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ACTIVE BAIAHEIRIC COITROL.OP BEA! IRANSVERSB VIBRATION

BY
Mostafa S.A. Habib

A DISSERIAIION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Mechanical Engineering
Michigan State University
East Lansing, Michigan 48824

1987

ms IMO m 01'
MW! mm

BY
Mostafa S. A. Habib

An active parametric vibration control theory was developed
which uses parametric fluctuation to control the beam transverse
vibration. The beam was modelled as non-linear, dynamic, simply-
supported Bernoulli-Euler beam using'the extended Hamilton's
principle. The closed- loop system was deduced using the direct method
of Liapunov from which the control algorithm for asymptotic stability

was derived.

The closed-loop system model was reduced to a nonhomogeneous
wave equation for the longitudinal vibration u(x,t) subject to the
nonhomogeneous boundary conditions which could be solved analytically
using the finite Fourier transform, and a nonlinear fourth order
parabolic equation in the transverse vibration y(x,t) which was

approximated using finite difference method.

A prototype control system was designed and constructed to
demonstrate and verify the approach and to evaluate its performance.
The basic measured quantities were the transverse vibration y(x,t) ,

the acceleration (or the displacement) of the end point of the beam

Mostafa 8 .A. Hebib

and the exciting force. Analog integrator and differentiator circuits
were designed and built to implement the control algorithm.

Both the simulation and the prototype control system were
tested and compared to evaluate stability of the transient vibration
and dynamic motions due to external disturbances. The comparison of
the simulation with experiment results showed good agreement. The
significant increases in stability of the test beam were measured and

feasibility of employing active parametric vibration control

demonestrated .

Thisistocertifythatthe

dissertation entitled

WPWCWOPMWSEWIOH

presented by

Mostafa S .A. Habib

has been accepted towards fulfillment
of the requirements for

 

Ph.D. 44min Mechanical Engineering

 

Major rofcssor

Date ' /

MS U i: an Affirmative Acn‘oaz Equal Opportunity Insulation 0- 12771

iv

The author wishes to express his sincere appreciation and
gratitude to his major advisor Professor Clark J. Radcliffe for his
continued suport in the form of knowledge, enthusiasm, and guidance
during the course of this research. Professor Radcliffe deserves much
credit for his contributions during my graduate study and also for his

friendship and painstaking review of my dissertation.

The efforts of the other members of my doctoral Guidance
Committee, Drs. David Yen, Hassan Khalil, and Allan Maddow are greatly
appreciated. Their comments concerning this dissertation were very
valuable. They provided guidance and are examples for me to follow.
Thanks are also due to my fellow students in the laboratory for their

good advice and encouragement.

Finally the author is indebted to the government of Egypt,
which provided the fellowship support necessary for completion of this

dissertation.

IABLE OF CONTENTS

LIST OF TABLES .............................................
LIST OF FIGURES .............................................
NOMENCLATURE ..............................................
CHAPTER 1 - INTRODUCTION .................................
1.1 Active vibration control .....................
1.2 A new approach .............................
1.3 Scope of dissertation ......................
CHAPTER 2- PROBLEM FORMULATION AND ACTIVE
PARAMETRIC VIBRATION CONTROL THEORY .........
2.1 Problem formulation ........................
2.2 Stability analysis ..........................
2.2.1 Liapunov functional for asymptotic stability
CHAPTER 3 - ANALYTICAL-NUMERICAL SOLUTION ................
3.1 Analytical solution for beam axial motion ....
3.2 Numerical solution of the parapolic equation
3.3 Steps of solution ..........................
3.4 Simulation results and discussion ...........
3.4.1 Control force ..............................
3.4.2 Internal energy and work done ...............
CHAPTER 4 - EXPERIMENTAL FACILITIES, PROCEDURES AND RESULTS
4.1 Simply-supported beam test stand ............
4.2 Active control prototype ...................
4.2.1 Main instrumentation circuits ...............
4.2.2 End beam motion measurement .................

vi

........

.........

........

16
17
3O
3O
35
4O
41
42
43
79
79
84
84

87

PAGE

4.2.2a End beam acceleration measurement ............. 87
4.2.2b End beam displacement measurement ............. 92
4.2.3 Transverse displacement measurement ............. 95
4.2.4 Beam excitation ..................................... 95
4.2.5 Control force measurement .......................... 96
4.3 Experimental test results and discussions ............ 96

4.4 Comparison of the uncontrolled ideal beam with

the modelled beam with respect to natural
frequencies and modulus of elasticity ................ 115
CHAPTER 5 - COMPARISON OF EXPERIMENTAL WITH SIMULATION RESULTS .. . 118
5.1 Comparison considerations ........................... 118
5.1.1 Damping model for the uncontrolled beam .............. 119
5.1.2 Open-loop medal frequencies .......................... 120
5.2 Comparison cases .................................... 122
CHAPTER 6 SUMMARY AND CONCLOSIONS ............................. 128
APPENDICES .................................................... 130
APPENDIX A Displacement and Forces Measurements and Calibrations .. 131
A1 Displacement Sensor Calibration .......................... 131
A2 Forces Measurements and Calibration .................... 133
APPENDIX B Real Time Record For the Steady-State

Second-Mode Test Results For a - 35.46 N .............. 143
APPENDIX C Time Domain Test Results For Two-Mode Excitation ......... 148
LIST OF REFERENCES ................................................. 152

vii

Table

Table

Table

Table

Table

Table

Table

Table

LIST OF TABLES
PAGE

A subset of the simulation results for the
initial displacement for a - 40 N
for test case 1 ......................................... 49

Effect of the control gain a on the amplitude ratio an,
the energy ratio en and the efficiency factor n for

the initial displacement test results for test case 1 ....,51

Effect of the control gain a on the amplitude ratio aj,
the energy ratio eJ and the efficiency factor n

for initial velocity test results for test case 2 ........ 55

Effect of the control gain a on.the amplitude ratio an,
the energy ratio cn and the efficiency factor n

for initial displacement of an infinite number of

modes test results for test case S ........................ 66

Effect of the control gain a on the amplitude ratio a ,

J

the energy ratio e and the efficiency factor n for

J
initial impulse of an infinite number of modes for

test case 6 .............................................. 70

Effect of the control gain a on the amplitude ratio aj,
the energy ratio ej and the efficiency factor n for

internally unstable system test results for test case 7 . .76

Effect of the control gain on the transient
motion amplitudes test results ........................... 103

Effect of the control gain on the steady state

first mode response for test case 2 .................... 107

iix

Table

Table

Table

Table

Table

Table

Table

Table

Effect of the control gain on the steady state

second mode response for test case 2 ..............

Effect of the control gain on the steady state

tow-mode excitation ...............................

Effect of the control gain on the transfer
function at the first four modal frequencies

for the random noise excitation test results .......

Comparison of the uncontrolled natural

frequencies and Young's modulus of the ideal

beam with those of the modelled beam .................

Open-loop natural frequencies of the experimental

and simulation results of the beam. .............

Experimental and simulation results for

transient motions of the beam .......................

Experimental and numerical amplitude test
results for the steady-state motion of the

beam. ............................................

A subset of the measured displacement, exciting
force and control force histories for the

steady state second mode test result for a -

35.46 N .........................................

ix

PAGE

..... 107

..... 110

..... 114

..... 116

..... 121

..... 123

..... 126

..'...1az.

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

.1a

.1b

.2a

.2b

.2c

.2d

.3a

.3b

LIST OF FIGURES

Flexible beam model

Beam configuration

Displacements and forces acting on an

infinitesimal element of length Ax ..................

Modified bang-bang control force
Active vibration control

Computational molecules for the finite
difference approximation

Response of the transverse vibration at x-2/2
for initial displacement test results for various

control gains for test case 1

The required control forces, P(t) for test

case 1

The internal energy of the system, E(t) for

various control gains for test case 1

The work done by the control force, P(t)for

various control gains for test case 1

Response of the transverse vibration at x-l/Z
for initial velocity test results for various

control gains for test case 2

The required control forces, P(t) for test

case 2

...... 8

...... 8

..... 26

..... 44

..... 44

..... 46

..... S3

..... S3

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

.3c

.3d

.4a

.4b

.4c

.4d

.5a

.5b

.5c

.6a

.6b

PAGE

The internal energy of the system, E(t) for
various control gains for test case 2 ................. 54

The work done by the control force, P(t)for

various control gains for test case 2 .................. 54

Response of the transverse vibration at x-2/2
for resonance excitation test results for

various control gains for test case 3 ................. 57

The required control forces, P(t) for test

case 3 ................................................ 57

The internal energy of the system, E(t) for

various control gains for test case 3 .................. 58

The work done by the control force, P(t)for

various control gains for test case 3 .................. 58

Response of the transverse vibration at x-2/2
for steady-state excitation test results for

various control gains for test case 4 .................. 61

The required control forces, P(t) for test

case 4 ................................................ 62

Effect of the control action on the internal
energy of the system for test case 4 ................... 62

Response of the transverse vibration at x-2/2 for
initial displacement of an infinite number of modes

for various control gains for test case 5 ................ 64

The required control forces, P(t) for test

case 5 ................................................ 65

xi

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

4.1

4.2

4.3

4.4

PAGE

Internal energy and work done for test case 5 ........... 65
Response of the transverse vibration at x-l/2 for

initial impulse of an infinite number of modes

for various control gains for test case 6 .............. 68
The required control forces, P(t) for test

case 6 ................................................ 69
Internal energy and work done for test case 6 ........... 69
Response of the transverse vibration at x-2/2

for internally unstable system test results

for various control gains for test case 7 .............. 72
Curves of response and control force for a -40 N

for test case 7 ........................................ 74
The required control forces, P(t) for test

case 7 ................................................ 74
Internal energy for test case 7 ....................... 75
The work done for test case 7 ......................... 7S
Prototype active vibration controller showing

the simply-supported beam test stand and the

main instrumentation devices .......................... 81
Prototype active vibration controller showing

the end motion sensors and the structure of the

moving beam end and the control actuator ................ 83
Measurement and control flow diagram ................ 86
Integrator and analog switch circuit .................. 88

xii

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

.5a

.5b

.5c

.7a

.7b

.10

.ll

.12

.13

PAGE

Integrator transfer function test results .............. 90
Integrator test signal results at 20 H2 .......... 91
Integrator test signal results at 200 Hz .............. 91
Differentiator and analog switch circuit .............. 93
Differentiator transfer function test results ............ 94
Differentiator test signal results at 20 H2 ............ 94
Sensors and electromagnets locations ................... 97
Transient response first-mode test results at

x - 367.2 mm for the uncontrolled beam ................ 99
Transient vibration, first-mode test results

for control gain a - 10.63 N.

a)the response at x - 367.2 mm

b)the control force P(t) ............................ 101
Transient vibration, first-mode test results

for control gain a - 13.98 N.

a)the response at x - 367.2 mm

b)the control force P(t) ............................. 101
Transient vibration, first-mode test results

for control gain at a - 58.67 N.

a)the response at x - 367.2 mm

b)the control force P(t) ............................. 102
Steady state second-mode test results for a - 35.46 N.

a) the exciting force.

b) the response at x - 367.2 mm.

c) the control force ................................. 106

xiii

Figure 4.14

Figure 4.15

Figure 4.16

Figure 5.1

Figure

Figure

Figure

Figure

Figure

Figure

A1

A2

A3

A4

A5

A6

PAGE

FFT of the steady state response of the two-
mode test results ..................................... 109

Typical test results of the transfer function

of the beam. The displacement sensor located

at x - 367.2 mm and the exciting force applied

at x - 82.55 mm.

a) the transfer function for a - O

b) the transfer function for a - 13 N ................. 112

Random noise excitation test results for the

controlled beam for various control gains ............. 113

Results of experiment and simulation for
steady-state first mode response for

comparison case 2 .................................... 125
Displacement sensor calibration results . ................ 132

Four-arm strain gauge bridge for measuring the

control force.

a) bridge circuit.

b) strain gauge orientation ............... . ........... 134

Four-arm strain gauge bridge for measuring the

exciting force.

a) bridge circuit.

b) strain gauge orientation. ......................... 135

Strain gauge calibration for the axial control
force ................................................ 137

Magnet drive signal calibration results for

the axial control force ................................ 138

Strain gauge calibration results for the

exciting force ........................................ 140

Figure A7 Magnet drive signal calibration results.for
the exciting force .................................... 141
Figure C1 Steady state two-mode test results for the

uncontrolled beam.
a) the exciting force.
b) the response at x - 367.2 mm ....................... 149

Figure C2 Steady state two-mode test results for control
gain a - 41.7 N.
a) the exciting force.
b) the response at x - 367.2 mm
c) the control force ................................ 150

Figure C3 Steady state two-mode test results for control
gain a - 50 N.
a) the exciting force.
b) the response at x - 367.2 mm
c) the control force ................................... 151

1.5

p-p(X.t)
P-P(t)

2
.............................. cross-sectional area of beam, m

................ speed of sound in beam material (-JE/p ) m/sec

......................... transverse vibration amplitude ratio.

