Nodal Control of Vibrating Structures: Beam

A Thesis

Submitted to the Graduate Faculty of the Louisiana State University and

Agricultural and Mechanical College in partial fulfillment of the

requirements for the degree of Master of Science in Mechanical Engineering

in The Department of Mechanical Engineering

by

Akshay Nareshraj Singh

B.E. in Mechanical Engineering Maharaja Sayajirao University, 1999

December 2001

To my parents Alka and Nareshraj Singh and my younger brother Abhishek

ii

Cataloging Abstract

Vibration control is an important engineering problem and many methods for both active and passive vibration absorption have been developed. This thesis deals with developing a method to achieve nodal control at the point of excitation in a Bernoulli-Euler beam. It is established that, for a uniform Bernoulli-Euler beam, the steady state motion at the point of excitation can be absorbed by means of a control force determined from displacement information at the point of application. A closed form solution for the control gain is presented and a criterion for implementing the control by active and passive means is developed. The result for the control gain is generalized for the case of a non-uniform beam. Chapter 4 shows through some examples that the theory can be also applied to eliminate the steady state motion at any desired location other than the point of excitation. Analysis is also performed to determine the optimal control force and investigate the stability of the overall system. Several controllability graphs are shown and meaningful conclusions are drawn from these graphs. An experiment is designed to validate the proposed theory and display its practicality. A uniform steel beam supported at two locations is tested. Modal testing is performed to extract natural frequencies in order to characterize the system and assist in formulation of an appropriate mathematical model. The steel beam is then excited by a known harmonic force supplied by a vibration exciter and a spring, with suitable spring constant obtained by performing the control gain calculations on the model, is used to absorb the motion at the free end. It is confirmed, that the theory developed in this thesis produces accurate results, and that it can serve as a vital tool in developing practical solutions to structural control problems. Akshay Nareshraj Singh, B.E., Maharaja Sayajirao University, 1999 Master of Science, Fall Commencement, 2001 Major: Mechanical Engineering Nodal Control of Vibrating Structures: Beam Thesis directed by Associate Professor Yitshak Ram Pages in thesis, 105. Words in abstract, 297

iii

Acknowledgements

The guidance and support provided by my major professor Dr. Yitshak Ram is

acknowledged. Thanks go to Dr. Michael Khonsari and Dr. Su-Seng Pang for serving

on my graduate committee and evaluating my thesis.

The intellectual input of my colleagues Kumar Vikram Singh, Jaeho Shim, Sumit

Singhal and Madhulika Sathe, as well as the assistance of Mr. Ed Martin in constructing

the experiment is also acknowledged.

Last but not the least, my gratitude goes to my aunty Mrs. Anita Singh and uncle Dr.

Vijay Singh who have always been there for me, and to my parents Alka and Nareshraj

Singh who have made me what I am today.

The research was supported in part by a National Science Foundation research grant

CMS-9978786.

iv

Table of Contents

Dedication..ii Cataloging Abstract.iii Acknowledgements ........................................................................................................ iv Table of Contents............................................................................................................ v List of Tables................................................................................................................. vii List of Figures .............................................................................................................. viii List of symbols ................................................................................................................ x Abstract ......................................................................................................................... xii Chapter 1: Introduction................................................................................................. 1 Chapter 2: Literature Survey and Background .......................................................... 6

2.1 Introduction ...................................................................................................... 6 2.2 Passive vibration control .................................................................................. 6 2.3 Active vibration control.................................................................................... 9

Chapter 3: Beam: Theory and Background .............................................................. 11 3.1 Introduction .................................................................................................... 11 3.2 Equation of motion for a non-uniform beam.................................................. 12 3.3 Natural frequencies and modeshapes ............................................................. 15 3.4 Steady state response and natural frequencies for a uniform clamped cantilever beam............................................................................................... 17 3.5 Summary......................................................................................................... 23

Chapter 4: The Control Gain ...................................................................................... 24 4.1 Introduction .................................................................................................... 24 4.2 Dynamic absorption in a uniform beam and a formula for the control gain ............................................................................................... 24 4.3 Analysis of results .......................................................................................... 29 4.4 Some illustrations ........................................................................................... 36 4.5 Summary......................................................................................................... 44

Chapter 5: Stability and Optimality ........................................................................... 45 5.1 Introduction .................................................................................................... 45 5.2 Stability analysis............................................................................................. 45 5.3 Optimality....................................................................................................... 48 5.4 Summary......................................................................................................... 50

Chapter 6: Experimental Verification........................................................................ 52 6.1 Introduction .................................................................................................... 52 6.2 Proposed model for the experiment................................................................ 52

v

6.3 Determination of natural frequencies ............................................................. 53 6.4 Determination of the control gain................................................................... 64 6.5 Control system design .................................................................................... 66 6.6 Procedure........................................................................................................ 70 6.7 Experimental result......................................................................................... 70 6.8 Validation of the proposed theory .................................................................. 70 6.9 Summary......................................................................................................... 72

Chapter 7: Conclusions and Recommendations........................................................ 73 7.1 Conclusions .................................................................................................... 73 7.2 Recommendations for future research............................................................ 75

References ..................................................................................................................... 77 Appendix A.................................................................................................................... 79

Matlab Programs ........................................................................................................ 79

Appendix B.................................................................................................................. 100 Eigenvalues............................................................................................................... 100

Vita............................................................................................................................... 105

vi

List of Tables

TABLE 4.1: Comparison of control gain formulae........................................................ 35

TABLE 4.2: Comparison of control gain formula for static case .................................. 35

TABLE 6.1: Comparison of natural frequencies............................................................ 63

TABLE 1 (Appendix B): Eigenvalues ..................................................................... 100 TABLE 2 (Appendix B): Eigenvalues for a=0.25 .................................................. 101 TABLE 3 (Appendix B): Eigenvalues for a=0.5..................................................... 102 TABLE 4 (Appendix B): Eigenvalues for a=

21 ................................................... 103

TABLE 5 (Appendix B): Eigenvalues for a=2/3..................................................... 104

vii

List of Figures

FIGURE 1.1: A schematic for active control ................................................................... 2

FIGURE 1.2: Vibration control of a harmonically excited beam .................................... 4

FIGURE 2.1: Single-degree-of-freedom dynamic absorber ............................................ 7

FIGURE 3.1: Equation of motion for a beam ................................................................ 13

FIGURE 3.2: Uniform cantilever beam subject to harmonic excitation ........................ 17

FIGURE 3.3: Steady state amplitude for a uniform cantilever beam............................. 22

FIGURE 4.1: Vibration control of a harmonically excited beam .................................. 25

FIGURE 4.2: Plot of control gain against ............................................................. 29 FIGURE 4.3: Plot of control gain against - (superimposed) ................................. 30 FIGURE 4.4: Clamped and clamped-double-hinged uniform beam.............................. 31

FIGURE 4.5: Static deflection of a clamped-hinged beam............................................ 32

FIGURE 4.6: Illustration demonstrating control gain calculations................................ 37

FIGURE 4.7: Controlled uniform beam......................................................................... 39

FIGURE 4.8: Uncontrolled uniform beam..................................................................... 41

FIGURE 4.9: Controlled uniform beam......................................................................... 42

FIGURE 4.10: Implementation of nodal control............................................................ 43

FIGURE 5.1: Passively controlled uniform beam.......................................................... 46

FIGURE 5.2: Stability analysis and equivalent stiffness ............................................... 47

FIGURE 5.3: Plots of control gain, control force and inverse of static deflection against the beam span x for cases with excitation frequency of (a) 10= , (b) 20= , (c) 30= and (d) 80= ................................ 49 FIGURE 6.1: Mathematical model of the test beam used in the experiment................. 53

FIGURE 6.2: Dimensions of the test beam used in the experiment............................... 57

viii

FIGURE 6.3: Clamping details ...................................................................................... 58

FIGURE 6.4: Impact Hammer ....................................................................................... 59

FIGURE 6.5: Accelerometer .......................................................................................... 59

FIGURE 6.6: Modal Analysis ........................................................................................ 61

FIGURE 6.7: VirtualBench DSA display ...................................................................... 63

FIGURE 6.8: Controlled test beam ................................................................................ 64

FIGURE 6.9: Control gain and control force variation along the beam span ................ 65

FIGURE 6.10: Test beam modeshape before and after control ..................................... 66

FIGURE 6.11: Spring housing configuration................................................................. 67

FIGURE 6.12: Attaching the vibration exciter to the beam........................................... 67

FIGURE 6.13: Schematic of the experimental setup ..................................................... 68

FIGURE 6.14: Attaching the springs and the shaker ..................................................... 69

FIGURE 6.15: Experimental setup................................................................................. 69

FIGURE 6.16: Active Vibration Control ....................................................................... 71

ix

List of symbols

)(, xAA area of cross-section

iiii DCBA and ,,, constant coefficients for each i

A square matrix

b force vector

control gain static deflection

)(, xEE the Youngs modulus of elasticity

pe pth unit vector

)(tf harmonic force

non-dimensional parameter )(, xII moment of inertia

pk spring stiffnes of the primary system

sk spring stiffnes of the secondary system

L length of the beam

eigenvalues of clamped-hinged beam pm mass of the primary system

sm mass of the secondary system

),( txM bending moment

eigenvalues of clamped-double-hinged beam )(, x density

x

BS bending stiffness

)(tu control force

),( txV shear force

)(xv shape function

),( txw deflection

frequency of excitation n natural frequencies

z displacement vector

xi

Abstract

Vibration control is an important engineering problem and many methods for both

active and passive vibration absorption have been developed. This thesis deals with

developing a method to achieve nodal control at the point of excitation in a Bernoulli-

Euler beam. It is established that, for a uniform Bernoulli-Euler beam, the steady state

motion at the point of excitation can be absorbed by means of a control force

determined from displacement information at the point of application. A closed form

solution for the control gain is presented and a criterion for implementing the control by

active and passive means is developed. The result for the control gain is generalized for

the case of a non-uniform beam. Chapter 4 shows through some examples that the

theory can be also applied to eliminate the steady state motion at any desired location

other than the point of excitation. Analysis is also performed to determine the optimal

control force and investigate the stability of the overall system. Several controllability

graphs are shown and meaningful conclusions are drawn from these graphs.

