on

the research objective. Kang (2000) gave a detailed review on the publications relevant to

heavy vehicle dynamics and stability for rigid cargo vehicles. According to this review,

degrees o f freedom of vehicle models vary considerably, depending upon the number of

axles and units o f the vehicle combinations, the analysis objectives and simplifying

22


assumptions, as well as the various operating conditions. Several earlier reviews on this

topic were listed, including Dugoff and Murphy (1971), Vlk (1982), Nalcez and Genin

(1984) and Fancher (1985). Researches were divided into several catalogues by Kang

(2000). The first one is the yaw and lateral directional dynamic response where both

linear and nonlinear yaw plane models had been developed. The yaw plane models can

be used to study several topics, such as the rearward amplification, dynamic off-tracking

and yaw and lateral stability limits. The second one is the roll dynamics analyses of

heavy vehicles where roll plane models were necessary to evaluate the roll stability limits

of the vehicles. In these models, linear or nonlinear vertical and roll stiffness of

suspensions, lateral stiffness of tires, and torsional compliance of the vehicle structures

and the articulation mechanisms were considered. Static rollover threshold under steady

turning manoeuvres and dynamic roll characteristics under transient directional

manoeuvres were studied in many investigations. The third is the 3D yaw and roll plane

analyses where the strong coupling between the yaw and roll were considered. The fourth

is the directional response under braking and turning. Among different vehicle models,

Kang (2000) described in detail the Phase IV model, which was considered as the most

comprehensive vehicle dynamics model developed for analysis of yaw, roll and lateral

stability of a heavy vehicle. The model integrated the properties of braking and an anti-

lock braking system, nonlinear cornering properties of tires using lookup tables,

nonlinear force-deflection properties of suspension springs, properties of the articulation

mechanism and driving/braking torque. The model was developed to analyze different

vehicle combinations comprising of up to three units and ten axles, such as trucks, tractor

23


assumptions, as well as the various operating conditions. Several earlier reviews on this

topic were listed, including Dugoff and Murphy (1971), Vlk (1982), Nalcez and Genin

(1984) and Fancher (1985). Researches were divided into several catalogues by Kang

(2000). The first one is the yaw and lateral directional dynamic response where both

linear and nonlinear yaw plane models had been developed. The yaw plane models can

be used to study several topics, such as the rearward amplification, dynamic off-tracking

and yaw and lateral stability limits. The second one is the roll dynamics analyses of

heavy vehicles where roll plane models were necessary to evaluate the roll stability limits

of the vehicles. In these models, linear or nonlinear vertical and roll stiffness of

suspensions, lateral stiffness of tires, and torsional compliance o f the vehicle structures

and the articulation mechanisms were considered. Static rollover threshold under steady

turning manoeuvres and dynamic roll characteristics under transient directional

manoeuvres were studied in many investigations. The third is the 3D yaw and roll plane

analyses where the strong coupling between the yaw and roll were considered. The fourth

is the directional response under braking and turning. Among different vehicle models,

Kang (2000) described in detail the Phase IV model, which was considered as the most

comprehensive vehicle dynamics model developed for analysis o f yaw, roll and lateral

stability o f a heavy vehicle. The model integrated the properties o f braking and an anti

lock braking system, nonlinear cornering properties of tires using lookup tables,

nonlinear force-deflection properties o f suspension springs, properties o f the articulation

mechanism and driving/braking torque. The model was developed to analyze different

vehicle combinations comprising of up to three units and ten axles, such as trucks, tractor

23


semi-trailer combinations, doubles, and triple combinations, with a maximum of 71

DOFs (Kang, 2000).

The dynamic behaviour of tank vehicles is greatly affected by the liquid motion in

the tanks if they are partially filled. Influence of liquid sloshing on tank trucks can be

found on many different aspects, such as the vehicle dynamics and stability, ride quality

and vehicle structure integrity. Among these topics, vehicle dynamics and stability,

especially the lateral stability, were paid the most attention.

Investigations on liquid cargo vehicles started as early as the 1970s. Bauer (1972)

pointed out that the unstrained free surface of a liquid has an alarming propensity to

undergo a large excursion for even very small motions of the container. This fact may

endanger the stability, as well as the riding and manoeuvring quality of the vehicle

considerably. Unfortunately, the suggested theoretical solutions based on the linearization

assumption can only be used for the rectangular and upright cylindrical container, and

cannot be applied to the horizontal cylindrical road container in tank vehicles. Also, the

mechanical model deduced from the rectangular and upright cylindrical containers could

hardly be directly applied to the road tanks either.

Slibar and Troger (1977) studied the steady-state behaviour of a truck-trailer

system carrying liquid cargo. The liquid sloshing was modeled by a spring-mass system

in the roll plane. This is the earliest research that employed the equivalent mechanical

model to describe the liquid behaviour in the partially filled tanks in road transportation.

The steady-state frequency responses of the yaw and roll deflections of the tractor, dolly

and trailer were investigated.

24


semi-trailer combinations, doubles, and triple combinations, with a maximum of 71

DOFs (Kang, 2000).

The dynamic behaviour of tank vehicles is greatly affected by the liquid motion in

the tanks if they are partially filled. Influence o f liquid sloshing on tank trucks can be

found on many different aspects, such as the vehicle dynamics and stability, ride quality

and vehicle structure integrity. Among these topics, vehicle dynamics and stability,

especially the lateral stability, were paid the most attention.

Investigations on liquid cargo vehicles started as early as the 1970s. Bauer (1972)

pointed out that the unstrained free surface of a liquid has an alarming propensity to

undergo a large excursion for even very small motions of the container. This fact may

endanger the stability, as well as the riding and manoeuvring quality o f the vehicle

considerably. Unfortunately, the suggested theoretical solutions based on the linearization

assumption can only be used for the rectangular and upright cylindrical container, and

cannot be applied to the horizontal cylindrical road container in tank vehicles. Also, the

mechanical model deduced from the rectangular and upright cylindrical containers could

hardly be directly applied to the road tanks either.

Slibar and Troger (1977) studied the steady-state behaviour o f a truck-trailer

system carrying liquid cargo. The liquid sloshing was modeled by a spring-mass system

in the roll plane. This is the earliest research that employed the equivalent mechanical

model to describe the liquid behaviour in the partially filled tanks in road transportation.

The steady-state frequency responses o f the yaw and roll deflections of the tractor, dolly

and trailer were investigated.

24


Research on lateral stability of partially loaded liquid vehicles was reported by

Strandberg (1978). The influence on the overturning and skidding stability of road

tankers from large amplitude sloshing was experimentally quantified. Liquid force

measurements in lateral oscillated model tanks were evaluated by simplified vehicle

models on a hybrid computer. The deteriorations of the cornering capacity due to

dynamic liquid motions were found. A simplified vehicle model without roll and yaw

was adopted. The effect of baffles and cross walls was also studied.

Bauer (1981) studied the dynamic behaviour of an elastic separating wall in

vehicle containers, based on the potential theory. The horizontal cylindrical tank divided

by an elastic separating wall was considered to be subjected to a small excitation in the

longitudinal direction with a harmonic form. The separating wall was modeled by a

membrane and a thin elastic plate. However, it was required that the tank be completely

filled. The derivation of the solutions would be impossible for a partially filled tank due

to the lack of axisymmetry.

Early investigations on the influence of liquid sloshing on the road vehicle

dynamics have two characteristics. The first one is that some suggested theories and

methods could only be useful for some simple tank configurations, such as the

rectangular or upright cylindrical containers, for which the linear solutions based on the

potential theory could be obtained. They are not applicable to partially filled horizontal

cylindrical containers. The second characteristic is that some schemes are so simple that

many important operation conditions could not be studied by these methods. Many

research results in this area were obtained in recent studies, since the late 1980s and early

1990s, most of which were conducted by researchers in CONCAVE Research Centre at

25


Research on lateral stability of partially loaded liquid vehicles was reported by

Strandberg (1978). The influence on the overturning and skidding stability of road

tankers from large amplitude sloshing was experimentally quantified. Liquid force

measurements in lateral oscillated model tanks were evaluated by simplified vehicle

models on a hybrid computer. The deteriorations o f the cornering capacity due to

dynamic liquid motions were found. A simplified vehicle model without roll and yaw

was adopted. The effect of baffles and cross walls was also studied.

Bauer (1981) studied the dynamic behaviour o f an elastic separating wall in

vehicle containers, based on the potential theory. The horizontal cylindrical tank divided

by an elastic separating wall was considered to be subjected to a small excitation in the

longitudinal direction with a harmonic form. The separating wall was modeled by a

membrane and a thin elastic plate. However, it was required that the tank be completely

filled. The derivation o f the solutions would be impossible for a partially filled tank due

to the lack of axisymmetry.

Early investigations on the influence o f liquid sloshing on the road vehicle

dynamics have two characteristics. The first one is that some suggested theories and

methods could only be useful for some simple tank configurations, such as the

rectangular or upright cylindrical containers, for which the linear solutions based on the

potential theory could be obtained. They are not applicable to partially filled horizontal

cylindrical containers. The second characteristic is that some schemes are so simple that

many important operation conditions could not be studied by these methods. Many

research results in this area were obtained in recent studies, since the late 1980s and early

1990s, most o f which were conducted by researchers in CONCAVE Research Centre at

25


Concordia University, Canada. Some important conclusions were worked out, especially

on lateral stability and rollover of partially filled tank vehicles. In the later 1990s and

early 2000s, publications by other researchers on liquid cargo vehicles could also be

found. Most of the studies on this topic were focused on mathematical models and

methods.

Experimental tests on liquid cargo vehicles can provide an effective way to study

the vehicle dynamics of the coupled liquid-vehicle system. Experimental tests can be

conducted on both real size tanks and scaled tank models. However, there are some

obvious disadvantages for both. For example, for real size tank vehicle tests, there are

some problems, such as high experimental cost, safety concerns, poor repeatability of test

results, and difficulties in data analysis due to the system complexity. For tests conducted

on scaled tank models and real vehicles, they can hardly be used to study the influence of

liquid motion on the vehicle dynamics and the coupled liquid-vehicle system. As a matter

of fact, liquid sloshing only has significant influence when the weight of the liquid cargo

is comparable to the total weight of the vehicle itself. Therefore, a scaled tank model test

can be used to obtain some important results of liquid behaviour under different

operations. However, it is not easy to study the coupled system dynamics.

A review of recent investigations on liquid cargo vehicles was conducted and

shown according to the different liquid methods adopted, such as the mass centre method,

2D dynamic model, and equivalent mechanical model.

(1) Mass centre method

26


Concordia University, Canada. Some important conclusions were worked out, especially

on lateral stability and rollover of partially filled tank vehicles. In the later 1990s and

early 2000s, publications by other researchers on liquid cargo vehicles could also be

found. Most of the studies on this topic were focused on mathematical models and

methods.

Experimental tests on liquid cargo vehicles can provide an effective way to study

the vehicle dynamics of the coupled liquid-vehicle system. Experimental tests can be

conducted on both real size tanks and scaled tank models. However, there are some

obvious disadvantages for both. For example, for real size tank vehicle tests, there are

some problems, such as high experimental cost, safety concerns, poor repeatability of test

results, and difficulties in data analysis due to the system complexity. For tests conducted

on scaled tank models and real vehicles, they can hardly be used to study the influence of

liquid motion on the vehicle dynamics and the coupled liquid-vehicle system. As a matter

of fact, liquid sloshing only has significant influence when the weight o f the liquid cargo

is comparable to the total weight o f the vehicle itself. Therefore, a scaled tank model test

can be used to obtain some important results o f liquid behaviour under different

operations. However, it is not easy to study the coupled system dynamics.

A review of recent investigations on liquid cargo vehicles was conducted and

shown according to the different liquid methods adopted, such as the mass centre method,

2D dynamic model, and equivalent mechanical model.

(1) Mass centre method

26


In earlier studies on the influence of liquid motion on vehicle dynamics, the load

shift caused by the liquid motion inside the tanks were usually calculated by the mass

centre of the liquid bulk, based on the assumption that the liquid would behave as a rigid

body. The mass centre of the liquid bulk could be determined based on the assumption

that the liquid free surface could be replaced by a flat line for 2D situations and a flat

surface for 3D situations. Therefore, only the geometry of the liquid bulk, which could be

determined by constant lateral and/or longitudinal accelerations, would be needed under

different operations, such as turning and/or braking.

Rakheja et al (1988) studied the rollover immunity level of articulated tank

vehicles with partial loads. The static roll plane model was established by considering the

vertical and lateral translation of the liquid cargo due to the vehicle roll angle and lateral

acceleration during steady turning. Ranganathan et al (1990) studied the influence of

liquid load shift on the dynamic response of a tractor semi-trailer by integrating the 2D

static liquid model that was described by the mass centre of liquid bulk in the roll plane

into the 3D vehicle model. Ranganathan et al (1993a) studied the directional response of

a B-train vehicle during turning, lane change and evasive manoeuvre by the same

method. The liquid motion was approximated in the lateral direction by the liquid bulk

that was assumed to move as a rigid body.

Popov et al (1996) carried out a numerical analysis to optimize the shape of

elliptical road containers to minimize the peak overturning moment based on the static

mass centre position. The container was subjected to constant lateral acceleration. It was

stated that the optimal height/width tank ratio decreased with the magnitude of lateral

27


In earlier studies on the influence of liquid motion on vehicle dynamics, the load

shift caused by the liquid motion inside the tanks were usually calculated by the mass

centre o f the liquid bulk, based on the assumption that the liquid would behave as a rigid

body. The mass centre of the liquid bulk could be determined based on the assumption

that the liquid free surface could be replaced by a flat line for 2D situations and a flat

surface for 3D situations. Therefore, only the geometry o f the liquid bulk, which could be

determined by constant lateral and/or longitudinal accelerations, would be needed under

different operations, such as turning and/or braking.

Rakheja et al (1988) studied the rollover immunity level o f articulated tank

vehicles with partial loads. The static roll plane model was established by considering the

vertical and lateral translation o f the liquid cargo due to the vehicle roll angle and lateral

acceleration during steady turning. Ranganathan et al (1990) studied the influence of

liquid load shift on the dynamic response of a tractor semi-trailer by integrating the 2D

static liquid model that was described by the mass centre of liquid bulk in the roll plane

into the 3D vehicle model. Ranganathan et al (1993a) studied the directional response of

a B-train vehicle during turning, lane change and evasive manoeuvre by the same

method. The liquid motion was approximated in the lateral direction by the liquid bulk

that was assumed to move as a rigid body.

Popov et al (1996) carried out a numerical analysis to optimize the shape of

elliptical road containers to minimize the peak overturning moment based on the static

mass centre position. The container was subjected to constant lateral acceleration. It was

stated that the optimal height/width tank ratio decreased with the magnitude of lateral

27


acceleration, and an elliptical container was less stable that a rectangular container of the

same capacity.

Ranganathan and Yang (1996) investigated the braking characteristics of a tractor

semi-trailer vehicle by incorporating the liquid load shift occurring within the partially

filled tank. It was assumed by the authors that the transient wave effect could be

neglected during constant deceleration manoeuvres, and the mass centre of liquid bulk

was obtained by calculating the gradient of the free surface of liquid subjected to constant

accelerations. Therefore, the dynamic axle loads could not be obtained by this totally

static approach. Wang et al (1996) used the static model to get the optimal partition

location for the compartmented tanks subjected to constant accelerations in the

longitudinal direction. The moments caused by braking were calculated based on the

mass centre of the liquid bulk expressed by the geometric relationship.

Kang et al (2002) studied the influence of the cargo load shift on the dynamics of

a tractor semi-trailer under braking and turning. The tanks had circular cross sections.

The liquid bulk in the roll and pitch plane under combined steering and braking was

derived as a function of the longitudinal and lateral accelerations. The corresponding load

shift was expressed by the instantaneous mass centre coordinates of the liquid bulk and

the mass moments of inertia of the liquid bulk. This static model was then integrated into

a dynamic vehicle model to simulate the dynamic response in terms of the load shift,

forces and moments induced by the cargo shift, and directional and roll response.

Rekheja et al (2002) carried out the same investigation for a tractor semi-trailer, in which

the tank has cross sections other than a circular cross section. In these studies, the mass

centre of the liquid bulk was solved based on the geometric relationship with the

28


acceleration, and an elliptical container was less stable that a rectangular container o f the

same capacity.

Ranganathan and Yang (1996) investigated the braking characteristics o f a tractor

semi-trailer vehicle by incorporating the liquid load shift occurring within the partially

filled tank. It was assumed by the authors that the transient wave effect could be

neglected during constant deceleration manoeuvres, and the mass centre o f liquid bulk

was obtained by calculating the gradient of the free surface o f liquid subjected to constant

accelerations. Therefore, the dynamic axle loads could not be obtained by this totally

static approach. Wang et al (1996) used the static model to get the optimal partition

location for the compartmented tanks subjected to constant accelerations in the

longitudinal direction. The moments caused by braking were calculated based on the

mass centre o f the liquid bulk expressed by the geometric relationship.

Kang et al (2002) studied the influence of the cargo load shift on the dynamics of

a tractor semi-trailer under braking and turning. The tanks had circular cross sections.

The liquid bulk in the roll and pitch plane under combined steering and braking was

derived as a function of the longitudinal and lateral accelerations. The corresponding load

shift was expressed by the instantaneous mass centre coordinates o f the liquid bulk and

the mass moments of inertia o f the liquid bulk. This static model was then integrated into

a dynamic vehicle model to simulate the dynamic response in terms of the load shift,

forces and moments induced by the cargo shift, and directional and roll response.

Rekheja et al (2002) carried out the same investigation for a tractor semi-trailer, in which

the tank has cross sections other than a circular cross section. In these studies, the mass

centre o f the liquid bulk was solved based on the geometric relationship with the

28


assumption that the liquid free surface could be an inclined flat plane when subjected to

the accelerations in both the longitudinal and lateral directions. Cargo load shift has also

been used in analyzing the braking/accelerating operation effects to structural strength of

tank vehicle subframe by a B-train tank truck model (Xu and Dai, 2004).

It is obvious that the static mass centre model neglects the transient liquid motion

and can hardly show dynamic vehicle behaviour. However, due to it simplicity, the mass

centre model was widely used in early studies on liquid cargo vehicles.

(2) 2D Dynamic liquid models

To overcome the problems caused by the mass centre model of liquid sloshing,

several dynamic liquid models have also been developed. The commonly used dynamic

liquid model was based on fluid mechanics equations, such as the Navier-Stokes

equations and the potential theory. When the liquid motion is described in 2D space, the

dynamic liquid model is established in either lateral direction or longitudinal direction.

Popov et al (1993b) investigated the dynamics of 2D liquid sloshing in horizontal

cylindrical containers of a circular cross section subjected to a suddenly applied constant

lateral acceleration to simulate the steady turning manoeuvre of a tank truck. It was

assumed that liquid motions of different cross sections had the same behaviour. The

transient response of the liquid was obtained by a numerical solution of the

incompressible 2-D Navier-Stokes, continuity, and the free-surface differential equations

using the Marker-and-cell technique (Harlow and Welch, 1965). Considerable

endeavours were made in dealing with the boundary conditions on the curved walls by

29


assumption that the liquid free surface could be an inclined flat plane when subjected to

the accelerations in both the longitudinal and lateral directions. Cargo load shift has also

been used in analyzing the braking/accelerating operation effects to structural strength of

tank vehicle subframe by a B-train tank truck model (Xu and Dai, 2004).

It is obvious that the static mass centre model neglects the transient liquid motion

and can hardly show dynamic vehicle behaviour. However, due to it simplicity, the mass

centre model was widely used in early studies on liquid cargo vehicles.

(2) 2D Dynamic liquid models

To overcome the problems caused by the mass centre model o f liquid sloshing,

several dynamic liquid models have also been developed. The commonly used dynamic

liquid model was based on fluid mechanics equations, such as the Navier-Stokes

equations and the potential theory. When the liquid motion is described in 2D space, the

dynamic liquid model is established in either lateral direction or longitudinal direction.

Popov et al (1993b) investigated the dynamics o f 2D liquid sloshing in horizontal

cylindrical containers o f a circular cross section subjected to a suddenly applied constant

lateral acceleration to simulate the steady turning manoeuvre of a tank truck. It was

assumed that liquid motions of different cross sections had the same behaviour. The

transient response o f the liquid was obtained by a numerical solution of the

incompressible 2-D Navier-Stokes, continuity, and the free-surface differential equations

using the Marker-and-cell technique (Harlow and Welch, 1965). Considerable

endeavours were made in dealing with the boundary conditions on the curved walls by

29


interpolation (Popov, 1991, Popov et al, 1993c). The influence of the viscosity, input

acceleration and fill level on the liquid motion was considered.

This numerical scheme was also used in the investigation of directional response

of tank vehicles by Sankar et al (1992). The 2D dynamic liquid sloshing model was

integrated into a 3D vehicle model to solve the vehicle responses in the time domain. The

oscillatory phenomena of lateral acceleration and roll angle were shown.

Longitudinal dynamic vehicle behaviour was also attempted by some researchers.

Popov et al (1993a) studied the dynamics of liquid sloshing in compartmented and

baffled road containers for braking and turning operations under prescribed accelerations.

The authors employed the same solution method (Popov, 1991) for the dynamic free

surface problem for rectangular road containers. Rumold (2001) simulated the braking

characteristics of partially filled tank vehicles of two axles and two identical

compartments under constant braking torques. The sloshing liquid dynamics were

determined by solving the instationary, incompressible Navier-Stokes equations under

consideration of free surfaces, by applying the finite volume approach and Volume-of-

Fluid method (Hirt and Nichols, 1981). It was shown that loss of controllability is more

likely for tank vehicles with a liquid load than for vehicles with equivalent rigid cargo. It

is very interesting that the rectangular compartments were also chosen. Since the liquid

motion was only calculated in the longitudinal direction and the 2D vehicle model was

established in the pitch plane, the simulation was conducted in exactly a 2D plane.

Currently there are very few reports on the extension from the existing numerical

methods to the 3D horizontal cylindrical containers due to the difficulties in capturing the

free surface in a time-varying 3D area. Aliabadi et al (2003) performed a comparison of

30


interpolation (Popov, 1991, Popov et al, 1993c). The influence o f the viscosity, input

acceleration and fill level on the liquid motion was considered.

This numerical scheme was also used in the investigation o f directional response

of tank vehicles by Sankar et al (1992). The 2D dynamic liquid sloshing model was

integrated into a 3D vehicle model to solve the vehicle responses in the time domain. The

oscillatory phenomena of lateral acceleration and roll angle were shown.

Longitudinal dynamic vehicle behaviour was also attempted by some researchers.

Popov et al (1993a) studied the dynamics of liquid sloshing in compartmented and

baffled road containers for braking and turning operations under prescribed accelerations.

The authors employed the same solution method (Popov, 1991) for the dynamic free

surface problem for rectangular road containers. Rumold (2001) simulated the braking

characteristics o f partially filled tank vehicles o f two axles and two identical

compartments under constant braking torques. The sloshing liquid dynamics were

determined by solving the instationary, incompressible Navier-Stokes equations under

consideration o f free surfaces, by applying the finite volume approach and Volume-of-

Fluid method (Hirt and Nichols, 1981). It was shown that loss o f controllability is more

likely for tank vehicles with a liquid load than for vehicles with equivalent rigid cargo. It

is very interesting that the rectangular compartments were also chosen. Since the liquid

motion was only calculated in the longitudinal direction and the 2D vehicle model was

established in the pitch plane, the simulation was conducted in exactly a 2D plane.

Currently there are very few reports on the extension from the existing numerical

methods to the 3D horizontal cylindrical containers due to the difficulties in capturing the

free surface in a time-varying 3D area. Aliabadi et al (2003) performed a comparison of

30


numerical analyses of fluid mechanics and mechanical models. The Navier-Stokes

equations have been solved by the finite element method to measure the accuracy of the

pendulum model considering liquid sloshing in a tanker truck during constant

acceleration turning. Parallel supercomputers with 96 fast processors and element-level

computations with millions of elements were used in the finite element analysis.

According to the results in this paper, both methods were in relatively good agreement

when the fuel inside the tanker was low. The difference between the amplitude and

frequency of sloshing was significant when there was a significant amount of fuel inside

the tanker. The braking characteristics in the longitudinal direction were also studied by

calculating the liquid forces on the tanker. In the comparison, the pendulum was so

established that all liquid mass was supposed to contribute to the pendulum mass, and the

pendulum had the same length irrespective of the liquid fill level. This was different from

the commonly used models, in which only part of the liquid mass could anticipate the

sloshing. However, the liquid model itself is one of the few really 3D models. Although it

was stated that such a method could provide the most accurate simulation results, the

extremely high computational cost needed by this scheme makes it difficult to be

performed on normal personal computers.

(3) Equivalent mechanical model

Equivalent mechanical models were developed as an alternate approach in

studying pure liquid sloshing (Abramson and Silverman, 1966). Spring-mass-damper

systems and pendulum systems are two equivalent mechanical models representing the

dynamic behaviour of liquid in moving containers. The parameters of the spring-mass-

31


numerical analyses o f fluid mechanics and mechanical models. The Navier-Stokes

equations have been solved by the finite element method to measure the accuracy of the

pendulum model considering liquid sloshing in a tanker truck during constant

acceleration turning. Parallel supercomputers with 96 fast processors and element-level

computations with millions of elements were used in the finite element analysis.

According to the results in this paper, both methods were in relatively good agreement

when the fuel inside the tanker was low. The difference between the amplitude and

frequency of sloshing was significant when there was a significant amount o f fuel inside

the tanker. The braking characteristics in the longitudinal direction were also studied by

calculating the liquid forces on the tanker. In the comparison, the pendulum was so

established that all liquid mass was supposed to contribute to the pendulum mass, and the

pendulum had the same length irrespective o f the liquid fill level. This was different from

the commonly used models, in which only part of the liquid mass could anticipate the

sloshing. However, the liquid model itself is one of the few really 3D models. Although it

was stated that such a method could provide the most accurate simulation results, the

extremely high computational cost needed by this scheme makes it difficult to be

performed on normal personal computers.

(3) Equivalent mechanical model

Equivalent mechanical models were developed as an alternate approach in

studying pure liquid sloshing (Abramson and Silverman, 1966). Spring-mass-damper

systems and pendulum systems are two equivalent mechanical models representing the

dynamic behaviour o f liquid in moving containers. The parameters of the spring-mass-

31


damper system or the pendulum system can be determined in such a way that the centre

of gravity, the force and moment resultants, oscillation frequencies and mass and inertial

properties of the mechanical system could be equivalent to those of the fluid system.

The parameters for the equivalent mechanical systems can be found in Chapter 6

in Abramson's monograph for a number of tank shapes, such as rectangular, cylindrical

and ellipsoidal. Further development on pursuing more accurate mechanical models and

parameters to simulate complicated nonlinear characteristics of sloshing problems were

later conducted by several researchers. Sayer and Baumgarten (1981) studied the

nonlinear fluid oscillations in spherical containers by a pendulum model. To compensate

for the strong boundary curvature of the spherical container, a cubic spring was included

to provide sufficient elasticity. The proper values for the coefficient of the cubic spring

were found in an experimental observation. Kana (1987) developed a compound

pendulum model to predict rotary slosh of liquid in a scale model Centaur propellant tank

at low fill levels. A portion of the liquid acted as a spherical pendulum that experienced

rotary motion throughout a frequency range below, at, and above first mode resonance.

The remainder of the fluid acted as an ordinary linear pendulum. Kana (1989) further

developed a combined spherical pendulum and linear pendulum system to produce the

same dynamic in-line and cross-axis reaction weight as liquid exhibiting rotary liquid

sloshing. Experimental measurements were used to get the pendulum parameters. It was

found that a constant-parameter combined system model could not be used to represent

typical rotary slosh over the entire frequency range. The author developed a model with

some parameters constant, and others were allowed to vary as necessary to match the

force data.

32


damper system or the pendulum system can be determined in such a way that the centre

o f gravity, the force and moment resultants, oscillation frequencies and mass and inertial

properties o f the mechanical system could be equivalent to those o f the fluid system.

The parameters for the equivalent mechanical systems can be found in Chapter 6

in Abramson’s monograph for a number of tank shapes, such as rectangular, cylindrical

and ellipsoidal. Further development on pursuing more accurate mechanical models and

parameters to simulate complicated nonlinear characteristics o f sloshing problems were

later conducted by several researchers. Sayer and Baumgarten (1981) studied the

nonlinear fluid oscillations in spherical containers by a pendulum model. To compensate

for the strong boundary curvature o f the spherical container, a cubic spring was included

to provide sufficient elasticity. The proper values for the coefficient o f the cubic spring

were found in an experimental observation. Kana (1987) developed a compound

pendulum model to predict rotary slosh of liquid in a scale model Centaur propellant tank

at low fill levels. A portion o f the liquid acted as a spherical pendulum that experienced

rotary motion throughout a frequency range below, at, and above first mode resonance.

The remainder o f the fluid acted as an ordinary linear pendulum. Kana (1989) further

developed a combined spherical pendulum and linear pendulum system to produce the

same dynamic in-line and cross-axis reaction weight as liquid exhibiting rotary liquid

sloshing. Experimental measurements were used to get the pendulum parameters. It was

found that a constant-parameter combined system model could not be used to represent

typical rotary slosh over the entire frequency range. The author developed a model with

some parameters constant, and others were allowed to vary as necessary to match the

force data.

32


Instead of using analytical and experimental methods, Salem (2000) numerically

developed a 2D trammel pendulum model to simulate liquid sloshing in a 2D elliptical

tank using the FEM package- LS_Dyna. Based on the assumptions of small angle

oscillation and a flat liquid free surface, the pendulum parameters were obtained by

matching the first natural frequency, forces and moments characteristic of the pendulum,

and those calculated from the liquid motion simulation. The elliptical tankers with

different aspect ratios and different fill levels were considered.

In recent years, the equivalent mechanical models were applied to liquid cargo

tank vehicles in studying the vehicle dynamics and responses by different vehicle models.

Khandelwal and Nigam (1982) performed the simulation of the dynamic response of a

railway wagon carrying liquid cargo in a rectangular container on a random, uneven

railway track. The pendulum model was adopted to describe the dynamic behaviour of

the liquid sloshing. The system equations were established by the Lagrange's equations

and solved by the Runge-Kutta method in the time domain. The wagon was idealized by

a heave model and a heave-pitch model. This is the earliest study that used the equivalent

pendulum model for liquid sloshing in road transportation. This is also one of the very

few investigations that included uneven road excitations.

Ranganathan et al (1993b) established a pendulum analogy model and applied this

model in studies on the directional response of tank vehicles. It was assumed that only a

part of the entire liquid mass created the sloshing effects. The rest of the liquid bulk was

considered to be attached to the tank. The model parameters were obtained by setting the

fundamental frequency and the forces arising from the mechanical system to be the same

as those derived from the dynamic fluid equations (Budiansky, 1960). The 2D

33


Instead o f using analytical and experimental methods, Salem (2000) numerically

developed a 2D trammel pendulum model to simulate liquid sloshing in a 2D elliptical

tank using the FEM package- L S D yna. Based on the assumptions o f small angle

oscillation and a flat liquid free surface, the pendulum parameters were obtained by

matching the first natural frequency, forces and moments characteristic o f the pendulum,

and those calculated from the liquid motion simulation. The elliptical tankers with

different aspect ratios and different fill levels were considered.

In recent years, the equivalent mechanical models were applied to liquid cargo

tank vehicles in studying the vehicle dynamics and responses by different vehicle models.

Khandelwal and Nigam (1982) performed the simulation of the dynamic response of a

railway wagon carrying liquid cargo in a rectangular container on a random, uneven

railway track. The pendulum model was adopted to describe the dynamic behaviour of

the liquid sloshing. The system equations were established by the Lagrange’s equations

and solved by the Runge-Kutta method in the time domain. The wagon was idealized by

a heave model and a heave-pitch model. This is the earliest study that used the equivalent

pendulum model for liquid sloshing in road transportation. This is also one o f the very

few investigations that included uneven road excitations.

Ranganathan et al (1993b) established a pendulum analogy model and applied this

model in studies on the directional response of tank vehicles. It was assumed that only a

part of the entire liquid mass created the sloshing effects. The rest of the liquid bulk was

considered to be attached to the tank. The model parameters were obtained by setting the

fundamental frequency and the forces arising from the mechanical system to be the same

as those derived from the dynamic fluid equations (Budiansky, 1960). The 2D

33


mechanical model describing the liquid behaviour in the lateral direction was then

integrated into a 3D five-axle tractor semi-trailer to study the roll angle and lateral

accelerations of the tanks. The authors stated that the pendulum model approach provided

a simple but accurate model of representing the fluid slosh within partially filled tank

vehicles.

Rangananthan et al (1994) established a model of a partially filled liquid tank

vehicle to study the dynamic characteristics during straight-line braking with constant

accelerations. The liquid motion was described by a spring-mass model. The oscillatory

response of the normal load on the tractor front and rear axles and the trailer axles was

shown after the tank vehicle came to a complete stop, representing the dynamics of the

fluid motion within the partially filled tank.

To study the dynamic behaviour of a truck carrying two spherical tanks, Ibrahim

et al (1998a) used the single degree of freedom pendulum model to represent the liquid

motion in the longitudinal direction. Governing equations of the whole system were

obtained by the Lagrange's method and solved by the Runge-Kutta method.

Considering the asynchronism of the liquid motion in the lateral direction in

different cross sections of the tank, Mantriota (2003) developed the elementary pendulum

model to describe the dynamics of the tank vehicles in the yaw direction. The liquid

motion was simulated by infinitesimal pendulums with the same parameters in the

transversal direction. In this model, it was assumed that the longitudinal motion could be

neglected. Liquid motions in pitch and roll planes were also ignored. Small amplitude

motion assumption was also needed to make use of the linear pendulum theory.

Equivalent spring-mass model has also been employed in studying the nonlinear impact

34


mechanical model describing the liquid behaviour in the lateral direction was then

integrated into a 3D five-axle tractor semi-trailer to study the roll angle and lateral

accelerations o f the tanks. The authors stated that the pendulum model approach provided

a simple but accurate model of representing the fluid slosh within partially filled tank

vehicles.

Rangananthan et al (1994) established a model of a partially filled liquid tank

vehicle to study the dynamic characteristics during straight-line braking with constant

accelerations. The liquid motion was described by a spring-mass model. The oscillatory

response o f the normal load on the tractor front and rear axles and the trailer axles was

shown after the tank vehicle came to a complete stop, representing the dynamics o f the

fluid motion within the partially filled tank.

To study the dynamic behaviour o f a truck carrying two spherical tanks, Ibrahim

et al (1998a) used the single degree o f freedom pendulum model to represent the liquid

motion in the longitudinal direction. Governing equations o f the whole system were

obtained by the Lagrange’s method and solved by the Runge-Kutta method.

Considering the asynchronism of the liquid motion in the lateral direction in

different cross sections o f the tank, Mantriota (2003) developed the elementary pendulum

model to describe the dynamics o f the tank vehicles in the yaw direction. The liquid

motion was simulated by infinitesimal pendulums with the same parameters in the

transversal direction. In this model, it was assumed that the longitudinal motion could be

neglected. Liquid motions in pitch and roll planes were also ignored. Small amplitude

motion assumption was also needed to make use o f the linear pendulum theory.

Equivalent spring-mass model has also been employed in studying the nonlinear impact

34


behaviour of tank vehicles (Xu and Dai, 2003, Dai and Xu, 2005) and ride quality

problem (Xu et al. 2004).

Obviously, the accuracy of the results obtained depends on the validity of the

pendulum model or the mass-spring-damper model used to replace the liquid. Parameters

of mechanical systems can only be reliable for limited tank configurations and tank

motions. For many tanks, the parameters of equivalent systems are not even available.

Except for the spherical pendulum model used for liquid sloshing in spherical containers,

all pendulum models and mass-spring-damper models can only be used to replace the

liquid sloshing in a 2D plane. For arbitrary liquid motion, it is impossible to find a

suitable mechanical model to simulate sloshing in a 3D space. For example, it is

impossible to use either a 2D planar pendulum or a 2D planar mass-spring-damper model

in a tank subjected to both lateral and longitudinal excitations.

Equivalent mechanical models have such advantages that they can be easily

integrated into the structure systems to study the dynamics of the coupled liquid-structure

systems, without the need for solving the fluid mechanics equations. Analytical solutions

for the structure dynamics could be pursued for some systems. Parametric studies of the

structure system can also be carried out much more easily. This provides great

convenience for researchers and engineers in both dynamics analysis and structural

design.

2.5 Summary

Based on the review conducted above, a list of problems in the research on liquid

sloshing and liquid cargo vehicles is summarized below.

35


behaviour of tank vehicles (Xu and Dai, 2003, Dai and Xu, 2005) and ride quality

problem (Xu et al. 2004).

Obviously, the accuracy of the results obtained depends on the validity o f the

pendulum model or the mass-spring-damper model used to replace the liquid. Parameters

o f mechanical systems can only be reliable for limited tank configurations and tank

motions. For many tanks, the parameters o f equivalent systems are not even available.

Except for the spherical pendulum model used for liquid sloshing in spherical containers,

all pendulum models and mass-spring-damper models can only be used to replace the

liquid sloshing in a 2D plane. For arbitrary liquid motion, it is impossible to find a

suitable mechanical model to simulate sloshing in a 3D space. For example, it is

impossible to use either a 2D planar pendulum or a 2D planar mass-spring-damper model

in a tank subjected to both lateral and longitudinal excitations.

Equivalent mechanical models have such advantages that they can be easily

integrated into the structure systems to study the dynamics of the coupled liquid-structure

systems, without the need for solving the fluid mechanics equations. Analytical solutions

for the structure dynamics could be pursued for some systems. Parametric studies of the

structure system can also be carried out much more easily. This provides great

convenience for researchers and engineers in both dynamics analysis and structural

design.

2.5 Summary

Based on the review conducted above, a list of problems in the research on liquid

sloshing and liquid cargo vehicles is summarized below.

35


1. The liquid sloshing problem was extensively studied for rectangular and upright

cylindrical tanks. However, liquid sloshing in tanks with curved walls, such as

horizontal cylindrical tanks, was only studied in limited investigations.

2. The modal method was applied for most analytical solutions for rectangular and

upright cylindrical tanks, based on the potential theory. For tanks with non-vertical

walls, either inclined straight walls or curved walls, the modes have a time-varying

domain. This makes the modal approach inapplicable even for linear problems.

3. Analytical solutions for dynamic liquid behaviour in horizontal cylindrical tanks

under forced oscillations are not available even for linearized problems. Numerical

investigations on sloshing problems in 2D horizontal circular tanks have been carried

out by several studies using different discretization approaches, such as the boundary

element method and finite difference method. However, the available numerical

methods and algorithms that deal with sloshing problems in 2D circular tanks can

hardly be easily extended to corresponding 3D problems.

4. A combination of 3D vehicle models with 2D dynamic liquid motion was used to

simulate vehicle dynamics by only considering the lateral liquid motion, with the

assumption that liquid at all cross sections behaves identically in the transversal

direction. Considerable approximation could be introduced and the research is

constrained to simulate the steady turning operation because of the inability of

describing the liquid motion in 3D space.

5. Researches on liquid cargo vehicles in the longitudinal direction are often carried out

on rectangular tanks instead of horizontal cylindrical ones due to the lack of an

effective algorithm to solve the liquid sloshing problems in horizontal cylindrical

36


1. The liquid sloshing problem was extensively studied for rectangular and upright

cylindrical tanks. However, liquid sloshing in tanks with curved walls, such as

horizontal cylindrical tanks, was only studied in limited investigations.

2. The modal method was applied for most analytical solutions for rectangular and

upright cylindrical tanks, based on the potential theory. For tanks with non-vertical

walls, either inclined straight walls or curved walls, the modes have a time-varying

domain. This makes the modal approach inapplicable even for linear problems.

3. Analytical solutions for dynamic liquid behaviour in horizontal cylindrical tanks

under forced oscillations are not available even for linearized problems. Numerical

investigations on sloshing problems in 2D horizontal circular tanks have been carried

out by several studies using different discretization approaches, such as the boundary

element method and finite difference method. However, the available numerical

methods and algorithms that deal with sloshing problems in 2D circular tanks can

hardly be easily extended to corresponding 3D problems.

4. A combination of 3D vehicle models with 2D dynamic liquid motion was used to

simulate vehicle dynamics by only considering the lateral liquid motion, with the

assumption that liquid at all cross sections behaves identically in the transversal

direction. Considerable approximation could be introduced and the research is

constrained to simulate the steady turning operation because of the inability of

describing the liquid motion in 3D space.

5. Researches on liquid cargo vehicles in the longitudinal direction are often carried out

on rectangular tanks instead of horizontal cylindrical ones due to the lack of an

effective algorithm to solve the liquid sloshing problems in horizontal cylindrical

36


tanks in 3D space. At the same time, many important operation conditions, such as

straight line driving, turning at non-constant speed and radius, lane change and double

lane change, turn-in-braking, and driving on uneven roads still cannot be

systematically studied by coupled dynamic liquid and vehicle models. Obviously, an

effective algorithm to describe the liquid motion inside 3D horizontal cylindrical

tanks needs to be developed to solve some of these problems.

6. Due to its simplicity, the mass centre model was widely used in early studies on

liquid cargo vehicles for both 2D and 3D cargo load shift. However, the mass centre

model does not consider the dynamic behaviour of the liquid motion. The

introduction of the mass centre model into a dynamic vehicle model makes it

unsuitable to simulate the vehicle behaviour for both transient operations and

unsteady state conditions.

7. Equivalent mechanical models such as pendulum models and mass-spring-damper

models have such advantages that they can be easily integrated into the structure

systems to study the dynamics of the coupled liquid-structure systems without the

need for solving the fluid mechanics equations. The longitudinal dynamics of liquid

cargo vehicles, such as the ride quality of these vehicles under straight-line driving

and the influence of the liquid impact on vehicles under rough road excitations, which

were not studied before, can be investigated by using the equivalent mechanical

models.

37


tanks in 3D space. At the same time, many important operation conditions, such as

straight line driving, turning at non-constant speed and radius, lane change and double

lane change, tum-in-braking, and driving on uneven roads still cannot be

systematically studied by coupled dynamic liquid and vehicle models. Obviously, an

effective algorithm to describe the liquid motion inside 3D horizontal cylindrical

tanks needs to be developed to solve some of these problems.

6. Due to its simplicity, the mass centre model was widely used in early studies on

liquid cargo vehicles for both 2D and 3D cargo load shift. However, the mass centre

model does not consider the dynamic behaviour o f the liquid motion. The

introduction o f the mass centre model into a dynamic vehicle model makes it

unsuitable to simulate the vehicle behaviour for both transient operations and

unsteady state conditions.

7. Equivalent mechanical models such as pendulum models and mass-spring-damper

models have such advantages that they can be easily integrated into the structure

systems to study the dynamics of the coupled liquid-structure systems without the

need for solving the fluid mechanics equations. The longitudinal dynamics of liquid

cargo vehicles, such as the ride quality o f these vehicles under straight-line driving

and the influence o f the liquid impact on vehicles under rough road excitations, which

were not studied before, can be investigated by using the equivalent mechanical

models.

37


CHAPTER 3 DYNAMIC LIQUID MOTION IN 2D

HORIZONTAL TANKS

3.1 Introduction

The liquid motion in 2D circular tanks is a simplification of liquid motion in 3D

horizontal cylindrical tanks, based on the assumption that the liquid has exactly the same

behaviour at different cross sections of a cylindrical tank. This can be true when the

liquid inside the cylindrical tanks is only subjected to lateral excitation. For the liquid

motion in cylindrical tanks employed in road transportation, when the vehicle is in a

steady turning operation and, to some extent, in a lane change operation, the application

of a 2D model can give an approximation without consideration of the coupling effect

between the longitudinal and lateral liquid modes, if the effect of the longitudinal load

can be neglected. The analysis can then be significantly simplified by solving the liquid

motion in 2D space in the transverse direction. This approach had been adopted by

almost all past investigations in the study of directional response and lateral stability of

tank vehicles.

A review of past investigations shows that it is necessary to develop mathematical

methods that can easily manage the boundary conditions on the curved walls, and update

the free surface in the time-varying domain without the need for complicated algorithms.

In this chapter, a new mathematical method is developed for studying the dynamic liquid

behaviour in partially filled horizontal circular tanks, for above-mentioned reasons. The

governing equations of the sloshing problem in 2D circular tanks are based on the

potential flow theory. The derivations of the rearranged governing equations are provided

38


CHAPTER 3 DYNAMIC LIQUID MOTION IN 2D

HORIZONTAL TANKS

3.1 Introduction

The liquid motion in 2D circular tanks is a simplification o f liquid motion in 3D

horizontal cylindrical tanks, based on the assumption that the liquid has exactly the same

behaviour at different cross sections of a cylindrical tank. This can be true when the

liquid inside the cylindrical tanks is only subjected to lateral excitation. For the liquid

motion in cylindrical tanks employed in road transportation, when the vehicle is in a

steady turning operation and, to some extent, in a lane change operation, the application

of a 2D model can give an approximation without consideration o f the coupling effect

between the longitudinal and lateral liquid modes, if the effect o f the longitudinal load

can be neglected. The analysis can then be significantly simplified by solving the liquid

motion in 2D space in the transverse direction. This approach had been adopted by

almost all past investigations in the study o f directional response and lateral stability of

tank vehicles.

A review of past investigations shows that it is necessary to develop mathematical

methods that can easily manage the boundary conditions on the curved walls, and update

the free surface in the time-varying domain without the need for complicated algorithms.

In this chapter, a new mathematical method is developed for studying the dynamic liquid

behaviour in partially filled horizontal circular tanks, for above-mentioned reasons. The

governing equations o f the sloshing problem in 2D circular tanks are based on the

potential flow theory. The derivations of the rearranged governing equations are provided

38


step-by-step, utilizing three continuous coordinate transformations. The adopted

assumptions for all transformations are given. The limits and characteristics of the

method are also discussed. The efficiency of the proposed method is demonstrated by

numerical results for harmonic and transient liquid responses in 2D circular tanks under

different lateral excitations. The natural frequencies of liquid motion inside elliptical

tanks can also be solved using this newly developed method.

3.2 Mathematical model using potential flow theory

The nonlinear liquid motion in a 2D circular tank is shown in Figure 3.1. The

liquid is assumed to be inviscid and incompressible, and the liquid motion is assumed to

be irrotational. A Cartesian coordinate system, xty, is fixed on the tank, with its origin at

the middle point of the still free surface. (x, ,t) is the free-surface elevation above still

liquid level, d is the still liquid depth, R is the radius of the tank, and co is the distance

between the origin and the centre of the tank. The tank is subjected to a lateral

displacement, Dx. Assume the local velocity potential is q).(xl ,yo t). The governing

equation of liquid motion is given by the Laplace equation.

Figure 3.1 Sketch of liquid sloshing in a circular tank

39


step-by-step, utilizing three continuous coordinate transformations. The adopted

assumptions for all transformations are given. The limits and characteristics of the

method are also discussed. The efficiency o f the proposed method is demonstrated by

numerical results for harmonic and transient liquid responses in 2D circular tanks under

different lateral excitations. The natural frequencies of liquid motion inside elliptical

tanks can also be solved using this newly developed method.

3.2 Mathematical model using potential flow theory

The nonlinear liquid motion in a 2D circular tank is shown in Figure 3.1. The

liquid is assumed to be inviscid and incompressible, and the liquid motion is assumed to

be irrotational. A Cartesian coordinate system, x y u is fixed on the tank, with its origin at

the middle point of the still free surface. £,{xx, t) is the ffee-surface elevation above still

liquid level, d is the still liquid depth, R is the radius o f the tank, and Co is the distance

between the origin and the centre o f the tank. The tank is subjected to a lateral

displacement, Dx. Assume the local velocity potential is (p{xx, y x, t ) . The governing

equation o f liquid motion is given by the Laplace equation.

-C o

► Dx

Figure 3.1 Sketch of liquid sloshing in a circular tank

39


(3.1)

The kinematic boundary condition on the free surface is:

= av _aco at ay, ax, ax,

The dynamic boundary condition on the free surface is:

av i[( ac0\ 2at 2 a.x,

\

(3.2)

— g — Axxi (3.3)

On the rigid wall, the normal velocity components are zero.

aq3 —o an (3.4)

In the above equations, g is the acceleration of gravity, n is the normal vector, and

t is the time. The initial values of velocity potential and free surface height are set to zero,

which corresponds to still liquid at the beginning.

The following quantities are introduced for generating dimensionless governing

equations.

X ,

R ° R

A Ax =—' , tVg I R, (0'= r c°

g RaigR (3.5)

In the above equations, h is the dynamic liquid height, and co is the excitation

frequency. Using Eq. (3.5) and omitting the primes, the governing equations (3.1)-(3.4)

can be rewritten in the following nondimensional forms.

a2 (0 a2 co = 0

ax? ay?

40

(3.6)


fix, fiy,


fi£ _ d(p dcp dE,dt fiy, fix, fix,


(3.1)

(3.2)

dcpdt

f dcp^

v& i J+

dcp

V ^ i J- g 4 - (3.3)

On the rigid wall, the normal velocity components are zero.

^ = 0dn

(3.4)

In the above equations, g is the acceleration of gravity, n is the normal vector, and

t is the time. The initial values of velocity potential and free surface height are set to zero,

which corresponds to still liquid at the beginning.

The following quantities are introduced for generating dimensionless governing

equations.

R y i -A, /,'4 .c0=%R R R R R

A'x = — , t ' = t J g / R , (p'= Z— S R jg R

, (O '' =co

(3.5)

In the above equations, h is the dynamic liquid height, and co is the excitation

frequency. Using Eq. (3.5) and omitting the primes, the governing equations (3.1)-(3.4)

can be rewritten in the following nondimensional forms.