......................................... differential operator

................................... damping cofficient Kg m/sec
2
.......... Young's modulus for the modelled beam material, N/m

2
............. Ybung's modulus for the ideal beam material, N/m
............................ internal energy of the system N mm

.................................................. energy ratio

........................................... transverse load N/m

.............. modal natural frequency of the modelled beam, Hz

................. modal natural frequency of the ideal beam, Hz
............. principal moment of inertia of beam cross section

with respect to direction of bending m4
................. finite difference node at xi- iAx and tj- jAt
.................... '...............................mode number
............................ longitudinal vibration mode number
................................................... beam length
............................. transverse vibration cycle number
........................................ axial parametric force
................................................. control force

.................................................. state VGCtOI'

...................................................... time Sec

xvi

T-T(t) ................................................ kinetic energy

u~u(x,t) .......................... axial displacement of beam (positive

.......................... u corresponds to extension of beam)
U(A,t),U(n,t) ........................ finite Fourier transform for u(x,t)
VdV(q) ........................................... Liapunov functional
WdW(t) ................................................ work done N mm
x ............................................ axial co-ordinate m
y-y(x,t) ................................. lateral deflection of beam mm
a ....................................... z ........ control gain N
1 ........................................ scalar positive number
6 ........................................... difference operator
A ..................................... total axial deflection mm
c ............. Bang-Bang control force modification factor m/Sec
C ................................................... vector e R4
n ............................................ energy loss factor
A ....................... eigenvalue of the wave equation problem
f ......................................... - .......... vector 6 R‘
p ................... mass per unit volume of beam material Kg/m3
po,p1 ....................................................... metrics
r ................................................. time variable
w ................. natural frequency of the transverse vibration

of the beam rad/Sec

xvii

MON
1 . 1 Active Vibration Control

Active vibration control of destributed parameter systems is an
active area. One application is active control of large space
structures [1-3] . Large space structures are generally lightly damped
due to low structural damping in the material. Performance
requirements for shape, orientation, alignment and pointing accuracy
require the use of active vibration control because these systems have

low frequency flexural modes.

The vibration of a distributed parameter system (DPS) is
governed by one or more coupled partial differential equations
(PDE's) [4] whose coefficients or parameters are, in general,
functions of spatial variables and time. Three current approaches to
control the vibrations of DPS include: Modal Active Control [5-7]
Spectral Active Control [8] and Distributed Parameter Feedback [9,101.
Modal Active Control uses a finite number of modes to describe the
motion of the vibrating DPS. A spatial description of each mode,
e.g., its eigenfunction in this finite set is used to separate the
total motion of the system into the motion of each of these modes.
Because motion of the system in modes not included in the controlled
modes always occurs, this motion results in truncation errors in the
observation algorithm referred as observation spillover [5]. Spectral
Active Control separates the motions of the modes using their

different eigenvalues instead of the eigenfunctions used in the Modal

Active Control. This method also suffers from spillover problems
associated with modes at identical eigenvalues or at eigenvalues above
the operating eigenvalues of the controllers. Spatial filtering has
been employed with this method to control frequency domain spillover
and improve controller performance. The third approach is Distributed
Parameter Feedback control, in which the system is treated as if it
has infinity number of modes. To this date there are much theoretical

work but few applications in the literature [11] .

Active parametric control discussed here uses controlled
parameter fluctuation to control transverse vibration. It is well
known that transverse and longitudinal vibrations of a beam are
coupled [12]. If a transversally vibrating rod is subjected to time
dependent axial force; P(t), Figure 1.1 at its moving boundary, a
parametric time varying force p(x,t) will be produced. If the force
P(t) is applied with an appropriate control algorithm, the induced
parametric time varying force will work as an active vibration
control. Ball and Slemrod [20] and Ball, Marsden and Slemrod [21]
studied the abstract problem of controlling a semilinear evolution
equation and applied the formalism to the case of a Bernoulli-Euler
beam with parametric force p(x,t)-p(t). They proved the
controllability for finite-dimensional observations (y,ay/at) provided

the initial data are active in all modes.

M)

71? x—J u(x,t), amp“)

p(x,t)

 

 

 

 

 

 

Figure 1.1 Flexible beam model

lcz Aplmwluuummch

This dissertation presents a theoretical and experimental study
for the application of the active parametric vibration control on a
beam. The beam is modelled as a nonlinear, dynamic, simply-supported
Bernoulli-Euler beam using the extended Hamilton principle. The
direct method of Liapunov is applied to develop a control algorithm
for asymptotic stability of the system. The control algorithm enables
one to map from observing and controlling a theoretically infinite
number of points of the domain into observing and controlling just one
point to stabilize the system. Since no truncation associated with
the control algorithm, the control does not suffer from the spillover
problems. To demonstrate the effectiveness of the approach, a
numerical finite difference approximation and an analytical solution

are used to solve the closed-loop control system PDE's

Based on success with control simulations, a prototype control
system was constructed to evaluate the performance of the active
parametric vibration control system on a simply supported beam. The
comparison of experimental system response with the simulation showed
good agreement between the analytical and experimental results.
Significant increases in stability of the test beam were measured and
the feasibility of employing active parametric vibration control

demons trated .

1.3 Scope of Dissertation

This thesis is divided into six chapters. Following this
introductory chapter, chapter 2 will develop the mathematical model of
the beam and derive the control law. The mathematical model will be
derived using Hamilton's principle and the control law from the direct
method of Liapunov which will lead to a closed-loop system model.
Chapter 3 will develop the analytical-numerical solution of this
closed-loop system model. Seven simulation test cases are given to
demonstrate the effectiveness of active parametric control. Chapter 4
will present the experimental facilities, procedures and results. The
simply supported beam test stand design is presented. The actuator
mechanisms, sensor systems and associated electronic circuits are
presented. Typical experimental procedures and results are then
discussed. Chapter 5 compares simulation and experiment results.

Chapter 6 will sumarize this dissertation's contributions.

m2

mam m1. All) ACTIVE PWC VIBRATION comer. THEORY

Active parametric control is a new method for controlling the
transverse vibration of an elastic beam. This chapter discusses the
theoretical basis of this active control method after developing a
mathematical formulation of the problem. The beam is modelled as a
modified nonlinear, dynamic, Euler-Bernoulli beam using Hamilton's
principle. The direct method of Liapunov is used to prove asymptotic
stability, and a closed-loop system of equations are deduced. Since
the main difficulty in applying the direct method of Liapunov is to
choose a Liapunov functional, the energy integral procedure is given
and compared to the time derivative of the Hamiltonian. In chapter 3,
the closed-loop system will be simulated and results from the

simulation will be discussed.

2 . l Probl- For-duties!

To develop a mathematical model of a structure two approaches
may be used. In the first approach, the problem is formulated in
terms of differential equations, which describe the local behavior of
a typical infinitesimal region, and include auxiliary conditions on
the motion. In the second approach, which will be used here, a
variational formulation called "the principal of least action" is

postulated which is valid over the whole domain of the structures.

6

Hamilton's principle is an example of a variational formulation which
reduces the problems of dynamics to the investigation of a scalar
integral which does not depend on the coordinates used. The condition
:rendering the value of the integral stationary leads to all the
equations of motion with their admissible boundary conditions. Once
Hamilton's principle is formulated, the total energy of the system can
be found which is very helpful in choosing a Liapunov functional. We
will consider the vibration of the beam due to excitation and
elasticity in the transverse and longitudinal directions. Our
mathematical model includes three unknown space and time-dependent
quantities which characterize the beam motion in a plane: the
transverse vibration y(x,t), the axial vibration u(x,t) and the axial
parametric force p(x,t) (Figure 2.1). The following derivation
involves the usual strength of material assumptions as to linearly
elastic material behavior, small displacements, and uniform
geometrical and physical properties, but neglects rotary inertia,
effects due to shear strains and passive dampings. The parameters of
the beam are the mass density p, the moment of inertia I, the modulus
of elasticity E, the cross sectional area A and the length of the beam

2.

The mathematical statement [22-25] of the extended Hamilton's

principle is:

t2
I(6T+5W)dt-O (2.1)
t:1

Where T is the kinetic energy and W is the the work function. The

yogi)

EI
L P(t)
u(x,t),
i x I p(x,t) I
I‘ l D

Figure 2.1a Beam configuration

 

 

 

 

 

 

___.,._|

*dX'I-A—I-

 

 

Figure 2.lb Displacements and forces acting on an
infinitesimal element of length Ax

kinetic energy of the entire beam is:

“run-é orpa[m] dx+éoIlpA[MJ-Q]dx (2.2)

To evaluate the work done by the axial force p(x,t) the total axial
displacement A of the right hand and of element dx should be
estimated. This total axial motion is due to the axial elastic
elongation [au(x,t)/ax]dx and the change in the horizontal projection
of the element ds due to bending which was initially dx, (ds - dx)
(Figure 2.1), i.e.;

3%)dx+12- [Md-‘1] dx (2.3a)

where the assumption has been made that the displacements are
sufficiently small that in the binomial expansion only the first two
terms can be retained. The axial strain-displacement relation is

given by

10

alasl_iu§x.sl+;[u§§_m]2 (2.31:)

X

For the purpose of derivation we assume that p(x,t) is a tensile
force. We note that the force p(x,t) acts against A, so that the work

is negative. It follows that the work function is

1 2 2
2 ° 6x

I 2 I
-% I 9.11.51 dx + 1““) y(x.c> dd: (2.4)
0 EA 0

Where the first, second and third terms in the RHS of (2.4) represent
the work done by the bending moment, the axial force and the

transverse load respectively. The variation in kinetic energy is

1 2
6T-IpAuL(6y)dx+JpAiua—(6u)dx
0 0

at at at at
30
t2 t2 1 23.2 3 0.9.6.
cf 51' dt-CI [ OI pAat a(6y)dx+oI pAat ac(5“)d"] dt

t, t

2 2
_ 121. 6.39.
OJ [ t I pA at at(6y) dt +t I pA at at(6n) dt ] dx
1 1

11

 

1 3.1 t32 I32 2 1% t2
_I[pAat8y -JIpA[‘a—y—2]5ydt+pAat8u
0 t1 t1 at t1
t2 2
- I pA.a—% 6u dt ] dx
c, a:
t2
2 2 l 2
--J' [Ipia—Jgsydx+ pAL%6udx]dt (2.5)
t1 ° at 0 at

The boundary terms vanish because, by definition, 6y(x,t) and, 6u(x,t)

are zero at t - t1 and t2. The virtual work can be written

1 2 2 2 z
5w--I EIL§97(5y)dx-Ig-A-spdx+I f6ydx
° 8x 6x ° °

using (2.3b) to find 6p(x,t)/EA yields

1 2' 2 1
sw--I 313+3-5-(5y)dx-Ip§—(5u)dx
° 8x 8x ° 3x

1 l
' I} p d1. %'(6y) dx + I f 6y dx
0 x 0

Integrating by parts yields

12

 

 

 

 

 

 

£2 a. 2 23: 2 ’2 ‘ ’2
6W--EI 2 (6y)|+EI ,Syl -IEIa—¥6ydx-p6u
a x 8x a 6x 0 0 6x
in _ fix a.
[Jax Sudx pax 8y 0+0] a,‘(paxl6ydx
1
+ I f 6y dx
0
I‘[ 6.! 23: a. ‘
-- EI ‘- “[p ]- f]6ydx-EI 2 (6y)|
0 8x 8x 8x 8x 6x 0
8’ 22 an 1
..X - -
+[EI a, pax]8 +0! axSudx pSu o (2.6)
x
Introducing (2.5) and (2.6) in (2.1) we obtain
t32 1 s 2
-I I[EIQ-3;[pg§}+pAL¥-f]sydx
t1 ° 6x at
3.1.2 as ’2 2‘3: 31 ‘
-.1..[ ] ..1.[-. ].,
3x 6x 0 8x 6x 0
‘ an 33. ’2
+I[ -pA 2]8udx+p6u dt-O (2.7)
o ax at 0

The integral must vanish for any arbitrary values of 6y, Hay/BX). and

6u, which obey the essential boundary conditions. Because each term

13

above is independent and the variations are arbitrary, each term in
the equation must vanish. The first and fourth terms are integrals

which yield the Euler equations

6 2
fl_Y .fl_. fl! fl_¥ _
EIaX,-ax[pax]+pAat, f (2.8)
2
'gfi+"":%'° . (2.9)

which are the differential equations of motion for the beam.
Furthermore, if we consider the boundary terms in (2.7), the nature of

arbitrary variations yields

 

 

2 2
51%581] -o (2.10a)
ax x °
3 2
[3114} - p31] 6y -o (2.1%)
6x x °
2
p Su - 0 (2.11)
0

 

Equations (2.10) allow the possibilities that either

2
21% -0 or s[g-§]-o atx-0,2 (2.12a)
6x

and that

l4

3
£1 §-¥ - p gi" 0 or 6y - 0 at x - o, 2 (2.12b)
‘

Admissible variations are those for which 6(3y/ax) and 6y vanish at
the boundaries, e.g. admissible functions, y(x,t) always satisfy the
low order boundary conditions. Equation (2.11) allows the possibility

that either 8u or p vanishes at either end; i.e.;
p-0 or 6u-0 atx-0,£ (2.13)
If the beam is clamped at the end x - 0, the boundary condition is
u( 0.t ) - 0 (2.14a)

and p(x,t) can be any force. If a force P(t) is applied at the end

x - l, we will have
p(2,t) - P(t) (2.14b)

and u(x,t) must satisfy the displacement condition at x - 1 exactly.