An experiment is designed to validate the proposed theory and display its practicality.

A uniform steel beam supported at two locations is tested. Modal testing is performed

to extract natural frequencies in order to characterize the system and assist in

formulation of an appropriate mathematical model. The steel beam is then excited by a

known harmonic force supplied by a vibration exciter and a spring, with suitable spring

constant obtained by performing the control gain calculations on the model, is used to

absorb the motion at the free end. It is confirmed, that the theory developed in this

xii

thesis produces accurate results, and that it can serve as a vital tool in developing

practical solutions to structural control problems.

xiii

Chapter 1: Introduction

Most mechanical components are subject to vibrations, which, depending on

circumstances may be desirable or undesirable. On one hand, vibrations of guitar

strings produce wonderful music and on the other hand, vibrations in an automobile

may cause excessive discomfort and fatigue to the driver.

This thesis is concerned with the case of undesirable vibrations and in particular in

developing methods for elimination of steady state response. Components are designed

to withstand definite levels of vibrations. Design, in vibrations, is used to denote an

educated method of choosing and adjusting the physical parameters of a vibrating

system in order to obtain a more favorable response [1]. Modifications of physical

parameters namely mass, damping, and stiffness, in order to improve the vibrational

response of the system fall in the category of passive control. The most common

passive control device is a vibration absorber, which manifests in the form of layers of

damping material added to vibrating structures. Passive control may also involve

changing values of mass and stiffness and hence is also referred as redesign.

Chapter 2 describes a passive control device called single-degree-of-freedom dynamic

absorber. A single-degree-of-freedom dynamic absorber is made up of single mass and

spring and may or may not have a damper. Essentially, the dynamic vibration absorber

introduces additional degree of freedom in the original dynamic system, which results

in a different steady state response. The values of mass, stiffness and damping can be

modified to tune the response of the resulting system to desired levels.

1

However, altering the physical parameters of the system may not always yield a desired

response. In these situations, one has to try implementing active control. Active

control uses external active device, called an actuator, which assists in shaping the

system response. The actuator (e.g. a piezoelectric device, a hydraulic piston, or rack

and pinion) is capable of applying control force to the system under consideration. The

control force is determined based on a mathematical rule, which operates on the system

response measured in realtime by a sensor. The mathematical rule used to apply the

force from the sensor measurement is called the control law.

Dynamic System

Actuator

Sensor

ProcessorDynamic System

Actuator

Sensor

Processor

FIGURE 1.1: A schematic for active control

The system comprising both, the actuator and the sensor together with the electronic

circuitry that reads the sensor output and calculates corresponding input to the actuator

is called the control system [5]. Figure 1.1 shows a schematic for implementing active

control.

2

Much work has been done on tuning vibrational response of multi degree of freedom

systems by applying results obtained from the study of dynamic absorbers. A single-

degree-of-freedom dynamic absorber is attached to continuous system like a beam to

control single a mode of vibration under the influence of harmonic excitation. Aida,

Toda, Ogawa and Imada in [11] and Kawazoe, Kono, Aida, Aso and Ebisuda in [12]

discuss a beam type vibration absorber capable of suppressing several vibration modes

of beams.

The subject investigated in this thesis is elimination of steady state vibration at a desired

location in beams by means of nodal control. The developed theory will provide a

strong foundation for realizing realistic and convenient methodologies in control

applications in cases like surgical procedures, drilling and turning operations etc.

However, one of the many direct applications of this method is structural vibration

control in an aircraft wing. Several measurements such as vibrational response, air

temperature, wind velocity etc are required in order to monitor flight conditions in an

aircraft. These data also assist the pilot in flying the aircraft. Sensors and data

collection circuitry form an entire network of the electrical wiring all in and around the

airplane body. Data acquisition devices are also located on the wing of the airplane.

Shielding of these devices from undesirable vibration of the wing is critical in order to

avoid noise in the gathered data and prevent damage to electrical wiring. Exclusion of

steady state vibration at the locations of these devices provides the motivation for this

investigation.

3

Consider a non-uniform Bernoulli-Euler beam of length . Suppose that the beam is

excited by a harmonic force

L

( ) ttf cos= , as shown in Figure 1.2(a).

x

( ) ttf cos=

( )txw ,

L

(a) Uncontrolled beam

x

( ) ttf cos=

( )txw ,

L

(b) Controlled beam

( ) ( )tawtu ,=a

x

( ) ttf cos=

( )txw ,

L

(a) Uncontrolled beam

x

( ) ttf cos=

( )txw ,

L

(b) Controlled beam

( ) ( )tawtu ,=a

FIGURE 1.2: Vibration control of a harmonically excited beam

The steady state motion of a prescribed point of the beam may be vanished by applying

a concentrated control force u at ),( ta ax = as shown in Figure 1.2(b).

The work here focuses on determining a closed form solution for the control gain that absorbs the motion of the beam at Lx = . Chapter 4 describes the method in detail and also provides criterion to determine the type of the control i.e. active nodal control

or passive nodal control. It also provides an illustration to show that motion can be

4

absorbed not only at the point of excitation but also at any other point along the beam

span. Chapter 5 discusses the criterion for stability of the controlled system and

optimality in relation to the control force. In order to validate the findings, an

experiment is designed exhibiting nodal control by means of a passive element, a

spring, in a steel beam under the influence of harmonic excitation. It is thus shown that

the approach presented in this thesis is highly practical.

5

Chapter 2: Literature Survey and Background

2.1 Introduction

Some fundamental knowledge and some key results found in the literature survey

related to vibration control are presented here. The survey covers topics related to the

design of both active and passive vibration absorbers.

2.2 Passive vibration control

In 1911 Frahm invented a device for stabilization of rocking oscillations of ships [4].

This device is now known as a dynamic absorber. The dynamic absorber is extremely

simple in principle and has large practical applications. For example Lee, Nian, and

Tarng in [6], Al-Bedoor, Moustafa, and Al-Hussain in [9], Yamashita, Seto, and Hara in

[10] describe design of a dynamic vibration absorber for vibration control in turning

operations, synchronous motor-driven compressors, and piping systems, respectively.

Theory of single-degree-of-freedom dynamic absorber

The dynamic vibration absorber is an additional mass-spring system, which is

appropriately chosen to neutralize the steady state force acting on a particular degree of

freedom. Consider the single-degree-of-freedom system shown in Figure 2.1(a), under

the harmonic excitation of ( ) tFtf sin0= . Let this system be called the primary system. Upon a harmonic excitation, the system vibrates with two frequencies, the

frequency of excitation , and the natural frequency of the system,

6

,ppn mk= (2.2.1) where subscript p denotes the parameters associated with the primary system. The

objective is to eliminate the forced component of vibrations. This is implemented by

attaching additional single-degree-of-freedom mass-spring system, which is called the

secondary system, with the mass and the spring stiffness , to the primary system.

The global system is shown in Figure 2.1(b).

sm sk

(a) Primary system

(b) Global system

( )tx p

( ) tFtf sin0=

pmpk

( )txs

sksm

( )tx p

( ) tFtf sin0=

pmpk

(a) Primary system

(b) Global system

( )tx p

( ) tFtf sin0=

pmpk

( )tx p

( ) tFtf sin0=

pmpk

( )txs

sksm

( )tx p

( ) tFtf sin0=

pmpk

FIGURE 2.1: Single-degree-of-freedom-dynamic absorber

7

The mathematical model of the global system has the form

,sin00

0 0 tF

xx

kkkkk

xx

mm

s

p

ss

ssp

s

p

s

p

=

++

&&&&

(2.2.2)

which can also be written as

( )

=+=++

.0sin0

sspsss

sspsppp

xkxkxmtFxkxkkxm

&&&&

(2.2.3)

Both masses vibrate with forced harmonic vibrations of the form

( )( )

==

,sinsin

tXtxtXtx

ss

pp (2.2.4)

where the constants , are the amplitudes of the forced component of vibration.