S > f i>dxf dyx

= 0 (3.6)

40


.a 4 _aco at ay, ax, ax,

ac, = _1 at 2

i aco \2 ( \2 -

ax1 .a-Y1)

ao an

— — Axx,

(3.7)

(3.8)

0 (3.9)

Replacing x, and y, by x and y, Eqs. (3.6) to (3.8) are rewritten as follows.

a2q)

+a2`)

. 0

aX2 ay 2

a = 4 _aco a at ay ax ax

4 =_ 1 at 2

(4)2 r aq) , 2

ax j + ay, _

(3.10)

(3.11)

— — Axx (3.12)

3.3 Mathematical method

When the governing equations of the liquid motion are expressed by the potential

theory for rectangular and upright cylindrical tanks, the modal approach can usually be

used to reduce the original free-boundary problem to an infinite-dimensional system of

nonlinear ordinary differential equations (modal system), where the unknowns are

generalized coordinates describing nonlinear evolution of natural modes. The forces and

moments acting on the tanks may also be explicitly expressed in terms of the generalized

coordinates. Multimodal modeling is usually employed instead of infinite-dimensional

modeling. With the increase in the number of modes, it can be adopted for simulations of

41


dcpdt

c^L _dcp__dcp_ d%_ dt dy{ dx[ cbc,

dcp'2

+ fd(pTdx J

dcp0

dn

Replacing x { andy, by x andy, Eqs. (3.6) to (3.8) are rewritten as follows.

d 2tp d 2tpdx2 ay2

d£ _ dcp dcp d% dt dy dx dx

dcpdt

dcp v dx j

+dcp

j- £ - A x X

(3.7)

(3.8)

(3-9)

(3.10)

(3.11)

(3.12)

3.3 Mathematical method

When the governing equations of the liquid motion are expressed by the potential

theory for rectangular and upright cylindrical tanks, the modal approach can usually be

used to reduce the original free-boundary problem to an infinite-dimensional system of

nonlinear ordinary differential equations (modal system), where the unknowns are

generalized coordinates describing nonlinear evolution of natural modes. The forces and

moments acting on the tanks may also be explicitly expressed in terms o f the generalized

coordinates. Multimodal modeling is usually employed instead of infinite-dimensional

modeling. With the increase in the number o f modes, it can be adopted for simulations of

41


realistic liquid motion. It can also be used for cases when the linear sloshing problem has

no analytical solutions, and the modes are approximated by a numeric-analytical method.

However, as pointed out by Faltinsen and Timokha (2002), it should be noted that

the modal method is a possible basis for studying sloshing with shallow, intermediate and

finite depth. It cannot describe overturning and breaking waves. The tank should have no

roof. More importantly, the tank walls in the equilibrium position must be vertical at the

mean free surface. For tanks with non-vertical walls, either inclined straight walls or

curved walls, the modes have a time-varying domain. This means the modal approach is

inapplicable even for linear sloshing problems. The actual liquid motion near the walls

cannot be described by using the modal approach and corresponding mathematical

treatments.

It should be noted that statements made by some researchers in their studies

suggesting that their methods could be used to solve sloshing problems for tanks of

arbitrary shapes could mislead readers. For example, Komatsu (1987) proposed a method

for calculating the nonlinear dynamic behaviour of liquid in tanks. The formulation used

the orthogonality of the linear mode shapes and the numerical perturbation technique.

The problem was reduced to the non-linear coupled ordinary differential equations

describing the timewise trend. The author stated that this method could deal with any

arbitrarily shaped tanks. However, because the linear solutions were pre-required in this

method, it actually cannot deal with arbitrarily shaped tanks. In fact, the method was

generally a modal method. The linear solutions were only analytically available for

rectangular and upright cylindrical containers. For tanks with non-vertical walls, it is

difficult to obtain linear solutions by either analytical or numerical method. The

42


realistic liquid motion. It can also be used for cases when the linear sloshing problem has

no analytical solutions, and the modes are approximated by a numeric-analytical method.

However, as pointed out by Faltinsen and Timokha (2002), it should be noted that

the modal method is a possible basis for studying sloshing with shallow, intermediate and

finite depth. It cannot describe overturning and breaking waves. The tank should have no

roof. More importantly, the tank walls in the equilibrium position must be vertical at the

mean free surface. For tanks with non-vertical walls, either inclined straight walls or

curved walls, the modes have a time-varying domain. This means the modal approach is

inapplicable even for linear sloshing problems. The actual liquid motion near the walls

cannot be described by using the modal approach and corresponding mathematical

treatments.

It should be noted that statements made by some researchers in their studies

suggesting that their methods could be used to solve sloshing problems for tanks of

arbitrary shapes could mislead readers. For example, Komatsu (1987) proposed a method

for calculating the nonlinear dynamic behaviour of liquid in tanks. The formulation used

the orthogonality of the linear mode shapes and the numerical perturbation technique.

The problem was reduced to the non-linear coupled ordinary differential equations

describing the timewise trend. The author stated that this method could deal with any

arbitrarily shaped tanks. However, because the linear solutions were pre-required in this

method, it actually cannot deal with arbitrarily shaped tanks. In fact, the method was

generally a modal method. The linear solutions were only analytically available for

rectangular and upright cylindrical containers. For tanks with non-vertical walls, it is

difficult to obtain linear solutions by either analytical or numerical method. The

42


efficiency of the method had only been demonstrated by nonlinear liquid motion in

rectangular and axisymmetric upright cylindrical containers. As a matter of fact, sloshing

problem depends largely on the shapes and orientations of the tanks. It is commonly

recognized that orthogonal linear modes do not exist for tanks with curved walls. A

universal method for liquid sloshing problems cannot be pursued due to mathematical

difficulties.

Due to the unavailability of analytical solutions, a numerical method should be

used to solve the above governing equations. Obviously, the boundary conditions on the

curved walls, the nonlinearity caused by the boundary conditions on the free surface, as

well as the time-varying integration domain for the time-varying free surface are the

major difficulties in obtaining the numerical solution of sloshing problems in circular

tanks.

Existing numerical schemes for 2D sloshing problems in circular tanks usually

directly discretize the governing equations in the 2D circular area and have the following

difficulties. First, when the governing equations are described by Navier-Stokes equations

and discretized by the finite difference method, the boundary conditions on the curved

walls are quite difficult to obtain in those cells that are enclosed by curved edges and

straight edges. Computer algorithms for considering all different configurations in

interpolation for pressure and velocity components on the curved cell edges are extremely

intricate. When the governing equations are described by the potential theory and

discretized by the boundary element method, some existent algorithms for the smoothing

of the free surface position cannot be directly used in the time-varying domain caused by

the curved walls. This could be solved by changing to a polar coordinate system, which

43


efficiency of the method had only been demonstrated by nonlinear liquid motion in

rectangular and axisymmetric upright cylindrical containers. As a matter o f fact, sloshing

problem depends largely on the shapes and orientations of the tanks. It is commonly

recognized that orthogonal linear modes do not exist for tanks with curved walls. A

universal method for liquid sloshing problems cannot be pursued due to mathematical

difficulties.

Due to the unavailability o f analytical solutions, a numerical method should be

used to solve the above governing equations. Obviously, the boundary conditions on the

curved walls, the nonlinearity caused by the boundary conditions on the free surface, as

well as the time-varying integration domain for the time-varying free surface are the

major difficulties in obtaining the numerical solution o f sloshing problems in circular

tanks.

Existing numerical schemes for 2D sloshing problems in circular tanks usually

directly discretize the governing equations in the 2D circular area and have the following

difficulties. First, when the governing equations are described by Navier-Stokes equations

and discretized by the finite difference method, the boundary conditions on the curved

walls are quite difficult to obtain in those cells that are enclosed by curved edges and

straight edges. Computer algorithms for considering all different configurations in

interpolation for pressure and velocity components on the curved cell edges are extremely

intricate. When the governing equations are described by the potential theory and

discretized by the boundary element method, some existent algorithms for the smoothing

of the free surface position cannot be directly used in the time-varying domain caused by

the curved walls. This could be solved by changing to a polar coordinate system, which

43


in turn makes the governing equations much more complicated due to the lack of

axisymmetry in the free surface problem. Second, although the Volume of Fluid method

for free surface updating has been established and developed for many years, the

algorithms based on solving the volume fraction of the liquid in the free surface cells are

usually quite complicated for both theoretical development and programming. Third, the

above two difficulties become much more obvious when 3D sloshing in the horizontal

cylindrical tanks is to be solved.

Therefore, it is necessary to develop numerical methods that can easily deal with

the boundary conditions on the curved walls, update the free surface in the time-varying

domain without the need for complicated algorithms, as well as be extended to the 3D

problems with computational efficiency. In the following sections, a new numerical

scheme is developed for these purposes. The governing equations are rearranged by three

continuous coordinate mappings before discretization, which can avoid all of the above

difficulties in solving the sloshing problems in horizontal circular tanks.

3.3.1 First transformation

In solving the boundary conditions on curved walls, the interpolation method is

commonly employed if the governing equations are directly discretized from Eqs. (3.9) to

(3.12) in the Cartesian coordinate system. Considerable work should be carried out on the

potential expression on curved walls for different configurations. If the polar coordinates

were adopted, derivation of governing equations and implementation of calculation

would become complicated. Due to the lack of axisymmetry of the free surface problem

in partially filled tanks, the adoption of a polar coordinate system for 2D problems would

44


in turn makes the governing equations much more complicated due to the lack of

axisymmetry in the free surface problem. Second, although the Volume o f Fluid method

for free surface updating has been established and developed for many years, the

algorithms based on solving the volume fraction o f the liquid in the free surface cells are

usually quite complicated for both theoretical development and programming. Third, the

above two difficulties become much more obvious when 3D sloshing in the horizontal

cylindrical tanks is to be solved.

Therefore, it is necessary to develop numerical methods that can easily deal with

the boundary conditions on the curved walls, update the free surface in the time-varying

domain without the need for complicated algorithms, as well as be extended to the 3D

problems with computational efficiency. In the following sections, a new numerical

scheme is developed for these purposes. The governing equations are rearranged by three

continuous coordinate mappings before discretization, which can avoid all o f the above

difficulties in solving the sloshing problems in horizontal circular tanks.

3.3.1 First transformation

In solving the boundary conditions on curved walls, the interpolation method is

commonly employed if the governing equations are directly discretized from Eqs. (3.9) to

(3.12) in the Cartesian coordinate system. Considerable work should be carried out on the

potential expression on curved walls for different configurations. If the polar coordinates

were adopted, derivation o f governing equations and implementation of calculation

would become complicated. Due to the lack o f axisymmetry of the free surface problem

in partially filled tanks, the adoption o f a polar coordinate system for 2D problems would

44


not be more convenient. At the same time, additional work needs to be done in updating

the free surface in the time-varying integration domain.

To overcome all of these difficulties caused by the curved walls, the first

transformation is employed using the following equations.

x (3.13) a = ,

Ail — (y — co )2 = y

The area enclosed by the tank is transformed from an xy coordinate system to an

a/3 coordinate system, which is shown in Figure 3.2.

A iS

a

Figure 3.2 First coordinate transformation

For the the top and bottom quadrant points, there is a singularity if Eq. (3.13) is

directly used. The transformation is then supplemented by the following condition.

—1<<—a<_1, at fi = co ± 1 (3.14)

To accomplish this transformation, two assumptions have been made. First, the

free surface will never pass the bottom quadrant point. Second, the free surface will never

climb over the top quadrant point. In real situations, these would occur when the fill level

is very low or very high under large excitations. Although these two assumptions set

45


not be more convenient. At the same time, additional work needs to be done in updating

the free surface in the time-varying integration domain.

To overcome all o f these difficulties caused by the curved walls, the first

transformation is employed using the following equations.

a - P = y (3.13)V M t -C q)2

The area enclosed by the tank is transformed from an xy coordinate system to an

a f lcoordinate system, which is shown in Figure 3.2.

Figure 3.2 First coordinate transformation

For the top and bottom quadrant points, there is a singularity if Eq. (3.13) is

directly used. The transformation is then supplemented by the following condition.

- 1 < « < 1 , at /? = c0 ± 1 (3.14)

To accomplish this transformation, two assumptions have been made. First, the

free surface will never pass the bottom quadrant point. Second, the free surface will never

climb over the top quadrant point. In real situations, these would occur when the fill level

is very low or very high under large excitations. Although these two assumptions set

45


certain limits on the presented scheme, the current study does not deal with these extreme

situations.

The governing equations (3.9)-(3.12) are then transformed into the a/3 coordinate

system by using Eq. (3.13).

ao = _i[ E

at 2 1

B1

32A 02, 02A

afbB • '

1 a a 2 + B ' + B ' + B = 0

2 &tap 3 8,62 4 act

_ 1

1-(fi-00)2+ Ea(flcoA2 {i 0 — coyt , B 2 a2 1 ((flfi —C C00) )2

B3 = 1 , B4 = a 1 + 20 — c0 )2

1— (6 — co)2

871 =G•ao

at ' • —aa +G2

00

ap

a ri 1 +

46 — co ) G = I G = 1

as VI _ (fi• _ co )2 1—(p — c0)2, 2

50 2 ± E2. 50 2 ± E3 ' 80 80' + E4 . 50 ± E 50 E6

a) .8,6 ) ,aa ap , 0a 5• a/340

(3.15)

(3.16)

(3.17)

(3.18)

ri — E7 a

(3.19)

Et1 [a(B — co)12

1, E 3E 224 — co )

+

= =

, , 1 — (fi — CO ) 2 [1 — (fi — Co ) 2 f 1- (p — Co ) 2

E 4 = 0 , E 5 = 0 , E 6 = 1 , E7 = A„ •• ill — (fl — co )2 (3.20)

Bottom: 50 50 /' 12 (3.21) • a

+ -,6 =0

I1= a( 6) —c°)2 12 1 (3.22) -- , 1 — (p —Co)

46


certain limits on the presented scheme, the current study does not deal with these extreme

situations.

The governing equations (3.9)-(3.12) are then transformed into the a p coordinate

system by using Eq. (3.13).

1 d a 2 2 dadp 3 dp3 a / 5 2 4 da(3.15)

Bx =[ a { p - c 0 )]2 2 a ( p - c 0)

B3 = 1, Ba =a

i P - c j f(3.16)

<h l = g a t + G mdt d a dp

dr] I | a ( p ~ c 0)

- { P - c J l - ( ^ - c 0):2 . G 2 =1

(3.17)

(3.18)

d l _ _ J_ dt ~ 2

E\ ’ + e 2 • + ^3-r dtj>_ d f

d a dp+ E4 - ^ - + E 5 ~ - E 6 ■T ] -E 7 -a

4 d a 5 d p 6 ' 7

(3.19)

£> =J , N y g -ô )]2 E =1 E _ 2 a ( p ~ c 0)

l - ( / 7 - c 0)2 l - ( / ? - c 0):

E 4 = 0 , E 5 = 0 , E 6 = 1 , E 7 = A x ^ I - ( P - c J

Bottom: /, • — + / 2 • ^ = 0d a dp

(3.20)

(3.21)

_ a { p - c 0)

2(3 .22)

46


a0 _(P_a = 0 Walls: HI . T t- + H2 ap

aHI =

1—(p — co)2 , H2 =

fl

— co

(3.23)

(3.24)

After the first coordinate transformation, the working domain becomes a

rectangular area, i.e., —1 _.01 1, —,c1 /3 5 2 —d. The curved walls are replaced by

vertical walls. The difficulty in dealing with the boundary conditions on the curved walls

is completely avoided at the price of more complicated governing equations.

3.3.2 Second transformation

In order to solve the boundary conditions on the time-varying free surface, free

surface capturing, smoothing of the free surface and volume correction after the position

updating are commonly adopted. However, the algorithm for updating the free surface

and the programming are usually intricate.

To overcome the difficulties in dealing with nonlinear boundary conditions on the

free surface, the sigma-transformation is adopted. By transforming the physical liquid

domain in a rectangular tank into a rectangular region bounded by horizontal and vertical

sides, the solution in the transformed computational domain exactly fits the free surface

boundary. Therefore, the algorithm is quite stable. The complicated algorithm commonly

adopted in other numerical schemes for free surface capturing can be avoided. Free

surface smoothing and volume correction will only be needed for steep waves of high

nonlinearities. The second transformation is expressed as follows.

47


After the first coordinate transformation, the working domain becomes a

rectangular area, i.e., -1 < a < 1, - d < J3 <2 - d . The curved walls are replaced by

vertical walls. The difficulty in dealing with the boundary conditions on the curved walls

is completely avoided at the price of more complicated governing equations.

3.3.2 Second transformation

In order to solve the boundary conditions on the time-varying free surface, free

surface capturing, smoothing of the free surface and volume correction after the position

updating are commonly adopted. However, the algorithm for updating the free surface

and the programming are usually intricate.

To overcome the difficulties in dealing with nonlinear boundary conditions on the

free surface, the sigma-transformation is adopted. By transforming the physical liquid

domain in a rectangular tank into a rectangular region bounded by horizontal and vertical

sides, the solution in the transformed computational domain exactly fits the free surface

boundary. Therefore, the algorithm is quite stable. The complicated algorithm commonly

adopted in other numerical schemes for free surface capturing can be avoided. Free

surface smoothing and volume correction will only be needed for steep waves o f high

nonlinearities. The second transformation is expressed as follows.

47


i \ X = a, Y=-1+ 2 (/3+d)

h(a ,t)

........a.m.._

46.---

a

MIIMIOP

1 Y

X

Figure 3.3 Second coordinate transformation

(3.25)

After the second transformation, the area enclosed by the rigid walls and free

surface is transformed from the afl coordinate system to an XY coordinate system, which

is shown in Figure 3.3. Because the dynamic liquid height h is a function of time t and

horizontal coordinate a, the vertical horizontal Y after the second transformation will also

be a function of time t and horizontal coordinate X, which makes the mesh in the xy

coordinate system change with time, and the governing equations become more

complicated.

The governing equations are then transformed into the XY coordinate system by

using Eq. (3.25).

2 0 2 20 c

a +c +C., +c •—+c —=o

' ax2 2 a

aXaY a aY2 4

ao ax 5

ao ay

(3.26)

ay 2 ay)2 (

h 2 ay

ax 1 - - ) B

1 1i

C, =B,, C2 = 131 . (2 —) + B2 • ( h ) , C3 = 13, (ax)

+B2 • — 1ax 172)'

( 2 all) (all a2Y)+B2 (— h2

—ax j+ B4 CaTc ) ,ax2

48

(3.27)


X = a, Y = - 1 +h{a,t)

( p + d ) (3.25)

Figure 3.3 Second coordinate transformation

After the second transformation, the area enclosed by the rigid walls and free

surface is transformed from the af3 coordinate system to an X Y coordinate system, which

is shown in Figure 3.3. Because the dynamic liquid height h is a function o f time t and

horizontal coordinate a, the vertical horizontal Y after the second transformation will also

be a function of time t and horizontal coordinate X, which makes the mesh in the xy

coordinate system change with time, and the governing equations become more

complicated.

The governing equations are then transformed into the X Y coordinate system by

using Eq. (3.25).

C, d 20 d 2o dO1 d x 2 2

+ C->----------1- C , ------T- + C , -------b C , ----- — 0dXdY 3 BY3 4 ax BY

(3.26)

C\ =B . , C2 = B ,BY_8 X

+ B-, , C3 — B\\ n j

BY y B X ,

+ *r 2 B Y "'

h BX+ 5 3 -

J \ n j

C4 = B , , C5 = B, f d 2 y ) + B 2 ■( 2 BH']

+ B. ■------------[d X 2J

2 L h 2 BX) 4 lax) (3.27)

48


51-1 ac =Ki . ao +lcat ax 2 aY

Ki =GI , K2 = G, • ( ay` ,ax ,

ac _ 1 [ ji r ao )2 +.12 r 2+.13.rao at at 2 aX

àY, ax aY)

ao ao+ J 4 •—+ J 5 • ay J 6 •H — J 7 • AT

ax

, ay,2 J 1 = El , J 2 = El +E2 • (-4-- + E • —

\ aX j h 2 , 3

J 3 = Ei •(2 ay` ax

2 + E3 *(77), J 4 = 0, J 5 = —

Bottom:

( ay 2`

ax h,

ay at '

J6

= E6 '

J7

= E7

1. r alc. T aCD n1 • — + L 2 - = V ax aY

ay 2L1 =11 , L2 = 1.1 *( ax )+12 •

hj

N1•P-41)

+N2 -a4:1)

= 0 Walls: ax aY

NI = 11,, N2 =111( ay`,ax ,

+H2 •(-2 )

(3.28)

(3.29)

(3.30)

(3.31)

(3.32)

(3.33)

(3.34)

(3.35)

The application of the second transformation introduces the following

relationships, taking into account that the vertical coordinate is a function of time and the

horizontal coordinate.

ay ax -

Y+1 (au) a2y 2(Y+1) ( afi`2 Y+1 7 a2H` h ax )' ax2 h 2 \ ax , h ax2 J

ay Y +1 (al at h at

49

(3.36)


dH „ 5 0 ^ 5 0 — Ki •------h Kj * —dt dX 2 dY

K, = G,, K 2 = G{ ■ + G2 •f - 1l a * ; 4 yk y

5 0dt

^ 5 0 ^ 25X

+ J^ 5 0 ^ 2

5 0

v 5 7 y+ ./-)

5 0 5 05W 57

5 0+ J 4 + J 5 J 6 - H - J 7 - X

dx

J \ = E \ , J 2 = E \ ■

dY

f

y d X j

f A \+ E 2 ■

k« 2j+ e 3 -

dY 2d x ' H

J 3 = E r( ? dY_) { dXj

r?L \h j

Bottom:

57/ = 0 T I = F I - F,i7 4 w , */ 5 — ^ ^ 6 6 1 J 1 ~dt

50 50L, b T-, ■ — — 0

1 5J7 2 57

L\ — I \ , L 2 — I x -r 57 A

v5Zy+ / 2 •

'2 ^

U y

Walls:5 0 5 0

N , + N 2 = 01 dx 2 dY

N\ - H x, N 2 - H { ■ + H 2 ■r 2 s

[ d X j 2 ^ y

(3.28)

(3.29)

(3.30)

(3.31)

(3.32)

(3.33)

(3.34)

(3.35)

The application of the second transformation introduces the following

relationships, taking into account that the vertical coordinate is a function o f time and the

horizontal coordinate.

dY 7 + 1 ( d H ^ 527 _ 2 (7 + i; ( d H \ 2 7 + 1 ( d 2H \dX h la*J ’ d X 2 h2 \ dX j h [ d X 2 J

57dt

7 + 1 rd H '

s. dt V(3.36)

49


The second transformation is exactly the so-called sigma-transformation, which

was first proposed by Phillips (1957) for numerical forecasting. It has recently been

applied in the research of hydrodynamics and sloshing in combination with different

numerical methods. For example, it has been used in research on pressures on dams by

Navier-Stokes equations and the finite difference method (Chen, 1994), 2-D sloshing in

rectangular containers by the potential theory and the pseudospectral method (Chern et al,

1999), 3-D standing and impulse waves in an upright cylindrical container with a central

cylindrical inclusion by the potential theory and the spectral method (Chern et al, 2001),

2-D sloshing in rectangular containers by the potential theory and the finite element

method (Turnbull et al, 2003), and 2-D sloshing in rectangular containers by the potential

theory and the finite difference method (Frandsen, 2003). However, an application of

this approach in analyzing the liquid motion in a horizontal cylindrical tank is not found

in the current literature.

To perform the second transformation, another assumption is needed. Since the

free surface height is expressed as a single value function of the horizontal coordinate and

stretched in the direction of the vertical coordinate, overturning waves and breaking

waves cannot be described. Therefore, it is assumed that overturning waves and breaking

waves are not involved in the current study. This again sets certain limits on the proposed

method. This extreme situation is not considered in this research. It should also be noted

that the application of the second transformation, which is one type of the boundary fitted

method, makes it unsuitable for large excitations. It is well known that the simple

algebraic mappings are not robust when the free surface is quite steep. Therefore, high

nonlinearity such as a wave with large steepness is not considered in this study. The

50


The second transformation is exactly the so-called sigma-transformation, which

was first proposed by Phillips (1957) for numerical forecasting. It has recently been

applied in the research o f hydrodynamics and sloshing in combination with different

numerical methods. For example, it has been used in research on pressures on dams by

Navier-Stokes equations and the finite difference method (Chen, 1994), 2-D sloshing in


1999), 3-D standing and impulse waves in an upright cylindrical container with a central

cylindrical inclusion by the potential theory and the spectral method (Chem et al, 2001),

2-D sloshing in rectangular containers by the potential theory and the finite element

method (Turnbull et al, 2003), and 2-D sloshing in rectangular containers by the potential

theory and the finite difference method (Frandsen, 2003). However, an application of

this approach in analyzing the liquid motion in a horizontal cylindrical tank is not found

in the current literature.

To perform the second transformation, another assumption is needed. Since the

free surface height is expressed as a single value function of the horizontal coordinate and

stretched in the direction of the vertical coordinate, overturning waves and breaking

waves cannot be described. Therefore, it is assumed that overturning waves and breaking

waves are not involved in the current study. This again sets certain limits on the proposed

method. This extreme situation is not considered in this research. It should also be noted

that the application o f the second transformation, which is one type o f the boundary fitted

method, makes it unsuitable for large excitations. It is well known that the simple

algebraic mappings are not robust when the free surface is quite steep. Therefore, high

nonlinearity such as a wave with large steepness is not considered in this study. The

50


method developed here will be used to simulate liquid sloshing under normal highway

operation conditions when lateral accelerations are within the rollover threshold.

After the second coordinate transformation, the liquid domain becomes a square

area, i.e., —1 X Y 1. The time-varying curved free surface is replaced by

the fixed straight-line boundary. Dealing with the boundary conditions on the time-

varying free surface, for which capturing and smoothing of the free surface and volume

correction after the position updating are usually needed, is completely avoided at the

price of more complicated governing equations.


After the second transformation, any numerical method can be employed to solve

the sloshing problem. In the current study, the finite difference method is adopted.

However, if the governing equations are discretized directly from Eqs. (3.26) to (3.35),

computational convergence can hardly be achieved, due to extremely large aspect ratios

of the mesh near the bottom when the grids in the XY coordinate system are transformed

back to the xy coordinate system. To overcome this difficulty, the third coordinate

transformation is employed by the following equations.

X* = X

Y =1 214B +1— (Y +1)/ 210 —1+ (Y + 1)/ 211

ln[(B +1)/(B —1)]

(3.37)

In the above equations, B is a parameter that can be adjusted to determine the

clustering of the grids near the bottom. In the solution, the singularity of the bottom

quadrant point is treated in an alternative way, in combination with the usage of the

parameter B. A tiny flat bottom is first assumed to replace the bottom quadrant point. The

51


method developed here will be used to simulate liquid sloshing under normal highway

operation conditions when lateral accelerations are within the rollover threshold.

After the second coordinate transformation, the liquid domain becomes a square

area, i.e., - \< X < 1, -1 < F < 1. The time-varying curved free surface is replaced by

the fixed straight-line boundary. Dealing with the boundary conditions on the time-

varying free surface, for which capturing and smoothing of the free surface and volume

correction after the position updating are usually needed, is completely avoided at the

price o f more complicated governing equations.


After the second transformation, any numerical method can be employed to solve

the sloshing problem. In the current study, the finite difference method is adopted.

However, if the governing equations are discretized directly from Eqs. (3.26) to (3.35),

computational convergence can hardly be achieved, due to extremely large aspect ratios

o f the mesh near the bottom when the grids in the X Y coordinate system are transformed

back to the xy coordinate system. To overcome this difficulty, the third coordinate

transformation is employed by the following equations.

X* - X... . , ln{|£ +1 - (y + 1)/ 2l/[fl -1 + (y + 1)/ 2P (3.37)

ln[(fi + l ) / ( 5 - l ) ]

In the above equations, B is a parameter that can be adjusted to determine the

clustering o f the grids near the bottom. In the solution, the singularity o f the bottom

quadrant point is treated in an alternative way, in combination with the usage o f the

parameter B. A tiny flat bottom is first assumed to replace the bottom quadrant point. The

51


exact solution can be achieved when the flat bottom approaches the bottom quadrant

point infinitely. In the numerical solution, the approximation can be achieved by making

the flat bottom small enough to meet the accuracy requirement. In all the calculation in

the following section, the approximation of the flat bottom is set to be 0.0001R, though it

is found that solution convergence could be obtained by 0.001R, which is much smaller

than the approximation made on curved walls by other numerical schemes, such as the

constant-element and linear-element boundary element methods. The parameter B is

adjusted according to this approximation.

If the fill level is quite high, clustering of the grids near the free surface is also

needed to gain accuracy and convergence. The third transformation can then be achieved

by modifying Eq. (3.37). If necessary, the clustering of the grids near the walls in the X

direction can also be implemented in the same way, though it is not adopted in this study.

Using the third transformation, the governing equations are expressed as:

a2 * a20*

* 52 . ao* a W, +W2 +Tf73 0* +P V4 * +W5 (I)* = 0 (3.38)

ax 2 ax*ay ay 2 ax ay

W3 = C 1 •

W 5 = C 1 •

WI = C 1 , W 2 = CI • (2 + C 2

i a Y *

a Y /

,- ,2 ay* 1 'ay* ay* -F C3 ' ay*\ 2+c2 , W4 = C4 , ax, ax ay \ ay , i I

I a2 Y* \

, ax 2 1 + C2

a 2 y*

axaY i + C 3

(a2y*) (ay* \ laY*)

ay2 + C 4 ax , ± C5 , ay (3.39)

ax* ao* ao*at

= PI ax*

+P2 aY *

52

(3.40)


exact solution can be achieved when the flat bottom approaches the bottom quadrant

point infinitely. In the numerical solution, the approximation can be achieved by making

the flat bottom small enough to meet the accuracy requirement. In all the calculation in

the following section, the approximation of the flat bottom is set to be 0.0001R, though it

is found that solution convergence could be obtained by 0.001 R, which is much smaller

than the approximation made on curved walls by other numerical schemes, such as the

constant-element and linear-element boundary element methods. The parameter B is

adjusted according to this approximation.

If the fill level is quite high, clustering o f the grids near the free surface is also

needed to gain accuracy and convergence. The third transformation can then be achieved

by modifying Eq. (3.37). If necessary, the clustering o f the grids near the walls in the X

direction can also be implemented in the same way, though it is not adopted in this study.

Using the third transformation, the governing equations are expressed as:

d20* d20* d20 * dO* dO*Wl - ^ r + W2 - . . +W3 - ^ t + W4 - ^ t + W5 - ^ - v = 0 (3.38)

1 ax dX dY dY dX 5 dY

wx= c {, w 2 = c x8 X\

+ C,/ v d F y

W3 = C rr dY*vv d X y

+ C2 •r dY* d Y*A

dX dY+ C3 -

r d Y * ^ 2

v dY jw = c, rr * — v 4 ,

w5 = c , . f a 2 r l + c , •f d 2Y* "1+ C 3 -f d 2Y*) + < v r dY* "1+ c5 ■( dY* ^

U x 2 J [dXdF J d Y 2 \ U1 J U f J (3.39)

dH* n dO* n dO*■ = P,- — r + P, •-

dt dX dY(3.40)

52


p = KI , P2 = K1gay .

ax + K2 •aY'

• ay

at 2 ax* - a Y * - ax* a Y * = 0, i + 0, i + 0,

ate ate` 'ate..•\2, * N2 racD* 1 acD

acD* acD* +o • +05 • OH• * —0 -X 7 4 ax* ay* 6

= , 02 = J 1 •

✓ * -\ 2 ay ay ay* ay" + * +J3 • ax 1 ay, ax ay,

• ay* ay' ay* ay* 0 3 =J, • 2 / +J3 ' \ , 04 = 0 , 05 =J5 '\• ax ay ay at

Bottom:

Q1 = , Q2 =L, •

06 = J6 , 0 7 =

acD* a Ql * +Q2

ate`= 0ax ay

ay* ax

+ L2

Walls: RI • ac= +R2 aCD

= 0 ax* ay*

R,= N,, R2 = NI • ( ay* \

ax + N2

ay* ay

ay* ay

(3.41)

(3.42)

(3.43)

(3.44)

(3.45)

(3.46)

(3.47)

After the third coordinate transformation, the working domain is still in a square

box, i.e., —1 x* 51, —1 Y* 1 . The application of the third transformation

introduces the following relationships, taking into account that the vertical coordinate

again is a function of time and the horizontal coordinate.

ay* ay* ray a2Y* a2Y* rayi ay* ( a2y` ax ay ,ax) ' axe ay2 aY ax2

53


px= k x, p2 = k f a y * l + k 2 ■f 57* A[ s * J L I 57 J

50Bt

1 O, ■

f 50* ^2

f2

( 50* 50* 1+ * + On '2 U * ' J L 5 7 ' J [dX' 5 7 'J

50* 50*+ 0 A- ^ T + 0 5 - ^ v - 0 6 -H - 0 7 X

4 8X 5 57 6 7

0, = Jx, 0 2 = J, f BY*2

f2

r 57* 57*+ J 2 ■ + A ■[ s x ) 1 5 7 J ^5W 57 J

0 3 = J X f BY*) f 5 7 *1+ Jn 'I M jJ

CD

,o. = o , a = a

° 6 = J 6 ’ ° 7 = J 1

r b y*^v 5 7 y

Bottom:50

0 , - ^ + 02 ' 5W 2

5057*

= 0

0i - A ’ 02 — A + L7 •[ax) 2 I 57 J

Walls: A - ^ + ^ - ~ = 05A

/2 ,=W1, /22 = W 1-

57

r 57*^ + n 2 ■f 5 7 *1L [ 5 7 J

(3.41)

(3.42)

(3.43)

(3.44)

(3.45)

(3.46)

(3.47)

After the third coordinate transformation, the working domain is still in a square

box, i.e., - 1 < X * < 1 , — 1 < 7* < 1 . The application of the third transformation

introduces the following relationships, taking into account that the vertical coordinate

again is a function o f time and the horizontal coordinate.

57 57dX 57

^ 5 7 ^ \dX ;

5 7 5 7BX BY2

r BY^\d X j

+ -5757

f 5 2f A

v5.W y

53


a2ye = a 2y* (ay) ay* ay ail axay ay2 ax at ay* ( at )

ax _51-1* 82H _32H* ax axe' ' ax2 ax*2

(3.48)

3.4 Numerical method

After the rearrangement by the three coordinate transformations, the governing

equations are solved by the finite difference method, which is easy on discretization and

programming. The metrics in Eqs. (3.38), (3.40), (3.42), (3.44) and (3.46) for velocity

potential can be discretized by the second finite difference. The metrics inside the domain

are represented by the second central finite difference (Hoffmann, 1989, Anderson,

1995).

ago' 4)* —24)* + 41:0*

ax' = (Ax*)2

,320* — 20;4 + (I),* (AY* )2 ay *2

a2c,* ax*ay*

i+i,J+1+ 4AX*AY*

ac* _ 0,.+1,; (13,_1, ax* 2AX*

_ OY* 2AY*

(3.49)

(3.50)

(3.51)

(3.52)

(3.53)

In the above equations, i and j are the indices of the grid in x* and y* directions.

On the boundaries, the metrics on the left wall and the bottom are represented by the

second front finite difference.

54


a 2r* a 2r* ^ a r ^ ar*

#

1 r a r^ ja x a r

i

CD [ a x j ’ 8 t a r U J

8 H 8 H * 8 2H 8 2H8x 8x’ ax2 a x

*2 (3.48)

3.4 Numerical method

After the rearrangement by the three coordinate transformations, the governing

equations are solved by the finite difference method, which is easy on discretization and

programming. The metrics in Eqs. (3.38), (3.40), (3.42), (3.44) and (3.46) for velocity

potential can be discretized by the second finite difference. The metrics inside the domain

are represented by the second central finite difference (Hoffmann, 1989, Anderson,

1995).

a2o* _o*+1J - 2o ’ .+o;_1>y.ax’2 (ax*)2

a2o* o ’,+1 - 2 0 ’ . + o ’y._, ar’2 (at*)2

a 2o* ®;-u+.ax 8 Y 4AX AT

ao’ < i j - K u ax’ 2 AX*

ao’ _ o*y+l - o ’

(3.49)

(3.50)

(3.51)

(3.52)

(3.53)ar 2A y

In the above equations, i and j are the indices of the grid in X* and T* directions.

On the boundaries, the metrics on the left wall and the bottom are represented by the

second front finite difference.

54


al)* — 34:13*,/ + 4(I)* 1ax* 2AX*

ao* 4o*;2 - (ID*13

ay* - 2AY*

(3.54)

(3.55)

The metrics on the right wall and the top are represented by the second back finite

difference.

ao* -(-30N+.,; + 40,1 ax* - 2AX*

ate' = — 3€1) +1 + m )

ay* 2AY*

(3.56)

(3.57)

The metrics in Eq. (3.48) for the liquid height can be discretized by the second

finite difference.

au* H* — H*= i+i ax* 2AX*

a2 H* H:+1 — 2H: + 1/:_1

ax * 2

au* -3H; + 4H; — H;

ax* 2AX* (left node)

* * +,ail* -(-31- i N + 4oN - (1)*N_ I ) - (right node) ax* 2AX*

(3.58)

(3.59)

(3.60)

(3.61)

In the above equations, N and M are the total numbers of cells in A" and Y*

directions.

At the beginning of each time step, the potential function and the liquid height at

the free surface are first calculated for this time step by the Adams-Bashforth method.

The Euler method or the Runge-Kutta method can be used for the first two steps to start

the calculation. The potential function is then updated inside the liquid domain by the

55


a$* - 3 o ; .+ 4 o ; . - o 3/.d X * 2 AX*

ao* - 3 0 * , +4®*2 - 0 * 3

(3.54)

(3.55)8Y 2A Y

The metrics on the right wall and the top are represented by the second back finite

difference.

a®* -(-3®,., j +«>».,-® »-Jax* 2 AX*

ao* - ( - 3 0 * M+1+ 4 0 ^ - 0 * ^ , )

(3.56)

(3.57)a r 2A7

The metrics in Eq. (3.48) for the liquid height can be discretized by the second

finite difference.

dH* _ H*m

ax* 2 AX*

a2//* h m - i h , + //,„ax*2 (a x ’)2

dH _ - 3 H x + 4H 2 ~ h ; ~d)C~ 2 AX*

m ' - ( - 3 # ; +1+ 4 q ; - q ; . , ) ax* 2 AX*

(3.58)

(3.59)

(left node) (3.60)

(right node) (3.61)

In the above equations, N and M are the total numbers o f cells in X and Y

directions.

At the beginning of each time step, the potential function and the liquid height at

the free surface are first calculated for this time step by the Adams-Bashforth method.

The Euler method or the Runge-Kutta method can be used for the first two steps to start

the calculation. The potential function is then updated inside the liquid domain by the

55


second order finite difference, as well as on the rigid walls. The linear algebraic equations

for the potential function are solved iteratively by the SOR (successive over relaxation)

method. Due to the adoption of the finite difference method and the clustering of the

grids on the lower part, more computational time may be needed than that for the

boundary element method. The calculation of the coefficients in the governing equations

after the coordinate transformations doesn't take too much time. Most of the time is used

by the iterations of the linear algebraic equations. The basic procedures can be illustrated

by Figure 3.4.

Update free surface

2❑d order central FD 3rd order Adams- Bas hfo rth Euler method to start

Solve potential inside and on rigid walls

2nd order central FD SOR method to iterate the linear algebraic equations

Figure 3.4 Numerical procedures

The detailed solution procedures are given below:

I. Specify the system parameters, including tank geometric parameters, liquid fill levels,

etc.

II. Specify the operating parameters, including grid sizes, simulation time, time step,

convergence criterion c, parameters used in transformations, the relaxation factor, etc.

56


second order finite difference, as well as on the rigid walls. The linear algebraic equations

for the potential function are solved iteratively by the SOR (successive over relaxation)

method. Due to the adoption of the finite difference method and the clustering of the

grids on the lower part, more computational time may be needed than that for the

boundary element method. The calculation o f the coefficients in the governing equations

after the coordinate transformations doesn’t take too much time. Most o f the time is used

by the iterations o f the linear algebraic equations. The basic procedures can be illustrated

by Figure 3.4.

2nd order central FD 3 rd order Adams-Bashforth Euler method to start

2nd order central FDSOR method to iterate the linearalgebraic equations

Update free surface

Solve potential inside and on rigid walls

Figure 3.4 Numerical procedures

The detailed solution procedures are given below:

I. Specify the system parameters, including tank geometric parameters, liquid fill levels,

etc.

II. Specify the operating parameters, including grid sizes, simulation time, time step,

convergence criterion s, parameters used in transformations, the relaxation factor, etc.

56


III. Specify the excitations for the specific operation: amplitude and frequency for sway,

final steady acceleration and input time for turning, the excitation function for lane

change and double lane change, etc.

IV. Specify initial values for liquid height and velocity potential.

V. Calculate Y*, Y, ay*

and a2y* , X*, X, a, x,

ax* and a' x* aY ay' ax ax2 •

VI. Calculate the initial pressures on the walls if the forces and moments are needed.

VII. Start the calculation in the time domain: if Time < TotalTime

i. Calculate the liquid forces (if needed):

1. Calculate liquid forces in x and y direction for the last time step based on the

liquid pressure distributions.

ii. Update the free surface:

ao* ao* art ari 2. Calculate * , * , * and — on the free surface.

ax aY ax ax

3. For each node on the free surface, calculate

ay ay ay* aY* (1) , — according to Eq. (3.36) and — according to Eq.(4.38);

ax at ax ' at

(2) the coefficients in Eqs. (3.20), (3.31), (3.43), (3.18), (3.29) and (3.41).

4. Calculate ate

and ail*

according to Eqs. (3.42) and (3.40). at at

5. Update the velocity potential and the liquid height on the free surface by:

(1) the Euler method for the first two time steps;

(2) the Adams-Bashforth method after the first two time steps.

6. (Only necessary for situations where convergence is difficult) Smooth the

velocity potential and liquid height on the free surface.

57


III. Specify the excitations for the specific operation: amplitude and frequency for sway,

final steady acceleration and input time for turning, the excitation function for lane

change and double lane change, etc.

IV. Specify initial values for liquid height and velocity potential.

„ ^ __ ar* J a2r* ^ v ex* J d2x 'V. Calculate Y , Y , a n d — ,X ,X. a , x , a n d —.

ar a r2 d x d x 2

VI. Calculate the initial pressures on the walls if the forces and moments are needed.

VII. Start the calculation in the time domain: if Time < TotalTime

i. Calculate the liquid forces (if needed):

1. Calculate liquid forces in x and y direction for the last time step based on the

liquid pressure distributions.

ii. Update the free surface:

2. Calculate , ^ r and on the free surface.d x ar e x s x

3. For each node on the free surface, calculate

ar ar ar* ar*(1)— , — according to Eq. (3.36) a n d , according to Eq.(4.38);

dX dt dX dt

(2) the coefficients in Eqs. (3.20), (3.31), (3.43), (3.18), (3.29) and (3.41).

ao* dH*4. C alculate a n d according to Eqs. (3.42) and (3.40).

dt dt

5. Update the velocity potential and the liquid height on the free surface by:

(1) the Euler method for the first two time steps;

(2) the Adams-Bashforth method after the first two time steps.

6. (Only necessary for situations where convergence is difficult) Smooth the

velocity potential and liquid height on the free surface.

57


iii. Solve potential inside and on the rigid walls:

aH* 52H* aft ail 7. Calculate and

ax* ' ax*2 ' ax ax

8. Calculate the coefficients

(1) in (3.22), (3.33), (3.45) for the bottom;

(2) in (3.16), (3.27), (3.39) for the inside;

(3) in (3.23), (3.35), (3.47) for the bottom for the left and right wall.

9. Solve the linear algebraic equations for the velocity potential by SOR.

If Convergence < c and Iteration number < Maximum iteration number

Stop the iteration and go to step 10;

If Convergence > c and Iteration number < Maximum iteration number

Continue the iteration;

If Convergence > c and Iteration number > Maximum iteration number

Stop and report the problem even if the convergence still could be

achieved by increasing the maximum iteration number.

10. Calculate the pressure distribution (if forces and moments are needed) on the

walls for the next time step.

11. Repeat steps 1-10 until the end of the simulation time.

iv. Posprocessing:

12. Save data;

13. Plot the figures for liquid height and liquid forces.

As pointed out previously, due to the application of the second transformation, i.e.

the boundary-fitted method, there is no need to deal with the boundary conditions on the

58


iii. Solve potential inside and on the rigid walls:

„ ^ , dH* 82H* BH* , dH7. Calculate — - , ----- —, and — .

dX dX dX dX

8. Calculate the coefficients

(1) in (3.22), (3.33), (3.45) for the bottom;

(2) in (3.16), (3.27), (3.39) for the inside;

(3) in (3.23), (3.35), (3.47) for the bottom for the left and right wall.

9. Solve the linear algebraic equations for the velocity potential by SOR.

If Convergence < s and Iteration number < Maximum iteration number

Stop the iteration and go to step 10;

If Convergence > e and Iteration number < Maximum iteration number

Continue the iteration;

If Convergence > s and Iteration number > Maximum iteration number

Stop and report the problem even if the convergence still could be

achieved by increasing the maximum iteration number.

10. Calculate the pressure distribution (if forces and moments are needed) on the

walls for the next time step.

11. Repeat steps 1-10 until the end o f the simulation time.

iv. Posprocessing:

12. Save data;

13. Plot the figures for liquid height and liquid forces.

As pointed out previously, due to the application of the second transformation, i.e.

the boundary-fitted method, there is no need to deal with the boundary conditions on the

58


time-varying free surface by using free surface smoothing and volume correction after the

free surface updating is finished. However, free surface smoothing could be helpful for

cases in which convergence is difficult to obtain. In step 6 of the above procedures, the

liquid height can optionally be smoothed by using the following equations (Longuet-

Higgins and Cokelet, 1975, Ortiz, 1996).

(11H, + 12H,+1 — 6H,+2 — 4111+3 + 31/„+4 )/ 16 i =1

+ 8H, + 6H„+, — H,+3 )116 i = 2

+1011, + —H,+2)/16 3 i N —1

(3H,+, +8H, +6H,_, —H,_3 )/16 i =N

(11H, +12H,_ — 6111_2 — 4H1_3 + 3H,_4 )/16 i = N+1

(3.58)

The velocity potential on the free surface can be smoothed in the same way by

using the above equations if necessary.

Compared with some other numerical schemes for sloshing problems in 2D

circular tanks, there are some advantages to the current scheme. The current method does

not need to deal with the boundary conditions on the time varying curved walls and free

surface. There is no need for capturing or smoothing of the free surface and performance

of volume correction, which are commonly used in the existing approaches. Complicated

algorithms for interpolation on rigid walls, as well as updating the free surface, are

completely avoided. These make the algorithm efficient and stable. The governing

equations are arranged so that the programming is easy. Replacement of different

transformation equations for the first transformation can be easily carried out in the same

way for tanks with arbitrary wall shapes. In fact, even in cases where the mathematical

expressions for the walls are not available, the metrics can still be obtained by the second

order central finite difference. Replacement of different transformation equations for the

59


time-varying free surface by using free surface smoothing and volume correction after the

free surface updating is finished. However, free surface smoothing could be helpful for

cases in which convergence is difficult to obtain. In step 6 o f the above procedures, the

liquid height can optionally be smoothed by using the following equations (Longuet-

Higgins and Cokelet, 1975, Ortiz, 1996).

( I I HI +12HM - 6 H M - 4 H M +3 HIJ / 1 6 i = 1

H, = <! ( - //,_2 + 4//,_, +107/,. + 4H m - H i+1)/16 3 < i < N - 1 (3.58)

(3W,.„+8ff,+6/fM- t f M)/16 i = N (l 1H, +12H,_, - 6H,_2 - 4 + 3//._4) /16 / = ,V +1

The velocity potential on the free surface can be smoothed in the same way by

using the above equations if necessary.

Compared with some other numerical schemes for sloshing problems in 2D

circular tanks, there are some advantages to the current scheme. The current method does

not need to deal with the boundary conditions on the time varying curved walls and free

surface. There is no need for capturing or smoothing of the free surface and performance

of volume correction, which are commonly used in the existing approaches. Complicated

algorithms for interpolation on rigid walls, as well as updating the free surface, are

completely avoided. These make the algorithm efficient and stable. The governing

equations are arranged so that the programming is easy. Replacement of different

transformation equations for the first transformation can be easily carried out in the same

way for tanks with arbitrary wall shapes. In fact, even in cases where the mathematical

expressions for the walls are not available, the metrics can still be obtained by the second

order central finite difference. Replacement o f different transformation equations for the

59


third transformation can also be done without much extra modification needed for

program codes in adjusting the grids distribution.

3.5 Results and discussion

3.5.1 Sloshing in circular tanks under harmonic excitations with small amplitudes

In this section, the presented method is first used to simulate the sloshing problem

in a horizontal circular tank subjected to lateral harmonic motion with small amplitudes.

The tank is half-full. The displacement and acceleration of the tank are described by sine

functions.

D, = Do sin wt A, = —Dow2 sin ox (3.59)

In the above equations, Do = 0.002 is the amplitude of the tank displacement. co

is the excitation frequency.

To obtain necessary computational accuracy and solution convergence, numerical

trials have been carried out to find the proper grid size and time step. The grid size is

finally set to be 41x81 and the time step is set to be 0.005 for all cases in order to meet

the accuracy requirement, although coarser meshes and a larger time step can also be

used to get the same accuracy for cases where nonlinearity is weak or the excitation

frequencies are not close to the natural frequency. The convergence criterion of the

velocity potential is set to be 10-8.

Figure 3.5 presents the nondimensional liquid heights on the left and right walls

and at the middle point in a two-dimensional circular tank subjected to harmonic motion

under small amplitudes and different frequencies. It can be seen from Figure 3.5(a) that

the liquid heights on the walls are sine waves when the excitation frequency is far from

60


third transformation can also be done without much extra modification needed for

program codes in adjusting the grids distribution.


3.5.1 Sloshing in circular tanks under harmonic excitations with small amplitudes

In this section, the presented method is first used to simulate the sloshing problem

in a horizontal circular tank subjected to lateral harmonic motion with small amplitudes.

The tank is half-full. The displacement and acceleration of the tank are described by sine

functions.