Equations (2.10 to 2.13) represent the admissible boundary conditions.
The equations (2.12a) require' either the vanishing of the bending
moment or requires exact satisfaction of the slope boundary conditions
at each end. The equations (2.12b) require either the vertical force
is zero or that admissible functions exactly satisfy the deflection
boundary conditions at each end. Restricting ourselves to simply
supported beam, the mathematical model is given by equations ( 2.8,

2.9 and 2.3b) with the boundary conditions given by first of (2.123),

15

the second condition of (2.12b) and ( 2.14a, b). Now we can write the

governing equations of motion of the open-loop system:

0 2
E1 §—¥ - g; [ p 3% ] + pA a4.}- f(x,t) (2.153)
x at
2
‘ g: + P‘ 2.? ‘ ° (2.15b)
at
2
EX _ 3%., I [ g: ] (2.15c)

with the appropriate boundary conditions

Y(0.t) " 303.13) " 0 (2-158)
2 2
2.2:9;31 _ 2.:i§.t) - o (2.16b)
x x
u(0,t) - o (2.16c)
2
EA [ anéfieil + g [—3¥§i*Sl ] ] - P(t) (2.16d)

and the initial conditions

y(x,0) - f1(x) o s x s 2 (2.17a)

16

aggm - f,(x) o s x s 2 (2.17b)
u(x,0) - g1(x) 0 s x s 1 (2.17c)
“fie-Q)- -0 05x52 (2.17d)

where f1(x) and f ,(x) are the initial displacement and initial

velocity distributions in the y-direction respectively. The function

g1(x) is the initial displacement distribution in the x-direction.

Equation (2.15a) expresses equilibrium of the beam in the transverse
direction, equation (2.15b) expresses equilibrium of the beam in the
axial direction and equation(2.15c) is the axial strain displacement
relation. Equations (2.15) with. their boundary and initial
conditions, represent a complete system of equations for determining
the three unknowns y(x,t), u(x,t) and p(x,t). Versions of this system
of equations can be found in the literature [26-31 especially 29]
concerned with the bending of columns under dynamically applied axial
loads. The system of equations derived here from the energy
functional indicates the accuracy of the functional which will be used

to derive a parametric control law.

2 . 2 Stability Analysis

The active parametric vibration control theory is based upon
using one of the time dependent distributed parameters to control
transverse displacement under specific control law. It is clear that

equation (2.15a) contains the parametric force p(x,t) as a

17

coefficient, our objective is to find a control law by which p(x,t)
can be manipulated and transform the open-loop system into an
asymptotically stable system. For the analysis of the stability of
distributed parameter systems the direct method of Liapunov is used.
We define a Liapunov functional which properly describes a kind of
energy distribution of the system, and it is the purpose of the direct
method to indicate whether the energy is always decreasing to zero. If
this is the case then the system is asymptotically stable. The

necessary theorem concerning stability of a partial differential

equation system has been given by Zubov [32] and Wang [33,34]. The
essence of such a theorem is to extend the Liapunov stability theory
from a finite dimensional space to a space of infinite dimensions and
the realm of partial differential equations. As with the simpler n-

dimensional space method, the determination of the Liapunov functional

V is the main difficulty [35] .

2.2.1 Liapunov functional for asymtotic stability

Leipholz [36] showed the close connection between Liapunov's
stability criterion and the classical energy ( Hamiltonian, H )
criterion, for autonomous, dynamic, continuous systems. He proved
that for a conservative system, if V is chosen as the Hamiltonian,

then

fl-fl-
dc dt 0 (2.18)

and for nonconservative system,

18

d! 513
dt‘dt- VOIQQ‘NO (2.19)

where Q. is the vector of generalized forces, ('1 is the derivative of

the state vector q with respect to t and V0 is the volume of the

system. For a non-conservative system, the stability problem is more

complicated. Leipholz's work showed that even then it might be

advisable to use H as V [36,37]. Here, we choose V as the total

internal energy of the system:

V(q) - I - w

.. 10I1[pA[g§]z+pa[§§]2+-EI [3} 2
2
+gx]dx (2.20)

where f(x,t) - 0. By using (2.15c) and if the velocities Y - Y(x,t)

and U - U(x,t) are introduced where;

Y(x,t) - fixiggil . U(x,t) - i“§§*§1

(2.20) becomes

2 2 2
v(q)-%I[pAY2+pA02+EI[i-%]
0

2
D—
+ EA ldx (2.21)
ax

19

It is also convenient to introduce the notation of vector q,

qT-[Ypr] (2.22)

Then we can say that the vector q - 0 corresponds to the undeformed
equilibrium position of the beam. The sign of V(q) and of its time
derivative will be investigated [38]. Let us also introduce the

auxiliary vectors f and 5 as

{T - [ (lo :2! {So {4 1! 6T- [ £12 £29 £39 6‘ ] (2°23)

so that we can introduce also the metric p1(§,€),

l 2 2
P1(§.€ )-{ 1/2°I[2A(§1- 6;) + pA(§'1- 61)

2

2
+ 31 [ a"2(9'3' 53) ] d3 )1/2 (2°24)
ax

It then follows that

1 1 2 2 flax 2 1/2
21(q.0) - [ E I [ pA Y + pA.U + EI( 2 ) ] dx (2.25)
° ax
Thus p1(q,0) is a measure of the distance between the equilibrium

state q -0 and the deformed state q #0, further, if p1(q.0) is small,

then each of the terms

20

2 2 2 2 2 £24 7
IpAY dx, J‘an dx and IEI[ 2]

o o 0 6x
must then be small, as the integrand of p1(q,0) is sum of these non-

negative terms. It is clear that the metric “((36) satisfies the

following properties [ 39]

(1) PALE) Z 0. 91636) - 0 if and only if S' - 5

(11) P1(§.E) - 21“.!) (Symon?)

(111) 91(93):) 5 216.6) + 22105.2() (the triangular immunity)
where x is a four dimensional vector. So the chosen metric p1(q,0) is

s
a metric space in the Euclidean space R . To prove the stability of

the equilibrium state q-O we must prove that

(a) V(q) is positive definite with respect to the metric p1(q,0);

(b) V(q) admits an infinitely small upper bound in the neighborhood of

q-O; and

(c) investigate the sign of dV(q)/dt.

a) To prove that V(q) is positive definite; comparing (2.21) with

(2.25) yields

2
V(q) 2 «. p.<q.0) 2 o (2.26)

21

where at, is positive constant - l in this case. The equality holds

2
only when q - 0. Therefore V(q) is positive definite since p1(q,0) is

positive definite .

b) To prove that V(q) admits an infinitely small upper bound in the

neighborhood of q - 0 ; equivalently this requires that

2
WC!) 5 1 po(q.0) (2.27)

where y is positive constant. The reason for this is to allow
continuity of the solution of the PDEs (equations(2.15 and 2.16))with
respect to the initial data (2.17). Thus by limiting the size of the

initial disturbances f1(x) and f,(x) , a bound can be placed on the
response. If we choose

2
1 z 1 and Po (q.0) - V(q.0) (2.28)

then (2.27) is true and requirement (b) is satisfied. The relation

between the metrics .pland pois given by

91 " Po ' 2' dx (2.29a)

It is clear that p; depends continuously on p0 and since

N‘s-

de>0 for p>0 and t
0 EA

IV
0

22

therefore p0 > ,01 for all t _>. 0

then by (2.26 to 2.29) we get

2 2
p: ((1.0) S V(Q) S ‘1 Po ((1.0)

c) Finally we need to examine the time derivative of V(q) ,

V(Q) - V(Yxx 9 Y: 9 p9 ut)

2121.91-21 2n,2xils,ax22,ax
at ayt at ap at au

Q) Q:
"L“

(2.29b)

(2.30)

(2.31)

substituting (2.20) into (2.31) and making use of the boundary

conditions (2.16) yields

man) 23:.
(it

2
-‘J [El yxxyxxt+pAytytt+EA +pAuu

2 I l
'J [ EI yxxdyxt + J ”A ytyredx +01 p[ “an:+ yxyxt ] dx

2
+0". pA “tutt dx

after integration by parts,

23

fl 2

I 2
dt 'J BI yaomx yt dx +0J. ”A yr ytt+ (J p(“xt + yx yxt) dx

2
+01 pA ututtdx
When the appropriate boundary conditions are applied,

2
511’- -
dc 0I[E1yxxxx [pyx]x+pAytt]ytdx

2

 

2
+OI[pAutt-px] utdx-i-put 0

After making use of the equation of motion,

fi§ - p(x,t) “: (2.t) - P(t) ut<2.c) (2.32)

which is of the form of (2.19).

The sign of p(2,t) ut(2,t) is not known in general for all

t> 0, hence, the sign of dV(q)/dt undetermined. The stability of the
system can not be judged unless the sign of dV/dt is guaranteed by
some relationship between the applied force, P and velocity on the end

of the beam.

Definition: Let d be a positive real number. The neighborhood S(o,d)

of q - 0 is defined as the set of q which belongs to the admissible

24

states for the system for which 0 s p°(q,0) < d where po(q,0) is a

metric measuring the distance between the equilibrium state q - 0 and
the deformed state q - q. Now let us state the stability theorem by

Zubov [32-34]:

In order for the solution q - 0 of the boundary value problem

to be stable with respect to p0 and p], where p1 depends continuously
on p0, it is necessary and sufficient that in a sufficiently small

neighborhood S(o,d) of q - 0 there exists a functional V having the

following properties when qu(0,d):

1) V is positive definite with respect to pl;

2) V admits an infinitely small upper bound with respect to po;

0 o
3) V(q(t,q )) is non-increasing for t z 0, whenever q 2 S(o,d). If,
in addition , there exists a d' , 0 < d' s d, such that.

o o
4) V(q(t,q ))-o 0 as t -’ 0 whenever q 2 S(o,d'), then q - O is
asymptotically stable with respect to p0 and ,01

Properties 1 and 2 require

2 2
p; (9.0) s V(q) s 1 po(q.0)

Which is inequality (2.30) given as a result of the discussion in

subsection 2.2.1 items a and b. And if we force

P(t) ut(2,t) < 0 for all t > 0 (2.33)

25

Then dV/dt < 0 i.e. negative definite, therefore properties 3 and 4
are proven,and the system given by (2.15.2.16 and 2.17)is
asymptotically stable subject to (2.33). Now any active control

algorithm given by:

P(t) - s ( ut (1.t)) (2.34)

where g ( u: (2,t)) is some function that depends on au(2,t)/6t, and

satisfies (2.33) will yield asymptotic stability. In- the au(2,t)/at -
P(t) plane, any force in the second or fourth quadrant results in
closed-loop asymptotic stability of the beam system. The control
force shown in Figure 2.2 satisfies this condition. Figure 2.3 shows

the structure of the active vibration control using this force.

It is worthwhile to mention that if we already have the
mathematical model (2.15 to 2.17) and we wish to construct the energy
integral of the given system by which a Liapunov functional and its
total time derivative can be deduced easily, the following procedure
may be suggested:

Multiply (2.15a) across by ay/at, integrate over 2 and use the

boundary conditions to get;

£2.11 [H [iifwfiflz] 9

2 2 2 2
_ _ :23: 1.1L _ _ .1. L 521
J p ax axat dx 2 J [ ] dx (2.36s)

26

P(t)

 

 

 

 

 

 

 

 

 

 

 

 

 

(
I f at“:
a
Figure 2.2 Modified bang-bang control force
r‘C‘t)=0 e r Poul , utflft)
.' £32. BEAM -—’

3. ’1 *
\’0'
13
l

 

 

e=-ut(l,‘t)

Figure 2.3 Active vibration control

27

By making use of (2.15c), equation (2.36a) becomes

2

Mm? ”Main ].

---‘-J‘v2:[%:-2%:1«

l l 3
-- Lu LL
Jmac‘k" Paxacd"
, - 2.
at 0 EA dx + I’ p axat dx (2.36b)

Now multiply (2.15b) across by au/at, integrate over 1 and use the

boundary conditions; yielding
1a. ‘ 232 ‘2222
2atoJ‘pA[at] dx-OI dx

2
- P(t) 3312.4). J “- dx (2.37a)

From which

2 2 2
LL ML). .1. L is
J p axac dx - P(t) ac - 2 at J pA [ 1 dx (2.37b)

Substituting from (2.37b) into (2.36b) yields

28

:z—fiI [21 (£32 ”+M[%%)’+ém[s:]’] ax
-p(t)m , (2.38a)

If we take V(q) as

1 2 2 2
V(q)-ll [EI[3—¥ +pA 1“ +L+pA 1“ dx
20 ax? at EA at

(2.38b)
Then (2.38a) can be written as:
dt " P(t) gills). (2.38c)

By comparing (2.38c) with (2.32) we find that both the the
infinitesimal and the variational approaches yield the same result.
In fact this procedure yields a relationship between the rate of
change of the total internal energy of the system with respect to time

and the external power applied on the system which agrees with (2.19).

