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HANDLING CHARACTERISTICS OF TRACKED VEHICLES ON NON-DEFORMABLE SURFACES by Chi-Feng Chiang, B.Eng. A thesis submitted to the Faculty o f Graduate Studies and Research in partial fùlfillment of the requirements for the degree of Master o f Engineering Ottawa-Carleton lnstitute for Mechanical and Aerospace Engineering Department of Mechanicd and Aerospace Engineering Carleton University Ottawa, Ontario June 1999 O Copyright Chi-Feng Chiang, 1999

CHARACTERISTICS OF TRACKED VEHICLES ON€¦ · Tracked vehicle steering has been a subject of interest to the off-road vehicle industry ... Tracked Vehicles for Trapezoidal Load Distribution

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Page 1: CHARACTERISTICS OF TRACKED VEHICLES ON€¦ · Tracked vehicle steering has been a subject of interest to the off-road vehicle industry ... Tracked Vehicles for Trapezoidal Load Distribution

HANDLING CHARACTERISTICS OF TRACKED VEHICLES ON

NON-DEFORMABLE SURFACES

by

Chi-Feng Chiang, B.Eng.

A thesis submitted to

the Faculty of Graduate Studies and Research

in partial fùlfillment of

the requirements for the degree of

Master of Engineering

Ottawa-Carleton lnstitute

for Mechanical and Aerospace Engineering

Department of Mechanicd and Aerospace Engineering

Carleton University

Ottawa, Ontario

June 1999

O Copyright Chi-Feng Chiang, 1999

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National Library !*u of Canada Bibliothèque nationale du Canada

Acquisitions and Acquisitions et Bibliographie Services services bibliographiques

395 Wellington Street 395. nie Wellington OnawaON K I A M OnawaON K I A W Canada canada

The author has granted a non- exclusive licence aliowing the National Library of Cauada to reproduce, loan, distribute or seii copies of this thesis in microform, paper or electronic formats.

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Page 3: CHARACTERISTICS OF TRACKED VEHICLES ON€¦ · Tracked vehicle steering has been a subject of interest to the off-road vehicle industry ... Tracked Vehicles for Trapezoidal Load Distribution

Abstract

Tracked vehicle steering has been a subject of interest to the off-road vehicle industry

since it was invented. Over the years a large nurnber of models for tracked vehicie steering

have been proposed. In most of the rnodels developed so far. Coulomb's law of fiction is

used in the prediction of the interacting forces between the tracked vehicle and the ground

dunng steering. This means that the full fictional force will be developed as soon as a smdl

relative motion takes place between the track and the ground. However, experimental

evidence has shown that the shear stress developed on the track-ground interface is

dependent upon the shear displacement. and the maximum shear stress will be developed

onIy afier a certain shear displacement has taken place.

A detailed study of the mechanics of tracked vehicle steering on non-defonnable

surfaces. taking into account the effects of shear displacement on the development of shear

stress has been carried out. Expenmental studies on the shear stress-shear dispiacement

relationship for a track link with rubber pad. used in the M l 13 armoured personnel carrier,

on asphah have been conducted. Based on the results of both anaiytical and experimentai

studies, a general theory for the handling of tracked vehicles on non-deformable surfaces has

been developed.

The basic features of the general theory are substantiated by available field measured

data. It is show that predictions of sprocket torquss and moments of turning resistance as a

function of turning radius, based on the general theory, bear a close resemblance to the

available experimental data.

With experimental substantiation of its basic features. the general theory has been

Page 4: CHARACTERISTICS OF TRACKED VEHICLES ON€¦ · Tracked vehicle steering has been a subject of interest to the off-road vehicle industry ... Tracked Vehicles for Trapezoidal Load Distribution

employed to evaluate the effects of major design parameters on the maneuverability of

tracked vehicles on non-defonnable surfaces. It is found that among the design parameters

exarnined. track contact length and tread of the vehicle have significant influence on tracked

vehicle steering while track width and location of center of gravity have less significant

etTec ts.

it is believed that this research makes a contribution to a better understanding of the

handling of tracked vehicles on non-deformable surfaces and that it provides a basis for the

further study of tracked vehicle maneuverability on deforrnable terrain.

ii i

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Ackaowleàgements

The author wishes to extend his deepest gratitude to his thesis supervisor, Dr. J.Y.

Wong. for his invaluable guidance and suggestions throughout the study, as well as the

preparation of the thesis.

The assistance provided by Mr. Y.C. Wu in setting up the apparatus for investigating

track link-ground shearing charactenstics is also greatly appreciated.

Finally, the author wishes to thank his colleague. Dr. Peijun Xu. for his assistance in

conducting the experiments.

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Table of Contents

Page

Acceptance Sheet

A bs trac t

Ackno wledgements

Table of Contents

List of TabIes

List of Figures

List of Symbols

iv

v

. S .

vi11

xv

xxvi

1. Introduction 1

2. Review of the State-of-Art on the Study of the Handling of Tracked Vehicles 3

2.1 . Steeds' Mode1 3

2.2. Crosheck's Mode1 9

2.3. Kitano's Mode1 10

2.4. Ehlert's Mode1 11

3. A General Theory for the Mechanics of Steering of Tracked Vehicles

on Non-deformable Surfaces 16

3.1. Skid-Steering on Non-deformable Surfaces Including the Effect of

Trac k W idth 16

3.1.1. Shear Displacement of the Outer Track 16

3.1.2. Shear Displacement of the Imer Track 21

3.1.3. Kinetics of Tracked Vehicles during a Steady-State Tuniing Maneuver 23

3.2. Normal Pressure Distribution Under the Tracks 28

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4. Simulation of Handling Behaviour of Tracked Vehicles using the Proposed

General Theory

4.1. Basic Design Puameters of the Vehicle Simulated

4.2. Simulation Results

4.3. Simulation of Turning Resistance Moment

4.4. Summary

5. Evaluation of the Effects of Design and Operating Factors on the Handting

of Tracked Vehicles using the General Theory

5.1. Contact Length. L

5.2. Tread of Vehicle. B

5.3. Longitudinal Offset of Vehicle Center of Gravity. c,

5.4. Track Width. b

5.5. Height of Vehicle Center of Gravity, h

5.6. Summary

6. Enperimental Study of the Shearing Characteristics of a Representative

Track Link on Asphalt

6.1. Apparatus for Measuring the Shearing Characteristics

6.2. Analysis of Shear Stress-Shear Displacement Data

6.3. The Determination of Coeff~cient of Friction

7. Simulation of Steering Behaviour of an M l 13 Armoured Personnel Carrier

on Non-deformable Cround Based on Measured Shear Data

7.1. Basic Design Parameters of an M 1 13 Armoured Personnel Carrier

7.2. Simulation Results

7.3. Summary

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8. Discussion and Conclusions

8.1. Conclusions

8.2. Future Work

Bibliography

Appendix

-4. The Effects of Design and Operating Factors on the Handling of

Tncked Vehicles under Continuous Trapezoidal Load Distribution

over the Entire Track

B. The Effects of Design and Operating Factors on the Handling of

Tracked Vehicles for Trapezoidal Load Distribution on the Track

Pitch under each Roadwheel

C. The Effects of Design and Operating Factors on the Handling of

Tracked Vehicles for Concentrated Load under each Roadwheel

vii

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List of Tables

Table Description

Cornparison of steering models for tracked vehicles

Basic design parameters of the Jaguar tracked vehicle used in the simulation

Sprocket torques of the outer track at various turning radii with a speed

of 7.5 km/h

Sprocket torques of the inner track at various tuming radii with a speed

of 7.5 km/h

Sprocket torques of the outer track at various turning radii with a speed

of 14.3 km/h

Sprocket torques of the inner track ai various turning radii with a speed

of 14.2 kmh

Sprocket torques of the outer track at various tuming radii with a speed

of 2 1.3 km/h

Sprocket torques of the inner track at various turning radii with a speed

of21.3 km/h

Sprocket torques of the outer track at various turning radii with a speed

of 29 km/h

Sprocket torques of the inner track at various turning radii with a speed

of 29 km/h

Basic parameters of the M 1 1 3 tracked vehicIe used in the predictions

Sprocket torques at various turning radii with a vehicle speed of 7.5 km/h for an Ml 13 armoured personnel carrier

Sprocket torques at various tuming radii with a vehicle speed of 14.2 kmlh for an M 1 13 armoured personnel carrier

Sprocket torques at various tuming radii with a vehicle speed of 21.3 km, for an M 1 1 3 armoured personnel carrier

Sprocket torques at various tuming radii with a vehicle speed of 29 km/h

viii

Page

15

30

36

36

37

37

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for an M 1 13 armoured personnel carrier 1 04

Sprocket torques at various turning radii with different track-ground contact

lengths at a vehicle speed of 7.5 kmlh under continuous trapezoidal load

distribution over the entire track

Sprocket torques at various turning radii with different track-ground contact

lengths at a vehicle speed of 14.2 km/h under continuous trapezoidal load

distribution over the entire track

Sprocket torques at various tuming radii with different traçk-ground contact

lengths at a vehicle speed of 2 1.3 kmlh under continuous trapezoidal load

distribution over the entire track

Sprocket torques at various turning radii with di fferent track-ground contact

lengths at a vehicle speed of 29 km/h under continuous trapezoidal load

distribution over the entire track

Sprocket torques at various turning radii with difTerent treads at a vehicle

speed of 7.5 kmh under continuous trapezoidal load distribution over the

entire track

Sprocket torques at various turning radii with different treads at a vehicte

speed of 14.2 krn/h under continuous trapezoidal load distribution over the

entire track 123

Sprocket torques at various turning radii with different treads at a vehicle

speed of 2 1.3 km/h under continuous trapezoidal load distribution over the

entire track 124

Sprocket torques at various tuming radii with different treads at a vehicle

speed of 29 km/h under continuous trapezoidal load distribution over the

entire track

Sprocket torques at various tuming radii with different longitudinal CG

offsets at a vehicle speed of 7.5 km/h under continuous trapezoidai load

distribution over the entire track

Sprocket torques at various turning radii with different longitudinal CG

offsets at a vehicle speed of 14.2 kmih under continuous trapezoidal load

distribution over the entire track

Sprocket torques at various turning radii with different longitudinal CG

Page 11: CHARACTERISTICS OF TRACKED VEHICLES ON€¦ · Tracked vehicle steering has been a subject of interest to the off-road vehicle industry ... Tracked Vehicles for Trapezoidal Load Distribution

offsets at a vehicle speed of 2 1.3 km& under continuous trapezoidal load

distribution over the entire track

Sprocket torques at various tuming radii with different longitudinal CG

offsets at a vehicle speed of 29 km/h under continuous trapezoidal load

distribution over the entire track

Sprocket torques at various turning radii with different track widths at a

vehicle speed of 7.5 kmlh under continuous trapezoidal Ioad distribution

over the entire track

Sprocket torques at various tuming radii with different track widths at a

vehicle speed of 14.2 km/h under continuous trapezoidal load distribution

over the entire track

Sprocket torques at various tuming radii with different track widths at a

vehicle speed of 2 1.3 km/h undrr continuous trapezoidal load distribution

over the entire track

Sprocket torques at various turning radii with different track widths at a

vehicle speed of 29 km/h under continuous trapezoidal load distribution

over the entire track

Sprocket torques at various turning ndii with different CG heights at a

vehicle speed of 7.5 km/h under continuous trapezoidal load distribution

over the entire track

Sprocket torques at various turning radii with different CG heights at a

vehicle speed of 14.2 km/h under continuous trapezoidal load distribution

over the entire track

Sprocket torques at various turning radii with different CG heights at a

vehicle speed of 2 1.3 km/h under continuous trapezoidal load distribution

over the entire track

Sprocket torques at various turning radii with different CG heights at a

vehicle speed of 29 km/h under continuous trapezoidal load distribution

over the entire track

S procket torques at various turning radii with different track-ground

contact lengths at a vehicle speed of 7.5 km/h for trapezoidal load

distri bution on the track pitch under each roadwheel

Page 12: CHARACTERISTICS OF TRACKED VEHICLES ON€¦ · Tracked vehicle steering has been a subject of interest to the off-road vehicle industry ... Tracked Vehicles for Trapezoidal Load Distribution

B-2

B-3

B-4

B-5

B-6

B-7

B-8

B-9

B-IO

B-1 1

B-12

Sprocket torques at various tuming radii with diflkrent track-ground

contact lengths at a vehicle speed of 14.2 kmh for trapezoidal load

distribution on the track pitch under each roadwheel

Sprocket torques at various turning radii with different track-ground

contact lengths at a vehicle speed of 2 1.3 km/h for trapezoidal load

distribution on the track pitch under each roadwheel

Sprocket torques at various tuming radii with different track-ground

contact lengths at a vehicle speed of 29 km/h Tor trapezoidal load

distribution on the track pitch under each roadwheel

Sprocket torques at various turning radii with different treads at a

vehicle speed of 7.5 km/h for trapezoidal load distribution on the

track pitch under each roadwheel

Sprocket torques at various tuming radii with different treads at a

vehicle speed of 14.2 km/h for trapezoidal load distribution on the

trac k pi tc h under each roadwheel

Sprocket torques at various tuming radii with different treads at a

vehicle speed of 2 1 -3 km/h for trapezoidal load distribution on the

track pitch under each roadwheel

Sprocket torques at various tuming radii with different treads at a

vehicle speed of 29 km/h for trapezoidal load distribution on the

trac k pitc h under each roadwhecl

Sprocket torques at various turning radii with different longitudinal CG

offsets at a vehicle speed of 7.5 krri/h for trapezoidal load distribution

on the track pitch under each roadwheel

Sprocket torques at various tuming radii with different longitudinal CG

offsets at a vehicle speed of 14.2 km/h for trapezoidal load distribution

on the track pitch under each roadwheel

Sprocket torques at various tuming radii with different longitudinal CG

offsets at a vehicle speed of 2 1.3 krnh for trapezoidal load distribution

on the track pitch under each roadwhee1

Sprocket torques at various turning radii with different longitudinal CG

offsets at a vehicle speed of 29 km/h for trapezoidal load distribution

Page 13: CHARACTERISTICS OF TRACKED VEHICLES ON€¦ · Tracked vehicle steering has been a subject of interest to the off-road vehicle industry ... Tracked Vehicles for Trapezoidal Load Distribution

on the track pitch under each roadwheel

Sprocket torques at various tuming radii with different track widths at a

vehicle speed of 7.5 kmih for trapezoidal load distnbution on the track

pitch under each roadwheel

Sprocket torques at various turning radii with different track widths at a

vehicle speed of 14.2 km/h for trapezoidal load distribution on the track

pitch under each roadwheel

Sprocket torques at various tuming radii with different track widths at a

vehicle speed of 2 1.3 h / h for trapezoidal load distribution on the track

pitch under each roadwheel

Sprocket torques at various turning radii with diflerent track widths at a

vehicle speed of 29 kmh for trapezoidal load distribution on the track

pitch under each roadwheel

Sprocket torques at various turning radii with dii'ferent CG heights at a

vehicle speed of 7.5 km/h for trapezoidal load distribution on the track

pitch under each roadwheel

Sprocket torques at various turning radii with different CG heights at a vehicle speed of 14.2 km/h for trapezoidal load distribution on the track

pitch under each roadwheel

Sprocket torques at various turning radii uith different CG heights at a

vehicle speed of 2 1.3 h h for trapezoidal load distribution on the track

pitch under each roadwheel

Sprocket torques at various turning radii with different CG heights at a

vehicle speed of 29 kmh for trapezoidal load distribution on the track pitch under each roadwheel

Sprocket torques at various tuming radii with different track-ground

contact lengths at a vehicle speed of 7.5 kmfh for concentrated load

under each roadwheel

Sprocket torques at various tuming radii with different track-ground

contact lengths at a vehicle speed of 14.2 kmh for concentrated load

under each roadwheel

S procket torques at various turning radi i with di fferent track-ground

xii

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contact lengths at a vehicle speed of 2 1.3 km/h for concentrated load

under each roadwheel

Sprocket torques at various turning radii with different track-ground

contact lengths at a vehicle speed of 29 km/h for concentrated load

under each roadwheel

Sprocket torques at various turning radii with different treads at a vehicle

speed of 7.5 kmh for concentrated load under each roadwheel

Sprocket torques at various turning radii with different treads at a vehicle

speed of 14.2 km/h for concentrated load under each roadwheel

Sprocket torques at various turning radii with different treads at a vehicle

speed of 2 1.3 kndh for concentrated load under each roadwheel

Sprocket torques at varioüs tming radii with different treads at a vehicle

speed of 29 km/h for concentrated load under each roadwheel

Sprocket torques at various turning radii with different tongitudinai CG offsets

at a vehicle speed of 7.5 km/h for concentrated load under each roadwheel

Sprocket torques at various turning radii with different longitudinal CG offsets

at a vehicle speed of 14.2 km/h for concentrated load under each roadwheel

Sprocket torques at various tuming radii uith different longitudinai CG offsets

at a vehicle speed of 2 1.3 km/h for concentrated load under each roadwheel

Sprocket torques at V ~ ~ O U S turning radii with different longitudinal CG offsets

at a vehicle speed of 29 km/h for concentrated load under each roadwheel

Sprocket torques at various turning radii with different track widths at a vehicle speed of 7.5 krnh for concentrated load under each roadwheel

Sprocket torques at various tuming radii with different track widths at a

vehicle sjxed of 14.2 km/h for concentrated load under each roadwheel

Sprocket torques at various turning radii with different track widths at a

vehicle speed of 2 1.3 km/h for concentrated load under each roadwheel

Sprocket torques at various turning radii with different track widths at a

vehicle speed of 29 km/h for concentrated load under each roadwheel

Sprocket torques at various tuming radii with different CG heights at a

vehicle speed of 7.5 km.h for concentrated load under each roadwheel

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C- 18 Sprocket torques at various tuming radii with different CG heights at a vehicle speed o f 14.2 km/h for concentrated load under each roadwheel

C- 19 Sprocket torques at various turning radii with different CG heights at a vehicle speed of 2 1.3 km/h tôr concentrated load under each roadwheel

C-20 Sprocket torques at various turning radii with different CG heights at a ~~ehic le speed o f 29 km/h for concentrated load under each roadwheel

xiv

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List of Figures

Figure Description

Kinetics of a tracked vehicle during a steady-state turn

Track force ellipse from Micklethwait

Kinetics of tracked vehicles during a steady-state turn.

Fundamentais of tracked vehicle steering considered in Hock's mode1

Kinematics of a tracked vehicle during a turning maneuver

Kinematics of a tracked vehicle during a steady-state tum

A shear curve of a simple exponential fonn

Kinetics of a tracked vehicle during a steady-state turn

Normal pressure distribution

Sprocket torques vs theoretical turning radius for a Jaguar at a vehicle speed

of 7.5 km/h with K = 0.075 m and p = 0.9 during a steady-state turn

Sprocket torques vs theoretical tuming radius for a Jaguar at a vehicle speed

of 14.2 kmh with K = 0.075 m and p = 0.9 during a steady-state tum

Sprocket torques vs theoretical tuming radius for a Jaguar at a vehicle speed

of 21 -3 km/h with K = 0.075 m and C( = 0.9 during a steady-state turn

Sprocket torques vs theoretical tuming radius for a Jaguar at a vehicle speed

of 29 km/h with K = 0.075 m and p = 0.9 during a steady-state turn

Lateral shear stress distribution of the outer track dong the longitudinal

centerline of the track-ground contact area predicted fiom the proposed

general theory at a vehicle speed of 14.2 km/h with different tuniing radii

Lateral shear stress distnbution of the inner track dong the longitudinal

centerline of the track-ground contact area predicted fiom the proposed

general theory at a vehicle speed of 14.2 km/h with different turning radii

Lateral shear stress distnbution of the outer track dong the longitudinal

centerl ine of the track-ground contact area predicted from S teeds' mode1 (K = O m) at a vehicle speed of 14.2 km/h with different tuming radii

xv

Page

5

6

8

13

18

19

23

25

28

32

33

34

35

42

42

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Lateral shear stress distribution of the inner track along the longitudinal

centerline of the track-ground contact area predicted fiom Steeds' mode1

(K = O m) at a vehicle speed of 14.2 km/h with different turning radii

Coefficient of turning resistance vs tuming radius for a Jaguar at a vehicle

speed of 7.5 km/h with K = 0.075 m under continuous trapezoidal load

distribution over the entire track

Coefficient of tuming resistance vs turning radius for a Jaguar at a vehicle

speed of 14.2 km/h with K = 0.075 m under continuous trapezoidal load

distribution over the entire track

Coefficient of tuming resistance vs turning radius for a Jaguar at a vehicle

speed of 2 1.3 km/h with K = 0.075 rn under continuous trapezoidal load

distribution over the entire track

Coefficient of tuming resistance vs turning radius for a Jaguar at a vehicle

spsed of 29 kmh with K = 0.075 m under continuous trapezoidal load

distribution over the entire track

Sprocket torques vs tuming radius for a Jaguar at a vehicle speed of

7.5 km% with different track contact lengths during a steady-state twn

under continuous trapezoidal load distribution over the entire track

Sprocket torques vs turning radius for a Jaguar at a vehicle speed of

14.2 km/h with different track contact lengths during a steady-state turn under continuous trapezoidal load distribution over the entire track

Sprocket torques vs tuming radius for a Jaguar at a vehicle speed of

2 1.3 krnh with different track contact lengths during a steady-state turn

under continuous trapezoidal load distribution over the entire track

Sprocket torques vs tuming radius for a Jaguar at a vehicle speed of

29 km/h with different track contact lengths during a steady-state tuni

under continuous trapezoidal load distribution over the entire track

Lateral forces vs tuming radius for a Jaguar at a vehicle speed of

7.5 km/h with different track contact lengths during a steady-state

turn under continuous trapezoidal load distribution over the entire track

Lateral forces vs tuming radius for a Jaguar at a vehicle speed of

1 4.2 km/h with different track contact lengths during a steady-state

\

xvi

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turn under continuous trapezoidal load distribution over the entire track