Dx = D 0 sin cot Ax - -D 0 co2 sin cot (3.59)

In the above equations, D0 = 0.002 is the amplitude o f the tank displacement, co

is the excitation frequency.

To obtain necessary computational accuracy and solution convergence, numerical

trials have been carried out to find the proper grid size and time step. The grid size is

finally set to be 41x81 and the time step is set to be 0.005 for all cases in order to meet

the accuracy requirement, although coarser meshes and a larger time step can also be

used to get the same accuracy for cases where nonlinearity is weak or the excitation

frequencies are not close to the natural frequency. The convergence criterion o f the

velocity potential is set to be 10'8.

Figure 3.5 presents the nondimensional liquid heights on the left and right walls

and at the middle point in a two-dimensional circular tank subjected to harmonic motion

under small amplitudes and different frequencies. It can be seen from Figure 3.5(a) that

the liquid heights on the walls are sine waves when the excitation frequency is far from

60


the natural frequency, which is 1.1644 or 1.169 (Lamb, 1945, Budiansky, 1960, and

Solaas and Faltinsen, 1997). When the excitation frequencies are close to the natural

frequency, i.e., Figure 3.5(c) and 3.5(e), the beat phenomena are obviously shown. When

the frequency is quite close to the natural frequency, i.e., Figure 3.5(d), the amplitude

grows monotonically with time. For other frequencies, i.e., Figure 3.5(b), the response is

the result of a combination of the excitation frequency and the natural frequency. For

most cases, the curves for the liquid height of the middle point at the free surface show

that the oscillation of the liquid height at the middle point is very small.

One criterion by which to verify the results is whether or not the conservation of

mass can be assured due to the assumption of incompressibility of liquid. The integral of

the free surface elevation over the free surface can be used for this purpose (Solaas and

Faltinsen, 1997). In the program, the volume error, Ve, is discretized on the free surface in

the a/3 coordinate system from the following expression.

Ve = 11 rida (3.60)

The volume errors corresponding to Figure 3.5 are obtained using the above

equation, and shown in Figure 3.6. Although smoothing of the free surface and volume

correction are not adopted in the current scheme, the results are quite satisfactory.


near resonance

When the tank is subjected to a harmonic excitation with a finite amplitude near

resonance, the liquid behaviour is different because of the strong nonlinearity and

participation of higher modes. Figure 3.7 illustrates some free surface profiles from the

61


the natural frequency, which is 1.1644 or 1.169 (Lamb, 1945, Budiansky, 1960, and

Solaas and Faltinsen, 1997). When the excitation frequencies are close to the natural

frequency, i.e., Figure 3.5(c) and 3.5(e), the beat phenomena are obviously shown. When

the frequency is quite close to the natural frequency, i.e., Figure 3.5(d), the amplitude

grows monotonically with time. For other frequencies, i.e., Figure 3.5(b), the response is

the result o f a combination of the excitation frequency and the natural frequency. For

most cases, the curves for the liquid height of the middle point at the free surface show

that the oscillation o f the liquid height at the middle point is very small.

One criterion by which to verify the results is whether or not the conservation of

mass can be assured due to the assumption o f incompressibility o f liquid. The integral of

the free surface elevation over the free surface can be used for this purpose (Solaas and

Faltinsen, 1997). In the program, the volume error, Ve, is discretized on the free surface in

the a p coordinate system from the following expression.

The volume errors corresponding to Figure 3.5 are obtained using the above

equation, and shown in Figure 3.6. Although smoothing of the free surface and volume

correction are not adopted in the current scheme, the results are quite satisfactory.


near resonance

When the tank is subjected to a harmonic excitation with a finite amplitude near

resonance, the liquid behaviour is different because o f the strong nonlinearity and

participation o f higher modes. Figure 3.7 illustrates some free surface profiles from the

(3.60)

61


Nondimensional liquid height

x10-4

1

0

a 20 40 60 80

Nondimensional time


X10-3

2

0

-2

-40

.•4 A • A t i I P e 1 I e I f I 1 % 1 li I 1i i I

%

I.I

I%

I. 1 I 1 it IA II II 4

20


0 20

40

40

i al A f I I i 1 ir I i i I i i

1

I I it

Ir I

V 1 I i

I1C

60 80

Nondimensional time

60 80

Nondimensional time

(a)o)=0.25

(b)o)=0.75

(c)w=1.00


amplitudes

left middle - - - - right

62



0

1

40 60 800 20

(a)co=0.25

Nondimensional time


■3xlO'

2

0V'

2 1 J

420 40 60 800

(b)co=0.75

Nondimensional time


0.01

If

- 0.01

0 20 40 60 80

(c)<d=1.00

Nondimensional time


amplitudes

left m id d le right

62



0:1

0.05

0

-0.05

-0.1 0 2.0


40 60 80

Nondimensional time

Nondimensional time

(d)w=1.16

(e)w=1.40


amplitudes (continued)

left middle - - - - right

63



0.1

0.05

-0.05

- 0.140 8060

Nondimensional time

(d)co=1.16


0.02

0.01

0

- 0.01

- 0.02

M l ' A « A /

' A ' m l f t w m l !

h aAAAA

n W ' A l l fk •

i I V\ t 1 1 » 1 1 § f r A 1 I I VI il [ i M J ,1 / u> a i H. w ' M i

V VV“■ i

i l p U S I l !/ I Af i1

v - v y V s

•

11 1

0

(e)co=1.40

20 40 60 80Nondimensional time


amplitudes (continued)

left m id d le right

63


Volume error

x10-8

2

0

-2

0

Volume error

Volume error

20 40 60 80

Nondimensional time

20

20

40 60 80 Nondimensional time

40 60 80

Nondimensional time

(a)w=0.25

(b)co=0.75

(c)co=1.00


with small amplitudes

64


Volume error

2

(a)co=0.250

2

20 600 40 80

Nondimensional time

Volume error

xlO'

20 40 60 80 Nondimensional time

Volume error

xlO'

20 40 60 80

Nondimensional time


with small amplitudes

64


Volume error

20 40 60 80

Nondimensional time

Volume error

(e)co=1.40

Nondimensional time


with small amplitudes (continued)

simulation results. The amplitude of the tank motion is 0.1, and the frequency is 0.9694.

Other parameters are the same as those in section 3.5.1. The higher modes can be seen in

the free surface curves. The volume error is also shown for each time point. Under

65


Volume error

Nondimensional time

Volume error

(e)oo=1.40

60 80

Nondimensional time

Figure 3.6 Volume error o f calculating liquid motion subjected to harmonic excitations

with small amplitudes (continued)

simulation results. The amplitude of the tank motion is 0.1, and the frequency is 0.9694.

Other parameters are the same as those in section 3.5.1. The higher modes can be seen in

the free surface curves. The volume error is also shown for each time point. Under

65


continuous excitation, the motion of the liquid becomes violent. The volume error will

then increase. The iteration finally loses convergence, as would be expected.

0 0 0 t = 3, Ve= 0.019 t = 4, Ve = 0.011 t= 5, Ve= 0.049

0 0 t = 6, Ve= 0.126 t = 7,Ve= 0.046 t = 8, Ve = 0.101


near resonance with finite amplitude

For extremely large amplitudes, Tosaka and Sugino (1990) simulated the sloshing

in a partially filled circular tank subjected to forced horizontal acceleration, where the

amplitude of the harmonic motion of the tank could be as large as the radius of the tank.

Simulation results showed that the liquid could climb up the wall and occupy the top part

of the tank. This is out of the scope of the present research.

66


continuous excitation, the motion of the liquid becomes violent. The volume error will

then increase. The iteration finally loses convergence, as would be expected.

t = 4, V e - = 0.011 t = 5, V e = 0.049t = 3, V e = 0.019

t = 7 , V e = 0.046 t = 8, V e ~ 0.101t = 6, V e = 0.126


near resonance with finite amplitude

For extremely large amplitudes, Tosaka and Sugino (1990) simulated the sloshing

in a partially filled circular tank subjected to forced horizontal acceleration, where the

amplitude o f the harmonic motion o f the tank could be as large as the radius o f the tank.

Simulation results showed that the liquid could climb up the wall and occupy the top part

o f the tank. This is out of the scope of the present research.

66


3.5.3 Transient liquid oscillations in circular tanks

In most applications of road transportation, the input accelerations do not

necessarily need to be harmonic. To study the transient liquid responses in circular tanks,

three different input types are considered, and sloshing problems under these input types

are simulated. The system parameters are the same as those in Section 3.4.1. The first

type is shown in Figure 3.8(a), where the lateral acceleration applied builds up to its final

steady value, Ao, in an input time, to. This can be used to simulate the acceleration input

for a liquid vehicle during a turning operation.

Figure 3.9 shows the nondimensional liquid height on the right wall of a circular

tank during turning subjected to different steady accelerations with the same input time,

10.0. It can be found that the liquid height decreases during the input time. After the final

steady acceleration is reached, the liquid heights begin to oscillate around their

equilibrium positions, which correspond to the static positions that can be obtained by

isobars. The oscillation amplitude increases with the increase in the steady acceleration.

Therefore, the forces and moments on the tank and supporting structures caused by

turning would also have oscillatory characteristics, which cannot be ignored when

structural integrity and fatigue are considered, especially under large turning

accelerations. When the lateral acceleration for the turning operation for liquid cargo

vehicles is higher than 0.3g, it is very possible that rollover of the partially filled vehicle

could happen, and the conditions for sloshing problems will not exist. Calculations also

show that when the input acceleration is higher than 0.3g, the convergence of the iteration

can become very slow and difficult, which is basically caused by the application of the

67


3.5.3 Transient liquid oscillations in circular tanks

In most applications of road transportation, the input accelerations do not

necessarily need to be harmonic. To study the transient liquid responses in circular tanks,

three different input types are considered, and sloshing problems under these input types

are simulated. The system parameters are the same as those in Section 3.4.1. The first

type is shown in Figure 3.8(a), where the lateral acceleration applied builds up to its final

steady value, A o , in an input time, t o - This can be used to simulate the acceleration input

for a liquid vehicle during a turning operation.

Figure 3.9 shows the nondimensional liquid height on the right wall o f a circular

tank during turning subjected to different steady accelerations with the same input time,

10.0. It can be found that the liquid height decreases during the input time. After the final

steady acceleration is reached, the liquid heights begin to oscillate around their

equilibrium positions, which correspond to the static positions that can be obtained by

isobars. The oscillation amplitude increases with the increase in the steady acceleration.

Therefore, the forces and moments on the tank and supporting structures caused by

turning would also have oscillatory characteristics, which cannot be ignored when

structural integrity and fatigue are considered, especially under large turning

accelerations. When the lateral acceleration for the turning operation for liquid cargo

vehicles is higher than 0.3g, it is very possible that rollover o f the partially filled vehicle

could happen, and the conditions for sloshing problems will not exist. Calculations also

show that when the input acceleration is higher than 0.3g, the convergence of the iteration

can become very slow and difficult, which is basically caused by the application o f the

67


Non

dim

ensi

onal

acc

eler

atio

n

Non

dim

ensi

onal

acc

eler

atio

n

(b)

0.2

0.1

-0.1

to Nondimensional time

-0.2 0

0.2

0.1

0

-0.1

-0.2. 0

(c)

10 20 30 40 50 Nondimensional time

20 40 60 80 Nondimensional time

Figure 3.8 Acceleration input

(a) turning (b) lane change (c) double lane change

68


o

<D

CTJfio

t oNondimensional time

2 0.2

8 0.1

I °Wge - o . ico

£ - 0.230 40 5

Nondimensional time

I 0.2

^ rv§ 0C/3

I -0 -1■oco£ - 0.2

4020Nondimensional time

Figure 3.8 Acceleration input

(a) turning (b) lane change (c) double lane change

68


second transformation, a type of boundary fitted method. It is known that the simple

algebraic mappings are not robust when the free surface is steep or has multiple values.

Therefore, the method can only be used to investigate liquid motion under normal

highway operation conditions.

Non

dim

ensi

onal

liq

uid

heig

ht

0

-0.05

-0.1

-0.15

-0.2

-0.25

-0.3

-0.35 0 5 10 15 20

Nondimensional time


final accelerations

Ao= 0 .05 — -- — A0 = 0 .10 Ao= 0.15

Ao — 0.20 Ao — 0.25 Ao = 0.30

To find out the influence of the input time during turning, the simulation has been

done using a steady acceleration, 0.1, at different input times. The nondimensional input

time range, 2.5 to 12.5, corresponds to a dimensional time range, 0.8 to 4.0 seconds, for a

69


second transformation, a type of boundary fitted method. It is known that the simple

algebraic mappings are not robust when the free surface is steep or has multiple values.

Therefore, the method can only be used to investigate liquid motion under normal

highway operation conditions.

-0.05

- 0.1

-0.15cd G o

• (/> ca.£ -0.25CO£ -0.3

- 0.2

-0.3520

Nondimensional time


final accelerations

A o = 0 . 0 5 ------------------0.10 A o = 0.15

A o ~ 0 .2 0 -------- A 0 = 0.25 ............ ^ 0=0.30

To find out the influence of the input time during turning, the simulation has been

done using a steady acceleration, 0.1, at different input times. The nondimensional input

time range, 2.5 to 12.5, corresponds to a dimensional time range, 0.8 to 4.0 seconds, for a

69


tank with a 2 m diameter. The results are shown in Figure 3.10. For different input times,

the liquid heights begin to oscillate at different times. The oscillation amplitude decreases

with the increase of the input time. For a smaller input time, the curves clearly show that

higher modes are excited and superposed on the lowest mode. Although the equilibrium

positions are the same, the oscillatory behaviour of the liquid is harmful. Therefore, a

large input time is better from the point of view of the vehicle stability and structural

integrity. The last case, to = 0, is an extreme situation, and represents a suddenly applied

acceleration without an input time. This causes the largest oscillatory amplitude and

should be avoided in the operation.

Non

dim

ensi

onal

liq

uid

heig

ht

0

-0.05

-0.1

-0.15

0 5 10 15 20 25 30 35

Nondimensional time


under different input time

to = 12.5 — — — to = 7.5 to — 2.5 to = 0.0

70


tank with a 2 m diameter. The results are shown in Figure 3.10. For different input times,

the liquid heights begin to oscillate at different times. The oscillation amplitude decreases

with the increase o f the input time. For a smaller input time, the curves clearly show that

higher modes are excited and superposed on the lowest mode. Although the equilibrium

positions are the same, the oscillatory behaviour of the liquid is harmful. Therefore, a

large input time is better from the point of view of the vehicle stability and structural

integrity. The last case, to = 0, is an extreme situation, and represents a suddenly applied

acceleration without an input time. This causes the largest oscillatory amplitude and

should be avoided in the operation.

2 -0.05rs

I -0 1C/3

C3us

-3

I -°15

0 5 10 15 20 25 30 35

Nondimensional time


under different input time

to = 12.5 — — - t0 =7.5 ---------- to =2.5 t0= 0 .0

70


The other two input types for transient response study are shown in Figure 3.8 (b)

and (c). In Figure 3.8 (b), the input acceleration is expressed using a single period sine

function. This is used to simulate the lateral acceleration input for a road vehicle during a

lane change operation. In Figure 3.8 (c), the input acceleration is described using a

combination of sine functions, which can approximate the lateral acceleration input for a

road vehicle during a double lane change operation. The wave developments under these

two types of input are shown in Figure 3.11(a) and Figure 3.11 (b). Since the input

frequency is much smaller than the liquid sloshing modes, the wave in the tank develops

in step with the input acceleration. The natural modes are actuated and shown in the

figures. After the operation is finished, the liquid height oscillates around the equilibrium

position. Because the liquid in the double lane change operation is actuated much more

than in the lane change, the oscillation amplitude after the double lane change is much

larger than after the lane change, even though the amplitude of the input acceleration is

larger in the lane change.

It is interesting that in some early vehicle stability researches, liquid motion in

tanks was sometimes modeled by mass centre models, which assumed that the liquid kept

a perfectly inclined flat surface during operations. This assumption may cause two

concerns. First, for many operations, the free surfaces have complicated shapes and

cannot be simply considered as a straight line. Also, the mass centre of the liquid bulk

would be different. This phenomenon has been shown numerically (Popov et al, 1993b),

as well as experimentally (Rakheja et al, 1992). Second, even the liquid motion is

approximated by the first natural liquid mode, for which the free surface could be

approximated by a straight line, the dynamic feature of liquid motion makes the flat free

71


The other two input types for transient response study are shown in Figure 3.8 (b)

and (c). In Figure 3.8 (b), the input acceleration is expressed using a single period sine

function. This is used to simulate the lateral acceleration input for a road vehicle during a

lane change operation. In Figure 3.8 (c), the input acceleration is described using a

combination o f sine functions, which can approximate the lateral acceleration input for a

road vehicle during a double lane change operation. The wave developments under these

two types of input are shown in Figure 3.11(a) and Figure 3.11 (b). Since the input

frequency is much smaller than the liquid sloshing modes, the wave in the tank develops

in step with the input acceleration. The natural modes are actuated and shown in the

figures. After the operation is finished, the liquid height oscillates around the equilibrium

position. Because the liquid in the double lane change operation is actuated much more

than in the lane change, the oscillation amplitude after the double lane change is much

larger than after the lane change, even though the amplitude of the input acceleration is

larger in the lane change.

It is interesting that in some early vehicle stability researches, liquid motion in

tanks was sometimes modeled by mass centre models, which assumed that the liquid kept

a perfectly inclined flat surface during operations. This assumption may cause two

concerns. First, for many operations, the free surfaces have complicated shapes and

cannot be simply considered as a straight line. Also, the mass centre o f the liquid bulk

would be different. This phenomenon has been shown numerically (Popov et al, 1993b),

as well as experimentally (Rakheja et al, 1992). Second, even the liquid motion is

approximated by the first natural liquid mode, for which the free surface could be

approximated by a straight line, the dynamic feature of liquid motion makes the flat free

71


surface have an oscillatory behaviour. This in turn makes the mass centre of the liquid

bulk have an oscillatory behaviour. Therefore, the dynamic liquid motion can hardly be

described by the liquid bulk. N

ondi

men

sion

al l

iqui

d he

ight

(a)

0 Xi

Xi 20

20 40

Nondimensional time

40

Nondimensional time

Figure 3.11 Wave profile in a horizontal circular tank

during (a) lane change (b) double lane change

72

60

80


surface have an oscillatory behaviour. This in turn makes the mass centre o f the liquid

bulk have an oscillatory behaviour. Therefore, the dynamic liquid motion can hardly be

described by the liquid bulk.

-1 0 Nondimensional time

§ -0.4

Nondimensional time

Figure 3.11 Wave profile in a horizontal circular tank

during (a) lane change (b) double lane change

72


In the above simulations of the transient response of liquids, it is shown that the

oscillatory motions of liquids continue due to the lack of damping in the current study. In

a real situation, the liquid will eventually calm down if there is no new excitation. The

time required for the transient effect to die out depends on the viscosity of the liquid

being carried.

The importance of viscosity in the liquid sloshing problems depends on the

different excitation types. The effect of viscosity is more significant near tank walls and

near a resonant frequency during liquid sloshing. The influence of fluid viscosity can be

considered without any approximation if the sloshing problems are described by the

Navier-Stokes equations. For example, a straightforward investigation was conducted by

Popov et al (1993b) to study the influence of liquid viscosity on the dynamics of road

tanks subjected to suddenly applied lateral accelerations, based on the Navier-Stokes

equations. The effect was shown in terms of the Reynolds number. It was found that the

Re number had no influence on the magnitudes and frequencies of the sloshing

parameters in the range 107-105. In the range 105-103, the influence was rather small. For

the range Re < 103, the difference in amplitudes and frequencies rapidly increased in such

a way that more viscous liquids vibrated more slowly, with a smaller amplitude and with

stronger decay. It was concluded that for the majority of practical cases, covered by Re >

105, the main sloshing parameters, such as the magnitudes and frequencies of the sloshing

forces and moments, were almost independent of the Re number (Popov et al, 1993b).

However, when the sloshing problems are modeled by the potential flow theory,

the liquid viscosity cannot be directly included. As long as the tank holding the liquid is

smooth inside and not very small, viscosity plays a minor role in determining the sloshing

73


In the above simulations of the transient response of liquids, it is shown that the

oscillatory motions of liquids continue due to the lack of damping in the current study. In

a real situation, the liquid will eventually calm down if there is no new excitation. The

time required for the transient effect to die out depends on the viscosity o f the liquid

being carried.

The importance of viscosity in the liquid sloshing problems depends on the

different excitation types. The effect of viscosity is more significant near tank walls and

near a resonant frequency during liquid sloshing. The influence o f fluid viscosity can be

considered without any approximation if the sloshing problems are described by the

Navier-Stokes equations. For example, a straightforward investigation was conducted by

Popov et al (1993b) to study the influence o f liquid viscosity on the dynamics o f road

tanks subjected to suddenly applied lateral accelerations, based on the Navier-Stokes

equations. The effect was shown in terms of the Reynolds number. It was found that the

Re number had no influence on the magnitudes and frequencies o f the sloshing

parameters in the range 107-105. In the range 105-103, the influence was rather small. For

the range Re < 103, the difference in amplitudes and frequencies rapidly increased in such

a way that more viscous liquids vibrated more slowly, with a smaller amplitude and with

stronger decay. It was concluded that for the majority o f practical cases, covered by Re >

105, the main sloshing parameters, such as the magnitudes and frequencies o f the sloshing

forces and moments, were almost independent o f the Re number (Popov et al, 1993b).

However, when the sloshing problems are modeled by the potential flow theory,

the liquid viscosity cannot be directly included. As long as the tank holding the liquid is

smooth inside and not very small, viscosity plays a minor role in determining the sloshing

73


behaviour. Hence it is usually neglected without sacrificing the essential features of the

phenomenon. In cases where the damping effect is expected in the simulation, a

commonly adopted method is to introduce a dissipation mechanism into the description

of the fluid instead of solving the full Navier-Stokes equations. By introducing a fictitious

term into the Euler equation, a modified Bernoulli equation can be obtained with a

modified Rayleigh damping term (Faltinsen, 1974 and Faltinsen, 1978). In this way, the

dynamic condition on the free surface, i.e., Eq. (3.12), can be modified as:

act ,__1 aq3N2at 2 ax

(a1

2— — Axx — pc,

a (3.61)

The modified Rayleigh damping term, ,uco, is included to simulate the viscosity of the

liquid in order to study the damping effect in the liquid motion. The way to decide the

value of p can be found in Faltinsen (1974), Faltinsen (1978) and Ortiz and Barhorst

(1998).

3.6 Liquid motion in 2D elliptical tanks

3.6.1 Statement of liquid motion in 2D elliptical tanks

Tanks with elliptical cross sections are also widely used in road transportation

industry. There are very few reports on sloshing studies for this specific configuration in

the current literature. The first natural mode of liquid motion was investigated by Salem

(2000) using an equivalent mechanical pendulum by linearization under the assumption

of a flat free surface and small oscillation angle. The analogy was established in the

transverse direction and therefore had a 2D solution. Liquid motion under finite

excitations in the roll plane and 3D liquid behaviour were not considered.

74


behaviour. Hence it is usually neglected without sacrificing the essential features o f the

phenomenon. In cases where the damping effect is expected in the simulation, a

commonly adopted method is to introduce a dissipation mechanism into the description

o f the fluid instead of solving the full Navier-Stokes equations. By introducing a fictitious

term into the Euler equation, a modified Bernoulli equation can be obtained with a

modified Rayleigh damping term (Faltinsen, 1974 and Faltinsen, 1978). In this way, the

dynamic condition on the free surface, i.e., Eq. (3.12), can be modified as:

dtpdt

r dcp^ \ d x j

+r dcpv

- £ - A xx-n<p (3.61)

The modified Rayleigh damping term, /J.(p, is included to simulate the viscosity o f the

liquid in order to study the damping effect in the liquid motion. The way to decide the

value o f n can be found in Faltinsen (1974), Faltinsen (1978) and Ortiz and Barhorst

(1998).

3.6 Liquid motion in 2D elliptical tanks

3.6.1 Statement of liquid motion in 2D elliptical tanks

Tanks with elliptical cross sections are also widely used in road transportation

industry. There are very few reports on sloshing studies for this specific configuration in

the current literature. The first natural mode of liquid motion was investigated by Salem

(2000) using an equivalent mechanical pendulum by linearization under the assumption

o f a flat free surface and small oscillation angle. The analogy was established in the

transverse direction and therefore had a 2D solution. Liquid motion under finite

excitations in the roll plane and 3D liquid behaviour were not considered.

74


The method established for 2D circular tanks in this chapter could be easily

extended to 2D elliptical tanks by simply modifying the working domain equation and

the first transformation. Actually, a tank with a circular cross section is a special case of

an elliptical cross section when the aspect ratio of the semi major axis to the semi minor

axis becomes one. The sketch of liquid sloshing in a 2D elliptical tank can be illustrated

by Figure 3.12.

2b

Figure 3.12 Sketch of liquid motion in an elliptical tank

The wall of the tank is described by the following elliptical curve.

2 +

1 x l VI b - CO )2 =1a 2 2

(3.62)

In the above equation, a and b are the lengths of the semi major axis and the semi

minor axis, respectively. co is the distance between the centre of the tank and the origin of

the coordinate system. When the governing equations of the liquid are normalized by the

length of the semi minor axis, i.e., b, the above equation will become:

75


The method established for 2D circular tanks in this chapter could be easily

extended to 2D elliptical tanks by simply modifying the working domain equation and

the first transformation. Actually, a tank with a circular cross section is a special case of

an elliptical cross section when the aspect ratio of the semi major axis to the semi minor

axis becomes one. The sketch of liquid sloshing in a 2D elliptical tank can be illustrated

by Figure 3.12.

2b

Figure 3.12 Sketch of liquid motion in an elliptical tank

The wall of the tank is described by the following elliptical curve.

i L + f o - C o ) =1 (3 62)a b

In the above equation, a and b are the lengths o f the semi major axis and the semi

minor axis, respectively, co is the distance between the centre of the tank and the origin of

the coordinate system. When the governing equations o f the liquid are normalized by the

length of the semi minor axis, i.e., b, the above equation will become:

75


2 X 2 ± (y — c0 )2 =1

a (3.63)

Note that a and co in this equation are the nondimensional quantities that are normalized

by b. The first transformation will then be performed using the following equations.

a= x

= Y (y — co )2

(3.64)

The singularity at the bottom point will be treated in the same way used for a

circular tank. By using the same method for the other two coordinate transformations and

the same numerical procedures, the sloshing problems in elliptical tanks can be solved

without any difficulties.

3.6.2 Natural frequencies

The natural frequencies of liquid sloshing in circular tanks have been well

evaluated by several researchers (Budiansky, 1960, McIver, 1989) under different fill

levels. However, there is no detailed study on the natural frequencies of liquid motion in

elliptical tanks. In this section, the natural frequencies in an elliptical tank are calculated

to verify the method developed in this chapter. When the natural frequency problem is to

be solved, the linearized Bernoulli equation on the free surface can be obtained as follows

(Budiansky, 1960).

a =

ay (3.65)

In the above equation, xis the eigenvalue. The frequency is to be solved using:

(3.66)

76


Note that a and Co in this equation are the nondimensional quantities that are normalized

by b. The first transformation will then be performed using the following equations.

The singularity at the bottom point will be treated in the same way used for a

circular tank. By using the same method for the other two coordinate transformations and

the same numerical procedures, the sloshing problems in elliptical tanks can be solved

without any difficulties.


The natural frequencies o f liquid sloshing in circular tanks have been well

evaluated by several researchers (Budiansky, 1960, Mclver, 1989) under different fill

levels. However, there is no detailed study on the natural frequencies o f liquid motion in

elliptical tanks. In this section, the natural frequencies in an elliptical tank are calculated

to verify the method developed in this chapter. When the natural frequency problem is to

be solved, the linearized Bernoulli equation on the free surface can be obtained as follows

(Budiansky, 1960).

In the above equation, k is the eigenvalue. The frequency is to be solved using:

x (3.64)a =

(3.66)

76


Since the natural frequency problem is fixed in the domain which the liquid

initially occupies, only the first and the third transformations in this chapter are used for

the linearized Bernoulli equation on the free surface. Discretization of the governing

equations leads to a generalized eigenvalue problem.

Tx = /Mx (3.67)

In the above equations, Wand 0 are coefficient matrices and x is the eigenvector.

To verify the method developed in this chapter for the eigenvalue problem, the first three

eigenvalues for a half-filled circular tank have been solved by trying different mesh sizes.

Figure 3.13 illustrates the eigenvalue results calculated by different grid numbers in both

horizontal and vertical directions. The results show that the eigenvalues eventually

converge when the cell numbers are 40x80. The eigenvalues converge to the values

calculated by other methods, which can be shown by the last points (McIver, 1989) for

corresponding curves.

Eigenvalues in a partially filled circular tank at different fill levels have been

solved in the same way. The results for a tank of unit radius are shown in Figure 3.14.

The first and third eigenvalues are the first and second antisymmetrical modes. The

second eigenvalue is the first symmetrical mode. The eigenvalues for points on the

dashed line in this figure are taken from McIver (1989), in which the linearized wave

theory was used to solve the eigenvalues of free oscillations numerically by integral

equations in a bipolar coordinate system for circular tanks. The eigenvalues for points on

the solid line are calculated using the current method. It can be seen that the results in this

study are in good agreement with the results from McIver (1989).

77


Since the natural frequency problem is fixed in the domain which the liquid

initially occupies, only the first and the third transformations in this chapter are used for

the linearized Bernoulli equation on the free surface. Discretization o f the governing

equations leads to a generalized eigenvalue problem.

V Z = «®Z (3.67)

In the above equations, W and <9 are coefficient matrices and x is the eigenvector.

To verify the method developed in this chapter for the eigenvalue problem, the first three

eigenvalues for a half-filled circular tank have been solved by trying different mesh sizes.

Figure 3.13 illustrates the eigenvalue results calculated by different grid numbers in both

horizontal and vertical directions. The results show that the eigenvalues eventually

converge when the cell numbers are 40x80. The eigenvalues converge to the values

calculated by other methods, which can be shown by the last points (Mclver, 1989) for

corresponding curves.

Eigenvalues in a partially filled circular tank at different fill levels have been

solved in the same way. The results for a tank o f unit radius are shown in Figure 3.14.

The first and third eigenvalues are the first and second antisymmetrical modes. The

second eigenvalue is the first symmetrical mode. The eigenvalues for points on the

dashed line in this figure are taken from Mclver (1989), in which the linearized wave

theory was used to solve the eigenvalues of free oscillations numerically by integral

equations in a bipolar coordinate system for circular tanks. The eigenvalues for points on

the solid line are calculated using the current method. It can be seen that the results in this

study are in good agreement with the results from Mclver (1989).

77


Eigenvalue 5

4 -

3 -

2 -

1

_ A' - A —

AL .... —_ 41— -- ill— -- il— -- il— -- 0— -- 0

E"'

A

•

• 4 0 • 0 4 4 • 4

,---, Cr% CO

% 0 0 0 0 0 0 0 ,—. CV c Sr Le) ko r- co ,_, N N N N N N N t_.

0 4-1 0 Lr-, c, in c> a) •—• ,—. CV CV Or) c ,cr

1---1 0

Figure 3.13 Eigenvalue in a half-full circular tank

first - - - - second third

Ei genvalue 9

Figure 3.14 Eigenvalue in a circular tank

current study---- McIver (1989)

78


Eigenvalue5

4

3

2

1C\00

o o o O o o oCM CO M" n kO r- 00

K N xO m O m o m o1 i—■1 CM CM Ci

3

Figure 3.13 Eigenvalue in a half-full circular tank

f i r s t second .......... third

Eigenvalue9

8

7

6

5

4 Third

3

Second2

First0

0 0.5Fill level

Figure 3.14 Eigenvalue in a circular tank

current study M clver(1989)

78


The natural frequencies in an elliptical tank depend not only on the fill level of the

liquid inside the tank, but also the aspect ratio of the tank, i.e., nondimensional a, the

length ratio of the semi major axis to the semi minor axis. Tables 3.1 - 3.5 list the first

five eigenvalues of liquid motion in elliptical tanks with different aspect ratios, which are

from 0.25 to 2.0, and different fill levels, which are from 0.1 to 0.9. Corresponding

figures of the eigenvalues are shown in Figures. 3.15 to 3.19. The following conclusions

can be made using these figures.

(1) For a given liquid fill level, the eigenvalues decrease with the increase in the aspect

ratio for all five eigenvalues.

(2) For a given aspect ratio, the first and second order eigenvalues increase with the

increase in the fill level when the aspect ratio is large.

(3) For a given small aspect ratio, the first and second order eigenvalues have a minimum

value at middle fill levels. The eigenvalues have larger values at lower and higher

liquid fill levels.

(4) The trend in (3) is especially true for all aspect ratios for higher order eigenvalues,

i.e., the third, fourth and fifth order eigenvalues shown in Figures. 3.17 to 3.19.

Table 3.1 First eigenvalue of liquid motion in an elliptical tank

Aspect ratio

Fill level 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0.25 8.9399 7.2483 6.5484 6.2648 6.2489 6.4824 7.0468 8.2354 11.3177 0.5 3.4324 3.1481 3.0146 2.9859 3.0509 3.2242 3.5603 4.2203 5.877 0.75 1.7507 1.7524 1.7805 1.8398 1.9399 2.1004 2.3637 2.8433 3.9977

1 1.0442 1.0966 1.1627 1.2462 1.3559 1.5077 1.7348 2.1239 3.0218 1.25 0.6903 0.7429 0.8101 0.8922 0.9964 1.1355 1.3365 1.6691 2.412 1.5 0.4902 0.5331 0.5925 0.6654 0.7582 0.8821 1.06 1.3508 1.9889 1.75 0.3667 0.3996 0.4501 0.5126 0.593 0.7015 0.8583 1.1152 1.6759

2 0.2853 0.3095 0.3525 0.4056 0.4746 0.5689 0.7067 0.9347 1.4344

79


The natural frequencies in an elliptical tank depend not only on the fill level of the

liquid inside the tank, but also the aspect ratio o f the tank, i.e., nondimensional a, the

length ratio o f the semi major axis to the semi minor axis. Tables 3.1 - 3.5 list the first

five eigenvalues o f liquid motion in elliptical tanks with different aspect ratios, which are

from 0.25 to 2.0, and different fill levels, which are from 0.1 to 0.9. Corresponding

figures o f the eigenvalues are shown in Figures. 3.15 to 3.19. The following conclusions

can be made using these figures.

(1) For a given liquid fill level, the eigenvalues decrease with the increase in the aspect

ratio for all five eigenvalues.

(2) For a given aspect ratio, the first and second order eigenvalues increase with the

increase in the fill level when the aspect ratio is large.

(3) For a given small aspect ratio, the first and second order eigenvalues have a minimum

value at middle fill levels. The eigenvalues have larger values at lower and higher

liquid fill levels.

(4) The trend in (3) is especially true for all aspect ratios for higher order eigenvalues,

i.e., the third, fourth and fifth order eigenvalues shown in Figures. 3.17 to 3.19.

Table 3.1 First eigenvalue of liquid motion in an elliptical tank

Aspectratio

Fill level0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0.25 8.9399 7.2483 6.5484 6.2648 6.2489 6.4824 7.0468 8.2354 11.31770.5 3.4324 3.1481 3.0146 2.9859 3.0509 3.2242 3.5603 4.2203 5.877

0.75 1.7507 1.7524 1.7805 1.8398 1.9399 2.1004 2.3637 2.8433 3.99771 1.0442 1.0966 1.1627 1.2462 1.3559 1.5077 1.7348 2.1239 3.0218

1.25 0.6903 0.7429 0.8101 0.8922 0.9964 1.1355 1.3365 1.6691 2.4121.5 0.4902 0.5331 0.5925 0.6654 0.7582 0.8821 1.06 1.3508 1.9889

1.75 0.3667 0.3996 0.4501 0.5126 0.593 0.7015 0.8583 1.1152 1.67592 0.2853 0.3095 0.3525 0.4056 0.4746 0.5689 0.7067 0.9347 1.4344

79


Table 3.2 Second eigenvalue of liquid motion in an elliptical tank

Aspect ratio

Fill level 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0.25 19.377 15.1262 13.4367 12.7131 12.5693 12.9337 13.9427 16.131 21.8291 0.5 8.4012 7.0927 6.4945 6.2473 6.2438 6.4759 7.0247 8.168 11.0869

0.75 4.6759 4.3095 4.1192 4.0594 4.1181 4.3129 4.7081 5.4946 7.4624 1 2.931 2.8904 2.8899 2.9332 3.0339 3.2172 3.5384 4.1446 5.6302

1.25 1.9935 2.0553 2.1368 2.2353 2.3637 2.5451 2.8246 3.3245 4.5191 1.5 1.4407 1.5248 1.636 1.7592 1.9009 2.0829 2.3393 2.7704 3.7734

1.75 1.0905 1.169 1.2863 1.4149 1.5636 1.7428 1.9829 2.3676 3.2365 2 0.8565 0.9184 1.0345 1.1599 1.3058 1.4805 1.708 2.0594 2.8305

Table 3.3 Third eigenvalue of liquid motion in an elliptical tank

Aspect ratio

Fill level 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0.25 30.0199 23.0409 20.3296 19.1575 18.8853 19.386 20.8534 24.0743 32.5144 0.5 13.848 11.1251 9.9747 9.4833 9.4055 9.7005 10.4759 12.1357 16.4253

0.75 8.1943 7.05 6.4862 6.2464 6.2432 6.4727 7.0163 8.1489 11.0381 1 5.3598 4.938 4.7002 4.6082 4.6524 4.8523 5.2785 6.142 8.3204

1.25 3.7448 3.6428 3.5961 3.6044 3.6865 3.8731 4.2291 4.9283 6.6761 1.5 2.7518 2.7814 2.8416 2.9176 3.0307 3.2129 3.5237 4.1128 5.5716

1.75 2.1036 2.1806 2.2961 2.4146 2.5514 2.7338 3.0149 3.5259 4.7779 2 1.6602 1.7465 1.8877 2.03 2.1831 2.3675 2.6284 3.0825 4.1796

Table 3.4 Fourth eigenvalue of liquid motion in an elliptical tank

Aspect ratio

Fill level 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0.25 40.518 30.9186 27.2134 25.604 25.2092 25.8496 27.7785 32.0327 43.2131 0.5 19.2002 15.0776 13.4166 12.7027 12.5605 12.9209 13.9182 16.0769 21.6752

0.75 11.8376 9.7322 8.7987 8.4006 8.3513 8.6227 9.3116 10.771 14.5154 1 8.0395 6.9933 6.4636 6.2389 6.2419 6.4704 7.0037 8.1092 10.9218

1.25 5.7748 5.3009 5.0356 4.9294 4.9702 5.1752 5.613 6.5043 8.7544 1.5 4.3263 4.1481 4.0605 4.0434 4.1139 4.3063 4.6833 5.4303 7.3057

1.75 3.3527 3.3193 3.3479 3.3965 3.497 3.682 4.0152 4.6596 6.2678 2 2.6732 2.7016 2.8034 2.9025 3.0264 3.2091 3.511 4.079 5.4875

80


Table 3.2 Second eigenvalue o f liquid motion in an elliptical tank

Aspectratio

Fill level0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0.25 19.377 15.1262 13.4367 12.7131 12.5693 12.9337 13.9427 16.131 21.82910.5 8.4012 7.0927 6.4945 6.2473 6.2438 6.4759 7.0247 8.168 11.0869

0.75 4.6759 4.3095 4.1192 4.0594 4.1181 4.3129 4.7081 5.4946 7.46241 2.931 2.8904 2.8899 2.9332 3.0339 3.2172 3.5384 4.1446 5.6302

1.25 1.9935 2.0553 2.1368 2.2353 2.3637 2.5451 2.8246 3.3245 4.51911.5 1.4407 1.5248 1.636 1.7592 1.9009 2.0829 2.3393 2.7704 3.7734

1.75 1.0905 1.169 1.2863 1.4149 1.5636 1.7428 1.9829 2.3676 3.23652 0.8565 0.9184 1.0345 1.1599 1.3058 1.4805 1.708 2.0594 2.8305

Table 3.3 Third eigenvalue o f liquid motion in an elliptical tank

Aspectratio

Fill level0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0.25 30.0199 23.0409 20.3296 19.1575 18.8853 19.386 20.8534 24.0743 32.51440.5 13.848 11.1251 9.9747 9.4833 9.4055 9.7005 10.4759 12.1357 16.4253

0.75 8.1943 7.05 6.4862 6.2464 6.2432 6.4727 7.0163 8.1489 11.03811 5.3598 4.938 4.7002 4.6082 4.6524 4.8523 5.2785 6.142 8.3204

1.25 3.7448 3.6428 3.5961 3.6044 3.6865 3.8731 4.2291 4.9283 6.67611.5 2.7518 2.7814 2.8416 2.9176 3.0307 3.2129 3.5237 4.1128 5.5716

1.75 2.1036 2.1806 2.2961 2.4146 2.5514 2.7338 3.0149 3.5259 4.77792 1.6602 1.7465 1.8877 2.03 2.1831 2.3675 2.6284 3.0825 4.1796

Table 3.4 Fourth eigenvalue of liquid motion in an elliptical tank

Aspectratio

Fill level0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0.25 40.518 30.9186 27.2134 25.604 25.2092 25.8496 27.7785 32.0327 43.21310.5 19.2002 15.0776 13.4166 12.7027 12.5605 12.9209 13.9182 16.0769 21.6752

0.75 11.8376 9.7322 8.7987 8.4006 8.3513 8.6227 9.3116 10.771 14.51541 8.0395 6.9933 6.4636 6.2389 6.2419 6.4704 7.0037 8.1092 10.9218

1.25 5.7748 5.3009 5.0356 4.9294 4.9702 5.1752 5.613 6.5043 8.75441.5 4.3263 4.1481 4.0605 4.0434 4.1139 4.3063 4.6833 5.4303 7.3057

1.75 3.3527 3.3193 3.3479 3.3965 3.497 3.682 4.0152 4.6596 6.26782 2.6732 2.7016 2.8034 2.9025 3.0264 3.2091 3.511 4.079 5.4875


Table 3.5 Fifth eigenvalue of liquid motion in an elliptical tank

Aspect ratio

Fill level 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0.25 51.0438 38.8114 34.1073 32.0601 31.5451 32.3309 34.734 40.0494 54.0709 0.5 24.4979 19.0289 16.8609 15.9223 15.7148 16.1422 17.3673 20.0413 27.0098 0.75 15.4459 12.3841 11.1019 10.5507 10.4556 10.7699 11.6095 13.4119 18.0669

1 10.7815 9.0123 8.2033 7.8579 7.8241 8.0831 8.7282 10.0909 13.5859 1.25 7.936 6.9452 6.4449 6.2338 6.2413 6.4682 6.9955 8.0912 10.8883 1.5 6.0582 5.5334 5.254 5.142 5.1816 5.3887 5.8369 6.7531 9.0832 1.75 4.7594 4.505 4.3867 4.353 4.42 4.6147 5.0065 5.7939 7.7903

2 3.831 3.7255 3.7233 3.7524 3.8438 4.0313 4.3814 5.0722 6.8184

Eigenvalue

15 ''''-

10 ''''

5 - 111111 140 a-,,A101010 1°41.14 01

M I •_ I I I I I I I I 1" -_,dollow' WO -41% 101.1 44%••••4-- -41,1-4 01*--"

0.5 Fill level 1.5

1

Aspect ratio

0.5

Figure 3.15 First eigenvalue of liquid motion in an elliptical tank

81


Table 3.5 Fifth eigenvalue of liquid motion in an elliptical tank

Aspectratio

Fill level0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0.25 51.0438 38.8114 34.1073 32.0601 31.5451 32.3309 34.734 40.0494 54.07090.5 24.4979 19.0289 16.8609 15.9223 15.7148 16.1422 17.3673 20.0413 27.0098

0.75 15.4459 12.3841 11.1019 10.5507 10.4556 10.7699 11.6095 13.4119 18.06691 10.7815 9.0123 8.2033 7.8579 7.8241 8.0831 8.7282 10.0909 13.5859

1.25 7.936 6.9452 6.4449 6.2338 6.2413 6.4682 6.9955 8.0912 10.88831.5 6.0582 5.5334 5.254 5.142 5.1816 5.3887 5.8369 6.7531 9.0832

1.75 4.7594 4.505 4.3867 4.353 4.42 4.6147 5.0065 5.7939 7.79032 3.831 3.7255 3.7233 3.7524 3.8438 4.0313 4.3814 5.0722 6.8184

Eigenvalue

0.50.5Fill level

Aspect ratio

Figure 3.15 First eigenvalue o f liquid motion in an elliptical tank


Eigenvalue

30 ''''''

Fill level Aspect ratio

Figure 3.16 Second eigenvalue of liquid motion in an elliptical tank

Eigenvalue ''''

#1110*Iri i

I I IA I *b Ill I I P -4 I lit. I I I I I I • -4 I PA P O

'41 11111F111.1W-

40 ---

20

0.5

Fill level

1.5 1 0.5

Aspect ratio

Figure 3.17 Third eigenvalue of liquid motion in an elliptical tank

82


Eigenvalue

30 >---

20

0.50.5

Aspect ratioFill level

Figure 3.16 Second eigenvalue o f liquid motion in an elliptical tank

Eigenvalue

4 0 ~ r-"

0.50.5


Figure 3.17 Third eigenvalue of liquid motion in an elliptical tank

82


0.5

Fill level

1.5 1Aspect ratio

0.5

Figure 3.18 Fourth eigenvalue of liquid motion in an elliptical tank

Eigenvalue

60 --------r.

—

Fill level Aspect ratio

Figure 3.19 Fifth eigenvalue of liquid motion in an elliptical tank

3.7 Summary

A new mathematical method used to solve the dynamic liquid behaviour in

partially filled 2D horizontal tanks has been developed in this chapter. The governing

83


Eigenvalue

6 0 -r —

0.50.5


Figure 3.18 Fourth eigenvalue of liquid motion in an elliptical tank

Eigenvalue

60 t —

0.50.5


0

Figure 3.19 Fifth eigenvalue o f liquid motion in an elliptical tank

3.7 Summary

A new mathematical method used to solve the dynamic liquid behaviour in

partially filled 2D horizontal tanks has been developed in this chapter. The governing

83


equations for the liquid motion in a tank are manipulated using continuous coordinate

transformations. The first transformation saves the performance of interpolation of

boundary conditions on the curved walls, which was required by the traditional methods.

The application of the second transformation, which is a kind of boundary-fitted method,

changes the working domain to a fixed area, avoiding the complex algorithm for free

surface updating and volume correction. When the governing equations are solved using

the finite difference method, the third transformation is adopted in order to gain

computational convergence and stability. This new approach has shown a good stability

and easiness in programming. Sloshing problems in 2D circular tanks subjected to

harmonic motions with small and finite amplitudes are simulated to show the efficiency

of the new method. Transient responses of the liquid in the road tanks have been studied

in detail under turning, lane change and double lane change manoeuvres. The natural

frequencies of liquid motion in 2D elliptical tanks with different aspect ratios and under

different liquid fill levels have been solved using the current method for the first five

liquid modes.

More importantly, the new method has an excellent capability of being extended

to solve liquid motion in 3D horizontal cylindrical tanks subjected to excitations in both

longitudinal and lateral directions. It can also be extended to solve the liquid behaviour in

tanks with arbitrary walls by simply changing the equations for the first and third

transformation according to specialties of the problems. The present approach shows

evident convenience in implementing the research of non-overturning and non-breaking

wave motions inside horizontal circular and elliptical tanks. This approach provides a

useful tool for determining resultant the liquid-structure interactions in road

84


equations for the liquid motion in a tank are manipulated using continuous coordinate

transformations. The first transformation saves the performance o f interpolation of

boundary conditions on the curved walls, which was required by the traditional methods.

The application o f the second transformation, which is a kind of boundary-fitted method,

changes the working domain to a fixed area, avoiding the complex algorithm for free

surface updating and volume correction. When the governing equations are solved using

the finite difference method, the third transformation is adopted in order to gain

computational convergence and stability. This new approach has shown a good stability

and easiness in programming. Sloshing problems in 2D circular tanks subjected to

harmonic motions with small and finite amplitudes are simulated to show the efficiency

o f the new method. Transient responses o f the liquid in the road tanks have been studied

in detail under turning, lane change and double lane change manoeuvres. The natural

frequencies o f liquid motion in 2D elliptical tanks with different aspect ratios and under

different liquid fill levels have been solved using the current method for the first five

liquid modes.

More importantly, the new method has an excellent capability o f being extended

to solve liquid motion in 3D horizontal cylindrical tanks subjected to excitations in both

longitudinal and lateral directions. It can also be extended to solve the liquid behaviour in

tanks with arbitrary walls by simply changing the equations for the first and third

transformation according to specialties o f the problems. The present approach shows

evident convenience in implementing the research o f non-overturning and non-breaking

wave motions inside horizontal circular and elliptical tanks. This approach provides a

useful tool for determining resultant the liquid-structure interactions in road

84


transportation and other relevant engineering fields. The development of the method for

liquid motion in 3D cylindrical tanks and the application of the new method in studying

3D liquid dynamics will be conducted in the following chapter.

85


transportation and other relevant engineering fields. The development o f the method for

liquid motion in 3D cylindrical tanks and the application of the new method in studying

3D liquid dynamics will be conducted in the following chapter.