In this chapter the theoretical basis of the active parametric
control theory for controlling an elastic beam was presented. The
mathematical model was derived using Hamilton's principle which led to
a modified nonlinear, dynamic, Bernoulli-Euler beam. Also asymptotic
stability using the direct method of the Liapunov was proven from

which the closed-loop control system was deduced. The energy integral

29

as an alternative procedure and the time derivative of the internal

energy led to the same result.

mm-mxcu. SOLUTION

To see the effectiveness of the approach, the closed-loop
control system is simulated. An analytical-numerical solution of the
closed-loop control system is presented in section 3.1 and 3.2
followed by section 3.3 which discusses the steps of solution. The
numerical-analytical results of the closed-loop system simulation are
given and discussed in section 3.4. The simulation results show the
effectiveness of the closed-loop control law derived in chapter 2 from

the Liapunov function for the beam.
3 . 1 Analytical Solution For Bes- Axial lotion

This subsection presents the analytical solution of the wave
equation which gives beam axial motion. The two governing equations
of motion of the closed-loop system are coupled and nonlinear in-
general, and it has not been possible to find an analytical solution
to the beam equations. However, an analytical solution for the wave
equation can be obtained from which the parametric force p(x,t)
coupling the two equations can be evaluated and used in the
approximate solution of the parabolic equation modelling beam

transverse motion .

The axial force p(x,t) may be eliminated when combining

(2.15b) and (2.15c) to yield
30

31

2
2
“a - a 3.1: -max (3.1a)
a: 3:
where a2 - E ,
p

2 2
_a_ :31
¢(x.c) 2 [ax] (3.11.)

This is recognized as a nonhomogeneous wave equation for u, subject to
the nonhomogeneous boundary conditions of (2.16d). A formal solution
of(3.la) can be obtained using Finite Fourier Transform [14]. Let the

Finite sine transform of u(x,t) be defined by

1
U(x,t) - I u(x,t) sin Xx dx (3.2a)
o

2
where A is the eigenvalue to be determined. In addition, the Fourier

Lanna.

sine transform of 2 is given by
8:

1 2
U(2)(A t) - I sin Ax dx
2 2
° 8x

as

- sin Ax

2 l
- A I QB cos Ax dx
6x 0

0 6x

 

32

I
U(2)(.\,t) - flgfi‘n sin A! - M u(x,t) cos Ax ] o

2 l
- A Iu(x,t) sin Ax dx
0

2
- dig-h”- sin A2 - Au(£,t) cos A! -A U(x,t) (3.2b)

where the boundary conditions have been used. Since u(1,t) is not

available we search for eigensolutions with cos A1 - O, which leads to
A -( 2n -1)12'7; n - 1,2,3, ..... (3.2c)

It follows that

sin A! - sin (2n - 1) g - (_1)n-l ; n - 1,2,... (3.2d)
Transforming (3.1a) we obtain
2 2
Wig-51+[(2n- 1)fl] U(n,t)
2!
dt
1
- 2
-(-1)“ 1 a 1332:2241 -( 2n-l ) 12L; J ¢(x,t) cos(2n - 1) fix dx

+ ¢(2,c)(-1)“‘1 (3.2e)

33

since

1 l l
I a‘é:*£l sin Ax dx - - A OI ¢ (x,t) cos A dx + ¢(2,t)(-l)n'
0

therefore

n-l

t
' inlllll g;
«(2n-l) J 3:: sin(2n-1) 22 (t-r)dr

U (n,t) -
l c n 1 rs
- a oI sin(2n -l) 21 (t-r)°I ¢(x,r) cos(2n - l) 21 x dxdr

+ wtor ¢(1,r)sin(2n-1) 12‘} (t-7)dr
+ U(n,0) cos(2n -1) 3 c (3.2f)
2
where U(n,0) - OI u(x,0) sin At
The corresponding inverse formula of (3.2a) is

u(x,t) - % E U(n,t)sin(2n - 1) £3 (3.3a)

n-l

which yields

34

-l
2 n 22:21)}.
u(x,t) - l E [ sin(2n - l) 21 [ «(2n-1)

n-l
”I d! (I, r ) sin(2n-1)¥1(t- r)dr

t
-i “I sin(2n -l)? £(t-r) 0!! ¢(x, r)cos(2n - 1) £5 xdxdr

a«(2n-1)1°JI ¢<1 ')81n(2n-1)¥1(t- -,)d,

+ U(n,0) cos(2n -l) g: t ] ] (3.3b)

Therefore the distributed parameter force (2.3b) is given by

p(x, c) - EA 39+ +2 EA[ 3% ]

- EA a“ +‘EA (3.4a)

ax+ a2

Where au/ax is given by

35

Q n-l
2:93_§ ““2“,”23 [24.341.—
n-l

2

OIC[£_ALL)_.;.[§%£LLI] ] sin(2n -1)fi dr

t 1
- 211;} I I sin(2n -1) 3 (t-r) ¢(x,r) cos(2n - 1) if x dx dr
a! o o

2

2 I “n-l c
+ l- I ‘(1.T)81n(2n-1) n (t’7)df + -
a 0 22 ‘2“:111

U(n,0) cos(2n -l) 3 t ] (3.4b)

3 .2 Numerical Solution of the Parabolic Equation.

Equation (2.15a) is a nonlinear parabolic equation. Since no analytic
solution is known for this equation with its boundary conditions and
initial values, an approximate solution is developed here. It is
assumed that the x-t solution domain is covered by a uniform
rectangular lattice with Ax and dt denoting space and time increments,
respectively. Furthermore, it is assumed that the point considered is
the point x - iAx, t - jAt of the solution domain. Then if we agree

to denote y( iAx, jAt) by yi j we have the following finite difference

approximation.

2
1

a_x .___ ( 2
2 - 2 y - 2y . + y _ ) + o(At )

36

1% 2 J: a: + 0(At2) (3.58)
31'- At
6x2 " E (YB-1.1 ' 2’1.) + Y1-1’J ) + 0(AX )
éJ,§+om’) 05m
Ax
‘ l 2
2.! _ - ) + 0(Ax )

— 4 +6 -4 +
3x4 Ax‘ (Y1+2,j y1+1,j Y1,j Y1-1,j Y1-2,j

9 -1, a; + o(Ax‘) (3.5c)
Ax

2; [ P‘x"’ gi'] ' 5; [ ”1.1 6; ] ’1.1

1.51

Ax x [ 91,1 [ Y1+1/2 j - yi_1/2 J ] + och’)

‘.:;2 [ p1+1/2,j [ y1+1,j ' Y1,j ]

2
‘ p1-1/2.j [ ’1.J ' ’1-1.J ] + °‘Ax ’ (3'5d)

1
L
where 6x yi,j ' Ax [ yi+l/2,j - y1-1/2,j ]

Making use of (3.5) , the finite difference approximation for (2.15a)

takes the form

37

El A; ‘
Y1,j +1' 2Y1,j ' Y1,j-1 ‘ pA Ax? ] 5x Y1,J
_l_.As_
- pA AX: - At [p1+1/2vj [y1+lsj - yivj]

' Pia/2.1 [’12 ' ’14.: H ' 3% EL:

2 2
+ o (Ax ,At )

2 2
1r Y1-2,j * [ 4°1‘ + °2rP1-1/2,j] yi-l,j

5'0

2
[ ' 6°1r * °2rP1+1/2,j * °2rP1-1/2.J *2 ] Y1,J

2 2
[ 4c1r + pi+l/2,j °2r ] yi+l,j ‘ c1r Y1+2,j

- y1’1_1 (3.6a)

where r - 2 , c1 - and c2 -.

A; El
Ax

The boundary and initial conditions of (2.16a,b) and (2.17a,b) require

yi’o - f1 (iAx), yi’o - yi’-1 +At f2(iAx) (3.6b)

38

You, -0 o yN,J -0 (3.66)

yN+1,j' ' yN-1,j . Y.1,J - - y1,j (3.6d)

where N is the number of mesh divisions in the x-direction; i.e.

NAx - 1. If piil/Z are known equations (3.6) yield a value of y at a

point in the j + 1 row in terms of already known values in the j-l and
j-rows. Thus the entire y-mesh can be computed, and the solution can
be generated one row at a time. Figure 3.1 illustrates the
computational molecules for the explicit difference approximation

of (3.6).

To study the convergence of the finite difference
approximation, it is sufficient to study the stability and
consistency. To study the stability of (3.6), p(x,t) is assumed to be
constant. Now since the stability bounds are not affected by the

lower-order terms; i.e., in which piil/Z j arecoefficients of them,

the nature of the problem assumes that the vibration is governed

mainlygby the beam stiffness EI, rather than the axial force pi+1/2 j’

Fourier stability method yield [16]

2
4 [ ]

0 < s l
pA Ax2

or

39

couueaaxoudmo ooaouomm«n

ouwcwm one now moaaooaoa amaoaueusaaoo

H.m ouswwm

 

 

 

 

 

 

35 Lo

«Ax<v\u< I u

 

 
     

a

3.. 5..

     

no net-u
+
.
Ck DUI“

Xd

x4

:1.

 

 

 

 

I—F

1+?

1.-

 

 

3V

2V

“is

1/2
[‘5] (3.7)
Ax

EI

NIH

This stability bound was found to give good results.

The numerical approximation was derived from the PDE, i.e.
(2.15a), and since that the truncation errors go to zero as
At, Ax -' 0 no matter how this limit is taken, the explicit finite
difference approximation (3.6) is consistent with the differential
equation (2.15a), and as long as (3.7) holds, the system is stable.
By Lax's theorem; [14,17,40 and 41] stability and consistency imply

convergence, provided that the parabolic PDE is well posed.
3.3 Steps of Solution

The analytical-numerical solution presented in the previous
sections are given in order to find the response of the beam at any
point x under the action of a control force P(t) - g(au(£,t)/at) and
some initial and working data. This section presents the steps of

solution to achieve this purpose below

1) Select initial values f1(x), f2 (x) in (3.6b), working conditions

f(x,t) in (3.6a) and a control gain a.

2) Calculate the corresponding response of the initial values

using (3.6b), i.e. i - 1,2,3, ..... N.

2 yi,o 9
3) Calculate the end velocity of the beam using the

approximation:

wwsszan] ..

4) Calculate the control force P(t) using (2.34) subject to

(2.33) .

5) Find the distributed parameter force p1+1/21 , i.e.,

[p((ii1)Ax,jAt)] which requires that P(t) [ - g(6u(£, t)/at ]

and ay(2, t)/6x to be known at t - jAt using (3.4) with truncation

number of 20 modes.

6) Calculate the response y1 1+1 using (3.6)

7) Increase j by l and repeat steps 3-6
3.4 Sinnlation Results and Discussion

This section. contains the numerical-analytical results of the
closed-loop control system simulation. The simulation is designed to
illustrate of the following aspects of the. controlled system's

dynamics :

1- The stability.
2- The transient motions due to initial data.

3- The dynamic system response due to external disturbances.

42

The simulation were run on a Prime 750 computer at the Albert
H. Case Center for Computer-Aided Design at Michigan State University.

The calculations for the test case are carried out using a mesh of

2 2
N -15 nodes in the x-direction and ‘AE-; - .219 sec/m [.0203 sec/ft ].
(AX)

The steel beam material properties and dimensions for laboratory

tests, which will be presented in the following chapter are:

2 2 1 2
2.10(10 ) kg/mm [3(10 )1b/in ]

m
I

s s

p - 8304 kg/m [.3lb/in ]

2 - .61 m [2ft]

A - 50.8mm width x 1.588mm thickness [Zin x .062Sin]

For a particular control gain a, the results of the simulations

are the response at a point x1 - iAx, the required control force P(t),

the internal energy and the work done by the external disturbance.
3.4.1 Control Force.

For our control system; the asymptotic stability is guaranteed
when (2.34) and (2.33) are satisfied. The function g in (2.34) may be

chosen such that the relationship between g and ut(2,t) lies in the
second or fourth quadrant in the g-ut(£,t) plane. In the test cases
the modified bang-bang control force as shown in Figure 2.2 is chosen.

-3
In all simulation test cases 5 - 3(10 ) m/sec.

43

3.4.2 Internal Energ and Work Done.

The relationship between the internal energy E(t) and the work
done by the control force P(t) can be deduced by integrating (2.38c)

from 0 to t:

t
E(t) - 3(0) - J P(r) “£43- (17 (3.9a)

2 2 2

E(t)-%°II[EI [ii-1f 2+pA[‘g'§] +DEZ+pA[g%] ]dx

(3.9b)

The work done by an external force f(x,t) can be added to

(3.9a) to yield
1: t 1
E(t) - 2(0) - I P(r) We: dr +1 I f(x,r) alga-L)- dxdr
0 o 0

(3.9c)
Test Case 1: Initial Displacement.

In this case the initial displacement is, y(x,0) - sin(«x/£)mm,
which is the first eigenfunction of the beam. Figure 3.2a shows the
response of the middle ofthe beam, y(2/2,t) versus time and control
gains or - 0,40,80 and 108 N. In Figure 3.2a the curve for a: - 0
(control is off) is the response of the free stable vibration. The
frequency of this response is 9.654 Hz. which is the first natural
frequency of the beam. The curves for or - 40,80 decay rapidly in the

first two cycles of oscillation. For at - 108 N the motion decreases

44

 

 

 

 

 

 

 

 

 

 

 

 

 

 

AEEV 9&2» .mmzoammm

TIME, t (Sec)

Response of the transverse vibration at x-1/2

Figure 3.2a

for initial displacement test results for various

control gains for test case 1

 

 

 

 

 

 

 

 

 

 

 

0.
u — 1 - 1 J d J- - u all
a .
. u
T . .. .
2 m m
1
rs m r\ ( m '8.
\II nU I I nu
r N \\J5 1 m
( ..Ib = u
N \
w a -mfl a m ..s.
i a aJQ\ . nu
a u u
r
.
.
.
I u ..4.
u 0
c
.
u f
.
.
.
.. u 2
u so
llllllllll 4 - .
. IIIIIIIIII . _ u
n I, J H "
.........s..............u. ..... . .
. _ u o.
d d d - 4 d 1n q d - d O
nu nu nu nu nu nu nu
A; no 2* 2% no 04
1 _ _ 4.