Substituting (2.2.4) in (2.2.3) gives

sp XX ,

( )

=+=++

.020

2

sspsss

sspsppp

XkXkXmFXkXkkXm

(2.2.5)

Since is to be eliminated, substituting pX 0=pX in the above set of equations yields

=+=

.020

ssss

ss

XkXmFXk

(2.2.6)

The second equation in (2.2.6), gives

,2s

s

mk= (2.2.7)

and the first equation in (2.2.6) provides the amplitude of vibration of , sm

.0s

s kF

X = (2.2.8)

Hence, it is concluded that the vibratory motion of the primary mass can be eliminated

provided that the stiffness and the mass values for the secondary mass are chosen such

8

that they satisfy (2.2.7). In this case the secondary mass vibrates out of phase to the

external harmonic excitation and the spring force exactly contradicts the harmonic

force, causing the forced motion of to vanish. pm

This idea can also be extended to a multi-degree-of-freedom system. Ram and Elhay in

[8], show that a multi-degree-of-freedom dynamic absorber may absorb the steady state

motion associated with the harmonic excitation of several frequencies. There are,

however, some limitations associated with practical implementation of the dynamic

absorber. Firstly, it is not always feasible to attach the absorber to the specific degree of

freedom of which the motion is to be absorbed. Secondly, application of dynamic

absorber increases the dimension of the system, and hence introduces new natural

frequencies that may interfere with other excitations. Thirdly, the theory of dynamic

absorbers for damped systems is not fully developed. It is, therefore, not clear how the

dynamic absorption phenomenon may be used in eliminating the steady state motion of

a damped system that is excited by a harmonic force [7]. Herzog in [16] investigates

the topic of performance degradations of dynamic systems implementing passivity-

based control. He has analyzed the topic in terms of flatness of response of the

controlled system in the vicinity of the natural frequency of the dynamic absorber.

2.3 Active vibration control

As described earlier in Chapter 1, in some cases implementing the active vibration

absorber is imperative. Nishimura, Yoshida and Shimogo in [17] have studied optimal

design method of the active dynamic vibration absorber for multi-degree-of-freedom

9

systems. The method was also validated by performing numerical simulations and an

experiment on a 3-degree-of-freedom building like structure. Aida, Toda, Ogawa and

Imada in [11] and Kawazoe, Kono, Aida, Aso and Ebisuda in [12] demonstrate control

of several mode shapes of a beam by using a beam type vibration absorber with

boundary conditions same as the main beam. It is shown that for specific vibration

modes, mode equations of a beam with beam-type dynamic absorber are approximate

equivalents of the motion of two-degree-of-freedom system. Hence, the dynamic

absorber system can be tuned by Den Hartog method.

Gaudreault, Liebst, Bagley in [18] present four techniques for combining active

vibrational control and passive viscous damping. The motivation behind the work is

some findings, which reveal that the passive damping can reduce the amount of active

damping needed to control structural vibrations. However, inappropriate design of the

passive damping can produce contrary results in that it may increase system reaction

times, reducing control effectiveness.

The partial-pole assignment problem is addressed by Ram in [15]. The paper

determines the force required to place a few poles of the spectrum while leaving the rest

unchanged and the conditions under which the solution is unique. The work of Ram in

[3] lays the foundation for this thesis. The paper provides a closed form control gain

solution for absorbing the harmonic response at a desired location in an axially

vibrating rod and analyzes the stability of the controlled system.

10

Chapter 3: Beam: Theory and Background

3.1 Introduction

In order to determine the dynamic behavior of mechanical systems, one needs to

develop an appropriate mathematical model. Mechanical systems can be modeled as

lumped-parameter systems, where it is assumed that the motion of the system is

governed by the mass of the system concentrated at a particular point. The modeled

system is known as a lumped parameter system, which has finite number of lumped

masses connected to each other by means of springs and dampers. Even though a

discrete model provides an acceptable solution to the system, it is not capable of

accounting for the flexibility of various structures. Engineering problems such as

swaying of tall buildings, torsional and bending vibrations of shafts and vibrations in

wings of aircraft demand an insight into elastic behavior of structural members. These

elastic systems consist of continuous mass and elasticity throughout their span. Hence,

these systems are modeled assuming that the mass of the system is distributed in the

entire system as infinitesimally small elements. Such a model for a mechanical system

is known as a distributed parameter model. There are only few distributed parameter

systems such as beams, bars, strings and plates that have closed form solutions.

However, study of these systems provides understanding of behavior and modeling of

most complex systems, which cannot be solved in a closed form manner.

This chapter deals with study of vibration of continuous beams. The equation of motion

for a beam is described and a closed form solution is provided. Investigation is done in

terms of natural frequency and steady state response by considering the case of a

11

cantilever beam subjected to dynamic excitations. This chapter forms the foundation to

the problem investigated in the thesis. The source of most of the material presented in

Sections 3.2 and 3.3 is [2].

3.2 Equation of motion for a non-uniform beam

Consider a non-uniform Bernoulli-Euler beam of length as shown in Figure 3.1(a).

The transverse vibrations are denoted as

L

( )txw , . The cross-sectional area is ( )xA , modulus of elasticity is ( )xE , density is ( )x , and moment of inertia is ( )xI . The external force applied to the beam per unit length is denoted by . ( )txf ,

From strength of materials, the bending moment ( )txM , is related to the beam deflection by ( txw , )

( ) ( ) ( ) ( )22 ,,

xtxwxIxEtxM

= . (3.2.1) One can look on an infinitesimal element of the beam, shown in the Figure 3.1(b), and

determine the model of flexural vibrations. It is assumed that the deformation is small

enough such that the shear deformation is much smaller than . From Newtons

second law in the - direction,

( txw , )y

( ) ( ) ( ) ( ) ( ) ( ) ( )22 ,,,,,

ttxwdxxAxdxtxftxVdx

xtxVtxV

=+

+ . (3.2.2)

12

( )txf ,

( )txw ,

dxxL

x

y

( )txw ,

( )txf ,

Undeformed x-axis

x dxx +dx

( )txM ,

( ) ( ) dxx

txMtxM + ,,

( )txV ,

( ) ( ) dxx

txVtxV + ,,

(a)

(b)

( )txf ,

( )txw ,

dxxL

x

y ( )txf ,

( )txw ,

dxxL

x

y

( )txw ,

( )txf ,

Undeformed x-axis

x dxx +dx

( )txM ,

( ) ( ) dxx

txMtxM + ,,

( )txV ,

( ) ( ) dxx

txVtxV + ,,

(a)

(b)

FIGURE 3.1: Equation of motion for a beam

13

Here is the shear force at the left end of the element , ( txV , ) dx ( ) ( )dxtxVtxV x ,, + is the shear force at the right end of the element . The term on the right hand side of

the equality sign is the inertia force of the element. The sum of the moments on the

element yields

dx

( ) ( ) ( ) ( ) ( ) ( )[ ] 02

,,,,,, =+

++

+ dxdxtxfdxdxx

txVtxVtxMdxx

txMtxM . (3.2.3)

Here the right hand side in the equation vanishes, since it is assumed that the rotary

inertia of the element is negligible. Simplification of this expression yields, dx

( ) ( ) ( ) ( ) ( ) 02

,,,, 2 =

++

+ dxtxfdx

xtxVdxtxVdx

xtxM . (3.2.4)

Since is small, and hence dx ( )2dx is negligible. The above expression takes the form

( ) ( )x

txMtxV = ,, . (3.2.5)

This expression relates shear force and the bending moment. Substituting (3.2.5) in

(3.2.2) gives

( )[ ] ( ) ( ) ( ) ( )22

2

2 ,,,t

txwdxxAxdxtxfdxtxMx

=+ . (3.2.6)

Substituting (3.2.1) in (3.2.6) and dividing by yields dx

( ) ( ) ( ) ( ) ( ) ( ) ( )txfx

txwxIxExt

txwxAx ,,, 22

2

2

2

2

=

+

. (3.2.7) If no external force is applied ( )txf , is zero. The equation of motion of beam for

is then given as 0 ,0 >

The above expression (3.2.8) is a fourth order partial differential equation, which

governs the vibration of a non-uniform Bernoulli-Euler beam. If the parameters ,

, and

)(xE

)(xA )(xI )(x are constant then (3.2.8) can be further simplified to give

( ) ( ) ,0,, 44

22

2

=+

x

txwct

txw (3.2.9)

where

.A

EIc = (3.2.10)

3.3 Natural frequencies and modeshapes

The equation of motion (3.2.9) contains four spatial derivatives and two time

derivatives. Hence, in order to determine a unique solution for , four boundary

conditions and two initial conditions are needed. Usually, the values of displacement

and velocity are specified at time

),( txw

0=t , and so the initial conditions can be given as, 0)0,( and ,0)0,( == xwxw & , (3.3.1)

where dots denote derivates with respect to time. The method of separation of variables

is used to determine the free vibration solution,

)()(),( tTxVtxw = . (3.3.2) Substituting (3.3.2) in (3.2.9) and rearranging yields

( ) ( ) .)(

1)(

22

2

4

42

==dt

tTdtTdx

xVdxV

c (3.3.3)

Here, is a positive constant. The above equation can now be written as two

ordinary differential equations as shown below.