Lateral forces vs turning radius for a Jaguar at a vehicle speed of

2 1.3 kmh with different track contact lengths during a steady-state

turn under continuous trapezoidal load distribution over the entire track

Lateral forces vs tuming radius for a Jaguar at a vehicle speed of

29 km/h with different track contact lengths during a steady-state

turn under continuous trapezoidal load distribution over the entire track

Moments of turning resistance vs turning radius for a Jaguar at a vehicle

speed of 7.5 km/h with different track contact lengths during a steady-state

turn under continuous trapezoidal load distribution over the entire track

Moments of turning resistance vs turning radius for a Jaguar at a vehicle

speed of 14.2 kmh with different track contact lengths during a steady-state

turn under continuous trapezoidal load distribution over the entire track

Moments of turning resistance vs turning radius for a Jaguar at a vehicle

speed of 2 1.3 km/h with different track contact lengths during a steady-state

turn under continuous trapezoidal Ioad distribution over the entire track

Moments of turning resistance vs tuming radius for a Jaguar at a vehicle

speed of 29 krnfh with different track contact lengths dunng a steady-state

turn under continuous trapezoidal load distribution over the entire track

Ratio of tuming resistance moment vs contact length L for a Jaguar at a

vehicle speed of 7.5 kmh with different turning radii during a steady-state

turn under continuous trapezoidal load distribution over the entire track

Ratio of turning resistance moment vs contact length L for a Jaguar at a

vehicle speed of 14.2 km/h with different turning radii during a steady-state

turn under continuous trapezoidal load distribution over the entire track

Ratio of turning resistance moment vs contact length L for a Jaguar at a

vehicle speed of 2 1 -3 km/h with di fferent tuming radii during a steady-state

turn under continuous trapezoidal load distribution over the entire track

Ratio of tuming resistance moment vs contact length L for a Jaguar at a

vehicIe speed of 29 km/h with different turning radii during a steady-state

turn under continuous trapezoidal load distribution over the entire track

Sprocket torques vs tuming radius for a Jaguar at a vehicle speed of

xvii

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7.5 krnh with different vehicle treads during a steady-state turn under

continuous trapezoidal load distribution over the entire track

Sprocket torques vs turning radius for a Jaguar at a vehicle speed of

14.2 km/h with different vehicle treads during a steady-state turn under

continuous trapezoidal load distribution over the entire track

Sprocket torque vs turning radius for a Jaguar at a vehicle speed of

2 1 -3 krnh with different vehicle treads during a steady-state turn under

continuous trapezoidal load distribution over the entire track

Sprocket torques vs tuming radius for a Jaguar at a uehicte speed of

29 km/h with different vehicle treads during a steady-state turn under

continuous trapezoidal load distribution over the entire track 57

Lateral forces vs turning radius for a Jaguar at a vehicle speed of 7.5 km/h with different vehicle treads during a steady-state turn under continuous

trapezoidal load distri bution over the entire track 58

Lateral forces vs tuming radius for a Jaguar at a vehicle speed of 14.2 km/h with different vehicle trcads during a steady-state turn under continuous

trapezoidal ioad distribution over the entire track 58

Lateral forces vs turning radius for a Jaguar at a vehicle speed of 21 -3 km/h

with different vehicle treads during a steady-state turn under continuous

trapezoidal load distribution over the entire track 59

Laterai forces vs turning radius for a Jaguar at a vehicle speed of 29 km/h with different vehicle treads during a steady-state turn under continuous

trapezoidal load distribution over the entire track

Moments of tuming resistance vs tuming radius for a Jaguar at a vehicle

speed of 7.5 km/h with different vehicle treads during a steady-state tum

under continuous trapezoidal load distribution over the entire track

Moments of turning resistance vs turning radius for a Jaguar at a vehicle

speed of 14.2 km, with different vehicle treads during a steady-state turn

under continuous trapezoidal load distribution over the entire track

Moments of turning resistance vs turning radius for Jaguar at a vehicle

speed of 2 1.3 kmh with di fferent vehicle treads during a steady-state turn

under continuous trapezoidal load distribution over the entire track

xviii

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Moments of tuming resistance vs t m i n g radius for a Jaguar at a vehicle

speed of 29 lun/h with diflkrent vehicle treads during a steady-state tum

under continuous trapezoidal load distribution over the entire track

Ratio of magnitude of the sprocket toques vs vehicle tread B for a Jaguar at

a vehicle speed of 7.5 km/h with different tuming radii during a steady-state

turn under continuous trapezoidal load distribution over the entire track

Ratio of magnitude of the sprocket toques vs vehicle tread B for a Jaguar at

a vehicle speed of 14.2 km/h with different turning radii during a steady-state

tum under continuous trapezoidal load distribution over the entire track

Ratio of magnitude of the sprocket toques vs vehicle tread B for a Jaguar at

a vehicle speed of 21 -3 km/h with different turning radii during a steady-state

turn under continuous trapezoidal load distribution over the entire track

Ratio of magnitude of the sprocket toques vs vehicle tread B for a Jaguar at

a vehicle speed of 29 km/h with different tuming radii during a steady-state

turn under continuous trapezoidal load distribution over the entire track

Sprocket torques vs turning radius for a Jaguar at a vehicle speed of

7.5 km/h with different longitudinal CG offsets dwing a steady-state

turn under continuous trapezoidal load distribution over the entire traçk

Sprocket torques vs tuming radius for a Jaguar at a vehicle speed of

11.2 km/h with different longitudinal CG offsets during a steady-state

turn under continuous trapezoidal load distribution over the entire track

Sprocket torques vs turning radius for a Jaguar at a vehicle speed of

2 1.3 km/h with different longitudinal CG offsets during a steady-state

turn under continuous trapezoidal load distribution over the entire track

Sprocket torques vs turning radius for a Jaguar at a vehicle speed of

29 km/h with different longitudinal CG offsets during a steady-state

turn under continuous trapezoidal load distribution over the entire track

Lateral forces vs tumine radius for a Jaguar at a vehicle speed of 7.5 km/h with different longitudinal CG offsets during a steady-state turn under

continuous trapezoidal load distribution over the entire track

Lateral forces vs tuming radius for a Jaguar at a vehicle speed of 14.2 kmh with different longitudinal CG offsets during a steady-state turn under

xix

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continuous trapezoidal load distribution over the entire track 67

Lateral forces vs turning radius for a Jaguar at a vehicle speed of 21 -3 km/h with different longitudinal CG offsets dunng a steady-state hm under continuous trapezoidal load distribution over the entire track 68

Lateral forces vs turning radius for a Jaguar at a vehicle speed of 29 km/h with different IongitudinaI CG offsets during a steady-state t m under

continuous trapezoidal load distribution over the entire track 68

Moments of turning resistance vs turning radius for a Jaguar at a vehicie speed

of 7.5 km/h with different longitudinai CG offsets during a steady-state turn under continuous trapezoidal load distribution over the entire track 69

Moments of turning resistance vs turning radius for a Jaguar at a vehicle speed

of 14.2 kmh with different longitudinal CG offsets during a steady-state h m

under continuous trapezoidal load distribution over the entire track 69

Moments of turning resistance vs tuming radius for a Jaguar at a vehicle speed

of 21.3 km/h with different longitudinal CG offsets during a steady-state turn under continuous trapezoidal load distribution over the entire track 70

Moments of turning resistance vs tuming radius for a Jaguar at a vehicle speed

of 39 km/h with different longitudinal CG offsets during a steady-state turn

under continuous trapezoidal load distribution over the entire track

Ratio of tuming resistance moment vs longitudinal CG offset c, for a Jaguar at

a vehicle speed of 7.5 km/h with different tuming radii during a steady-state

turn under continuous trapezoidal load distribution over the entire track 71

Ratio of tuming resistance moment vs longitudinal CG offset c, for a Jaguar at

a vehicle speed of 14.2 kmh with different tuming radii during a steady-state

turn under continuous trapezoidal load distribution over the entire track 71

Ratio of turning resistance moment vs longitudinal CG offset c, for a Jaguar at

a vehicle speed of 2 1.3 km/h with different tuming radii during a steady-state

turn under continuous trapezoidal load distribution over the entire track 72

Ratio of turning resistance moment vs longitudinal CG offset cy for a Jaguar at

a vehicle speed of29 k m h with different turning radii during a steady-state

turn under continuous trapezoidal load distribution over the entire track 72

Sprocket torques vs tuming radius For a Jaguar at a vehicle speed of

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7.5 km/h with different track widths during a steady-state tum under

continuous trapezoidal load distribution over the entire track

Sprocket torques vs tuming radius for a Jaguar at a vehicle speed of

14.2 km/h with different track widths during a steady-state turn under

continuous trapezoidal load distribution over the entire track

Sprocket torques vs turning radius for a Jaguar at a vehicle speed of

2 1.3 km/h with different track widths during a steady-state turn under

continuous trapezoidal load distribution over the entire track

Sprocket torques vs turning radius for a Jaguar at a vehicle speed of

29 km/h with different track widths during a steady-state turn under

continuous trapezoidal load distribution over the entire track

Lateral forces vs turning radius for a Jaguar at a vehicle speed of 7.5 krnh with different track widths during a steady-state tum under continuous

trapezoidal load distribution over the entire track

Lateral forces vs tuming radius for a Jaguar at a vehicle speed of 14.2 kmh with different track widths during a steady-state turn under continuous

trapezoidal load distribution over the entire track 77

Lateral forces vs turning radius for a Jaguar at a vehicle speed of 21.3 kmh with different track widths during a steady-state turn under continuous

trapezoidal load distribution over the entire track

Lateral forces vs t m i n g radius for a Jaguar at a vehicle speed of 29 km/h with different track widths during a steady-state tum under continuous

trapezoidal load distribution over the entire track

Moments of turning resistance vs turning radius for a Jaguar at a vehicle

speed of 7.5 km/h with different track widths during a steady-state turn

under trapezoidal load distribution over the entire track

Moments of turning resistance vs tuming radius for a Jaguar at a vehicle

speed of 14.2 kmh with different track widths during a steady-state turn

under trapezoidal load distribution over the entire track

moments of turning resistance vs turning radius for a Jaguar at a vehicle

speed of 2 1.3 km/h with different track widths during a steady-state turn under trapezoidal load distribution over the entire track

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Moments of tuming resistance vs turning radius for a Jaguar at a vehicle

speed of 29 kmh with different track widths during a steady-state turn under trapezoidal load distribution over the entire track

Ratio of tuming resistance moment vs track width b for a Jaguar at a

vehicle speed of 7.5 km/h with different turning radii during a steady-state

turn under continuous trapezoidal load distribution over the entire track

Ratio of turning resistance moment vs track width b for a Jaguar at a vehicle speed of 14.2 km/h with different turning radii during a steady-state

turn under continuous trapezoidal load distribution over the entire track

Ratio of turning resistance moment vs track width b for a Jaguar at a

vehicle speed of 2 1.3 km/h with different turning radii during a steady-state

turn under continuous trapezoidal load distribution over the entire traçk

Ratio of turning resistance moment vs track width b for a Jaguar at a vehicle speed of 29 km/h with different turning radii during a steady-state

tum under continuous trapezoidal load distribution over the entire track

Sprocket torques vs tuming radius for a Jaguar at a vehicie speed of

7.5 km/h with different CG heights during a steady-state tum under

continuous trapezoidal load distribution over the entire track

Sprocket torques vs tuming radius for a Jaguar at a vehicle speed of

14.2 km/h with different CG heights during a steady-state tum under

continuous trapezoidal load distribution over the entire tmck

Sprocket torques vs tuming radius for a Jaguar at a vehicle speed of

2 1.3 km/h with different CG heights during a steady-state turn under

continuous trapezoidal load distribution over the entire track

Sprocket torques vs tuming radius for a Jaguar at a vehicle speed of

29 Iun/h with different CG heights during a steady-state turn under

continuous trapezoidal load distribution over the entire track

Lateral forces vs turning radius for a Jaguar at a vehicle speed of

7.5 km/h with different CG heights during a steady-state turn under

continuous trapezoidal load distribution over the entire track

Lateral forces vs turning radius for a Jaguar at a vehicle speed of

14.2 km/h with different CG heights during a steady-state tum under

xxii

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continuous trapezoidal load distribution over the entire track

Lateral forces vs tuming radius for a Jaguar at a vehicle speed of

2 1.3 km/h with different CG heights during a steady-state turn under

continuous trapezoidal load distribution over the entire track

Lateral forces vs turning radius for a Jaguar at a vehicle speed of

29 kmh with different CG heights during a steady-state turn under

continuous trapezoidal load distribution over the entire track

Moments of turning resistance vs turning radius for a Jaguar at a vehicle

speed of 7.5 km/h with di fferent CG heights during a steady-state tum

under continuous trapezoidal load distribution over the entire track

Moments of turning resistance vs tuming radius for a Jaguar at a vehicle

speed of 14.2 km/h with different CG heights during a steady-state turn

under continuous trapezoidal load distribution over the entire track

Moments of tuming resistance vs turning radius for a Jaguar at a vehicle

speed of 2 1.3 km/h with different CG heights during a steady-state tum

under continuous trapezoidal load distribution over the entire track

Moments of turning resistancc vs turning radius for a Jaguar at a vehicle

speed of 29 km/h with different CG heights during a steady-state tum

under continuous trapezoidal load distribution over the entire track

Ratio of turning resistance moment vs CG height h for a Jaguar at a vehicle

sperd of 7.5 km/h with different turning radii during a steady-state tum

under continuous trapezoidal load distribution over the entire track

Ratio of turning resistance moment vs CG height h for a Jaguar at a vehicle

speed of 14.2 km/h with different tuming radii dunng a steady-state tum

under continuous trapezoidal load distribution over the entire track

Ratio of turning resistance moment vs CG height h for a Jaguar at a vehicle

speed of 2 1.3 kmh with different tuming radii dunng a steady-state tum

under continuous trapezoidal load distribution over the entire track

Ratio of turning resistance moment vs CG height h for a Jaguar at a vehicle

speed of 29 km/h with different turning radii dunng a steady-state turn under

continuous trapezoidal load distribution over the entire track

Schematic of the test setup for the measurement of track shear force and shear

xxiii

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displacement on the asphalt

Block diagram of the data acquisition system for track shear force-shear

displacernent tester

Shear force-shear displacement curve at normal load 890 lb with sliding

speed O. 100 m/s

Shear force-shear displacement curve at normal load 890 lb with sliding

speed 0.145 m/s

Shear force-shear displacement curve at normal load 1000 lb with sliding

speed 0.100 m/s

Shear force-shear displacement curve at normal load 1000 lb with sliding

speed 0.145 m/s

Shear force-shear displacement c u v e at nomai load 1000 Ib with sliding

speed 0.195 m/s

Shear force-shear displacement curve at normal load 1 ZOO lb with sliding

speed 0.1 00 m/s

Shear force-shear displacement c w e at normal load 1 ZOO lb with sliding

speed 0.145 m/s

Shear force-shear displacement curve at normal load 1200 lb with sliding

speed 0.195 m/s

Shear force-shear displacement curve at normal load 1400 lb with sliding

speed 0.100 m / s

Shear force-shear displacement curve at normal load 1400 lb with sliding

speed 0.1 45 d s

Shear force-shear displacement curve at normal load 1400 lb with sliding

speed 0.195 m/s

The relation of maximum shear force and normal load

Sprocket torques vs tuming radius for an M 1 13 at a vehicle speed of

7.5 km/h with K = 0.0183 m and p = 0.684 during a steady-state tum

Sprocket torques vs tuming radius for an M 1 13 at a vehicle speed of 14.2 krn/h with K = 0.01 83 m and p = 0.684 during a steady-state tum

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Sproc ket torques vs tuming radius for an M 1 1 3 at a vehicle speed of

2 1 -3 h / h with K = 0.0 183 m and p = 0.684 during a steady-state turn

Sprocket torques vs tuming radius for an M 1 13 at a vehicle speed of

29 km/h with K = 0.01 83 m and p = 0.684 during a steady-state tum

Lateral force of the outer track vs turning radius for an M 1 13 at a vehicle speed

of 7.5 kmh with K = 0.0 183 rn and p = 0.684 dunng a steady-state tum

Lateral force of the imer track vs tuming radius for an M 1 13 at a vehicle speed

of 7.5 kmfh with K = 0.01 83 m and p = 0.684 dunng a steady-state turn

Laterai forces vs tming radius for an M 1 13 at a vehicle speed of 14.2 krnh with K = 0.01 83 m and p = 0.684 during a steady-state tum

Lateral forces vs turning radius for an M 1 13 at a vehicle speed of 2 1.3 km/h with K = 0.01 83 m and p = 0.684 during a steady-state turn Lateral forces vs tuming radius for an M 1 13 at a vehicle speed of 29 km/h

with K = 0.0 183 m and p = 0.684 during a steady-state turn

Moment of turning resistance of the outer track vs tuming radius for an

M 1 13 at a vehicle speed of 7.5 kmlh with K = 0.01 83 m and p = 0.684 dunng a steady-state turn

Moment of turning resistance of the inner tarck vs turning radius for an

M 1 13 at a vehicle speed of 7.5 kmh with K = 0.0 183 m and p = 0.684

during a steady-state turn

Moments of turning resistance vs tuming radius for an Ml 13 at a vehicle speed

of 1 - 4 2 km/h with K = 0.0 183 m and p = 0.684 during a steady-state turn

Moments of tuming resistance vs tuming radius for an Ml 13 at a vehicle speed

of 2 1.3 krn/h with K = 0.01 83 rn and p = 0.684 during a steady-state tum

Moments of turning resistance vs tuming radius for an M 1 13 at a vehicle speed

of 29 km/h with K = 0.01 83 m and p = 0.684 during a steady-state turn

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List of Symbols

slip radius of the i ~ e r track

slip radius of the outer track

track width

tread of vehicle

laterai offset of vehicle CG to the geometrical center of the vehicie

longitudinal offset of vehicle CG to the geometrical center of the vehicle

cone index of the soi1

track numeric of the j" track

coefticient of pull-slip equation

coefficient of pull-slip equation

coefficient of pull-slip equation

coefficient defining the extemal motion resistance

coefficient defining the extemal motion resistance

coefficient of extemal motion resistance

inner track longitudinal force

outer track longitudinal force

lateral force acting on the inner track

laterai force acting on the outer track

longitudinal force acting on the inner track

longitudina! force acting on the outer track

height of vehicle CG

track slip dong the longitudinal direction

shear displacement

shear displacement of the inner track

xxvi

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shear displacement of the outer track

X direction of shear displacement on the inner track in contact with the ground

X direction of shear displacement on the outer track in contact with the ground

Y direction of shear displacement on the i ~ e r track in contact with the ground

Y direction of shear displacement on the outer track in contact with the gound

track exponent

shear deformation parameter

trac k-ground contact length

track pitch

turning moment acting on the imer track

tuming moment acting on the outer track

moment of turning resistance acting on the inner track

moment of tuming resistance acting on the outer track

moment of turning resistance

normal pressure distribution

normal pressure distribution on the inner track

normal load distribution under the i" roadwheel and j* track

normal pressure distribution on the outer track

pitch radius of the sprocket

turning radius of the vehicle

lateral distance between the center of turn and vehicle CG

extemal motion resistance of j" track

turning radius at which both track forces are zero

external motion resistance

theoretical turning radius

extemal motion resistance on the imer track

external motion resistance on the outer track

xwii

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offset of turning center

time

mean speed of vehicle center of gravity

theoretical velocity of the inner track

sliding velocity of the i ~ e r track

sliding velocity of the outer track

lateral sliding velocity of a point (x,. y-) with respect to moving h m e of reference

iateral sliding velocity of a point (x,. y,) with respect to moving frame of reference

longitudinal sliding velocity a point (x,. y,) with respect to moving h e of

re ference.

longitudinal s!iding velocity of a point (x,. y,) with respect to moving frame of

reference

X direction of sliding velocity of a point (x2. y?) with respect to fixed fkme of

re ference

X direction of sliding velocity of a point (x,. y,) with respect to fixed fhme of

re ference

Y direction of sliding velocity of a point (x?. yi) with respect w fixed frame of

re ference

Y direction of sliding velocity of a point (x,. y,) with respect to fixed fi-ame of

re ference

theoretical velocity of the outer track

relative velocity of point ot, in y, direction with respect to O,

sliding velocity of point O[, along the longitudinal direction

absolute velocity of point O, in y, direction

absolute velocity of point O, in y, direction

vehicle weight

normal load acting on the inner track

normal load under the ilh roadwheel and jh track

xxviii

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CL'

normal load acting on the j" track

normal load acting on the outer track

angle defining the direction of centrifuga1 force

angle between the resulting sliding velocity and the x, axis

angle between the resulting sliding velocity and the xZ axis

direction angle of shear force acting on the inner track

direction angle of shear force acting on the outer track

goodness-O f- fi t

yaw angle

angular speed of the vehicle

coefficient of fiction

coefficient of fiction along lateral direction

coefficient of fiction along longitudinal direction

coefficient of turning resistance

maximum coefficient of turning resistance

shear stress

calculated sprocket torque

shear stress on the imer track

measured sprocket torque

maximum shear stress

shear stress on the outer track

rotating speed of the sprocket driving the imer track

rotating speed of the sprocket driving the outer track

xxix

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Chapter 1

Introduction

Tracked vehicles are widely used in construction, agriculture and military operattions.

While many studies (Bekker. 1956. 1960: Wong. 1989. 1993) on the mobility of tracked

vehicles have been perfonned. research on the handling behavior of tracked vehicles, has not

yet received the sarne level of attention.

Handling characteristics of tracked vehicles are quite different fiom those of wheeled

vehicles. Among the various methods that can accomplish the steering of a tracked vehicle,

such as skid-steering. steering by articulation. and curved track steenng, skid-steering is the

most widely used. In skid-steering. the tractive force of the outer track is increased and that

of the inner track is reduced to create a turning moment to overcome the moment of turning

resistance due to the skidding of the tracks on the ground and the rotational inertia of the

vehic te in gaw (Wong, 1993). Since the moment of tuming resistance is usually considerable,

braking of the imer track is ofien required in making a tum. This results in a reduction in the

forward thrust. Furtherrnore, skidding of the track on defonnable terrain causes additional

sinkage. Over weak terrain. these factors combined often lead to irnmobilization.

Skid-steering has been a subject of research for a considerable period of tirne. Steeds

( 1 950) exarnined the mechanics of skid-steering on rigid surfaces. His pioneering work laid

the foundation for the subsequent studies of this topic. In recent years* Weiss (1971), Kitano

et al. (1 976. 1977, 1992), and Ehlert et al. (1992) have expended considerable effort on the

fùrther study of the mechanics of steering of tracked vehicles, and a number of models have

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been developed. It should be noted however, that in most of the previous studies on s t e e ~ g ,

the shear stress developed between the track and the ground is assumed to obey Coulomb's

law of friction. This implies that the full shear stress will be developed as soon as a srnall

relative motion between the track and the ground takes place. Experimental evidence (Wong,

1989. 1993) has shown that, the shear stress developed on the traçked vehicle-ground

interface is dependent upon the shear displacement. This means that the shear stress will

reach its maximum value T~, only after a certain shear displacement has k e n developed.

This research focuses on ihe detailed examination of the mechanics of vehicle-terrain

interaction on non-deformable surfaces during tuming maneuvers, taking into account the

effects of shear displacement on the developrnent of shear stress. Experimental studies have

been carried out to measure the shearing characteristics of representative track Iinks of

armoured personnel carriers on asphalt in the laboratory. Based on the results of both

analytical and experimental studies. a computer mode1 for simutating the steering behaviour

of tracked vehicles under steady-state conditions over non-defonnable surfaces has been

deveIoped. It is believed that this study makes a contribution to a better understanding of the

handling behaviour of tracked vehicles on non-deformable surfaces, and provides a basis for

the further study of tracked vehicle maneuvembility on deformable terrain.

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C hapter 2

Review of the State-of-Art on the Study of the Handling of Tracked Vehicles

As mentioned previously. this research focuses on the handling characteristics of

tracked vehicles on non-deformable surfaces. As the forces and moments generated on the

track-ground interface control the tuming behavior. the study of the interaction between the

track and the ground is of importance to the understanding of the handling behavior.

Previous studies on tracked vehicle steering will be reviewed.

2.1. Steeds' Mode1

Steeds (1950) was one of the pioneers who conducted a systematic study of skid-

steering of tracked vehicles. Most of the subsequent studies on this topic were based on his

approac h. A brief outline of his method of approach is given below.

A. Kinematics

Consider a tracked vehicle in a tuming maneuver at low speeds with centrifuga1 force

negiected. as shown in Fig.2- 1. If the hull is turning about the center O with an angular speed,

6 . then the mean speed. V' of the vehicle a$ center of gravity (CG) is given by

V = R&

where R is turning radius.