EV 8A 5%... .5528

TIME, t (Sec)

Figure 3.2b

The required control forces, P(t) for test

case 1

45

monotonically with increasing t; i.e. , the active control overdamps
the vibration in this case. For at - 80 N the vibration almost dies
out in two cycles (the amplitude is .001mm), for a: - 40 N it almost

dies out in 4 cycles (the amplitude is - .059 m). Figure 3.2b shows
the required control force versus time for o: - 0,40,80 and 108 N. It
is clear from Figure 3.2b that the end velocity of the beam

au(£,t)/at reflects the internal energy content E(t) and lutl falls in

the interval (+e,-e) faster for bigger at. This is not the case for

a: - 108 N it seems that the control action freezes the initial shape
of the beam by slowing down the conversion of the initial energy, i.e.
the potential energy, into kinetic energy and dissipate the energy in
such way that it does not allow for oscillation, until the energy

dissipated.

The internal energy E(t) given by (3.9b) is plotted versus time
for a: - 0,150.80 and 108 N. in Figure 3.2c. For a: - O the system is
conservative, also it is clear in Figure 3.2c that at the early times
i.e. 0<t<.025 sec. the smaller the gain, the bigger the energy
dissipation is and at later times the energy dissipation is
proportional to the control gain. Except for a - 108 N (the
overdamped gain) which verifies the freezing phenomena mentioned

above .

The work done by the control force p(t) is shown in
Figure 3.2d. Comparing the corresponding curves in Figure 3.2c and
3.2d for the same or, one can see that (3.9a) is satisfied for each or
which proves the energy balance of the system at any time t. Again,
contrary to other «'5 the work done by p(t) for or - 108 N is less for

0<t<lsec.

46

 

 

 

 

 

0.5 . . . , . . , .
E qE/XV " J ~“‘\V 'fi‘ q'x A“, , G.“ V", 1“. fi‘ ”,1“. v" .“V’
E 0.3-". =0 (N) -
g \ fl-ma (N).
ED- . “X ‘. an
83 0,1- a=40 (N) _
E / ...................
«=80 (N)
-0.1 . , . . . . - . .
0.0 0.2 0.4 0.5 0.8 1.0

TIME, t (sec)

Figure 3.2c The internal energy of the system, E(t) for

various control gains for test case 1

 

 

 

 

 

 

0.5 . , .- . - ‘ .
A
E nnnnnnnnn
E .........................
.2, ,. ...... \ _
LLJ a=108 (N)
I:
8
x “=40 (N) _
D:
C)
g d

—O.1 ' I ' I t I ' I I

0.0 0.2 0.4 0.6 0.8 1,0

TIME, t (sec)

Figure 3.2d The work done by the control force, P(t)for

various control gains for test case 1

A subset of the simulated test results for c: - 40 N is shown in

table 3.1. Table 3.2 shows the effect of the control gain a: on the

amplitude ratio an - yn/yn.1 the energy ratio en - J En /En-1 and the

efficiency factor n - (E(O) - E(tn)/ E(O), where n

is the amplitude number. Notice the closed connection between the

energy ratio en and the amplitude ratio au for low values of control
gain or. For low a: it is clear that the amplitude ratio an and the
energy ratio en are almost constant and equal for the first three
cycles. The observation on the amplitude ratio an agrees with

Flouquet Liapunov theorem for this kind of system [26 and 42]

In this test case, i.e., case of controlling the transient
motion due to an initial displacement of the first mode, the control
algorithm could successfully transform the stable system into an
asymptotically stable system, and for a: - 108 N the vibration

monotonically decreases with time.

Table 3.1 A subset of the

tinut sec:

0.00397003916

0.02127304006
0.02362‘7967:
0. 02973574806
0.0342367002:
0.03861763366
0.04294860364
0.04727933914
0.03161030716

0.0019271634-
0.0062381233.
0.09000907634
0.09492002436
0.09723097236
0.1033019221b

 

0.11637‘i7721
0.1209037420.
0. 1232366900.
0.12936763296
0.13207830606
13822954090

0.

0. 1‘236048206
0.14609144490
0.1312224070.
0.1333333409-
0.13900430336
0.16421326676
0.1633461990:
0.1723771627-
0.17720009386
0.1813390337.
0.1830700216.

0.19020093474
0.1943317176-

 

0. 22917732136
0.2333104942;
0.237341‘2734
g.2¢21722902:
0.2300342362;
0.23316321936
0. 23949621204
0.26382714316
0.2681330732.
0.2724390709-

 

0. 239!

0. 2741 4379606

0. 290474729 1 a

0.3023037213-

0.3071366349.

0.31146738770
.31379838066
.32012931376
. 44604468.
.32379142956

for test case

y(I/Z.t) m

0.9743104313

-0.4196333614
.0.

-0.1 6
-0.1462930110
-o.1033493679
-0.056769177g

0.0062494111
0.1223309274
0.1474183260
0.1576147774
0.1347400037
0.1434462409
0.1304047390

-0. 0934970366
—0. 0993943099
-0.0966361016
-0.0399131396
-0.0796339113
-o.0663=3261:
-0 0304339326

0323322793

-0.

-0 0134003933.

0.0062633496.
232740160.-

0.
0. 0413106690

0342639014.

0.
0. 0614219973
0. 06 26444221
0. 0601907149
0. 0333919822
0. 0434343613
0. 0393394274
0. 0292020626

48

simulated test results for x - 40 N

l.

P(t) N

0.3390076037

EH) N'nmm

0.0042643017

.0201322702
0.0401223???
0.0733203360

0.1060836730
0.1326690216

0.2238601010

 

.J33947037Q
. 134-4471403
0.3361469307

.33'0366433

0.5439479470
0. 3443936773

 

 

0.3601701836
0.3609439996
0.3616133090

0.3630423469
0. 3637140723
0.3640986631
0.3643662930
0.3646706411
0.3649789691
0.3632274874
0.3633384330
0.3634341690

3
0.3633744693
0:3660662327

0:3661903143'
0.3663026690
332'

0.3664271

0.3663446043‘

E(t) N mm

053601737103
0.2303043387

. 9
0.2618064830
0.2356384993
0.2184076903

 

0.1373080692
0.1264070370

- 0.1138338712

0.1017307111
0.0719743776
0.0060384974

0047723011
0. 0820000917
0.0733963327
0.0674976707
0.0613130406
0.0391989979
0.0372336300
0.0336331074
0.0407331733

0.0332018130
0. 0:

02137123937
0. 0320667103
0.0291331942
0.0260820940
0. 024023 4770
0.0234711170
0.0223116071

 

0.0113034236
0.0101339437
0.0094736321
0.0072368246

3
020031340313
0.0023664332
8.0023613373

0.0013362301
0. 0012333309
0.0011117432
0. 0009943112

Table 3.1

tinul sen:

0.33910009044
0. 3634390233.
0.3677699366.
0.3721009493.
0. 3764310024.
0.300762 0133.
0.3030930002.
8.3094247413.

723040.

0. 441396 1 768.
0. 4437271099.
0. 4300380430.
0. 4343090337.
0. 4307199600.
0. 46303090 19.
0.4673010946.
0.4717120277.
0.4760437600.

.480374 7333.
0. 4047036066.
0.4090366197.
0.4933676124.

0.6319300070.
0. 6362090003.
0.6406199932.
0.6449300667.
0.6492010394.
0.6336120321.
0. 6379437236.
0.6622747103.
0.6666037110.

0.6732673771b
0.6793984307.
0. 6339294434.
0. 08826043610
0.6923913093.
0.6967223022.
0.7012332949.
0.7033841684.
0.7099131611.

(Continued)

49

7(1/2,t) mm 2(a) a

6

-0.

0176762640
”00 4620623
.0069471363

.0134302163
.0193302743
0233220310
0230373979
0246330310

.0231293961

0207231119

0090130074

'0. 0132220910

'0.
'0.
-0.

0.
0.
0.
0.
0.
0.

.0
.a
-o.
-0.
-0.
.0

0133397396
0139903490
0134920013
142270364

24023092
0100930622
0073919129
0044014443
0012149010

0106334336

.0046310373
.0020033733
0007932723

.0012634343

0032324334
.0049490293
.0062704397
.0070949420
.0073377390
0071666092
0064374794
0034193949
0040360473
°°3212227g
oo 2

0011587276
0023336963
0043902192
0033377969
0062974747

.0063731760

éééssssséé

.ééééééééfifififi

 

#98568
E

1323

éééé
333%

‘HJ N'um1

0.3666360127'
0.3666096224

0.3667022306
3667436030
0.3660139246
3.3660913629
0

.3669413712
.3669331343

.3670101762
3670360466
0.3671064973

.3671314900

0. 3674033020
0. 3674904704
0.3674920346
0.3674930930
0.3674933699
0.3674933300
0.3674902706
0.3673011992
0.3673031662
0.3673030210
0.3673030013
0.3673044179
0.3673039676

.3673070134

0
o.
8.3673091063

0.3673099013
0.3673110340
0.3673124049
0.3673132990
0.3673136366
0.3673136366
0.3673139346
0.3673147293
0.3673136032
0.3673163300
0.3673164300

0.3673174713
0.3673102462
0.3673107026
0.3673109614
0.3673109614
0.3673191402
0.3673196171
0.3673202131
0.3673203700
0.3673206900

E(t) N’mm

0.0009032673
0.0000377919
0.0000300969
0.0000100066
0.0007300930
0.0006602316
0.0006070200
0.0003927070
0.0003691936
0.0003301976
0.0 0040117332
0.0004312026
0.0003066329
0.0003346739
0.0003403440.
0.0003373736
0.0003176096
0.0002069230
0.0002372074
0.0002397024
0.0002336677
0.0002241163

407
0.0001013093
0.0000960072
0.0000933903
0.0000901317
0.0000022301
0.0000737400
0.0000639930

0.0000603093
0.0000304710

0.0000317140
0.0000297790
0.0000204993
0.0000200930
0.0000278307
0.0000271346
0.0000230003
0.0000243340
0.0000236311
0.0000234337
0.000023360:
0.000023400:
0.000022670:
0. 0000216704
0. 0000209444
0.000020669‘
0. 0000203227
.0000201209

“' ' __-

 

.0000100901.
.0000176344
.0000170362
.0000163701
”000016373

Table 3.1

tin. soc

142461330.
103770273.
7229000200.
.727239012 .
.7313690062.
.7339000709.
.7402310716.
.7443627431.
.7400937370.
.7332247303.
.7373336040.
.7610063967.
.7662173094.
"03404629.
.7740794336.
104403

‘30

OSHDOCHDOEHDOOMHWOC

$3;

0.7963341006.
0.0000631733.
0.0031961660.
0.0093270393.
0.0130300322.
0.0101009037.
0.0223190904.
0.0260300911.
0.0311017646.
0.0333127373.
0.0390437300.
0. 0441746233.
0.0403036162.
0.0320366009.
0.0371674024.
0. 0614904731.
0.0630294670.
0.0701603413.
0.0744913340.
0. 0700223267.
0. 0031332001.
0.0074041920.
.0910131033.
0961460390.
9004770317.
9040000444.
9091309179.
9134699106.
9170009033.
9221317760.
9264627693.
9307937622.
9331246337.
9394336204.
9437066211.
0.9401174946.
0.9324404073.
9367793600.
0.9611103333.
0.9634413462.
0.9697722197.
0.9741032124.
0.9704342031.
8.9027630703.
0

0

9990999990999.

.9070960712.
.9914270639.
.9937379374.

(Continued)

Y(I/Z . t) m

-0.0063929660
.0037930141
.0040422069
0033969042
0021340463
0003379667
0.0010931303
0.0026461060
0.0040022926
0.0030346023
003 7'72092
.0039013643
.0030223007
.0032003971
.0044072121
.0032630003
.0019161239
.0004406272
'0.0010406797
0024709762
0037119324
0046739360
‘0.0032927320
-0.0033270734

66:56.6

IOOOMHNDOO

25.36

.0029970747
.0017419413
.0.0003730641
0.0010139039
0.0023339003
0.0034030163
0.0043712077

0.0030226720
.0043493004
.0037034026
.00277am

.0013900036
.0003127749

-0.0009916937
-0.0022271623

IDOCHDO

“.12.
‘000
1:3
OWJO
6g::
68::
IHOO

.0047269110
.0042739907
0033409633
0023932710
0014739072
0002399607

6666666666

0.0009731492'

0.0021372307
0031406739
0039297091
0044273417
0046104373
0044760043
0040441491
0033406276
0024302630
0013706179
0002133339

0999999999

P(c) R

-3.7066
-1.1346

WDNm

0.3673206900
0.3673209204
0.3673212060

0.3673227163
0.3673230742
0.3673233722
0.3673234310
3673234310
3673236106
3673230490
3673242066
3673244431
.3673243047
.367 243047
.3673246239
.3673240623
.3673231603
0. 36732 3339 1'.
0. 3673233391
0.3673233907
0.3673233179
0.3673236960
0.3673239332
0.3673261140
0.3673261736
0. 3673261736

3673 262332

0. 3673264120
0. 3673266304
0.3673267696
0.367326
0.3673260292
0.3673269403
Q 3673270677
0. 3673273061
3. 3673274233
0

 

.3673274049
.3673274049
0.3673274049

3673203370
3673207366
3673207962
367320033

0.3673200330
0.3673209134
3673290346
0.3673290942
0.3673292134
0.3673292730

9.09.00

P

MC) 8 m

.0000162643
.0000160072
.0000133362
.0000130716
.0000147637
.0000147166
.0000147902
.0000147440
.0000144331

66
8C!
68
HH
0‘!