2

15

( ) ,0)(444

= xVdx

xVd (3.3.4)

( ) ,0)(222

= tTdt

tTd (3.3.5) where,

EIA

c

2

2

24 == . (3.3.6)

The solution to (3.3.5) can be given as

tBtAtT sincos)( += . (3.3.7) The constants A and B can be evaluated using two initial conditions given by (3.3.1),

and the solution to (3.3.4) is assumed as,

sxexV =)( . (3.3.8) Substituting (3.3.8) in (3.3.4) and simplifying furnishes,

044 = s . (3.3.9) The roots of (3.3.9) are

iss == 4,32,1 , , (3.3.10) hence, the solution to (3.3.4) can be given as

xexexeexV xixixx +++= 4321)( . (3.3.11) Equation (3.3.11) can also be expressed alternatively as

xxxxxV coshsinhcossin)( 4321 +++= , (3.3.12) where are different constants, which can be evaluated using the four

boundary conditions. The natural frequency of the beam can be therefore computed

from (3.3.6) as

4321 and ,,

16

( ) 42422 ALEI

AlEIL

AEI

=== , (3.3.13) where, is a non-dimensional parameter.

3.4 Steady state response and natural frequencies for a uniform clamped cantilever beam The dynamic behavior of a beam can be determined by analyzing the case of a uniform

clamped cantilever beam shown in Figure 3.2.

x

( ) ttf sin=( )tbw ,

L

a

x

( ) ttf sin=( )tbw ,

L

a

FIGURE 3.2: Uniform cantilever beam subject to harmonic excitation

The boundary conditions at the clamped end are no displacement and no slope, which

can be given as,

( ) 0,0 =tw , (3.4.1) and,

( ) 0,0 =

xtw , (3.4.2)

respectively. The boundary conditions at the free end are no bending moment and no

shear force, represented by

17

( ) ( ) ( ) 0,22

=

xtLwxIxE , (3.4.3)

and,

( ) ( ) ( ) 0,22

=

xtLwxIxE

x. (3.4.4)

The steady state response ( )tbw , , measured at bx = , of the beam under the influence of harmonic excitation of ( )t tf sin= at some other position ax = , is described by a Green function. The Green function is a function of

Frequency of excitation Position of excitation ax = , and Position of interest where the response is to be measured . bx =

The beam is separated in two parts, ax 0 , and a Lx < , and denoted by

( ) ( )( )

At ax = , the deflection, the slope and the moment are the same for both parts of the beam. The shear force differs by EI1 . These four conditions represent the matching

conditions at ax = and can be described by ( ) ( )tawtaw ,, 21 = , (3.4.9)

( ) ( )x

tawx

taw

= ,, 21 , (3.4.10)

( ) ( )2

22

21

2 ,,x

tawx

taw

= , (3.4.11)

and

( ) ( )EIx

tawx

taw 1,,3

23

31

3

=

. (3.4.12) Separation of variables gives

( ) ( ) txvtxw ii sin, = , 2,1=i . (3.4.13) Substitution of (3.4.13) in (3.4.6) and (3.4.7) yields

0 12

1 = vAvEI , ax

The equations (3.4.14) and (3.4.15) can be written as

0 14

1 = vv , ax

The determinant of is determined for different values of A . The values of which make singular are designated as A n . Then the natural frequencies of the beam n are

( ) 42422 ALEI

ALEIL

AEI

nnn === , (3.4.24) where is a dimensionless parameter. The fundamental natural frequency of a cantilever beam leads to 875.11 = . Denoting EI , the bending stiffness of the beam, by , the first natural frequency for the cantilever beam can be expressed as, BS

41 5156.3 ALSB

= . (3.4.25) Now,

zAb 1= (3.4.26) allows determination of the values for constants for i . iiii DCBA and ,,, , 2,1=

The steady state amplitude at any point other than the point of excitation is given by the

Green function as below

( ) ( )( )

where a is the point of harmonic excitation and b is the point of measurement of the

steady state amplitude. The system parameters and ,,, LIAE can be chosen arbitrarily. The figure shows 875.11 = , 694.42 = , 854.73 = , and 995.104 = . Hence, for known values of and ,,, LIAE the first four natural frequencies of the cantilever beam can be given by (3.4.24).

(e)

7.85 7.855 7.86

0

)(bw

0 2 4 6 8 10 12

0

)(bw

0 2 4 6 8 10 12

0

)(bw

(b)

(d)

0 2 4 6 8 10 12

0

)(bw

0 2 4 6 8 10 12

0

)(bw

(a)

(c)

(e)

7.85 7.855 7.86

0

)(bw

0 2 4 6 8 10 12

0

)(bw

0 2 4 6 8 10 12

0

)(bw

0 2 4 6 8 10 12

0

)(bw

(b)

(d)

0 2 4 6 8 10 12

0

)(bw

0 2 4 6 8 10 12

0

)(bw

(a)

(c)

FIGURE 3.3: Steady state amplitude for a uniform cantilever beam

22

However, the cantilever beam, being a distributed parameter system, has infinite

number of natural frequencies, which can be approximated by the formula ( )2

12 n for

[5]. 5>n

Cases (a) and (c) represent the response at a collocated point, i.e., the point of excitation

and the point of measurement are the same. In cases (b) and (d) the response is at a

non-collocated point, i.e., the point of excitation and the point of measurement are

different.

In Figure 3.3(c) the pole in the neighborhood of 854.73 = is not observed. This is because the pole and the zero are very close to each other. A magnified view of the

region marked by circle in Figure 3.3(c) is shown in Figure 3.3(e). Here, it is clear that

the pole and the zero are extremely close to each other.

3.5 Summary

One popular approach in controls is pole-zero cancellation. Here, the idea is to place

some zeros on some poles to reduce vibration in the rod. However, a major drawback

of the method is that one has to be extremely careful in placing the zeros, because the

slightest error in positioning of a zero can lead to a non-vanishing pole, which will

make the system unstable.

23

Chapter 4: The Control Gain

4.1 Introduction

This chapter deals with development of the theory for elimination of steady state

response at a prescribed location for the case of a harmonically excited Bernoulli-Euler

cantilever beam. A closed form expression for the control gain is established for the

uniform cantilever beam and the results are then generalized to the case of a non-

uniform beam. Several graphs indicating the control gain requirement with the change

in excitation frequency are shown. Investigation is performed on the obtained results to

draw meaningful conclusions and provide groundwork for further development of the

proposed theory.

4.2 Dynamic absorption in a uniform beam and a formula for the control gain Consider a uniform beam of length , modulus of elasticity L E , density , cross-sectional area , and moment of inertia A I . The axial distance from the supported end

of the beam is x . Suppose that the beam is excited by a harmonic force ( ) ttf cos= , as shown in Figure 4.1(a).

24

FIGURE 4.1: Vibration control of a harmonically excited beam

x

( ) ttf cos=

( )txw ,

L

(a) Uncontrolled beam

x

( ) ttf cos=

( )txw ,

L

(b) Controlled beam

( ) ( )tawtu ,=a

x

( ) ttf cos=

( )txw ,

L

(a) Uncontrolled beam

x

( ) ttf cos=

( )txw ,

L

(b) Controlled beam

( ) ( )tawtu ,=a

Then as shown in Chapter 3, the steady state transverse vibrations of the beam are

governed by the differential equation

022

4

4

=+

twA

xwEI , Lx

and

( ) tx

tLwEI cos,33

= . (4.2.5)

The conditions (4.2.2) and (4.2.3) describe no-displacement and no-slope at 0=x , condition (4.2.4) imposes no bending moment at Lx = and condition (4.2.5) shows that shear induced by the exciting force is tcos . The steady state motion of a prescribed point of the beam may be vanished by applying a concentrated control force at ax = ,

( ) ( )tawtau ,, = , (4.2.6) where is a constant. The partial differential equation for the controlled system is then

( ) waxtwA

xwEI =

+

2

2

4

4

, Lx

( ) ( )( )

( ) 02 =Lv . (4.2.18) Hence, the four conditions in (4.2.14), the first three conditions of (4.2.15), and the

condition (4.2.18), can be written in matrix form

bAz = , (4.2.19) where

=A

LshLchLsLcLchLshLcLs

LchLshLcLsachashacasachashacas

ashachasacashachasacachashacasachashacas

3333

2222

22222222

000000000000

00000000001010

,

[ ]TDCBADCBA 22221111=z , (4.2.20) 8

1 ebEI

= , (4.2.21)

where is the ppeth unit vector, and s, c, sh, and ch represent sine, cosine, hyperbolic

sine and hyperbolic cosine, respectively. Once is found the shape functions z ( )xiv , , are determined by (4.2.16). The fourth condition expressed by (4.2.15) can then

be resolved for

2,1=i , i.e.

( ) ( ) ( )( )

=

avavavEIa

1

21, . (4.2.22)

The values of the control gain for various points of excitation a , as function of the exciting frequency

8.0 ,6.0 ,4.0 ,2.0= are shown in Figures 4.2(a)-4.2(d).

LAIE and ,,, can be chosen arbitrarily.