Since the tracks are attached to the vehicle hull. the tracks are also rotating with the

3

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sarne angular speed, 6 . If a track develops a fonvard thmst. then it will slip backward with

respect to the ground. The magnitude of the sliding velocity of the outer track, Vjo, can be

expressed by :

vI0 =a,$ (2-2)

where a, is called the slip radius of the outer track. which is the distance fiom the centerline

of outer track to the instantaneous center O' of that part of the outer track in contact with the

ground. 0' must lie on the line passing through the vehicle Nming center and perpendïcular

to the longitudinal centerline of vehicle hull. The instantaneous center O' is in fact the center

about which the part of the outer track in contact with the ground rotates. Consequently, the

direction and magnitude of the sliding velocity on the ground of a point on the outer track

c m be determined from the instantaneous center 0'. For instance, the direction of the sliding

velocity is perpendicular to the line connecting the point in question and 0' and its sense is

determined by the direction of the angular velocity. 6 . as s h o w in Fig.2-1. The magnitude

of the sliding velocity is equal to the product of 6 and the distance between the point in

question and 0'. Note that when the track skids forward relative to the ground. the sliding

veiocity. V,,. is negative (Le.. skid). and the instantaneous center 0 will lie between the

centerline of the track and the vehicle turning center, as s h o w in Fig.2-1. In this case, the

inner track develops a braking force.

From the above analysis, it can be seen that the speed of point Q on the outer track in

contact with the gound relative to the coincident point on the vehicle hull is (R + ~ / 2 ) 4 +

V,,. where B is the tread of the vehicle (i.e. the distance between the longitudinal centerlines

of the left- and right-hand tracks). This is equal to the theoretical speed of the outer track ro,,

where r is the pitch radius of the sprocket. and o, is the rotating speed of sprocket driving the

outer track. This is expressed by

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Similady, for the i ~ e r track. the corresponding equation is

r q = ( R - B / ~ - ~ , W (2-4)

where a, is the slip radius of the imer track, assurning that the imer track skids, Le. sliding

fo ward .

Fig.2-1 Kinernatics of a tracked vehicle during a steady-state him.

B. Kinetics:

In analyzing the forces and moments generated by track-ground interaction, there are

three assumptions in Steeds' model:

1. The ground is non-deformable and the interacting forces between the tmck and the

ground obey Coulomb's law of friction: that is. the fictional force is proportional to the

normal load and acts in the opposite direction of the relative motion of the track with

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respect to the ground as shown in Fi@-2. If the friction coefficient in the track

longitudinal direction, h. is different from that of in the lateral direction, pK, that is the

fiction is anisotropic. then the direction of the resultant shear force is modified as

suggested by Micklethwait (1944) and shown in Fig.2-2.

Direction of resu l tan t shear force

according to Y Micklethwait

Direction of resultant shear force according to

law

P ~ P ~ Y (p is normal load per unit track length)

Fig. 2-2 Track force ellipse from Micklethwait.

2. Normal pressure distribution on the track-yound contact area is considered to be

uni form.

3. The width of the track is neglected.

Based on Coulomb's law of isotropie friction, the forces and moments developed by

the track can be formulated as follows (see Fig.2-3). It should be noted that when the effect

of centrifuga1 force is taken into account. the tuming center O is at a distance, %, fiom the

center of gravity in the longitudinal direction. as shown in Fig.2-3, in order to satisfy the

condition of dynamic equilibrium in the lateral direction.

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where Wo .Wi, F,,,. FKi. F,,. F?,. Mer, MTi. Ôo. and 6i are the normal loads, lateral forces,

longitudinal forces. moments of turning resistance. and direction angles of shear forces

acting on the outer and inner tracks. respectively (subscript O indicates outer track, and i

indicates inner track). p is the coefficient of friction. L is the track-ground contact length,

and so is the offset of turning center frorn the transverse centerline of the track-ground

contact area (the center of gravity is assumed to be located at the center of the track), as

shown in Fig.2-3.

As a result. under a steady-state turn the following equilibriurn equations can be

O btained.

W' WV' J R L ~ ~ F,, + F,, = - Co$ =

gR B ~ '

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where R is the extemal motion resistance. and j3 is the angle defining the direction of the

centrifuga1 force with respect to the x-axis, as shown in Fig.2-3.

In Steeds' model. a set of monographs was provided, and the iteration method was

suggested to obtain the three unknowns &. ai and s,. Although, the model can be extended to

anisotropic case if coefficients of m'ction dong the longitudinal and lateral directions are

known. isouopic Coulomb fnction was used in the study. It should also be noted that there

was no measured data to support his analysis.

Y

Fig.2-3 Kinetics of tracked vehicles during a steady-state tum.

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2.2. Crosheck's Model

Based on the study of skid-steering by Steeds, Crosheck (1975) provided a model

which is claimed to be applicable to deformable surfaces. in his model, the following

assumptions were made:

1 . The fiction between the track and the ground is anisotropic as suggested by Micklethwait.

The coefficient of friction. pi,. along the longitudinal direction is expressed by

where El. Ez and E; are coefficients for defining the longitudinal coefficient of friction,

derived from the pull-slip test of the track. and are 0.95. -0.1, and -1.0 for the particular

track on the specific terrain cited in the reference (Crosheck. 1975), respectively, and i, is

track slip along the longitudinal direction. In the lateral direction, full skid was assumed.

Therefore. the lateral coefficient of friction. p,. is equal to the value of H, at iy = 1. Cnj is

the track nurneric of the j" track and is expressed by

ClbL C", = -

*, where C, is the cone index of the soil. b is the track width, L is the track contact length,

Wj is the normal load acting on the j" track.

2. Triangular normal pressure distribution along track-terrain interface is considered.

3. The motion resistance is expressed by

where and E4 and Es are 0.45 and 0.045. for the particular track on the specific terrain

cited in the reference (Crosheck. 1 975). respectively.

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4. The w-idth of the track is neglected.

Based on these assumptions and following the sarne approach to kinetic analysis as that

of Steeds. the forces and moments in a steady-state nirn can be detemined. In this model,

bulldozing effect due to skidding of the track on the terrain m d actual normal pressure

distribution have not k e n taken into consideration. In addition. no cornparison with

esperimentai resul ts was provided.

2.3. Kitano's Model

Similar to Crosheck's analysis. Kitano (1 976. 1977, 1992) provided a model for

predicting steering behavior of a tracked vehicle on level ground. and was considered to be

useful by Ehlert et al. (1992). In this model. the following assumptions were made:

1. The friction between the track and the ground is Coulomb friction, and it is anisotropic as

suggested by Micklethwait. The coefficients of friction dong longitudinal and lateral

directions. p? and p,. respectively. were derived from pull-slip relation for the track as

follows:

where El = 0.44. and Ez = 20.0 for a representative hard ground.

2. Normal load is concentrated under each road wheel.

3. Transverse distribution of normal load per unit track width is assumed to be of

trapezoidal form, which can be expressed by

1 -5 w, At middie of the track. p,, (O) = -

b

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and at the edges.

where pij and Wu are the normal load F r unit width and normal load under the i'

roadwheel and j track. respectively.

4. Extemal motion resistance of the jth track. R,. is assumed to be proportional to normal

Ioad.

R I =f;W, (2- 1 9)

where f, and W, are the coefficient of extemal motion resistance and normal load on the

jIh tnck. respectively.

Kitano's model is usefùl in predicting the steeiing behavior of tracked vehicles on hard

ground and is substantiated by Ehlert et al. ( 1992) from the field measurement.

2.4. Ehlert's Model

In order to establish an adequate simulation model for tracked vehicle handling, three

different analytical models. including Kitano's model, were investigated by Ehiert et al.

( 1992). After certain modifications and improvernents. the results were verified by field

measurements. primarily on hard surfaces (Schmid. 1984). Although Kitano's mode1 offers

the possibility of predicting the tuming radius for a given set of inputs, it is time-consurning

and is not suitable for on-line test stand simulation. As a result, a model based on the

assumptions similar to those originally proposed by Hock (1970) was developed by Ehlert et

al. ( 1 992).

1. Center of gravity is located at the center of the vehicle.

2. Ground pressure on the track-ground contact area is uniformly distributed.

3. Outer track longitudinal force. F,. imer track longitudinal force, Fi, and moment of

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turning resistance. M,. are predicted without considering the effects of centrifuga1 force,

and they are expressed by (as s h o w in Fig.2-4.)

W - L F, =-•

4B P w

W - L M W =-.

4 Ci,

where p, is the coefficient of turning resistance and is a iünction of theoretical tuMng

radius. Rh. which is the tuming radius if track slip is equal to zero. The parameters p,

and Rrh. are defined as

where p,,, is the maximum coefficient of tuming resistance. B is the tread of the vehicle.

RI. is the t m i n g radius at which both track forces are approaching zero, k is called track

exponent and is a function of Rb. and Vo and Vi are theoretical speeds of outer and imer

tracks when track slip is equal to zero. respectively. It should be pointed out that k is an

ernpirical parameter and is derived from experimental data.

4. Extemal motion resistance is a function of the theoretical tuming radius, Rh.

5. The radius enlargement factor, which is the ratio of actual turning radius, R, and

theoretical turning radius. Rih (as shown in Fig.2-4(c)). is taken to be 1.8.

The basic features of the mode1 have k e n substantiated by experimental data obtained

on non-deformable surfaces and is quite accurate in the prediction of sprocket torques for on-

line simulations.

12

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Fig.2-4 Fundarnentals of tracked vehicle steering considered in Hock' s mode1 (Ehlert er al., 1 992).

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The basic features of existing models for skid-steering of tracked vehicles are

summarized in Table 2- 1.

From the brief review given above. it can be said that most of the previous studies

focus on tracked vehicle steering on @id surfaces. and are based on simplifjing assumptions,

such as the normal pressure on the track k i n g either uniformly distributed or concentrated

under the roadwheels, and the negligible effects of track width on the moment of tuming

resistance. Furthemore. in most of the models developed, it is assurned that during steering

the shear forces on the track-ground interface obey Coulomb's law of friction. This impiies

that full shear force will be developed as soon as a small relative motion between the track

and the ground takes place. As noted previously, experimental evidence indicates that the

shear force developed on the track-ground interface is related to the shear displacement.

In this research. the et'fects of shear displacement on the development of shear force

will be taken into account. The effect on the handling behaviour of major vehicle design

parameters. such as track width. and the location of the center of gravity of tracked vehicles

on non-deformable surfaces will be examined in detail.

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Table 2- 1 Cornparison of steering models for trackcd vchiclcs

Factors iiot iiiçluded Mode1

Steeds

[1950]

lrosheck

[ 19753

Kitano

[ 1975, 1977, 19921

Ehlert

il9921

Kiiieiiiatics and kiiictics of skid-steering on deforniable

terrain.

Track width effect.

Features coiisidered

Kitieiiiatics of stcady-statc tiirtiiiig oii level grauiid. Basic mcchanics of skid-steeriiig,

Coulomb friction representation of track-groiind

shearing.

O Uiiiform normal pressure distribution under thc tracks. Centrifugal force and motion resistance.

O Kinematics of steady-state turiiiiig on soft terrain.

O Anisotropic Coiiloiiib friction rcpreseiitatioii of track- groiiiid shearing.

a Location of C.G. of the vehicle, Triaiigular normal pressure distribution under the tracks. Centrifugal force and motion resistance.

O Kiiietics and kineiiiatics of skid-stcering on level ground. O Normal pressure conceiitrated under eacli roadwheel, an(

lateral pressure distribution of trapezoidal form.

O Anisotropic Coulomb friction representation of track- ground shearing.

O Centrifugal force and motion resistance.

0 Track tension.

Uni form ground pressure on track-ground contact area. O Longitudinal tractive forces and turning moments

dependent on the tuming resistance cwficient ~ r , which was a function of theoretical turning radius.

O Extemal motion resistance also a function of theoretical

turning radius.

O Centrifuga1 force.

Kinetics of skid-steering oii

deforniable surfaces. 1 'l'rack width effect.

Rciiiarks

No cxperiiiiental validatioii of analytical flndings.

Iteration method of solution.

No experiineiital validation of

Kinematics and kinetics of skid-steering on defonnable

terrain.

D Kinematics and kinetics of skid-steering on deformable

terrain, D Track width effect

a Experimental results on rigid surfaces provided to verif) simulation data.

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Chapter 3

A General Theory for the Mechanics of Steeriag of

Tracked Vehicles on Non-deformable Surfaces

In most of the models described previously. shear force between the track and the

ground is assumed to obey Coulomb's law of friction and the coefficient of friction is either

isotropic or anisotropic. As mentioned previously. experimental evidence (Wong, 1989.

1993) indicates that the shear stress developed on the track-ground interface is dependent

upon the shear dis placement. In addition, the effects of track width on steenng have not k e n

considered in most of the previous studies. In this chapter. a general theory is developed

based on a detailed analysis of skid-steering on non-deformable surfaces. taking into account

the effects of the shear stress-shear displacement relationship and of track width.

3.1. Skid-Steering on Non-deformable Surfaces Including the Effect of Track Width

3.1.1. Shear Displacement of the Outer Track

Consider that a tracked vehicle with track width, b, is in a steady-state tum about O. as

shown in Fig.3-1 and Fig.3-2.

Let O, be the origin of a M e of reference (x,. y,) fixed to and moving with the vehicle

hull and located on the longitudinal centerline of the outer track and at a distance, s,,, fiom

the center of gravity (CG) of the vehicle. as shown in Fig.3-1. The offset distance, so, will be

deterrnined later from the dynamic equilibrium of the vehicle in the lateral direction during a

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turn. As the vehicle hull is rotating about turning center O with an angular speed, 6 , the

absolute velocity of O:. V,,,,. in the y, direction can be expressed by

where R' is the lateral distance between the center of turn O and center of gravity of the

vehicle. and is equal to R-Cosp (or JR' - s j ). as shown in Fig.3-1, R is the tuming radius of

the vehicle. c, is the lateral distance between the vehicle CG and loagitudinal centerline of

vehicle hull (or the lateral offset of the center of gravity with respect to the geometrical

center o f the vehicle), and tread. B. is the distance between the centerlines of the outer and

inner tracks.

A point O,, on the outer track in contact with the ground coincident with 0, has a

relative velocity. V,,,,,, with respect to O,. which is expressed by

Vil'Ol = f o o (3-2)

where r is the pitch radius of the sprocket and o, is the angular speed of the sprocket of the

outer track.

As a result. the sliding velocity. V,,,. of point O,, on the ground dong the longitudinal

direction of the outer track is expressed by

Consider an arbitrary point defined by (x,. y,) on the outer track in contact with the

ground. Since the track is rotating with the vehicle at angular speed, 4 . during a steady tum,

the relative velocity components of the point (x,. y,) with respect to ot, in the longitudinal

and lateral directions of the track are given by x ,& and y,& respectively.

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Hull

Fig.3-l Kinematics of a tracked vehicle during a tuming maneuver.

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Fig.3-2 Kinematics of a tracked vehicle during a steady-state tum.

Based on the above analysis. the sliding velocity of a point defined by (x,, y,) on the

outer track in contact with ground. with respect to the fixed M e of reference (X. Y) c m be

expressed by (Fig.3- 1 )

X component of sliding velocity:

V,,, = -V ,,,, Sin4 + ro,Sin+ - x ,&~in$ - y,&os4

= -[(R'+B/~ +c, + x , ) 4 - ro,]~in$ - y,@os4

Y cornponent of sliding velocity:

V,,, = V ,,,,, C o 3 - ro,Cos4 + x ,(Cos+ - y ,+Sin+

= [(RI+ BI2 + c, + x, )Q - ru, ]COS+ - y ,&sin+

The angle, 4. as shown in Fig.3-1 is the angular displacement of the vehicle and can be

19

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determined by the integration of the yaw velocity. 4 . with respect to the time. t, it takes for

the point (x,. y,) to mvel from the initial point of contact at the front of the track (at y, = U2

+ C- - so). that is

and

where c,. is longitudinal distance between CG and lateral centerline of the vehicle hull, or the

longitudinal offset of the center of p v i t y with respect to the geornehic center of the vehicle

(see Fig.3- 1 ).

-4s a result. the shear displacement. jso. at a point (x,, y,) on the outer track in contact

with the ground along the X direction with respect to fixed frame of reference (X, Y) can

therefore be determined by

and shear displacement, jY,. along the Y direction

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B [ ( ~ , ' 2 + c , -so

= ( R f + l + c K + x , s i n faO ]-($+cy - S . ) +

The resultant shear displacement. j,. of a point at (x,. y ,) on the outer track in contact

with the ground is given by

3.1.2. Shear Displacement of the Inner Traok

Similarly let o2 be the origin for a frame of reference (x,, yJ fixed to and moving with

the vehicle hull and be located on the longitudinal centerline of the imer track. The absolute

velocity of O,. VeY2. in the y, direction can be expressed by

Following a similar approach described in 3.1.1, the sliding velocity of a point (x,, y 3

on the inner track in contact with the ground with respect to fixed h e of reference (X, Y)

can be expressed by (see Fig.3- 1 )

X component of sliding velocity:

V,,i = -V,2,2Sin+ + ro,Sin+ - x,&sin+ - y , & ~ o s +

= -[(R'-B/Z + c , + x2)& - r o , ] ~ i n + - y ,&os4

Y component of sliding velocity:

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= [(RI-8/2 + c, + x 2 ) h - rw , ]~os+ - y , & ~ i n 4

where ai is the i ~ e r irack sprocket angular speed.

However. the contact time, t. for the point defined by (x,, y2) will be equal to (W2 + c, - s, - yL)/raI. and. dt = - dy J rmi. Therefore. the shear displacement, jxi, on the imer track of the

vehicle along the X direction with respect to fixed frame of reference (X. Y) can therefore be

deterrnined by

and shear displacement, jvi. along the Y direction

The resultant shear displacement. ji, of a point at (x', y,) on the imer tmck in contact

with the ground is given by

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3.1.3. Kinetics of Tracked Vehicles during a Steady-State Turning Maneuver

As noted above. the shear stress developed on the track-ground interface is dependent

upon the shear displacement. On a non-defomable surface, in most cases the shear stress, T.

initial I y increases rapidly with an increase in the shear displacement, j, and then approaches a

constant value with a M e r increase in shear displacement, as show in Fig.3-3. This type

of shear stress-shear displacement relationship rnay be described by the following

exponential equation:

T=T,,(i-e-' K ) = p - p - ( ~ - e-JJl() (3- 1 7)

where K is the shear deformation parameter, and may be considered as a measure of the

magnitude of the shear displacement required to develop the maximum shear stress, p is the

normal pressure. and p is the coefficient of fiction between the track and the ground. K may

be determined by the intersection of the line tangent to the curve at the origin and the

horizontal line representing the maximum shear stress. as shown in Fig.3-3.

Accordiag to Coulomb's L a w

Actual Sbear Curve

1 ' I I I ' I 3 , -, I

Shear Dispîacement

Fig.3-3 A shear curve of a simple exponential form.

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Therefore. the shear force developed on an element dA of the track in contact with the

ground c m be expressed as:

on the outer track d ~ , = s , d ~ = pop(l - e-jO " )- d~ (3-1 8)

on the inner track dF, = =,dA = p,p(l - e - ~ . ' " ) - d ~ (3- 19)

where t,, t,. p, and p, are the shear stress and normal pressure on elements of the outer and

i ~ e r tracks. respective1 y.

It should be noted that the assumption used in Steeds' mode1 that the shear force

between the track and the ground obeys Coulomb's law of friction is a special case of

equation (3-17). Coulomb's law assumes that full frictional force is reached as soon as a

small relative motion between the track and the ground takes place. This is equivaient to the

value of K in equation (3-1 7) equal to zero. This indicates that equation (3-17) is a more

general representation of the charactenstics of the shear force between the track and the

ground. C

As s h o w in Fig.3-4. the longitudinal forces. F,, and Fyi, acting on the outer and imer

tracks relative to the moving Frame of reference. can be expressed respectivAy by

where 6, and S2 are the angles between the resultant sliding velocities of the points on the

outer and inner tracks and the lateral directions of the tracks (Le., x, and x2 axes),

respectively. It should be noted that following Coulomb's law of fiction, the shear force

acting on the track will be in the opposite direction of the resultant sliding velocity.

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Fig.3-4 Kinetics of a tracked vehicle during a steady-state tum.

The lateral forces. Fxo and FKi, acting on the outer and i ~ e r tracks, are given

respectively by

The tuming moments, ML, and ML,. acting on the outer and imer tracks respectively,

due to the longitudinal shear forces acting on the outer and inner tracks respectively with

respect to the longitudinal centerline o f vehicle hul1 can be expressed by

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The moments of turning resistance. M T ~ and MTi, of the tracks due to the moments of

the lateral forces acting on the outer and inner tracks respectively, with respect to the line

passing through center of tum O and perpendicular to vehicle longitudinal axis c m be

expressed by

In order to determine 6, and 6?. the longitudinal sliding velocities, V,, and V,,,, of the

track elements on the outer and inner tracks with respect to the moving fiames of reference

(xI. y l ) and (x:. y,). respectively. are first calculated as follows (see Fig.3-2)

The iateral sliding velocities. V,, and V,,,. of the track elements on the outer and imer

tracks c m be expressed by

Therefore, the 6, and 62 c m be defined, respectively, by the following equations

26

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coso, = vp - y Z i

- J = =z)+-ro,~ +(y,+y

From the above analysis. one can derive the equilibrium equations for the tracked

vehicle during a steady-state tum as follows (Fig.3-4)

CF, = O w'

FK0 + F, + - CosP = O (3-36) !ZR

whsre Rt, and Rt, are external motion resistance on the outer and imer tracks, respectively.

It should be noted that the forces and moments are functions of the theoretical speeds,

ro, and roi, and the offset. so. With the other parameters known or given, such as tread, B,

shear deformation parameter. K. track-gound contact length. L, fonvard speed, V, tuming

radius. R. weight. W. track width. b. longitudinal offset. c,. and lateral offset, c,. of the

center of gravity with respect to geometrical center of the vehicle, CG height, h, coefficient

of motion resistance. f, and coefficient of friction. p. these three unknown parameters, ro,

ru, and so. c m be determined by solving the above three simukaneou equations. Thus al1 the

forces and moments during a given steady-state tum can be completely defined.

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3.2. Normal Pressure Distribution under the Tracks

First consider the lateral load transfer of the imer and outer tracks due to the lateral

component of the centrifuga1 force s h o w in Fip.3-j(a). From force and moment equilibrium.

the normal load on the outer <rack. W,. and that on the imer track, Wi, c m be expressed by:

Y=-+ \z [iz2 -- CosP - " 1 B

Wl =-- 2 (BT2 -- COSP - c W ) B

( c )

Fig.3-5 Normal Pressure distribution.

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Then consider the longitudinal load transfer due to the longitudinal component of the

centrifuga1 force and the longitudinal offset of vehicle CG as shown in Fig.3-5(b). Assuming

that the normal load distribution under the outer track has the form of a trapezoid, then the

normal pressure under the outer track. p,(y,). is given by

Similariy. normal pressure under the inner track. pi(y2). can be expressed as

If the normaf load is assumed to be supported only by the track link with pitch, L,,

under each roadwheel as shown in Fig.3-j(c). then p,(y,) and piO'z) can respectively be

espressed by

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Chapter 4

Simulation of Handling Behaviour of Tracked Vehicles

using the Proposed General Theory

In order to demonstrate the utility of the general theory described above, the steering

behavior of a military tracked vehicle was simulated and the results were compared with

avaiIable msasured data (Ehlert er al.. 19%).