.0000133630
.0000134762
.0000132097
.0000129713
0.0000126330
0000124304
0000124103
0000124737
0000124430
.0000122307
0.0000119361
0.0000116960
0.0000113790
0.0000113063
0.0000113627
0.0000111240
0.0000100700

OCMHNDOOCKHNOOOOOD

0.9.099

0.0000100403
0.0000099240
0.0000097303
0.0000093304

..°
0
0
0
o
0
g

 

0.0000006399
0.0000003122
0.0000004347
0.0000004437
0.0000004090
0.0000004023
0.0000003033
0.0000002400
0.0000001191
0.0000000319
0.0000000022
0. 0000079193
0. 0000077964
0.0000076779
0. 0000076196
0. 0000076339

51

Table 3.2 Effect of the control gain a on the amplitude ratio

the energy ratio enand the efficiency factor n for

 

 

for initial displace-ant test results for teat case 1
gain n tn yn an an n 8
a N sec mu mm/mn N um
10 1 .1057 .81548 .81548 .2573 .836 30.08
2 .21077 .653619 .8015 .18131 .839 50.73
3 .31255 .535909 .81991 .1241 .827 66.28
20 1 .10791 .6218 .622 .1485 .635 59.65
2 .21185 .4187 .673 .0717 .695 80.51
3 .31688 .2766 .661 .031 .657 91.58
40 1 .112 .39 .39 .059 .40 86.41
2 .2205 .157 .40 .0094 .399 97.45
3 .33 .062 .395 .0014 .386 99.62
80 1 .168 ..064 .064 .0021 .075 99.43
2 .328 .008 .125 .000024 .0081 99.9
3 .433 .0053 .662 .00 .00 100

 

108 .......... motion ~ 0,

no oscilation.

 

52
Test Case 2: Initial Velocity.

This case excites the first mode. Impulsive loads are common
for mechanical structural elements like beams, yielding an initial
kinetic energy to the beam. An impulse was simulated as an initial
velocity. An example of such a simulation is ay(x,0)/at -
.13331n(1rx/£) m/sec. Figure 3.3a shows the response of the middle of
the beam for at - 0,10, and 20 N. The curve for a: - 0 is the response
of the free stable vibration. The frequency of this response is
9.654 Hz which is the first natural frequency of the beam. Table 3.3

shows the effect of or on the amplitude ratio aj - yJ / yj_2, energy

ratio:

ej- JEJ / Ej-2 and the efficiency factor n - (E(O) ‘ E(tj))/ E(O),

where j is the amplitude numbers Comparing the values of the
amplitude ratio and the energy ratio at the same gain, in tables (3.2

and 3.3), they are very much close.

Figure 3.3b shows the required control force for car—10 and 20 N.
The stabilizing effect of the control is evident again. The internal
energy E(t) given by(3.9b) is plotted in Figure 3.3c for a: - 0,10
and 20 N and can be compared with the work done by p(t) Figure 3.3d.
The energy balance (3.9a) is satisfied for each a. In this test case;
i.e. case of controlling the transient motion due to an initial
velocity of the first model, the control algorithm could again
successfully transform the stable system into an asymptotically

stab1e system.

RESPONSE, y(l/2,t) (mm)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

3.0 u I I 1
. a=0 (N
20‘ {If 3 if“: .0 A {“1
4 i ' w ,‘1 'i "I,
1 .0" gm ‘
' ’ ‘1 a!“ ,I‘\ ’\ -
3 I ' 1 ll 1 I ,“
0.0“ L” ' ‘ 9“?“
‘ 1 K t “a": “j x " 1“, ~41
...1 .0.) ‘51,? V I l —l
a i I i I ‘2 3 E i g '1
. l ! i ' I I ‘
“2'0“ -. .3 ‘J ‘1; ‘0’ it} ‘6' V 1",“
. «=10 (N) "
—300 I [ fi 1 v I T f 1
0.0 0.2 0.4- 0.6 0.8 1.0

TIME, t (Sec)

Figure 3.3a Response of the transverse vibration at x-1/2

CONTROL FORCE. P(t) '(N)

for initial velocity test results for various

control gains for test case 2

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

40 . . . . , T .
a=20(N)
20‘ ” nwn / r-
THW W “T WW F
.L'W”‘""” "=4 hi"
.1”.
.IJULIULJULML Us “Wit!
_20-LULL_LLJLJLL L LJLL'
‘ a=10(N)
"40 I r r I ' T ' I '
0.0 0.2 0.4- 0.6 0.8 1.0

TIME, t (Sec)

Figure 3.3b The required control forces, P(t) for test

case 2

 

 

4
q
s 0“ 0“ 0“ f‘ "\ '\\ ’u‘ ’~ ' “ ..‘ q
«I \u’ ‘0, \J' ‘5'. .Vo .l. 1“; xv. “ I ‘31
q
q
-
«4
q
.4
.3
a=20 (N ‘
-
1
d
1
\fi
— ‘ ‘ ~ " ‘0 '- s .4

 

1

 

 

 

1.0

 

 

 

 

 

 

 

1.0

. *8 r1 rlrrrI—TITUUI—rIVITIIITIUIUrrUrUTU rll 'UVUTrIT
0.0 0.2 0.4 0.6 0.8
TIME, t (sec)
Figure 3.30 The internal energy of the system, E(t) for
various control gains for test case 2
ITIITIUIIIIVITTFrj'rI rrrTI rtii[UTIIrTTTFIIFUIIrrqu
. .
2.2- 7
E I r
z i
V —.
L1J i
Z .
o_ 1
C3 .
X 1
0:
O 2
3 -
—O.2 TTTIITYTUITUTIIUrr‘I'IrTrrTTrIIr'TYITIIII'II'3‘1‘
0.0 0.2 0.4 0.6 0.8
TIME, t (sec)
Figure 3.3d The work done by the control force, P(t)for

various control gains for test case 2

Table 3.3 Effect of the control gain.¢ on the amplitude ratio

 

 

J
the energy ratio ejand the efficiency factor n for
initial velocity test results for test case 2.
gain tJ yJ aJ EJ eJ n t
a N sec mm mm/mm N mm
10 .02517 -2 1359 1.694
.07714 1 9036 1.3475
.12911 -1 7241 .8072 1.1187 .812 40.5
.18108 1 5475 .813 0.9055 .819 51.8
.23305 -1.39735 .8104 0.7411 .813 60.6
.2850 1.242 .804 0.596 .811 68.3
.3370 -1.12196 .803 0.4861 .81 74.1
20 .0235 -2.026 1.53
.07633 1.6037 0.965
.1291 ~1.2994 .641 0.632 .634 66.4
.1819 1.03096 .643 0.4048 .647 78.5
.23468 -0.83564 .643 0.26651 .649 85.8
.2834 0.67515 .654 0.17594 .658 90.6

 

56

Test Case 3: Resonance.

In this case the system which is initially at rest is excited

by applying a disturbing force f(x,t) - .146 sin w1t N/m, where 031 is

the fundamental frequency (- 60.66 rad./sec). Figure 3.4a shows the
response at or - 0,10,20 and 40 N. For a: - 0, the growth is linear
with t as is expected for resonance. For a: - 20 - 40 N the amplitudes
are kept bounded, and the amplitude value decreases with the increase
of the control gain 0:. Figure 3.4b shows the required control force.
The internal energy of the system is shown in Figure 3.4c. For at - 0
the internal energy increases unboundedly with t but for a: - 20 - 40 N
it is kept bounded. Figure 4d shows the work done by p(t) and the
energy balance of the system as given by (3.9c) for a: - 20 N. It
should be noticed that the work done by p(t) is equal to the work done

by f(x,t) Figure 3.4d and this limits the response amplitude.

57

[A
a
C)
l

d

.I

.1

.1

'I

.1

J

J

a
.1

1

 

1°

C)

.1111! j;
9
II
0

A q
z q
V

'o

RESPONSE, y(l/2,t) (mm)
I
h) C)
C) C)

II

 

 

A
C)
“an“; I n 11:11: u 1 luunu I run
(_
< .
( '

(

I n

I
9"
C)
‘1
1
.1
4

J

 

UUTrIIfi'r'rTr

"”01?" 0.8 1.0
TIME, t (Sec)

Figure 3.4a Response of the transverse vibration at x-1/2

.0
C)
S:) .
N:
C)
4;

for resonance excitation test results for

various control gains for test case 3

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

aom......,.-..-....,.....m.,--.....m.-.-..r..q

E 60 “=40 (N) a=20 (n) -E
C :
\_/ 4{) f“ F~ _ '5 q r1 1 m P' n "1 q r1 n r“ r ~1 ~ rt
CL I ‘ ' ‘ / : ' ' !«
‘20'40 r14“ M m-
DJ I .
0 Fr" >- 11D!" B b1 ”'1 *1 U M I" U P1 I1 I- M -
c5 0 , I ‘

..J m - I—I L1 LI ..I .- .a .4 I- .4 w

”-20 ULUUUUEIIIMUUUIIJUULI
E3 '1 ‘fl I L\\ 1" i
(1:: ...444:) LJ '.7 LJ ..' LA {.5 1.4 L.J .4 .ua\\<:;\\:J ..I .J 1.. La hi on «t
E I
(::) ...ESI:) Clr===:1.() <EBQ1) .:

C)

—ao.fifi....rr.........,......m,.r.......,.........
0. 0.2 0.4- 0.6 0.8 1.0

TIME, t (Sec)

Figure 3.4b The required control forces, P(t) for test

case 3

 

 

 

 

 

TIME, 1: (sec)

The internal energy of the system, E(t) for

various control gains for test case 3

 

 

 

 

-0
-0-
--O
a
a .....
n-‘
-0.
.”

 

 

A
E
E
Z:
>-
CD
0:
DJ
2:
UJ
Figure 3.4c
1
2.21
A II
E .
E 1-31
z 1
"’ 1.41
2% .
O 1.0;
D i
x l
D: 0.6.
O I
3 0.2-3
—0.2‘.3.
0.0.
Figure 3.4d

IIIIIIIII

TIME, t (sec)

The work done by the control force, P(t)for

various control gains for test case 3

59

Test Case 4: Steady State Time Response.

To study the steady state time response of the been, a damping
term should be added to the 138 of (2.15a). The damping model chosen
is the structural damping (3.7). The basic property of structural
damping is that the amplitudes of normal modes of vibration are
attenuated at rates which are proportional to the oscillation
frequencies. The structural damping of a DPS is found to be

consistent with the model [19]:
1/2
By - CB yt +p A yct - 0

Where C is the damping coefficient and B is the operator EIyxxxx

1 2
defined on an appropriate domain and B / is the unique positive

definite square root of B. Therefore (2.15a) becomes

0 3
E1 6.1.3.2519. _ c 6.132115). , g; [p(x,t)gfiait).
a x a x a t
2
+pAa-19L-‘l- f(x,t), 0<x< 2 (3.10)
a c

In this case the forcing function,

f(x,t) - 4.38 sin(wlt) N/m
where an is the fundamental natural frequency (- 60.66 rad/sec), and

C - .267 kg m/sec which was chosen to produce fast steady-state
nesponse not as a model of the experimental test system. Figure 3.5a

shows the time response of the closed-loop system for zero initial

60

conditions. For a: - 0, due to the passive damping the amplitude is
limited to 1.55m. For a: - 40 N two control actions are considered,
the first is that for t > 0 the control is turned on and the second is
that the the control is turned on for t 2 1.115 sec. In both cases
the amplitude is reduced from 1.55 mm to .545 mm. Figure 3.5b shows
the control forces p(t) versus t. The internal energy is shown in
Figure 3.5c for both cases. The internal energy for or - 0 is constant
for t 2 1.1 due to the passive damping. For a: - 40 N the largest
dissipation is due to the active control action as shown in the
figure. From figures 3.5a and 3.5c we conclude that the desired
steady state response may be obtained regardless of the time the
control action starts. The open-loop steady state internal energy
is .958 (N mm) and the closed loop steady state internal energy is

.13 (N m) which yields a steady state 1) - 86.64 %.