28

(b)

(a)

0 2 4 6 8 10 12-8

-4

0

4

8

0 2 4 6 8 10 12-8

-4

0

4

8

(c) (d)

0 2 4 6 8 10 12-8

-4

0

4

8

0 2 4 6 8 10 12-8

-4

0

4

8

310

310

310

310

(b)

(a)

0 2 4 6 8 10 12-8

-4

0

4

8

0 2 4 6 8 10 12-8

-4

0

4

8

(c) (d)

0 2 4 6 8 10 12-8

-4

0

4

8

0 2 4 6 8 10 12-8

-4

0

4

8

(b)

(a)

0 2 4 6 8 10 12-8

-4

0

4

8

0 2 4 6 8 10 12-8

-4

0

4

8

(c) (d)

0 2 4 6 8 10 12-8

-4

0

4

8

0 2 4 6 8 10 12-8

-4

0

4

8

310

310

310

310

FIGURE 4.2: Plot of control gain against

4.3 Analysis of results

Large amount of information can be derived from the above graphical results. The plots

for are singular at certain values of . Also, the positions of these singularities change with the change in control location ax = . However, Figure 4.3 shows the above four graphs superimposed on each other. It can be seen from these graphs that

the zeros of ( ) ,a are invariant of the control location. Further analysis is performed in order to understand this behavior.

29

0 2 4 6 8 10 12

-8000

-4000

0

4000

8000

0 2 4 6 8 10 12

-8000

-4000

0

4000

8000

0 2 4 6 8 10 12

-8000

-4000

0

4000

8000

FIGURE 4.3: Plot of control gain against - (superimposed) Consider the clamped-hinged uniform beam shown in Figure 4.4(a). The natural

frequencies of this beam are indeed the zeros of ( ) ,a . Also, the singularities in ( ) ,a are the natural frequencies of the beam shown in Figure 4.4(b) which is

clamped in the left end and hinged both at ax = and Lx = .

From the above arguments and from [19] and [3], the control gain ),( a can also be determined from the following formula

( )

=

=

=1

2

1

2

1

1,

i i

i ica

, (4.3.1)

30

(a) Clamped-hinged uniform beam

L

(b) Clamped-double-hinged uniform beam

L

a

(a) Clamped-hinged uniform beam

L

(b) Clamped-double-hinged uniform beam

L

a

L

a

FIGURE 4.4: Clamped and clamped-double-hinged uniform beam

where is a constant, c i denotes the eigenvalues of the clamped-hinged beam shown in Figure 4.4(a) and i denotes the eigenvalues of the clamped-double-hinged beam in Figure 4.4(b). Constant c can be evaluated by considering the static case i.e. when

0= , applied to the case shown in Figure 4.4(a). Substituting 0= in (4.3.1) gives ( ) ca =0, . (4.3.2)

Denote by the static deflection at ax = of the clamped-hinged beam shown in Figure 4.5(a) due to a collocated unit static load 1=U . This is a static problem where the deflection of any point on the beam is no longer a function of time , and

can be denoted by . Hence the mathematical model of the beam shown in Figure

4.5(a) can be described by

),( txw

(w

t

)x

31

LxUaxdx

wd

0)(22

=dx

Lwd . (4.3.7)

The beam is separated in two parts as described below.

[ ]TDCBADCBA 22221111=z , (4.3.14) 6

1 ebEI

= . (4.3.15)

For each value of the constants can be evaluated from . Then the static

deflection

a bAz 1= can be given as

2,1 ,23 =+++= iDaCaBaA iiii . (4.3.16) By the linearity of the problem (4.3.3), the static deflection due to a force applied to

the beam shown in Figure 4.5(b) is

F

F . Hence, invoking the control law (4.2.6) gives ( ) FaF 0,= , (4.3.17)

which yields

1=c , (4.3.18)

by virtue of (4.3.2). The control gain is therefore

( )

=

=

=1

21

2

1

1,

i i

i ia

. (4.3.19)

Note that the control gain given by (4.3.19) is identical to that expressed in (4.2.22).

Table 4.1 shows the comparison between the values of control gain as determined from (4.3.19), using different numbers of eigenvalues and , and that determined from (4.2.22), for .10=

34

TABLE 4.1: Comparison of control gain formulae

64.7608271.9743164.52690245.735447

67.2449174.8187566.70656254.081601

64.9247872.1497964.68074246.365423

64.7489071.9736864.52439245.69129as determined from (4.2.22)

64.7803572.0130564.55730245.821325

a=2/3a=a=0.5a=0.25

as determined from (4.3.1)No. of eigenvalues

64.7608271.9743164.52690245.735447

67.2449174.8187566.70656254.081601

64.9247872.1497964.68074246.365423

64.7489071.9736864.52439245.69129as determined from (4.2.22)

64.7803572.0130564.55730245.821325

a=2/3a=a=0.5a=0.25

as determined from (4.3.1)No. of eigenvalues

( ) 5.05.0

It is evident that with increase in number of eigenvalues used to calculate from (4.3.19) the comparison converges to the exact value determined from (4.2.22). Again,

the accuracy of the numerical value of from (4.3.19) is largely influenced by the accuracy in the numerical value of eigenvalues and . In calculations for in Table 4.1 the eigenvalues and have been determined by solving the transcendental eigenvalue problem as shown in [13]. The eigenvalues are listed in Appendix B. This

argument can be further strengthened by comparing values of for the static case calculated at . 25.0=a

TABLE 4.2: Comparison of control gain formula for static case

364.0889364.0927

as determined from (4.3.1)as determined from (4.2.22)

364.0889364.0927

as determined from (4.3.1)as determined from (4.2.22)

35

Here, 364.0889 is the exact value for . The inaccuracy in the value determined from (4.2.22) is due to singularity of in (4.2.19) for A 0= . Hence, 364.0927 is the value of control gain for 00001.0= . Also, it is important to note that the boundary condition for the beam shown in Figure 4.5(a) is clamped-hinged. This is because the

desired objective is to achieve no steady state displacement at the free end in the beam

shown in Figure 4.1(a).

4.4 Some illustrations

Some simple examples depicting the calculation procedure are presented for better

understanding of the theory developed in the previous sections. Example 1 shows the

stepwise numerical calculations of the control gain, the control force and the steady

state displacement at some positions of interest. Example 2 demonstrates that control

can be achieved in order to eliminate the steady state motion at a point other than the

free end by making a small modification to the theory. Simulated mode shapes are

shown in example 3, which display beam mode shapes before and after implementation

of the nodal control.

Example 1:

A uniform Bernoulli-Euler beam is described by

035 22

4

4

=+

tw

xw , ,20 (4.4.1)

( ) 0,0 =tw , ( ) 0,0 =

xtw , ( ) 0,22

2

=

xtw , and ( ) t

xtw 10sin2,23

3

=5 (4.4.2)

Hence, EI = bending stiffness = 5, and A = 3. Amplitude of harmonic force = 2.

36

x

( ) ttf 10sin2=

( )txw ,

L

x

( ) ttf 10sin2=

( )txw ,

L

FIGURE 4.6: Illustration demonstrating control gain calculations

Determine control gain and control force needed to eliminate steady state displacement

at the free end.

From separation of variables

( ) ( ) txvtxw 10sin, = , (4.4.3) and,

605/310/ 224 === EIA . (4.4.4) Hence, (4.4.1) gives

060 = vv , (4.4.5) and, (4.4.2) gives

( ) ( ) ( ) 0200 === vvv , ( ) 522 =v . (4.4.6) The beam is separated into two parts, 0 75.0 x , and 275.0 < x , denoted as

( ) ( )( )

Boundary condition are

( ) ( ) ( ) 0200 211 === vvv , ( ) 5222 =v , (4.4.10) and

( ) ( )75.075.0 21 vv = , ( ) ( )75.075.0 21 vv = , ( ) ( )75.075.0 21 vv = , (4.4.11) and

( ) ( )( ) ( )75.075.075.05 121 vvv = . (4.4.12) The conditions in (4.4.11) and (4.4.12) are the matching conditions at . The

general solution of (4.4.8) and (4.4.9) is given as

75.0=x

( ) ( ) ( ) ( ) ( ) xDxCxBxAxv iiiii 41414141 60cosh60sinh60cos60sin +++= , (4.4.13) where i . The desired objective is to achieve no displacement at the free end, i.e., 2,1=

( ) 0,2 =tw , (4.4.14) or equivalently,

( ) 022 =v . (4.4.15) The following problem is solved to determine the coefficients .2.1for , ,,, =iDCBA iiii

bAz = , (4.4.16) where

=A

364.2818447.2818164.14252.160000680.1012650.1012839.5089.50000

736.130732.130754.0657.00000710.31750.30826.3735.6710.31750.30826.3735.6049.11394.11420.2375.1049.11394.11420.2375.1094.4970.3494.0 0.870-4.094 3.970 0.494-0.870

00000783.202.78300001010

,

[ ]TDCBADCBA 22221111=z , (4.4.17)

38

852 eb = . (4.4.18)

,0144.0 ,0032.0 ,0054.0 ,0032.0 ,0054.0 21111 ===== ADCBA 0686.0 and ,0687.0,0125.0 222 === DCB ,

(4.4.19)

( ) ( ) ( )75.060cos0032.075.060sin0054.075.0 4/14/11 +=v ( ) ( )75.060cosh0032.075.060sinh0054.0 4/14/1 + , (4.4.20)

( ) 0.0022 75.01 =v . (4.4.21) The control gain is determined as

( ) ( ) ( )( ) .5.178175.075.075.0510,75.0

1

21 ==v

vv (4.4.22)

Since is negative, the control can be implemented by means of spring and the system is stable.