85


CHAPTER 4 LIQUID MOTION IN 3D HORIZONTAL

CYLINDRICAL TANKS

4.1 Introduction

In the current literature, 2D sloshing models have been combined with 3D vehicle

models in almost all studies on vehicle dynamics for liquid cargo vehicles. For the lateral

liquid models, the liquid motion is described only in the transverse direction with the

assumption that the liquid in all cross sections of the cylindrical tank behaves identically

from the head to the end of the tank. This may be reasonable if the tank only suffers the

excitation in the transverse direction. However, for tanks used in road vehicles, this

cannot be true. In the turning, lane change and double lane change operations, the mass

centre of the tank is subjected to the translational and rotational accelerations in all six

directions. Even when the tank is considered to be turned along a perfect circular curve

with a constant forward speed, where the mass centre of the tank is only subjected to a

constant centrifugal acceleration, the liquid particles at different positions in the tank can

become subject to different lateral and longitudinal accelerations. Therefore, even for this

case, the above assumption cannot show the real motion of the liquid. For turning along

arbitrary trajectories and the lane change and double lane change, the mass centre of the

tank is subjected to both tangential acceleration and centrifugal acceleration. The motion

of the liquid particles should then be determined according to these time varying

accelerations. The assumption that the liquid has the same behaviour at different cross

sections neglects the longitudinal effect exerted on the liquid. Also, the application of the

same constant lateral acceleration on all cross sections cannot show the lateral

86


CHAPTER 4 LIQUID MOTION IN 3D HORIZONTAL

CYLINDRICAL TANKS

4.1 Introduction

In the current literature, 2D sloshing models have been combined with 3D vehicle

models in almost all studies on vehicle dynamics for liquid cargo vehicles. For the lateral

liquid models, the liquid motion is described only in the transverse direction with the

assumption that the liquid in all cross sections o f the cylindrical tank behaves identically

from the head to the end of the tank. This may be reasonable if the tank only suffers the

excitation in the transverse direction. However, for tanks used in road vehicles, this

cannot be true. In the turning, lane change and double lane change operations, the mass

centre o f the tank is subjected to the translational and rotational accelerations in all six

directions. Even when the tank is considered to be turned along a perfect circular curve

with a constant forward speed, where the mass centre o f the tank is only subjected to a

constant centrifugal acceleration, the liquid particles at different positions in the tank can

become subject to different lateral and longitudinal accelerations. Therefore, even for this

case, the above assumption cannot show the real motion of the liquid. For turning along

arbitrary trajectories and the lane change and double lane change, the mass centre o f the

tank is subjected to both tangential acceleration and centrifugal acceleration. The motion

of the liquid particles should then be determined according to these time varying

accelerations. The assumption that the liquid has the same behaviour at different cross

sections neglects the longitudinal effect exerted on the liquid. Also, the application of the

same constant lateral acceleration on all cross sections cannot show the lateral

86


acceleration differences among the liquid particles. For some situations, turning, lane

change and double lane change operations are often performed along with braking or

accelerating. For these cases, the influence of the tangential acceleration is much more

significant. For all of the above situations, the forces and moments obtained from the

liquid motion model considered only in the lateral direction are undoubtedly unable to

accurately reflect the true liquid action on the tank and the vehicle. When using the

resultant tank motion as the input to calculate the liquid motion for the next time step, the

simulation can hardly be reliable. The simulation results will, in the long run, derivate

from the true response.

During ideal straight line driving, the tank is subjected to translational

accelerations in the longitudinal and vertical directions, as well as rotational acceleration

in the pitch plane. In this case, the longitudinal liquid model is needed. It can be found in

the current literature that rectangular tanks, which are not common in road transportation

engineering, have been selected by researchers to study the vehicle dynamics in the

longitudinal direction, due to the difficulties of solving the governing equations in the 3D

space. For the rectangular tanks, existing algorithms for 2D liquid free surface problems,

such as the MAC method and VOF method, could be applied. However, because of the

existence of curved walls in the horizontal cylindrical tanks used by most liquid cargo

vehicles, the above schemes could hardly be extended to solve the liquid-vehicle system,

because the liquid motion in the longitudinal direction in cylindrical or elliptical tanks is

a 3D problem even for perfect straight line driving, braking and accelerating.

Obviously, the lack of an effective algorithm to solve the liquid sloshing problems

in horizontal cylindrical tanks in 3D space is the main reason that most studies on liquid

87


acceleration differences among the liquid particles. For some situations, turning, lane

change and double lane change operations are often performed along with braking or

accelerating. For these cases, the influence of the tangential acceleration is much more

significant. For all o f the above situations, the forces and moments obtained from the

liquid motion model considered only in the lateral direction are undoubtedly unable to

accurately reflect the true liquid action on the tank and the vehicle. When using the

resultant tank motion as the input to calculate the liquid motion for the next time step, the

simulation can hardly be reliable. The simulation results will, in the long run, derivate

from the true response.

During ideal straight line driving, the tank is subjected to translational

accelerations in the longitudinal and vertical directions, as well as rotational acceleration

in the pitch plane. In this case, the longitudinal liquid model is needed. It can be found in

the current literature that rectangular tanks, which are not common in road transportation

engineering, have been selected by researchers to study the vehicle dynamics in the

longitudinal direction, due to the difficulties of solving the governing equations in the 3D

space. For the rectangular tanks, existing algorithms for 2D liquid free surface problems,

such as the MAC method and VOF method, could be applied. However, because o f the

existence of curved walls in the horizontal cylindrical tanks used by most liquid cargo

vehicles, the above schemes could hardly be extended to solve the liquid-vehicle system,

because the liquid motion in the longitudinal direction in cylindrical or elliptical tanks is

a 3D problem even for perfect straight line driving, braking and accelerating.

Obviously, the lack of an effective algorithm to solve the liquid sloshing problems

in horizontal cylindrical tanks in 3D space is the main reason that most studies on liquid

87


vehicle dynamics have been constrained to lateral response under steady turning.

In this chapter, an effective method is established to study the liquid sloshing

problem in 3D horizontal cylindrical tanks based on the method developed in the

previous chapter for liquid motion inside 2D circular tanks. To overcome the difficulties

caused by the cylindrical surface and hemispherical surface of a tank, governing

equations based on the potential flow theory are transformed using continuous coordinate

mappings. For cylindrical tanks with flat heads and non-flat heads, a transformation is

first carried out in the axial direction before the stretch in the transversal plane is

performed. The boundary-fitted method is then used for changing the transformed

domain into a fixed domain. Linearized Bernoulli equation is solved as a generalized

eigenvalue problem to determine the natural frequencies of liquid sloshing. Transient

liquid motion is simulated for the tank subjected to a longitudinal acceleration input to

study the accelerating/braking operations. The liquid forces and moments caused by the

liquid motion are calculated by the integration of the liquid pressure distribution on the

tank walls. The influence of liquid motion on compartmented tanks is also investigated.

4.2 Statement of the problem

A 3D partially filled horizontal cylindrical tank is shown in Figure 4.1. A

Cartesian coordinate system, xiyizi, is fixed on the tank, with its origin located at the

geometrical centre of the still free surface of the liquid. (x1,z, , t) is the free-surface

elevation of a point (x1,z,) above the still liquid level. t is the time and d is the still liquid

depth. The tank is subjected to a longitudinal acceleration, Az, by which the

accelerating/braking operation conditions can be simulated. 2c is the length of the tank

88


vehicle dynamics have been constrained to lateral response under steady turning.

In this chapter, an effective method is established to study the liquid sloshing

problem in 3D horizontal cylindrical tanks based on the method developed in the

previous chapter for liquid motion inside 2D circular tanks. To overcome the difficulties

caused by the cylindrical surface and hemispherical surface o f a tank, governing

equations based on the potential flow theory are transformed using continuous coordinate

mappings. For cylindrical tanks with flat heads and non-flat heads, a transformation is

first carried out in the axial direction before the stretch in the transversal plane is

performed. The boundary-fitted method is then used for changing the transformed

domain into a fixed domain. Linearized Bernoulli equation is solved as a generalized

eigenvalue problem to determine the natural frequencies o f liquid sloshing. Transient

liquid motion is simulated for the tank subjected to a longitudinal acceleration input to

study the accelerating/braking operations. The liquid forces and moments caused by the

liquid motion are calculated by the integration o f the liquid pressure distribution on the

tank walls. The influence o f liquid motion on compartmented tanks is also investigated.

4.2 Statement of the problem

A 3D partially filled horizontal cylindrical tank is shown in Figure 4.1. A

Cartesian coordinate system, is fixed on the tank, with its origin located at the

geometrical centre of the still free surface of the liquid. %{xx, z x,t) is the free-surface

elevation o f a point (x, ,z ,) above the still liquid level, t is the time and d is the still liquid

depth. The tank is subjected to a longitudinal acceleration, Az, by which the

accelerating/braking operation conditions can be simulated. 2 c is the length of the tank

88


with flat heads, and the length of the cylindrical section of the tank with hemispherical

heads. R is the radius of the tank and the radius of the hemispherical heads, and co is the

vertical distance between the origin of the coordinate system and the longitudinal

symmetric axis of the cylindrical tank.

Figure 4.1 Sketch of horizontal cylindrical tanks

(a) flat heads (b) hemispherical heads

The liquid is assumed to be incompressible, and the liquid motion is assumed to

be irrotational. Let the local velocity potential be q,, which is a function of the location

and time. The governing equation of liquid motion inside the tank can be given by the

Laplace equation.

89


with flat heads, and the length of the cylindrical section of the tank with hemispherical

heads. R is the radius o f the tank and the radius o f the hemispherical heads, and co is the

vertical distance between the origin of the coordinate system and the longitudinal

symmetric axis o f the cylindrical tank.

Figure 4.1 Sketch of horizontal cylindrical tanks

(a) flat heads (b) hemispherical heads

The liquid is assumed to be incompressible, and the liquid m otion is assumed to

be irrotational. Let the local velocity potential be (p, which is a function o f the location

and time. The governing equation o f liquid motion inside the tank can be given by the

Laplace equation.

89


a2 v ± a2co ± a2 v = 0

ax; ay, az


a _4 4 ta 4 a , (yi _ ) at ay, ax, ax, az, az,


ac — 1 __ at 2 ax,,

ao \ 2 ( \2 aci + +

\aY1 ) Paz, }( aq' 2 — Azz, —g — MD, (y1 = 6

(4.1)

(4.2)

(4.3)

The modified Rayleigh damping term, ,ugo, which is explained in the previous chapter, is

included to simulate the viscosity of the liquid and damping effects on the liquid motion.

g is the acceleration of gravity. On the rigid tank walls, the normal velocity is zero.

aq' =o an (4.4)

In the above equation, n is the normal vector on the rigid walls, i.e., the

cylindrical wall, flat or hemispherical heads. The initial values of the velocity potential

and the free surface height are set to be zero, corresponding to the initial still liquid. The

Bernoulli equation is used to obtain the liquid pressure distribution, p, on the tank walls.

P 4 -\ 2 aco

P at 2 ax 1 \

( 2 / \ 2} aco aq, — Az z l — gyp — P+ +

\.. aYi \, ) aZ1 O/

)

In the above equation, p is the density of the liquid being carried.

(4.5)

4.3 Mathematical approach

4.3.1 Continuous coordinate mappings

One important feature of the numerical scheme developed in Chapter 3 is its good

90


d 2<P d 2(p d 2<pdxf dyx dzx


dt; _ dcp dcp dt; dcp dt; , _ \dt dyx dxi dxx dzx dzx ’


(4.1)

(4.2)

dcpdt

/ d^ 2

\ dx\ j

r dkp^2

V î j+

' d(p 2

v& i j~ A Z\ - g £ - w > U =4) (4.3)

The modified Rayleigh damping term, /icp, which is explained in the previous chapter, is

included to simulate the viscosity o f the liquid and damping effects on the liquid motion.

g is the acceleration o f gravity. On the rigid tank walls, the normal velocity is zero.

^ = 0dn

(4.4)

In the above equation, n is the normal vector on the rigid walls, i.e., the

cylindrical wall, flat or hemispherical heads. The initial values o f the velocity potential

and the free surface height are set to be zero, corresponding to the initial still liquid. The

Bernoulli equation is used to obtain the liquid pressure distribution, p, on the tank walls.

P_P

d(p 1 dt 2

' chp^ KdxXJ

dcp

\ p y \ j+

/ dcp 2

KdzXJ A z x - g y x-p(p (4.5)

In the above equation, p is the density of the liquid being carried.

4.3 Mathematical approach

4.3.1 Continuous coordinate mappings

One important feature of the numerical scheme developed in Chapter 3 is its good

90


capability of extension. Currently there are very few reports on the extension from the

existing numerical methods to 3D horizontal cylindrical tanks, due to the difficulties in

capturing the free surface in a time-varying 3D area. When the procedure of free surface

updating is extended to 3D geometry, it can be extremely complicated, and the algorithm

calls for considerable programming sophistication. Furthermore, the stability of the

program can sometimes hardly be guaranteed. The advantages of the current method will

be more obvious for the reason that easy extension can hardly be achieved by some of the

other numerical methods

For numerically solving the governing equations in Section 4.2, discretization in

the 3D cylindrical space is necessary. However, for performing the discretization,

intricate algorithms are usually employed to deal with the time varying boundary

conditions on the curved tank walls. Moreover, the management of the boundary

conditions on the time-varying free surface, as well as the time-varying integration

domain for the free surface also needs to be considered. In this research, to overcome all

these difficulties caused by the cylindrical walls and hemispherical walls and to apply the

boundary-fitted method, the governing equations are first rearranged by using continuous

coordinate mappings with the following transformations before discretization.

x=xl , y=y1, z=

a= x

z1

8 . 11R 2 + xi2 + ( y1 + co )2 + c

AIR' — (y —c0 )2 '

(4.6)

fl = y, r = z (4.7)

, \ X=a 2 , Y=-1+ , , (fi+d), Z=7

10,y,t) (4.8)

By Eq. (4.6) and Eq. (4.7), the 3D cylindrical space enclosed by the tank walls is

91


capability of extension. Currently there are very few reports on the extension from the

existing numerical methods to 3D horizontal cylindrical tanks, due to the difficulties in

capturing the free surface in a time-varying 3D area. When the procedure o f free surface

updating is extended to 3D geometry, it can be extremely complicated, and the algorithm

calls for considerable programming sophistication. Furthermore, the stability of the

program can sometimes hardly be guaranteed. The advantages o f the current method will

be more obvious for the reason that easy extension can hardly be achieved by some of the

other numerical methods

For numerically solving the governing equations in Section 4.2, discretization in

the 3D cylindrical space is necessary. However, for performing the discretization,

intricate algorithms are usually employed to deal with the time varying boundary

conditions on the curved tank walls. Moreover, the management o f the boundary

conditions on the time-varying free surface, as well as the time-varying integration

domain for the free surface also needs to be considered. In this research, to overcome all

these difficulties caused by the cylindrical walls and hemispherical walls and to apply the

boundary-fitted method, the governing equations are first rearranged by using continuous

coordinate mappings with the following transformations before discretization.

By Eq. (4.6) and Eq. (4.7), the 3D cylindrical space enclosed by the tank walls is

z.(4.6)

a = f3 = y, y = z (4.7)

(4.8)

91


transformed from the xiy iz 1 coordinate system to an afly coordinate system, which results

in a working domain enclosed by a cubic box. By Eq.(4.6), a stretch is performed in the

axial direction, which is especially important for tanks with hemispherical heads. The

parameter 8is expressed as:

8= {oi flat heads

hemispherical heads (4.9)

By Eq.(4.7), a stretch is performed to avoid the complex algorithm for

interpolating the boundary conditions on curved tank walls. In order to avoid singularities,

the transformation is constrained by the following condition.

—1<ct 1, at /3=c0 ±R (4.10)

In Eq.(4.8), h is the total liquid height from the bottom to the transient free surface

in the afly coordinate system. Equation (4.8) reflects the boundary-fitted method, by

which the transformed liquid domain is further transformed into another cubic box in the

XYZ coordinate system, in which the vertical coordinate value, Y, changes with the

horizontal coordinate values, X and Z, and the time, t. Once again, it should be noted that,

for applying the boundary-fitted scheme, the above development is based on the

assumption that the liquid height is expressible as a single-value function of the location.

This implies that overturning waves and breaking waves will not occur during the motion

of the liquid tank, and the liquid will never hit the top of the tank or leave the bottom of

the tank in the case that sudden change of boundary conditions occurs. These

assumptions are commonly accepted by other researchers in this field. In fact, there still

exist no effective ways to numerically simulate these extreme cases by solving fluid

mechanics equations (Ibrahim et al, 2001).

92


transformed from the x y xz x coordinate system to an afiy coordinate system, which results

in a working domain enclosed by a cubic box. By Eq.(4.6), a stretch is performed in the

axial direction, which is especially important for tanks with hemispherical heads. The

parameter S is expressed as:

By Eq.(4.7), a stretch is performed to avoid the complex algorithm for

interpolating the boundary conditions on curved tank walls. In order to avoid singularities,

the transformation is constrained by the following condition.

In Eq.(4.8), h is the total liquid height from the bottom to the transient free surface

in the afiy coordinate system. Equation (4.8) reflects the boundary-fitted method, by

which the transformed liquid domain is further transformed into another cubic box in the

XYZ coordinate system, in which the vertical coordinate value, Y, changes with the

horizontal coordinate values, X and Z, and the time, t. Once again, it should be noted that,

for applying the boundary-fitted scheme, the above development is based on the

assumption that the liquid height is expressible as a single-value function o f the location.

This implies that overturning waves and breaking waves will not occur during the motion

o f the liquid tank, and the liquid will never hit the top of the tank or leave the bottom of

the tank in the case that sudden change o f boundary conditions occurs. These

assumptions are commonly accepted by other researchers in this field. In fact, there still

exist no effective ways to numerically simulate these extreme cases by solving fluid

mechanics equations (Ibrahim et al, 2001).

f lat heads hemispherical heads

(4.9)

- 1 < « < 1 , at fi = c0 ± R (4.10)

92


4.3.2 Formulae derivation

Applying Eqs. (4.6) and (4.7) to the governing equations (4.1)-(4.4) leads to the

following rearranged equations in the afiy coordinate system. The Laplace equation

becomes:

B• 82(4 +B• '32° +B• a2° +B 82° +B5• a20

• +B6•

a2A

Y' + B • B 8 • = 0 1 a ct 2 2 as2 3 ay2 4 a a ap may aaay 7 a a a,/

(4.11)

In the above equation, the coefficients are expressed by the metrics relating the

coordinate systems.

B 3

= (aa

+ (ace

+2 j 2

ax — ay

az " aa 801\

8X1 \ 8X 831

-N 2

, B2 =(—afi a y

7 az \ 2 \ 2 \ 2 2

aZ aZ ay a/3`+ + - • B 4 =

2(aa

ax ay, j az, j az ay ay

az ay az 'aa ay B

5 = 2—

ay, —az

' B6 = 2 ay, ay az

B 8

, B7 =

7 \2

7 aZ

\ 2

8X \ aYi

+2 az ' aa ale

ax, ax ay

raa 2 az a2a +

13Y .1 ax, axay

ay

az

The kinematic boundary condition on the free surface is transformed into:

=G• ° +G . a°at ' aa 816 3 ar

G =-8a +—aa ace) a77 ( ay az G _afi

ay ax aaax ) ay az ax, ) 2- ay

az 5a ari - ay aZ • 2. aZ 2

ax, ax aa az ` ax, az,

93

ari}ay

(4.12)

(4.13)

(4.14)


4.3.2 Formulae derivation

Applying Eqs. (4.6) and (4.7) to the governing equations (4.1)-(4.4) leads to the

following rearranged equations in the afiy coordinate system. The Laplace equation

becomes:

da dp dy dadp dpdy dady d a dy(4.11)

In the above equation, the coefficients are expressed by the metrics relating the

coordinate systems.

5 ,= ( d a )2

+ 1 ^

1 a

2+ 2

dz ' rda da?, B2 =

' d p 'Vdx J Kfy ) ^ dx dy j

S3 =' d z V r d z vva*iy

B5 = 2

r d z ' 1

v ^ i j

dz dy dyx dz

v& iv

' dy_<l ydz

, B, = 2r d a d p '

dy dry+ 2

dzdxx

d a dp dx dy

2 ^ -da d y '

, Bn =

s 11 2 o dz + 2 ----

d 2adyx d z y 7 /

y d y ) dx | dxdy ’

Bq =' d z ' 2

ydX , ,+

r d z ' 2

\ f y \ j

dydz

(4.12)

The kinematic boundary condition on the free surface is transformed into:

drjdt

d(j)= G , 1- G1 1- G , -----

d(/> , ^ d<j>da dp dy

(4.13)

G, = — +da da I dr]dy dx \ da

dr} i dy dz dy v dz dxx j • ^ 2 =

dPdy ’

g 3 =( d y \Kdz ,

r d z ' dz d a djj dy dxx dx d a dz

' d z ' 2

y dxXJ+

r d z ' 1

ydzxJdr\dy

(4.14)

93


The dynamic boundary condition on the free surface is transformed into:

•\ ao =_ 1 {E

, • ( -00 )2 +E 2.(a0 at 2 as afl)

E _ r aay + i ox ) r aa

)2 E = ay 2

E4 = 2 7 aa ap`

ay ay /

2 + E3 • ( '* 2 E a2° E E

6

a2d)Y" +E

} 7ay I

+ 4 aaap

• a2° + 5 apay

+ aaar

2 r IV \ —

, E3 .aY i

/ \ 2 r 2 2

az az j az ) + +aX 1 / \ ay, az,

az (as ay az (as ay` ,E =2— — —+ 2 — — —

5 ax aX az) ayi ay az j

az afl ay E6

ay az = 2 — [— —2±),E7 = - A zZi - gq — 44

i ay

(4.15)

paz ) ry

'

(4.16)

In the above equations, 0 is the corresponding potential and ri is the

corresponding liquid height in the afly coordinate system. The boundary conditions on

the walls parallel to the aj3, ay and fiyplanes are changed to the following equations.

ao 00 ao D

' • aa

+ D2

• ap

+ D3

• —ay

= 0

r a 0 , r a 0 , r3 •

ay 80

i • -- ri • -- ri • - =v

ap

a0 a0 a0 H1 •aa

— + H2 • ap — + I - I3 •ay

• — =0

D, = 8 •{ a

a

a aa / x1 + — 011 — c0 5 )1 D2 — • r ay , -.5 —co)

a D3 = 8 • { Oz x ± —oz (y, — co)+ az

• (z ± c)} • i-/ + (1— 8)• alax, ' ay, az, 1 az az

/1 = sa /2 =

94

_ ay az az ay,

(4.20)

(4.21)


The dynamic boundary condition on the free surface is transformed into:

1 II 1 1 -

A F'i ■{d<)2

+ E 2 • f d<f>~)2

+ £ 3 •fdt 2 { d a ) I W ) [ d y )

+ £d 2(j)

+ E<d 2(j)

4 dad[3 5 dfidy+ e 6 -

d 2<j>+ £ 7

dady

(4.15)

E x =f da} 2

\ d x jr d a }

\ d y s, e 2 =

r dp} 2.^3 = ydXxj

+ f d z '* +V& 1 J

dydz

E4 =2 f da 8 0} dy dy

E6 = 2dzdyx

,E S = 2 — dx

dp dy dy dz

i V

d a dy dx dz

+ 2 d a dydyx Kdy dz

, £ 7 = - A tz x- g r \ - (4.16)

In the above equations, (j) is the corresponding potential and 77 is the

corresponding liquid height in the aPy coordinate system. The boundary conditions on

the walls parallel to the ap, ay and Py planes are changed to the following equations.

d</> d(/> d<t>D , h A !■ D, • — — 0

da dp J dy

/ r 3 £ + v M + / 3 . M = 0da dp dy

H, ■ - x - + / / , • — + H, • — = 0d<i>

da dp J dy

(4.17)

(4.18)

(4.19)

A = S } ' + } ■ i.l'i - c„)}, A -Co)}

dz dz dydx dy dz. dz

dydz

(4.20)

j da _ dp _ dy dz 1 dy ’ 2 dy ’ 3 dz dyx

(4.21)

94


as act Hl = 1 — co ), H2 =-8/3 • (y, — co)

ax ay ay

H3 = az az (

ax, x1+ y1— c")

ay, k

oy

az (4.22)

In Eq.(4.20), the sign ± is used for two hemispherical heads. The governing

equations can be further transformed by applying the boundary-fitted method, i.e., Eq.

(4.8).

020 520 520 a2C13 a20 +c2 +C3 +c4axe aY2 az2 axaY

+C5 aYaZ

a20+ c6 ± •— + • + C9 —

axaz ax aY az

CI = B1 ,

\ 2 ayy ( ay 2 B5C2 = B1 •(

ax, + B2 • (—

:2 + B3 r az aX ) h1

(4.23)

aY 2) + B6

ray ay

az h ) (ax az ) '

aY aY C3 =B3 , C4 = 131 .(2—

ax)+ B4 • ( —

h) ± B6 (az),

ay C5 = B3 + B5 • + B

6 c6 B , C, = B7 , • (2

—az) (a/71

ax )'

C8 =BI .( 8Y +B 3 • a2

B 4 • ( 2

•

ax B 2 ari) (2

ax az) ax 022 azh25

. (

a2Y+ B,

( aY) ay)

axaz \ ax )+B

8 az

C9 = B 8 (4.24)

aH , ao K + K3

a(130

ay

(4.25) = 1 at 1 •

+ • —ao —ax 2 aY az

= Gi , K 2 = •(Tlayc)+ G2 (-1,-; +G

3 ' K

3 = G

3 (4.26) •

az

95


In Eq.(4.20), the sign ± is used for two hemispherical heads. The governing

equations can be further transformed by applying the boundary-fitted method, i.e., Eq.

(4.8).

a 2o a 2o a2o a 2o a2o c , + a • —t- + C-, + c •— + c<8 X l 8 Y

OXdZ

dZ 2

ae x 8 ' ay

dXdY

+ a . ^ + c 7 . ^ + c 8 . ^ + c 9 - ^

a raz

az= o

(4.23)

a x y+ B-,

\ h 2 jAa y vv.azy + ^ 4 ‘

' a r 2 " d x ' hy

+ b 5 ■r dY 2 \ „ f a y ay^

• - + b„ •az a a x az

' 3 ~ 3 ’ 4

( dY \ "2"+ b6 -B, • 2 ---- + b a •I

I s x j4 0 l a z j

Q = 5 3

CD f 2 l2 — + Br • ■+* B6 ■I 5 Z J j 0 l a x j , C6 — B6, C7 — B7,

ayva x y

+ V^ a y vva z y - b 4 -

2 d H ' ( 2 d H N --------------r ax / V a az

r a2y ^ + b 7 ■( ay^+ B. •

f d Y )—

[ax az J 1 U x J 0 I d Zj

C9 — ^8

at a x ay az

(4.24)

(4.25)

X ^ G ^ X ^ G , ( 8Y) + G, •'2" + G3 • 'a y "

U x J Z Khj j ,a z ,, X3 = G 3 (4.26)

95


80 { (50 ) 2 (34 2 ao

J + at 2 I * (ax ) j2 a Y) j3 •

+J, ay +J8

J 2 - E i •

r ay )2

,ax + E 2 •

( 4"\

ft j

+ E 3

J 1 , E,,

( 4520 \ ,axay 1

+ J 5

( akp ayaz

J 6

a2,1),axaz

(4.27)

off 2

2 ay) r 2 ay) ( ay ay) ,7

L'4 h ax )± • --h az )± ,ax az )'

(

J 3 = E 3 , J 4 = El • 2 ay\ 2) ( ay` ax) + E4 •

(—h + E

6

•

az,

ay\ 2 a Y a Y J S =E3 •(2 +E5 • +E6 (—ax j,J 6 =E6 , =— ,J8 =E, (4.28)

az h i at

In the above equations, 0 is the corresponding potential and H is the

corresponding liquid height in the XYZ coordinate system. The boundary conditions on

the walls parallel to the XY, XZ and YZ planes are changed to the following equations.

acD acD acD Mi .— + M2 • —

a Y + M3 —

az =

ax 0

, i

ae. azi) L • — 1.2 • — /.3 —

ac =

ax a Y az

M, M2 , D,

al) a0 +N2 • — +N3 ac =0 ax a Y az

(a axn D2 1+D3 \a an , M3 = D3

( aY I I , L2 =I, • ,ax )+ I 2 PI+ 13 .1 ay\az

,L3 =1317)

96

(4.29)

(4.30)

(4.31)

(4.32)

(4.33)


dt 2 ' 1

aov5X y

aoy d Y ;

+ J^ a o vva z y

+ J 4 ■r a 2o A

8 XdY+ J 5 ■

C \a odYdZ + J 6 '

/ 2,*x Aa o a x a z ,

r 5 0 r+ .Z7 b Jq1 b y 8

(4.27)

J\ = E X,

J 2 - E\ •r d Y vKdXj

+ E 2 •K ' l 2 J

+ is 3 •

ayva z y + A -

r i a y A v/* a x y

+ a^ 2 a y N\ h d Z j

+ Aay a r a x az

1 - F J - F*j j ^ 3 3 ^ 4 ^1' dY^v a x y \ h j

+ E<f dY \ d Z j

J S = E 3 ‘^2— V a z y + A -

' 2 N+ VV" /

"a y " va x y

,y6=£6, y 7= ~ , y 8=£7 (4.28)dt

In the above equations, 0 is the corresponding potential and H is the

corresponding liquid height in the XFZ coordinate system. The boundary conditions on

the walls parallel to the XY, X Z and YZ planes are changed to the following equations.

a o ao aoM, • - — + M7 • - — b M, • —— = 0ax ay az

(4.29)

a o _ ao aoL. ------ b Z,7 b L-, -----— 0a x

ao

ay J az

ao aoX, + N i + N i = 0ax ay az

M, = Z),, M 2 = D] • " ay "va x ,

+ a • f - 1 + A '^ a y Ava z y

, AZ3 = Z)3

(4.30)

(4.31)

(4.32)

L\ — 11 , L 2 — Z[ •r dY^ va x y

+ z2 •" 2 "

U , + Vr d Y )vazy A3=/ 3 (4.33)

96


ay NI = Hi , N2 = H

' •(

ax)+ H

2 • i 2 \

\ j/ j+ H3

a (—

ayz ),N3 = H, (4.34)

It can be seen that after continuous coordinate mappings, the governing equations

do not need to be directly discretized in the physical domain. The boundary conditions on

the curved tank wall surfaces are transformed into the computational domain at the price

of more involved governing equations. It should be noted that no approximation in

dealing with the boundary conditions on the curved walls is applied in performing the

above development. This avoids employing the conventional complex algorithms for

managing boundary conditions on the curved tank walls and the liquid free surface. This

significantly reduces the programming sophistication. The application of the boundary-

fitted method introduces the following relationships, taking into account that the vertical

coordinate is a function of time and the other two coordinates.

aY ail (y +1) ay aH (y +1) a2Y —(ax)

H)2 2(y +1) a2H (y +1) ax ax h ' az az h 'axe ax h2 ax2 h '

a2y (aH )2 2(Y+1) a2H (Y+1) a2y (ail a 2(Y+1) 52H (Y+1) az2

- az h2 az2 h 'axaz ax az) h 2 axaz h '

8Y _ aH (Y+1) at at h

(4.35)

4.3.3 Numerical method

After the rearrangement using the above continuous mappings, the governing

equations are ready to be solved by an appropriate numerical method based on the

discretization of continuous systems. In this study, the finite difference method is adopted.

To gain computational convergence and stability, another transformation is performed to

97


n x = h x, n 2 = h x-" d F N

+ h 2 ■' 2 s

+ H 3 ■{ d Y 12 J [ d Z ), n 3 = h 3 (4.34)

It can be seen that after continuous coordinate mappings, the governing equations

do not need to be directly discretized in the physical domain. The boundary conditions on

the curved tank wall surfaces are transformed into the computational domain at the price

o f more involved governing equations. It should be noted that no approximation in

dealing with the boundary conditions on the curved walls is applied in performing the

above development. This avoids employing the conventional complex algorithms for

managing boundary conditions on the curved tank walls and the liquid free surface. This

significantly reduces the programming sophistication. The application o f the boundary-

fitted method introduces the following relationships, taking into account that the vertical

coordinate is a function of time and the other two coordinates.

8 Y _ 8 H (F + l) 8 Y _ 8 H (F + l) 8 2Y _ f 8 H ) 2 2(Y + 1) d 2/ / (F + l)8 X 8 X

8 Y 8 Z 2

r 8 H} 2

k 8 Z j

8 Z 8 Z h 8X 2

2 (r + l) 8 2H ( f + i ) 8 2Y8 Z h 8 X 8 Z

8 Y _ _ _ 8 H_ (F + l) dt dt h

y d X y

8 H 8 H 8 X 8 Z

8 X 2

2(F + l) 8 2H (F + l)8 X 8 Z h

(4.35)

4.3.3 Numerical method

After the rearrangement using the above continuous mappings, the governing

equations are ready to be solved by an appropriate numerical method based on the

discretization of continuous systems. In this study, the finite difference method is adopted.

To gain computational convergence and stability, another transformation is performed to

97


Eqs. (4.23), (4.25), (4.27), (4.29), (4.30) and (4.31) to stretch the grid in the vertical

direction by the following relationship.

,

In `Bg

+ 1 — (Y + 1) I kg)

(Bg —1+ (Y +1)I k‘g ) =1— 2Ag + 1

In(Bg +1) ' f \

2(A g —1) 0 ' g —1)

(4.36)

In the above equations, coefficients Ag and Bg are to be adjusted to control the grid

distribution near the bottom and free surface. kg is a constant that can be calculated with

given Ag and Bg.

kg = 2

2Bg

1— Bg +1+0 g +1)I(Bg — viig-i)

(4.37)

Let (13* (X ,Y , Z * ,t) be the velocity potential and H * (X * , Z * , t) the free surface

liquid height in the X*Y*Z* coordinate system. Application of Eq. (4.36) leads to the final

governing equations.

a 2 (1 )* a 2 * 82 cp* a 2,13* a 2,13*

W • +w2 • +w3 +w +w ax .2 aY*2 az .2 4 ax*aY* 5 aY*az*

2 * ao* ao* ao* +w • 6 ax*az

*+W' ax* +W8 a

* +W9 a*

=0

W 2 -= C i •

+ C 5

W, = C,

( * • \ 2 ( * )2 aY aY ' aY* 12

2 • 3 • pax + c + c , ,ay az'ay* aY* + C 'aye ay' ,aY az I

6 ax az 1

98

+ c 41 ay* ay*1

ax ay ) ,

(4.38)


Eqs. (4.23), (4.25), (4.27), (4.29), (4.30) and (4.31) to stretch the grid in the vertical

direction by the following relationship.

, Z * = Z (4.36)X = X , Y = I - 2 A +( b +1)

2 ( A , - l ) lr,f c ^ T )

In the above equations, coefficients Ag and Bg are to be adjusted to control the grid

distribution near the bottom and free surface. kg is a constant that can be calculated with

given Ag and Bg.

k„ = 25(4.37)

g

l - B* + u

Let ®*(x*,r*,Z*,f) be the velocity potential and H*[x* ,Z * ,t) the free surface

liquid height in the X*r*Z* coordinate system. Application of Eq. (4.36) leads to the final

governing equations.

a2®* a 2®* a 2®* TT/ a 2®* a 2®W, + W , r- + W, T- + W , ; r + W.1 ax*2 2 ar*2 3 az*2 4 ax*ar* 5 a r ’az*

a2®* a®* a®* a®*+ w<- . . * +w1~ + w , - - — + w9- - — = 0e x az a x a r az

(4.38)

fr ,=c,r d x * v

\ dX ;

w2 = c x2

f2

f a r ’ ^2

f a r ’ a r ’ "j+ C, • + C3 • + c 4 *U x J U y J [ a z J s CD

ar* ar ar az

«\+ Q •

* \ar* ar ax az

98


W3 = C3

W5 = C3 • 2

T478 = CI

'az' ,W4 = C, 2

'aY* az az az ,

+ C5

•ay* ax" • ax ax

az* ay*` az ay

+ C4

+ C6 •

'ax*aY*

ax ay ,

•ay* az*, ax az

( * *\ ax aY +C6

ax az

W6 =

a2x* a2z* ax w7 =c • +c • -pc axe az2 ax'

ax* az' ax az,

a2Y* a 2 y* a2Y* a 2 Y * a 2 Y * a 2 Y *

ax2 y2 az 2 + C4 aXa Y+ C5 a YaZ

+ C6 axaz

+C2 + C3

ay* ay* ay* +c, +c 8 + C9ax aY az

az*W9 = C9

az

aH*=P ao* + P • ao*+P ao*

• at ' ax* 2 aY* 3 az*

ax* ay* aY* ay* az P=K P =K • +K +K• P =K

' ax ' 2 I ax 2 aY 3 az ' 3 3 az

at 2 'ate* ao* + 05 ,ay* az* ,

'ate* ao* + 06 (ao

,ax* ate*

) * az*

7 + 0 80 Y

,ax* aY*

( * \ 2 ( * \ 2

+04

• 0* )2 +492 + o 3

ate* ax* aY* az*

(4.39)

(4.40)

(4.41)

0 2 = J, •

, ay* 12 (ayt y

,ax) + j2 +j3

0 l az. 1 23 = az '04=f'

(4.42)

ax*`2oi =J ' (ax ,

* ` 2

(ay* ay * ray * ay * ) ( ay * ay] • ay + j4 * ± 15* ay az )± j6 ax az az , , ax ay , 2(ax* ay*)

+J4 (ax* ay*)+J6 (

ax* ay*) ax ax ax ay ax az ,

99


w3 = c 3-âz* ^2 v a Z y

, W . = C x-2 dY' dX* dX dX

+ C4 •* \ax* d r

dX dY+ C,

r dX* d Y ^

J

W5 = C3 • 2r dY* dZ*A

dZ dZ+ C5 -

Aaz* dY *A dZ dY

+ C,f dY* aZ*A

dX dZ,w6 = c 6

dX dZ

r d X ' dZ* ^

v dX dZ

w7 = c x + C3 . ^ + C7 •—7 1 a z 2 3 a z 2 7 ax

a2r* a 2r* a2r* a 2r* a 2r* a 2r* - c, • + c , • + c , • + c • + c , • + c .

a x 2 a r 2 a z 2 dXdY dYdZ axaz

+ c , - — + c„- — + c 0 a rax a r

wg = c 9-

az

az*az

a#* n ao* „ ao* „ ao* = p •— - t + p , - — ^ + p , -

dt ax ar* az

= X, . — , p , = K t - ^ r + X, - ^ - + K ^ ^ — ,P ,= K dZ1 1 a x ’ 2 ax a r az az

(4.39)

(4.40)

(4.41)

ao <V/ » A2ao

ax*

/ \ 2 aova r y

^ a o * ^ 2vaz y

dt 2 ( ao* ao* "i f a o ’ ao* 'I f ao* ao* ^+ Oa * + Oc * + Os *

% CD [ar* a z ’ J [ax* a z ’ J

ao+ o 7 ~ + o 8

7 a r 8

(4.42)

Ox= J x ax*v a x y

o 2Aa r* ^ 2v a x y

+ «/*> 'dy*" '2v a r y

r dY^ 2

\ d Z y+ Ji

r dY' a r * A ax a r +J<

r 8 Y* ar*^ a r az

Aar* ar*" ax az ,

o 3 = j 3 -âz*^2v a z y

, 0 4 = J X-2r a x ’ ar* 'I f ax* ar* 'I f ax* ar* 'I+ J 4 ‘ + J 6 '

[ a x ax J [ a x a r J [ a x az J

99


OZ* OY*O5 = J

3 • 2( . az)+ J

5

'az* ay'\ az ay j+.16

'az* ay*) n T (ax* az* 3 3-0'6 - a 6 3

az ax ax az

ay* aY* 0, = J 7 ay at , 0 8 = J 8 (4.43)

The boundary conditions on the walls parallel to the X*Y* plane, X*Z* plane and

I/Z* plane are changed to the following.

a o* ate* a es* S'

+s +s =0 1 ax* 2 ay' az*

, ao* , aci* acb* v, ax*

+Q2 ay* +Q3 az* = 0

ao* ao* ao* R1 ax* + R2 ay* + R3 az*

=0

ax* ay* ay ay az s,=m, • , S2 = MI • +M2 +M3 , S3 = M3ax ax OY az az

ay* ay* ay* az* Q, = LI • —

ax* , Q2 = L, • +L2' +L3. , Q3 = L3ax ax aY az az

ax aY* ay* ay* az*R, = N 1 — , R2 = N, • + N2 + N3 , R3 = N3

ax ax aY az az

(4.44)

(4.45)

(4.46)

(4.47)

(4.48)

(4.49)

The following relationships can be obtained with the application of the boundary-

fitted method.

ay* ay* ay ay* ay* ay ay* ay* aY = ax ay ax 5 3 az ay az at ay at 5

a2Y* _ a2Y* ( ay ) +

2 ay* a2Y a2Y* = a2Y* ( 3)12 +

ay* a2Yax2 aY2 .aAT aY ax2 ' az2 aY2 .3z ) ay aZ2 '

a2Y* a2Y* axaz = aY2

( ay ay) ay* a2Y ax az + ay axaz'

100


Os = J 3-2r dZ* 3Y*A

dZ dZ+ Jc f dZ' ar*A

dZ dY + -Vf dZ* dY*A

8 Z dX ’Of, — Jf,r dx' d z ^

dx dz

O - J .2L . . 2L o = j'-/7 ~ J l _ 9^8 ^8 (4.43)dY dt

The boundary conditions on the walls parallel to the X*Y* plane, X*Z* plane and

Y*Z* plane are changed to the following.

ao*ao ao s r ^ + s 2 - ^ 1 ax 2 ar + s 3 ■az = 0

_ ao* ao* ao*Q\ ’ — r Qi — *■ Q-i — *■_ 0

1 dX 2 dY 3 a z

ax ar* az*

(4.44)

(4.45)

(4.46)

_ ax _ .. ar ar .. ar _ .. az5, = M , -------,S 2 - M , -------- yM 2 ------- h M ,------ , S , = M ,------ (4.47)1 1 ax ax 2 ar 3 az 3 3 az

_ r ax* T ar* r ar* , ar* r az*Q\ — L\ > Q? — A 1- L2 1- L , , Q-, — L-,------1 ' a x 2 1 ax 2 ar 3 az 3 3 az

= n x■— ,R 2 = N i • — + x 2 1 1 ax 2 1 ax 2az*d . . 1- N-,------ , R-, — N-,------ar 3 az 3 3 az

(4.48)

(4.49)

The following relationships can be obtained with the application of the boundary-

fitted method.

ar _aT ar ar _aT ar ar _aT ardX ~ dY ' d X ’ dZ ~ dY ' d Z ’ dt ~ dY ' dt ’

a2r* a2r* farY . ar* a2r a2r* a2r* farY ar* a2rax2 ar2

ar* a2r a2r* _ a2r* "ar tUxJ ar ax2 ’ az2 ar2 UzJ ar dz2

a2r* _a2r* far ar\ ar a2r axaz_ ar2 'laxaz J+ ar ’axaz

100


a2y* a2y* ay 52y* 52y* ay

axay ay2 ax' ayaz ay2 az'

aH air ax* aH ail az* ax ax* ax' az az* az'

52x 52x* ax*'2 ax* 52x*` 52x 52x* (at"' ax* (52Z*(4.50)

ax2 ax*2 ax +

ax*

ax2 az2 az*2 az , +

az*

,az2

4.3.4 Calculation procedures

The calculation procedures are exactly the same as those described in Section 3.3,

except that all quantities should be calculated in both lateral and longitudinal directions,

as the liquid motion is modeled in 3D space. The metrics of the velocity potential in the

rearranged governing equations are now written using the followings expressions.

a24)* (-13,7+,,k ax

*2 = (Ax *Y 520* 0:,j •+1,k —20:,J.k + (1)i,j-1,k =

ay *2 ry

520* cD*. — 2cD1*. + cDis,j,k ,j,k-1

az *2 (A7*Y * a20* 1:13 * * * i+1, j+1,k + 43 —43.i-1,j-1,k i+1,j-1,k — 43

aX * ar 4AX*AY*

azo* cD* + cD* — cD* — cD*i+1,j,k+1 i-1,j,k-I i+1,j,k-1 i-1,j,k+1

ax*az* 4AX*A7*

(13a2cD * * + * . — cro* . 4:)i,J+1,k+1 — o i*,j-1,k+1 ar aZ * 4A Y*A7*

101

(4.51)

(4.52)

(4.53)

(4.54)

(4.55)

(4.56)


8 2Y 8 2Y dY 8 2Y d 2Y dYa XdY d Y 2 8X ’ 8 Y8 Z d Y 2 8 Z

8 H 8 H * dX* 8 H 8 H * 8 Z*s x d x * a x ’ az az* az ’

8 2h 8 2h * fa x * va x 2 ax*2 v a x y

a //ax*

^ a 2x * A a x 2

a 2/ / a 2/ / r 8 Z *^2

s z 2 az* v az ,+-

a //az*

( a 2z*^ v a z 2 y

(4.50)

4.3.4 Calculation procedures

The calculation procedures are exactly the same as those described in Section 3.3,

except that all quantities should be calculated in both lateral and longitudinal directions,

as the liquid motion is modeled in 3D space. The metrics of the velocity potential in the

rearranged governing equations are now written using the followings expressions.

a2o* _ o *+U;4-2(D*7, , + o *_1i (4 5i)

(4.52)

(4.53)

(4.54)

(4.55)

(4.56)

ax*2 (AX*)2

a2®* _ ®U+U - 2< , * + ® u -uar*2 (AT*)2

a 2®* + <M -.az*2 (az*)2

aX*aZ* 4AX*A 7*

a2®* _ + K um-x ~ ,4+1

ax*az*

a2®*ar*az*

4 AX AZ

® i , . / + l , 4 + l + ® i , y - l , 4 - l ^ i .j + 1,4 -1 ^ i ' J - 1 , 4 + 1

4AF AZ

101


ac* ax* 2AX*

a _ i,j+1,k (1);,j-1,k

ay* 2AY*

ac)* _€13

az* — 2AZ*

On the boundaries:

ao* + 40*2,J,k -03,J,k ax* 2AX*

* * ao _ — 30*N+1,J,k + 4(1) N,J,k — 0 ;1_1, ,k) ax* 2AX*

ao* 30, Ik + 40,2,k 0,3,k a Y* 2AY*

ao = - 443:,M,k -413i,M-1,k)

ay* 2AY*

ao* _ + 4o*„.,,2 az* 2AZ*

ate* 3(1)* + 407 — (1)t.

i,j,L+1 t,j,L ,j,L-1

az* 2AZ*

(4.57)

(4.58)

(4.59)

(4.60)

(4.61)

(4.62)

(4.63)

(4.64)

(4.65)

The metrics of the liquid height in the rearranged governing equations are written

by the followings expressions.

ax 11:+i k — H: k , - ,

ax* 2AX*

a2 H* = H: +1,k 211,:k H: Lk

ax*2 (Ax*)2

—

aH + 4.1-1; k — H3* k

ax* 2AX*

102

(4.66)

(4.67)

(4.68)


d x 2 AX

5 0 O iy+l k — O, j_x k8 Y 2AY

5 0 _ Q;j,*+i ~ Oi,y,t-i a z ^ - 2AZ*

On the boundaries:

a o ’ - 3 0 ^ + 4 0 ^ - 0 âx* 2 AX*

c"t.’ - (- - ^V iJ t )ax* 2 AX*

ao* _ - 3 0 * u + 40* 2i - 0 *3

ar 2a f

5 0 _ ~ (~ 3 Q / A/+l t + 4 Q ;M t ~Q),a/-i,^) ar* “ 2Ar*

ao* - 3Q/JJ + 40*ji2 - O/j.3 az* 2AZ*

ao* - ( - 3 0 * j .i+1+ 4 0 ’M - 0 * 7.i . 1)

(4.57)

(4.58)

(4.59)

(4.60)

(4.61)

(4.62)

(4.63)

(4.64)

(4.65)az 2AZ

The metrics of the liquid height in the rearranged governing equations are written

by the followings expressions.

a //’ _ h ;+1, - H*_l kax* 2 AX*

a 2//*ax*2 (ax*)2

a//* _ -3H*l k + 4 H ’2 k - H l k ax* 2 AX*

102

(4.66)

(4.67)

(4.68)


OH* ( 3H- N. 4eN,k eN-1,k ax* 26,A' *

all* —

, I I k+1 H ,k-1

az* 2AZ*

a211* *k+l - 2H ,*k =

az*2 (Az * )2

_ + 4HL — H i: 3

az* 2A7*

art _—(-31-1,7,L+1+ az* 2A7*

(4.69)

(4.70)

(4.71)

(4.72)

(4.73)



Knowledge of the natural frequencies of liquid sloshing in horizontal cylindrical

tanks is important in designing and manufacturing liquid cargo tanks and their supporting

structures. Identification of the natural frequencies may help to avoid the oscillation that

leads to large-amplitude fluid motion, which causes vehicle instability and even structural

failure. The mathematical method for the natural frequencies of liquid motion in 2D

circular and elliptical tanks developed in the previous chapter can be extended to solve

the natural frequencies of liquid motion in 3D cylindrical tanks. A linearized Bernoulli

equation on the free surface and the governing equations inside the liquid domain and on

the rigid walls lead to a generalized eigenvalue problem. The procedures used in the

previous chapter can be easily applied by increasing the third dimension in the

longitudinal direction and solving this problem in a 3D space.

Theoretical solutions for longitudinal sloshing frequency and liquid motion have

103


(4.69)dX 2AX

(4.70)8Z* 2AZ*

(4.71)az*2 (a z *)2

(4.72)

(4.73)dZ 2AZ



Knowledge o f the natural frequencies o f liquid sloshing in horizontal cylindrical

tanks is important in designing and manufacturing liquid cargo tanks and their supporting

structures. Identification o f the natural frequencies may help to avoid the oscillation that

leads to large-amplitude fluid motion, which causes vehicle instability and even structural

failure. The mathematical method for the natural frequencies o f liquid motion in 2D

circular and elliptical tanks developed in the previous chapter can be extended to solve

the natural frequencies o f liquid motion in 3D cylindrical tanks. A linearized Bernoulli

equation on the free surface and the governing equations inside the liquid domain and on

the rigid walls lead to a generalized eigenvalue problem. The procedures used in the

previous chapter can be easily applied by increasing the third dimension in the

longitudinal direction and solving this problem in a 3D space.