RESPONSE, y(I/2,t) (mm)

61

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

screw.-.“..-...fi......,... WWW
2.0.1. For t>0 «=40 (N)
1.0-: I I I I
j I 3" ." II I". I I"
0.0-é
: J U L V U
4
-—1.0-
3 I I I I
-2.0-3
: For t>1.115 (sec) a=40 (N)
—3.0‘r....e.jr.-...-, ....... V. ....,e......,
0.0 0.4 0.8 1.2 1.6 2.0
TIME, t (Sec)
Figure 3.5a Response of the transverse vibration at x-2/2

for steady-state excitation test results for

various control gains for test case 4

62

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

so. ....... r ....... , ....... , ....... , ......
’2 I For t>0 «=40 (N)
z—\ : I
1” 401w 1 ’F I: r H M II I I .
CL' ' a I ‘3 I ” I 1
- 20-1 = i :1
23‘ III I I
0: .2 , ,
E 3 I M”
..20- a K” i
S 1 a ‘ I” l I I
a: ..44): 5.1I... AIJJ dd tab In .Iu II III III I I III
’2 3 1
. -50-‘ ‘
8 ‘ For t>1.115 (sec) «=40 (N)
—80‘ ...T. ................... -

 

0.0 ‘0141
TIME, t (Sec)

Figure 3.5b The required control forces, P(t) for test

case 4

 

 

 

 

 

A

E

E

I:

V

>-

CD

0:

DJ

2:

UJ
1 For t>0 «=40 (N) «

—O.2 ....... ,-...-.., ...... , ....... , .......

0.0 0.4 0.8 1.2 1.6 2.0

TIME, t (sec)

Figure 3.5c Effect of the control action on the internal

energy of the system for test case 4

63

Test Case 5: Initial Displacement of an Infinite Rider of Nodes.

In this test case the initial data is given by y(x,0) -
21.527x(x-£) m where x is in meters. This case excites an infinite
number of modes. The control action is set "on” for internal energy
E(t) 2 .3 8(0). Figure 3.6a shows the response for a: - 0, 10,
and 30 N. Since the control action is based on internal energy; The
amplitudes of a: - 10, 20 and 30 N dropped to .8m. for E(t) z .3E(0)

for each a. It is the equality starts to be satisfied at t - t1, one .
can see from Figure 3.6b that E(t1)-.3E(0) is satisfied after 3/4 of a

cycle for or - 30, and after 4 1/2 cycles for a: - 10 N. The internal
energy and the work done are shown in Fig.(3.6c) in which the energy
balance is satisfied for each at. Table 3.4 shows the effect of a: on

an, en and n in the first three cycles for each or.

64

I I I I I I 047' 1’ I I I I I I U I YT I 1’ U 1 I

r T i U I I I I

‘7 I

TI I’ I I ii I r I I IV I T I

  

 

 

du-dmmq—I-qdd-d-Imm

 

0000000000000000000
.0:
OOOOOOOOOO “HQ
I 0 00.0000. 0
...
.
sn:.
OOOOOOOOOOOOOOOO
00000000000000000
If
0 000000
000000000000
) 0000000000000
0000000
e
N ooooooooooooo
IIIIII 90".
0.. 0.... 00000000000000
O ooooooo
eeeeeeeee
‘fl'.’
\ 00000000000000
00.. I'll. O
. ..J
O
0.. 0.
Ole OOe ID.
as 000000000
8 000000
m ::::::
0000000000
..:w ................
0‘.
.0 00000000000000000
600000000000
ab. 0.00.
8.8. A:

.
0.......:...". . .

 

 

U I I I I I I I If I I

V’U'Vr'ri‘rr'll'II’TIT‘I’TUIIIII‘IIIITTUU

0.8 1.

0.6
TIME, t (Sec)

0.4-

0.2

 

 

D
3

dcldd-quuuufifidq—unJ

_
nu nu nu AN nu
04 1: AU «fl

AEEV 9Q? .mmzoammm

 

Response of the transverse vibration at x-£/2 for

Figure 3.6a

initial displacement of an infinite number of modes

for various control gains for test case 5 .

65

 

 

 

 

 

 

 

 

 

 

  

 

 

 

 

 

 

6C)1 ......... , ......... ‘ ......... , ......... ‘ .........
2 3 i
v 40': a: 10 (N) a=30 (N) E
8 3 n m 1
°-_ 20-: .
LLJ. a .
o g F '1 '7 :
“5 °‘2 .1 h :
_J —2o-2 -
z -40-§ -2
CD = 3
0 a
—50 ........ , ......... . ......... . ........ mfimm‘
0.0 0.2 0. 0.6 0.8 1.0
TIME, t (Sec)
Figure 3.6b The required control forces, P(t) for test
case 5
3.0 1*fi W I r Fr I 'Trrr IF ‘ l ' ' j
3 =30 (N) a=0 (N) i
E 2.0-3 . j
E 5* WD, a=30 (N)
i Z—
3: 1 .9 i
>' 1'01 .’ ND 10 (N)
4 a—
CD I t. ’
E 0.0-g a=10 (N)
—4.O1T ....... 1, ....... “j 111111111 1 ......... r ......... I
0.0 0.2 0.4 0.6 0.8 1.0
TIME, t (sec)
Figure 3.6c Internal energy and work done for test case 5

Table 3.4 Effect of the control gain.¢ on the amplitude ratio an,
the energy ratio enand the efficiency factor n for

initial displace-ant for an.infinite nuiber of nodes

for test case 5.

 

 

gain n tn yn an En en n I
a N . sec mm mm/mm N mm
10 1 .103 -1.741 .8705 1.343 0.86 25.8
2 .2135 -1.37 .7869 0.9272 0.831 48.77
3 .3110 -1.17 .854 0.6818 0.857 62.33
20 l .103 -l.4l8 .709 0.983 0.737 45.7
2 .2038 -0.993 .70 0.580 0.768 67.95
3 .311 -0.8427 .848 0.438 0.864 75.8
30 1 .103 -l.174 .587 0.759 0.647 58.1
2 .2038 '-0.84 .715 0.508 0.818 71.9

3 .311 -O.777 .925 0.427 0.917 76.4

 

67

Test Case 6: Initial Iqulse Excitation of an

Infinite Mr of Hades.

The impulse in this case excites an infinite number of modes.

This impulse may be simulated as an initial velocity of the form
8y(x,0)/at - l.094x(£-x) m/sec

where x in meter, 0 s x z 2 . In this case. because the numerical
differentiation process for estimating 8u(1,t)/8t causes numerical
instability when the energy level is less than.0026 E(O), the control
action is set "on" for internal energy E(t) z 0.0026E(0).
(Figures 3.7a, b and c) show the response of the middle of the beam,
the control force and the internal energy and the work done versus

time for a: - 0, 10 and 30 N respectively. It is clear from
1
figures 3.7 b,c that E(tl) - .0026E(0) is satisfied in less than 5;

cycles for a: - 30. Also the energy balance is satisfied for all t _>. 0

as shown in Figure 3.7c.

Table 3.5 shows the effect of control gain a: on the amplitude

ratio aj, the energy ratio e_1 and the efficiency factor n. It is

clear from table 3.5 that for certain at, the amplitude ratio

aj- yj / yj_2 and the energy ratio ej- JEJ /Ej-2 are equal up to the

first decimal point, and these ratios decrease with increase of or,
where j is the amplitude number. In this test case the control action

could reduce the intial impulse energy to .0026 of its initial value

1
in less than 5'; cycles for or - 30 N.

RESPONSE, y(I/2,t) (mm)

—‘ 03 LA
C) C) C)
nnlLliiiLanLLbinLJAIn111111

P
O

I
'o

I
N
o

-30

'01

68

 

 

rTrIrIITrlIIrTrIIIIIIIIrIIIITIrIITTIIrrrIIIrIrI—

a=0 (N)

a=30 (N)

e.~“..
.0...

so. ......
‘11"iuj '44 1 1J4 L 1 1

Q“......

3‘

 

I

€3.11:
. " .0 m g . fl .0
.LLJ.LLJALL4J.M&3QW£

 

 

  
 

 

:
34
w

.otqov'" ‘
..zr-‘.m .

V.

 

Figure 3.7a

Frrrrtri rrrrrrrrrrirrrfrltri trrrfirtrrrrFrI—Irrrtr'

0 0.2 0.4 0.6 0.8 1.0
TIME, t (Sec)

Response of the transverse vibration at x-l/Z for
initial impulse of an infinite number of modes

for various control gains for test case 6

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

50:- ........ , ......... r ......... , ......... , .......

2 i

v 40; a=1o (N) a=30 (N) 3

c : I 3

V i1 F F I‘ F I“ I1 I1 i

0: 201; // H F -;

8 SF 4 1 I 11.FJUUH.

o: 0-; I

a guJuUu I

_, -2o-E -

a iL-I J LI J _. ... ..I L L. 0 3

'2 —4o-i 4:

o a a

o z
—50 ism” ..“WWW”....,-.....-..,.-....-.

0.0 0.2 0.4 0.6 0.8 1.0

TIME, t (Sec)

Figure 3.7b The required control forces, P(t) for test

case 6

 

LL‘A

—L
C)
'11::

 

 

 

A
E
E

2

V

>- ' 1

c9

0:

UJ

2

DJ

9
o

 

 

 

L14 LA L1 141 lillJl Al

 

II‘rIIIIIlTIIIIII711IIIIjIIrI'Ir‘IIIIIrle‘rIIIII’T

0.4 0.6 0.8 1.0
TIME, t (sec)

I
90
o
P
[0

Figure 3.7c Internal energy and work done for test case 6

Table 3.5 Effect of the control gain a on the amplitude ratio
the energy ratio

initial impulse excitation of an infinite number of

70

e and the efficiency factor a for

.1

modes for testcase 6.

 

 

J

 

 

 

 

“h 5 ‘1 ’J ‘J '1 ‘1 " ‘
e 3 sec - -/-a l -
0 21.74 1.123 0

10 1 .023 1.631 1.0170 9.33
2 .0771 °1.4923 .0376 0.02119 .034 27.0

3 .129 1.333 .007 0.6624 .007 41.12

4 .101 -1.1927 .00 0.3364 .000 32.32
.233 1.0701 .0007 0.4320 .000 61.33

6 .203 -0.9746 .010 0.3490 .0073 60.91

7 .337 0.0637 .003 0.2019 .0071 73.02

20 1 .0219 1.376 0.942 16.27
2 .0771 -1.276 .733 0.6011 .731 33.02

3 .12911 1.023 .63 0.3906 .644 63.20

4 .101 -0.0172 .64 0.2336 .623 77.20

3 .2290 0.6606 .646 0.170 .639 04.09

6 .203 -0.341 .662 0.1003 .631 90.36

7 .3402 0.4301 .631 0.0701 .642 92.99

30 1 .021 1.319 0.062 23.34
2 .0771 -1.0003 .623 0.430 .624 61.07

3 .1291 0.770 .312 0.2271 .313 79.01

4 .101 -0.333 .300 0.120 .323 09.33

3 .233 0.4039 .319 0.0630 .326 94.4

6 .02002 -0.2912 .326 0.0327 .322 97.09

7 .343 0.211 .322 0.0173 .327 90.44

 

71

Test Case 7: Internally Unstable System.

This test case is considered to give more insight in order to
understand the control and to show that it can transform an internally
unstable system into an asymptotically stable one. The system is
internally distabilyzed by replacing a positive C in (3.10) by a
negative value, C - -.00445 Kg m/sec. An initial displacement,
y(x.0) - 1.5 sin(arx/l) mm, which is the first eigenfunction of the
beam is applied. Figure 3.8a shows the response of the middle of the
beam, y(1/2,t) versus time for 0: - 0, 10, 20 and 100 N. In
Figure 3.8a the curve for a - 0 (control is off) is the response of
the free unstable, self excited vibration. The frequency of this
response is 9.654 H2 which is the the first self excited frequency of

the beam.

The frequency of the response For ariO varies with time within
one cycle of oscillation. It oscillates about the frequency of the
uncontrolled (a: - 0) system, therefore every half cycle of the
response curve, there are two different zones: slow and fast.
Figure 3.8b is a plot of the response and the control force for

a-40 N. In Figure 3.8b t c:and t are the time of any point xel taken

1:
to reach the equilibrium position from the maximum amplitude of
oscillation and from the equilibrium position to the maximum amplitude
of oscillation respectively. It should be noticed from Figure 3.8b

that during tcand tt, P(t) is compression and tension respectively

which gives a physical interpretation of the control mechanism,
e.g. ,when the distance between the beam supports tends to get longer,

the control force is compression and vice versa. In this case the

72

 

IIrIrIIrTII

IIIIIrIrIrfifrIIIIII‘IIIIIIIIIIIIIIIII

dIdIu.I—uudquqddI-GIFI-ed-dude-q-Jheedl‘efi

0‘

 

 

 

 

I I
I0 I00
.... ---uu.
.................. u. - --
”3.000000090030030...“
”000000000000 III 000 0
I\\ | OHIV
De. OIIeOIItoflI I
in“! n.
I’OOOIOII‘IIOHV I. N
’ Il‘OIOOIIIOIIiIOIeOIOq (
v 00000000900000
. IIIIIIIII II“. ~ 00
efl ..... F\ _ O
eeooIIIe. IeeII I0“! , 0
I ‘flHIIOIOIIOOI 1
N v u. .. ..
I .4. “NOIIOeeIOeIII .—
.l\ ........ u.
nu... . . a
0 0 660003,”!-
-— Qfinf’tlile’
a 0F»H.\O\IO¢OIIII"
e 63
e e

IIrrrIrI—IIIrII‘IFrIrrIIIIIIIIrTrIrTTTIIIIIrIrI'III

 

 

3.5

I
4
.1

2.5
1.

e55 QQE .mmzoammm

 

 

0.8 1.0

0.6

0.4-
TIME, t (Sec)

0.2

0.0

Response of the transverse vibration at x-1/2

Figure 3.8a

for internally unstable system test results

for various control gains for test case 7

73

numerical values of tc and tt give the frequencies «ac-7.35 Hz
and «ac-11.36 Hz respectively, i.e. , during half of a cycle of the
transverse vibration, the frequency of oscillation varies from wcto

wt. This fluctuation shows that there are softening and stiffening

actions due to the parametric force p(x,t) generated by the control
action P(t) .