4.1. Basic Design Parameters of the Vehicle Simulated

The basic design parameters of a military tracked vehicle, known as a Jaguar, used in

the simulations are given in Table 4-1 (Foss. 1993-94).

rable 4-1 Basic design parameters of the Jaguar tracked vehicle used in the simulation. - - - - -

vehicle weight, kg

Height of CG, m Sprocket radius, m (Scaled from the drawing)

Track-ground contact length. m Longitudinal CG location from centerline of vehicle hult, rn

Track width. m

Track pitch, m

Tread of vehicle. m

; hearing characteristics

Shear deformation parameter. m Coefficient of friction between the track and the ground, Coefficient of external motion resistance,

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The coefficient of friction, p. and shear deformation parameter, K, are dependent upon

properties of the track-ground interface. such as the matenal of tmck pad and the type of

ground. Since the values of p and K were not given in the reference (Ehlert et al.. 1992), the

values used in the simulations were derived fiorn fitting the simulated results to the

rneasured data using a optimization procedure. The goodness-of fit, c, achieved is 88.58%

and is defined by

where 5, and is the rneasured sprocket torque from the reference (Ehlert et al., 1992), N is

the nurnber of data points. and Tc is the calculated sprocket torque based on the model

developed.

4.2. Simulation Results

By solving the three simultaneous equations from (3-36) to (3-38), the circumferential

speed of sprocket on the outer track. ru,. and that on the iriner track, ro;, and the offset, %,

c m be obtained. The rneasured sprocket torque (Ehlen el al., 1992), the predicted data with

different types of normal load distribution under the tracks. as well as the predicted results

from Steeds' model (K = O m) are s h o w in Figs.4-1 to 4-4. Tables 4-2 to 4-4 show a

cornparison of the predictions obtained from Steeds' mode1 and those obtained fiom the

proposed mode1 with the measured data at different fonvard speeds.

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: Measured (Speed Range: 5.8 - 9.2 kmh)

- - - - : Continuous trapezoidal load distribution over the entire track (Goodness-of-fit 90.55%)

---O. : Trapezoidal load distribution on the track pitch under

each road w heel (Goodness-of-fit 85.67%) ----. : Concentrated load under each roadwheel

(Goodness-of-fit 82.54%) --- -- : Steeds' mode1 under trapezoidal load distribution (K = O m)

Outer Track

Inner Track

- - - - , - , - - . - - - - - - - - - O - .

l I 1 1 1 1 1 1 I 1 1 I I I I I I I 1 I I ~ I I

1 10 100 1000

Theoretical Turning Radius (m)

Fig.4-1 Sprocket torques vs theoretical tuming radius for a Jaguar at

a vehicle speed of 7.5 kmh with K = 0.075 m and p = 0.9 during a steady-state tum.

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: Measured (Speed Range: 1 1.9 -16.5 km&) - - - - : Continuous trapezoidal lord distribution over the

entire track (Goodness-of-fit 88.65%) ---o. : Trapezoidal load distribution on the track pitch under

each roadw heel (Goodness-of-fit 86.17%) ----- : Concentrated load under each roadwheel

(Goodness-of-fit 84.67%) - - - - - : Steeds' model under trapezoidal load distribution (K = O m)

-26000 1 I I I 1 1 I I I I I I I 1 1 1 1 I I I I I I I I

1 10 100 1 000

Theoretical Turning Radius (m)

Fig.4-2 Sprocket torques vs theoretical tuming radius for a Jaguar at a vehicle speed of 14.2 kmh with K = 0.075 m and p = 0.9 during during a steady-state turn.

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: Measured (Speed Range: 18.6 - 24.0 kmR)

- - - - : Continuous trapezoidal load distribution over the

entire track (Goodness of Fit 88.3 1 %) ----. . . Trapezoidal load distribution on the track pitch under

each roadwheel (Goodness-of-fit 72.51%) - O * - . 0 . Concentrated load under each roadwheel

(Goodness-of-fit 73.93%) --*--a . Steeds' model uader trapezoidal load distribution (K = O m)

Outer Track

tnaer Track

1 10 f 00 1000

Theoretical Turning Radius (m)

Fig.4-3 Sprocket torques vs theoretical turning radius for a Jaguar at a vehicle speed of 2 1.3 km/h with K = 0.075 m and p = 0.9 uring a steady-state tum.

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: Measured (Speed Range: 25.8 - 32.2 k m )

- - - - : Continuous trapezoidal load distribution over the

entire track (Goodness-of-fit 83.96%) -O--. . Trapezoidal load distribution on the t n c k pitch under

each roadwheel (Goodness-of-fit 83.71 %) -O--- . Concentrated load under each roadwbeel

(Coodness-of-fit 83.54%) - - * O - . Steeds' model under trapezoidal load distribution (K = O m)

1 10 100 11000

Theoretical Turning Radius (m)

Fig.4-4 Sprocket torques vs theoretical tuming radius for a Jaguar at a vehicle speed of 29 km/h with K = 0.075 m and p = 0.9 during a steady-state turn.

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Table 4-2 Sprocket torques of the outer track at various turning radii with a speed of 7.5 kmh. -- -p

Theoretical

tuming

radius (m)

- -- - -

Predicted by the proposed general theory I I

(kN.m)

rable 4-3 Sprocket t o q u e s of the inner track at various turning radii with a speed of 7.5 kmh.

Theoretical

turning

radius (m)

Predicted by mode I

kN.m

23.898

18.947 23.89 1

12.895 23.88 1

6.71 1 23.870

3.816 23.866

Measured

(kN.m)

error

-9.58%

-26.1%

-85.2'/0

-256%

-525%

Predicted by Steeds mode1

Continuous

trapezoidal load over

the entire track

kN,m

20.8 14

17.875

13.345

7.682

4.860

kN.m 1 error

kN.m

20.6 12

18.149

14.189

8.561

5.462

error

4.56%

-5.66%

-3.59%

-15.5%

727.4%

Predicted by the proposed general theory

Trapezoidal load on

the tract pitch under

each roadwheel

Concentrated load

under each

roadw heel

error

-5.48%

-4.2 1%

+ 10.0%

+27.6%

+43.1%

kN.m 1 error 1 kN.m 1 enor 1 kN.m 1 e m r

- - - - - - -

Continuous

trapezoidal load over

the entire track

L

kN.m

19.156

17.437

13.926

8.522

5.468

error

- 12.2%

-7.97%

+7.96%

+27.0°/0

+43.3%

-

Trapezoidal load on

the mck pitch under

each roadwheel

Concenrrated load

under each

roadwheel

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Table 4 4 Sprocket torques of the outer track at various turning radii with a speed of 14.2 kmh.

Predicted by the proposed generai theory 1 1 Theoretical

turning

radius (m)

Measured

(kN.m)

Predicted by Steeds'

mode1

Continuous

trapezoidal load over

the entire track

Table 4-5 Sprocket torques of the inner track a t various turning radii with a speed of 14.2 kmh.

- 3 1 .-- 735

18.947

12.895

6.711

3.8 16

I Predicted by the proposed general theory I

Trapezoidal load on the track pitch wider

each roadwheel

Theoretical

Concentrated load

under each roadwhee l

--

Concentrated load 1

kN.m

23.566

23.825

23.919

23.925

23.916

Continuous Trapezoidal load on Measured Predicted by Steeds'

trapezoidal load over the track pitch under (kN.m) mode1

the entire track eac h roadwheel

error

- 1 1 -0Yo

-25.7%

-85.5%

+256%

-527%

turning

radius (rn)

kN.m

19.709

17.168

11.796

7.347

4.673

under each roadwheel I

error

-7.14%

-9.39%

-0.77%

-9.48%

-22.5%

' k ~ . m error k ~ . m e m r

kN.m

20.1 69

17.956

13.949

8.348

5.324

error

- 16.6%

-3 1.7%

kN.m

19.060

17.375

13.728

8.323

5.339

error

-4.98%

-5.23%

+8.17%

+24.4%

+39.5%

emor

- 10.2%

-8.30%

+6.46% 1

+24.0%

+39.9??

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Table 4-6 Sprocket torques of the outer track at various tuming radii with a speed of 2 1.3 kmh. -- -

Predicted by the proposed general theory

Theoretical

tuming

radius (m)

TabIe 4-7 Sprocket torques of the inner track at various tuming radii with a speed of 2 1.3 kmh.

turning

radius (m)

Measured

(kN.m)

1 O

2 O

50

1 O 0

Measured

(kNm)

- 10.658

15.789

12.500

6.7 1 1

3.816

Predicted by the proposed general theory I I

Predicted by Steeds'

mode1

kN.m

23.158

23.78 1

23.963

23.968

Continuous 'redicted by Steeds'

trapezoidal load over mode1

the entire track

error

-46.7%

-90.2%

~ 2 5 7 9 6

-54596

1

kN.m error kN-m error

1

k N m

15.802

1 1.895

6.765

4.294

Trapezoidal load on 1 Concentrated load (

Continuous

trapezoidal Ioad over

the entire track

error

-0.08%

4.84%

-O.80'!4

712.5%

the track pitch under 1 under each I

kN.m

17.254

13.5 14

each roadwheel roadwheel

Trapezoidal load on the mck pitch under

each roadwheel

error

+9-28%

+8.1 1%

kN.m

16.865

13.356

Table 4-8 Sprocket torques of the outer track at various tuming radii with a speed of 29 k d h .

Concentrated load

under each

roadwheel

7.974

5.031

error

+6.8 1 %

~ 6 . 8 5 %

Theoretical

tum ing

radius (m)

1 1 1 Predicted by the proposed general theory 1

i l 8.8% ) 7.962 1 +18.6%

+3 1.8%

. . -

Continuous

trapezoidal load over

the entire track r

5.055

Measured

(kN.m) Predicted by Steeds'

mode 1

Trapezoidal load on

the track pitch under

each roadwheel

Concentrated load

under each

roadwheel

20

50

100

12.895

6.71 1

3.816

kN.m

23.189

23.932

24.012

error

+79.8%

+257%

+529?6

kN.m

11.444

6.543

4.1 13

error

- 1 1.3%

-2.50%

+7.78%

kN.m

13.233

7.777

4.815

kN.m

13.197

7.804

4.836

error

12.62%

+15.9%

+26.2%

error

+2.34%

+16.3%

+26.7%

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Table 4-9 Sprocket torque of the inner track at various turning radii with a speed of 29 km/h.

1 1 1 Predicted by the proposed general îheory 1 Theoretical

tuming

radius (rn)

It can be seen fiom Figs.4-1 to 4-4 and Tables 4-2 to 4-9 that the relationships between

sprocket torques and tuniing radius at various forward speeds predicted by the proposed

general theory bear a close resemblance to those measured. whereas the relationships C

predicted by Steeds' model are far from the measured data particularly at larger tuming radii.

This can be explained by the difference in the laterai shear stress distribution predicted by the

proposed general theory and by Steeds' model. as shown in Figs.4-5 to 4-8.

20

5 0

1 00

in Steeds' model. the maximum shear stress is developed whenever a small relative

motion between the track and the ground takes place. As a result. there is a large lateral shear

stress developed even at the front contact point of the track. where a track element just

cornes into contact with the ground. as shown in Figs.4-7 and 4-8. On the other hand, the

pro posed general theory takes into account the s hear stress-shear displacement relationship.

Consequently at the fiont contact point of the track. the shear displacement and hence the

shear stress is zero. as indicated in Figs.4-5 and 4-6. It should also be pointed out that the

lateral shear stress distribution predicted by Steeds' model varies little with the increase in

turning radius. except at very small turning radius. such as R = 10 m shown in Figs.4-7 and

4-8. This is the reason that the sprocket torques predicted by Steeds' model vary Iittle with

Measured (kN.m)

-9.367

4.2 1 I

-1.316

Predicted by Steeds' model

kN.m

-20.200

-2 1.165

Concentrated load under each roadw heel

Continuous

trapezoidal load over the entire track

error

7116%

4 0 3 %

Trapezoidal Ioad on the track pitch under

each roadwheel

kN.m

-8.409

-3.764

error

-10.2%

- 10.6%

kN.m

-10.379

-5.029

-;.196/o -21.261

error

+10.8%

+19.4%

-2.068 -1516OA -1.358

kN.m

-10.362

-5.060

+S7.I%

error

+10.6%

+20.2%

-2.090 +58.80/0

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turning radius beyond a certain value. as shown in Figs.4-1 to 4-4. On the other hand, the

magnitude of the lateral shear stress predicted by the proposed general theory decreases with

the increase in t m i n g radius. As a result. the sprocket torques predicted by the proposed

gsneral theory also decrease with the increase in tuming radius.

I t c m . therefore. be concluded that the trac k-ground shearing characteristics.

specifically the shear stress-shear displacement relationship. must be taken into account in

predicting the handling behaviour of tracked vehicles. The normal pressure distribution is

s h o ~ n to have a certain influence on the relationships between sprocket torques and tuming

radius. It appears that among the three types of normal pressure distribution examined, the

continuous trapezoidal tàrm of normal pressure distribution provides the closest agreement

with the measured data.

4.3. Simulation of Turning Resistance Moment

For the computation of the tuming resistance moment. several analytical models were

developed in the past. The basic model is described in Section 2.4. In this model. the

interacting forces and moments between the tracks and the ground can be defined and

espressed in terms of coefficient of turning resistance. p,. as shown in equations (2-20) and

(2-2 1). It should be noted that p, in these equations is considered to be a fhction of track

exponent. k. which varies with turning radius. k is essentially an empirical parameter which

c m only be denved from experimental data (Ehlen et al.. 1992).

As described previously. using the general theory presented in Chapter 3. the moments

of turning resistance for the outer and inner tracks can be predicted analytically using

equations (3-26) and (3-27). respectively. Consequently. the equivalent coeficient of turning

resistance. p,, in equation (2-20c) can be derived quantitatively by equating the sum of the

moments of tuming resistance, Mr, and MTi- calculated fiom equations (3-26) and (3-27) to

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.MN in equation (3-20c). The equivalent coeffkient of turning resistance, p, (for C( = 0.9),

using this method as a Cunction of turning radius is shown in Figs.4-9 to 4-12. The equivalent

coefficient of turning resistance. p,. derived from experimental data reported by Ehlert et al.

( 1 992) is also shown in the figures. It was calculated using equation (3-38), with measured

data on sprocket torques and external motion resistance and the calculated centrifuga] force

based on the forward speed and turning radius. It is shown that there is a reasonably close

agreement between the measured md predicted results obiained using the general theory

presented in Chapter 3.

4.4. Summary

It is shoun that there is a reasonably close agreement between the sprocket torques and

moments of turning resistance predicted using the general theory and the available measured

data. The variations of sprocket torques and equivalent coefficient of turning resistance with

turning radius at various forward speeds bear a close resemblance to those measured. On the

other hand. the predictions based on Steeds' mode1 are far from measured data. The mode1

proposed by Elhert et al includes a number of empirical coeflficients. such as track exgonent,

k. turning radius. Rk, where track forces approaching zero. etc. To obtain these values

requires extensively field tests. It can be concluded that the general theory developed offers a

significant improvement over al1 previous models.

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rca r Outcr Track front

- - --- ; .

rcar lnncr Track + front

-1.90 -0.96 0.00 0.96 1.90 Location of Con tact Length (m)

Fig.4-5 Lateral shear stress distribution of the outer track along

the longitudinal centerline of track-ground contact arca

predicted from the proposed general theory at a vehicle

speed of 14.2 kmlh with different turning rridii.

-1.90 4.96 0.00 0.96 1.90 Location of Contact l m g t h (m)

Fig.4-6 Lateral shear stress distribution of the inner track along

the longitudinal centcrline of track-ground contact area

predictcd from the proposed general theory at a vehicle

speed of 14.2 kmlh with different tuming radii.

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+ + +: Merisuretl data (Speecl range: 5.14 - 9.2 knilh) + + + : Mcasurcd datu (Spccd range: 11.9 -16.5 kmlh)

1 10 100 1000 1 10 100 1000 Theoretical Turning Radius (m) Theoretical Turning Radius (m)

Fig.4-9 CoefFicient of tuming resistance vs tuming radius Fig.4- 1 O Coefficient of tuming resistance vs tuming radius

for a Jaguar at a vehicle speed of 7.5 kmk with for a Jaguar at a vehicle speed of 14.2 kmlh with

K = 0.075 m under continuous trapezoidal load K = 0.075 m under continuous irapezoidal load

distribution over the entire track. distribution over the entire track.

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Chapter 5

Evaluation of the Effects of Design and Operating Factors on

the Handling of Tracked Vehicles using the General Theory

In the previous chapter. a general theory for tracked vehicle steering on non-

deforrnable surfaces has k e n established. The predicted results of a particular vehicle during

a steady-state turning maneuver compare favourably with available field measurements.

With the basic features of the proposed theory substantiated, in this chapter, the effects on

handling of major vehicle design parameters, such as track-ground contact length, L, tread, B.

track width. b. longitudinal offset. c,, of vehicle CG with respect to the geometrical center of

the vehicle. and CG height. h. wiil be examined in detail.

5.1. Contact Lengtb, L

Using the tracked vehicle with design parameters given in Table 4-1 as a basis, the

effects of track-ground contact length. L, on steering behavior were exarnined using the

simulation mode1 developed. The sprocket torques under continuous trapezoidal normal load

distribution over the entire track contact length (as shown in Fig.3-5(b)) increase with the

increase of L as shown in Figs.5-1 to 5-4. This is because the moment of tuming resistance

generated during a tuming maneuver also increases with the increase of contact length, L, as

shown in Figs.5-9 to 5-12. The lateral force will be more or less the same with different

contact lengths as shown in Figs.5-5 to 5-8, since the centrifuga1 force during the tuming

maneuver does not change with L. The sprocket torques with different contact lengths and

vehicle speeds at various twning radii are tabulated in Tables A-1 to A-4 in Appendix A.

46

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: Hcfcrencc (Contact Lcngth 1, = 3.8 m)

- - - :L=1.90m - - - : L = 2 . 8 5 m

30000L---, Outer Track

P---/#- lnner Track

10

Fig.5-3 Sprocket t

100 1000 Turning Radius (m)

orqi ues vs turning radius for a Jaguar at a

vehicle speed of 2 1.3 kmh with di fferent track contact

lengths during a steady-state turn undcr continuous

trapezoidal load distribution over the entire track.

: Hcfcrencc (Coiitact Lcngt h L = 3.8 ni)

- - - : L = 1.90 m --• : L = 2.85 m

30000

Outer Track -- --\. --.

Fig.54 Sprocket torques vs tuming radius for a Jaguar at a

vehicle speed of 29 kmlhr with different irack contact

lengths during a stead y-state turn under cont inuous

trapezoidal load distribution over the entire track.

- lnner Track

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 I

10 IO0 1000 Turning Radius (m)

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: Rcfcrcncc (Contact Length L = 3.8 m)

- - - : L=1.90m - = : Id=2.85m

----O . L = 4.75 m ----- : L = 5.70 m

10 100 1000 Turaimg Radius (m)

Fig.5-7 Lateral forces vs tuming radius for a Jaguar ai a vehicle speed of 21.3 kmni with different track contact lengths during a steady-state tum under continuous trapezoidal load distribution over the entire track.

: Hekrcnce (Contact Length L = 3.8 m)

- - - : L = 1.90 m - . : L = 2.85 m

10 100 1000 Turning Radius (m)

Fig.5-8 Lateral forces vs tuming radius for a Jaguar at a vehicle speed of 29 kmlh with different track contact lengths during a steady ostate tum under continuous trapezoidal toad distribution over the entire track.

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: Rcference (Con tact Lcngth L = 3.8 m)

- - - : L= 1.90m ----.: L=2.85m

- 0 - * . L = 4.75 ni ----- : L = 5.70 m

1 10 100 1 O00

Turning Radius (m)

Fig.5-9 Moments of tuming resistance vs turning radius for a Jaguar at a vehicle speed of 7.5 kmlh with different track contact Iengths during a steady-state turn under continuous trapezoidal load distribution over the entire

track,

O -

A

E 0 8 aoooo- e a Y V)

'1 * Da .O -120000- e e, O

8 3 -18000& E

E

: Heference (Contact Length L = 3.8 m)

- - - : L = I . W m ----m:L=2.8Sm

- - * . L = 4.75 m -O - - - : L = 5.70 m

Turning Radius (m)

Fig.5-IO Moments of tuming resistance vs turning radius for a Jaguar at a vehicle speed of 14.2 kmlh with different track contact lengths during a steady-state tum under continuous trapezoidal load distribution over the entire

track,

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The effects of track-ground contact length, L. on the moment of turning resistance are

surnmarized in Figs.5- 13 to 5- 16. It shows the variation of the ratio of the moment of himing

resistance (Le., the surn of MT, and MTi at a particular contact length to that at the reference

length of 3.8 m) with the track contact length, L, at various turning radii. At the reference

contact length of 3.8 m, the ratio is equal to one.

The same trend is also observed for normal load distributed only on the track pitch (L,

= 15 cm) under each roadwheel (as shown in Fig.3-5(c)), as well as those for normal load

coricentratrd at a point (L, = 1 cm) under each roadwheel. These results are shown in Tables

B-1 to B-4 in Appendix B, and in Tables C-l to C-4 in Appendix C, respectively.

5.2. Tread of Vehicle, B

The s procket torques under con t inuous trapezoidal normal load distri bution over the

entire track increase with the decrease of B as shown in Figs.5-17 to 5-20. This is because

u-ith the decrease of B. the turning moment formed by the longitudinal shear forces on the

outer and inner tracks decreases. due to the decrease in the moment m. However, the lateral

force and moment of turning resistance generated during the tuniing maneuver remain

approximately the same. as shown in Figs.5-21 to 5-28. To balance the moment of tuming

resistance. the sprocket torques have to be increased with the decrease of B. The sprocket

torques with different treads and vehicle speeds at various turning radii are tabulated in

Tables A-5 to A-8 in Appendix A.

The effects of vehicle tread, B, on magnitude of the surn of the sprocket torques on the

outer and inner tracks are surnmarized in Figs.5-29 to 5-32. It shows the variation of the ratio

of the sum of sprocket torques at a particular vehicle tread. B, to that at the reference tread of

2.54 m. At the reference tread of 2.54 m. the ratio is equal to one.

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0.95 1.90 2.85 3.80 4.75 5.70 0.95 1.90 2.85 3.80 4.75 5.70 Contact Length L (m) Contact Length L (m)

Fig.5-13 Ratio of turning resistance moment vs contact length L for Fig.5-14 Ratios of turning resistance moment vs contact length L for

a Jaguar at a vehicle speed of 7.5 kmlh with diffèrent a Jaguar at a vehicle speed of 14.2 kmlh with different

tuming radii under contiriuous trapezoidal load distribution tuming radii under continuous trapezoidal load distribut ion

over the entire track. over the entire track.

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0.95 1.90 2.85 3.80 4.75 5.70 Contact Length L (m)

0.95 1.90 2.85 3.80 4.75 5.70 Contact LRngtb L (m)

Fig.5-15 Ratio of turning resistance moment vs contact length L for Fig.5-16 Ratio of turning resistance moment vs contact length L for

a Jaguar at a vehicle specd of 2 1.3 kinlh with diffcrent a Jaguar at a vehicle speed of 29 kmlh with different

turning radii under continuous trapezoidal load distribution turning radii under continuous trapezoidal load distribution

over the entire track. ovcr the cntire track.