The control force is given by

( ) ( ) ( ) tvtwtu 10sin75.05.1781,75.05.1781,75.0 1== . (4.4.23) ( ) ttu 10sin-3.9018,75.0 = . (4.4.24)

x

( ) ttf 10sin2=( )txw ,

L

5.1781=Kx

( ) ttf 10sin2=( )txw ,

L

5.1781=K

FIGURE 4.7: Controlled uniform beam

The steady state displacement at the free end is as required and is given as

39

( ) ( ) ( )260cos0125.0260sin0144.02 4/14/12 =v ( ) ( )260cosh0686.0260sinh0687.0 4/14/1 + . (4.4.25)

( ) ( ) 010sin101.7764 10sin2),2( -152 == ttvtw . (4.4.26)

Example 2:

The control can also be implemented so as to eliminate steady state motion at any other

prescribed point, for example at 5.0=x . This can be mathematically expressed as ( ) 0,5.0 =tw , (4.4.27)

or equivalently,

( ) 05.01 =v . (4.4.28) The sixth row in matrix , in A bAz = changes to accommodate for the condition in (4.4.26). The new matrix is given by

=A

364.2818447.2818164.14252.160000680.1012650.1012839.5089.50000

0000135.2886.1178.0984.0710.31750.30826.3735.6710.31750.30826.3735.6049.11394.11420.2375.1049.11394.11420.2375.1094.4970.3494.0 0.870-4.094 3.970 0.494-0.87000000783.202.78300001010

,

( ) 0.0046 75.01 =v . (4.4.29) The control gain is

( ) .-958.111010,75.0 = (4.4.30) The control force is given by

( ) ( ) ( ) tvtwtu 10sin75.0-958.111,75.0-958.111,75.0 1== (4.4.31)

40

( ) ttu 10sin-4.3685,75.0 = (4.4.32) The steady state displacement at 5.0=x is, as desired

( ) ( ) 010sin101.4728 10sin5.0),5.0( -151 == ttvtw . (4.4.33)

Example 3:

Consider a uniform Bernoulli-Euler beam

022

4

4

=+

tw

xw , ,10 (4.4.34)

The position of excitation is at 8.0== bx . The desired objective is to eliminate steady state motion at the free end. The mode shapes of the beam before and after

implementing nodal control at position 3.0== a , ,,

x

,

are plotted. For the case with the

uncontrolled beam, the coefficients 2,1for =iDC iiBA ii are determined and the

mode shapes can be determined from (4.4.35) given below.

x

( ) ttf sin=( )txw ,

L

b

x

( ) ttf sin=( )txw ,

L

b

FIGURE 4.8: Uncontrolled uniform beam

( ) xDxCxBxAxv iiiii coshsinhcossin +++= , i . 2,1= (4.4.35)

41

x

( ) ttf sin=( )txw ,

L

( ) ( )tawtu ,=

b

a

x

( ) ttf sin=( )txw ,

L

( ) ( )tawtu ,=

b

a

FIGURE 4.9: Controlled uniform beam

For the case with the controlled beam, the coefficients are

determined and the mode shapes are again determined from (4.4.35). The mode shapes

before and after implementing nodal control are shown in Figure 4.10. It is clear from

the figures that the condition of no motion at the free end is achieved as desired.

3.2,1for , ,,, =iDCBA iiii

42

Before Control After Control

0 0.2 0.4 0.6 0.8 1-0.008

-0.004

0

0.004

0.008

Mode 2

0 0.2 0.4 0.6 0.8 1-0.008

-0.004

0

0.004

0.008

Mode 1

x x

)(xv)(xv

0 0.2 0.4 0.6 0.8 1-0.004

-0.002

0

0.002

0.004

Mode 3

0 0.2 0.4 0.6 0.8 1-0.004

-0.002

0

0.002

0.004

Mode 2

x x

)(xv)(xv

0 0.2 0.4 0.6 0.8 1-0.004

-0.002

0

0.002

0.004

Mode 4

0 0.2 0.4 0.6 0.8 1-0.004

-0.002

0

0.002

0.004

Mode 2

x x

)(xv)(xv

0 0.2 0.4 0.6 0.8 1-0.08

-0.04

0

0.04

0.08

Mode 1

0 0.2 0.4 0.6 0.8 1-0.08

-0.04

0

0.04

0.08

Mode 1

x x

)(xv)(xv

Before Control After Control

0 0.2 0.4 0.6 0.8 1-0.008

-0.004

0

0.004

0.008

Mode 2

0 0.2 0.4 0.6 0.8 1-0.008

-0.004

0

0.004

0.008

Mode 1

x x

)(xv)(xv

0 0.2 0.4 0.6 0.8 1-0.008

-0.004

0

0.004

0.008

Mode 2

0 0.2 0.4 0.6 0.8 1-0.008

-0.004

0

0.004

0.008

Mode 1

x x

)(xv)(xv

0 0.2 0.4 0.6 0.8 1-0.004

-0.002

0

0.002

0.004

Mode 3

0 0.2 0.4 0.6 0.8 1-0.004

-0.002

0

0.002

0.004

Mode 2

x x

)(xv)(xv

0 0.2 0.4 0.6 0.8 1-0.004

-0.002

0

0.002

0.004

Mode 3

0 0.2 0.4 0.6 0.8 1-0.004

-0.002

0

0.002

0.004

Mode 2

x x

)(xv)(xv

0 0.2 0.4 0.6 0.8 1-0.004

-0.002

0

0.002

0.004

Mode 4

0 0.2 0.4 0.6 0.8 1-0.004

-0.002

0

0.002

0.004

Mode 2

x x

)(xv)(xv

0 0.2 0.4 0.6 0.8 1-0.08

-0.04

0

0.04

0.08

Mode 1

0 0.2 0.4 0.6 0.8 1-0.08

-0.04

0

0.04

0.08

Mode 1

x x

)(xv)(xv

0 0.2 0.4 0.6 0.8 1-0.08

-0.04

0

0.04

0.08

Mode 1

0 0.2 0.4 0.6 0.8 1-0.08

-0.04

0

0.04

0.08

Mode 1

x x

)(xv)(xv

FIGURE 4.10: Implementation of nodal control

43

4.5 Summary

The problem of absorbing steady state motion at the free end of a cantilever beam under

the influence of harmonic excitation is investigated. It has been shown that for a fixed

excitation frequency and point of application of control force there exists a unique

control gain that absorbs the harmonic motion at the free end. A closed form solution for the control gain is given. This result is better than the one given by Ram in

[3] because it does not require infinite product of eigenvalues.

44

Chapter 5: Stability and Optimality

5.1 Introduction

The concept of stability explains the limitations on motion of a vibrating system. A

stable system vibrates in specific bounds while an unstable system has an unbounded

motion. The study of stability also assists in tuning the response of a dynamic system

so that it remains in desired limits. Control systems design involves determining

various parameters like and . Desired overshoot, settling time, response levels,

system robustness, cost of implementing control and simplicity of implementing and

monitoring the control are some of many important factors governing the choice of

these parameters. Study of optimality provides an insight into smart selection of these

physical parameters. Conditions ensuring the stability of the controlled system are

determined in this chapter. Graphs demonstrating regions of active and passive control,

stability and optimal control locations are shown and discussed.

cm, k

5.2 Stability analysis

If the control gain is negative then the control can be implemented by means of a spring of constant = , which should be attached between point and the ground as shown in Figure 5.1.

a

45

x( ) ttf cos=( )txw ,

L

FIGURE 5.1: Passively controlled uniform beam

In this case the system is stable. Note that there are many frequency intervals for which

the control gain is negative. Since the beam in Figure 4.4(b) is obtained by imposing a

single constraint to the beam shown in Figure 4.4(a), the eigenvalues i interlace the eigenvalues i in the sense that

L32211 if kk

Corollary 2

The controlled system is stable if and only if

1< (5.2.4)

(a) Controlled beam

F

x ( )txw ,

L

x

1

xF

(b) Equivalent lumped parameter model

for displacement at x = a

FIGURE 5.2: Stability analysis and equivalent stiffness

Proof: By definition from (4.3.16), static deflection is positive. Hence, inequality (5.2.4) holds for negative . This is the passively controlled case where the control can be implemented by attaching a spring as discussed above, for which the system is

obviously stable. Instability may arise only for the case where is positive. Consider

47

now the case when is positive. From the linearity of the differential operator in (4.2.1) it follows that there is a linear relation between force and deflection in the beam

i.e. without the control, ),( tauF = . Hence the deflection at ax = in the controlled beam can be considered as governed by the sum of two springs, one negative with

constant applied by the control, the other positive with constant 1 applied by the flexibility of the beam, as shown in Figure 5.2(a) and Figure 5.2(b), where x is an infinitesimal element of the beam. The system is stable if and only if the equivalent

spring is positive, i.e., 0>1 , and otherwise unstable. This is precisely the inequality (5.2.4).