Theoretical solutions for longitudinal sloshing frequency and liquid motion have

103


not been found in the current literature. An effective calculation method for the

longitudinal cylindrical tanks was proposed by Kobayashi et al (1989) by substituting an

equivalent rectangular tank for a horizontal cylindrical tank. Frequency sweep tests

showed the effectiveness of this method. By putting the area of a rectangle equal to the

cross section of the still liquid in the cylinder, the equivalent liquid level can be

determined by the following expression (Kobayashi et al, 1989).

1 1 \ — OR)+ rc12 H = - R)+ e 2 2 ild(2R— d))

(4.74)

For a cylindrical tank with flat heads, the equivalent length is the original length

of the tank (Kobayashi et al, 1989).

Le = 2c (4.75)

For a cylindrical tank with hemispherical heads, the equivalent length is

determined as follows (Kobayashi et al, 1989).

Le =2c +2V I S (4.76)

In the above equation, V is the liquid volume in the hemispherical head and S is

the cross section area of the liquid in the cylindrical section. Referring to the sloshing

theory in rectangular tanks, the longitudinal natural frequency of a cylindrical tank can

then be approximated as follows.

co, = gg

i = 1,2, • • • (4.77) Le

A, =itanh Le

(4.78)

In liquid sloshing analysis, the first natural frequency is usually important because

higher modes usually make rather small contributions to the amplitude. For the 3D

104


not been found in the current literature. An effective calculation method for the

longitudinal cylindrical tanks was proposed by Kobayashi et al (1989) by substituting an

equivalent rectangular tank for a horizontal cylindrical tank. Frequency sweep tests

showed the effectiveness o f this method. By putting the area o f a rectangle equal to the

cross section o f the still liquid in the cylinder, the equivalent liquid level can be

determined by the following expression (Kobayashi et al, 1989).

For a cylindrical tank with flat heads, the equivalent length is the original length

o f the tank (Kobayashi et al, 1989).

4 = 2c (4.75)

For a cylindrical tank with hemispherical heads, the equivalent length is

determined as follows (Kobayashi et al, 1989).

Le =2c + 2 V I S (4.76)

In the above equation, V is the liquid volume in the hemispherical head and S is

the cross section area o f the liquid in the cylindrical section. Referring to the sloshing

theory in rectangular tanks, the longitudinal natural frequency o f a cylindrical tank can

then be approximated as follows.

4 = i tanh *n^ e- (4.78)

In liquid sloshing analysis, the first natural frequency is usually important because

higher modes usually make rather small contributions to the amplitude. For the 3D

104


frequency problem, the first eigenvalue in the longitudinal direction has been picked up

from the eigenvalue results calculated by the current method. Grid sizes in the lateral

cross section of the tank are the same as those used in the calculation of 2D circular and

elliptical tanks. Grid sizes in the axial direction depend on LoD, which is defined as the

ratio of the tank length of the cylindrical section, 2c, to the tank diameter, D. In the

simulation, the grid sizes have been set as 41 and 81 for the ratio of 1 and 2. These grid

sizes have been adopted for both natural frequency calculation and transient liquid

simulation.

The results are shown in Figure 4.2 for two different LoD values under different

fill levels. The solid lines are obtained directly from the equivalent equation, i.e., Eq.

(4.78), and the dots in the figures show the results calculated using the current method for

cylindrical tanks with different heads. It can be seen that the calculated results agree with

results of the equivalent equation.

4.4.2 Transient liquid dynamics

Transient liquid motion in cylindrical tanks is important for the dynamics analyses

of tanks and supporting structures. Accelerating/braking operations are critical to vehicle

stability and structural integrity due to the sudden change of vehicle accelerations. As

pointed out previously, because of the existence of curved walls in horizontal cylindrical

tanks used by most liquid cargo vehicles, the liquid motion in the longitudinal direction

in cylindrical or elliptical tanks is exactly a 3-D problem even for perfect straight line

driving, braking and accelerating.

To simulate the transient liquid behaviour inside cylindrical tanks during

105


frequency problem, the first eigenvalue in the longitudinal direction has been picked up

from the eigenvalue results calculated by the current method. Grid sizes in the lateral

cross section of the tank are the same as those used in the calculation of 2D circular and

elliptical tanks. Grid sizes in the axial direction depend on LoD, which is defined as the

ratio o f the tank length of the cylindrical section, 2c, to the tank diameter, D. In the

simulation, the grid sizes have been set as 41 and 81 for the ratio o f 1 and 2. These grid

sizes have been adopted for both natural frequency calculation and transient liquid

simulation.

The results are shown in Figure 4.2 for two different LoD values under different

fill levels. The solid lines are obtained directly from the equivalent equation, i.e., Eq.

(4.78), and the dots in the figures show the results calculated using the current method for

cylindrical tanks with different heads. It can be seen that the calculated results agree with

results o f the equivalent equation.

4.4.2 Transient liquid dynamics

Transient liquid motion in cylindrical tanks is important for the dynamics analyses

of tanks and supporting structures. Accelerating/braking operations are critical to vehicle

stability and structural integrity due to the sudden change of vehicle accelerations. As

pointed out previously, because of the existence of curved walls in horizontal cylindrical

tanks used by most liquid cargo vehicles, the liquid motion in the longitudinal direction

in cylindrical or elliptical tanks is exactly a 3-D problem even for perfect straight line

driving, braking and accelerating.

To simulate the transient liquid behaviour inside cylindrical tanks during

105


accelerating/braking operations, the longitudinal acceleration is applied as the body force

on the liquid particles. The transient liquid free surface can be directly obtained during

the iterations. To calculate the forces and moments caused by liquid motions, the pressure

distribution of liquid on the tank walls is first calculated using the Bernoulli equation, i.e.,

Eq. (4.5). The same transformations used for the governing equations can be used for the

Bernoulli equation, except that all the coefficients in different transformation are

determined with their values on the walls. The forces and moments can then be quantified

by integrating the pressure distribution on the walls.

1.2

0.9 0.6

(a)

0.3

0

4 8 I 12

I I 16 20 j 0

1.2

0.9 - (b) 0.6 -

0.3 -

0

0 7 14 21 28 35

Le /He


tank

(a) LoD = 1 (b) LoD = 2

• flat heads • hemispherical heads

106


accelerating/braking operations, the longitudinal acceleration is applied as the body force

on the liquid particles. The transient liquid free surface can be directly obtained during

the iterations. To calculate the forces and moments caused by liquid motions, the pressure

distribution of liquid on the tank walls is first calculated using the Bernoulli equation, i.e.,

Eq. (4.5). The same transformations used for the governing equations can be used for the

Bernoulli equation, except that all the coefficients in different transformation are

determined with their values on the walls. The forces and moments can then be quantified

by integrating the pressure distribution on the walls.

1 .2 ----------------------------0.9 - 0.6 -

0.3 -‘0 -

^ 0 4 8 12 16 20

1.2 -

0.9 - 0.6 -

0.3 -i-0 -

0 7 14 21 28 35

h i n .


tank

(a) LoD = 1 (b) LoD = 2

• flat heads ■ hemispherical heads

106


Fl = f(pds)ii , A = f(pds)(7- x ii) (4.79)

During accelerating or braking in straight-line driving, the liquid is only subjected

to the longitudinal acceleration, and the force acting in this direction is considered. The

axis for moment calculation is selected to be in the x1 direction and through the middle

point at the tank bottom (Figure 4.3). When the method developed is integrated into a

vehicle system, the axis can be selected at a corresponding location.

z1

Figure 4.3 Force and moment calculation by fluid dynamics

Mass centre

Figure 4.4 Force and moment calculation by mass centre

107


F; = ^(pds)n , M, = ^{pds\r x n) (4.79)

During accelerating or braking in straight-line driving, the liquid is only subjected

to the longitudinal acceleration, and the force acting in this direction is considered. The

axis for moment calculation is selected to be in the x\ direction and through the middle

point at the tank bottom (Figure 4.3). When the method developed is integrated into a

vehicle system, the axis can be selected at a corresponding location.

Figure 4.3 Force and moment calculation by fluid dynamics

Mass centre

Figure 4.4 Force and moment calculation by mass centre

107


The liquid free surface, forces, and moments caused by liquid motion have also

been calculated by the mass centre model (Xu and Dai, 2004), in which the mass centre is

determined on the assumption that the free surface can be described by an inclined flat

surface, and that liquid bulk behaves like a rigid body (Figure 4.4). The free surface

gradient is simply considered as a constant under a given acceleration.

tan 0 = A, /g (4.80)

The axial force and moment about the selected axis are expressible as follows.

F. = mi Az , M1 = mi Az • A + mig • f 1 (4.81)

In the above equations, in/ is the total liquid mass inside the tank, and y, and f l

are the distances between the mass centre and the selected axis for moment calculation.

Since only geometrical information is needed to get the locations of the mass centres of

the liquid bulk inside the tank, they are directly obtained using a 3D solid modelling

program for different tanks, fill levels and accelerations. In the following analyses, the

force and moment values calculated by the mass centre model are plotted with straight

lines for various situations for the purpose of comparison.

Due to the wide use of partitions in liquid cargo tanks, the numerical simulation

implementing the present method is carried out for liquid motion in compartmented tanks

with different tank configurations, fill levels, as well as acceleration values. The analysis

is first conducted on a basic compartment for different situations, and then for tanks with

different configurations. The forces and moments and free surfaces are determined in the

simulations with various parameters. The liquid inside the tank is water for all studies. All

the tanks have the diameter of 2 meters. For tanks with hemispherical heads, the diameter

of the heads is the same as a diameter of the cylindrical section. The time step varies from

108


The liquid free surface, forces, and moments caused by liquid motion have also

been calculated by the mass centre model (Xu and Dai, 2004), in which the mass centre is

determined on the assumption that the free surface can be described by an inclined flat

surface, and that liquid bulk behaves like a rigid body (Figure 4.4). The free surface

gradient is simply considered as a constant under a given acceleration.

tan 6 = A j g (4.80)

The axial force and moment about the selected axis are expressible as follows.

FZ| = m,Az, M, = m,Az -y, + m ,g - z l (4.81)

In the above equations, mi is the total liquid mass inside the tank, and y, and z,

are the distances between the mass centre and the selected axis for moment calculation.

Since only geometrical information is needed to get the locations o f the mass centres of

the liquid bulk inside the tank, they are directly obtained using a 3D solid modelling

program for different tanks, fill levels and accelerations. In the following analyses, the

force and moment values calculated by the mass centre model are plotted with straight

lines for various situations for the purpose o f comparison.

Due to the wide use of partitions in liquid cargo tanks, the numerical simulation

implementing the present method is carried out for liquid motion in compartmented tanks

with different tank configurations, fill levels, as well as acceleration values. The analysis

is first conducted on a basic compartment for different situations, and then for tanks with

different configurations. The forces and moments and free surfaces are determined in the

simulations with various parameters. The liquid inside the tank is water for all studies. All

the tanks have the diameter o f 2 meters. For tanks with hemispherical heads, the diameter

o f the heads is the same as a diameter o f the cylindrical section. The time step varies from

108


0.001s to 0.005s, depending on different situations. The results are recorded for every

0.01 seconds. Instead of applying the acceleration by a step function, a ramp function, by

which the acceleration increases from zero to its steady value within 0.1s, is adopted to

gain convergence and to create a scenario reflecting real operations.

(a)

10000

8000 6000

4000 2000

0

0

10000

8000 V 6000

4000 2000

0

0 (b)

2 4 6

Time(s)

8 10

2 4 6

Time (s)

8

Figure 4.5 Force and moment under different accelerations

(a) force (b) moment

0.1g 0.2g

10

The influences of the acceleration on liquid behaviour are shown by calculated

forces and moments in Figure 4.5, where two acceleration values, 0.1g and 0.2g, have

been selected for a tank compartment with the LoD of 1. The tank is half-full. When the

109


0.001s to 0.005s, depending on different situations. The results are recorded for every

0.01 seconds. Instead of applying the acceleration by a step function, a ramp function, by

which the acceleration increases from zero to its steady value within 0.1s, is adopted to

gain convergence and to create a scenario reflecting real operations.

<L)OUo

(a)

100008000600040002000

0

2 6 80 4 10

Time(s)

a&+->C<DaO

(b)

100008000600040002000

00 2 4 6 8 10

Time(s)

Figure 4.5 Force and moment under different accelerations


O.lg ------- 0.2g

The influences o f the acceleration on liquid behaviour are shown by calculated

forces and moments in Figure 4.5, where two acceleration values, O.lg and 0.2g, have

been selected for a tank compartment with the LoD of 1. The tank is half-full. When the

109


tank is subjected to a suddenly applied acceleration, the liquid inside the tank undergoes

oscillatory motion, which significantly changes the pressure distributions on the tank

walls. This causes oscillatory forces and moments on the tank. It is quite clear that larger

acceleration causes larger forces and moments in both mean values and extreme values.

The mean values of forces and moments for 0.2g are twice as large as those for 0.1g, as

shown in Figure 4.5. For these two cases, the acceleration values, forces, and moments

appear with the same waveform, in which the first liquid mode is dominating.

(a)

(b)

t=4.2s

t=5.4s

t=4.2s

t=5.4s

t=4.5 s

t=5.7s

t=4.5 s

t=5 .7 s

t=4.8s

t=6.0s

t=4.8s

t=6.0s

t=5.1s

t=6.3s

t=5.1s

t=6.3s

Figure 4.6 Free surface development under different accelerations

(a) AZ 0.1g (b) AZ 0.2g

110


tank is subjected to a suddenly applied acceleration, the liquid inside the tank undergoes

oscillatory motion, which significantly changes the pressure distributions on the tank

walls. This causes oscillatory forces and moments on the tank. It is quite clear that larger

acceleration causes larger forces and moments in both mean values and extreme values.

The mean values o f forces and moments for 0.2g are twice as large as those for O.lg, as

shown in Figure 4.5. For these two cases, the acceleration values, forces, and moments

appear with the same waveform, in which the first liquid mode is dominating.

t=4.2s(a)

t=5.4s

t=4.5s

t=5.7s

t=4.8s

t=6.0s

t=5.1s

t=6.3s

t=4.2s(b)

t=5.4s

t=4.5s t=4.8s

\

t=5.1s

t=5.7s t=6.0s t=6.3s

Figure 4.6 Free surface development under different accelerations

(a) Az= O.lg (b) Az= 0.2g

110


Corresponding free surface development is illustrated in Figure 4.6 at different

times. It can be found that the free surfaces evolve in a similar manner for both

acceleration values, except that the liquid heights on the walls are higher under larger

acceleration. It can be seen from the figure that at most of the times, the free surfaces are

no longer inclined flat surfaces, even though the first natural liquid mode is dominating.

Higher modes are superposed on the first mode, a situation which is clearly shown by the

free surface shapes. For such a short tank compartment, the liquid motion takes the form

of standing waves.

The influence of fill levels on the liquid motion inside the same tank is studied by

comparing the forces and moments on the tank walls when the fill levels are 0.4D and

0.6D. This comparison is exhibited in Figure 4.7. As can be seen from Figure 4.7, the

force and moment have larger mean values at a fill level of 0.6D due to more liquid

inside the tank at a higher fill level. At the same time, the variations of force and moment

at 0.6D have smaller periods due to the increase in frequency caused by the higher fill

level.

Figure 4.8 shows the forces and moments of three different tanks to investigate

the influence of the hemispherical heads on the transient liquid behaviour. The tanks are

all half-full and the acceleration is 0.1g. Due to the different lengths of the tanks, the

periods of the three tanks are different. The longest tank has the largest period. The mean

values and the alternating values of oscillatory forces and moments are also different. The

values for the second and third tanks are much larger than those of the first tank, due to

their larger liquid volumes. It can be found from this figure that the force and moment

characteristics of the tank with hemispherical heads are much closer to those of the tank

111


Corresponding free surface development is illustrated in Figure 4.6 at different

times. It can be found that the free surfaces evolve in a similar manner for both

acceleration values, except that the liquid heights on the walls are higher under larger

acceleration. It can be seen from the figure that at most o f the times, the free surfaces are

no longer inclined flat surfaces, even though the first natural liquid mode is dominating.

Higher modes are superposed on the first mode, a situation which is clearly shown by the

free surface shapes. For such a short tank compartment, the liquid motion takes the form

of standing waves.

The influence of fill levels on the liquid motion inside the same tank is studied by

comparing the forces and moments on the tank walls when the fill levels are 0.4D and

0.6D. This comparison is exhibited in Figure 4.7. As can be seen from Figure 4.7, the

force and moment have larger mean values at a fill level o f 0.6D due to more liquid

inside the tank at a higher fill level. At the same time, the variations o f force and moment

at 0.6D have smaller periods due to the increase in frequency caused by the higher fill

level.

Figure 4.8 shows the forces and moments o f three different tanks to investigate

the influence of the hemispherical heads on the transient liquid behaviour. The tanks are

all half-full and the acceleration is O.lg. Due to the different lengths o f the tanks, the

periods o f the three tanks are different. The longest tank has the largest period. The mean

values and the alternating values o f oscillatory forces and moments are also different. The

values for the second and third tanks are much larger than those o f the first tank, due to

their larger liquid volumes. It can be found from this figure that the force and moment

characteristics o f the tank with hemispherical heads are much closer to those of the tank

111


with a LoD of 2 and flat heads. The free surface profiles of the liquid in the tank with

hemispherical heads are demonstrated in Figure 4.9 at various times. Compared to the

free surface shapes in tanks with flat heads, the free surfaces inside the tank with

hemispherical heads are much flatter. The influence of the higher modes is quite weak

due to the existence of the curved head walls. For this situation, the free surfaces can be

approximately replaced by oscillatory inclined flat surfaces.

6000

g 4000

° 2000

0

(a)

8000

6000

40 4000

2000

0

0 2 4 6

Time(s)

8 10

0 2 4 6 (b)

Time(s)

Figure 4.7 Force and moment under different fill levels


0.4D 0.6D

112

8 10


with a LoD o f 2 and flat heads. The free surface profiles o f the liquid in the tank with

hemispherical heads are demonstrated in Figure 4.9 at various times. Compared to the

free surface shapes in tanks with flat heads, the free surfaces inside the tank with

hemispherical heads are much flatter. The influence o f the higher modes is quite weak

due to the existence of the curved head walls. For this situation, the free surfaces can be

approximately replaced by oscillatory inclined flat surfaces.

6000

g, 4000<DO° 2000

0 2 4 6 8 10

Time(s)

8000

; 6000

£ 4000

S 2000

2 4 6 8 100

Time(s)

Figure 4.7 Force and moment under different fill levels


0.4D 0.6D

112


10000

a ., 8000 - 6000

,‘5° 4000 -w

2000

0

0

(a)

25000 r. 20000 15000

g 10000 5000

0

0 (b)

LoD=1,flat

t=7.4s

2 4 6

Time(s)

8 10

2 4 6

Time(s)

Figure 4.8 Force and moment for different tank shapes


LoD=1,hemi spherical LoD=2, flat

t=7.8s t=8.2s

8

t=8.6s

10

Figure 4.9 Free surface development in a tank with hemispherical heads

113


100008000600040002000

<DO

0 2 4 6 8 10(a) Time(s)

_ 25000J 20000

15000 10000

% 5000

Cb)

0

0 8 102 4 6

Time(s)

Figure 4.8 Force and moment for different tank shapes


LoD= l,flat - LoD= 1 hemispherical — — LoD=2, flat

r

t=5.8s t=6.2s t=6.6s t=7.0s

t=7.4s t=7.8s t=8.2s t=8.6s

Figure 4.9 Free surface development in a tank with hemispherical heads

113


The mobility of liquid cargo causes different dynamic vehicle behaviour from that

of rigid cargo, and poses challenges to researchers and engineers in vehicle stability

analysis and structure design. When the tank is subjected to a suddenly applied

acceleration, the oscillatory liquid motion causes oscillatory forces and moments of

considerable magnitudes, as shown in the above analysis and corresponding figures.

Compared to rigid cargo vehicles, the induced oscillatory forces and moments have

harmful influence on vehicle stability and controllability. They also exert cyclical

loadings on the supporting structures, which is one of the main reasons for the reduction

of the fatigue life of tank vehicle structures. It can be clearly seen from the previous

discussion that the mass centre model only shows the mean values of the oscillatory

forces and moments. It completely ignores the dynamic behaviour of the liquid and

cannot be used to describe the transient liquid motion during accelerating/braking

operations.

Numerical simulation is also conducted to study the influence of the time used for

applying the acceleration. In this case, the time is expressed by a ramp function. The

acceleration builds up to its final steady value within 4 seconds. It can be found from

Figure 4.11 that the force and moment also take the form of a ramp function and oscillate

around their equilibrium positions. At the same time, the amplitudes of the oscillations

are much smaller than those of a sudden applied acceleration. After the acceleration

reaches its steady value, the oscillation is so small that it can be neglected. Therefore,

when the acceleration is smoothly applied within a long enough time, the liquid can be

finally simplified as a rigid body by only considering the first liquid mode without

oscillation, i.e., fixed inclined flat surface. The load shift may then be obtained by using

114


The mobility of liquid cargo causes different dynamic vehicle behaviour from that

o f rigid cargo, and poses challenges to researchers and engineers in vehicle stability

analysis and structure design. When the tank is subjected to a suddenly applied

acceleration, the oscillatory liquid motion causes oscillatory forces and moments of

considerable magnitudes, as shown in the above analysis and corresponding figures.

Compared to rigid cargo vehicles, the induced oscillatory forces and moments have

harmful influence on vehicle stability and controllability. They also exert cyclical

loadings on the supporting structures, which is one o f the main reasons for the reduction

o f the fatigue life o f tank vehicle structures. It can be clearly seen from the previous

discussion that the mass centre model only shows the mean values o f the oscillatory

forces and moments. It completely ignores the dynamic behaviour o f the liquid and

cannot be used to describe the transient liquid motion during accelerating/braking

operations.

Numerical simulation is also conducted to study the influence o f the time used for

applying the acceleration. In this case, the time is expressed by a ramp function. The

acceleration builds up to its final steady value within 4 seconds. It can be found from

Figure 4.11 that the force and moment also take the form of a ramp function and oscillate

around their equilibrium positions. At the same time, the amplitudes o f the oscillations

are much smaller than those of a sudden applied acceleration. After the acceleration

reaches its steady value, the oscillation is so small that it can be neglected. Therefore,

when the acceleration is smoothly applied within a long enough time, the liquid can be

finally simplified as a rigid body by only considering the first liquid mode without

oscillation, i.e., fixed inclined flat surface. The load shift may then be obtained by using

114


the mass centre model. However, the accelerating/braking operations usually happen in a

very short time. The acceleration applied is usually not a constant. The transient liquid

oscillation is actually unavoidable and should be included in the vehicle dynamics study

and structural integrity analysis. In fact, the present method provides the availability to

assess liquid motions under both the transient acceleration and steady state condition. It is

therefore a useful tool for the dynamic analysis of the tank subjected to liquid sloshing.

5000

a ., 4000 6 3000

O"- 2000 1-1-1

1000 0

0 (a)

5000

4000 3000

'Id 2000 1000

0

(b) 0

2 4 6

Time (s)

8 10

2 4 6

Time (s)

8 10

Figure 4.11 Influence of input time


suddenly applied acceleration smoothly applied acceleration

Characteristics of forces and moments in compartmented tanks are much more

115


the mass centre model. However, the accelerating/braking operations usually happen in a

very short time. The acceleration applied is usually not a constant. The transient liquid

oscillation is actually unavoidable and should be included in the vehicle dynamics study

and structural integrity analysis. In fact, the present method provides the availability to

assess liquid motions under both the transient acceleration and steady state condition. It is

therefore a useful tool for the dynamic analysis of the tank subjected to liquid sloshing.

5000 _ 4000 § 3000 “ 2000

1000

uOP4

4 6 8 100 2

Time(s)

5000 4000

_ 3000 | 2000 J 1000

0 2 4 6 8 10

Time(s)

Figure 4.11 Influence o f input time


suddenly applied acceleration smoothly applied acceleration

Characteristics o f forces and moments in compartmented tanks are much more

115


complicated. Figure 4.10 shows the simulation of a 2-compartment tank and 3-

compartment tank. The acceleration is 0.1g. The tank configurations and liquid levels are

illustrated in Figure 4.10(a). For a tank with two identical compartments, Figure 4.10(b)

shows that the axial force has the smallest variation when the two compartments have

different fill levels, i.e., case I. When the liquid levels in the two compartments are

different, the liquid frequencies are different, as shown in Figure 4.7. The asynchronous

liquid motion helps to decrease the magnitude of the varying force when the resultant

liquid force is obtained by combining the forces in both compartments. When the two

compartments have the same liquid level, i.e., case II, the synchronicity of the liquid

motion makes the force larger than that of case I, even though the tanks have the same

volume of liquid for these two cases. It can also be found that a tank with the same

volume but without a partition, i.e., case III, has the largest force, which makes it a bad

choice for carrying liquid product. It is known that the free liquid surface motion and the

liquid impact can be more severe longitudinally than laterally if no transverse partitions

are introduced (Ibrahim et al 2001). Therefore, compartmented tanks are widely used in

road transportation industry.

For a 3-compartment tank, Figure 4.10(c) shows that the axial force has the

smallest magnitude when the three compartments are filled to different levels, i.e., case

IV. For case VI, the tank is composed of a short compartment with an LoD of 1 and a

long compartment with an LoD of 2. Although the liquid frequencies are different, the

axial force has not been depressed by the asynchronous liquid motion, because the axial

force caused by the liquid in the longer compartment dominates. It should be noted that if

the liquid forces are calculated by the mass centre model, the results would be the same

116


complicated. Figure 4.10 shows the simulation o f a 2-compartment tank and 3-

compartment tank. The acceleration is O.lg. The tank configurations and liquid levels are

illustrated in Figure 4.10(a). For a tank with two identical compartments, Figure 4.10(b)

shows that the axial force has the smallest variation when the two compartments have

different fill levels, i.e., case I. When the liquid levels in the two compartments are

different, the liquid frequencies are different, as shown in Figure 4.7. The asynchronous

liquid motion helps to decrease the magnitude o f the varying force when the resultant

liquid force is obtained by combining the forces in both compartments. When the two

compartments have the same liquid level, i.e., case II, the synchronicity o f the liquid

motion makes the force larger than that o f case I, even though the tanks have the same

volume o f liquid for these two cases. It can also be found that a tank with the same

volume but without a partition, i.e., case III, has the largest force, which makes it a bad

choice for carrying liquid product. It is known that the free liquid surface motion and the

liquid impact can be more severe longitudinally than laterally if no transverse partitions

are introduced (Ibrahim et al 2001). Therefore, compartmented tanks are widely used in

road transportation industry.

For a 3-compartment tank, Figure 4.10(c) shows that the axial force has the

smallest magnitude when the three compartments are filled to different levels, i.e., case

IV. For case VI, the tank is composed o f a short compartment with an LoD o f 1 and a

long compartment with an LoD o f 2. Although the liquid frequencies are different, the

axial force has not been depressed by the asynchronous liquid motion, because the axial

force caused by the liquid in the longer compartment dominates. It should be noted that if

the liquid forces are calculated by the mass centre model, the results would be the same

116


for different compartment configurations, as shown by the straight lines in the figures.

(a)

0.4D 0.6D

I

0.4D 0.5D 0.6D

10000

, .., 8000

' 6000

o'ci,

4000 P-1

2000 0

0 (b)

IV

0.5D

II

0.5D

0.5D 0.5D 0.5D

V

0.5D

III

0.5D 0.5D

VI

15000

10000 . U ,... ra-e ° 5000

(c)

0

2

case I

4 6

Time(s)

case II

8

case III

10

0 2 4 6 8

Time(s) case IV - case V case VI

Figure 4.10 Axial forces in compartmented tanks

(a) tank configurations (b) 2-compartment (c) 3-compartment

117

10


for different compartment configurations, as shown by the straight lines in the figures.

0.5D 0.5D0.6D0.4D

(a) I II HI

0.6D0.4D 0.5D 0.5D 0.5D 0.5D 0.5D 0.5D

IV V VI

■DOuO[J-i

100008000600040002000

06 80 2 4 10

(b)

case ITirne(s) case II case III

Q JOU.o[J-i

(c)

15000

10000

5000

06 8 100 2 4

case IVTime(s)

■ case V case VI

Figure 4.10 Axial forces in compartmented tanks

(a) tank configurations (b) 2-compartment (c) 3-compartment

117


4.5 Summary

In this chapter, a new mathematical method is developed to study the liquid

dynamics in partially filled 3D horizontal cylindrical tanks based on the method

developed for 2D circular and elliptical tanks in the previous chapter. The governing

equations based on potential flow theory, including the Laplace equation and boundary

conditions on the free surface and curved walls of a tank, are rearranged by continuous

coordinate mappings, such that the difficulties of direct discretization for numerical

calculation are avoided. The efficiency of this method has been approved by solving the

natural frequencies of a generalized eigenvalue problem using the linearized Bernoulli

equation. The transient liquid motion and corresponding liquid forces and moments

acting on the tank walls have been calculated for the tank subjected to longitudinal

acceleration input.

When the tank is subjected to a suddenly applied acceleration during

accelerating/braking operations, the liquid inside the tank undergoes severe sloshing,

which causes oscillatory forces and moments on the tank. The liquid free surface, liquid

forces, and moments under different acceleration values and liquid levels are investigated

in detail with the method developed. The influence of the existence of hemispherical

heads is studied by comparing the forces and moments to those with flat heads. The

configuration of compartmented tanks and liquid distribution inside different

compartments have a significant influence on the forces and moments, which have been

simulated and analysed under different situations for a 2-compartment tank and a 3-

compartment tank.

The methodology developed solves the transient liquid motion in a completely 3D

118


4.5 Summary

In this chapter, a new mathematical method is developed to study the liquid

dynamics in partially filled 3D horizontal cylindrical tanks based on the method

developed for 2D circular and elliptical tanks in the previous chapter. The governing

equations based on potential flow theory, including the Laplace equation and boundary

conditions on the free surface and curved walls o f a tank, are rearranged by continuous

coordinate mappings, such that the difficulties o f direct discretization for numerical

calculation are avoided. The efficiency of this method has been approved by solving the

natural frequencies o f a generalized eigenvalue problem using the linearized Bernoulli

equation. The transient liquid motion and corresponding liquid forces and moments

acting on the tank walls have been calculated for the tank subjected to longitudinal

acceleration input.

When the tank is subjected to a suddenly applied acceleration during

accelerating/braking operations, the liquid inside the tank undergoes severe sloshing,

which causes oscillatory forces and moments on the tank. The liquid free surface, liquid

forces, and moments under different acceleration values and liquid levels are investigated

in detail with the method developed. The influence o f the existence o f hemispherical

heads is studied by comparing the forces and moments to those with flat heads. The

configuration o f compartmented tanks and liquid distribution inside different

compartments have a significant influence on the forces and moments, which have been

simulated and analysed under different situations for a 2-compartment tank and a 3-

compartment tank.

The methodology developed solves the transient liquid motion in a completely 3D

118


manner for horizontal cylindrical tanks with flat heads and hemispherical heads. This

methodology can be used to evaluate liquid motion in tanks of arbitrarily shaped walls,

such as tanks with elliptical cross sections and tanks with other types of heads. It can also

be easily integrated into coupled liquid-structure system to study the vehicle system

dynamics. This provides the availability of a systematic analysis of the tank vehicle

structures subjected to liquid sloshing and other loadings.

119


manner for horizontal cylindrical tanks with flat heads and hemispherical heads. This

methodology can be used to evaluate liquid motion in tanks o f arbitrarily shaped walls,

such as tanks with elliptical cross sections and tanks with other types o f heads. It can also

be easily integrated into coupled liquid-structure system to study the vehicle system

dynamics. This provides the availability o f a systematic analysis o f the tank vehicle

structures subjected to liquid sloshing and other loadings.

119


CHAPTER 5 INFLUENCE OF LIQUID MOTION ON RIDE

QUALITY OF LIQUID CARGO TANK VEHICLES

5.1 Introduction

The issue of ride comfort is of great concern because exposure to high levels of

vibration will cause driver fatigue, which in turn can have a harmful influence on health

problems and driving safety. Ride quality problems have been studied for many years.

Some basic theories can be found in Wong (1993). A literature survey especially on heavy

vehicle ride comfort has recently been conducted by Jiang et al (2001). A general concept

of vibration-related health problems, ride comfort assessment criteria and methods and

methodology of using computer simulation to analyze ride comfort have been discussed.

Seven vehicle models, five driver/seat models, and detailed modeling techniques have

been reviewed. However, nothing has been discussed regarding ride comfort for liquid

cargo vehicles.

The ride quality of tractor semi-trailers carrying rigid cargo has attracted the

attention of several researchers in recent years (Vaduri and Law 1993, Elmadany, 1987).

However, a systematic assessment of the influence of liquid sloshing on ride quality in

partially filled tank vehicles is still outstanding in the current literature. For most tank

trucks, the total payload of the liquid cargo accounts for a large portion of the total

vehicle weight. The liquid motion within the partially filled tanks has a negative

influence on the driver's ride quality. The influence is much greater when other factors,

such as the articulation of tractor semi-trailer and B-train tank trucks, are taken into

120


CHAPTER 5 INFLUENCE OF LIQUID MOTION ON RIDE

QUALITY OF LIQUID CARGO TANK VEHICLES

5.1 Introduction

The issue o f ride comfort is of great concern because exposure to high levels of

vibration will cause driver fatigue, which in turn can have a harmful influence on health

problems and driving safety. Ride quality problems have been studied for many years.

Some basic theories can be found in Wong (1993). A literature survey especially on heavy

vehicle ride comfort has recently been conducted by Jiang et al (2001). A general concept

o f vibration-related health problems, ride comfort assessment criteria and methods and

methodology o f using computer simulation to analyze ride comfort have been discussed.

Seven vehicle models, five driver/seat models, and detailed modeling techniques have

been reviewed. However, nothing has been discussed regarding ride comfort for liquid

cargo vehicles.

The ride quality of tractor semi-trailers carrying rigid cargo has attracted the

attention o f several researchers in recent years (Vaduri and Law 1993, Elmadany, 1987).

However, a systematic assessment of the influence o f liquid sloshing on ride quality in

partially filled tank vehicles is still outstanding in the current literature. For most tank

trucks, the total payload of the liquid cargo accounts for a large portion of the total

vehicle weight. The liquid motion within the partially filled tanks has a negative

influence on the driver’s ride quality. The influence is much greater when other factors,

such as the articulation of tractor semi-trailer and B-train tank trucks, are taken into

120


consideration. The liquid motion depends on different operation conditions, which makes

the ride quality analysis different from that of rigid cargo vehicles.

Although the mathematical method developed in the previous chapter can be

applied to study the vehicle dynamics by integrating the fluid mechanics solution into the

vehicle system, it is basically an approach to analyzing the liquid motion in a temporal

domain, and not suitable for investigation of ride quality in a frequency domain. To study

the ride quality problem of liquid cargo vehicles, equivalent mechanical models should

be employed. In this chapter, a multi-degree-of-freedom pitch plane vehicle model

representing the dynamic response of partially filled compartmented tank vehicles is

developed and analyzed in the frequency domain to assess the effect of liquid sloshing on

ride quality. The dynamic liquid motion within the tank is modeled by a linear spring-

mass system. The input to the model is a user-specified power spectral density of the

vertical road irregularities. The power spectral density of the vertical and longitudinal

driver seat accelerations is simulated and compared with that of rigid cargo vehicles. The

influence of liquid fill level, vehicle speed, the suspension system, and road condition on

the ride quality of the tank vehicles is also investigated.

5.2 Vehicle model

The tractor semi-trailer for this study is modeled as shown in Figure 5.1. The

vehicle is considered to be traveling over an uneven road at a constant forward speed. It

is assumed that the left and right wheels of the vehicle experience identical excitations.

Vibration corresponding to the model is thus constrained to the pitch plane. The tractor

and the semi-trailer structure without payload are treated as perfect rigid bodies. The

121


consideration. The liquid motion depends on different operation conditions, which makes

the ride quality analysis different from that o f rigid cargo vehicles.

Although the mathematical method developed in the previous chapter can be

applied to study the vehicle dynamics by integrating the fluid mechanics solution into the

vehicle system, it is basically an approach to analyzing the liquid motion in a temporal

domain, and not suitable for investigation o f ride quality in a frequency domain. To study

the ride quality problem of liquid cargo vehicles, equivalent mechanical models should

be employed. In this chapter, a multi-degree-of-freedom pitch plane vehicle model

representing the dynamic response of partially filled compartmented tank vehicles is

developed and analyzed in the frequency domain to assess the effect o f liquid sloshing on

ride quality. The dynamic liquid motion within the tank is modeled by a linear spring-

mass system. The input to the model is a user-specified power spectral density o f the

vertical road irregularities. The power spectral density of the vertical and longitudinal

driver seat accelerations is simulated and compared with that o f rigid cargo vehicles. The

influence o f liquid fill level, vehicle speed, the suspension system, and road condition on

the ride quality of the tank vehicles is also investigated.

5.2 Vehicle model

The tractor semi-trailer for this study is modeled as shown in Figure 5.1. The

vehicle is considered to be traveling over an uneven road at a constant forward speed. It

is assumed that the left and right wheels o f the vehicle experience identical excitations.

Vibration corresponding to the model is thus constrained to the pitch plane. The tractor

and the semi-trailer structure without payload are treated as perfect rigid bodies. The

121


tractor and semi-trailer are allowed to translate in the forward and vertical directions, and

to pitch about the fifth wheel, which is modeled as a pin connection. The vehicle is

supported by three axle suspension systems. The wheels and axles are supported through

the tire springs and dampers by the road. The tires of the vehicle are assumed to remain in

contact with the road surface at all times. The seat suspension is also modeled as a linear

spring with a damper. The tank includes four compartments, as illustrated in Figure 5.1.

As verified in the previous chapter, the natural frequencies of the liquid motion in

the longitudinal direction can be effectively calculated by using an equivalent calculation

method to make use of the theory of the liquid motion response in rectangular tanks. This

equivalent method was originally established by Kobayashi et al (1989) with frequency

sweep testing. Calculated results of natural frequencies, sloshing wave heights and

sloshing forces were in good agreement with the experimental ones. Another similar

method, a summation technique, was developed by Ranganathan et al (1994) to calculate

the parameters of the spring-mass model according to the theory of sloshing response in

rectangular tanks. Although the mathematical method developed in the previous chapter

can be used in dynamic liquid motion and liquid-structure system study, it can only solve

the problems in the time domain. For the study of ride quality of the liquid cargo tank

vehicles, it is necessary to find the influence of the liquid motion on the vehicle system in

the frequency domain, and compare with the ISO standard. The equivalent mechanical

model of liquid motion, i.e., the spring-mass system, can actually be a useful tool for this

purpose. In this chapter, the equivalent method generated by Kobayashi et al (1989) is

used to study the ride quality of partially filled tank vehicles. When the liquid motion is

represented by the linear mechanical model and only the fundamental mode is

122


tractor and semi-trailer are allowed to translate in the forward and vertical directions, and

to pitch about the fifth wheel, which is modeled as a pin connection. The vehicle is

supported by three axle suspension systems. The wheels and axles are supported through

the tire springs and dampers by the road. The tires of the vehicle are assumed to remain in

contact with the road surface at all times. The seat suspension is also modeled as a linear

spring with a damper. The tank includes four compartments, as illustrated in Figure 5.1.

As verified in the previous chapter, the natural frequencies o f the liquid motion in

the longitudinal direction can be effectively calculated by using an equivalent calculation

method to make use o f the theory o f the liquid motion response in rectangular tanks. This

equivalent method was originally established by Kobayashi et al (1989) with frequency

sweep testing. Calculated results o f natural frequencies, sloshing wave heights and

sloshing forces were in good agreement with the experimental ones. Another similar

method, a summation technique, was developed by Ranganathan et al (1994) to calculate

the parameters of the spring-mass model according to the theory o f sloshing response in

rectangular tanks. Although the mathematical method developed in the previous chapter

can be used in dynamic liquid motion and liquid-structure system study, it can only solve

the problems in the time domain. For the study o f ride quality o f the liquid cargo tank

vehicles, it is necessary to find the influence of the liquid motion on the vehicle system in

the frequency domain, and compare with the ISO standard. The equivalent mechanical

model of liquid motion, i.e., the spring-mass system, can actually be a useful tool for this

purpose. In this chapter, the equivalent method generated by Kobayashi et al (1989) is

used to study the ride quality of partially filled tank vehicles. When the liquid motion is

represented by the linear mechanical model and only the fundamental mode is

122


considered, the liquid can be divided into two parts. One part is fixed to the tank and

moves as a rigid mass, while the other part moves as a point mass.

The displacements of the component of the vehicle system are denoted by the

independent generalized coordinates Yrp measured from the position of static equilibrium.

Yrp = [zs Zrl 191 Zil 1 ;12 Zt21 0 2 Xri • • • Xe, • • 1T (5.1)

Other dependent generalized coordinates, xs , xr2 , zr2 , x111 , .X,12 , x, 21 and ze, , are

determined by the rigid body motions, as indicated in the Figure 5.1. The dimensions of

the model are also shown in Figure 5.1. The descriptions of the generalized coordinates

are listed in Table 5.1. The masses and moments of inertia of the vehicle components,

together with the spring and damping components, will be discussed later.

The artificial soft spring and damper ( kf and cf ) attached to the tractor restrain

the otherwise semi-definite system and represent the behaviour of free rolling tires of a

vehicle traveling at a constant forward speed on a straight road. The parameters of the

spring-mass model of the liquid motion, along with the mass and moment of inertia of the

fixed part can be calculated from the equations given by Abramson and Silverman

(1966).

123


considered, the liquid can be divided into two parts. One part is fixed to the tank and

moves as a rigid mass, while the other part moves as a point mass.

The displacements o f the component o f the vehicle system are denoted by the

independent generalized coordinates Yrp measured from the position o f static equilibrium.

Y r p ~ Z r \ Z l l l Z t l 2 Z t 2 \ @ 2 X r \ X e i ” ' ] ( 5 - 1 )

Other dependent generalized coordinates, xs , x r2, z r2, xlU , xtn , xt2l and zei, are

determined by the rigid body motions, as indicated in the Figure 5.1. The dimensions of

the model are also shown in Figure 5.1. The descriptions of the generalized coordinates

are listed in Table 5.1. The masses and moments o f inertia o f the vehicle components,

together with the spring and damping components, will be discussed later.

The artificial soft spring and damper ( k f and cf ) attached to the tractor restrain

the otherwise semi-definite system and represent the behaviour o f free rolling tires o f a

vehicle traveling at a constant forward speed on a straight road. The parameters o f the

spring-mass model of the liquid motion, along with the mass and moment o f inertia o f the

fixed part can be calculated from the equations given by Abramson and Silverman

(1966).

123


Reproduced w

ith permission o

f the copyright owner.

Further reproduction prohibited w

ithout permission.

zrll

z11

I Cf.

k11

aeia2

r

ms

i ks c L _

mi

• • •

/ 0i m 20 / 20

I 1zr 1

Cei

1121P mor

Xr ei

r 2I r2

1

mr11

b

k 12 C12 Xr12 Xrll m 12

Cri1 Zi12 r12 Cr12

Z12

a30

2

a3 a4

b2

C21 21 Xr21 m 21

Z121 Kr 21 Cr21

Figure 5.1 Pitch plane model of the tractor semi-trailer

124

Reproduced w

ith permission of the copyright ow

ner. F

urther reproduction prohibited without perm

ission.

Table 5.1 Description of the generalized coordinates of the vehicle model

Symbol Description

zs Vertical displacement of seat

Zrl Vertical displacement of mass centre of tractor

Angular displacement of tractor

Znn 1 Vertical displacement of front axle of tractor

ZI12 Vertical displacement of rear axle of tractor

Zt21 Vertical displacement of semi-trailer axle

0 2 Angular displacement of semi-trailer

Xrl Horizontal displacement of mass centre of tractor

x ei Horizontal displacement of ith sloshing mass

xs Horizontal displacement of seat

Xr2 Horizontal displacement of mass centre of semi-trailer

Zr2 Vertical displacement of mass centre of semi-trailer

Horizontal displacement of front axle of tractor

x,12 Horizontal displacement of rear axle of tractor

Horizontal displacement of semi-trailer axle

Zei Vertical displacement of ith sloshing mass

Z11 Road profile at the front axle of tractor

Z12 Road profile at the rear axle of tractor

Z21 Road profile at the semi-trailer axle

125


Table 5.1 Description of the generalized coordinates o f the vehicle model

Symbol Description

Vertical displacement of seat

Vertical displacement of mass centre of tractor

Angular displacement o f tractor

z m Vertical displacement o f front axle o f tractor

Z t \ 2 Vertical displacement of rear axle o f tractor

Z t 2 \ Vertical displacement o f semi-trailer axle

e2 Angular displacement o f semi-trailer

X r X Horizontal displacement o f mass centre of tractor

X e i Horizontal displacement of z'th sloshing mass

X s Horizontal displacement of seat

X r 2 Horizontal displacement o f mass centre of semi-trailer

Z r 2 Vertical displacement o f mass centre o f semi-trailer

XI\\ Horizontal displacement o f front axle of tractor

X t \ 2 Horizontal displacement of rear axle o f tractor

X t 2 1 Horizontal displacement o f semi-trailer axle

2 el Vertical displacement o f z'th sloshing mass

z xx Road profile at the front axle of tractor

Z X 2 Road profile at the rear axle o f tractor

Z 2 X Road profile at the semi-trailer axle

125


The governing equations of the vehicle system are derived using the following

Lagrange's equations.

d a a - = Q, j =1,•••,n (5.2) dt - a Y rj

In the above equations, L is the lagrangian, the difference between the kinetic and

potential energy at an arbitrary instant, yrj is the jth generalized coordinate of the system,

and Q, is the jth generalized force.

The motions of the system are assumed to be small such that the sines of the

angles of rotation may be taken equal to the angles themselves, and cosines of the angles

may be taken as unity. The linearized governing equations of the coupled system can be

expressed as follows.

msEs +csi s —cs2r,—csase9,+kszs —kszri —ksasOl = 0 (5.3)

(mri mr2 E me, )ri (mr2 E me, )a 2 5.11 (mr2a3 E meiae,

+ (Cs + CI C12 ± C21 )i ri (csas — la, + c12a2 e2ia2 + c2, (a3 + ajd2 — cs±, — c11±111—c12±112—C21-2t21

k 2i )Zri (ksas — la, + ki2a2 + k2ia2

+ k2i (a3 + a4 )02 — ksz, k2,z,21 = 0

mr2a2 E meiaeia2 msbs2 mr2bi2 + mil IN + mti2N m121b12 + E meib,2

(5.4)

±(mr2a2+ E meiaei [Mr2a2a3 E meicie2, — mr2bib2 — 4)— Im„beib,182

(msbs — mr — mi. ikt — mt. 2b4 — mt2ibi — E /17,4 )1 r1 — ineib lkq

+(csas — + c12a2 + c2,a2 + (csas2 +clog; cuai c2iaZ )41 — csasi s Cl2a 2±t12 c21a2E,21 c2ia2(a3 a4

+ ki2a2 + k2ia2 )zr, + (ksas2 + ki k21ai )91 — ksaszs + kllaizti — k12a2; 12 — k21(22zt21 + k21a2(a3 + a4 = 0

(5.5)

mmE111 (c11 cti 1a1191 (k11 km )zti = kll1z11 +ct11 11 (5.6)

126


The governing equations of the vehicle system are derived using the following

Lagrange’s equations.

d_dt dy v J rJ y

81 =Qri J = l - , n (5.2)dy,

In the above equations, L is the lagrangian, the difference between the kinetic and

potential energy at an arbitrary instant, y . is the yth generalized coordinate o f the system,

and Qrj is theyth generalized force.