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: Rcfercncc (Trcad B = 2.54 ni) : Hcfcrcncc (Trcad B = 2.54 m)

lnner Track

10 100 1000 Turning Radius (m)

Fiy.5-19 Sprocket torques vs tuming radius for Jaguar at a vehicle speed of 21.3 km/h with different vehicle

treads during a steady-state tum under continuous trapezoidal load distribution over the entire track.

1 , Outer Track

I inner Track

10 100 1000 Turning Radius (m)

Fig.5-20 Sprocket torques vs tuming radius for a Jaguar at a vehicle speed of 29 kmh with different vehicle

treads during a steady-state turn under continuous trapezoidal load distribution over the entire track.

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: Rcfcrence (Trelid B = 2.54 m)

- - - :B=1.90rn - : H = 2 . 1 7 m

- : B=3.04m - : B=3.80m

Outer Track

1 10 100 1000 Turning Radius (m)

Fig.5-21 Lateral forces vs turning radius for a Jaguar at a

vehicle speed of 7.5 km/h with different vehicle treads during a steady-state turn under continuous

trapezoidal load distribution over the entire track.

: Hcfcrcncc (Trcud B = 2.54 m)

- - - :B=1.90m ---:8=2.17m

- : B=3.04m : B=3.80 m

Outcr Track

d

1 10 100 1000 Turning Radius (m)

Fig.5-27 Lateral forces vs turning radius for a Jaguar at a

vehicle sped of 14.2 km/h with different vehicle treads during a steady-state turn under continuous

trapemidal load distribution over the entire trnck.

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: Hcference (Tread B = 2.54 ni)

--- :B=1.90m : B = 2 . 1 7 m

- - : H=3.O4 m : B =3.80 m

Outer Track 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 I 1 1 1 1 1 1 1 1

1 10 100 1000 Turning Radius (m)

Fig.5-25 Moments of tuming resistance vs tuming radius

for a Jaguar at a vehicle speed of 7.5 kmh with different vehicle treads during a steady-state turn

under continuous tmpezoidal load distribution over the entire track.

: Hefcrence ('Trclid 13 = 2.54 m)

- - - : B = 1.90ni --:B=2.17m

: B = 3.04 m - : B = 3.80 m

Inter Track

1 10 100 1000 Turning Radius (m)

Fig.5-26 Moments of tuming resistance vs tuming radius

for a Jaguar at a vehicle speed of 14.2 kmlh with different vehicle treads during a steady-state tum under continuous trapezoidal load distribution over the eniire track.

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: Refcrence (Trclid II = 2.54 m)

- - - : II = 1.90 m - 4 B = 2.17 m

10 100 1000 Turning Radius (m)

Fig.5-27 Moments of turning resistance vs tuming radius for a Jaguar at a vehicle speed of 21.3 km/h with different vehicle treads during a steady-state turn under continuous trapezoidal load distribution over the entire track.

: Hefcrcnce ( ï r c a d R = 2.54 m)

- - - :R=l.!Nhn - - - = : B = 2 . 1 7 m

- : R = 3 . 0 3 m : B = 3 . 8 0 m

10 100 1000 Turning Radius (m)

Fig.5-28 Moments of tuming resistance vs tuming radius

for a Jaguar at a vehicle spced of 29 km/h with different vehicle treads during a steady-state tum undcr cont inuous trapezoidal load distri but ion over the entire track.

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1.90 2.38 2.85 3.33 3.80 Con tact Lengt h L (m)

1.90 2.38 2.85 3.33 3.80 Contact Length L (m)

Fig.5-29 Ratio o f magnitude o f the sprocket torques vs vehicle Fig.5-30 Ratio o f magnitude o f the sprocket torques vs vehicle

tread B for a Jaguar at a vehicle speed o f 7.5 km/h with tread B for Jaguar nt a vehicle speed of 14.2 kmh with

di fferent turning radi i under continuous trapczoidal load different turning rcidii under continuous trapezoidal load

distribution over the entire track. distribution over the cntire track.

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1 .BO 2.38 2.85 3.33 3.80 Contact LRngth L (m)

1.90 2.38 2.85 3.33 3.80 Contact Length L (m)

Fig.5-31 Ratio of magnitude of the sprocket torqucs vs vehiclc Fig.5-32 Ratios of magnitude of the sprocket torques vs vehicle

tread B for a Jaguar ai a vehicle speed of 2 1.3 kiii/îi with tread B for a Jaguar at a vehicle speed o f 29 kmlh with

di fferent tuming radi i under cont inuous trapçzoidal load different tuming radii undçr continuous trapezoidal load

distribution over the entire track. distribution over thc entirc track.

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The same trend is also observed for normal load distributed only on the track pitch (L, =

15 cm) under each roadwheel (as shown in Fig.3-5(c)). as well as those for normal load

concentrated at a point (L, = 1 cm) under each roadwheel. These results are shown in Tables

B-5 to B-8 in Appendix B, and in Tables C-5 to C-8 in Appendix C, respectively.

5.3. Longitudinal Offset of Vehicle Center of Graviîy, cr

Track forces are related to the shear displacement as mentioned previously. Ln general

the shear displacement increases from the front to the rear of the track-ground contact length.

Wirh the shifiing of the center of gravity of the vehicle to the rear or with the increase of cy (cy

is positive from the center of track contact length to the rear), the normal pressure under the

track is higher in the rear than in the front. Therefore. the vehicle will develop higher shear

force and as a result. track forces increase with the increase of c,. As a result, the sprocket

torques and moments of turning resistance will also increase with the increase in cy as shown

in Fig.5-33 to Fig.5-44. The sprocket torque at different c, values and vehicle speeds at

various tuming radii are tabulated in Table A-9 to Table A- 12 in Appendix A.

The same trend is also observed for normal load distributed only on the track pitch (L, =

t 5 cm) under each roadwheel (as shown in Fig.3-S(c)), as well as those for normal load

concentrated at a point (L, = 1 cm) under each roadwheel. These results are shown in Tables

B-9 to B- 1 2 in Appendix B. and in Tables C-9 to C- 12 in Appendix C, respectively.

The effects of the longitudinal offset of the center of gravity, c,, on the turning

resistance moment are summarized in Figs.5-45 to 5-48. It shows the variation of the ratio of

the turning resistance moment (i.e.. the tuming resistance moment at a particular value of cY to

that at c, = O m) with the value of c,. At c, = O m, the ratio is equal to one.

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: Referencc (Longitudinal CG offset c, = O m)

- - - : C, = -0.20 in --- : c, = -0.40 m

----: C, = 0.20 m ----- : C, = 0.40 m

i 10 100 1000 Turning Radius (m)

Fig.5-33 Sprocket torques vs tuming radius for a Jaguar at a vehicle speed of 7.5 kmh with different longitudinal

CG offsets during a steady-state turn under continuous trapezoidal load distri but ion over the ent ire track.

: Rcfcrcnce (Longitudinal CG offsct c, = O m)

- 2 4 0 0 0 0 1

1 10 100 1000 Turning Radius (m)

Fig.5-34 Sprocket torques vs turning radius for a Jaguar at a vehicle speed of 14.2 kmlh with diffcrent longitudinal CG offsets during a steady-state turn under continuous

trapezoidal load distri but ion over the entire track.

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: Reference (Longitudinal CG offset c,. = O m)

10 100 1000 Turning Radius (m)

Sprocket torques vs turning radius for a Jaguar ai a

vehicle speed of 2 1.3 km/h with different longitudinal

CG offsets durinp a steady-state tum under continuous

trapezoidal load distribution over the entirc track.

: Heferençe (Longitudinal CG offset c, = O m)

-- - : c,. = -0.20 m I . . c,. = -0.40 m

---- : C, = 0.20 n> ----- : C, = 0.40 m

1 lnner Track

10 100 tooo Turning Radius (m)

Sprocket torques vs turning radius for a Jaguar at a vehicle speed of 29 kmlh with different longitudinal

CG offsets during û steady-state tum under continuous

trapezoidal lood distribution over the entire track.

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: Rcfcrcncc (Longitudinal CG offset c, = O 111) : Helerence (Longitudinal CC offset c, = O ni)

- - - : c, = -0.20 m - 0 . c,. = -0.40 m - - - : c,. = -0.20 m - 0 - 0 . c, = -0.40 m

----: C, = 0.20 m ----- : C, = 0.40 m ---- : C, = 0.20 m ----- : C, = 0.40 m

240001 Outcr Trirck

1 I O 100 1000 Turning Radius (m)

Outcr ~ r v c k

1 10 100 1000 Turning Radius (m)

Fig.5-37 Lateral forces vs tuming radius for a Jaguar at a vehicle Fig.5-38 Lateral forces vs turning radius for a Jaguar at a vehicle speed o f 7.5 kmlh with differcnt longitudinal CG offsets speed of 14.2 kmlh with diffcrcnt longitudinal CG offsets during a steady-state tum under continuous trapezoidal during a stcady-state turn undcr continuous trapezoidal Ioad distribution ovcr the entire track. load distribution over thc cntire track.

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: Reference (Loiigituclinal CC offset cy = O ni) : Referençe (Longitudinal CG offset c,. = 0 m)

- 9 - : c, = -0.20 m . O . C , = -0.40 m - - - : c,. = -0.20 m -.-# , c,. = -0.40 m

---- : C, = 0.20 m ----- : C, = 0.40 m ---- : c,. = 0.20 m ----- : C, = 0.40 m

10 100 1000 Turning Radius (m)

10 100 100 Turning Radius (m)

Fig.5-39 Lateral forces vs tuming radius for a Jaguar at a vehicle Fig.5-40 Lateral forces vs turning radius for a Jaguar ai a vehicle

speed of 2 1.3 km/h with different longitudinal CG offsets speed of 29 kmlh with different longitudinal CG offsets

during a steady-state tum under continuous trapezoidal during a steady-state tum und& continuous trapezoidal

load distribution over the entire track. load distribution over the entire track.

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: Heferencc (Longitudinal CG offset c, = O m) : Refercnïc (Longitudinal CC offset c,. = O ni)

-- --- : cy = -0.40 m - - - : c,. = -0.20 m - 7 - - : c, = -0.20 m . c,. = -0.40 in

---- : C, = 0.20 m ----- : C, = 0.40 m ---- : c,. = 0.20 m ----- : C, = 0.40 m

Fig.54 Moments of tuming resistance vs tuming radius for a

Jaguar at a vehicle speed of 7.5 kmlh with different

longitudinal CG offsets during a steady-state tum under

continuous trapezoidal load distribution over the cntire

track.

-100000 1 1 1 1 1 1 1 1 r r I I I I I I ~ I i r r i r i i l I

Fig.5-42 Moments of tuming resistancc vs tuming radius for a

Jaguar at a vehiclc speed of 14.2 km/h with different

longitudinal CG offsets during a steady-state turn under

continuous trapezoidal load distribution over the entire

track.

1 10 100 1000 Turning Radius (m)

-160000 Outer Track 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 I I I I I ~ I 1

1 10 100 1000 Turning Radius (m)

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: Refercnce (Longitudinal CG offset ç, = O m)

-- - : c, = -0.20 m I . . c, = -0.40 m

---- : C, = 0.20 ni ----- : C, = 0.40 ni

lnner Track/ ./* 4

Track

10 100 1 O00 Turning Radius (m)

Fig.5-43 Moments of turning resistance vs turning radius for a

Jaguar at a vehicle speed of 21.3 kmlh with different longitudinal CG offsets during a steady-state turn under continuous trapczoidal load distributioii over the entire track.

: Kcfcrcnçe (Longitudinal CC offset c, = O m)

- 9 - : c, = -0.20 ni -.-O . C, = -0.40 m

-a-- : c,. = 0.20 ni ----- : C, = 0.40 m

Outer Track

10 100 1000 Turning Radius (m)

Fig.5-44 Moments of turning resistance vs turning radius for a

Jaguar at a vehicle speed of 29 kmlh with differeiit longitudinal CG offstes during a steady-state tum uiider continuous trapezoidal load distribution over the entire track.

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-0.40 -0.20 0.00 0.20 0.40 Longitudinal CG Offset c, (m)

Fig.5-45 Ratio o f turning resistance moment vs longitudinal CG

offset c, for a Jaguar at a vehicle speed of 7.5 kmlh with

different turning radii under continuous trapezoidal load

distribution over the entire track.

0.5

-0.40 -0.20 0.00 0.20 0.10

Longitudinal CG Offset ce, (m)

Fig.546 Ratio o f tuming resistance moment vs longitudinal CG

offset ç, for a Jaguar at a vehicle speed o f 14.2 kmlh with

different tuming radii under continuous trapezoidal load

distribution over the entire track.

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-0.40 -0.20 0.00 0.20 0.40 Longitudinal CG Offset c, (m)

4.40 4.20 0.00 0.20 0.40 Longitudinal CG Offset c, (m)

Fig.5-47 Ratio of turning resistance moment vs longitudinal CG Fig.5-48 Ratio o f turning resistance moment vs longitudinal CG

offset c, for a Jaguar at a vehicle speed o f 2 1.3 kmh with offset c, for a Jaguar at a vehicle speed of 29 kmlh with

different tuming radii under continuous trapewidal load di fferent turning radii under continuous trapezoidal load

distribution over the cntire track. distribution over the cntire track.

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5.3. Track Width, b

The effects of track width, b. on the sprocket torque, lateral force and moment of

turning resistance during a steady-state tum under continuous trapezoidal normal load

distribution over the entire track are s h o w in Figs.5-49 to 5-60. It is noted that the traçk

width does not have much influence on the tracked vehicle steering on the nondefonnable

surfaces. The sprocket torques with different track widths and vehicle speeds at various

turning radii are tabulated in Tables A-1 3 to A-16 Appendix A.

The same trend is also observed for nonnal load distributed only on the track pitch (L,

= 15 cm) under each roadwheel (as shown in Fig.3-5(c)), as well as those for normal Ioad

concentrated at a point (Lp = 1 cm) under each roadwheel. These results are s h o w in Tables

B- 13 to B- 16 in Appendix B, and in Tables C- 13 to C- 16 in Appendix C, respectively.

The effects of track width on the turning resistance moment are sumrnarized in Figs.5-

6 1 to 5-64. It shows the variation of the ratio of the turning resistance moment (Le., the

turning resistance moment at a particular track width. b. to that at a track width, b9 of 0.45 m)

with track width. It appears that the track width has insignificant effect on the tuming

resi stance moment for the vehicle simulated.

5.5. Height of Vehicle Center of Gravity, h

The effects of height of vehicle center of gravity, h, on sprocket torques during a

turning maneuver under continuous trapezoidal nonnal load distribution over the entire track

are insignificant. as shown in Figs.5-65 to 5-68. The same observation can be made for both

IateraI forces and turning resistance moments on both outer and imer tracks of the vehicle.

except at a small turning radii, as s h o w in Figs.5-69 to 5-76.

The sprocket torques at different CG heights and forward speeds at various turning

73

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radii are tabulated in Tables A- 17 to A-20 in Appendix A.

The same trend is also observed for normal load distributed only on the track pitch (L,

= 15 cm) under each roadwheel (as s h o w in Fig.3-5(c)), as well as those for normal load

concentrated at a point (L, = 1 cm) under each roadwheel. These results are shown in Tables

B- 17 to B-20 in Appendix B. and in Tables C- 17 to C-20 in Appendix C, respectively.

The effects of height of vehicle center of p v i t y on the turning resistance moments are

summarized in Figs.5-77 to 5-80. It appears that the ratio of the himing resistance moment

(i-e.. the tuming resistance moment at a particular CG height to that at a height of 1.3 m)

does not v u y significantly with the height of center of gravity.

5.6. Summary

Based on the parametric studies described above. it can be concluded that arnong the

vehicle design parameters, the vehicie contact length. L. and tread, B, have significant effects

on the steering of tracked vehicles. Increasing the track contact length, L, causes a

considerable increase in the turning resistance moment. On the other hand. reducing the tread.

B. would cause an increase in the sprocket torques and track longitudinal forces. It is found

that track width. b. longitudinal offset, c,, and CG height, h, of the vehicle have insignificant

effects on steering behaviour.

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: Hcfcrcncc (Traïk witlth b = 0.45 m) : Hcfcrcncc (Track width b = 0.45 m)

1 10 100 1 O00 Turning Radius (m)

1 loner Track

1 10 100 1000 Turning Radius (m)

Fig.5-49 Sprocket torques vs tuming radius for a Jaguar at a Fig.5-50 Sprocket torques vs tuming radius for a Jaguar at a

vehicle speed of 7.5 km/h with different track widths vehicle speed of 14.2 kmlh with different track widths

during a steady-state turn under continuous trapezoidal during a steady-statc tum undcr continuous trapezoidal

load distribution over the entire track. load distribution ovcr the entire track.

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: Hcfcrencc (Track width b = 0.45 m) : Rcference (Track width b = 0.45 m)

I I - * . b = 1.35 m

200001

Outer Track

h

2 l0OW- 8 - 3 P i: Y

O - Y 8 L P Fn

40000- lnner Track

10 100 1 O00 Turning Radius (m)

10 100 ~ 0 0 0 Turning Radius (m)

Fig.5-51 Sprocket torques vs turning radius for a Jaguar at a Fig.5-52 Sprocket torques vs tuming radius for a Jaguar at a

vehicle speed of 21.3 kmh with different track widths vehicle speed of 29 kmRi with different crack widihs during a steady-state tum under continuous trapezoidal during a steady-state tum under continuous trapezoidal load distribution over the entire track. load distribution over the entire track.

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: Hcfcrcnce (Track width b = 0.45 m)

- - - :b=O.Olm : b = 0 . 9 0 m

- : b = 1.35 m

1 10 100 1000 Turning Radius (m)

Fig.5-53 Lateral forces vs tuming radius for a Jaguar at a vehicle

speed of 7.5 kmh with different track widths during a

steady-state tum under continuous trapezoidal load

: Hefercnce (Truck width h = 0.45 ni)

1 10 100 1000 Turning Radius (ni)

Fig.5-54 Lateral forces vs turning radius for a Jaguar at a vehicle

speed of 14.2 km/h with different track widths during a

steady-state tum under continuous trapezoidal load

distribution over the entire track. distribution over the entire track.

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: Heferencc (Track width b = 0.45 m)

10 100 1000 Turning Radius (m)

Fig.5-55 Lateral forces vs tuming radius for a Jaguar at a vehicle speed of 2 1.3 k m h with different track widths during a steady-state tum under continuous trapezoidal load distribution over the entire track.

: Heferencc (Track width b = 0.45 m)

--- : h = 0.01 m --- : b = 0.90 m

- : b = 1.35 m

10 100 1000 Turning Radius (m)

Fig.5-56 Lateral forces vs tuming radius for a Jaguar at a vehicle speed of 29 kmni with different track widths during a steady-state turn under continuous trapezoidal load distribution over the entire track.

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: Wcfcrcnce (Track witlth h = 0.45 m) : Wefcrence (Track width h = 0.45 m)

Fig.5-57 Moments of tuming resistance vs tuming radius for a Fig.5-58 Moments of tuming resistance vs tuming radius for a Jaguar at a vehicle speed of 7.5 km/h with different Jaguar at a vehicle speed of 14.2 krnh with different track widths during a stcady-state turn under continuous track widths during a steady-state turn under continuous trapemidal load distribution over the entire track. trapzoidal load distribution over the entire track.

-100000

1 10 100 1000 1 I O 100 1000

Turning Radius (m) Turning Radius (m)

Outer Track ' 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 7 1 1 1 1 1 1 1 I -t60000 1 1 1 1 1 1 1 1 1 1 I I I ~ I I I 1 1 1 1 1 1 1 1

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: Refercncc (Track w idth 1) = 0.45 m) : Referencc (Truck width h = 0.45 m)

10 100 1000

Turning Radius (m)

460000 ; ' , 1 1 1 1 1 1 1 1 1

10 100 1000 Turning Radius (m)

Fig.5-59 Moments of tuming rcsistance vs tuming radius for a Fig.5-60 Moments of tuming rcsistance vs turning radius for a

Jaguar at a vehicle speed of 2 1.3 kmlh with di ffercnt Jaguar at a vehicle speed of 29 kmlh with different track track widths during a steady-state turn undcr continuous widths during a steady-state tum under continuous trapezoidal load distribution over the entire track. trapezoidal load distribution ovcr the entire track.

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0.00 0.45 0.90 1.35 Track Widtb b (m)

Fig.5-63 Ratios o f tuming resistance moment vs track width b for a Jaguar at a vehicle specd of 21.3 km/h with

di fferent turning radii under cont inuous trapezoidal

load distribution over the entire track.

0.00 0.45 0.90 1.35 Track Width b (m)

Fig.5-64 Ratio of turning resistance moment vs track width b

for a Jaguar at a vehicle speed o f 29 kinh with

different turning radii under continuous trapezoidal

load distribution over the entire track.

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: Hcfcrcnce (CG hcight h = 1.30 m)

--- : h = 0 . 8 0 m O - : h = 1 . 8 m

1 10 100 1000 Turning Radius (m)

: Hcfcrcnce (CG hcight h = 1.30 m)

--- : h=Q.$Om --- : h = 1 . 8 m

1 lnner Track

1 10 100 1000 Turning Radius (m)

Fig.5-65 Sprocket torques vs tuming radius for a Jaguar at a Fig.5-66 Sprocket torques vs tuming radius for a Jaguar at it

vehicle speed of 7.5 km/h with different CG heights vehicle speed of 14.2 km/h with different CG heights during a steady-state turn under continuous trapezoidal dwing a steady-state turn under continuous trapezoidal load distribution over the cntire track. load distribution over the entire track.

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: Hcfcrcncc (CG hcight h = 1.30 m) : Hcferencc (CC hcight h = 1.30 m)

10 100 1000 Turning Radius (m)

I lnner Trrick

10 100 1000 Turning Radius (m)

Fig.567 Sprocket torques vs tuming radius for a Jaguar at a Fig.5-68 Sprocket torques vs tuming radius for a Jaguar at a

vehicle speed of 21.3 krnh with difl'erent CG heights vehicle speed of 29 kndh with different CG heights

during a steady-state turn under continuous trapezoidal during a steady-state turn under continuous trapezoidal

load distribution over the entire track. toad distribution over the entire trrick.

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: Hefercnce (CG height h = 1 -30 m)

0 0 0 : h=0.80m --- : h = 1.8m

\ Outer Track

lnner Track \\

: Hefcrcncc (CG height h = 1.30 m)

--- : h=O.SOm ---: h = 1.8m

1 10 100 1000 Turning Radius (m)

1 10 100 1000 Turning Radius (m)

Fig.5-69 Lateral forces vs tuming radius for a Jaguar at a vehicle Fig.5-70 Lateral forces vs turning radius for a Jaguar at a vehicle speed of 7.5 km/h with different CG heights during a speed of 14.2 km/h with different CG heights dwing a steady-state tum under continuous trapezoidal load steady-state tum under continuous trapezoidal load

distribution over the entire track. distribution over the entire track,

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: Refcrcnce (CG hcight h = 1 J O m)

10 100 1000 Turning Radius (m)

: Refcrencc (CG hcight h = 1.30 m)

--- : h = 0.80 m ---: h = 1.8 m

10 100 Io00 Turning Radius (ai)

Fig.5-71 Lateral forces vs tuming radius for a Jaguar at a vehicle Fig.5-72 Lateral forces vs tuming radius for a Jaguar at a vehicle

speed of 21.3 k m h with different CG heights during a speed of 29 kmh with different CG heights during a

steady-state tum under continuous trapezoidal load steady-state turn under continuous trapezoidal load distribution over the entire track. distribution over the cntire track.