1

5.3 Optimality

The topic of optimality discussed here is based on the minimal control force

requirement. Several graphs are shown in Figure 5.3. Considerable information can be

drawn from these graphs in terms of understanding stability, optimality, and type of

control. The graphs are a plot of control force u , control gain ),( ta and inverse of

static deflection i.e. over the span of beam described by (4.2.1) through (4.2.5) for

four different values of excitation . Here, LIE , A and ,, can be chosen arbitrarily.

48

x0 0.2 0.4 0.6 0.8 1

-2

0

4

8

-10

-5

0

5

10

0 0.2 0.4 0.6 0.8 1x

0 0.2 0.4 0.6 0.8 1

-2

0

2

4

x0 0.2 0.4 0.6 0.8 1

-5

0

5

x

(a) (b)

(d)(c)

control force control gain /1

310

310 310

310

x0 0.2 0.4 0.6 0.8 1

-2

0

4

8

-10

-5

0

5

10

0 0.2 0.4 0.6 0.8 1x

0 0.2 0.4 0.6 0.8 1

-2

0

2

4

x0 0.2 0.4 0.6 0.8 1

-5

0

5

x

(a) (b)

(d)(c)

control force control force control gain control gain /1 /1

310

310 310

310

FIGURE 5.3: Plots of control gain, control force and inverse of static deflection against the beam span x for cases with excitation frequency of (a) 10= ,

(b) 20= , (c) 30= and (d) 80= The following observations can be made from the graphs.

1. The inverse of static deflection is always positive (consistent with the

definition).

Static deflection is positive by definition and hence the inverse of static deflection is

clearly positive.

49

2. Inverse of static deflection is unbounded at 0=x and . 1=xStatic deflection is 0 at (clamped end) and at 0=x 1=x (desired objective). Hence, the inverse is .

3. Control force is unbounded near 0=x . At location just next to 0=x the control gain requirement is very large and the displacement is almost zero. Hence the control force is large. At the control

force should be zero because the displacement is zero.

0=x

4. The control force is -1 at 1=x . Since the excitation amplitude at 1=x is unity, the control force at is 1. 1=x

5. The control gain is positive in some regions and negative in some others.

The control gain follows Corollary 1. Again if 0 , active control has to be implemented.

6. The control gain is unbounded at 0=x and 1=x . It is not possible to determine control gain at 0=x numerically because in (4.2.19) becomes singular. However, using pseudo inverse one can calculate the control gain. It

is obvious that infinite amount of control gain is needed to achieve the state of no steady

state motion at the free end. Also, if control is implemented at the end

A

1=x , the control gain required is infinity because the displacement is zero and the control force

which is the product of control gain and steady state amplitude at the end has to be 1.

5.4 Summary

The sign of the control gain i.e., positive or negative, determines the type of control to

be implemented i.e., active or passive. Equations (5.2.2) and (5.2.3) in Corollary 1

50

establish the conditions for the sign of the control gain . If < 0, the control can be realized by a passive element i.e., spring of constant =k . However, if > 0, then the control has to be implemented by active means. Corollary 2, through (5.2.4),

establishes the conditions for the controlled system to be stable. The optimal control

force requirements may be obtained from the graphs shown in Figure 5.2.

The issue of robustness is more critical than the issue of optimality of control force,

because at the free end the control force requirement is minimum, but the control gain

required is infinity. A passively controlled system is more robust in comparison to a

system controlled by active means. It is obvious from the graphs that the flatness of the

control gain graph over the beam span is a direct indication of robustness. Also with

increase in the frequency of excitation the flatness of the control gain graphs goes on

decreasing and also there are few locations where control can be implemented by

passive means and active control is impending.

51

Chapter 6: Experimental Verification

6.1 Introduction

The analysis so far has been purely theoretical and has shown that the steady state

motion at a prescribed location on a beam can be eliminated by describing the problem

appropriately and then performing simple mathematical manipulations. The analysis

also helps in determining the value of necessary the control gain, the magnitude of the

control force, the type of control i.e. active or passive and the location of control. The

ultimate objective however, is to be able to implement the theory in practical

applications. A small experiment is carried out in order to validate the method

developed and results derived. This chapter describes the mathematical model, the

experimental setup and the results obtained from the experiment.

6.2 Proposed model for the experiment

In order to validate the theory a simple experiment is constructed so as to mimic the

configuration shown in Figure 6.1. The beam is under the influence of a harmonic

excitation at cx = . The objective is to implement nodal control by using a spring so as to eliminate the steady state displacement at the free end i.e. at . This is a simple

configuration and can be easily modeled from the information available from previous

analysis and illustrations.

Lx =

52

( ) ttf sin=x

L

( )txw ,

c

b

( ) ttf sin=x

L

( )txw ,( )txw ,

c

b

FIGURE 6.1: Mathematical model of the test beam used in the experiment

It is imperative that there is significant resemblance in terms of dynamics between the

experimental model constructed and the schematic model shown in Figure 6.1. For this

reason modal analysis is performed on a model of a double-simply-supported beam, as

shown in Figure 6.6. First few natural frequencies of the beam are extracted and

compared to natural frequencies of the beam in Figure 6.1 obtained analytically.

6.3 Determination of natural frequencies

(A) Analytical determination of natural frequencies.

The natural frequencies for the double-simple-supported beam can be extracted

analytically by solving transcendental eigenvalue problem from [13]. However, more

simply, one can write the equation to the beam and impose the boundary conditions (in

this case double-simple-supports and free end) and solve for natural frequencies by

solving a linear algebraic problem as demonstrated in Chapter 3, section 3.4. The

results are not as accurate as one obtained by solving the transcendental eigenvalue

53

problem, but provide significantly close approximation. The governing differential

equation for the beam is given by

022

4

4

=+

twA

xwEI , Lx

( ) 0,1 =taw , ,( ) 0,2 =taw ( ) ( )xtaw

xtaw

=

,, 21 , ( ) ( )222

21

2 ,,x

tawx

taw

= . (6.3.9)

From separation of variables

( ) ( ) txvtxw ii sin, = , 2,1=i . (6.3.10) Direct substitution of (6.3.10) in (6.3.6) and (6.3.7) gives,

0 2 = ii AvvEI , 2,1=i . (6.3.11) Again,

EIA 24 = . (6.3.12)

Hence,

0 4 = ii vv , 2,1=i . (6.3.13) Accordingly, the boundary conditions are

( ) ( ) ( ) ( ) ,0 ,0,00,00 2211 ==== LvLvvv (6.3.14) and the matching conditions are

( ) ( ) ( ) ( ) ).( ),(,0,0 212121 avavavavavav ==== (6.3.15) The solution to (6.3.13) is

( ) xDxCxBxAxv iiiii coshsinhcossin +++= , 2,1=i (6.3.16) where, for ,,, iii CBA ,iD 2,1=i are constants. Equation in (6.3.16) and the conditions

in (6.3.14) and (6.3.15) together give a set of algebraic equations, which are represented

here in matrix form

bAz = (6.3.17) where,

55

=

aaaaaaaaaaaaaaaa

aaaaaaaa

LLLLLLLL

coshsinhcossincoshsinhcossinsinhcoshsincossinhcoshsincos

coshsinhcossin00000000coshsinhcossin

sinhcoshsincos0000coshsinhcossin0000

0000101000001010

A ,

, [ ]TDCBADCBA 22221111=z (6.3.18) and

[ ]T00000000=b . (6.3.19) Clearly there are infinite numbers of solutions for for i for every

position

,,, iii CBA ,iD 2,1=

ax = of the simple support. The values of , which make the matrix singular, give the eigenvalues

Z

for the Bernoulli-Euler beam by the following relationship

AEI

4= . (6.3.20)

The natural frequencies of the beam are then given by taking the square root of the

eigenvalues and dividing them by 2 , i.e.,

2=nf . (6.3.21)

The physical parameters namely, modulus of elasticity E , the cross-sectional area ,

density

A

and moment of inertia I are assumed to be invariant along the beam span. Parameters and A I are determined from the physical dimensions of the beam.

56

h

w

AISI 1005Steel

L

h

w

AISI 1005Steel

L

FIGURE 6.2: Dimensions of the test beam used in the experiment

Beam Dimensions:

mm 0.1727=L , mm 8.50=w ,

mm 0.6=h . Hence,

2mm 8.30468.50 === hwA , (6.3.22) and

433

mm 914.40 12

68.5012

=== hwI . (6.3.23)

The beam is made up of AISI 1005 Steel.

Beam properties:

GPa 200=E . -3m-Kg 7870= .

57

Simulation is performed in MATLAB (Appendix A Program 11) for the double-simple-

supported case discussed above and also for the clamped-simple-supported case.

(B) Determination of natural frequencies from modal test.