The motions of the system are assumed to be small such that the sines of the

angles o f rotation may be taken equal to the angles themselves, and cosines of the angles

may be taken as unity. The linearized governing equations o f the coupled system can be

expressed as follows.

m z + c z - c z , - c a d , + k z — k z , —k a 0 , = 0 (5.3)s s s s s r 1 5 s i s s s r 1 s s 1 V /

(mrt + mr2 + X ^ e iK l + ( m r 2 + + ( m r 2 a 3 +

+ ( c s + c xx + c n + c 2 i K i + ( C A - c xxa x + c n a 2 + c 2 xa ^ x

+ c2i(u3 + aA)d2 - c szs — cxxznx — cl2zt 12 — c2lzl2l (5.4)+ (K + k\ i + k]2 + klx )zrX + (ksas - kx xax + kx2a2 + k2]a2)0X + k2X{a2 + a4)02 — kszs — kxxztxx — kx2ztX2 — k2xzt2x = 0

[lri + mr2a\ + + ™A2 + mrlbx + mtX xb\ + mtnb] + mt2xbx )&x

+ ( " L 2 « 2 + + Y , m e i a l ~ m r 2 h x h + “ b X ) ~ Z m e P e P x

+ (msb, - mr2bt - mtXXb4 - m,12Z>4 - m(21Z>, - J X a K i " 2 X A * e/

+ (<A - c„fl, + c12a2 + c21a2)zrl + ( c ^ 2 + cna 2 + c12u2 + c21a22)0,

~ C s a s ^ s C \ \ a \ Z l \ \ ~ ~ C X 2 a 2 ^ l X 2 ~ ^ 2 \ a 2 Z l 2 X “*“ ^ 4 ) ^ 2

+ (* A - A + k n a 2 + k 2Xa 2 ) z rX + ( k s a ] + A:nu,2 + £12a 2 + k 2Xa 22 ] d x

- K a s z s + k xxa xz n x - k x2 a 2 z m ~ k 2 xa 2 z m + M 2 f a + a 4 ) 0 2 = 0(5.5)

m,xxzax +(cn +ctXX)ztXX - c xxzrl +cxxax0x +(kxx + klXX)ztXX - k xxzrX +kxxaxOx = knxzxx +ctxxzx(5.6)

126


Mtl 2 t12 + ( C.12 + c 112 )±tI2 Cl2 r1 C12a 2° 1 ( k 12 k tI2 ) Zt12 k 12Zr1 k 12a 2e 1 = k tI2Z12 c tI2 12

(5.7)

m t21 t21 ( c 21 ± )t 21 — C2I i rl c n a ze i c 2I +6 /4 ) 4.92

( k 2I kt21);21 2k i Zri k 2ia 20 1 k 2 (a 3 + a.4 )0-

L = k t21 Z 21 c a1i 21

I./r2 m r2a 3 mejae2i E meibe2, m r2b m t2I ( b 4 — b l ) 212

[mr2a2a3 meiaeia2 — mr2b1b2 + mi214(b4 — b1 )— E (mr2a3 +E 1;iaei)1

+{mr2b2 — ma1(b4 — b1)+ E rrzeibei liri +Im eibeik i

c21(23 ± a 4 ) /-1 c2Ia2(a3 + a4)0, — c2i(a3 + a ) 4 ' j ai ± C21(23 4- '24 ) 262

+ k 2, (a 3 + ri k 21a 2(a 3 + a 4 ) , — k 2, (a 3 + ); 21 + [k21(a 3 +a 4 — meig(bei—Imeigxei = 0

(5.8)

2)192

(5.9)

(Ms + Mrl Mr2 Mtl I ± Mt12 ± Mt 21 ± E ( Msb s Mr2b 1 Mt2lb 4 Mil2b 4 maibiA kr2b 2 m t 21 (b4 b1)+ E meibei F 2 4- E Meii ei C f .xri k f Xri = 0

(5.10)

meibei + ineAl —meiblel+ meibeio2 + ei + keixei — meig02 = 0, i = 1,2,• • • (5.11)

In the above equations, m r2 and / r2 are the combined mass and moment of inertia

of the empty semi -trailer and all fixed parts of the liquid.

mr2 = m zo E moi

11.2 =' 20 + {(b2 b 20 + (a3 a 30 in2o E fro, + [(b0, — b2 + (a 3 — a,,) 2

(5.12)

m o,} (5.13)

The position of the centre of gravity for the combined mass can be determined by

the following equations.

b z = (b zom zo +yboimoi)/(m2.± E m 01 )

a 3 = ( a 30 m 20 E aeimoi )A m 20 E mOi

(5.14)

(5.15)

The governing equations of the system defined can be concisely expressed in

matrix form as:

127


m t\2 Zt\2 + i C\2 + Cn2Xl2 C\2^r\ + c \2a 2@\ + ( 12 + 12 ) rl2 ^ \2 Z r\ + k \2 ° 2 ^ \ k t\2Z \2 + Ct\2Z \2

(5.7)

W (2 1 ^ ( 2 1 + ( C 21 + C l 2 l ) Z t 2 1 ~ C 2 \ Z r \ + C 2 \ a 2 ^ \ ~ C 2 l ( f l 3 + f l 4 ) ^ 2 ^ g ^

+ ( & 21 ^ k l 2 i ) z , 2 i ~ k 2 \ Z r \ — k l x a 2 O x — k 2 \ i . a 2 + ^ 4 ) ^ 2 — ^ 7 2 \ Z 2 \ ^ C t 2 \ Z 2 \

[7,2 + m n a \ + Y j m eia l i + Y j m e.b l + m r2b 2 + ™ t2 \ ib 4 “ b \ f J 2

Wr2a2fl3 + T j m eia eia 2 ~ m r2b A + *”,21^4 “ A)" Z ™«AA]^1 + (Wr2«3 + Z W*fl*K l

m r2b 2 ~ m m ( b 4 - A) + Z ™ A f c l + Z ™ A A

C 21 ( ^ 3 ^ 4 ) ^ r l C 2 1 ^ 2 ( f l 3 ^ 4 ) ^ 1 _ C 2 I ( ^ 3 ^ 4 ) ^ / 2 l C 21 ( f l 3 ^ 4 ) ^ 2

+ £21(a3 + a4)zH + £ 21a2(a3 + a4)6> - * 2 i ( « 3 + a 4 ) z m + [*2i(«3 + o 4)2 - Z w«-gfe/ A ) k

- Z WrfS*e*=°(5.9)

k + Wr, + ror2 + /»„, + mn2 + mt2l + £ mei)xrl + (msbs - mr2bx - mt2lb4 - mn2b4 - ml2xbx )0X

+ [™r2b 2 - m ,2 \{b 4 ~ b l ) + Z W A & + Z V e i + C f K \ + k f X r\ = 0

( 5 . 1 0 )

m elX a + m j r l ~ m eib A + m c A A + C eiK i + K i* e i ~ ™ e iS d 2 = 0> * = A V •1 1)

In the above equations, mr2 and I r2 are the combined mass and moment o f inertia

of the empty semi-trailer and all fixed parts o f the liquid.

m r2 = m 20 + Z m 0i (5-12)

7,2 =ho+[(b2 - b2of +(a3 - a2o)2\ n20 + T JÔi + [(bOi-b2 f + (a2 - aei)2\ m0i] (5-13)

The position o f the centre o f gravity for the combined mass can be determined by

the following equations.

b 2 = (h io W io + Z 60,™0, )/ko + Z OTo/) (5-14)

« 3 = ( « 3 0 W 20 + Z ^ - m 0 1 ) / k o + Z ' ” 0i ) ( 5 ‘ 1 5 )

The governing equations o f the system defined can be concisely expressed in

matrix form as:

127


M r f rp ± C r icp ± K r Yrp = B ir L I rp ± B 2rU rp (5.16)

In the above equation, Mr, Cr and Kr are the mass, damping and stiffness matrices.

Urp is the vector of instantaneous values of vertical displacements of the road profile at

each axle location.

211T U rp = Z[z11 z 12 (5.17)

The road profile at the tractor rear axle and the semi-trailer axle can be expressed

in terms of the road profile at the tractor front axle, z,1(t), as:

Z12 ( t ) = Z I1 ( t — t l ) (5.18)

z21(t) = zii(t — t2) (5.19)

In the above equations, the delay times, t, and t2 , can be decided by the speed of

the vehicle, v.

t, = (a, +a2 )/v (5.20)

t2 = (a, + a2 + a3 + a4 )/v (5.21)

5.3 Analysis procedure

In order to obtain the eigenvalues, eigenvectors and transfer functions of the

system, standard frequency analysis techniques are employed in the computer simulation.

Then, the natural frequencies, damped natural frequencies, damping ratios and mode

shapes for the system can be determined from the eigenvalues and eigenvectors.

Taking the Laplace transform of Eq. (5.16) and using the linearity of the

transform, the following equation can be obtained.

(Mrs 2 + C rS + Kr )Y,p(s)= (Ars + B 2r )U rp (S) (5.22)

128


M X P +CrYrp + K J rp= B [rUrp+B2rUrp (5.16)

In the above equation, Mr, Cr and Kr are the mass, damping and stiffness matrices.

Urp is the vector of instantaneous values of vertical displacements o f the road profile at

each axle location.

Urp=[zu z l2 z 2J (5.17)

The road profile at the tractor rear axle and the semi-trailer axle can be expressed

in terms o f the road profile at the tractor front axle, z ,, ( t ) , as:

z i2(t) = z n (* -* ,) (5.18)

z 2\(t) = z u ( t - t 2) (5.19)

In the above equations, the delay times, t] and t2, can be decided by the speed of

the vehicle, v.

ty - (a, +a2) /v (5.20)

t2 = (ax + a2 + a 3 +a4) /v (5.21)

5.3 Analysis procedure

In order to obtain the eigenvalues, eigenvectors and transfer functions of the

system, standard frequency analysis techniques are employed in the computer simulation.

Then, the natural frequencies, damped natural frequencies, damping ratios and mode

shapes for the system can be determined from the eigenvalues and eigenvectors.

Taking the Laplace transform of Eq. (5.16) and using the linearity o f the

transform, the following equation can be obtained.

(hi,*1 + C rS + K X ( s ) = ( K s + Blr)Ure(s) (5-22)

128


In the above equation, U rp (s) is the Laplace transform of U rp (t).

U rp (S)=-- Frp (s)zil (s) (5.23)

Frp (s)=[1 e-st' e-'21 T (5.24)

The transfer function vector for the system can be achieved by solving Eq. (5.22).

I', (s) / zl , (s) =(1 rs2 + C rs + Kr ) 1 *(1211rs+B2r )* F rp (S) (5.25)

Thus the transfer functions of the generalized coordinates of their linear

combinations can be obtained from Yrp (s)/ z11 (s). Note that Yrp is a vector, the elements

of which are the transfer functions of the independent coordinates. This implies that the

transfer functions of the independent coordinates can be obtained directly; for example,

the first element is for the vertical seat acceleration. The transfer functions of dependent

coordinates can also be obtained using the linear combinations of the independent

coordinates.

The frequency response for any transfer function can be obtained by substituting

i2z f (f is the frequency in units of Hz) for s in the transfer function and varying the

frequency over the range of interest.

The power spectral density of a given output variable, Sy, may be expressed as:

S y (f)=1G y (i2ir f)1 2 S zu (f) (5.26)

In the above equation, G y (i2rc f) is the frequency response of the given output,

yr, in response to the road input, z11 , and Szii (f) is the power spectral density function of

the elevation of the surface profile.

129


In the above equation,!/ is) is the Laplace transform of Urp(t).

= / / » „ ( * ) (5-23)

Fn,(s)=[l e-'1' e - ' ^ Y (5.24)

The transfer function vector for the system can be achieved by solving Eq. (5.22).

y „ (s )/ Z|,M = { M y + C,s + AT,)'1 • (Blrs + f l j * F j s ) (5.25)

Thus the transfer functions of the generalized coordinates o f their linear

combinations can be obtained from Yrp{s)lzu (s). Note that Yv is a vector, the elements

o f which are the transfer functions o f the independent coordinates. This implies that the

transfer functions o f the independent coordinates can be obtained directly; for example,

the first element is for the vertical seat acceleration. The transfer functions o f dependent

coordinates can also be obtained using the linear combinations o f the independent

coordinates.

The frequency response for any transfer function can be obtained by substituting

i2n f ( / is the frequency in units of Hz) for s in the transfer function and varying the

frequency over the range of interest.

The power spectral density o f a given output variable, Sy, may be expressed as:

S , ( f ) = \ G y{ i 2 x f f s , a ( f ) (5.26)

In the above equation, Gy {i2n / ) is the frequency response o f the given output,

y r, in response to the road input, zn , and 5lZ i ( / ) is the power spectral density function of

the elevation o f the surface profile.

129


In this study, the vehicle is subjected to disturbances from road irregularities.

When the surface profile is regarded as a random function, it can be characterized by a

power spectral density function. The relationship between the power spectral density and

the spatial frequency for the road profile can be approximated by (Wong 1993) the

following expression.

szIl (ci) = c„friv, (5.27)

In the above equation, Csp and Nr are constants. The power spectral density can

be expressed in terms of the temporal frequency in Hz instead of the spatial frequency n

as follows.

S zii (f)= S z (S2)/ v (5.28)

Two sets of values for Cs/0 and Al", in Eq. (5.27) for the smooth highway and the

highway with gravel are reproduced from Wong (1993) and listed in Table 5.2. The

influence of liquid sloshing on ride quality can then be investigated by studying the PSDs

of the seat accelerations in both vertical and longitudinal directions.

Table 5.2 Values of Csp and Nr for the power spectral density function for various road

surfaces

(Wong, 1993)

Description Nr Csp (In 2 / cycles I m)

Smooth highway 2.1 4.8 x10-7

Highway with gravel 2.1 4.4 x10 -6

130


In this study, the vehicle is subjected to disturbances from road irregularities.

When the surface profde is regarded as a random function, it can be characterized by a

power spectral density function. The relationship between the power spectral density and

the spatial frequency for the road profile can be approximated by (Wong 1993) the

following expression.

In the above equation, Csp and Nr are constants. The power spectral density can

be expressed in terms of the temporal frequency in Hz instead o f the spatial frequency Q

as follows.

Two sets o f values for Csp and Nr in Eq. (5.27) for the smooth highway and the

highway with gravel are reproduced from Wong (1993) and listed in Table 5.2. The

influence o f liquid sloshing on ride quality can then be investigated by studying the PSDs

o f the seat accelerations in both vertical and longitudinal directions.

Table 5.2 Values o f Csp and Nr for the power spectral density function for various road

surfaces

(5.27)

(5.28)

(Wong, 1993)

Description Nr Csp (m2 / cycles / m)

Smooth highway 2.1 4.8 x 10"7

Highway with gravel 2.1 4.4 x 10~6

130


Table 5.3 Masses and moments of inertia of tractor semi-trailer components

(Ranganathan et al, 1993, Elmadany, 1987, Vaduri and Law, 1993)

Symbol Description Value Unit

Ins Mass of seat/driver 102 kg

Mel Mass of tractor 6440 kg

m20 Mass of empty semi-trailer 15000 kg

mtll Mass of tractor front axle 553.8 kg

mt12 Mass of tractor rear axle 1112 kg

mt 21 Mass of semi-trailer axle 1334.4 kg

Iri Moment of inertia of tractor 10000 kg • m 2

-120 Moment of inertia of empty semi-trailer 152000 kg • m 2

Table 5.4 Dimensions of tractor semi-trailer (m)


Symbol Value Symbol Value

as 0.5 a3 + a4 8.0

bs 0.5 b4 1.0

a1 1.6 a el 1

a 2 2.4 ae2 3

bt -0.3 ae3 5

a30 4.0 ae4 7

b20 0 D 2.44

131


Table 5.3 Masses and moments o f inertia o f tractor semi-trailer components



ms Mass of seat/driver 102 kg

mrX Mass o f tractor 6440 kg

m20 Mass o f empty semi-trailer 15000 kg

mm Mass of tractor front axle 553.8 kg

mtM Mass o f tractor rear axle 1112 kg

ml2i Mass of semi-trailer axle 1334.4 kg

In Moment of inertia o f tractor 10000 kg • m

120 Moment o f inertia of empty semi-trailer 152000 kg ■ m

Table 5.4 Dimensions o f tractor semi-trailer (m)


Symbol Value Symbol Value

as 0.5 8.0

bs 0.5 K 1.0

1.6 ael 1

a2 2.4 ae2 3

-0.3 aei 5

aio 4.0 ae4 7

20 0 D 2.44

131


Table 5.5 Spring and damping coefficients of tractor semi-trailer components



k,

k tll

Spring coefficient of seat

Spring coefficient of tractor front axle

4027

1560000

N / m

N 1m

kl2 Spring coefficient of tractor rear axle 5250000 N / m

k 121 Spring coefficient of semi-trailer axle 5250000 N / m

k11 Spring coefficient of tractor front suspension 357000 N / m

ki2 Spring coefficient of tractor rear suspension 630000 N / m

k 21 Spring coefficient of semi-trailer suspension 630000 N / m

kf Spring coefficient of artificial spring 100 N / m

cs Damping coefficient of seat 256.5 N 1(m / s)

CtIl Damping coefficient of tractor front axle 700 N /(m / s)

C,12 Damping coefficient of tractor rear axle 1200 N 1(ml s)

Ct21 Damping coefficient of semi-trailer axle 1200 N 1(m / s)

C11 Damping coefficient of tractor front suspension 11500 N /(m I s)

C12 Damping coefficient of tractor rear suspension 29000 N 1(m / s)

C21 Damping coefficient of semi-trailer suspension 29000 N 1(m / s)

Cf Damping coefficient of artificial damper 10 N /(m I s)


The investigation of ride quality of partially filled liquid cargo vehicles driving

along a straight lane on random uneven road can be carried out with computer simulation

132


Table 5.5 Spring and damping coefficients o f tractor semi-trailer components



K Spring coefficient o f seat 4027 N / m

K n Spring coefficient of tractor front axle 1560000 N tm

K n Spring coefficient o f tractor rear axle 5250000 N / m

kt2\ Spring coefficient of semi-trailer axle 5250000 N / m

kn Spring coefficient o f tractor front suspension 357000 N / m

k\2 Spring coefficient o f tractor rear suspension 630000 N / m

k2l Spring coefficient o f semi-trailer suspension 630000 N / m

k f Spring coefficient of artificial spring 100 N / m

cs Damping coefficient o f seat 256.5 N /(m / s )

cm Damping coefficient o f tractor front axle 700 N /(m/ s)

Ct\2 Damping coefficient o f tractor rear axle 1200 N /(m l s)

Ct2\ Damping coefficient o f semi-trailer axle 1200 N /(m / s )

Cu Damping coefficient o f tractor front suspension 11500 N /(m/ s)

cX2 Damping coefficient o f tractor rear suspension 29000 N !{m / 5 )

C 2l Damping coefficient of semi-trailer suspension 29000 N l(ml s)

Cf Damping coefficient of artificial damper 10 N l(ml s)


The investigation of ride quality o f partially filled liquid cargo vehicles driving

along a straight lane on random uneven road can be carried out with computer simulation

132


employing the power spectral density of the driver seat accelerations in both vertical and

horizontal directions. The PSDs are determined by substituting the different frequency

values (in the calculation, 200 points from 0.1 Hz to 100 Hz are used) directly into the

expressions of the desired functions, such as the vertical and horizontal accelerations,

after Eq. (5.26) is obtained.

The candidate tank includes four separated compartments. Therefore, four degrees

of freedom are needed to describe the liquid motion of the tank vehicle, and it is assumed

that the four parts of liquid in different compartments have the same fill levels. The

parameters used in the simulation are listed in Tables 5.3, 5.4 and 5.5 (Ranganathan et al,

1993, Elmadany, 1987, Vaduri and Law, 1993). The symbol D, which is not shown in

Figure 5.1, represents the diameter of the tank.

5.4.1 Frequency characteristics of partially filled liquid cargo vehicles

The natural frequencies of the vehicle system are calculated for different fill

levels under the given tank configurations. The frequencies due to the liquid sloshing

modes are in the frequency range of 0.1-0.7 Hz. Table 5.6 lists the frequencies for two

different fill levels, 30% and 70%. As shown in the table, the frequencies due to the

liquid sloshing modes increase with the increase of the liquid fill levels, while the

frequencies due to the bounce and pitch modes of the tractor and semi-trailer decrease

with the increase in the fill levels. It should be noted that when the fill level increases, the

increased mass and moment of inertia will make the frequencies of the bounce and pitch

modes of the tractor and semi-trailer lower. At the same time, the frequencies of liquid

sloshing will increase because of the higher fill level. This will generate a coupled effect

133


employing the power spectral density of the driver seat accelerations in both vertical and

horizontal directions. The PSDs are determined by substituting the different frequency

values (in the calculation, 200 points from 0.1 Hz to 100 Hz are used) directly into the

expressions o f the desired functions, such as the vertical and horizontal accelerations,

after Eq. (5.26) is obtained.

The candidate tank includes four separated compartments. Therefore, four degrees

o f freedom are needed to describe the liquid motion of the tank vehicle, and it is assumed

that the four parts o f liquid in different compartments have the same fill levels. The

parameters used in the simulation are listed in Tables 5.3, 5.4 and 5.5 (Ranganathan et al,

1993, Elmadany, 1987, Vaduri and Law, 1993). The symbol D, which is not shown in

Figure 5.1, represents the diameter o f the tank.

5.4.1 Frequency characteristics of partially filled liquid cargo vehicles

The natural frequencies of the vehicle system are calculated for different fill

levels under the given tank configurations. The frequencies due to the liquid sloshing

modes are in the frequency range of 0.1-0.7 Hz. Table 5.6 lists the frequencies for two

different fill levels, 30% and 70%. As shown in the table, the frequencies due to the

liquid sloshing modes increase with the increase o f the liquid fill levels, while the

frequencies due to the bounce and pitch modes of the tractor and semi-trailer decrease

with the increase in the fill levels. It should be noted that when the fill level increases, the

increased mass and moment o f inertia will make the frequencies o f the bounce and pitch

modes of the tractor and semi-trailer lower. At the same time, the frequencies of liquid

sloshing will increase because o f the higher fill level. This will generate a coupled effect

133


on the frequency response, which makes the frequency distribution in the range

considered quite different from that of rigid cargo vehicles. The above frequency range,

which is determined by the tank configurations and fill levels, is quite close to the

frequencies due to the bounce and pitch modes, as well as the seat mode. The effect of

the integrated multi-degree-of-freedom system subjected to liquid slosh can make the ride

quality of liquid cargo vehicles quite different from that of rigid cargo vehicles in the low

frequency domain.

Table 5.6 Natural frequencies (Hz) of tractor semi-trailer

Mode description Fill level: 30% Fill level: 70%

Axle wheel hop 11.58,10.57,9.57 11.58,10.57,9.57

Bounce and pitch 1.59,1.38,1.04 1.56,1.16,0.79

Seat 0.99 0.99

Liquid motion 0.57,0.52,0.52,0.52 0.68,0.62,0.62,0.62

Artificial spring 0.0086 0.0069

5.4.2 Ride performance under variable fill conditions

The PSDs of seat accelerations in both directions of the partially filled tank

vehicle at three different fill levels are presented in Figures 5.2 through 5.4. The vehicle

is traveling at the speed of 100 km/h. It is shown in these figures that the frequency range

that will have the largest influence on the ride quality is mainly in the range of 0.4-2 Hz.

The amplitude of vertical acceleration is higher than that of the horizontal acceleration

when the frequency is less than 2 Hz, while the amplitude of the horizontal acceleration is

134


on the frequency response, which makes the frequency distribution in the range

considered quite different from that of rigid cargo vehicles. The above frequency range,

which is determined by the tank configurations and fill levels, is quite close to the

frequencies due to the bounce and pitch modes, as well as the seat mode. The effect of

the integrated multi-degree-of-ffeedom system subjected to liquid slosh can make the ride

quality o f liquid cargo vehicles quite different from that o f rigid cargo vehicles in the low

frequency domain.

Table 5.6 Natural frequencies (Hz) o f tractor semi-trailer

Mode description Fill level: 30% Fill level: 70%

Axle wheel hop 11.58,10.57,9.57 11.58,10.57,9.57

Bounce and pitch 1.59,1.38,1.04 1.56,1.16,0.79

Seat 0.99 0.99

Liquid motion 0.57,0.52,0.52,0.52 0.68,0.62,0.62,0.62

Artificial spring 0.0086 0.0069

5.4.2 Ride performance under variable fill conditions

The PSDs o f seat accelerations in both directions o f the partially filled tank

vehicle at three different fill levels are presented in Figures 5.2 through 5.4. The vehicle

is traveling at the speed of 100 km/h. It is shown in these figures that the frequency range

that will have the largest influence on the ride quality is mainly in the range of 0.4-2 Hz.

The amplitude o f vertical acceleration is higher than that of the horizontal acceleration

when the frequency is less than 2 Hz, while the amplitude of the horizontal acceleration is

134


PS

Ds

of s

eat a

ccel

erat

ions

, (m

/s2)

2/H

z

100

10 2

10 4

10-6

104

Vertical ---------..

ISO2631 reduced comfort boundaries.....--' . ..

1Hr, .- ....0-1Hr

.0.

Horizontal

V = 100km/h Fill level: 20%

Empty Slosh

100

Frequency, Hz

101

Figure 5.2 Influence of variable fill levels (20%)

higher than that of the vertical acceleration when the frequency is greater than 2 Hz. For

the vertical seat acceleration, when the fill level increases from 20% (Figure 5.2) to 50%

(Figure 5.3), one more peak appears in the curve comparing with the curve for the empty

vehicle, due to the greater contribution of liquid sloshing. When the fill level further

increases to 80% (Figure 5.4), the frequencies of the corresponding peaks move from the

right to the left. The maximum value of the peaks occurs at the leftmost peak at a low fill

level (20%, Figure 5.2) and at the rightmost peak at a high fill level (80%, Figure 5.4).

The maximum value of the peaks moves from the left to the right because the increased

fill level has an effect on the sloshing mode, as well as on the bounce and pitch modes.

135


Nffi

C/3Uou<L>

13ooacdd>C/3twOC/3

QooPh

IS02631 reduced comfort boundaries.,'

1 HrlH rVertical

\: 8Hr 8Hr

Horizontal

EmptySlosh

V = lOOkm/h Fill level: 20%

10 10 '

Frequency, Hz


higher than that o f the vertical acceleration when the frequency is greater than 2 Hz. For

the vertical seat acceleration, when the fill level increases from 20% (Figure 5.2) to 50%

(Figure 5.3), one more peak appears in the curve comparing with the curve for the empty

vehicle, due to the greater contribution o f liquid sloshing. When the fill level further

increases to 80% (Figure 5.4), the frequencies of the corresponding peaks move from the

right to the left. The maximum value o f the peaks occurs at the leftmost peak at a low fill

level (20%, Figure 5.2) and at the rightmost peak at a high fill level (80%, Figure 5.4).

The maximum value of the peaks moves from the left to the right because the increased

fill level has an effect on the sloshing mode, as well as on the bounce and pitch modes.

135


The maximum value of the peaks of partially filled vehicles decreases with the increase

in the fill level. As the fill level increases, the horizontal acceleration increases and one

more peak appears. The peak due to liquid sloshing moves from the left to right. At

higher fill levels, the amplitude of horizontal seat acceleration is higher than that of the

empty vehicle when the excitation frequency is less than 1 Hz, and is lower than that of

the empty vehicle when the excitation frequency is greater than 1 Hz. Therefore, the ride

quality of the liquid cargo vehicle is dependent upon the influence of the fill conditions

on both acceleration amplitudes and frequency distributions.

PS

Ds

of s

eat a

ccel

erat

ions

, (m

/s2)

Z/H

z

10o

10 2

10

0-6

Vertical

ISO2631 reduced comfort boundaries,,,--

1Hr .,---'1Hr

Horizontal


Empty Slosh

10-1

100

Frequency, Hz

101


136


The maximum value of the peaks of partially filled vehicles decreases with the increase

in the fill level. As the fill level increases, the horizontal acceleration increases and one

more peak appears. The peak due to liquid sloshing moves from the left to right. At

higher fill levels, the amplitude o f horizontal seat acceleration is higher than that of the

empty vehicle when the excitation frequency is less than 1 Hz, and is lower than that of

the empty vehicle when the excitation frequency is greater than 1 Hz. Therefore, the ride

quality o f the liquid cargo vehicle is dependent upon the influence of the fill conditions

on both acceleration amplitudes and frequency distributions.

N

5

e_oca

Jin13ooca"ca<DwomQ cn Oh

IS02631 reduced comfort boundaries.*W l H r *

\ lHr

o0

Vertical

8Hr 8 Hr•210

•40

Horizontal

EmptySlosh•6

10 V - lOOkm/h Fill level: 50%

l o l

Frequency, Hz


136


PS

Ds

of s

eat a

ccel

erat

ions

, (m

/s2)

2/H

z

100

10 2

10.4

10 6

101

Vertical

ISO2631 reduced comfort boundaries

1Hr

Horizontal

- Empty

V = 100km/h - Slosh

Fill level: 80%

1 0o

Frequency, Hz

101


Past studies on the ride quality of heavy vehicles carrying rigid cargo have shown

that a heavy vehicle ride is most sensitive to excitations of low frequency modes in the

range of 1-8 Hz (Gillespie, 1985) or 0.9-5.8Hz (Hassan and McManus, 2002). At these

frequencies, modes such as body bounce, pitch and roll are actuated. As can be seen from

the above analysis for liquid cargo vehicles, the effect of coupling of liquid motion

modes and other rigid body modes makes the frequency characteristics of the PSDs of

seat accelerations different from those of rigid cargo vehicles in the very low frequency

range. It is already known that the low frequency vibration modes of heavy vehicles have

a greater influence on the driver's perception of ride than the high frequency modes. In

137


N

EC

C/3co• rH

Idf-H<L>*3oocdod<uCOocoQC/3CL,

IS02631 reduced comfort boundaries. V v lH r1 X

010

Vertical

8 Hr 8Hr■20

410

Horizontal

EmptySlosh•6

10 V = lOOkm/h Fill level: 80%

l o l10 10 10

Frequency, Hz

Figure 5.4 Influence o f variable fill levels (80%)

Past studies on the ride quality of heavy vehicles carrying rigid cargo have shown

that a heavy vehicle ride is most sensitive to excitations of low frequency modes in the

range of 1-8 Hz (Gillespie, 1985) or 0.9-5.8Hz (Hassan and McManus, 2002). At these

frequencies, modes such as body bounce, pitch and roll are actuated. As can be seen from

the above analysis for liquid cargo vehicles, the effect of coupling o f liquid motion

modes and other rigid body modes makes the frequency characteristics o f the PSDs of

seat accelerations different from those o f rigid cargo vehicles in the very low frequency

range. It is already known that the low frequency vibration modes o f heavy vehicles have

a greater influence on the driver’s perception of ride than the high frequency modes. In

137


the above figures, the PSDs of seat accelerations in both directions are compared with the

ISO 2631 reduced comfort boundaries. The ISO guide has been converted into equivalent

spectral values (Smith, 1976). As indicated by Figure 5.2(20%), the seat accelerations in

both vertical and horizontal directions are below ISO 1-hour reduced comfort boundaries,

and above 8-hour reduced comfort boundaries. As the fill level increases, the maximum

value of the vertical acceleration decreases and moves from left to right, which is shown

in Figure 5.3(50%) and Figure 5.4(80%). It can be found that the maximum vertical

accelerations are above the ISO 2631 1-hour reduced comfort boundary. The ride quality

becomes worse even though the acceleration level decreases.

5.4.3 Ride performance under variable liquid types

The influence of the density of the liquid being carried upon ride quality is

investigated by changing the density of the liquid while keeping the liquid at the same fill

level. Figure 5.5 gives the PSDs of the seat accelerations in both vertical and horizontal

directions for three different densities when the vehicle speed is 100km/h and the fill

level is 70%. The tendency of the change in both acceleration amplitudes and frequency

characteristics is the same as that of the change due to variable fill conditions. For the

vertical seat acceleration, as the density increases, one more peak appears. When the

density further increases, the frequencies of the corresponding peaks move from the right

to the left. The maximum value of the peaks moves from the left to the right, and the

maximum value of the peaks decreases with the increase in liquid density. As the density

increases, the horizontal acceleration increases, and one more peak appears. As indicated

in Figure 5.5, the ride quality becomes worse, and is above the ISO 2631 1-hour reduced

138


the above figures, the PSDs of seat accelerations in both directions are compared with the

ISO 2631 reduced comfort boundaries. The ISO guide has been converted into equivalent

spectral values (Smith, 1976). As indicated by Figure 5.2(20%), the seat accelerations in

both vertical and horizontal directions are below ISO 1-hour reduced comfort boundaries,

and above 8-hour reduced comfort boundaries. As the fill level increases, the maximum

value o f the vertical acceleration decreases and moves from left to right, which is shown

in Figure 5.3(50%) and Figure 5.4(80%). It can be found that the maximum vertical

accelerations are above the ISO 2631 1-hour reduced comfort boundary. The ride quality

becomes worse even though the acceleration level decreases.

5.4.3 Ride performance under variable liquid types

The influence of the density o f the liquid being carried upon ride quality is

investigated by changing the density o f the liquid while keeping the liquid at the same fill

level. Figure 5.5 gives the PSDs of the seat accelerations in both vertical and horizontal

directions for three different densities when the vehicle speed is lOOkm/h and the fill

level is 70%. The tendency of the change in both acceleration amplitudes and frequency

characteristics is the same as that of the change due to variable fill conditions. For the

vertical seat acceleration, as the density increases, one more peak appears. When the

density further increases, the frequencies o f the corresponding peaks move from the right

to the left. The maximum value of the peaks moves from the left to the right, and the

maximum value of the peaks decreases with the increase in liquid density. As the density

increases, the horizontal acceleration increases, and one more peak appears. As indicated

in Figure 5.5, the ride quality becomes worse, and is above the ISO 2631 1-hour reduced

138


comfort boundary. It can be concluded that the ride quality of the liquid cargo vehicle

could depend upon the influence of the type of liquid being carried on both acceleration

amplitudes and frequency distributions.

PS

Ds

of s

eat a

ccel

erat

ions

, (m

/s2)

2/H

z

100

10 2

10 4

1a 6

.-------F'Vertical

IS02631 reduced comfort boundaries ..-d-• ....

Horizontal

' , 1.03e3 kg/m „ ..

,''V = 1001an/h — 0.69e3 kg/m -'- Fill level: 70% 0.31e3 1(4,/m

1Hr ........" ... ...0 1Hr

1 0- 1

10

Frequency, Hz

101

Figure 5.5 Influence of liquid densities

5.4.4 Ride performance under variable vehicle speeds

When investigating the effect of vehicle speed on the ride quality, the RMS (root

mean square) values of the seat accelerations are calculated with respect to various

vehicle speeds. The RMS values are obtained by integrating the spectra in the frequency

range of 0.1 to 40 Hz. Figure 5.6 illustrates the relative RMS values that are obtained by

139


comfort boundary. It can be concluded that the ride quality o f the liquid cargo vehicle

could depend upon the influence o f the type of liquid being carried on both acceleration

amplitudes and frequency distributions.

N

5

G.2cdjB13oocdodvC/5

oC/5

Qc/oPh

Vertical

IS02631 reduced comfort boundaries -

: 8 Hr 8 Hr

Horizontal

-l10


1.03e3 kg/m" 0.69e3 kg/m3 0.3 le3 kg/m3

10 10

Frequency, Hz

Figure 5.5 Influence o f liquid densities

5.4.4 Ride performance under variable vehicle speeds

When investigating the effect o f vehicle speed on the ride quality, the RMS (root

mean square) values of the seat accelerations are calculated with respect to various

vehicle speeds. The RMS values are obtained by integrating the spectra in the frequency

range o f 0.1 to 40 Hz. Figure 5.6 illustrates the relative RMS values that are obtained by

139


dividing the RMS values at different speeds by the RMS values at the speed of 40 km/h.

The fill level used for the calculation is 70%. The curves in Figure 5.6 indicate that the

RMS values increase with the increase in vehicle speed. However, the rate of increase in

the horizontal acceleration is much larger than that in the vertical acceleration. Also, the

RMS values do not change too much in the vertical direction. This reveals that the ride

quality is significantly affected by the increase of the vehicle speed in the horizontal

direction.

Rel

ativ

e R

MS

val

ues

of s

eat a

ccel

erat

ions

Speed, km/h

Figure 5.6 Influence of vehicle speed

5.4.5 Ride performance under variable road conditions

The influence of the road roughness on ride quality is also examined by choosing

different road conditions. The parameters for the smooth highway and highway with

gravel are listed in Table 5.2. Figure 5.7 shows the PSDs of the seat accelerations in both

directions for the vehicle traveling over the smooth highway and the highway with gravel

140


dividing the RMS values at different speeds by the RMS values at the speed of 40 km/h.

The fill level used for the calculation is 70%. The curves in Figure 5.6 indicate that the

RMS values increase with the increase in vehicle speed. However, the rate o f increase in

the horizontal acceleration is much larger than that in the vertical acceleration. Also, the

RMS values do not change too much in the vertical direction. This reveals that the ride

quality is significantly affected by the increase o f the vehicle speed in the horizontal

direction.

t /3flO

<DOocamcawc/3CmOC/3<D

la>oo

<D.>Mca

&

2.5

HorizontalVertical

Fill level: 70%

40 80 100

Speed, km/h

Figure 5.6 Influence of vehicle speed

5.4.5 Ride performance under variable road conditions

The influence o f the road roughness on ride quality is also examined by choosing

different road conditions. The parameters for the smooth highway and highway with

gravel are listed in Table 5.2. Figure 5.7 shows the PSDs of the seat accelerations in both

directions for the vehicle traveling over the smooth highway and the highway with gravel

140


when the vehicle speed is 100 km/h and the fill level is 70%. As indicated by Figure 5.7,

both the vertical and horizontal accelerations are well above the ISO 2631 1-hour reduced

comfort boundaries on the highway with gravel. As can be expected, as the road quality

deteriorates, ride quality is degraded.

10 2

le Vertical

ISO2631 reduced comfortboundaries

4 , . .. . . . - : -, - - •

1Hr

„. ..1Hr _,

r".

v ial ,8Hr 1414'

V

'tHorizontal


Smooth highway Highway with gravel

101 100

Frequency, Hz

101

Figure 5.7 Influence of road condition

5.4.6 Ride performance of different seat suspensions

Other parameters of the tractor semi-trailer system such as the seat suspension can

also have an influence on the ride performance. Figure 5.8 presents the PSDs of the seat

accelerations when a much harder seat suspension has been selected. The frequency due

to the seat mode increases to 3.18 Hz when increasing the stiffness of the seat spring to

40270 N/m and keeping other parameters unchanged. The results are compared with the

141


when the vehicle speed is 100 km/h and the fill level is 70%. As indicated by Figure 5.7,

both the vertical and horizontal accelerations are well above the ISO 2631 1-hour reduced

comfort boundaries on the highway with gravel. As can be expected, as the road quality

deteriorates, ride quality is degraded.

N

C/3c_oc3

jo73ooCCj<uC/3Cmo1/3

pmPh

10

IS02631 reduced comfort boundaries

o Vertical lH rlH r

10

8Hr■210

■40

Horizontal

Smooth highway I Highway with gravel

■60 V = lOOkm/h"

Fill level: 70%l o l

10 10 10Frequency, Hz

Figure 5.7 Influence of road condition

5.4.6 Ride performance of different seat suspensions

Other parameters o f the tractor semi-trailer system such as the seat suspension can

also have an influence on the ride performance. Figure 5.8 presents the PSDs o f the seat

accelerations when a much harder seat suspension has been selected. The frequency due

to the seat mode increases to 3.18 Hz when increasing the stiffness of the seat spring to

40270 N/m and keeping other parameters unchanged. The results are compared with the

141


soft seat suspension at the same fill level, 70%, and the same vehicle speed, 100 km/h. It

can be seen that the peaks of the vertical seat acceleration shift to the right, and the

acceleration amplitudes increase significantly with the increase in seat stiffness. However,

the value of the peak due to the liquid motion decreases. Therefore, when the liquid

motion modes and the seat mode are separated, the contribution of the liquid motion will

decrease. However, other modes will be actuated, which makes the ride quality much

worse. The PSDs of the vertical acceleration are well above the recommended ISO 2631

1-hour reduced comfort boundary. At the same time, the horizontal seat accelerations are

nearly the same as those of the soft seat suspension. Therefore, the adjustment of seat

suspension has no effect on the horizontal seat acceleration.

PS

Ds

of s

eat a

ccel

erat

ions

, (m

/s2)

Z/H

z

102

100

10-2

1a 4

10 6

Vertical

ISO2631 reduced comfort boundaries

•-• 1Hr .------- ... -

1Hr

Horizontal

- Hard - Soft


. . . . . . . . . . . . ,

1 04

10 101

Frequency, Hz

Figure 5.8 Influence of seat suspension

142


soft seat suspension at the same fill level, 70%, and the same vehicle speed, 100 km/h. It

can be seen that the peaks of the vertical seat acceleration shift to the right, and the

acceleration amplitudes increase significantly with the increase in seat stiffness. However,

the value of the peak due to the liquid motion decreases. Therefore, when the liquid

motion modes and the seat mode are separated, the contribution o f the liquid motion will

decrease. However, other modes will be actuated, which makes the ride quality much

worse. The PSDs of the vertical acceleration are well above the recommended ISO 2631

1-hour reduced comfort boundary. At the same time, the horizontal seat accelerations are

nearly the same as those of the soft seat suspension. Therefore, the adjustment o f seat

suspension has no effect on the horizontal seat acceleration.

10'2

IS02631 reducedcomfort boundariesVertical

Horizontal


10-l

10o

10Frequency, Hz

Figure 5.8 Influence o f seat suspension

142


5.5 Summary

In this chapter, the ride performance of partially filled compartmented tank

vehicles has been investigated using a linearized multi-degree-of-freedom dynamic

model. The liquid motion in the partially filled tank is described as a linear spring-mass

model. The power spectral density of the vertical and horizontal seat accelerations has

been utilized to study the influence of liquid motion on ride quality. Since the natural

frequencies due to the liquid motion modes are in the very low range and are quite close

to those of the bounce and pitch modes, the effect of the coupling of these vibration

modes makes the ride quality quite different from that of the rigid cargo vehicles in the

very low frequency range. Simulation results show that the acceleration amplitudes and

frequency distributions are significantly affected by fill level, vehicle speed, road

condition, and the type of liquid being carried.

143


5.5 Summary

In this chapter, the ride performance o f partially filled compartmented tank

vehicles has been investigated using a linearized multi-degree-of-freedom dynamic

model. The liquid motion in the partially filled tank is described as a linear spring-mass

model. The power spectral density o f the vertical and horizontal seat accelerations has

been utilized to study the influence o f liquid motion on ride quality. Since the natural

frequencies due to the liquid motion modes are in the very low range and are quite close

to those of the bounce and pitch modes, the effect o f the coupling of these vibration

modes makes the ride quality quite different from that o f the rigid cargo vehicles in the

very low frequency range. Simulation results show that the acceleration amplitudes and

frequency distributions are significantly affected by fill level, vehicle speed, road

condition, and the type of liquid being carried.

143


CHAPTER 6 INFLUENCE OF NONLINEAR IMPACT ON

LIQUID CARGO TANK VEHICLES

6.1 Introduction

Although the mathematical method developed in Chapter 4 can be employed to

study the longitudinal dynamics for coupled liquid-vehicle-road systems under normal

driving conditions, it cannot deal with nonlinear impact problems during rough road

driving. The impact problem of liquid sloshing cannot be analytically solved. Other

methods, such as experimental studies (Kobayashi et al, 1989, Ye, 1990, Ye and Birk,

1990), numerical simulations (Arai et al, 1994, Kim, 2001), and equivalent mechanical

models (Ibrahim et al, 2001), have been employed in past investigations. Arai et al (1994)

developed a numerical method to simulate the impact load on tank walls and ceilings by

using the Marker-and-Cell (MAC) method for 3D rectangular containers. Due to the

discrete modeling of flow in the tank during the liquid impact on the tank ceiling, the

sudden change of boundary conditions from the free surface to the rigid one was detected

only at selected points in space. Therefore, the numerical solution of pressure time

history consisted of a series of isolated pulses that were generated at discrete grid points.

The authors used a numerical approach that mitigated this artificial discrete pressure

pulse in order to overcome this problem. Kim (2001) simulated sloshing with an impact

on the ceiling in 2D and 3D rectangular containers based on the Navier-Stokes equations,

finite difference method and SOLA scheme (Hirt et al, 1975). The concept of a buffer

zone was adopted near the tank ceiling, where a mixed boundary condition of rigid wall

and free surface was imposed before an impact. In order to mitigate a series of discrete

144


CHAPTER 6 INFLUENCE OF NONLINEAR IMPACT ON

LIQUID CARGO TANK VEHICLES

6.1 Introduction

Although the mathematical method developed in Chapter 4 can be employed to

study the longitudinal dynamics for coupled liquid-vehicle-road systems under normal

driving conditions, it cannot deal with nonlinear impact problems during rough road

driving. The impact problem of liquid sloshing cannot be analytically solved. Other

methods, such as experimental studies (Kobayashi et al, 1989, Ye, 1990, Ye and Birk,

1990), numerical simulations (Arai et al, 1994, Kim, 2001), and equivalent mechanical

models (Ibrahim et al, 2001), have been employed in past investigations. Arai et al (1994)

developed a numerical method to simulate the impact load on tank walls and ceilings by

using the Marker-and-Cell (MAC) method for 3D rectangular containers. Due to the

discrete modeling o f flow in the tank during the liquid impact on the tank ceiling, the

sudden change of boundary conditions from the free surface to the rigid one was detected

only at selected points in space. Therefore, the numerical solution o f pressure time

history consisted of a series o f isolated pulses that were generated at discrete grid points.

The authors used a numerical approach that mitigated this artificial discrete pressure

pulse in order to overcome this problem. Kim (2001) simulated sloshing with an impact

on the ceiling in 2D and 3D rectangular containers based on the Navier-Stokes equations,

finite difference method and SOLA scheme (Hirt et al, 1975). The concept o f a buffer

zone was adopted near the tank ceiling, where a mixed boundary condition o f rigid wall

and free surface was imposed before an impact. In order to mitigate a series of discrete

144


impacts, the computed signal was averaged over several time steps. The simulation

results showed a favourable agreement of impact pressures as well as the global fluid

motion.

However, the above methods can only be effective for a fairly gentle touch of

liquid on the ceiling. As pointed out by Arai et al (1994), in the case of violent sloshing,

such as nearly flat impact, the combined boundary condition was not effective. Generally

speaking, the numerical simulation of impact sloshing problems needs an extremely fine

mesh and time step, which will cause an extremely high demand for CPU time and

memory. For large amplitude excitations, the liquid becomes violent, and the stability of

numerical simulation can hardly be achieved. Impact is a quite challenging area in

sloshing studies.

An equivalent mechanical impact model was adopted by Pilipchuk and Ibrahim

(1997) and El-sayad et al (1999). Instead of pursuing the sloshing load by fluid

mechanics equations, the authors adopted an analogous pendulum model to simulate the

strongly nonlinear motion, where the nonlinearity was mainly due to the rapid velocity

changes associated with hydrodynamic pressure impacts of liquid motion close to the free

surface. The fluid free surface was modeled as a pendulum that could reach the walls of

the tank. Instead of using the equations of linear relationship with constraints, the authors

employed impact characteristic functions such as power nonlinearity with a higher

exponent to produce the same effect as the impact system. When the liquid model

described by the pendulum with hydrodynamic impact was included in the structural

system, the system model was obtained as multi-dimensional vibration equations with

both impact nonlinear terms and other nonlinear terms such as geometrical nonlinearities.

145


impacts, the computed signal was averaged over several time steps. The simulation

results showed a favourable agreement o f impact pressures as well as the global fluid

motion.

However, the above methods can only be effective for a fairly gentle touch of

liquid on the ceiling. As pointed out by Arai et al (1994), in the case o f violent sloshing,

such as nearly flat impact, the combined boundary condition was not effective. Generally

speaking, the numerical simulation of impact sloshing problems needs an extremely fine

mesh and time step, which will cause an extremely high demand for CPU time and

memory. For large amplitude excitations, the liquid becomes violent, and the stability of

numerical simulation can hardly be achieved. Impact is a quite challenging area in

sloshing studies.

An equivalent mechanical impact model was adopted by Pilipchuk and Ibrahim

(1997) and El-sayad et al (1999). Instead of pursuing the sloshing load by fluid

mechanics equations, the authors adopted an analogous pendulum model to simulate the

strongly nonlinear motion, where the nonlinearity was mainly due to the rapid velocity

changes associated with hydrodynamic pressure impacts o f liquid motion close to the free

surface. The fluid free surface was modeled as a pendulum that could reach the walls of

the tank. Instead of using the equations of linear relationship with constraints, the authors

employed impact characteristic functions such as power nonlinearity with a higher

exponent to produce the same effect as the impact system. When the liquid model

described by the pendulum with hydrodynamic impact was included in the structural

system, the system model was obtained as multi-dimensional vibration equations with

both impact nonlinear terms and other nonlinear terms such as geometrical nonlinearities.

145


This method has been used to study the dynamics of elastically supported elevated water

towers by the multiple scale method (El-sayad et al, 1999) and a special saw-tooth time

transformation technique (Pilipchuk and Ibrahim, 1997).

For liquid motion in horizontal cylindrical tanks, it is already known that the free

liquid surface motion and the liquid impact can be more severe longitudinally than

laterally if no transverse baffles are introduced (Ibrahim et al, 2001). An experimental

study was conducted by Kobayashi et al (1989) to determine the liquid natural

frequencies and the resultant slosh forces under small and large slosh wave heights in

horizontal cylindrical tanks, both laterally and longitudinally. The measured longitudinal

slosh forces, including the impulsive forces, were much larger than the calculated ones of

the linear theory. Compared with the stability analysis and directional response

characteristics of heavy vehicles carrying liquid cargo in the roll plane vehicle model, the

influence of liquid sloshing in the pitch plane has only been investigated in limited

studies. Only linear mechanical models were employed to represent the complex liquid

motion, and constant deceleration braking characteristics were considered. Rough road

conditions have never been included in the past investigations.

Equivalent mechanical models can be used for liquid cargo vehicle systems in

simulating vehicle behaviour. In this chapter, a nonlinear impact mechanical model for

liquid sloshing in partially filled liquid tank vehicles has been developed to investigate

the longitudinal dynamic characteristics of tank vehicles during typical straight-line

driving. The dynamic fluid motion within the tank has been modeled by utilizing a

mechanical system that can describe the behaviour of the liquid motion as a spring-mass

sloshing model with an impact subsystem for longitudinal oscillations. Computer

146


This method has been used to study the dynamics of elastically supported elevated water

towers by the multiple scale method (El-sayad et al, 1999) and a special saw-tooth time

transformation technique (Pilipchuk and Ibrahim, 1997).

For liquid motion in horizontal cylindrical tanks, it is already known that the free

liquid surface motion and the liquid impact can be more severe longitudinally than

laterally if no transverse baffles are introduced (Ibrahim et al, 2001). An experimental

study was conducted by Kobayashi et al (1989) to determine the liquid natural

frequencies and the resultant slosh forces under small and large slosh wave heights in

horizontal cylindrical tanks, both laterally and longitudinally. The measured longitudinal

slosh forces, including the impulsive forces, were much larger than the calculated ones of

the linear theory. Compared with the stability analysis and directional response

characteristics o f heavy vehicles carrying liquid cargo in the roll plane vehicle model, the

influence o f liquid sloshing in the pitch plane has only been investigated in limited

studies. Only linear mechanical models were employed to represent the complex liquid

motion, and constant deceleration braking characteristics were considered. Rough road

conditions have never been included in the past investigations.

Equivalent mechanical models can be used for liquid cargo vehicle systems in

simulating vehicle behaviour. In this chapter, a nonlinear impact mechanical model for

liquid sloshing in partially filled liquid tank vehicles has been developed to investigate

the longitudinal dynamic characteristics o f tank vehicles during typical straight-line

driving. The dynamic fluid motion within the tank has been modeled by utilizing a

mechanical system that can describe the behaviour o f the liquid motion as a spring-mass

sloshing model with an impact subsystem for longitudinal oscillations. Computer

146


simulation of the tank vehicle under typical rough road conditions has been performed by

incorporating the forces and moments caused by liquid motion into the pitch plane

vehicle model. The fifth wheel loads and the normal axle loads have been computed

using the mechanical system approach in order to investigate the influence of liquid

motion.