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: Heference (CC height h = 1.30 m)

O - - :h=0.80m ---:h=1.8m

1 10 100 1000 Turaing Radius (m)

: Refcrencc (CC height h = 1.30 m)

lnner

-180000 1 1 1 1 1 1

1 10 100 1000 Turniag Radius (m)

Fig.5-73 Moments of tuming resistance vs turning radius for a Fig.5-74 Moments of tuming resistance vs turning radius for a

Jaguar at a vehicle speed of 7.5 km/h with different CG Jaguar at a vehicle speed of 14.2 km/h with different

heights during a steady-staie turn under continuous CG hcights during a steady-state turn under continuous

trapezoiâal load distribution over the entire track. trapemidal load distribution over the entire track.

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: Hcfercnïc (CG hcight Ii = 1.30 m)

--- :h=0.80m - - - :h=l .Um

Outer Track

10 100 1000 Turning Radius (m)

: Rcfcrencc (CG height h = 1 .JO m)

Fig.5-75 Moments of turning resistancc vs turning radius for a Fig.5-76 Moments of turning resistance vs turning radius for a Jaguar at a vehicle speed of 21.3 kinlh with different Jaguar at a vehicle speed of 29 km/h with different CG

CG heights during a steady-state turn under continuous heighis during a steady-state turn under çontinuous trapezoidal load distribution over the entire track. trapemidal load distribution over the entire track.

-200000 I 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

10 100 1000 Turning Radius (m)

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Chapter 6

Experimental Study of the Shearing Characteristics of a

Representative Track Link on AsphaIt

As mentioned previously. the shearing behaviour of a tracked vehicle is greatly

influenced by the shear force developed on the track-ground interface. A detailed

esperirnental study was canied out to measure the shearing characteristics between a

representative track link with rubber pad used on a curent amoured personnel carrier M 1 13

and asphalt in the laboratory.

6.1. Apparatus for Measuring tbe Sbearing Cbaracteristics

The apparatus used to measure the shearing characteristics. including the relation

between shear force and shear displacement. was a modification of an existing tire tester in

the Transport Technology and Research Laboratory. Carleton University. The apparatus

includes state-of-art data acquisition system. developed by Mr. Y.C. Wu for his Ph.D

research on the lateral dynamics of off-road tires.

The schematic of the test apparatus is illustrated in Fig.6-1. The major components of

the track link shear force-shear displacement tester include:

0 a horizontal motion carrier

0 a test bed on which an asphalt block was mounted

0 a double parallelograrn mechanism on which test track links were rnounted

0 a data acquisition system

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1

Horizontal Motion Caniage

1- Double Parallelogram

/ Mec hanism

Dead Weight

x-y

1 a I - 1

1 - O 3 1 v Load Cell

Track Supporting Frame

Linear Bearing

Asphalt Track Links

Test Bed

Fig.6-l Schematic of the test setup for the rneasurement o f track shear

force and shear displacement on the asphalt.

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two Lebow 6643 x-y load cells. and one displacement potentiometer.

Two test track links and their supporting h e are connected with a double

parallelogrm mechanism which is mounted on the horizontal motion carriage. The motion

carriage is driven by an electric motor through a chain drive mechanism. When the carriage

moves fonvard. shearing action between track links and the asphalt surface takes place. The

horizontal shear force was monitored by hvo Lebow 6643 x-y Ioad cells located at both lefi

and right hand sides of the carriage, as shown in Fig.6-1. A displacement potentiometer is

mounted to measure the horizontal travelling distance of the h e , which is the shear

displacement of the track links. These forces as well as horizontal displacement are al1

recorded by a data acquisition system (Fig.6-2).

Corn p u t e r

I - D i s p l a c e m e n t

P o t e n t i o m e t e r

Fig.6-2 Block diagram of the data acquisition system for the

shear force-shear displacement tester.

L e f t a n d R i g h t

x - y L o a d C e l l s

Tests were carried out under four normal loads on the track links of 890 , 1000 , 1200.

V

S t r a i n G a u g e

C o n d i t i o n e r A

w

A na Iogue/D i g i t C o n v e r t e r

D a t a A c q u i s i t i o n s o f t w a r e

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and 1400 1 b. and at three different speeds. 0.195. 0.145. and 0.100 m/sec. Seven test nins

were conducted for each forward speed-normal load combination.

6.2. Analysis o f Shear Stress-Shear Displacement Data

The most commonly used semi-empirical equation to describe the shear stress-shear

displacement relationship on the track-ground interface is equation (3-1 5). To determine the

optimum value of shear deformation parameter. K. which minimizes the error in curve fitting.

the following equation is used (Wong. 1983)

where r, is measured stress. 5, is mê'timum shear stress under a particular load, and j is

measured shear displacement.

To solve for an appropriate value of K, the following procedure is used.

1 ). The maximum value of shear stress. rmx. in equation (6-1) was first identified fiom

the measured data, as shown in Fig.6-3.

2). K is calculated using equation (6- f ).

3). A goodness-of-fit is calculated using equation (3-43).

Foltowing the above procedure. the average value of the shear deformation parameter,

K. for the M l 13 track link with rubber pad on asphalt is found to be 0.0183 m and the

goodness-of-fit ranges from 86.72% to 93.52%. The measured data and fitted curves using

the calculated K value are shown in Figs.6-3 to 6-13.

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0.0 O. 1 0.2 0.3 0.4

Shear Displacement (rn)

Fig.6-3 Shear force-shear displacement curve at normal load 890 Ib with sliding speed 0.100 mls.

Shear Displacemen t . .

Fig.6-4 Shear force-shear displacement curve at nomal load 890 Ib with sliding speed 0.145 mls.

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0.0 0.1 0.2 0.3 0.4 0.6

Shear Displacement (m)

Fig.6-5 Shear force-shear displacement curve ai normal load 1 O00 Ib with sliding speed 0.100 mls.

0.0 0.1 0.2 0.3 0.4 0.6

Shear Displacement (m)

Fig.6-6 Shear force-shear displacement curve at nomial load 1 O00 Ib with sliding speed 0.145 m/s.

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0.0 O. 1 0.2 0.3 0.4 0.6

Shear Displacement (m)

Fig.6-7 Shear force-shear displacement curve at normal load 1000 Ib with sliding speed 0.195 m/s.

0.0 0.1 0.2 0.3 0.4 0.5

Shear Displacement (m)

Fig.6-8 Shear force-shear displacement curve at normal load 1200 Ib with sliding speed O . I O m/s.

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0.00 0.10 0.20 0.30 O.* 0.60

Shear Displacement (m)

Fig.6-11 Shear force-shear displacement curve at normal load 1400 Ib with sliding speed O. 100 m/s.

0.0 0.1 0.2 0.3 0.4 0.6

Shear Displacement (m)

Fig.6- 1 2 Shear force-shear displacement curve at nonnal load 1400 Ib with sliding speed 0.145 m/s.

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0.0 0.1 0.2 0.3 0.4 0.5

Shear Displacement (m)

Fig.6- 13 Shear force-shear displacement curve at normal

load 1400 lb with sliding speed 0.195 m/s.

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6.3. The Determination of the Coefficient of Friction

From the measured data shown in Figs.6-3 to 6-13, the maximum shear force

developed under various normal loads can be identified. Their relationship is illustrated in

O 400 M)O 1200 100

Normal Lord (N)

Fig.6-14 The relation of maximum shear force and normal load.

The coeficicnt of fiction. p. between the track Iink with rubber pad and the asphalt

surface. c m be found frorn the slope of the fitted line shown in Fig.6-11.

Based on the experimental data shown. it was found the average value of p was 0.684.

It should be mentioned that the shear tests were conducted at three different values of sliding

velocity. 0.10.0.145. and 0.195 m/s. The test results show that within this range, the sliding

velocity has insignificant effect on the coefficient of fiction, p, and the shear deformation

parameter. K. Within the range of speeds used. the value of p varies fiom 0.664 to 0.737, and

K varies in the range of 0.0 167 to 0.0225 m.

In the simulation of the steenng behaviour of Ml 13 on asphait described in the next

chapter. the average value of the shear defornation parameter K = 0.0183 m and p = 0.684

were used.

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Chapter 7

Simulation of Steering Behaviour of an Ml13 Armoured

Personnel Carrier on Non-deformable Ground Based on

Measured Shear Data

7.1. Basic Design Parameters o f an M l 13 Armourd Personnel Carrier

Using the measured data on the shearing characteristics of an Ml 13 track link with

rubber pad on asphalt described in the previous chapter, the handling behaviow of an Ml 13

mored personnel carrier was predicted. The predictions were made using the general theory

described in Chapter 3. The basic vehicle and track parameten used in the predictions are

given in Table 7- 1 .

Table 7- 1 Basic parameters of the M 1 1 3 tracked vehicle used in the predictions

Sprung weight, kN 100.57 Unsprung weight, kN 10.08 ~ e i i h t of c.G.. rn Sprocket radius, m Trac k-ground contact length. m Longitudinal CG. location from centerline of vehicle hull, m Track width. m Track pitch Tread of vehicle, m

1 Shearing characteristic

Shear deformation parameter, m 0.0 183 Coeficient of friction between track-ground (asphalt) interface, 0.684 Coefficient of motion resistance, 0.03+0.000 1 5*V

*: The unit of forward speed V is in kilometers per hour.

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7.2. Simulation Results

The predicted sprocket torques. lateral forces. and moments of tuning resistance for the

M l 13 dunng a steady-state tm on an asphalt surface are shown in Figs.7-1 to 7-14. A

cornparison of sprocket torques predicted by Steeds' model and by the proposed model at

different vehicle speeds and normal load distributions are shown in Tables 7-2 to 7-9.

Table 7-2 Sprocket torques at various turning radii with a vehicle speed of 7.5 km/h for an M 1 13

armoured personnel carrier.

Tuming 1 Predicted by Steeds' 1 Predicted by proposed general theory (kN.m)

1 1 the entire track lpitch under each madwheel 1

(kN.m)

(m) ndius- l

Outer lnner Outer

Continuous trapezoidal

load distribution over

Table 7-3 Sprocket torques at various turning radii with a vehicle speed of 14.2 kmih for an MI 13

armoured personnel carrier.

Inner

-3.678

-3 -3 94

-2.486

-1 567

Trapezoidal load distribution on the track

(ml 1 1 load distribution over 1 disaibution on the oack 1 under each roadwheel 1

Concentrated load under each roadwheel

Tuming

radius

1 1 the entire track 1 pitch under each roadwheel 1 1

Outer

4.664

4.338

3 -496

2.557

I ~ e r

-3 -938

-3 -607

-2.76 1

-1.821

Outer

4.304

4.123

3.427

2.538

Predicted by Steeds'

(kN.m)

lnner

-3.588

-3.393

-2.692

- 1 -802

Predicted by proposed general theory (kN.m)

Continuous trapezoidal [ Trapezoidal load 1 Concentrated load

1 O

20

50

1 O0

Outer

3.590

4.630

4.621

4.61 1

lnner

-3.748

-3.863

-3.861

-3.85 1

Outer

4.336

4.102

3.175

2.268

Inner

-3.504

-3 .340

-2.4 16

- 1 -508

Outer

4.679

4.386

3.52 1

2.579

Inner

-3.854

-3.63 1

-2.765

- 1.820

Outer

4.503

4.256

Inner

-3.700

-3.507

3.479

2.569

-2.723

-1.810

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Table 7 4 Sprocket torques at various turning radii with a vehicle speed of 2 1.3 km/h for an M 1 13

armoured personnel carrier.

- m . - - - a - . - . a m . I I P L . - . n .- . I . .-

1 O

20

50

1 O0

I able 1-3 '4;proc)tet torques at vartous turning raaii witn a venicie speea or LY Kmn ror an M I i

Turning

radius

(ml

radius

im )

Predicted by Steeds'

model (kN-m)

Predicted by proposed general theory (kN.m)

Continuous trapezoidal load distribution over

I the entire track

1 theentiretrack lpitchwidereachroadwheell

armoured personnel carrier.

Trapezoidal load distribution on the track

Outer

4.5 18

4.259

3 -476

2.575

lnner

-2 -462

-2 -900

-2.257

-1 -377

pitch under each roadwheel

Outer

4.192

4.555

1.642

4.636

lnner

Concentrated load

under each roadwheel

Outer

4.380

4.24 1

3 -462

2.564

Turning

15

30

50

1 O0

inner

-3 .O25

-3.4 14

-2.680

- 1 -780

lnner

Predicted by Steeds'

model (kN-m) Predicted by proposed general theory (kN.m)

Outer

Concentrated Ioad under each roadwheel

Continuou traperoidal load distribution over

-2.88 1

-3.714

-3.855

Trapezoidal load distribution on the track

Outer

3.836

3 -670

3 .O87

2.353

Outer

3.732

4.474

4.621

4.656

3.769

3.738

3.043

1~

Outer

3.144

2.836

2.322

1.85 1

Inner

-2.346

-3.601

-3.797

-3.843

Inner

-1.707

-1 -941

-1.489

- 1 .O36

Imer

-2.346

-2.799

-2.264

-1 -541

-3.050

L

Outer

4.193

3.865

3.264

2.44 1

2.162

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: Cantinuous trspe~oitlal load distribution aver the entirc truck

- - - : Canccntratcd load under each rolidwheel

: Trapczoidal load distribution on the trcick pitch unrlcr each roadwhecl

- - - - : Stceds' madcl

-1 Outer Trwck Outer Track

-

-6000 1 lnnet Track lnner Track

I I 1 1 1 1 1 1 I 1 1 1 1 1 1 1 I I 1 1 1 1 1 1 ) 1

1 10 100 1000 Theoretical Turning Radius (m)

1 10 100 1000 Theoreticml Turning Radius (m)

Fig.7- 1 Sprocket torques vs tuming radius for an M 1 13 at Fig.7-2 Sprocket toques vs tuming radius for an Ml 13 at

a vehicle speed of 7.5 kmlh with K = 0.01 83 m a vehicle speed of 14.2 km/h with K = 0.0183 m

and p = 0.684 during a stcady-state tum. and p = 0.684 during a steady-state tum.

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: Cnntinuous trnpcroidal load distribution ovcr the cnqire track

- - - . : Concentratcd load under cach roadwhecl

- - : Trapzoidal loud distribution on the track pitch under eich rovdwhccl

- - - : Steeds' mode1

"1 Outer Track

Fig.7-3 Sprocket torques vs tuming radius for an MI 13 at a vchicle speed of 21.3 km/h with K = 0.01 83 m and )i = 0.684 during a steady-state turn.

-

-6000

"1 Outer Track

lnner Track

I 1 1 1 1 1 1 1 I 1 1 1 1 1 1 1 1 1 I 1 1 1 1 1 1 1 I

Inner Track

JO00 1 10 100 1000

Theoretical Turaiag Radiua (m)

1 10 100 1000

Theoretical Turning Radius (m)

Fig.7-4 Sprocket torques vs tuming radius for an M l 13 at a vehicle speed of 29 kmh with K = 0.01 83 rn and p = 0.684 duriny a steady-state tum.

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: Continuous trapczoidal loarl distribution ovcr the cntirc track

- - - : Conccntrwtcd toatl under elrch ralrdwheel

: TrapmidaI load distribution on the track pitch under each road~hccl

1 10 100 1000

Theoretical Turning Radius (m)

Fig.7-7 Lateral forces vs tuming radius for an M 1 13 üt ü

vehicle speed of 14.2 km/h with K = 0.0183 m

and y = 0.684 during a steady-state tum.

1 qo 100 1006

Theoretical Turning Radius (m)

Fig.7-8 Lateral forces vs tuming radius for an M 1 13 at a vehicle speed of 21.3 kmlh with K = 0.0183 m and p = 0.684 during a steady-state tum.

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: Continuous trapczoidal load distribution ovcr the cntire track

- - - : Concentratcd load undcr each roadwhccl

: Trupzoidal load distribulion on thc t r ~ c k pitch undcr euch roadwhcel

- - - - : Stceds' model

Fig.7-12 Moments of tuming resistance vs tuming radius Fig.7-13 Moments of tuming resistance vs tuming radius

for an MI 13 ai o vehicle speed of 14.2 k m h with for an Ml 13 at a vehicle speed of 21.3 k m h with

K = 0.01 83 m and p = 0.684 during a steudy-state K = 0.0183 m and )i = 0.684 during a steady-state

tum. turn.

-48m

1 10 100 1000 1 10 100 1OOO

Theoretical Turning Radius (m) Thearetical Turning Radius (m)

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 I 1 1 1 1 1 1 1 1 uooo 1 1 1 1 1 1 1 1 I 1 1 1 1 1 1 1 1 I 1 1 1 1 1 1 1 1 1

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7.3. Summary

It cm be seen from Figs.7-l to 7 3 and Tables 7-2 to 7-5 that the relationships between

the sprocket torque and tuming radius at various vehicle speeds have similar trend to those

predicted for the Jaguar vehicle described in Chapter 4. The predictions using Steeds' mode1

again show that the sprocket torques and moments of twning resistance at higher ~ i n g

radii remain almost constant. This indicates again that the track-ground shearing

c harac terist ics. speci fical 1 y the shear stress-shear displacernent relation. must be taken into

consideration in predicting the handling of tracked vehicles. It also appears that normal

pressure distribution also has certain influence on the relationships between sprocket torque

and turning radius.

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Chapter 8

Discussion and Conclusions

This research project focuses on a detailed study of the mechanics of steenng of

tracked vehicles on non-deformable surfaces. taking into account the effects of shear

displacement on the development of shear stress and of track width. Experimental studies

ha\.e been carried out to identi@ the shearing characteristics on asphalt of the track link with

the mbber pad used on the M l 13 annoured personnel carrier. A computer simulation model

for steering behavior of tracked vehicles under steady-state conditions over non-deforrnable

surfaces has been developed.

8.1. Conclusions

The handling characteristics of tracked vehicles predicted using the mode1 developed

in this project bear close resemblance to those measured in the field, as shown in Figs.4- t to

4-4 and in Tables 4-2 to 4-9. It is also shown that this model represents a significant

improvement on Steeds' model. This is because the model developed takes into account the

effect of shear displacement on the development of shear stress on the track-ground interface.

whereas Steeds' model assumes that full shear stress develops as soon as a small relative

motion between the track and the ground takes place. However, Steeds' model may be

considered a special case of the general theory presented in this thesis where the shear

defonnation parameter, K, is zero.

Using the model developed, the effects of design and operating parameters on the

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stéering of tracked vehicles have been systematically examined. It is found that increasing

tnck-ground contact length. L. the sprocket torques required in a steady-state h n g

maneuver increase significantly. This is caused by the considerable increase of the NNng

resistance moment.

It is also found that the sprocket torques required to execute a steady-state hirn increase

u.ith the decrease of tread. B. This is primarily due to the decrease in the moment arms of the

longitudinal track forces. which tom the tuniing moments and initiate the turn of the

vehicle.

Longitudinal offset of vehicle center of gravity, c,. also affects the sprocket torques

required during a tuming maneuver. General speaking, sprocket torques will increase with

the shifting of CG rearward. As a result of shifiing CG rearward the normal pressure at the

rear is higher than that at the front. This causes an increase in the tuming resistance moment

and hence sprocket torques.

in general. the height of vehicle CG and track width do not have significant effects on

the steering characteristics of tracked vehicles over non-deformable surfaces.

Experimental studies have also been carried out to identiQ the shearing characteristics

between the track link used in the armoured personnel carrier M 1 1 3 and asphalt. Based on

experimental data the steering characteristics of Ml 13 over asphalt are predicted using the

model developed. and compared with those predicted using Steeds' model.

In conclusion. the general theory for tracked vehicle steering on non-deformable

surfaces presented in this thesis is a significant improvement on existing models. The

predictions based on this model have a very similar trend to measured field data. It is

believed that the general theory developed provides a basis for the M e r study of tracked

vehicle maneuverability on deformable terrain.

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8.2. Future Work

Based on the experimental and analytical studies mentioned previously, the following

recommendations for future work can be made:

1 ). Further study on the factors affecting the shearing characteristics between the track and

the ground. such as sliding velocity. temperature. moisture content of the terrain, etc.

1). Initiation of systematic study of tracked vehicle steering on deformable terrain based on

the studies presented in the thesis. with the following factors included in the analysis:

O Vehicle sinkage on deformable terrain

a Shearing characteristics between the track and the deformable terrain

a Normal pressure distribution under the track on deformable terrain during a tuming

maneuver

O Bulldozing effect of the track in lateral skidding on turning resistance

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Bibliography

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Predicting the Steering Performance of Track-Laying Vehicles, Proceedings o f the 6th

International Conference o f the International Society for Terrain-Vehicle Systems, pp285-332.

Baladi. G.Y. and Rohani. B.. 198 1. Analysis o f Steerability of Tracked Vehicles: Theoretical

Pred ic t ions Versus Field Measurements. Proceedings of the 7m Internat ional Con ference of the

lnternational Society for Terrain-Vehicle Systems, pp. 1 175- 1220.

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Press.

Bekker. M.G.. 1960.Off-the-Road Locomotion. Ann Arbor, MI: University of Michigan Press.

Bekker. M.G.. 1969. Introduction to Temin-Vehicle Systems, Ann Arbor, MI: University of

Michigan Press.

Crosheck. J.E.. 1973. Skid-Steering o f Crawlers, SAE Paper No.750552, pp. 1 - 15.

Ehiert. W. Hug. B. and Schmid, I.C., 1992, Field Measurements and Analflical Models as a

Basis of Test Stand Simulation of the Turning Resistance o f Tracked Vehicles, Journal of

Terramechanics.Vol.39( 1 ), pp.57-69.

Eiyo, F. and Kitano, M., 1984. Study on Controllability and Stability o f High Speed Tracked

Vehicles. Proceedings o f the 8" International Conference of the International Society for

Terrain-Vehicle Systems, pp.789-8 19.

Foss. C.F.. 1993-94, Jane's Amour and Artillery, 1 4 ~ Ed.. Jane's Information Group Inc..

Alexandria, USA.

Hayashi. J. 1975, Practical Analysis o f Tracked Vehicle Steering Depending on Longitudinal

Track Slippage. Proceedings of the 5'h lnternational Conference o f the International Society for

Terrain-Vehicle Svstems. DD-493-5 1 2.

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I 1 . Kar. M.K., 1987, Production of Track Forces in Skid Steering of Military Tracked Vehicles,

Journal of Terramechanics. Vo1.23( 1 ), pp.75-86.

12. Karafiath, L.L.. 1981. Analytical Model for the Turning of Tracked Vehicles in Soft Soils.

Proceedings of the 6' international Conference of the lntemational Society for Terrain-Vehicle

Systems. pp. 1385-141 1.

13. Kitano. M. and Jyozaki, H., 1976. A Theoretical Analysis of Steerability of Tracked Vehicles.

Journal of Terramechanics. Vol. 13(2), pp.24 1-26?.

14. Kitano. M.. and Kurna M.,I977. An Analysis of Horizontal Plane Motion of Tracked Vehicles,

Journal of Terramechanics, Vol. 14(4), pp.2 1 1-225.