A test where the structure or component is vibrated with a known excitation, often out

of its normal service environment, which includes both the data acquisition, and its

subsequent analysis, is called modal testing [14]. It is performed in order to extract the

natural frequencies of the test beam. The complete experimental setup is shown in

Figure 6.6. The following material gives a brief description of the setup, necessary

equipment, procedure of performing the test and some necessary precautions.

(i) Test Beam

The test beam is fastened to a heavy metal frame as shown in Figure 6.4 using clamps

Bolt

Nut

Test Beam

Clamps

Bolt

Nut

Test Beam

Clamps

FIGURE 6.3: Clamping details

at two locations. The details of the clamping are shown in Figure 6.3.

(ii) Impact Hammer

Model 291M78-086C05 from PCB Piezotronics is used to cause an impact. It consists

of an integral ICP quartz force sensor mounted on the striking end of the hammerhead.

58

The frequency range is approximately 5 kHz and hammer range is approximately

22000N.

FIGURE 6.4: Impact Hammer

Its resonant frequency is near 28 KHz. Figure 6.4 shows a picture of the hammer and

the beam.

(iii) Accelerometer

FIGURE 6.5: Accelerometer

Quartz shear ICP accelerometer, Model 353B33, from piezoelectric is used in the test.

The range of frequencies is from 2 Hz to 4000 Hz with %5 and voltage sensitivity is 100mV per g .

59

(iv) Data Acquisition

NI-4551 from National Instrument is used for data collection. BNC 2140 also from

National Instruments is used to provide ICP power to the accelerometer and the hammer

and connect them to the computer via NI 4551. VirtualBench DSA (Dynamic Signal

Analyzer) is used for signal processing and monitoring.

(v) Connecting Cables

The connecting wires for hammer and accelerometer are recommended to have

impedance in the range of 50~75 .

Modal analysis technique

The natural frequencies of a system are a function of the physical parameters, material

properties and boundary conditions of the system. Hence, the frequency response

function of the test beam should remain invariant under ideal conditions irrespective of

the point of measurement and the point of excitation. The accelerometer is mounted on

the beam as shown in Figure 6.5. An impact is made at several locations along the span

of the beam. Again, frequency response is also obtained for several different locations

of accelerometer for same position of impact. The schematic shown in Figure 6.6 gives

a clear picture of the setup and data acquisition for modal analysis. The clamping of the

beam is very critical. Repeated impacts on the beam may cause loosening of the bolts

in the clamps. Also the heavy metal frame should be as rigid as possible and should not

move. These factors change the characteristics of the system and influence the

frequency response function.

60

Test

Bea

mIm

pact

Ham

mer

Acc

eler

omet

er

ICP

Powe

r U

nit

PC w

ith

Dyn

amic

Sig

nal

Ana

lyze

r

Hea

vyM

etal

Fram

e

Test

Bea

mIm

pact

Ham

mer

Acc

eler

omet

er

ICP

Powe

r U

nit

PC w

ith

Dyn

amic

Sig

nal

Ana

lyze

r

Hea

vyM

etal

Fram

e

Test

Bea

mIm

pact

Ham

mer

Acc

eler

omet

er

ICP

Powe

r U

nit

PC w

ith

Dyn

amic

Sig

nal

Ana

lyze

r

Hea

vyM

etal

Fram

e

FIGURE 6.6: Modal Analysis

61

Some important considerations for the Modal Test

It is essential to mount accelerometer securely on the structure to avoid relative motion between the accelerometer and the structure. Insecure mounting of

accelerometer can cause noise in the collected data for the system. In this test,

an accelerometer is first attached on the beam using a double sided scotch tape

as seen in Figure 6.5 and it is wrapped using a tape around the beam and the

accelerometer for secure mounting.

The location of accelerometer is decided after observation of the dynamic motion of beam from the side. The mode shape of slow natural frequency can

be observed looking at the motion from the side. The mounting position is then

decided avoiding the nodal points between free end and clamped support.

The impact of the hammer should be quick and sharp. Also for low frequency range analysis a soft hammer tip should be used. It should be insured that the

amplitude spectrum for the hammer is approximately flat in the frequency range

to be analyzed. This is shown in Display 2 of Figure 6.7.

Figure 6.7 shows the display of the VirtualBench DSA. Sharp peaks in the frequency

response shown in Display 3 are the natural frequencies of the test beam. Time

waveform of the accelerometer and the hammer is shown in Display 1 and the amplitude

spectrum is shown in Display 2.

62

FIGURE 6.7: VirtualBench DSA display

Table 6.1 shows a comparison of the natural frequencies for the two cases with natural

frequencies derived from modal analysis on the test beam.

TABLE 6.1: Comparison of natural frequencies

11.26

73.8570.9568.804

44.13

23.17

3.6

Modal Analysis of Test Beam

(Hz)

55.16

22.28

3.31

AnalyticalClamped Simple support

(Hz)

3.131

21.142

40.763

AnalyticalDouble Simple support

(Hz)

Natural frequency

11.26

73.8570.9568.804

44.13

23.17

3.6

Modal Analysis of Test Beam

(Hz)

55.16

22.28

3.31

AnalyticalClamped Simple support

(Hz)

3.131

21.142

40.763

AnalyticalDouble Simple support

(Hz)

Natural frequency

63

The double-simple-supported case is chosen because the approximation of natural

frequencies is closer to the experimental result. The frequency of 11.26 Hz is believed

to be coming from the overall system comprising the test beam and the heavy metal

frame.

6.4 Determination of the control gain

The control gain calculations are performed as discussed in Chapter 4, Section 4.2.

Actual test beam parameters and dimensions are used. A harmonic force of 45N and

frequency of 3.75 Hz acts exactly in the middle of the test beam span between the two

supports.

( ) ttf 56.23sin45=

x

m7272.1

0.327m ( ) ( )tawtu ,=0.654m

a

( ) ttf 56.23sin45=

x

m7272.1

0.327m ( ) ( )tawtu ,=0.654m

a

FIGURE 6.8: Controlled test beam

Graph of control gain variation with changing location of control along the beam span is

shown in Figure 6.9. It is clear from the graphs that is negative between the two supports (shown by triangles in the graph). Hence the control can be implemented by

means of a spring.

64

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6-100

-80

-60

-40

-20

0

20

40

60

80

100

control force in N control gain in N/mmx

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6-100

-80

-60

-40

-20

0

20

40

60

80

100

control force in N control gain in N/mmx

FIGURE 6.9: Control gain and control force variation along the beam span

In order to verify the accuracy of the calculations, simulation is done to plot the

modeshape of the test beam (Appendix A Program 14 and Program 15) before and after

implementing control. The Figure 6.10 shows that the steady state motion at the free

end is eliminated after implementing control. However, Figure 6.10(b) shows that there

is discontinuity in the shape of the beam. This helps in understanding the huge control

force requirement in the range 0.2m 0.3m and also 0.6m 0.7m.

65

0 0.4 0.8 1.2 1.6-2.5

-1.5

-0.50

0.5

x

)(xv

210

0 0.4 0.8 1.2 1.6-0.8

-0.4

0

0.4

0.8

x

)(xv

210

(a) Before Control (b) After Control

0 0.4 0.8 1.2 1.6-2.5

-1.5

-0.50

0.5

x

)(xv

0 0.4 0.8 1.2 1.6-2.5

-1.5

-0.50

0.5

0 0.4 0.8 1.2 1.6-2.5

-1.5

-0.50

0.5

x

)(xv

210

0 0.4 0.8 1.2 1.6-0.8

-0.4

0

0.4

0.8

x

)(xv

210

0 0.4 0.8 1.2 1.6-0.8

-0.4

0

0.4

0.8

0 0.4 0.8 1.2 1.6-0.8

-0.4

0

0.4

0.8

x

)(xv

210

(a) Before Control (b) After Control

FIGURE 6.10: Test beam modeshape before and after control

6.5 Control system design

The graph in Figure 6.9 shows that the most optimal place for placing the spring is at

0.4138m from the first support. This is because before 0.4138m the control force

requirement increases sharply. Also after 0.4138m the control gain requirement

increases sharply even though the control force requirement decreases. The necessary

controlling spring should have a spring constant of 59.34N/mm.

In practical situation a spring of 59.34N/mm positioned at 0.4138m from the first

support may not necessarily eliminate the motion of the free end. However, this

simulation serves as a guideline in choosing both the spring and the location of the

spring on the test beam. A very simple configuration is designed which is extremely

flexible, both in terms of housing springs of variable length and diameter, and also, in

66

positioning the springs at different locations along the span of the test beam between the

two supports. The configuration is shown in Figure 6.11.

BeamBeam

FIGURE 6.11: Spring housing configuration

The photograph shown in Figure 6.14(a) shows how the device can be easily clamped

on the heavy metal frame. The vibration exciter is attached to the test beam by the

configuration shown in Figure 6.12. This is also shown in Figure 6.14(b).

Beam

VibrationExciter

Beam

VibrationExciter

FIGURE 6.12: Attaching the vibration exciter to the beam

The complete experimental setup is shown in Figure 6.13. Photograph of the same is

shown in Figure 6.15.

67

3.6

50

H

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Spri

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Vib

ratio

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Test

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e3

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0

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3.6

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