6.2 Nonlinear impact model of liquid sloshing

Equivalent mechanical models have been proven to be simple and effective ways

to describe the liquid slosh because the equations of motion for point masses and rigid

bodies could be included more readily in the overall vehicle model than the equations for

a continuously deformable medium such as fuel oil. Linear analytical representations of

liquid sloshing by the pendulum theory and spring-mass theory were usually employed in

liquid cargo vehicle research. In this section, the spring-mass model including nonlinear

impact effect has been developed to investigate the dynamics of liquid sloshing under

large amplitude situations.

For sloshing in the longitudinal direction of horizontal cylindrical tanks,

Ranganathan et al (1994) developed a summation technique to calculate the parameters of

the spring-mass model according to theory of the slosh response in rectangular tanks.

Kobayashi et al (1989) established an equivalent calculation method to make use of the

theory of slosh response in rectangular tanks in seismic design research. Calculated

results of natural frequencies are in good agreement with the experimental ones. This

calculation method was introduced in Section 4.4.1 and has been applied in studying the

ride quality problem for tractor semi-trailer systems in Chapter 5.

147


simulation o f the tank vehicle under typical rough road conditions has been performed by

incorporating the forces and moments caused by liquid motion into the pitch plane

vehicle model. The fifth wheel loads and the normal axle loads have been computed

using the mechanical system approach in order to investigate the influence o f liquid

motion.

6.2 Nonlinear impact model of liquid sloshing

Equivalent mechanical models have been proven to be simple and effective ways

to describe the liquid slosh because the equations o f motion for point masses and rigid

bodies could be included more readily in the overall vehicle model than the equations for

a continuously deformable medium such as fuel oil. Linear analytical representations of

liquid sloshing by the pendulum theory and spring-mass theory were usually employed in

liquid cargo vehicle research. In this section, the spring-mass model including nonlinear

impact effect has been developed to investigate the dynamics o f liquid sloshing under

large amplitude situations.

For sloshing in the longitudinal direction o f horizontal cylindrical tanks,

Ranganathan et al (1994) developed a summation technique to calculate the parameters of

the spring-mass model according to theory o f the slosh response in rectangular tanks.

Kobayashi et al (1989) established an equivalent calculation method to make use o f the

theory o f slosh response in rectangular tanks in seismic design research. Calculated

results of natural frequencies are in good agreement with the experimental ones. This

calculation method was introduced in Section 4.4.1 and has been applied in studying the

ride quality problem for tractor semi-trailer systems in Chapter 5.

147


When the liquid motion is described by a linear mechanical model, the equation

of motion is:

d 2x,, dx,imn1 dt 2 +CO dt + KnIx n = —Mnla x (6.1)

In this equation, innl, cn1 and k,,1 are equivalent mass, damping and stiffness of the

equivalent mass-spring system, and zn is the displacement of the mass. ax is the applied

acceleration on the equivalent mass.

Under severe conditions, nonlinear liquid motion will appear due to rapid velocity

changes associated with hydrodynamic pressure impacts. For example, this strongly

nonlinear motion could happen when the exciting frequency is equal to or near the

sloshing frequency. The resulting forces and moments caused by the hydrodynamic

pressure impact will affect the responses of the tank vehicles, and are extremely

important in design of the supporting structures and internal components of the vehicle

tanks. Pilipchuk and Ibrahim (1997) studied the nonlinear liquid sloshing impact in

moving rectangular containers. The liquid sloshing was modeled by a pendulum

describing impacts with the container walls. The mathematic model included the

constraint that 18„l < Ono when considering the pendulum and the container walls as rigid

bodies, where on was the pendulum angle, and e, was the angular when the pendulum

reached the container walls. To solve the problem, impact characteristic functions could

be used to produce the same effect as the linear equation with constraints.

Similarly, it is assumed that Ixa l< x,,0 is the constraint for the linear spring-mass

system for liquid slosh, where zno = w/2 . w is the longitudinal length of the tank

148


When the liquid motion is described by a linear mechanical model, the equation

o f motion is:

d 2x n dxnm nl — j - + cnX — + k nXx n = - m nlax (6.1)

dt dt

In this equation, mn\,c„\ and kn\ are equivalent mass, damping and stiffness of the

equivalent mass-spring system, and x„ is the displacement o f the mass. ax is the applied

acceleration on the equivalent mass.

Under severe conditions, nonlinear liquid motion will appear due to rapid velocity

changes associated with hydrodynamic pressure impacts. For example, this strongly

nonlinear motion could happen when the exciting frequency is equal to or near the

sloshing frequency. The resulting forces and moments caused by the hydrodynamic

pressure impact will affect the responses o f the tank vehicles, and are extremely

important in design of the supporting structures and internal components o f the vehicle

tanks. Pilipchuk and Ibrahim (1997) studied the nonlinear liquid sloshing impact in

moving rectangular containers. The liquid sloshing was modeled by a pendulum

describing impacts with the container walls. The mathematic model included the

constraint that \en | < enQ when considering the pendulum and the container walls as rigid

bodies, where Qn was the pendulum angle, and was the angular when the pendulum

reached the container walls. To solve the problem, impact characteristic functions could

be used to produce the same effect as the linear equation with constraints.

Similarly, it is assumed that \xn \ < x n0 is the constraint for the linear spring-mass

system for liquid slosh, where xn0=w/2 . w is the longitudinal length o f the tank

148


compartment. The potential energy function is (Pilipchuk and Ibrahim 1997) expressed

as:

ri(xj=bnxn0 xn

2nn

-\ 2n„

(6.2)

In this equation, nn >> 1 is a positive integer and bn is a positive constant parameter. The

impact force is:

Fm =

dInxn

) =

bn

„ (6.3)

Thus the equation of motion (6.1) can be modified to include the impact effect.

d2 xn dxn—

nin1 dt 2 + cn1 dt + kn,Xn bn

( xn

\Xn0

\ 2n„-1

= —Mnl a x

The motion equation can be expressed in the non-dimensional form.

2 dx

c 24-a) + con x + qnxc2 n 1 = — ax

dt n

dx

dt xno

(6.4)

(6.5)

According to Dodge (Abramson and Silverman, 1966), even considering the

energy dissipation due to free surface effects, the total dissipation or damping is so small

in unbaffled tanks that practically no limit is placed on the slosh amplitude at resonance.

By neglecting the damping effect, the interaction between the liquid and the tank walls

can be expressed as:

Fs = Mn1Can2Xn bnXc 11" -1 (6.6)

bn =gmmnixn0 (6.7)

xc = Xn0

xn

149

(6.8)


compartment. The potential energy function is (Pilipchuk and Ibrahim 1997) expressed

as:

/ \2«„

2 n(6 .2)

n V nO /

In this equation, nn » 1 is a positive integer and b„ is a positive constant parameter. The

impact force is:

F _ d n k ) _ bim j r,dx„

(6.3)

Thus the equation o f motion (6.1) can be modified to include the impact effect.

md 2x dx

n\ + C n\ ------ + knXXn + bnd t 2 nx dt " "

/ ' \ 2 « „ - l

I ’± .

\ X nO J

~ mnxax (6.4)

The motion equation can be expressed in the non-dimensional form.

d 2x, dxc— f + 2 f r n —d t 1

2 2n -1 a rM + ^ n Xc +flnXc" = -----“dt X«0

(6.5)

According to Dodge (Abramson and Silverman, 1966), even considering the

energy dissipation due to free surface effects, the total dissipation or damping is so small

in unbaffled tanks that practically no limit is placed on the slosh amplitude at resonance.

By neglecting the damping effect, the interaction between the liquid and the tank walls

can be expressed as:

F s = m n \ ( ° 2n X n + ( 6 - 6 )

K = (6.7)

x„ = ■x,

(6 .8)n 0

149


con , ‘

icil

Inn] (6.9)

6.3 Tank vehicle model in the pitch plane

In this section, the longitudinal tank vehicle dynamics model will be established

by integrating the equivalent mechanical liquid sloshing model developed in Section 6.2

into the pitch plane model of partially filled tank vehicles under rough road conditions.

Figure 6.1 shows the model of a tractor semi-trailer in the pitch plane. The tank has Nn

compartments. It is assumed that the left and right wheels of the vehicle experience

identical excitations. The tractor and the semi-trailer structure without payload are treated

as rigid bodies. The tractor and semi-trailer are connected by the fifth wheel, which is

modeled as a pin connection. The aerodynamics effect and suspension dynamics are not

included in this model for simplicity (Wong, 1993, Ranganathan et al, 1994, Rumold,

2001).

Vn

an

(X,., Yr) (Xm , Y„,) (Xf , Yf)

Figure 6.1 Tractor semi-trailer model and motion profile

6.3.1 Horizontal accelerations of the tractor and the tank on rough roads

In past investigations of lateral stability analysis of tank vehicles, the liquid cargo

was usually assumed to be subjected to constant steer inputs or constant lateral

150


6.3 Tank vehicle model in the pitch plane

In this section, the longitudinal tank vehicle dynamics model will be established

by integrating the equivalent mechanical liquid sloshing model developed in Section 6.2

into the pitch plane model of partially filled tank vehicles under rough road conditions.

Figure 6.1 shows the model of a tractor semi-trailer in the pitch plane. The tank has Nn

compartments. It is assumed that the left and right wheels o f the vehicle experience

identical excitations. The tractor and the semi-trailer structure without payload are treated

as rigid bodies. The tractor and semi-trailer are connected by the fifth wheel, which is

modeled as a pin connection. The aerodynamics effect and suspension dynamics are not

included in this model for simplicity (Wong, 1993, Ranganathan et al, 1994, Rumold,

2001).

Vn

(Xr,Yr) (Xm,Ym) (Xf , Yj)

Figure 6.1 Tractor semi-trailer model and motion profile

6.3.1 Horizontal accelerations of the tractor and the tank on rough roads

In past investigations o f lateral stability analysis o f tank vehicles, the liquid cargo

was usually assumed to be subjected to constant steer inputs or constant lateral

150


accelerations, while in the braking characteristics analysis, the liquid cargo was subjected

to constant decelerations. One of the main factors that could cause severe liquid sloshing

in partially filled tank vehicles, the rough road condition, has never been included. In this

study, the liquid cargo is assumed to be subjected to the vehicle motion caused by rough

road conditions. Assume the tank vehicle is traveling over the rough road at a constant

horizontal speed U. The global coordinates, X, and Yn, are used to present the changes of

positions of the tank vehicle within the running time. The local coordinates, x, and yn, as

shown in Figure 6.1, are established on the tractor and the tank. If only one term

expression is considered, the road contour is approximated by:

=a„ 1 —cos t „WL,

(6.10)

In this equation, an is the amplitude of the road contour, and WL is the wavelength

of the road contour. Assume an and fin are the angles of the tank and the tractor with

respect to the X, direction.

= arctan Y — Y

X. — X,

— Y /3 „ arctan

X — X„,

(6.11)

(6.12)

In the above equations, (, cf,Yf), Y,n) and (Xr, Yr) are the tire contact points of the

tractor front axle, tractor rear axle and the semi-trailer axle, respectively. When

determining the coordinate values in the above equations, the following geometric

relationships should be satisfied.

AI f + frf = Lf (6.13)

AAA'. — X,Y+(Y. —1c)2 =1,

151

(6.14)


accelerations, while in the braking characteristics analysis, the liquid cargo was subjected

to constant decelerations. One of the main factors that could cause severe liquid sloshing

in partially filled tank vehicles, the rough road condition, has never been included. In this

study, the liquid cargo is assumed to be subjected to the vehicle motion caused by rough

road conditions. Assume the tank vehicle is traveling over the rough road at a constant

horizontal speed Un. The global coordinates, X„ and Yn, are used to present the changes of

positions o f the tank vehicle within the running time. The local coordinates, xn and y„, as

shown in Figure 6.1, are established on the tractor and the tank. If only one term

expression is considered, the road contour is approximated by:

Y'=a„' 2« r . Al - c o s (6.10)

WL

In this equation, an is the amplitude o f the road contour, and Wl is the wavelength

o f the road contour. Assume a„ and /?„ are the angles o f the tank and the tractor with

respect to the X n direction.

an = arctan — — — (6.11)

P n = arctan — ---- — (6.12)x f - x m

In the above equations, (Xf,YJ), (Xm,Ym) and (Xr, Yr) are the tire contact points o f the

tractor front axle, tractor rear axle and the semi-trailer axle, respectively. When

determining the coordinate values in the above equations, the following geometric

relationships should be satisfied.

f a , - x j +(¥,-¥„? =L{ (6.13)

y l (Xm - X r f + (Ym- Y r )2 =lr (6.14)

151


In the above equations, Lf and lr are the geometric dimensions shown in Figure 6.2

and Figure 6.3. Since an and fl are usually very small, the following relations can be

written for the tractor and the tank by neglecting the centrifugal effect.

Uf sin an =Vi cosan (6.15)

Uf =Uf cos an +Ili sin an (6.16)

Ur sin fi,, =v, cos fin (6.17)

u,. = U,. cos /fn + V, sin fin (6.18)

The subscripts f and r represent the tractor and the tank, respectively. Thus the

horizontal accelerations of the tractor and the tank in the local coordinate system can be

expressed by these two equations.

= du

f = U sin fi

" fx dt

ncos' 16 dt

a = dur = U sin a n dan

, dt n COS' a n dt

(6.19)

(6.20)

6.3.2 Equations of the semi-trailer

Following the modeling methods for tractor semi-trailers in the pitch plane by

Ranganathan et al (1994), Wong (1993) and Rumold (2001), the mathematical tank

vehicle model is developed in this section. With the consideration of rough road

conditions as the cause of excitation for liquid sloshing, the tank vehicle model is

established on a curved road, which is approximated by a cosine function, instead of a

flat road. Figure 6.2 shows the loading configuration of the semi-trailer. Only the

152


In the above equations, Z/and lr are the geometric dimensions shown in Figure 6.2

and Figure 6.3. Since an and are usually very small, the following relations can be

written for the tractor and the tank by neglecting the centrifugal effect.

Uf sina„ = Vf cosan (6.15)

uf - Uf cos an + Vf sin an (6.16)

Ur sin p n = Vr cos /?„ (6.17)

ur = Ur cos Pn + Vr sin /?„ (6.18)

The subscripts / and r represent the tractor and the tank, respectively. Thus the

horizontal accelerations of the tractor and the tank in the local coordinate system can be

expressed by these two equations.

d u f s in /? dBa fx= — L = Un----- fa .— Bjl /6 19)dt cos Pn dt

dur TT s in«„ da„arX = - r - = U n (6-20)

dt cos a„ dt

6.3.2 Equations of the semi-trailer

Following the modeling methods for tractor semi-trailers in the pitch plane by

Ranganathan et al (1994), Wong (1993) and Rumold (2001), the mathematical tank

vehicle model is developed in this section. With the consideration of rough road

conditions as the cause o f excitation for liquid sloshing, the tank vehicle model is

established on a curved road, which is approximated by a cosine function, instead of a

flat road. Figure 6.2 shows the loading configuration of the semi-trailer. Only the

152


mechanical model of the jth compartment is shown in the figure. Let cor be the tangent

angle of the road profile at the tire-ground contact point of the semi-trailer axle.

27ran 27rX,tan g = sin

Wr. (6.21)

The following three equations are established according to the equilibrium of the

forces in X„ and Y,, direction and the equilibrium of the moment.

a — Fr sin q),. — Rr cos gor + Fx — a cos a,,—EWoi —1--x cos a„+ EFs; cos a n = 0

g j=1 g J=1

Fr cosyor — Rr sin g + Fy — TY, — EWoi — W arx sin a„ j=1 J=1

. —E W f a sin a„ +ZF s, sin a„= 0 o ./=-1 g f=1

(FY cosa„— Fx sin a n k+1,.)—(Fx cosa„+ Fy sin a n )h,,„

+ Fr (cosgor cosa„+ sin g sin a n ) lg +R,.(cosgor sin a,— sin go,. cos a n ) lg

+(W, sin a n Xiiw +h2 )+W, + h2 )— (W, cos a n k g +1,,)

+ E(wo; sin a n )(11„,+1101)+ZW0i —a"(17,„+hoi)—E(w0; cos a n ) 1 1=1 j=1 j=1 N„ N„ N„

+ E(wii sin a ii(h 1 j) — E(w11cos a ,,X1 + x j) — (11 „ + ) = 0 i=i i=i j=i

(6.22)

(6.23)

(6.24)

Rr = frFr (6.25)

WO j = nin0j • g W lj = innlj • g (6.26)

The jth sloshing force on the tank, Fsi , can be found by solving the following

equations obtained from Eq. (6.5) to Eq. (6.9).

153


mechanical model o f the y'th compartment is shown in the figure. Let cpr be the tangent

angle o f the road profile at the tire-ground contact point of the semi-trailer axle.

tan (pr = sin 'n^ r (6.21)WL WL

The following three equations are established according to the equilibrium of the

forces in X n and Yn direction and the equilibrium of the moment.

a N" a N"- Fr s in (pr - Rr co s (pr + F X - fVt - nLc o sa n - 'Y 'fV 0, - SLc o sa n + Y f . c o s« n = 0

g M g M

(6 .22)

N „ N „

F, cos <?, - R , sin <pr + F , - W, - £ ,W0, - £ ,Wt, - W, ^ ;sin ,a ,7 =1 7 = 1 g

(6-23)" Z W oj — sin «„ + Z F s jsin a n = 0

7 = 1 g 7 = 1

far cos a n - F x s \n a n cos a n + Fy sin a n )hw

+ Fr (cos (pr cos a n + sin (pr sin a n) / + Rr (cos (pr sin a n - sin (pr cos a n) /

+ (Wt sin a n \ h w + h2 ) + Wt ^ (hw + h2) ■- (Wt cos a n \ l g + lw)§

N „ N n N „

+ Z S'm a n l K + f h j ) + ' L W 0 j — ( K + C 0 S a n) h7 = 1 7 = 1 g 7 = 1

N n N „ N „

+ £ (fV{J sin a n\ h w + hXJ)~ Z ( K j co sa n \ l j + x}) - £ K { K + K j ) = 07 = 1 7 = 1 7 = 1

(6.24)

R r = f rFr (6.25)

K j = ™„oj ■ g > W\j = mnXJ ■ g (6.26)

The yth sloshing force on the tank, Fsj, can be found by solving the following

equations obtained from Eq. (6.5) to Eq. (6.9).

153


lg

Figure 6.2 Loading configuration of the semi-trailer

2 d X d.Xci 2n-1 a rx

COnix ci + niXci — dt 2

+ 24"Joni dt X nO j

2n-1 F sj = Mn1 j CO2nj Xcj b nj X cj

b nj = njni nlj Xn0j

X njX ci =

X nO j

To solve Eq. (6.22) to Eq. (6.24), the following quantities are introduced.

154

(6.27)

(6.28)

(6.29)

(6.30)


r

Figure 6.2 Loading configuration o f the semi-trailer

d x . d x . , , , a - + 2C (D __ - + a> x + n x = __—dt1 ^ * dt + a ’ + '! Xmj

b nj = I n j ^ n X j ^ n O j

XnjX c j = —

X n O j

To solve Eq. (6.22) to Eq. (6.24), the following quantities are introduced.

154

(6.27)

(6.28)

(6.29)

(6.30)


=(w, sin a„ )02„, + h2 )+ W, + 10—(47,cosan k g +1,„)

+1(W o j sin an )(h,,+ hc,j)+1Wo jarx (h„, + I/01 )— E(wojeosan) lJ

J=1 g J=1

+I(W1 j sin an k,+11,j)-1(wl cosa,,Xli +xj)-1F si(k+h,j) j=1 1=1

= sin cor + f r coso,.

I N„ N„

.E. = wt +lwoiarx

cosan —IFsi cosa„ J--1 j g j=1

=—cosq,r + f r sin gis,

N„ , N„

=wt +DK; -kwu)+ W +Iwo.;j=1 j=1

E =(1,.+1g )cosan — hw sin an

= —(1, + lg )sin an —h„,cosan

(6.31)

(6.32)

(6.33)

(6.34)

N„

arx sin an —IFs., sin a n (6.35) J=1

(6.36)

(6.37)

G = [(cos car cos an + sin car sin an )+ f. (cos cor sin an co,. cos (in )]lg (6.38)

The semi-trailer axle normal force, the longitudinal and vertical fifth wheel load

can be obtained.

Fr = —ff•F—D•k—k A•F+C•E+G

(6.39)

Fx =71•Fr +B (6.40)

Fy = C • Fr +D (6.41)

155


R = (Wt s ina n\ h w + h2)+ ^ — (K + h2)-(jV, co sa n g + lw)g

+ Z K y sin a j h w + h0j)+ Y W 0J^ (hw + /*>,-)- Z K y co s««) h (6.31)j = 1 J = 1 £ / = !

N,

+ Z K Sin« J ^ + 0 ~ Z K ' c o sa J / . + X7) - Z ^ ( ^ + ^y)y=i y=i 7=i

B =N„ N„

wt + i w0j — cos«„ - I X cosa„7 = 1

C = - c o s (pr + f r s in <pr

N n f N n \ " n

D =Wt + Z (^ o y + W,j)+ Wt + Y W 0j — s i n - Z Fsjs i na nV 7 = 1 ) g7 = 1 7 = 1

E = ( l r + lg )cos a „ - hw sin a„

^ = - ( /r + ^ ) sin« i , - ^ cos a i.

(6.32)

(6.33)

(6.34)

(6.35)

(6.36)

(6.37)

G = [(cos <pr cos a n + sin cpr sin a n) + f r (cos (pr sin a n - sin (pr cos a n )]/g (6.38)

The semi-trailer axle normal force, the longitudinal and vertical fifth wheel load

can be obtained.

- B F - D E - R r ~ A - F + C -E + G

Fx = A • Fr + B

FV= C F + D

(6.39)

(6.40)

(6.41)

155


6.3.3 Equations of the tractor

The loading configuration of the tractor is shown in Figure 6.3. Let yof be the

tangent angle of the road profile at the tire-ground contact point of the tractor front axle,

and co,,, be the tangent angle of the road profile at the tire-ground contact point of the

tractor rear axle.

Yn

fin

Figure 6.3 Loading configuration of the tractor

tan go = 27-ca

sin 27TXf f WL WL

(6.42)

tan corn= 2ga

sin 27-rX.

(6.43) W W L L

For most of the tractor semi-trailers, the tractor rear wheel is driven. The

following equations are established according to the equilibrium of the forces in X, and

Yn direction and the equilibrium of the moment.

156


6.3.3 Equations of the tractor

The loading configuration of the tractor is shown in Figure 6.3. Let q> be the

tangent angle o f the road profile at the tire-ground contact point o f the tractor front axle,

and (pm be the tangent angle of the road profile at the tire-ground contact point of the

tractor rear axle.

<Pf

Figure 6.3 Loading configuration o f the tractor

2 m . 2.7lKftane?, = s in (6.42)

Vr WL WL

2 m . 27tXmtan (pm = s in ^ (6.43)

WL WL V 7

For most o f the tractor semi-trailers, the tractor rear wheel is driven. The

following equations are established according to the equilibrium of the forces in X n and

Yn direction and the equilibrium of the moment.

156


— Ff sin ypf — Rf cos col — Fx — Fm sin yom — Wk —arf cos fin + (Fd — Rm )cos yom = 0

(6.44)

Ff cos yo f — Rf sin yof — WK — Fy F„, cos q)„, — WK s.fx

sin fin + (Fd — Rm )sin q)„, = 0

(6.45)

(Fx cos fin + Fy sin fin + WK :2- H2 + (WK sin 13,,)H2 — O K cos )3 n )L,„ g

+[(F f cos yo f R f sin y)f )cos /3n + (Ff sin q)./. + R f cos q)./ )sin f = 0

(6.46)

= f„,Fm (6.47)

R f = ff Ff (6.48)

The tractor front and rear axle normal forces and driving force can be obtained as

follows.

Ft =

(Fx cos fin + Fy sin fin + (WK sin fin )H2 — (Wk cos fin )L,„ + Wk a

fx H2g

[(cos yof — f f sin yof )cos fin + (sin yof + f f cos yof )sin fin iLf

F„, = —{(Ff sin yof sin yom + Rf cos sin q). + Ff cos yof cos yom — Rf sin cos

+ (cos fi sin yom —sin fi cos yom )Wk 2c—fx

+ Fx sin q,„, —(Wk +Fy )coscpng

Fx + (F f sin gof R f COS f )± Fm sin yom + Wk afx cos i n

Fd = + cos q)„,

157

(6.49)

(6.50)

(6.51)


(6.44)

Ff co s<pf - R f sin(pf - W K - F Y + Fm co s<pm - WK ^ s i n J3n + {Fd - Rm)s in (pm = 0§

(6.45)

(Fx cos A + F, sin A )f f , + ^ t f 2 + (WK sin A )ff 2 - cos A Kg

[(/*) cos (pf - R f sin <pf )cos Pn + {Ff sin (pf + Rf cos (pf )sin P ^ f = 0+

R ,n = f m F m

(6.46)

(6.47)

R f = f f F f (6.48)

The tractor front and rear axle normal forces and driving force can be obtained as

follows.

( f , cos A + Fr sin A )H , + (WK sin A ) # 2 - cos A ) A + ^ ^F f -

[(cos (pf - f f sin (pf )cos (3n + (sin <pf + f f cos <pf )sin P n ]Lfg

(6.49)

Fm = -[(.F} sin (pf sin (pm + Rf cos (pf sin (pm + Ff cos (pf cos cpm - Rf sin (pf cos (pm)

+ (cos p sin cpm - sin p cos <pm )Wk — + FX sin cpm - (Wk + Fy) cos <pmg

(6.50)

Fx + (1Ff sin <pf + Rf cos <pf )+ Fm sin (pm + Wk ^ c o s Pn ______________ _________ _ ________ s __

c o s ^(6.51)

157



The longitudinal dynamic response of partially filled liquid cargo tank vehicles

under rough road conditions has been investigated. The tank vehicle is traveling from the

flat road to the rough road at a constant horizontal speed of 80 km/h at the starting time.

The circular cylindrical tank is divided into four compartments. The accelerations of the

tractor and the tank, the normal axle loads and the fifth wheel loads have been computed

using the model developed in Section 6.3. The simulation parameters of the road

conditions and the tank vehicle are listed in Table 6.1. To solve the nonlinear equation of

the sloshing model, the Runge-Kutta method has been employed. The nonlinear impact

model is compared with the linear spring-mass sloshing model. The dynamic responses,

including the fifth wheel loads and normal axle loads, are analyzed to study the influence

of the liquid sloshing under rough road conditions.

Table 6.1 Simulation parameters

Pry 75 m an: 1.5 m W, : 48069 N Wk : 52974 N

IL --17,0 : 1.0 nn: 50 p : 1032.6 kg/m3 FL, - FL4 : 1.2 m

D: 2m L,„: 4.2m Lf : 5.6m HI : 1.5m

H 2 : 1.2 m lr : 7.0 m h2 : 0.0m lc: 1.5 m

f f , f„,,fr : 0.01 ig : 0.5m lw : 3.5m


The horizontal accelerations of the tractor and the tank in the local coordinate

system can be determined by Eq. (6.19) and Eq. (6.20). The accelerations depend on the

158



The longitudinal dynamic response o f partially filled liquid cargo tank vehicles

under rough road conditions has been investigated. The tank vehicle is traveling from the

flat road to the rough road at a constant horizontal speed of 80 km/h at the starting time.

The circular cylindrical tank is divided into four compartments. The accelerations of the

tractor and the tank, the normal axle loads and the fifth wheel loads have been computed

using the model developed in Section 6.3. The simulation parameters o f the road

conditions and the tank vehicle are listed in Table 6.1. To solve the nonlinear equation of

the sloshing model, the Runge-Kutta method has been employed. The nonlinear impact

model is compared with the linear spring-mass sloshing model. The dynamic responses,

including the fifth wheel loads and normal axle loads, are analyzed to study the influence

o f the liquid sloshing under rough road conditions.

Table 6.1 Simulation parameters

WL: 75 m a„: 1.5 m Wt : 48069 N Wk : 52974 N

nnl-

orf1 n„: 50 P - 1032.6 kg/m3 F L X - F L a : 1.2 m

D: 2 m L w: 4.2 m L f 5.6 m H x: 1.5 m

h 2 : 1.2 m I/- 7.0 m h2, 0.0m K - 1.5 m

f f 0.01 h-- 0.5 m K-- 3.5 m


The horizontal accelerations of the tractor and the tank in the local coordinate

system can be determined by Eq. (6.19) and Eq. (6.20). The accelerations depend on the

158


road conditions, the motion and geometry of the vehicle. Since it is difficult to express

them analytically, the numerical method is used to obtain the global coordinates of the

tire-ground contact points, and then the accelerations. From Eq. (6.10), the period of the

road contour is 3.34s. Figure 6.4 shows the horizontal accelerations of the tractor and the

tank. The period of each is 1.67s, which is half that of the road contour. When the vehicle

travels on the rough road, the connecting line between the tire-ground contact points

forms a secant line of the curve. When the road contour is expressed by a single term

cosine function, the frequency of the change of the slope of this secant line is twice the

frequency of the curve. The sloshing masses will undergo harmonic excitation of which

the frequency is two times that of the road contour. At the same time, there is a phase

difference between the tractor and the tank, since the tractor and the tank pass the same

point at different times. This excitation, which is quite close to the fundamental sloshing

frequency under the given parameters, has been used to study the dynamic characteristics

of the tank vehicle in the neighbourhood of resonance.

5 10 15 20 25 Time (s)

Figure 6.4 Horizontal accelerations of the tractor and tank

tractor tank

159


road conditions, the motion and geometry o f the vehicle. Since it is difficult to express

them analytically, the numerical method is used to obtain the global coordinates of the

tire-ground contact points, and then the accelerations. From Eq. (6.10), the period o f the

road contour is 3.34s. Figure 6.4 shows the horizontal accelerations o f the tractor and the

tank. The period of each is 1.67s, which is half that o f the road contour. When the vehicle

travels on the rough road, the connecting line between the tire-ground contact points

forms a secant line o f the curve. When the road contour is expressed by a single term

cosine function, the frequency of the change of the slope of this secant line is twice the

frequency of the curve. The sloshing masses will undergo harmonic excitation of which

the frequency is two times that of the road contour. At the same time, there is a phase

difference between the tractor and the tank, since the tractor and the tank pass the same

point at different times. This excitation, which is quite close to the fundamental sloshing

frequency under the given parameters, has been used to study the dynamic characteristics

of the tank vehicle in the neighbourhood of resonance.

0.4

« 0.2

cq n

<D8 - 0.2 o <

-0.420

Time (s)

Figure 6.4 Horizontal accelerations o f the tractor and tank

tractor ------- tank

159


6.4.2 Comparison between linear model and nonlinear impact model

Figure 6.5 presents the sloshing mass displacements of the linear spring-mass

model and nonlinear impact model. The result of the linear model grows far out of the

compartment walls, which makes it unsuitable to simulate the large amplitude liquid

sloshing. On the contrary, the nonlinear impact model sets the constraint at the walls.

When n„ —) 00 , the limit of absolutely rigid bodies interaction can be realized. If the

exponent 2n,-1 is large but finite, then the interaction field is not absolutely localized at

the points xn = ±x,i0 . This means that the tank walls and the mass are not absolutely

rigid, and admit a small deformation about the points of contact at x„ = -±x,0 . The

computed maximum non-dimensional displacement is 1.041. Because of this, a finite

value of n„ seems more realistic than the rigid body limit.

• • 1 . _ 0 10 20 30 40 50 60

Time (s)

Figure 6.5 Nondimensional sloshing mass displacement (wavelength: 75m)

linear model — impact model

160


6.4.2 Comparison between linear model and nonlinear impact model

Figure 6.5 presents the sloshing mass displacements o f the linear spring-mass

compartment walls, which makes it unsuitable to simulate the large amplitude liquid

sloshing. On the contrary, the nonlinear impact model sets the constraint at the walls.

When nn —> go , the limit of absolutely rigid bodies interaction can be realized. If the

exponent 2nn-\ is large but finite, then the interaction field is not absolutely localized at

the points x n — ± xn0. This means that the tank walls and the mass are not absolutely

rigid, and admit a small deformation about the points of contact at xn = ± xn0. The

computed maximum non-dimensional displacement is 1.041. Because o f this, a finite

value o f n„ seems more realistic than the rigid body limit.

model and nonlinear impact model. The result o f the linear model grows far out of the

2

2 ■ ■ ■ ■ ■" 0 10 20 30 40 50 60

Time (s)


linear model impact model

160


When the excitation frequency is changed with the change of the wavelength of

the road contour from 75 m to 100 m, the sloshing mass displacements of the two models

shown in Figure 6.6 are nearly the same, since the sloshing excitation frequency is far

from the excitation frequency. Therefore, the nonlinear impact model is useful to predict

the dynamic characteristics for both small and large amplitude sloshing of liquid cargo

tank vehicles. The parameters, nn and /in, , need to be determined by experimental studies

for different tank and liquid properties.

Non

dim

ensi

onal

0.04

1 A

)1 11 MI 11 11 Ji !ill 111 I IPA 11111.

- Vij il ihnj qf h

-0.04 0 10 20 30 40 50 60

Time (s)


linear model — — — impact model

6.4.3 Dynamic fifth wheel loads

Figure 6.7 illustrates the dynamic horizontal and vertical loads at the fifth wheel.

Under the given road conditions, the loads at the fifth wheel oscillate around the static

loads that can be obtained by running the tank vehicle at the constant speed under the flat

road condition. The amplitudes of the loads in both directions increase with the increase

of the sloshing mass displacements. The forces experience some extremely high values

161


When the excitation frequency is changed with the change o f the wavelength of

the road contour from 75 m to 100 m, the sloshing mass displacements of the two models

shown in Figure 6 .6 are nearly the same, since the sloshing excitation frequency is far

from the excitation frequency. Therefore, the nonlinear impact model is useful to predict

the dynamic characteristics for both small and large amplitude sloshing o f liquid cargo

tank vehicles. The parameters, n„ and 77 ., need to be determined by experimental studies

for different tank and liquid properties.

0.04

0.02

-0.0420 30 40 50

Time (s)

Figure 6 .6 Nondimensional sloshing mass displacement (wavelength: 100m)

linear m o d e l impact model

6.4.3 Dynamic fifth wheel loads

Figure 6.7 illustrates the dynamic horizontal and vertical loads at the fifth wheel.

Under the given road conditions, the loads at the fifth wheel oscillate around the static

loads that can be obtained by running the tank vehicle at the constant speed under the flat

road condition. The amplitudes of the loads in both directions increase with the increase

o f the sloshing mass displacements. The forces experience some extremely high values

161


when the non-dimensional sloshing mass displacements are about 1, which means the

sloshing mass reaches the compartment walls. After that, the amplitudes of the forces

decrease to their initial values. A periodic impact will be observed while the excitation

continues. This impulsive characteristic of the fifth wheel loads comes from the

hydrodynamic pressure impact on the tank walls due to the strongly nonlinear motion of

the sloshing liquid. Because the sloshing impact can cause serious damages to the tank

structures, the periodic impulsive behaviour of the fifth wheel loads will have an adverse

influence on the integrity of the tank vehicle supporting structures, such as the main

frame and the sub-frame. A thorough understanding of the impulsive loads is useful for

fatigue life analysis and vehicle structure design. Figure 6.7 also shows that the ratio of

the peak value to its static value (the value at the starting time, since the tank vehicle is

running from the flat road) of the horizontal force is much greater than that of the vertical

force. The oscillating amplitude of the horizontal force is also much greater than the

vertical force. At the same time, unlike the static force whose direction is always forward,

the direction of the horizontal fifth wheel force could change periodically due to liquid

sloshing, which is shown by the negative values of the force in Figure 6.7. These indicate

that more attention should be paid to the influence of liquid sloshing in the longitudinal

direction.

6.4.4 Dynamic normal axle loads

Figure 6.8 presents the normal force of the tractor front, rear and semi-trailer axle

computed by using the same parameters as Figure 6.7. As can be expected, the axle loads

also have impulsive characteristics. The influence of the impact on the tractor front and

162


when the non-dimensional sloshing mass displacements are about 1, which means the

sloshing mass reaches the compartment walls. After that, the amplitudes o f the forces

decrease to their initial values. A periodic impact will be observed while the excitation

continues. This impulsive characteristic o f the fifth wheel loads comes from the

hydrodynamic pressure impact on the tank walls due to the strongly nonlinear motion of

the sloshing liquid. Because the sloshing impact can cause serious damages to the tank

structures, the periodic impulsive behaviour o f the fifth wheel loads will have an adverse

influence on the integrity of the tank vehicle supporting structures, such as the main

frame and the sub-frame. A thorough understanding of the impulsive loads is useful for

fatigue life analysis and vehicle structure design. Figure 6.7 also shows that the ratio of

the peak value to its static value (the value at the starting time, since the tank vehicle is

running from the flat road) o f the horizontal force is much greater than that o f the vertical

force. The oscillating amplitude of the horizontal force is also much greater than the

vertical force. At the same time, unlike the static force whose direction is always forward,

the direction of the horizontal fifth wheel force could change periodically due to liquid

sloshing, which is shown by the negative values o f the force in Figure 6.7. These indicate

that more attention should be paid to the influence o f liquid sloshing in the longitudinal

direction.

6.4.4 Dynamic normal axle loads

Figure 6.8 presents the normal force o f the tractor front, rear and semi-trailer axle

computed by using the same parameters as Figure 6.7. As can be expected, the axle loads

also have impulsive characteristics. The influence o f the impact on the tractor front and

162


rear axle normal forces is greater than that of the semi-trailer axle normal force. Since the

static liquid weight is mainly supported by the tractor rear and semi-trailer axle, the

average value of the tractor front axle normal force is much less than that of the tractor

rear and semi-trailer axle. Therefore, the investigation of the longitudinal stability of

partially filled tank vehicles under rough road conditions is focused on the tractor front

axle. Extreme cases can be seen in this figure. At the moment when the sloshing mass

reaches the compartment walls, the normal force of the tractor front axle becomes

negative, which in a real situation means the front wheel may lose contact with the road

surface at these particular instants. Though the force goes back to its equilibrium point

after the impacts, the possible liftoff of the front wheel may possibly cause instability of

tank vehicles under some extreme conditions, which may cause catastrophic accidents.

1

10 2.0 30 40 50 60 Time (s)

Figure 6.7 Fifth wheel loads

horizontal load vertical load

163


rear axle normal forces is greater than that o f the semi-trailer axle normal force. Since the

static liquid weight is mainly supported by the tractor rear and semi-trailer axle, the

average value o f the tractor front axle normal force is much less than that o f the tractor

rear and semi-trailer axle. Therefore, the investigation of the longitudinal stability of

partially filled tank vehicles under rough road conditions is focused on the tractor front

axle. Extreme cases can be seen in this figure. At the moment when the sloshing mass

reaches the compartment walls, the normal force o f the tractor front axle becomes

negative, which in a real situation means the front wheel may lose contact with the road

surface at these particular instants. Though the force goes back to its equilibrium point

after the impacts, the possible liftoff of the front wheel may possibly cause instability of

tank vehicles under some extreme conditions, which may cause catastrophic accidents.

1

^ 0.555

VO0S 01 -°-5

-i0 10 20 30 40 50 60

Time (s)

Figure 6.7 Fifth wheel loads

horizontal load ------- vertical load

163

■ : j

v - v w V v v y W .........


z

iG

a) 00

4.1

2.5 2

1.5

1 0.5 0

-0.5 -1

-1.50

e A ! • • f; A fl it A 4. NA; tIEV• IL/ I. • .iiii•il, •1.% ext t. Ai • i• v %.; • P xi 0 V

I I

10 20 30 40 50 60 Time (s)

Figure 6.8 Normal axle loads

tractor front axle tractor rear axle semitrailer axle

6.5 Summary

In this chapter, a non-linear impact mechanical system that describes the liquid

motion as a linear spring-mass system with an impact subsystem has been developed to

investigate the longitudinal dynamic behaviour of partially filled tank vehicles under

rough road conditions. Major conclusions include the following:

(1) Rough road conditions have been included as the cause of severe liquid sloshing.

When the exciting frequency is equal to or near the sloshing frequency, the periodic

input can cause severe nonlinear impact, which is not included under the assumptions

of flat road and constant acceleration in the past investigations by other researchers.

(2) The non-linear impact mechanical system has been developed to investigate tank

vehicle dynamics under severe liquid sloshing conditions. By setting constraints on

the sloshing mass displacement for large amplitude situations, the impact model is

useful for both small and large amplitude liquid sloshing.

164


2.52

^ 1.5£oo* 0.5

-1-1.5

0 10 20 30 40 50 60Time (s)

Figure 6.8 Normal axle loads

tractor front axle tractor rear axle semitrailer axle

6.5 Summary

In this chapter, a non-linear impact mechanical system that describes the liquid

motion as a linear spring-mass system with an impact subsystem has been developed to

investigate the longitudinal dynamic behaviour o f partially filled tank vehicles under

rough road conditions. Major conclusions include the following:

(1) Rough road conditions have been included as the cause o f severe liquid sloshing.

When the exciting frequency is equal to or near the sloshing frequency, the periodic

input can cause severe nonlinear impact, which is not included under the assumptions

of flat road and constant acceleration in the past investigations by other researchers.

(2) The non-linear impact mechanical system has been developed to investigate tank

vehicle dynamics under severe liquid sloshing conditions. By setting constraints on

the sloshing mass displacement for large amplitude situations, the impact model is

useful for both small and large amplitude liquid sloshing.

164


(3) Periodic impulsive behaviour of fifth wheel loads and normal axle loads is shown in

the neighbourhood of resonance. More attention should be paid to the fifth wheel load

in the horizontal direction in structure design and fatigue analysis, due to the drastic

changes in both the amplitude and the direction.

(4) Instability of the tank vehicle may occur under certain rough road conditions due to

the possible loss of tire-ground contact at the tractor front axle.

165


(3) Periodic impulsive behaviour o f fifth wheel loads and normal axle loads is shown in

the neighbourhood o f resonance. More attention should be paid to the fifth wheel load

in the horizontal direction in structure design and fatigue analysis, due to the drastic

changes in both the amplitude and the direction.

(4) Instability of the tank vehicle may occur under certain rough road conditions due to

the possible loss o f tire-ground contact at the tractor front axle.

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CHAPTER 7 CONCLUSIONS AND RECOMMENDATIONS

7.1 Conclusions

In this dissertation, the dynamic behaviour of liquid motion inside liquid cargo

vehicle tanks has been investigated in detail based on the newly developed mathematical

method especially for liquid motion in horizontal cylindrical tanks. Liquid cargo vehicle

dynamics in the longitudinal direction has been investigated by equivalent mechanical

models for some situations where the newly developed method cannot be used, i.e., the

ride comfort problem in the frequency domain and the nonlinear impact problem in the

pitch plane.

The research involves: the development of a new mathematical method and

corresponding numerical procedures for liquid motion inside 2D circular and elliptical

tanks; the development of a new mathematical method for liquid motion inside 3D

horizontal cylindrical tanks with flat heads and hemispherical heads; a dynamic liquid

behaviour study based on the newly developed methods for both 2D and 3D road tanks

under typical highway operation conditions, such as turning, lane change and

braking/accelerating; the investigation of the ride quality of liquid cargo tank vehicles in

the longitudinal direction by integrating the equivalent linear mass-spring systems into

the tractor semi-trailer model; the study of the influence of liquid impact by the

development of an equivalent nonlinear impact model.

Research results from the author's research activities during his Ph.D. program

have been published in the form of journal papers, conference papers and a technical

report ([19, 20, 21, 22, 23, 130, 131, 132, 133, 134, 135, 136, 137]), most of which are

166


CHAPTER 7 CONCLUSIONS AND RECOMMENDATIONS

7.1 Conclusions

In this dissertation, the dynamic behaviour o f liquid motion inside liquid cargo

vehicle tanks has been investigated in detail based on the newly developed mathematical

method especially for liquid motion in horizontal cylindrical tanks. Liquid cargo vehicle

dynamics in the longitudinal direction has been investigated by equivalent mechanical

models for some situations where the newly developed method cannot be used, i.e., the

ride comfort problem in the frequency domain and the nonlinear impact problem in the

pitch plane.

The research involves: the development o f a new mathematical method and

corresponding numerical procedures for liquid motion inside 2D circular and elliptical

tanks; the development of a new mathematical method for liquid motion inside 3D

horizontal cylindrical tanks with flat heads and hemispherical heads; a dynamic liquid

behaviour study based on the newly developed methods for both 2D and 3D road tanks

under typical highway operation conditions, such as turning, lane change and

braking/accelerating; the investigation o f the ride quality of liquid cargo tank vehicles in

the longitudinal direction by integrating the equivalent linear mass-spring systems into

the tractor semi-trailer model; the study of the influence of liquid impact by the

development of an equivalent nonlinear impact model.

Research results from the author’s research activities during his Ph.D. program

have been published in the form of journal papers, conference papers and a technical

report ([19, 20, 21, 22, 23, 130, 131, 132, 133, 134, 135, 136, 137]), most o f which are

166


directly from this dissertation. The major conclusions of the research presented in this

work are as follows.

• A new mathematical method to solve the dynamic liquid behaviour in partially

filled horizontal 2D circular and elliptical tanks has been developed to study the

lateral dynamics of liquid motion inside road tanks. The governing equations for

the liquid motion in a tank are manipulated with the continuous coordinate

transformations. The first transformation saves the performance of interpolation

of boundary conditions on the curved walls. The application of the second

transformation changes the working domain to a fixed area, avoiding the complex

algorithm for free surface updating and volume correction. When the governing

equations are solved using the finite difference method, the third transformation is

adopted to gain computational convergence and stability.

• Compared with some other numerical schemes for sloshing problems in 2D

circular tanks, there are some advantages to the current method. The current

method does not need to deal with the boundary conditions on the time varying

curved walls and free surface. There is no need for capturing or smoothing of the

free surface, and the performance of volume correction. Complicated algorithms

for interpolation on rigid walls and updating the free surface are completely

avoided. These make the algorithm efficient and stable. The governing equations

are rearranged so that the programming is easy. Replacement of different

transformation equations for the first transformation can be easily carried out in

the same way for tanks with arbitrary wall shapes. Replacement of different

167


directly from this dissertation. The major conclusions o f the research presented in this

work are as follows.

• A new mathematical method to solve the dynamic liquid behaviour in partially

filled horizontal 2D circular and elliptical tanks has been developed to study the

lateral dynamics o f liquid motion inside road tanks. The governing equations for

the liquid motion in a tank are manipulated with the continuous coordinate

transformations. The first transformation saves the performance of interpolation

o f boundary conditions on the curved walls. The application o f the second

transformation changes the working domain to a fixed area, avoiding the complex

algorithm for free surface updating and volume correction. When the governing

equations are solved using the finite difference method, the third transformation is

adopted to gain computational convergence and stability.

• Compared with some other numerical schemes for sloshing problems in 2D

circular tanks, there are some advantages to the current method. The current

method does not need to deal with the boundary conditions on the time varying

curved walls and free surface. There is no need for capturing or smoothing of the

free surface, and the performance o f volume correction. Complicated algorithms

for interpolation on rigid walls and updating the free surface are completely

avoided. These make the algorithm efficient and stable. The governing equations

are rearranged so that the programming is easy. Replacement o f different

transformation equations for the first transformation can be easily carried out in

the same way for tanks with arbitrary wall shapes. Replacement o f different

167


transformation equations for the transformations can also be done without much

extra modification needed for program codes in adjusting the grid distribution.

• More importantly, the new method has an excellent capability of extending to

solve liquid motion in 3D horizontal cylindrical tanks subjected to excitations in

both longitudinal and lateral directions. It has been concluded in the literature

review that the combination of 3D vehicle models with 2D dynamic liquid motion

have been used to simulate vehicle dynamics by considering only the lateral liquid

motion with the assumption that liquid at all cross sections behaves identically in

the transversal direction. The lack of an effective algorithm to describe the liquid

motion in 3D space constrained past researches to the steady turning operation. At

the same time, researches on liquid cargo vehicles in the longitudinal direction

were often carried out on rectangular tanks instead of horizontal cylindrical ones,

for the same reason. In this research, a new mathematical method has been

developed to study the liquid dynamics in partially filled 3D horizontal cylindrical

tanks based on the method developed for 2D circular and elliptical tanks. This

approach provides a useful tool for solving the liquid dynamics in horizontal

cylindrical road tanks in a completely 3D manner. It can be easily integrated into

coupled liquid-structure systems to study the vehicle system dynamics. This also

provides the availability of a systematic analysis of tank vehicle structures

subjected to liquid sloshing and other loadings.

• During turning operations, the newly developed method has been used to simulate

168


transformation equations for the transformations can also be done without much

extra modification needed for program codes in adjusting the grid distribution.

• More importantly, the new method has an excellent capability o f extending to

solve liquid motion in 3D horizontal cylindrical tanks subjected to excitations in

both longitudinal and lateral directions. It has been concluded in the literature

review that the combination o f 3D vehicle models with 2D dynamic liquid motion

have been used to simulate vehicle dynamics by considering only the lateral liquid

motion with the assumption that liquid at all cross sections behaves identically in

the transversal direction. The lack o f an effective algorithm to describe the liquid

motion in 3D space constrained past researches to the steady turning operation. At

the same time, researches on liquid cargo vehicles in the longitudinal direction

were often carried out on rectangular tanks instead of horizontal cylindrical ones,

for the same reason. In this research, a new mathematical method has been

developed to study the liquid dynamics in partially filled 3D horizontal cylindrical

tanks based on the method developed for 2D circular and elliptical tanks. This

approach provides a useful tool for solving the liquid dynamics in horizontal

cylindrical road tanks in a completely 3D manner. It can be easily integrated into

coupled liquid-structure systems to study the vehicle system dynamics. This also

provides the availability of a systematic analysis o f tank vehicle structures

subjected to liquid sloshing and other loadings.