! 5. Kitano. M. Watanabe. K. Takaba, Y. and Togo. K.. 1 988. Lane-Change Maneuver of High Speed

Tracked Vehicles. Journal Terramechanics. Vo1.25(2), pp.91-102.

16. K. Kitano. Watanabe. M. and Nagatomo, N.. 1990. Stability and Controllability of High-Speed

Tracked Vehicles-Linear Model and Vehicle Response. Proceedings of the 10" lntemational

Con ference of the International Society for Terrain-Vehicle Systems, pp.659-670.

17. Merritt. H.E.. 1939. Some Considerations Influencing the Design of High Speed Tracked

Vehicles, Proceedings of Institute of Automobile Engineers, Vol.33, pp.398-430.

18. Micklethwait. E.W.E., 1944, Soil Mechanics in Relation to Fighting Vehicles, Military Coli. of

Science. Chobham Lane. Chertsey,

19. Murakami. H. Watanabe, K. and Kitano, M.. 1992, A Mathematical Model for Spatial Motion

of Tracked Vehicles on Soft Ground, Journal of Terramechanics, Vo1.29( 1 ), pp.7 1-8 1.

20. Ogorkiewicz, R.M., 199 1, Technolow of Tanks. London: Jane's Information Group.

2 1 . Schmid. I.C., 1984, A tracked Vehicle Test Plant for the Simulation of Dynarnic Operation,

Proceedings of the 8" In ternational Con ference of the 1 ntemational Society for Temin-Veh icle

Systems. pp.835-853.

22. Shiller, Z and Serate, W. 1995, Trajectory Planning of Tracked Vehicles, Journal of Dynamic

Systems. Measurement, and Control. Vol. 1 1 7, pp.6 19-623.

23. Steeds, W., 1950, Tracked Vehicles (in Three Parts), Automobile Engineer, pp 1 43-1 48, pp 1 87-

190 and pp.2 19-222.

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24. Stoer. J. and Bulirsch, R.. 1980, Introduction to Numerical Analysis, Spronger-Verlag: New

York. Inc.

25. Stroud. A.H.. 1966. Gaussian Quadrature Formulas, Prentice-Hall, Inc.

26. Sugiyama. N. and Kondo. H., 1984, Basic Study on the Turning Resistance o f Track,

Proceedings of the 8& International Conference of the International Society for Terrain-Vehicle

Systems. pp.889-899.

27. Watanabe. K. Kitano, M. and Katahira, T.. 1993, Controllability and Stability of Tracked

Vehicles on lnclined Ground, Proceedings of the 1 2 ' ~ International Conference of the

international Society for Terrain-Vehicle Systems. pp.426-435.

28. Watanabe. Murakami. H. Kitano, M. and Katahira. T., 1993, Experimental Characterization of

Dynamic Soil-Track Interaction of Dry Sand, Journal of Terramechanics, Vol-30(2), pp.1 I l -

131.

29. Weiss. K.R., 197 1. Skid-Steering, Automobile Engineer, pp.22-25.

30. Wismer, R.D. and Luth. H.J., 1973, Off-Road Traction Prediction for Wheeled Vehicles.

Journal o f Terrarnechanics. Vol. 1 O(2). pp.49-6 1.

3 1 . Wong. J.Y. and Preston-Thomas. J.. 1983, On the Characterization of the Shear Stress-Shear

Displacement Relationship of Terrain. Journal of Terrarnechanics, Vol. 19(4), pp.225-234.

32. Wong. J .Y. and Preston-Thomas. J., 1988, Investigation into the Effects of Suspension

Characteristics and Design Parameters on the Performance of Tracked Vehicles using an

Advanced Computer Simulation Model, Proceedings of Institution of Mechanical Engineers.

V01.202(D3), pp.143-161.

33. Wong. J.Y .. 1989. Terramechanics and Off-Road Vehicles. Elsevier. Amsterdam.

34. Wong. J.Y.. 1993. Theory of Ground Vehicles, 2"' Edition, John Wiley & Sons, Inc.

35. Wong. J.Y.. 1994. Cornputer-Aided Methods h r Design Evaluation of Track Systems, SAE

Paper No.94 1675, pp. 1-9.

Page 150: CHARACTERISTICS OF TRACKED VEHICLES ON€¦ · Tracked vehicle steering has been a subject of interest to the off-road vehicle industry ... Tracked Vehicles for Trapezoidal Load Distribution

Appendix A.

The Effects of Design and Operating Factors oa the Handling of Tracîced

Vehicles under Continuous Trapezoidal Load Distribution over the entire track

Page 151: CHARACTERISTICS OF TRACKED VEHICLES ON€¦ · Tracked vehicle steering has been a subject of interest to the off-road vehicle industry ... Tracked Vehicles for Trapezoidal Load Distribution

A.1. Contact Lengtb, L

Table A-1 Sprocket torques at various turning radii with different track-ground contact lengths at a vehicle speed of 7.5 km/h under continuous trapezoidal load distribution over the entire track.

Table A-2 S procket torques at various turning radi i with di fferent track-ground contact lengths at a vehicle speed of 14.2 km/h under continuous trapezoidal load distribution over the entire track.

1

Tuming

Radius

(ml

5

1 O

20

50

1 O0

Turning

Rad i us

<ml

5

1 O

20

50

L = 1.90 m

N.m

L = 4.75 m.

N.m

Outer

7810

5460

3886

2407

1947

L = 3.8 m

(Reference)

N-m

L = 2.85 m

N.m

Outer lnner

1 O0

Outer

25438

24033

21689

15153

L = 1.90 m

N.m

L = 5.7 m.

N.m

Inner

-5191

-2959

-1195

83.70

544.6

Outer

21742

19740

15762

9509

6012

15598

12401

8665

4980

3363

Inner

-18997

-21082

-19125

-12594

Outer

8833

5483

3534

2291

Outer

28654

27295

23302

20740

14765

L = 4.75 m

N-m

lnner

-19110

-17273

-13286

-7021

-3522

-13022

-9920

-6181

-2491

-871.8

[mer

4010

-2470

-868.8

292.2

L = 2.85 m.

N.m

Inner

-20999

-24163

-23313

-18183

-12199

Outer

25777

24532

L = 5.7 m

N.m

1909

Outer

14837

11042

7431

4283

r

L = 3.8 m.

(Reference)

N.m

lnner

-23043

-22049

21866

15230

9983

Outer

28519

27670

664.5

lnner

-9919

-8137

-4804

-1707

Outer

20948

18883

15211

9144

1

lnner A

-25665

-25164 .

3049

Inner

-15451

-16047

-12641

-6578

-476.8

-23505

-18226

-12242

-19391

-12744

-7493

-7368 5798

25983

20712

14732

-3228 9936

Page 152: CHARACTERISTICS OF TRACKED VEHICLES ON€¦ · Tracked vehicle steering has been a subject of interest to the off-road vehicle industry ... Tracked Vehicles for Trapezoidal Load Distribution

Table A-3 Sprocket torques at various turning radii with different track-ground contact

lengths at a vehicle speed of 2 1.3 km/h under continuous trapezoidal load

Table A-4 Sprocket torques at various turning radii with different track-ground contact lengths at a vehicle speed of 29 kmh under continuous trapezoidal load

distribution over the entire track.

distribution over the entire track.

N.m 1 (Reference)

1

Outer [nner Outer Inner Outer Inner Outer Inner 1

-6132 15489 -10149 19678 -13972 22952 -16586

7358 4175 11765 -8684 17881 -14848 22854 -19772

-2346 8682 -5846 14625 -1 1833 20410 -17629

3404 1 -622.3 5396 -2628 9680 -6929 14790 -12048

Tuming

Radius

(m.)

1 O

20

50

100 ,

L = 1.90 m

N.m

L = 2.85 m.

N.m

Outer

7874

4720

2756

1085

Outer

1 1896

7918

43 15

lnner

-3096

-1638

-39.98

583.0

L = 4.75 m.

N.m

L = 3.8 m.

(Reference)

N.m

Inner

-7317

4936

-1617

Outer

22286

20887

14948

Outer

17482

14317

8581

5368

L = 5.7 m.

N.m

2932 -265.91

lnner

-17078

-18016

-12300

Inner

-12781

-11440

-5912

-2709

Outer

25615

25184

20688

14788 9818 , -7166 ,

Inner

-19660

-22224

-18045

-12140

Page 153: CHARACTERISTICS OF TRACKED VEHICLES ON€¦ · Tracked vehicle steering has been a subject of interest to the off-road vehicle industry ... Tracked Vehicles for Trapezoidal Load Distribution

A.2. Tread of Vehicle, B

Table A-5 Sproc ket torques at various turning radii with different treads at a vehicle speed of 7.5 km/h under continuous trapezoidal load distribution over the entire track.

Table A-6 Sprocket torques at various turning radii with different treads at a vehicle speed of 14.2 krnh under continuous trapezoidal load distribution over the entire track.

Outer

25877

23969

19946

12497

7845

inner

-23231

-21502

-17473

-10010

-5355

Outer

24047

22066

18010

11056

6938

Inner

-21412

-19601

-15536

-8569

4448

Outer

21742

19740

15762

9509

6012

Inner

-19110

-17273

-13286

-7021

-3522

Outer

19003

17063

13322

7968

5128

lnner

-16363

-14591

-10843

-5479

-2638

Outer

15692

13903

10554

6318

4201

lnner

-13029

-11419

-8071

-3828

-1711

Page 154: CHARACTERISTICS OF TRACKED VEHICLES ON€¦ · Tracked vehicle steering has been a subject of interest to the off-road vehicle industry ... Tracked Vehicles for Trapezoidal Load Distribution

Table A-7 Sprocket torques at various turning radii with different treads at a vehicle speed of 2 1.3 km/h under continuous trapezoidal load distribution over the entire track.

Turning

Radius (Reference)

N.m

Outer Inner

17482 -12781

14317 -11440

8581 -5912

Outer Outer Inner Outer 1 Inner Inner

1 able A-u sprocKet torques at various rurning raaii witn airrerent ueaas at a venlcie speea

of 29 km/h under continuous trapezoidal load distribution over the entire track.

Tum ing

Radius

(ml

B = 2.54 m

(Reference)

N.m 1

Outer Inner Outer Outer

18316 -12546 17225 15489

15786 -12745 13893 11765 r

Inner Outer lnner

11356 -5966

7805 -4627

5638 -2757

- - -

Inner Outer 1 Inner

Page 155: CHARACTERISTICS OF TRACKED VEHICLES ON€¦ · Tracked vehicle steering has been a subject of interest to the off-road vehicle industry ... Tracked Vehicles for Trapezoidal Load Distribution

A.3. Longitudinal Offset of Vehicle Center of Gravity, c,

Table A-9 Sprocket torques at various turning radii with different longitudinal CG offsets at

a vehicle speed of 7.5 km/h under continuous trapezoidai load distribution over the entire track.

Radius

(m)

(Re ference) 1 N.m 1 N.m I Inner Outer lnner

1

Inner Outer Outer Inner

Table A- 1 0 Sprocket torques at various turning radii with di fferent longitudinal CG offsets at

a vehicle speed of 14.2 km/h under continuous trapezoidal load distribution over

the entire track.

1 Radius 1 N.m 1 N.rn (Reference) 1 N.m 1 N.m l

Page 156: CHARACTERISTICS OF TRACKED VEHICLES ON€¦ · Tracked vehicle steering has been a subject of interest to the off-road vehicle industry ... Tracked Vehicles for Trapezoidal Load Distribution

Table A- 1 1 Sprocket torques at various turning radii with different longitudinal CG offsets at a vehicle speed of 2 1.3 km/h under continuous trapezoidal Ioad distribution over the entire track.

Turning

Radius

(m)

1 able A- 1 f Sprocket torques at vanous turning rad11 with dltterent Longitudinal Cti oïtsets at a vehicle speed of 29 km/h under continuous trapezoidal load distribution over the entire track.

Turning

Radius

(m.)

N.m I N.m ( (Reference) 1 N.m

c, = -0.4 m

Outer Inner & c, = -0.2 m

Outer

14156

10410

7781

4967

c ,=Om

lnner

-8606

-7298

4940

-2200

c,. = 0.2 m

Outer lnner Outer

15489

11765

8682

5396

Inner

-10149

-8684

-5846

-2628

Outer

15356

11511

8437

5248

15049

Inner

-10102

-8416

-5591

-2478

-9605

11371

8462

53 18

-8283

-5628

-2552

Page 157: CHARACTERISTICS OF TRACKED VEHICLES ON€¦ · Tracked vehicle steering has been a subject of interest to the off-road vehicle industry ... Tracked Vehicles for Trapezoidal Load Distribution

A.& Track Width, b

Table A- 1 3 Sprocket torques at various t u m i n g radii with di fferent track widtbs at a vehicie

speed of 7.5 km/h under continuous trapezoidal load distribution over the entire track.

1 Radius 1 N.m 1 (Rrference) I N.m I N.m

Table A- 13 Sprocket torques at various tuming radii with different track widths at a vehicle

5

1 O

20

50

speed of 14.2 km/h under continuous trapezoidal load distribution over the entire

Outer

21695

19660

15619

9327

trac k.

L

1 O0

Turning

Radius

(ml

5

10

20

50

1 O0

Inner

-19068

5868

Outer

-39 15 6937

b = 0.01 m

N.m

-4447 -3377

Outer

20883

18833

15099

897 1

5646

6012 -3522

lnner

-15432

-16007

-12530

-6405

-3076

b = 0.45 m

( Reference)

N.m

lnner

-17504

-13711

-7549

20353

16861

10795

6406

Outer

20948

18883

15211

9144

5798

-17877

-14385

-8307

Outer

-17194

-13143

-6839

lnner

-15451

-16047

-12641

-6578

-3228

b= 0.90 m

N.m

Outer Inner

22163 21742

19740

15762

9509

-17273

Outer

21185

19053

15551

966 1

6218

b = 1.35 m

N.m

lnner

21879 -19500 -19110

19975

lnner

-15565

-16192

-12978

-7096

-3649

Outer

21691

19441

16166

10464

6803

-19232

Inner

-15931

-16547

-13591

-7900

-4234

-13286 16190

-7021 ' 10037

Page 158: CHARACTERISTICS OF TRACKED VEHICLES ON€¦ · Tracked vehicle steering has been a subject of interest to the off-road vehicle industry ... Tracked Vehicles for Trapezoidal Load Distribution

Table A- 15 Sprocket torques at various tuming radii with different track widths at a vehicle

Tuming

Radius

(m.)

speed of 2 1.3 km/h under continuous trapezoidal load distribution over the entire

trac k.

N.m 1 (Reference) 1 N.m N.m

inner Outer lnner Outer lnner Outer Inner

- 1 2694 17482 -12781 17835 -13057 18409 -13539

- 1 1342 14317 -11440 14638 -11748 15218 -12309

-577 1 858 1 -59 12 9029 -636 1 9817 -7151

-2572 5368 -2709 5810 -3 153 6435 -3778

Table A- 16 Sprocket torques at various twning radii with different track widths at a vehicle speed of 29 km/h under continuous trapezoidal load distribution over the entire track.

Turning b=O.OI m

1 Radius ( N-m

b = 0.45 m b= 0.90 m b = 1.35 m

(Reference) N.m N.m

N.m

Page 159: CHARACTERISTICS OF TRACKED VEHICLES ON€¦ · Tracked vehicle steering has been a subject of interest to the off-road vehicle industry ... Tracked Vehicles for Trapezoidal Load Distribution

A S . Height of Vebicle Center of Cravity, b

Table A- 17 Sprocket torques at various twning radii uith different CG heights at a vehicle speed of 7.5 hdh under continuous trapezoidal load distribution over the entire track.

Radius

(ml

5

10

20

Table A- 18 Sprocket torques at various turning radii with different CG heights at a vehicle speed of 14.2 km/h under continuous trapezoidal load distribution

N.m

over the entire track.

Outer

2 1778

19747

Tuming

Radius

(m)

( Reference)

N.m

Inner

-19147

- 1 7281

lnner

Outer

2 1742

19740

N.m

( Reference)

lnner

-191 10

- 1 7273

- 13286

-702 1

-3522

15767

9506

6010

Outer

21681

19728

15767

9513

6014

Outer inner

20525 -14935

1 867 1 - 1 5822

Inner

- 19047

-17261

-13291

-7025

-3523

-13281 15762

-70 1 8 9509

-3520 1 60 12

Page 160: CHARACTERISTICS OF TRACKED VEHICLES ON€¦ · Tracked vehicle steering has been a subject of interest to the off-road vehicle industry ... Tracked Vehicles for Trapezoidal Load Distribution

Table A- 19 Sprocket torques at various turning radii with different CG heights at a

vehicle speed of 2 1.3 krn/h under continuous trapezoidal load distribution

over the entire track.

Table A-20 Sprocket torques at various turning radii with different CG heights at a

vehicle speed of 29 kmdh under continuous trapezoidal load distribution

Turning

Radius

(m)

1 O

20

50

1 O0

over the entire track.

h = 0 1 . 8 r n

N.m

Turning

Radius

(m)

15

30

50

1 O0

Outer

17565

1430 1

85 10

5328

h = 1.3 m

(Reference)

N.m

lnner

-12915

- 1 1425

-5840

2669

Outer

17482

1-4317

858 1

5368

h = 1.8 rn

N.m

h = 0.8 m

N. rn

lnner

-12781

- 1 1440

-59 12

-3709

Outer

17199

14268

8649

5407

Outer

15239

1 1648

8563 L

h = 1.3 rn

(Reference)

N.m

lnner

-12376

- 1 1385 A

-5980

2748

lnner

-9946

-8565

-5725

Outer

15489

1 1765

8682

Inner

-10149

-8684

-5846

-2628 532 1 -2553 5396

Page 161: CHARACTERISTICS OF TRACKED VEHICLES ON€¦ · Tracked vehicle steering has been a subject of interest to the off-road vehicle industry ... Tracked Vehicles for Trapezoidal Load Distribution

Appendix B.

The Effects of Design and Opernting Factors on the Handhg of Tracked

Vehicles for Trnpezoidal Load Distribution on the Track Pitch under each Roadwheel

Page 162: CHARACTERISTICS OF TRACKED VEHICLES ON€¦ · Tracked vehicle steering has been a subject of interest to the off-road vehicle industry ... Tracked Vehicles for Trapezoidal Load Distribution

B. 1. Contact Lengtb, L

Table B- 1 Sprocket torques at various turning radii with different traçk-ground contact lengths

at a vthicle speed of 7.5 kmh for trapezoidal load distribution on the track pitch

under eac h roadwheel

Outer Inner Outer lnner Outer

Turning L = 1.90 rn L = 2.85 m 1 Radius 1 N..

L = 3.8 m L = 4.75 m L = 5.7 rn

(Reference) 1 N.m 1 N m

I

L able B-2 Sprocket torques at various turning fadi1 w t n ditterent track-grouna contact lengfns

i Inner

-22819

-21231

Inner

-19282

-17495

at a vehicle speed of 14.2 km/h for trapezoidal load distribution on the track pitch

Outer

2533 1

23624

-19194 24609 -22162

Outer

28045

26139

-14349 21643

Inner

-25376

-23740

under each roadwheel

Turning

Radius

(m)

5

10

20

50 1

L=1.90m

N.m

L = 3.8 m

(Reference)

N.m

L = 5.7 m

N.m

Outer

9144

5865

3724

2355

Outer

21206

19414

16583

10730

6936

Outer

28877

26105

24752

21094

15947

L = 2.85 m

N.m

L = 4.75 m

N.m

lnner

-4418

-2926

-1080

223.5

616.7

lnner

-15864

-16774

-14083

-8178

-4370

lnner

-20776

-23174

-22254

-18555

-13386

Outer

1561 3

12311

8196

4699

3258

Outer

25467

23441

21727

16433

11294 1 O0

lnner

-10946

-9617

-5619

-2128

-686.9

lnner

-18914

-20697

-19240

-13891

-8731 1956

Page 163: CHARACTERISTICS OF TRACKED VEHICLES ON€¦ · Tracked vehicle steering has been a subject of interest to the off-road vehicle industry ... Tracked Vehicles for Trapezoidal Load Distribution

Table B-3 Sprocket torques at various turning radii with different track-ground contact lengths at a vehicle speed of 2 1.3 kmih for trapezoidal load distribution on the track pitch

under each roadwheel

Table B-4 Sprocket torques at various turning radii with different track-ground contact lengths

Tuming

Radius

(m)

at a vehicle speed of 29 kmih for trapezoidal load distribution on the track pitch

L = 1.90 m

N.m

under each roadwheel

Outer

1 IO0 1 21321 535.1 1 31371 -476.31 65871 -39361 113441 -87021 16127 1-134901

lnner

-3153

-1794

-116.3

L = 2.85 m

N.m

1 O

20

50

Turning

Radius

(m)

Outer

12752

9060

4819

7873

4836

2825

L = 1.90 m

N.m

L = 2.85 m

N.m

L = 3 . 8 m

(Reference)

N.m

lnner

-8325

-6192

-2141

15

30

50

1 O0

L = 4.75 rn

Outer

18226

16016

10337

L = 5.7 m

Inner

-13602

-13280

-7700

L = 5.7 m

N.m

l

L = 3 . 8 m

(Reference)

N.m

-19068

-21526

-18661

22221

21296

16501

L = 4.75 m

N.m

N.m

Outer

8290

5017

3639

N.m

i

Inner

-2517

-1716

-712.0

Outer

12091

8224

5876

-16933

-18548

-13888

Outer

22951

Outer

16390

Inner

-6725

-5116

-3024

25289

24368

21267

2544

Inner

-16016

Inner

-10960

11436

Outer

20092 --

Inner Outer

245.2

13641

IO428

-8703 16365 -13643 d

Inner

-14071

lnner Outer

-10676

-7649

-3762

19227

16517

3735

-13784 21287 -185601

-962.2 6513

Page 164: CHARACTERISTICS OF TRACKED VEHICLES ON€¦ · Tracked vehicle steering has been a subject of interest to the off-road vehicle industry ... Tracked Vehicles for Trapezoidal Load Distribution

B.2. Tread of Vehicle, B

I able 15-3 >proclcet torques ot vanous tuming radil wim ditterent treaas a[ a vemcie speea or 7.5 hh for trapezoidal load distri bution on the track pitch under each roadwheel.

- -

B = 2.54 m

(Retèrence)

Inner

Tuming

Radius

(ml

Inner Outer Inner Outer Imer Outer 1 r

B = 3.04 m

N.m

B = 1.90 m

N.m

Table 8-6 Sprocket torques of various turning radii with different treads at a vehicle speed of

B = 3.80 m

N.m

11.2 kmh for trapezoidal load distribution on the mck pitch under each roadwheel.

Tuming l- B = 2.54 m

(Reference)

N-rn

Radius

(ml

Outer Inner Outer Inner

18985 -13799 16271 -11105

16909 -14269 13622 -10957 1

Outer lnner lnner

Page 165: CHARACTERISTICS OF TRACKED VEHICLES ON€¦ · Tracked vehicle steering has been a subject of interest to the off-road vehicle industry ... Tracked Vehicles for Trapezoidal Load Distribution

Table B-7 Sprocket torques of various turning radii with different treads at a vehide speed of

Table B-8 Sprocket torques o f various turning radii with different treads at a vehicle speed o f 29 km/h for trapezoidal load distribution on the track pitch under each roadwheel.