• During turning operations, the newly developed method has been used to simulate

168


the oscillatory liquid motion that could not be described by mass centre models.

The oscillatory motion depends on the acceleration input time and the final

acceleration amplitude. The oscillation amplitude of the liquid motion increases

with the increase in steady acceleration. The oscillation amplitude decreases with

an increase of the input time. A suddenly applied acceleration without an input

time causes the largest oscillatory amplitude and should be avoided in the

operation. The dynamic liquid motion during a lane change operation and a

double lane change has also been calculated.

• Longitudinal dynamic liquid motion inside 3D horizontal cylindrical tanks has

been analyzed by the new method during accelerating/braking operations. When

the tank is subjected to a suddenly applied acceleration, the liquid inside the tank

undergoes oscillatory motion, which significantly changes the pressure

distributions on the tank walls. This causes oscillatory forces and moments on the

tank. Larger acceleration causes larger forces and moments in both mean values

and extreme values. Compared to the free surface shapes in tanks with flat heads,

the free surfaces inside the tank with hemispherical heads are much flatter. The

influence of higher modes is quite weak due to the existence of the curved head

walls. For compartmented tanks with different fill levels, the asynchronous liquid

motion helps to decrease the magnitude of the varying force when the resultant

liquid force is obtained by combining the forces in all compartments. Long tanks

without any partitions are harmful for longitudinal dynamics and are therefore a

bad choice for carrying liquid product.

169


the oscillatory liquid motion that could not be described by mass centre models.

The oscillatory motion depends on the acceleration input time and the final

acceleration amplitude. The oscillation amplitude of the liquid motion increases

with the increase in steady acceleration. The oscillation amplitude decreases with

an increase o f the input time. A suddenly applied acceleration without an input

time causes the largest oscillatory amplitude and should be avoided in the

operation. The dynamic liquid motion during a lane change operation and a

double lane change has also been calculated.

• Longitudinal dynamic liquid motion inside 3D horizontal cylindrical tanks has

been analyzed by the new method during accelerating/braking operations. When

the tank is subjected to a suddenly applied acceleration, the liquid inside the tank

undergoes oscillatory motion, which significantly changes the pressure

distributions on the tank walls. This causes oscillatory forces and moments on the

tank. Larger acceleration causes larger forces and moments in both mean values

and extreme values. Compared to the free surface shapes in tanks with flat heads,

the free surfaces inside the tank with hemispherical heads are much flatter. The

influence o f higher modes is quite weak due to the existence of the curved head

walls. For compartmented tanks with different fill levels, the asynchronous liquid

motion helps to decrease the magnitude o f the varying force when the resultant

liquid force is obtained by combining the forces in all compartments. Long tanks

without any partitions are harmful for longitudinal dynamics and are therefore a

bad choice for carrying liquid product.

169


• The ride comfort for vehicle operations is of great concern because an exposure to

high levels of vibration will cause driver fatigue, which in turn can have a harmful

effect on health problems and driving safety. The liquid motion within the

partially filled tanks has a negative influence on the driver's ride quality, a

problem which had never been studied. In this research, the ride performance of

partially filled compartmented tank vehicles in the longitudinal direction has been

investigated by a linearized multi-degree-of-freedom dynamic model. The power

spectral density of the vertical and horizontal seat accelerations has been utilized

to study the influence of liquid motion on the ride quality. The frequencies due to

the liquid sloshing modes are in the frequency range of 0.1-0.7 Hz, which is

determined by the tank configurations and fill levels and quite close to the

frequencies due to the bounce and pitch modes, as well as the seat mode. This

generates a coupled effect on the frequency response, which makes the frequency

distribution in the range considered quite different from that of rigid cargo

vehicles in the low frequency domain, i.e., 0.1-2Hz. Simulation results show that

the amplitudes and frequency distributions of seat accelerations in both directions

are significantly affected by the liquid fill level, vehicle speed, road surface

condition, the type of liquid being carried, and the seat suspension parameters.

• Under severe conditions, highly nonlinear liquid motion will occur due to rapid

velocity changes associated with hydrodynamic pressure impacts. One of the

main factors that could cause severe liquid sloshing in partially filled tank

vehicles is the rough road surface. When the excitation frequency of the road

170


• The ride comfort for vehicle operations is of great concern because an exposure to

high levels o f vibration will cause driver fatigue, which in turn can have a harmful

effect on health problems and driving safety. The liquid motion within the

partially filled tanks has a negative influence on the driver’s ride quality, a

problem which had never been studied. In this research, the ride performance of

partially filled compartmented tank vehicles in the longitudinal direction has been

investigated by a linearized multi-degree-of-ffeedom dynamic model. The power

spectral density o f the vertical and horizontal seat accelerations has been utilized

to study the influence of liquid motion on the ride quality. The frequencies due to

the liquid sloshing modes are in the frequency range o f 0.1-0.7 Hz, which is

determined by the tank configurations and fill levels and quite close to the

frequencies due to the bounce and pitch modes, as well as the seat mode. This

generates a coupled effect on the frequency response, which makes the frequency

distribution in the range considered quite different from that o f rigid cargo

vehicles in the low frequency domain, i.e., 0.1-2Hz. Simulation results show that

the amplitudes and frequency distributions o f seat accelerations in both directions

are significantly affected by the liquid fill level, vehicle speed, road surface

condition, the type o f liquid being carried, and the seat suspension parameters.

• Under severe conditions, highly nonlinear liquid motion will occur due to rapid

velocity changes associated with hydrodynamic pressure impacts. One o f the

main factors that could cause severe liquid sloshing in partially filled tank

vehicles is the rough road surface. When the excitation frequency o f the road

170


contour is equal to or near the sloshing frequency, the periodic input can cause

severe nonlinear impact, a problem which had never been considered in the past

investigations by other researchers, since the assumptions of flat road and

constant acceleration were adopted. In this research, a nonlinear impact

mechanical system that describes the liquid motion as a linear spring-mass system

with an impact subsystem has been developed to investigate the longitudinal


By setting constraints to the sloshing mass displacement for large amplitude

situations, the impact model is useful for both small and large amplitude liquid

motion. The periodic impulsive behaviour of fifth wheel loads and normal axle

loads is shown in the neighbourhood of resonance. More attention should be paid

to the fifth wheel load in the horizontal direction in structure design and fatigue

analysis due to the drastic change in both the amplitude and the direction.

As pointed out previously, mass centre models and 2D sloshing models have been

used in combination with 3D vehicle models in vehicle dynamics studies in the current

literature. The lack of 3D sloshing models in horizontal cylindrical tanks had prevented

further investigations attempting to obtain a thorough and comprehensive understanding

of liquid behaviour in partially filled tanks and liquid-vehicle systems subjected to liquid

sloshing and external excitations. The methodology developed in this research will

provide a completely 3D analysis for liquid motion in such tanks, as well as an effective

3D approach for liquid-vehicle coupling systems. Much more accurate and reliable

results are expected for topics that were previously studied by mass centre models and 2D

171


contour is equal to or near the sloshing frequency, the periodic input can cause

severe nonlinear impact, a problem which had never been considered in the past

investigations by other researchers, since the assumptions o f flat road and

constant acceleration were adopted. In this research, a nonlinear impact

mechanical system that describes the liquid motion as a linear spring-mass system

with an impact subsystem has been developed to investigate the longitudinal

dynamic behaviour o f partially filled tank vehicles under rough road conditions.

By setting constraints to the sloshing mass displacement for large amplitude

situations, the impact model is useful for both small and large amplitude liquid

motion. The periodic impulsive behaviour o f fifth wheel loads and normal axle

loads is shown in the neighbourhood o f resonance. More attention should be paid

to the fifth wheel load in the horizontal direction in structure design and fatigue

analysis due to the drastic change in both the amplitude and the direction.

As pointed out previously, mass centre models and 2D sloshing models have been

used in combination with 3D vehicle models in vehicle dynamics studies in the current

literature. The lack o f 3D sloshing models in horizontal cylindrical tanks had prevented

further investigations attempting to obtain a thorough and comprehensive understanding

o f liquid behaviour in partially filled tanks and liquid-vehicle systems subjected to liquid

sloshing and external excitations. The methodology developed in this research will

provide a completely 3D analysis for liquid motion in such tanks, as well as an effective

3D approach for liquid-vehicle coupling systems. Much more accurate and reliable

results are expected for topics that were previously studied by mass centre models and 2D

171


liquid models. At the same time, the research will be significantly expanded to areas that

cannot be studied by 2D sloshing models. The established methodology will provide a

useful tool for researchers in performing investigations on liquid behaviour and dynamics

of systems carrying and storing liquid. This will also benefit engineers in vehicle

structure designing and manufacturing.

It should also be pointed out that liquid sloshing is a strongly nonlinear

phenomenon, which depends largely on the tank geometry, liquid fill level, different

excitation amplitude and frequency. It is well recognized that it is difficult to analyze

liquid sloshing problems either analytically or numerically. As previously mentioned, the

research objective of this study is to develop an effective mathematical method that can

be used especially for liquid motion in horizontal cylindrical road tanks in studying liquid

cargo tank vehicle dynamics under normal highway operations. It is not the author's

purpose to develop a general computational fluid dynamics method. It is important to be

aware of the assumptions and limitations of the newly developed method when applying

it to dynamic problems of liquid cargo tank vehicles. Currently, for situations where the

newly developed method cannot be applied, the equivalent mechanical models can be

further developed and employed to study the dynamics of liquid cargo tank vehicles

under certain kinds of operations.

7.2 Recommendations for future work

This research has developed a mathematical method to solve liquid sloshing

problems for horizontal cylindrical tanks with different configurations for liquid cargo

tank vehicles. Equivalent mechanical models of liquid motion have also been applied and

172


liquid models. At the same time, the research will be significantly expanded to areas that

cannot be studied by 2D sloshing models. The established methodology will provide a

useful tool for researchers in performing investigations on liquid behaviour and dynamics

of systems carrying and storing liquid. This will also benefit engineers in vehicle

structure designing and manufacturing.

It should also be pointed out that liquid sloshing is a strongly nonlinear

phenomenon, which depends largely on the tank geometry, liquid fill level, different

excitation amplitude and frequency. It is well recognized that it is difficult to analyze

liquid sloshing problems either analytically or numerically. As previously mentioned, the

research objective o f this study is to develop an effective mathematical method that can

be used especially for liquid motion in horizontal cylindrical road tanks in studying liquid

cargo tank vehicle dynamics under normal highway operations. It is not the author’s

purpose to develop a general computational fluid dynamics method. It is important to be

aware o f the assumptions and limitations o f the newly developed method when applying

it to dynamic problems o f liquid cargo tank vehicles. Currently, for situations where the

newly developed method cannot be applied, the equivalent mechanical models can be

further developed and employed to study the dynamics of liquid cargo tank vehicles

under certain kinds of operations.

7.2 Recommendations for future work

This research has developed a mathematical method to solve liquid sloshing

problems for horizontal cylindrical tanks with different configurations for liquid cargo

tank vehicles. Equivalent mechanical models o f liquid motion have also been applied and

172


developed for two situations where the newly developed method cannot be used. It is

recommended to continue the research in the following areas.

1. Apply the method developed to tanks of arbitrary shapes employed in the road

transformation industry to study the dynamic liquid motion inside the road tanks.

Different transformations based on the tank shape expressions can be easily

carried out in the same way for tanks with arbitrary wall shapes in stretching the

tank areas to rectangular areas. Even in cases where the mathematical expressions

for the walls are not available, the metrics can be obtained numerically by the

second order central finite difference. This can be done without much extra

modification needed for program codes.

2. Apply the mathematical method to study the transient dynamic liquid motion

inside road tanks subjected to other highway operation conditions. The

mathematical method and corresponding numerical procedures can be applied to

liquid motion problems in horizontal cylindrical tanks in a real 3D manner. The

liquid dynamics under many important operation conditions, such as straight line

driving at varying accelerations, turning at non-constant speed and radius, lane

change and double lane change, and turn-in-braking, can be studied based on the

knowledge of the applied accelerations. Although the studies in this dissertation

focus on longitudinal dynamics, there is no difficulty in including both the lateral

acceleration and the longitudinal acceleration by the present method in order to

study the liquid behaviour in 3D space.

3. Apply the newly developed method to coupled dynamic liquid-vehicle systems.

Once the tank motions in three translatory and three rotational directions in the

173


developed for two situations where the newly developed method cannot be used. It is

recommended to continue the research in the following areas.

1. Apply the method developed to tanks o f arbitrary shapes employed in the road

transformation industry to study the dynamic liquid motion inside the road tanks.

Different transformations based on the tank shape expressions can be easily

carried out in the same way for tanks with arbitrary wall shapes in stretching the

tank areas to rectangular areas. Even in cases where the mathematical expressions

for the walls are not available, the metrics can be obtained numerically by the

second order central finite difference. This can be done without much extra

modification needed for program codes.

2. Apply the mathematical method to study the transient dynamic liquid motion

inside road tanks subjected to other highway operation conditions. The


liquid motion problems in horizontal cylindrical tanks in a real 3D manner. The

liquid dynamics under many important operation conditions, such as straight line

driving at varying accelerations, turning at non-constant speed and radius, lane

change and double lane change, and tum-in-braking, can be studied based on the

knowledge of the applied accelerations. Although the studies in this dissertation

focus on longitudinal dynamics, there is no difficulty in including both the lateral

acceleration and the longitudinal acceleration by the present method in order to

study the liquid behaviour in 3D space.

3. Apply the newly developed method to coupled dynamic liquid-vehicle systems.

Once the tank motions in three translatory and three rotational directions in the

173


coupled liquid-vehicle system are solved from the governing equations of the

vehicle system established by Newton's law of motion, Lagrange's method, or the

Hamilton principle, the forces and moments caused by the liquid pressure

distribution on the tank walls can be solved numerically based on the fluid

mechanics equations by using the mathematical method developed in this

research. They can be used to determine the motion of the vehicle components

and tanks for the next time step. The dynamics of the vehicle structure system,

which is usually described by a multi-DOF vibration system of rigid bodies linked

by linear or nonlinear elastic and damping elements, can be solved by numerical

methods, such as the Runge-Kutta method. The mathematical procedures

established in the newly developed method can then be used to study the

dynamics of the coupled liquid-vehicle systems under different normal operation

conditions.

4. Conduct structural integrity analysis using the structural dynamic loadings

calculated by the newly developed method. The design standards and safety

regulations for vehicles carrying dangerous goods have been outlined by the CSA

(Canadian Standards Association) B620-03 in Canada. This standard is to be used

for design, manufacturing, testing, inspection and maintenance for tanks and

vehicle structures. However, the stress calculation methods for both the liquid

tanks and frames or integral structural supports do not include the dynamic effect

caused by liquid sloshing under different operation conditions. Since the


liquid motion in horizontal cylindrical tanks and dynamics of coupled liquid-

174


coupled liquid-vehicle system are solved from the governing equations o f the

vehicle system established by Newton’s law o f motion, Lagrange’s method, or the

Hamilton principle, the forces and moments caused by the liquid pressure

distribution on the tank walls can be solved numerically based on the fluid

mechanics equations by using the mathematical method developed in this

research. They can be used to determine the motion of the vehicle components

and tanks for the next time step. The dynamics of the vehicle structure system,

which is usually described by a multi-DOF vibration system of rigid bodies linked

by linear or nonlinear elastic and damping elements, can be solved by numerical

methods, such as the Runge-Kutta method. The mathematical procedures

established in the newly developed method can then be used to study the

dynamics o f the coupled liquid-vehicle systems under different normal operation

conditions.

4. Conduct structural integrity analysis using the structural dynamic loadings

calculated by the newly developed method. The design standards and safety

regulations for vehicles carrying dangerous goods have been outlined by the CSA

(Canadian Standards Association) B620-03 in Canada. This standard is to be used

for design, manufacturing, testing, inspection and maintenance for tanks and

vehicle structures. However, the stress calculation methods for both the liquid

tanks and frames or integral structural supports do not include the dynamic effect

caused by liquid sloshing under different operation conditions. Since the


liquid motion in horizontal cylindrical tanks and dynamics o f coupled liquid-

174


vehicle systems in a real 3D manner, the corresponding dynamic loadings on the

vehicle structures can be used for the structural stress and strength analysis by

commercial finite element analysis software programs in order to assess and

improve existing structural design or provide guides for new structural design.

175


vehicle systems in a real 3D manner, the corresponding dynamic loadings on the

vehicle structures can be used for the structural stress and strength analysis by

commercial finite element analysis software programs in order to assess and

improve existing structural design or provide guides for new structural design.

175


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192


APPENDIX A: STRUCTURAL ANALYSIS OF A B-TRAIN

TANK TRUCK SUBFRAME SUBJECTED TO

BRAKING/ACCELERATING

A.1 Introduction

Generally, structural integrity problems of liquid cargo tank vehicles may include

any kind of structure strength and fatigue problems of both vehicle structures and tanks.

More failure has occurred on the vehicle structures than the tanks. For example, pictures

from MaXfield Inc., a Canadian tank truck manufacturer in Calgary, show that cracks in

the subframe, which is the important structure for supporting and connecting the front

tank and rear tank in B-train tank trucks, are one of the major failures that have happened

in the past. An experimental study was conducted (Olofsson et al, 1995) to measure the

acceleration time histories on the shell of a tank vehicle carrying liquid on different road

types and with different liquid levels in the tanks. Statistics of the different kinds of

failure that occurred in Sweden in 1990 were shown.

The assessment of the durability performance of vehicle structures was

traditionally carried out by test methods. In recent years, analytical fatigue life prediction

methods have been developed, which rely on numerical techniques. By combining the

vehicle motion simulation and finite element method with the traditional fatigue

technologies, several studies have conducted vehicle structure durability assessment and

fatigue analysis for the chassis component (Conle et al, 1991), suspension component

(Lee et al, 1995) and car body (Kuo and Kelkar, 1995) for rigid cargo vehicles or

193


APPENDIX A: STRUCTURAL ANALYSIS OF AB-TRAIN

TANK TRUCK SUBFRAME SUBJECTED TO

BRAKING/ACCELERATING

A .l Introduction

Generally, structural integrity problems of liquid cargo tank vehicles may include

any kind of structure strength and fatigue problems of both vehicle structures and tanks.

More failure has occurred on the vehicle structures than the tanks. For example, pictures

from MaXfield Inc., a Canadian tank truck manufacturer in Calgary, show that cracks in

the subframe, which is the important structure for supporting and connecting the front

tank and rear tank in B-train tank trucks, are one o f the major failures that have happened

in the past. An experimental study was conducted (Olofsson et al, 1995) to measure the

acceleration time histories on the shell of a tank vehicle carrying liquid on different road

types and with different liquid levels in the tanks. Statistics o f the different kinds of

failure that occurred in Sweden in 1990 were shown.

The assessment of the durability performance of vehicle structures was

traditionally carried out by test methods. In recent years, analytical fatigue life prediction

methods have been developed, which rely on numerical techniques. By combining the

vehicle motion simulation and finite element method with the traditional fatigue

technologies, several studies have conducted vehicle structure durability assessment and

fatigue analysis for the chassis component (Conle et al, 1991), suspension component

(Lee et al, 1995) and car body (Kuo and Kelkar, 1995) for rigid cargo vehicles or

193


passenger vehicles. Compared to the stability analysis and directional response

characteristics of heavy vehicles carrying liquid cargo, the influence of liquid motion on

the liquid cargo vehicle structure strength and fatigue were seldom investigated due to the

complexities and difficulties in describing the liquid motion and the liquid-vehicle

interactions.

Kang (2001) studied the influence of tank cross-sections on structural integrity by

using finite element analysis for the tank itself and the direct supporting structures that

were welded to the tank. The optimal and conventional tanks were modeled in ANSYS to

do preliminary stress analysis under static loading conditions, which included the weight

of the liquid cargo and the weight of the tank. The boundary conditions were applied

directly to the interface between the tank supports and trailer frame, assuming a

negligible influence of trailer frame flexibility. The von Mises stresses were evaluated

using the static structural analysis option of ANSYS. The stress distributions were

compared for different tank shapes and fill levels. A static finite element analysis was

conducted for the subframe by the author (Xu, 2003). Several types of loading conditions,

i.e., the static loading conditions caused by the weight of the liquid cargo and tanks, the

loading conditions caused by a lateral acceleration by the pendulum analog, and the

simplified loading conditions representing the influence of accelerating and braking, were

applied on corresponding supporting and connecting locations on the subframe. The von

Mises stress and deflection distributions were examined. Most dangerous areas were

identified. The FE analysis was carried out in ALGOR.

In the analysis of the tank vehicle structures, the subframes, which are usually

built with flat plates, are more likely to undergo elastic deformation. In this section, the

194


passenger vehicles. Compared to the stability analysis and directional response

characteristics of heavy vehicles carrying liquid cargo, the influence o f liquid motion on

the liquid cargo vehicle structure strength and fatigue were seldom investigated due to the

complexities and difficulties in describing the liquid motion and the liquid-vehicle

interactions.

Kang (2001) studied the influence of tank cross-sections on structural integrity by

using finite element analysis for the tank itself and the direct supporting structures that

were welded to the tank. The optimal and conventional tanks were modeled in ANSYS to

do preliminary stress analysis under static loading conditions, which included the weight

of the liquid cargo and the weight o f the tank. The boundary conditions were applied

directly to the interface between the tank supports and trailer frame, assuming a

negligible influence of trailer frame flexibility. The von Mises stresses were evaluated

using the static structural analysis option o f ANSYS. The stress distributions were

compared for different tank shapes and fill levels. A static finite element analysis was

conducted for the subframe by the author (Xu, 2003). Several types o f loading conditions,

i.e., the static loading conditions caused by the weight o f the liquid cargo and tanks, the

loading conditions caused by a lateral acceleration by the pendulum analog, and the

simplified loading conditions representing the influence o f accelerating and braking, were

applied on corresponding supporting and connecting locations on the subframe. The von

Mises stress and deflection distributions were examined. Most dangerous areas were

identified. The FE analysis was carried out in ALGOR.

In the analysis of the tank vehicle structures, the subframes, which are usually

built with flat plates, are more likely to undergo elastic deformation. In this section, the

194


structural stress analysis is conducted to study the effects of one typical vehicle operation,

accelerating and/or braking, on the subframe of a B-train tank truck. The transient cargo

load shift in the front and rear tanks of the B-train tank truck is calculated by the vehicle

pitch plane model to obtain the loading conditions on the subframe. The finite element

model of the subframe is established, and the finite element analysis is implemented. The

structural stress analysis is carried out under the prescribed accelerations and different fill

levels.

A.2 B-train tank truck model

Theoretically, during the service life of B-train tank trucks, loading conditions for

vehicle structures under all different kinds of operations could cause durability problems

of the structures. However, some operating conditions, such as stationary loading and

unloading, idle condition, and constant-speed driving, would not cause severe cyclic

loading on the vehicle structures. On the contrary, other conditions, such as repeated

accelerating and braking, and driving on rough roads and roads with unique flaws, should

be considered as the major contributors to cyclic loadings. The loading conditions on the

vehicle structures are much worse when the cargo load shifts and dynamic behaviour

under these conditions are included, due to the motility of the liquid cargo.

In this section, the loading conditions caused by one of the transient operations,

accelerating and braking, are considered to study the influence on the vehicle structures.

The tank vehicle is considered to be traveling over a flat road subjected to constant

accelerations. The motion is constrained to the pitch plane. Lateral turning acceleration

and the motion caused by the extremely uneven road in both longitudinal and lateral

195


structural stress analysis is conducted to study the effects of one typical vehicle operation,

accelerating and/or braking, on the subframe o f a B-train tank truck. The transient cargo

load shift in the front and rear tanks o f the B-train tank truck is calculated by the vehicle

pitch plane model to obtain the loading conditions on the subframe. The finite element

model o f the subframe is established, and the finite element analysis is implemented. The

structural stress analysis is carried out under the prescribed accelerations and different fill

levels.

A.2 B-train tank truck model

Theoretically, during the service life o f B-train tank trucks, loading conditions for

vehicle structures under all different kinds of operations could cause durability problems

o f the structures. However, some operating conditions, such as stationary loading and

unloading, idle condition, and constant-speed driving, would not cause severe cyclic

loading on the vehicle structures. On the contrary, other conditions, such as repeated

accelerating and braking, and driving on rough roads and roads with unique flaws, should

be considered as the major contributors to cyclic loadings. The loading conditions on the

vehicle structures are much worse when the cargo load shifts and dynamic behaviour

under these conditions are included, due to the motility o f the liquid cargo.

In this section, the loading conditions caused by one o f the transient operations,

accelerating and braking, are considered to study the influence on the vehicle structures.

The tank vehicle is considered to be traveling over a flat road subjected to constant

accelerations. The motion is constrained to the pitch plane. Lateral turning acceleration

and the motion caused by the extremely uneven road in both longitudinal and lateral

195


directions, which are also common, are not included for simplicity. Vehicle structures

experience large rigid body motion with small elastic deformations when the flexibility of

an individual component has no significant effect on the overall dynamic behaviour of

the vehicle system. The stresses are induced in the structures as a result of the small

elastic deformations. The liquid cargo vehicle studied in this section, which is shown in

Figure A.1, is an eight-axle B-train tank truck with a cylindrical front tank and a

cylindrical rear tank connected and supported by a subframe. The subframe is the

structure that suffers the cyclic loadings, and is to be studied using finite element

analysis.

Figure A.1 Schematic of a B-train tank truck

(a) accelerating (b) braking

Under the assumption of the rigid body vehicle model, the motion equations of

the whole system can be established for the tractor, front tank, rear tank, all axles and

196


directions, which are also common, are not included for simplicity. Vehicle structures

experience large rigid body motion with small elastic deformations when the flexibility of

an individual component has no significant effect on the overall dynamic behaviour of

the vehicle system. The stresses are induced in the structures as a result o f the small

elastic deformations. The liquid cargo vehicle studied in this section, which is shown in

Figure A .l, is an eight-axle B-train tank truck with a cylindrical front tank and a

cylindrical rear tank connected and supported by a subframe. The subframe is the

structure that suffers the cyclic loadings, and is to be studied using finite element

analysis.

Under the assumption of the rigid body vehicle model, the motion equations of

the whole system can be established for the tractor, front tank, rear tank, all axles and

(a;

Figure A .l Schematic o f a B-train tank truck

(a) accelerating (b) braking

196


suspensions according to the force and moment equilibrium requirements (Wong, 1993).

Because the vehicle experiences straight-line traveling during accelerating and braking

operations, the mass centres of liquid cargo in both the front and rear tanks shift in the

longitudinal direction. The free surface gradient of the liquid can be obtained by a basic

fluid mechanics equation under constant accelerations by Eq. (4.80). Since only

geometrical information is needed to get the mass centre locations of the liquid bulk

inside the tank, they can be obtained in a 3D solid modelling program for different tanks,

fill levels and accelerations.

In the following analysis, all geometric and physical information is from internal

drawings from MaXfield Inc., Calgary. By solving the motion equations under different

fill levels and accelerations, the loading conditions can be obtained and applied to the

corresponding attachment locations on the subframe in order to do the finite element

analysis.

A.3 Finite element model of the subframe

The subframe studied in this research has two symmetric longerons that are built

with flat plates. In the fifth wheel zone, several vertical flat plates are introduced for the

transverse connections of the longerons. Horizontal flat plates are added to the top of

these vertical plates to provide an installation area for the fifth wheel. In the front, two

pipes with rectangular hollow sections are used to provide connections for the longerons.

Vertical flat plates are also included in the lateral direction in the front to provide the

front tank with additional supports. At the bottom, horizontal plates are used to provide

the installation areas of the attachment for suspension brackets. On the top of the

197


suspensions according to the force and moment equilibrium requirements (Wong, 1993).

Because the vehicle experiences straight-line traveling during accelerating and braking

operations, the mass centres o f liquid cargo in both the front and rear tanks shift in the

longitudinal direction. The free surface gradient of the liquid can be obtained by a basic

fluid mechanics equation under constant accelerations by Eq. (4.80). Since only

geometrical information is needed to get the mass centre locations o f the liquid bulk

inside the tank, they can be obtained in a 3D solid modelling program for different tanks,

fill levels and accelerations.

In the following analysis, all geometric and physical information is from internal

drawings from MaXfield Inc., Calgary. By solving the motion equations under different

fill levels and accelerations, the loading conditions can be obtained and applied to the

corresponding attachment locations on the subframe in order to do the finite element

analysis.

A.3 Finite element model of the subframe

The subframe studied in this research has two symmetric longerons that are built

with flat plates. In the fifth wheel zone, several vertical flat plates are introduced for the

transverse connections o f the longerons. Horizontal flat plates are added to the top of

these vertical plates to provide an installation area for the fifth wheel. In the front, two

pipes with rectangular hollow sections are used to provide connections for the longerons.

Vertical flat plates are also included in the lateral direction in the front to provide the

front tank with additional supports. At the bottom, horizontal plates are used to provide

the installation areas of the attachment for suspension brackets. On the top of the

197


longerons, plates are used in the back to provide reinforcement. The geometric model and

finite element model of the subframe are shown in Figure A.2. The finite element model

of the subframe has been developed using ALGOR. The 3D plate/shell element type in

ALGOR is employed since the subframe is mainly built with welded flat plates of

different thickness. Loading conditions are applied to corresponding attachment locations

according to the calculated results from the vehicle model under different fill levels and

accelerations.

0/

(b)

Figure A.2 Subframe model

198


longerons, plates are used in the back to provide reinforcement. The geometric model and

finite element model o f the subframe are shown in Figure A. 2. The finite element model

o f the subframe has been developed using ALGOR. The 3D plate/shell element type in

ALGOR is employed since the subframe is mainly built with welded flat plates of

different thickness. Loading conditions are applied to corresponding attachment locations

according to the calculated results from the vehicle model under different fill levels and

accelerations.

Figure A. 2 Subframe model


(a) geometric model (b) finite element model

A.4 Results and discussion

Figure A.3 shows the load shift for different accelerations when the fill level is

30% in the front tank and 70% in the rear tank. As shown in the figure, when the vehicle

is accelerating, as the acceleration increases (indicated by the arrow), the centre of mass

in the front tank moves closer to the front part of the subframe, and the centre of mass in

the rear tank moves away from the rear part of the subframe. The load shift in the

longitudinal direction is much larger than that in the vertical direction, due to the

geometric shape of the tanks. Also, the load shifts at low fill levels are much larger than

at high fill levels. For a fully filled tank, no load shift will occur. The influence of liquid

motion does not exist for this situation. The load shifts for the same conditions under

braking are shown in Figure A.4. When the vehicle is braking, as the acceleration

increases, the centre of mass in the front tank moves away from the front part of the

subframe and the centre of mass in the rear tank moves closer to the rear part of the

subframe.

Figures A.5 and A.6 show the loads at the hitch point in both longitudinal and

vertical directions during accelerating and braking. The force in the longitudinal direction

increases with the increase in the acceleration and fill level. The negative sign means that

the force in the longitudinal direction is in the opposite direction when braking. The

increase is larger at a high fill level and acceleration. The vertical force increases with the

increase in total payload. During acceleration, the vertical force decreases with the

increase in acceleration because the cargo moves away from the fifth wheel zone. During

braking, the vertical force increases with the increase in acceleration because the load

199


(a) geometric model (b) finite element model

A.4 Results and discussion

Figure A.3 shows the load shift for different accelerations when the fill level is

30% in the front tank and 70% in the rear tank. As shown in the figure, when the vehicle

is accelerating, as the acceleration increases (indicated by the arrow), the centre o f mass

in the front tank moves closer to the front part o f the subframe, and the centre o f mass in

the rear tank moves away from the rear part o f the subframe. The load shift in the

longitudinal direction is much larger than that in the vertical direction, due to the

geometric shape o f the tanks. Also, the load shifts at low fill levels are much larger than

at high fill levels. For a fully filled tank, no load shift will occur. The influence o f liquid

motion does not exist for this situation. The load shifts for the same conditions under

braking are shown in Figure A.4. When the vehicle is braking, as the acceleration

increases, the centre o f mass in the front tank moves away from the front part o f the

subframe and the centre of mass in the rear tank moves closer to the rear part of the

subframe.

Figures A.5 and A.6 show the loads at the hitch point in both longitudinal and

vertical directions during accelerating and braking. The force in the longitudinal direction

increases with the increase in the acceleration and fill level. The negative sign means that

the force in the longitudinal direction is in the opposite direction when braking. The

increase is larger at a high fill level and acceleration. The vertical force increases with the

increase in total payload. During acceleration, the vertical force decreases with the

increase in acceleration because the cargo moves away from the fifth wheel zone. During

braking, the vertical force increases with the increase in acceleration because the load

199


shifts closer to the fifth wheel zone. The change in the loading conditions is influenced by

the acceleration, payload (fill level), and load shift.

1

1

2 3 4 5 6 7

Length (m)

0.6g

2 3 4 5 6 Length (m)

Figure A.3 Load shift during acceleration

(a) front tank, fill level: 30% (b) rear tank, fill level: 70%

(line — new liquid surface, point — new centre of mass)

200


shifts closer to the fifth wheel zone. The change in the loading conditions is influenced by

the acceleration, payload (fill level), and load shift.

xi.SP'5

(a)

2

1.5

1 O.lg

0.50.6g

03 5 6 70 1 2 4

Length (m)

O.lgS

• • •

0.6g0.5

(b)Length (m)

Figure A.3 Load shift during acceleration


(line - new liquid surface, point - new centre o f mass)

200


2

1.5 an

"U 1

0.5

0 0

(a) 1 2 3 4 5 6 7

Length (m)

Length (m)

Figure A.4 Load shift during braking


(line — new liquid surface, point — new centre of mass)

201


2

1 .5

O.lg1

0 . 5

0.6g0

0 1 2 3 4 5 6 7

Length (m)

O.lg

0.6g0.5

(b)Length (m)

Figure A.4 Load shift during braking


(line - new liquid surface, point - new centre o f mass)

201


(a)

(b)

0

0.8 0.6

0.4

Fill level 0.2

0

0.8 0.6

0.4

Fill level

x 105 Force (N)

x 105 Force (N)

-meL_M._•mW-111L11111‘.I N_ a.-1116.1.6.‘1W

0.4

e---___ --- '

0.2

0.2 0.4

Acceleration (g)

0.2

Acceleration (g)

Figure A.5 Forces at hitch point during acceleration

(a) longitudinal (b) vertical

202


(a) 5 Force (N)

0.80.6

0.4Fill level 0 2

0.40.2

Acceleration (g)

(b) .5 Force (N)

Fill level 0 2 0Acceleration (g)

Figure A. 5 Forces at hitch point during acceleration


202


(a)

(b)

x 1 0

0.8

0 0.8

0.6 0.4

Fill level 0.2

5 Force (N)

0.6 0.4

Fill level 0.2

x 105 Force (N)

0.2

0.6 0.4

Acceleration (g)

Acceleration (g)

Figure A.6 Forces at hitch point during braking


203


(a) Force (N)

0.2Fill level 0

Acceleration (g)

(b) . a5 Force (N) x 10 v ’

0.80.6

0.4Fill level ^

0.40.2

Acceleration (g)

Figure A.6 Forces at hitch point during braking


203


(a)

Force (N)

600

400

200

0 0.8

0.6 0.4

Fill level

(b) x 10

4 Force (N)

0.8 0.6

0.4

Fill level 0' 2.

0.2

0.6 0.4

0.2

Acceleration (g)

0.2

0.6 0.4

Acceleration (g)

Figure A.7 Force difference between the liquid cargo and equivalent rigid cargo

at hitch point during acceleration (a) longitudinal (b) vertical

204


(a)

Force (N)

600

400

200

0.80.6 0.40.4 0.20.2Fill level

Acceleration (g)

(b) 1 Force (N) x 10

0.8 0.60.6 / 0.4 <

Fill level

0.40.2

Acceleration (g)

Figure A.7 Force difference between the liquid cargo and equivalent rigid cargo

at hitch point during acceleration (a) longitudinal (b) vertical

204


To study the influence of the cargo load shift on the structure, it is assumes that

the vehicle with rigid cargo having the same weight is under the same fill level and

acceleration. The differences of the forces at the hitch point between the equivalent rigid

cargo and liquid cargo are shown in Figure A.7. The difference increases with an increase

in acceleration. It was also found that the difference has a larger value when the fill level

is between 40% and 60%. At low fill levels, the total cargo payload is low and the

influence is small, though the cargo load shift is large. At high fill levels, the shift of the

liquid cargo is quite small and the total influence is also small. The longitudinal force

difference is much smaller than the vertical force difference, because the load shift in the

vertical direction is much smaller than that in the longitudinal direction.

Though the influence of the load shifts on the structural loading conditions can be

obtained as above, the characteristics of the structural stresses can only be revealed by

finite element analysis. Figure A.8 shows an example of the stress distributions from the

analysis of the subframe during accelerating and braking with a fill level of 50% for both

the front tank and the rear tank. Areas with stresses under 30 MPa are greyed in this

figure. The von Mises stress distribution shows that large stress values occur at three

areas: the suspension attachment locations, the fifth wheel installation areas, and the front

parts of reinforcing plates on the top of the longerons. However, the normal stresses in

the longitudinal direction show that there are always compressive stresses in the first two

areas and they are not discussed because only tensile stresses initiate cracks in the

structures. It can found that the bending effect induced by the loading conditions on the

front part of the reinforcing plates on the top of the longerons is the main factor that

should be taken care of for structural integrity. Though the operations considered in this

205


To study the influence o f the cargo load shift on the structure, it is assumes that

the vehicle with rigid cargo having the same weight is under the same fill level and

acceleration. The differences of the forces at the hitch point between the equivalent rigid

cargo and liquid cargo are shown in Figure A.7. The difference increases with an increase

in acceleration. It was also found that the difference has a larger value when the fill level

is between 40% and 60%. At low fill levels, the total cargo payload is low and the

influence is small, though the cargo load shift is large. At high fill levels, the shift of the

liquid cargo is quite small and the total influence is also small. The longitudinal force

difference is much smaller than the vertical force difference, because the load shift in the

vertical direction is much smaller than that in the longitudinal direction.

Though the influence o f the load shifts on the structural loading conditions can be

obtained as above, the characteristics o f the structural stresses can only be revealed by

finite element analysis. Figure A.8 shows an example o f the stress distributions from the

analysis o f the subframe during accelerating and braking with a fill level o f 50% for both

the front tank and the rear tank. Areas with stresses under 30 MPa are greyed in this

figure. The von Mises stress distribution shows that large stress values occur at three

areas: the suspension attachment locations, the fifth wheel installation areas, and the front

parts o f reinforcing plates on the top o f the longerons. However, the normal stresses in

the longitudinal direction show that there are always compressive stresses in the first two

areas and they are not discussed because only tensile stresses initiate cracks in the

structures. It can found that the bending effect induced by the loading conditions on the

front part of the reinforcing plates on the top of the longerons is the main factor that

should be taken care of for structural integrity. Though the operations considered in this

205


study do not include all operations that may lead to structural failure, it is believed that

they are among the major causes. The following analyses focus on the influence of the fill

levels, accelerations and load shifts at these areas.

(a)

Load Case: 1 of 1 Maximum Value: 2.02589e+008 N/(mA2) Minimum Value: 0 N/(mA2)

(b)

Load Case: 1 of 1 Maximum Value: 1.25061e+008 N/LmA2i Minimum Value: -1.10376e+008 I\1/(mAz)

Stress von Mines

W(rn^2)

2.028e+008 :

1.688e+008 1.351 e+005

1.013 e+008

6.753e+007 3.3780+007 0

Stress Tensor X-X

Nkrn^2)

1.251e+008 8.582e+007

4.858 e+007

7.342e+006 -3.19e+007

-7.114e+007 -1.104e+008

Figure A.8 Stress distributions of the subframe

(Areas under 30 MPa are greyed. Circled areas are the ones that have large stress values.)

(a) von Mises stress (b) normal stress in x direction

206


study do not include all operations that may lead to structural failure, it is believed that

they are among the major causes. The following analyses focus on the influence o f the fill

levels, accelerations and load shifts at these areas.

L oad C ase : 1 of 1Maximum Value: 2 .0 2 5 8 9 e+ 0 0 8 N/(mA2) Minimum Value: 0 N/(mA2)

S t r e s s

v o n M is e s

N / ( m A2 )

2 . 0 2 6 e + 0 0 8

1 . 6 8 8 * + 0 0 8

1 . 3 5 1 * + 0 0 8

1 . 0 1 3 e + 0 0 8

6 . 7 5 3 e + 0 0 7

3 . 3 7 6 « + 0 0 7

0

(b)T e n s o r X - X

N/fmA2)

1 . 2 5 1 e + 0 0 8

8 , 5 8 2 e + 0 0 7

4 . 6 5 8 e + 0 0 7

7 . 3 4 2 e + 0 0 6

• 3 . 1 9 6 + 0 0 7

- 7 . 1 1 4 e + 0 0 7

- 1 . 1 0 4 e + 0 0 8

Load C ase: 1 of 1Maximum Value: 1.25061 e+008 IM/(mA2 Minimum Value: - 1 .10376e+ 008 W

Figure A.8 Stress distributions of the subframe

(Areas under 30 MPa are greyed. Circled areas are the ones that have large stress values.)

(a) von Mises stress (b) normal stress in x direction

206


In order to study the influence of acceleration, node 1844 on one side of the

subframe (Figure A.8 b), which has the largest von Mises stress in the above area, is

selected as the critical node under different conditions. The stress for this node is

calculated by taking the average value of all the elements in which this node appears. The

von Mises stresses under different accelerations during acceleration and braking are

recorded and plotted with the lines marked with diamonds in Figure A.9. As can be seen,

during braking, the von Mises stress at the critical node increases with the increase in

acceleration. During acceleration, the von Mises stress decreases with the increase in

acceleration. Actually, during acceleration, the longitudinal force at the hitch point

increases in the opposite direction of the vehicle and partly counteracts the bending effect

at the critical area. While during braking, the longitudinal force at the hitch point

increases in the same direction of the vehicle and enhances the bending effect at the

critical area. It can also be found that the von Mises stress during braking is much larger

than that of accelerating. It should be noted that the stresses under all conditions are

below the static yielding strength of the material. Therefore, the original design meets the

static strength requirement quite well. However, according to Figure A.9, the stress

differences between accelerating, driving at the constant speed, and braking are quite

significant. When considering the stress variations caused by repeated accelerating and

braking, which are quite common during normal driving, the alternating stress component

is also quite significant. Improvement of the original design can be achieved by further

reinforcement in the studied areas. Local structural redesign, such as the addition of

webs, could be employed to enhance the structural durability without making too much

change to the original design.

207


In order to study the influence of acceleration, node 1844 on one side of the

subframe (Figure A.8 b), which has the largest von Mises stress in the above area, is

selected as the critical node under different conditions. The stress for this node is

calculated by taking the average value of all the elements in which this node appears. The

von Mises stresses under different accelerations during acceleration and braking are

recorded and plotted with the lines marked with diamonds in Figure A.9. As can be seen,

during braking, the von Mises stress at the critical node increases with the increase in

acceleration. During acceleration, the von Mises stress decreases with the increase in

acceleration. Actually, during acceleration, the longitudinal force at the hitch point

increases in the opposite direction o f the vehicle and partly counteracts the bending effect

at the critical area. While during braking, the longitudinal force at the hitch point

increases in the same direction o f the vehicle and enhances the bending effect at the

critical area. It can also be found that the von Mises stress during braking is much larger

than that o f accelerating. It should be noted that the stresses under all conditions are

below the static yielding strength o f the material. Therefore, the original design meets the

static strength requirement quite well. However, according to Figure A.9, the stress

differences between accelerating, driving at the constant speed, and braking are quite

significant. When considering the stress variations caused by repeated accelerating and

braking, which are quite common during normal driving, the alternating stress component

is also quite significant. Improvement o f the original design can be achieved by further

reinforcement in the studied areas. Local structural redesign, such as the addition of

webs, could be employed to enhance the structural durability without making too much

change to the original design.

207


Von

Mis

es s

tres

s (N

/m2)

X 10$ 2.

2

1.8

1.6

1.4

1.2

1

0.8

0.6 -0.6

—With load shift $Without load shift

-0.4 -0.2 0.0 0.2

Acceleration (g)

0.4

Figure A.9 Von Mises stress at critical node

0.6

To further reveal the influence of the cargo load shift on the induced stresses, the

results of the equivalent rigid cargo are recorded to compare with those of the liquid

cargo, and are shown by the lines marked with circles in Figure A.9. It is obvious that

during braking, the von Mises stress at the critical node of the subframe under the

equivalent rigid cargo condition without cargo load shift is smaller than that under liquid

cargo condition with cargo load shift. During acceleration, the von Mises stress under the

equivalent rigid cargo condition is larger than that under liquid cargo condition. The

difference is especially significant during large accelerations. From the point of view of

failure criteria for metal fatigue in fatigue theories, though the increased stress during

braking is still below the static yielding strength under common operation conditions, and

the stress could even be lowered during acceleration, the stress increase during braking

208


■0— With load shift ■Q— Without load shift

£z

0.8

0.6 L -0.6 -0.4 - 0.2 0.0 0.2 0.4 0.6

Acceleration (g)

Figure A.9 Von Mises stress at critical node

To further reveal the influence of the cargo load shift on the induced stresses, the

results o f the equivalent rigid cargo are recorded to compare with those of the liquid

cargo, and are shown by the lines marked with circles in Figure A.9. It is obvious that

during braking, the von Mises stress at the critical node of the subframe under the

equivalent rigid cargo condition without cargo load shift is smaller than that under liquid

cargo condition with cargo load shift. During acceleration, the von Mises stress under the

equivalent rigid cargo condition is larger than that under liquid cargo condition. The

difference is especially significant during large accelerations. From the point of view of

failure criteria for metal fatigue in fatigue theories, though the increased stress during

braking is still below the static yielding strength under common operation conditions, and

the stress could even be lowered during acceleration, the stress increase during braking

208


and stress decrease during acceleration caused by the cargo load shift are extremely

harmful to the fatigue life of the structures. Under frequent acceleration and braking, the

combined effect of the acceleration and cargo load shift makes the alternating stress

component, which is the most important factor in determining the number of cycles of

load the material can withstand before fracture, much larger than that of the subframe

subjected only to acceleration. Therefore, the cargo load shift exerts an additional and

significant effect on the cyclic loadings on the subframe. It is detrimental to the structural

durability.

A.5 Summary

In this section, the influence of the accelerating and braking operations on the

structural strength of a B-train tank truck subframe has been investigated. The loading

conditions on an existing design of the subframe have been obtained using the pitch plane

vehicle model under constant accelerations. Finite element analysis of the subframe has

been implemented. The influence of the acceleration, fill level, and cargo load shift has

been studied. The analysis results show that the critical area is located on the front part of

the reinforcing plates on the top of the longerons, and that structural failure is mainly

caused by the bending effect at this area. Improvement of the existing design could be

achieved by further reinforcement at the studied locations. In addition to the inertia effect

induced by accelerations, the cargo load shift exerts an extra effect on the cyclic loadings

on the subframe. It has a detrimental influence on the structural durability.

As shown in Chapter 3 and Chapter 4, the results of the mass centre model only

show the mean values of the oscillatory forces and moments caused by the liquid motion

209


and stress decrease during acceleration caused by the cargo load shift are extremely

harmful to the fatigue life of the structures. Under frequent acceleration and braking, the

combined effect o f the acceleration and cargo load shift makes the alternating stress

component, which is the most important factor in determining the number o f cycles of

load the material can withstand before fracture, much larger than that o f the subframe

subjected only to acceleration. Therefore, the cargo load shift exerts an additional and

significant effect on the cyclic loadings on the subframe. It is detrimental to the structural

durability.

A.5 Summary

In this section, the influence o f the accelerating and braking operations on the

structural strength of a B-train tank truck subframe has been investigated. The loading

conditions on an existing design of the subframe have been obtained using the pitch plane

vehicle model under constant accelerations. Finite element analysis o f the subframe has

been implemented. The influence of the acceleration, fill level, and cargo load shift has

been studied. The analysis results show that the critical area is located on the front part of

the reinforcing plates on the top o f the longerons, and that structural failure is mainly

caused by the bending effect at this area. Improvement of the existing design could be

achieved by further reinforcement at the studied locations. In addition to the inertia effect

induced by accelerations, the cargo load shift exerts an extra effect on the cyclic loadings

on the subframe. It has a detrimental influence on the structural durability.

As shown in Chapter 3 and Chapter 4, the results of the mass centre model only

show the mean values of the oscillatory forces and moments caused by the liquid motion

209


immediately after the application of the acceleration. When the tank is subjected to a

suddenly applied acceleration, the oscillatory liquid motion causes oscillatory forces and

moments of considerable magnitudes, which are not only harmful to vehicle stability and

controllability, but also exert direct cyclical loadings on the supporting structures. The

stress analysis based on these cyclical loadings is very important for the fatigue life of

tank vehicle structures. To fully reveal the relationship between the liquid motion and the

structural strength and fatigue life, further studies are definitely necessary in order to

include the influence of the dynamic liquid behaviour under different operation

conditions on vehicle structures. With the integration of the method developed in the

previous chapters into dynamic vehicle models, structural analysis can be continued

based on the established finite element model and the loading conditions of the liquid-

vehicle-structure model.

210


immediately after the application of the acceleration. When the tank is subjected to a

suddenly applied acceleration, the oscillatory liquid motion causes oscillatory forces and

moments o f considerable magnitudes, which are not only harmful to vehicle stability and

controllability, but also exert direct cyclical loadings on the supporting structures. The

stress analysis based on these cyclical loadings is very important for the fatigue life of

tank vehicle structures. To fully reveal the relationship between the liquid motion and the

structural strength and fatigue life, further studies are definitely necessary in order to

include the influence of the dynamic liquid behaviour under different operation

conditions on vehicle structures. With the integration of the method developed in the

previous chapters into dynamic vehicle models, structural analysis can be continued

based on the established finite element model and the loading conditions o f the liquid-

vehicle-structure model.

210


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FLUID DYNAMICS IN HORIZONTAL CYLINDRICAL