2 1.3 km/h for trapezoidal load distribution on the track pitch under each roadwheel.

Turning

Radius

(m)

10

20

50

1 O0

B = 3-04 m

N.m

B = 3.80 m

N.m

Outer

16189

13305

7726

4719

Outer

13683

10267

5499

3684 A

B = 1.90 m

N.m

lnner

-11660

-10540

-5070

-2063

B = 2.54 m

(Reference)

N-rn

Inner

-9107

-7451

-283 1

-1025

Outer

20678

19863

1.103 1

9082

B = 2 . 1 7 m

N.m

Outer

18226

16016

10337

6587

Inner

-15586

-17115

- 1 1306

-6436

Outer

19749

18215

12355

7937

Inner

-13602

-13280

-7700

-3936

Inner

-14919

-15482

-9726

-5290

Page 166: CHARACTERISTICS OF TRACKED VEHICLES ON€¦ · Tracked vehicle steering has been a subject of interest to the off-road vehicle industry ... Tracked Vehicles for Trapezoidal Load Distribution

B.3. Longitudinal Offset of Vehicle Center of Gravity, c,.

Table B-9 Sprocket torques of various tuming radii with different longitudinal CG offsets at a vehicle speed o f 7.5 km/h for trapezoidal load distribution on the track pitch

Table B- 10 Sprocket torques of various turning radii with different longitudinal CG offsets

at a vehicle speed of 14.2 km/h for trapezoidal load distribution on the track pitch under each roadwheel.

Radius

(m.)

3

10

20

50

1 O0

N.m

Outer

16340

15671

12925

7521

4553

Inner

-11241

-13036

-10425

-4967

-1986

N.m

Outer

19190

lnner

-13913

(Re ference)

N m

Outer

21206

19414

16583

10730

6936

N.m

Inner

-15864

-16774

-14083

-8178

-4370

Outer

22546

20644

17892

11895

7755

17804

14897

9166

57iO

ïnner

-17229

-17991

-15391

-9344

-5189

-15175

-12397

-6613

-3143

Page 167: CHARACTERISTICS OF TRACKED VEHICLES ON€¦ · Tracked vehicle steering has been a subject of interest to the off-road vehicle industry ... Tracked Vehicles for Trapezoidal Load Distribution

Table B- 1 1 Sprocket torques of various turning radii with different longitudinal CG offsets at a vehicle speed of 2 1.3 km/h for trapezoidal load distribution on the track pitch under each roadwheel.

1 Radius 1 N.m N.m 1 (Reference) 1 N.m 1 N.m 1

at a vehicle speed of 29 km/h for trapezoidal load distribution on the track pitch under each roadwhesl.

Turning

Radius

(m)

Outer

20831

18404

12545

8246

Outer

16047

14405

9076

5503

c,=Om c, = 0.20 rn + = 0.40 rn

( Re ference) N.m N.m

lnner

-16251

-15660

-991 1

-5597

f

lnner

-11446

-11663

-6438

-285 1

Outer

18226

16016

10337

6587

I

Outer 2

19801

17313

11538

7534

lnner

-13602

-13280

-7700

-3936

lnner

-15187

-14574

-8902

-4885

Page 168: CHARACTERISTICS OF TRACKED VEHICLES ON€¦ · Tracked vehicle steering has been a subject of interest to the off-road vehicle industry ... Tracked Vehicles for Trapezoidal Load Distribution

BA. Track Width, b

Table B- 1 3 Sptoçket torques o f various turning radii with different track widths at a vehicle speed o f 7.5 km/h for trapezoidal load distribution on the track pitch under each

Table 8-1 4 Sprocket torques of various tuming radii with different track widths at a vehicle speed of 14.2 km/h for trapezoidal load distribution on the track pitch under each

roadwheel.

roadwheel.

h = 0.45 m

( Reference)

N-m

I

Turning

Radius

(m.)

b=0.01 m

N.m

b = 0.90 m

N.m

5

10

20

50

100

b = 1.35 m

N.m

Outer

21818

20049

17069

1 1299

7366

Tuming

Radius

(m.)

5

10

20

50

100

b = 1.35 m.

N.m

Outer

21988

20296

17480

11860

781 1

Inner

-19389

-17648

-14614

-8815

-4877

Outer

21736

19620

17084

Inner

-19538

-17890

-15024

-9376

-532 1

Inner

-16101

-16938

-14571

b=O.OI m .

N.m

b = 0.45 m

(Reference)

N .m

Outer

21655

19838

16711

10792

6960

Outer

21142

19387

16517

10604

6826

Outer

21206

19414

16583

10730

6936

b = 0.90 m.

N.m

lnner

-19243

-17443

-14258

-8308

-4470

Outer

21698

19892

16803

10925

7067

Inner

-15840

-16753

-14018

-8053

-4260

lnner

-15864

-16774

-14083

-8178

-4370

Outer

21402

19492

16776

11094 11659

7715

Inner

-19282

- 1 7195

-14349

-8441

1577

Inner

-15941

-16834

-14271

-8543 -9108

-5 150 7247 -4681 1

Page 169: CHARACTERISTICS OF TRACKED VEHICLES ON€¦ · Tracked vehicle steering has been a subject of interest to the off-road vehicle industry ... Tracked Vehicles for Trapezoidal Load Distribution

Table B-15 Sprocket torques of variaus tuming radii with different track widths at a vehicle

speed of 2 1.3 km/h for trapezoidal load distribution on the track pitch under each

Table B- 1 6 Sprocket torques of various tuming radii with di fferent track widths at a vehicle speed of 29 km% for trapezoidal load distribution on the track pitch under each

roadwheel.

Turning

Radius

(m)

10

20

50

1 O0

b = 1.35 m

N.m

b = 0.01 m

N.m

b = 0.45 m

Outer

17149 -

14412

1 1 176

7129 L

Outer

16287

13541

10333

6439

b = 0.90 m

Inner

-11543

-11425

-8394

-4378

lnner

-10881 .

-1Q579

-7554

-3688

( Retèrence)

N.m

N.m

Outer

16390

13641

10428

6513

Outer

16687 .

13937

10713

673 5

Inner

-10960

-10676

-7649

-3762

Inner

-11189

-10963

-7932

-3984

Page 170: CHARACTERISTICS OF TRACKED VEHICLES ON€¦ · Tracked vehicle steering has been a subject of interest to the off-road vehicle industry ... Tracked Vehicles for Trapezoidal Load Distribution

B.1. Height of Vehicle Center of Gravity, h

Table B- 1 7 Sprocket torques of various tuming radii with different CG heights ai a

vehicle speed of 7.5 km/h for trapezoidal load distribution on the track

pitch under each roadwheel

Radius 1 1 (Reference)

N.m

h = 1.8m h = 1.3 m Turning

Table B-18 Sprocket torques of various tuming radii with different CG heights at a

h = 0.8 m

5

1 O

20 - -

50

1 O0

vehicle speed of 14.2 km/h for trapezoidal load distribution on the track

Outer

2 1702

19892

16798 - - - - -

1092 1

7065

pitch under each roadwheel

Inner

- 19287

- 1 7496

-14334

-843 8

-4576

Turning

Radius

(m)

5

10

20

50

1 O0

Outer

2 1698

19892

16803 . - - .

10925

7067

Outer

2 1 669

19887

16808 ~-

10928

7068

fnner

- 19282

-1 7495

- 14349 - - -

-844 I

-4577

lnner

- 19250

- 1 7490

- 14354

-8444

-4579 L

h = 0.8 m

N.m

Outer

2 1325

19571

16573

10712

6927

inner

-15991

-16941

- 14073

-8161

-436 1

h = 1.3 m

( Reference)

N.m

h = 1.8 m

N.m

Outer

S 1206

19414

16583

10730

6936

Outer

2088 1

19162

16578

10747

6944

Inner

-1 5864

- 16774

- 14083

-8 178

-4370

In ner

-15418

- 1 6506

-14077

-8 195

-4378

Page 171: CHARACTERISTICS OF TRACKED VEHICLES ON€¦ · Tracked vehicle steering has been a subject of interest to the off-road vehicle industry ... Tracked Vehicles for Trapezoidal Load Distribution

Table B- 19 Sprocket torques of various turning radii with different CG heights at a vehicle speed of 2 1.3 k m , for trapezoidal load distribution on the track pitch under each roadwheel

Turning

Radius

(m)

h = 0.8 m.

N-m

Table B-20 Sprocket torques of various turning radii with different CG heights at a

vehicle speed of 29 km/h for trapezoidai load distribution on the track pitch under each roadwheet

h = 1.3 rn

( Reference)

N.m

Outer

1846 1

16088

10266

6543

1 Radius I N.m

h = 1.8 m.

N.m

Outer

17727

15845

10399

6628

Inner

- 13922

-13358

-7628

-3 893

lnner

- 12923

- 13099

-7762

-3978

(m)

15

30

50

I

Outer

18226

16016

10337

6587

lnner

- 13602

- 13280

-7700

-3936 i

Outer

16414

13630

10325

h = 1.3 m

(Reference)

N.m

h = 1.8m

N.m

Inner

-1 1078

- 1 0669

-7544

Outer

16390

13641

1 0428

Outer

16000

13572

10505

I

lnner

- 1 0960

- i 0676

-7649

Inner

-103 14

- 10597

-7727

Page 172: CHARACTERISTICS OF TRACKED VEHICLES ON€¦ · Tracked vehicle steering has been a subject of interest to the off-road vehicle industry ... Tracked Vehicles for Trapezoidal Load Distribution

Appendix C.

Tbe Effects of Design and Operating Factors on the Handling of Tracked

Vebicles for Conceatrated Load under eacb Roadwheei

Page 173: CHARACTERISTICS OF TRACKED VEHICLES ON€¦ · Tracked vehicle steering has been a subject of interest to the off-road vehicle industry ... Tracked Vehicles for Trapezoidal Load Distribution

C.1. Contact Lengtb, L

Table C- 1 Sprocket torques at various turning radii with di fferent track-ground contact lengths at a vehicle speed of 7.5 kmh for concentrated load under each roadwheel.

Tuming

Radius

(ml

5

1 O

20

50

1 O0

L = 1.90 m L = 2.85 m L = 3.8 m L = 4.75 m L = 5.7 m

N.m N.m ( Reference) N.m N.rn

Outer Inner Outer Inner Outer lnner Outer Inner Outer lnner

-20276 23560 -22382

-19571 23928 -21608

-18261 23361 -20926

1 ame L-1 sprocicer torques ar vanous rurning raaii witn airrerenr uacu-grouna contacr iengms at a vehicle speed of 11.2 km/h for concentrated load under each roadwheel.

Tuming

Radius

(m.)

N.m 1 N.rn 1 (Reference) 1 N.m

Outer Inner Outer

6251

3930

2470

2018

Inner

-3398

-1311

105.8 r

-553.9

Outer

12139

8112

4734

3364

Inner

-9504

-5532

-2164

-793.3

N.m

Outer

18732

16250

10688

6960

Outer

22182

20993

16169

11215

Inner

-16189

-13769

-8139

-4394

Inner

-19601

-18532

-13629

-8652

Page 174: CHARACTERISTICS OF TRACKED VEHICLES ON€¦ · Tracked vehicle steering has been a subject of interest to the off-road vehicle industry ... Tracked Vehicles for Trapezoidal Load Distribution

Table C-3 Sprocket torques at various tuming radii with different track-ground contact lengths

at a vehicle speed of 2 1.3 km/h for concentrated load under each roadwheel.

Tuming

Radius

1 ame L-u 3procKer torques at vanous tuming raaii wtn airrerenr uaclt-grouna conracr ienguis

(ml

1 O

20

50

IO0 1

at a vehicle speed of 29 k r n h for concentrated load under each roadwheel.

L=1.90m

N.m

Radius 1 N.m 1 N.m 1 (Reference) 1 N-m 1 N-III

L = 2.85 rn

N.m

Outer

9131

5514

3 130

2277 1

L = 3 . 8 m

(Reference)

Inner

-4542

-2517

-437.4

388.6 1

15

30

50

100

Outer Inner

N.m

Outer

18523

16100

IO496

6752 1

Outer

10129

5900

4174

2827

lnner

-14012

-13405

-7864

4104

13696

9573

5052

Inner

-4349

-2635

-1262

-40.19

-9412

-6755

-2382

Outer

13665

9064

6423

4028

3290 1 -630.8 1

Inner

-8311

-6002

-3589

-1260

Outer

17525

14219 -

10877

6844

lnner

-12014

-1 1289

-8111

-4097

Outer

20585

I9344

16628

Il567

Outer

22905

22590

21119

16333

lnner

-14331

-16436

-13906

-8838

Inner

-1555%

-19610

-18405

-13613

Page 175: CHARACTERISTICS OF TRACKED VEHICLES ON€¦ · Tracked vehicle steering has been a subject of interest to the off-road vehicle industry ... Tracked Vehicles for Trapezoidal Load Distribution

C.2. Tread of Vehicle, B

Table C-5 Sprocket torques at various tuming radii with different treads at a vehicle speed of

7.5 krnh for concentrated load under each roadwheel.

i ame L-O 3procket torques at vanous turning raciii wxtn dilterent treaas at a venicie speed ot

14.2 k m , for concentrated load under each roadwheel.

Turning

Radius

5

I O

20

50

1 O0

Radius1 N.m 1 (Reference)

B = 3.80 m.

N.m

B = 2.54 m.

(Reference)

B = 1.90 m.

N-m

B = 3.04 m.

N.m

B = 2.17 m.

N.m

Outer

22930

22147

19771

13591

8871

Outer Inner Outer Inner Outer lnner Outer Inner Outer Inner

Inner

-20703

-19783

-17323

-1 II08

-6381

5

10

Outer

21494

20631

18165

12245

7967

Inner

-19280

-18268

-15717

-9762

-5478

22751

21994

Outer

19638

18735

16255

10782

7029

Inner

-17426

-16371

-13806

-8299

4540

-16940

20

50

1 O0

Outer

17370

16500

14124

9290

61 I l

20042

13792

8992

Inner

-15135

-14130

-11673

-6807

-3622

Outer

14402

13764

11652

7710

5177

21919

-17567

- 1 1246

-6427

lnner

-12084

-11378

-9198

-5227

-2688

- 1 5879

-9783

-5444

18354

12330

8009

-16686

-19404 -18077

16250

10688

6960

20626

20328

18732

-13769

-8139

-4394

-15448

-16189

13687

8873

5857

18216

16275

-1 1186

-6320

-13557

-13718

9369

5490

-6803

-2920

-3291 3951

15510

-1381

-10949

12846 -10240

Page 176: CHARACTERISTICS OF TRACKED VEHICLES ON€¦ · Tracked vehicle steering has been a subject of interest to the off-road vehicle industry ... Tracked Vehicles for Trapezoidal Load Distribution

Table C-7 Sprocket torques at various tuming radii with difièrent treads at a vehicle speed of

Turning

Radius

(m)

I 1 O

20

50

1 O0

2 1.3 km/h for concentrated load under each roadwheel.

Outer tnner Outer Inner

Table C-8 Sprocket torques at various tuming radii with different treads at a vehicle speed of

29 km/h for concentrated load under each roadwheel.

Radius

1 O0

(Reference)

Outer Inner Outer inner

15478 -10345 13040 -7866

11873 -8914 9458 -6445

Page 177: CHARACTERISTICS OF TRACKED VEHICLES ON€¦ · Tracked vehicle steering has been a subject of interest to the off-road vehicle industry ... Tracked Vehicles for Trapezoidal Load Distribution

C.3. Longitudinal Offset Vehicle Center of Cravity, c,

Table C-9 Sprocket torques at various turning radii with dinèrent longitudinal CG offsets at

a vehicle speed of 7.5 km/h for concentrated load under each roadwheel.

Table C- 1 0 Sprocket torques at various twning radi i with di fferent longitudinal CG offsets at

a vehicle speed of 14.2 iunh for concentrated load under each roadwheel.

1 Radius 1 N.m 1 (Reference) 1 N.m [ N-m I

5

10

20

50

1 O0

Outer

15953

15100

12449

7199

4449

Inner

- 1 1341

-12610

-9974

-4646

-1882

Outer

18412

17094

14482

9098

5780

lnner

-13637

-14585

-12002

-6547

-3214

Outer

20328

18732

16250

10688

6960

Inner

-15448

-16189

-13769

-8139

-4394

Outer

21850

20103

17642

11867

7767

Outer

23035

21204

Inner

-16891

-17526

-15156

-9318

-5202

fnner

-17964

-18599

18700 -16205 k

12743

8345

-10192

5779

Page 178: CHARACTERISTICS OF TRACKED VEHICLES ON€¦ · Tracked vehicle steering has been a subject of interest to the off-road vehicle industry ... Tracked Vehicles for Trapezoidal Load Distribution

Table C-11 Sprocket torques at various turning radii with different longitudinal CG offsets at

Table C- 1 2 Sprocket torques at vanous turning radii with different longitudinal CG offsets at

a vehicle speed o f 21 -3 km/h for concentrated load under each roadwheel.

a vehicle speed of 29 km/h for concentrated load under each roadwheel.

Turning

Radius

(ml

Turning

Radius

(ml

15

30

50

1 O0

c, = -0.40 rn

N.m

Outer

Outer

19170

15260

1 1674

c , = -0.40 m

N.m

Inner

-9516

-9810

-5259

-2202

c, = -0.20 m

N.m

1 O

20

50

1 O0

Outer

13248

1 1069

8712

5595

Outer

16484

14487

9163

5612

13901

12513

7891

4852

c, = -0.20 m

N.m

Inner

-7529

-8094

-5938

-2847

lnner

c,=Om

( Re ference)

N.m

Outer

15566

12843

9923

6247

hner

-15125

-13112

-9640

Inner

c,=Om

(Reference)

N.m

Outer

20138

17471

11706

7656

Inner

-9980

-9899

-7157

-3500

Outer Outer

17525

14219

10877

6844

Inner

-14012

-13405

-7864

4104

-15597

-14761

-9075

-5009

c, = 0.20 m

N.m

21391

18658

12718

8354

-12029

-11797

-653I

-2962

Inner

-12014

-1 1289

-81 t 1

-4097

Outer

-16834 I

-15932

-10085

5706

c,, = 0.40 m

N.m

18523

16100

10496

6752

-13714

-12329

-8905

tnner Outer

20525

16056

12415

lnner

Page 179: CHARACTERISTICS OF TRACKED VEHICLES ON€¦ · Tracked vehicle steering has been a subject of interest to the off-road vehicle industry ... Tracked Vehicles for Trapezoidal Load Distribution

CA. Track Width, b

Table C- 1 3 Sprocket torques at variuus tuming radii with different track widths at a

vehicle speed 7.5 of km/h for concentrated load under each roadwheel.

1 Radius N.m 1 (Reference) I N.m l N.m

Table C-14 Sprocket torques at various tuming radii with different track widths at a vehicie speed of 14.2 km/h for concentrated load under each roadwheel.

5

1 O

20

50

1 O0

Tuming b = 0.01 m

Radius N.m

Ou ter lnner

Outer

19602

18678

16158

10646

6920

lnner

-17396

-16316

-13710

-8163

-443 1

Outer Inner Outer

f9743

18900

16531

11161

733 1

lnner

-17513

-16533

-14082

-8679

-4842

Outer

19919

19161

16950

1 1722

7775

19638

Inner

-17661

-16790

-14501

-9239

-5286

-17426

18735 -16371

16255

10782

7029

-13806

-8299

-4540

Page 180: CHARACTERISTICS OF TRACKED VEHICLES ON€¦ · Tracked vehicle steering has been a subject of interest to the off-road vehicle industry ... Tracked Vehicles for Trapezoidal Load Distribution

Table C-15 Sprocket torques at various turning mdii with different track widths at a

vehicle speed of 2 1.3 km/h for concentrated load under each roadwheel.

Table C- I 6 Sprocket torques at various turning radii with different track widths at a

vehicle speed of 29 kmh for concentrated load under each roadwheel.

Tuming

Radius

(ml

10

20

50

1 O0

Tuming

Radius

(m)

1 O

20

50

1 O0

b = 0.45 m

( Reference)

N.m

b=0.01 m

N.m

b = 0.90 m

N.m

Outer

18523

16100

10496

6752

Outer

18455

16052

10397

6648

b = 0.01 m

N-m

b = 1.35 m

N.m

Inner

-14012

-13405

-7864

4104

Inner

-13974

-13361

7766

-3999

Outer

17438

14130

1 0787

677 1

Inner

-11955

-11203

-8022

-4024

Outer

18725

16243

10791

7064

b = 0.35 m

(Reference)

N.m

b = 1.35 m

N.m

Outer

17525

14219

10877

6844

b = 0.90 m

N.m

Outer

18163

14904

11571

744 1

lnner

-12014

-11289

-8111

-3097

Outer

17775

14482

Il142

7064

,

lnner

-12447

-11952

-8802

-4693

Inner

-14326

-13763

-8652

4908

Outer

Inner

-12185

-11543

-8376

-4317

-14128

-13535

-8159

-44 1 6

19052

16486

11285

7555

Page 181: CHARACTERISTICS OF TRACKED VEHICLES ON€¦ · Tracked vehicle steering has been a subject of interest to the off-road vehicle industry ... Tracked Vehicles for Trapezoidal Load Distribution

CS. Height of Vehicie Center of Cravity, h

Table C- 17 Sprocket torques at various tuming radii with different CG heights at a

vehicle speed of 7.5 kmh for concentrated load under each roadwheel.

1 Radius I N.m 1 (Reference) I N.m

Turning h = 0.8 m

(ml

5

10

20

50

1 O0

h = 1.3 rn h = 1.8m

Outer Inner

N.m

Outer Outer Inner

- 1 7426

-16371

- 13806

-8299

4540

19654

18735

16250

10779

7028

Inner

19599

1873 1

16259

10784

703 1

-17380

- 16367

-13810

-8301

-454 1

- 1 7147

-16371

- 13802

-8296

-4539

1

1 9638

18735

I6255

IO782

7029

Page 182: CHARACTERISTICS OF TRACKED VEHICLES ON€¦ · Tracked vehicle steering has been a subject of interest to the off-road vehicle industry ... Tracked Vehicles for Trapezoidal Load Distribution

Table C- 19 Sprocket torques at various tuming radii with different CG heights at a vehicle speed of 2 1 -3 km/h for concentrated load under each roadwheel.

1 Tuming 1 h =O.8 rn 1 h = l . 3 m 1 h = 1.8m

Table C-20 Sprocket torques at various tuming radii with different CG heights at a vehicle speed of 29 km/h for concentrated load under each roadwheel.

Radius

(ml

10

20

50

1 O0

Tuming

Radius

(ml

15

30

50

1 O0

N.m

Outer

18713

16161

10432

6716

( Reference)

N.m

lnner

-14327

-13471

-7800

-4067

h = 0.8 m

N.m

Outer

18523

16100

10396

6752

N.m

h = 1.3 rn I h = 1.8m

( Reference) N.m

N.m

Outer

17443

14188

10769

6753

lnner

-13012

- 13405

-786.)

-4104

Outer

17953

1 593 5

10552

6787

Outer

17525

14219

10877

6844

Inner

-121 1 i

-1 1262

-8002

-4005

lnner

-13172

- 13228

-792 1

-4 139

lnner

-12014

-1 1289

-81 1 1

-4097

Outer

1701 7

Inner I

-1 1005

14162

10955

6929

-1 1220 1

-8191 1

-4182