Figure 3.8c shows the required control force versus time for
a-10, 20 and 100 N. The internal energy E(t) and the work done by the
control force p(t) are plotted versus t for the same control gains a:
in Figure 3.8d. An efficiency factor n - (E(O)-E(t))/E(0) giving
the fraction of the energy dissipated due to the control action during

the first cycle of oscillation, t, has been considered in order to

judge the efficiency of the approach in the first cycle. Table 3.6
shows the effect of the control gain a: on the amplitude ratio aj , the
energy ratio eJ and the efficiency factor n. In this test case, the
control algorithm could successfully transform the initially
internally unstable system into an asymptotically stable system
without exciting any other modes which might be excited easily since

the system is internally unstable and for m-lOO N the vibration

monotonically decreases with time.

This chapter has shown the effectiveness of the control law in
controlling beam transverse vibration through investigation of
stability, transient motions and controlled response from external
disturbances. The closed-loop system which was derived in chapter 2
from the Liapunov function for the beam, was reduced to a

nonhomogeneous wave equation for u(x,t) subject to the non homogeneous

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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3 E
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0.00 0.05 0.10 0.15 0.20 0.25 0.30
TIME (Sec)
Figure 3.8b Curves of response and control force for a -40
N for test case 7
120......”“q. ...r...
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Figure 3.8c The required control forces, P(t) for test

case 7

(N) 3030;: ‘IOHINOO

ENERGY (N mm)

WORK DONE (N mm)

75

 

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.I

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Figure 3.8d Internal energy for test case 7

 

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1 1
I q
1 1
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1 1
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Figure 3.8e The work done for test case 7

76

 

 

Table 3.6 Effect of the control gain.¢ on the amplitude ratio aj,
the energy ratio ejand.the efficiency factor n for
fbr internally unstable system for test case 7.
i t E t
5‘ “ J ’1 “J J ‘1 ”
a N sec mm mm/mm N mm
10 .0527 ~l.4l477 .74872 8.47
.1039 1.31884 .8792 .651616 .892 20.3
.157538 -1.235227 .8731 .57022 .87 30.29
.2086977 1.13517 .861 .481039 .86 41.19
.259857 -1.06915 .8655 .429797 .868 47.46
.31345 0.98863 .871 .3701688 .877 54.75
.36461 -0.92952 .8694 .3276178 .873 59.95
20 .0527 -l.26749 .596791 27.04
.106379 1.041047 .694 .403597 .702 50.66
.157538 -0.873227 .689 .287831 .694 64.81
.211134 0.72176 .693 .199689 .703 75.59
.262293 -0.610207 .699 .142664 .704 82.56
.31588 0.50948 .706 .09699 .697 88.26
.367048 -0.42654 .699 .070878 .705 91.34

 

77

boundary conditions which could be solved analytically using finite
Fourier transform and a nonlinear fourth order parabolic equation in

y(x,t) which was approximated by finite difference method.

The simulation was runon a prime 750 computer at the Albert
H. Case Center for Computer-Aided Design at Michigan State
University. The been chosen for the test had dimensions and
material properties appropriate for the laboratory tests which will
be given in chapter 4 and the simulations indicated asymptotically
stable response. Test cases 5 and 6 for infinite number of modes
demonstrated most of the initial energy was removed in the early
oscillating cycles. Test cases 3 and 4 showed resonant amplitude

could be limited by this control.
The single mode test results demonstrated:

1) asymptotically closed-loop stable transient response for at < 40 N

and monotonically decreasing response for control gain a: - 108 N.
2) the closed-loop control reduced the resonant response amplitude.

For test results exciting an infinite number of modes the control

action successfully:

1) reduced the amplitude from 2 m to 0.8 mm for control gains 10, 20
and 30 N for energy based control;i.e. control was set "on" for
E(t) z .3E(0) for the initial displacement y(x,0) - 21.527 x(x - 1) mm

where x in m,

78

2) reduce the initial impulse energy due to an initial velocity given
by 8y(x,0) / a t - 1.094(1 - x) m/sec where x in m, to .0026 of its

initial value for control gain a: - 30 N.

In both transient single mode and infinite number of modes

the closed connection between the energy ratio en and the amplitude
ratio an for low values of control gain a: was clear. For low a: it was
clear that the amplitude ratio an and the energy ratio en were almost

constant and equal for the first three cycles and depend only on on.
This observation has not been previously reported because none has

used this control before.

WFACIIITIES, PROCEDURES ANDRBSULIS.

A prototype control system was constructed to evaluate the
performance of the active parametric vibration control system on a
simply supported beam (Figure 4.1). The beam chosen for the
experimental tests has dimensions that give reasonable natural
frequencies. This chapter discusses the experimental control
evaluation in three main sections. In section 4.1 the construction of
the simply supported beam test stand is presented. The actuator
mechanisms and sensors and their associated circuits are presented in
section 4.2. Experimental results are given in section 4.3. The
modelled beam is compared with the ideal beam with respect to the
modal frequencies and the modulus of elasticity in section 4.4.
Significant increases in stability of the test beam were measured
which demonstrated the feasibility of employing active parametric
vibration control. The experimental results presented here will be

compared with simulation results in chapter 5.
4.1 SiQIy-Supported Beam Test Stand

In this section, the construction of the beam test stand to
assess control performance is described. The test stand had the

following requirements:
79

80

1- External damping due to frictions in the joints of the moving parts

should be minimized .

2- The actuators and sensors should have minimum interaction with the

beam dynamics .

3- The effect of gravity on the transverse and longitudinal vibration

should be eliminated.

Figure 4.1 shows the prototype active vibration controller.

The dimensions and physical properties for the steel beam are:

a s
p - 8304 kg/m [.3lb/in ]
1 - .61 m [2ft]
A - 50.8mm width x 1.588mm thickness [Zin x .06251n]

To eliminate the friction in the hinge junctions of the beam
supports while achieving near-zero bending moment at the ends of the

beam, i.e. yxx (0,t) - yxx (Lt) - 0; the end conditions were

approximated using very thin steel shims of thickness .01 inch which
were soldered in slots at the ends of the beam and tightly fixed to

the supports.

To realize the axial movement of the simply-supported beam end,
while keeping minimum interaction between the beam dynamics and the

supports; the other end of the corresponding shim was fixed in a rigid

81

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82

cross shaped section of plexi-glass (to reduce inertia)vfluch.was
supported on three deep groove ball bearings (to allow only axial
motion with minimum friction). To ensure proper contact between the
bearings and the plexi-glass support, one of the bearings was made
adjustable through a screw which drives two swivelling brackets and
tightened when adjusted. A steel strip of dimension 1.5 x 3.5 x.06
in. fixed by screws at the other end of the plexi-glass support, which
could be attracted by the electromagnet to implement the active
control as shown in Figure 4.2. The maximum control force applied by
the magnet was up to 90 N for .01 in. gap. The supports of the beam
were fixed through brackets to a machined 6.5 x 6.5 x 35.5 in. right
angle which was fixed to a heavy cast iron test stand through heavy
duty C-clamps. To excite the beam, two identical electromagnets were
fixed to the frame through brackets and centered at 82.55 mm. This
pair of magnets could excite the first five modes of the beam
efficiently. The effect of gravity on the transverse and longitudinal
vibrations was eliminated by allowing both y(x,t) and u(x,t) to be in

the horizontal plane .

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84
4.2 Active Control Prototype

In this section the experimental procedure and the measured
quantities are presented. This section is divided into five
subsections. Subsection 4.2.1 describes the main instrumentation
circuits,and subsections 4.2.2a and 4.2.2b describe the end beam
acceleration and displacement measurements respectively. The
transverse displacement measurement is given in subsection 4.2.3. The
beam excitation and the control actuators are presented in subsections

4.2.4 and 4.2.5 respectively.
4.2.1 lain Instrumentation Circuits

To evaluate the performance of the control law derived in
chapter 2 from the Liapunov function for the beam, the following

circuits constructed:

a) The transverse vibration measurement circuit.
b) Beam excitation and the exciting force circuits.

c) Beam control actuator circuits to implement the control law.

The transverse vibration was measured using an inductive noncontacting
displacement probe. The beam excitation circuit consisted of a pir of
electromagnets driven by a custom built magnet amplifier driven by two
wave functions or by a random noise generator. The exciting force was
measured using a strain gauge bridge circuit. The control actuator
was an electromagnet on the end of the beam excited by a magnet derive
amplifier driven by custom built analog circuits. The analog circuits

consisted of an integrator if the end beam motion was observed by an

85

accelerometer or a differentiator if the end beam motion was observed
by a noncontacting displacement probe. The output signal from either
the integrator or the differentiator represented the end beam velocity
(8u(1,t) /at). This velocity was fed into an analog switch deriving
the power amplifier that supplied current to the control
electromagnet. Figure 4.3 shows the measurement circuits and control

flow diagram .

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87

4.2.2 End Beam Notion Measurement

The control law derived in chapter 2 from the stability
analysis using the direct method of Liapunov gave the relation
between beam end velocity and the control force. A control force
opposing the velocity for all t > 0 was shown to yield asymptotic
stability. Two different methods to accurately observe the beam and
velocity were necessary. It was found in the transient motion test
experiments that the observed signal level produced by the
accelerometer was too small at low frequencies to be distinguished
from the noise. At low frequencies, the displacement of the end of
the beam was measured by one of the inductive noncontacting

displacement probes to improve the signal-to-noise ratio.

4.2.2a Ind Beam Acceleration Neasure-ent.

The closed-loop control law based upon observing the and beam
velocity and applying the actuating force accordingly. Because there
was no available velocity transducer, the acceleration was measured
and integrated by an analog integrator. Integrating acceleration

is a smoothing process, which reduces high frequency noise.

The end beam acceleration was measured with a piezoelectric
accelerometer and amplifier by PCB Pieonronics, Inc. model 482A10.
The measured signal was fed into the analog integrator (Figure 4.4) .
The output velocity was amplified and fed into an analog switch ;
which controlled the power amplifier type 2712 by Bruel 6: Kj aer which

drove the control electromagnet.

88

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To find the bandwidth of the integrator, the transfer function
of the integrator was measured over 0-200 Hz using the HP dynamic
analyzer as shown in Figure 4.5a which showed that the integrator was
acceptable over 20 - 200 Hz. Figures 4.5 b,c show two test signals at
20 Hz and 200 Hz and their integrator outputs which verify the

frequency response result.

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an ' ' ' ‘ T ‘ ' ' T

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Figure 4.5a Integrator transfer function test results

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92

4-2.2b had le- Displace-ent Keane-eat.

In the transient tests, an efficient way to find the beam end
velocity by observing the displacement was required. The displacement
was measured using inductive noncontacting probe model KD 2400 by
Kaman Sciences and the observed signal was fed into the analog
differentiator showed in Figure 4.6. The resulting velocity signal
was fed into the analog switch as in the integrator circuit. To find
the bandwidth of the differentiator, the transfer function of the
differentiator was measured from 0 to 50 Hz using the HP dynamic
analyzer as shown in Figure 4.7a. This test showed that the
differentiator was acceptable over a frequency range of (S- 25 Hz) .
Figure 4.7b shows a test signal at 20 Hz and the corresponding

differentiator output .

93

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95

4-2. 3 Transverse bisplacement Heasurement.

To evaluate control power, the transverse displacement of the
beam was measured using an inductive noncontacting displacement probe
model 102400 by Kaman Sciences Corporation, located at 367.2 mm from
the stationary end of the beam. This position was chosen to avoid

nodes in the first five modes. The output signal was fed into the

signal amplifier then into DEC 151/23+ after removing the DC component
as shown in Figure 4.3. The sensor had a sensitivity of .23 mm/V with
a resolution of .25 V (appendix A1) yielding a measurement range of

0-2.5 mm and accuracy i 0.06 on.

4.2.4 Ben Excitation.

To achieve minimum interaction between the beam dynamics and
the beam excitation force, two identical electromagnets opposing each
other and located at 82.55 mm from the stationary end of the beam were
used. The two electromagnets were driven either by a function
generator (Wavetek Model 180) or random noise generator (HP 54410 A
and 5423 A). The exciting force was calibrated using strain gauge
(type EA-13-1253T-120 by Micro-Measurements group, INC.) the maximum

force available was 5 N and the strain gauges had a sensitivity of

1.5 N/V for a linearized magnet output range of 0-.75 N with
resolution of .08 N. The exciting force calibration procedure and the

strain gauge arrangement are given in appendix A2.

96

4.2.5 Control Force Measurement

The axial force produced by the control electromagnet was
calibrated using a full strain gauge bridge (type EA-13-12SBT-l20 by
M-M). The strain gauges had a sensitivity of 36.1 [IN for magnet

drive output up to 90 N with resolution of .75 N (appendix A2).

4-3 Experimental Test Results and Discussions.

The prototype control system presented in the previous sections
was constructed to obtain experimental data to evaluate the
performance of the control law derived in chapter 2 by using the
direct method of Liapunov for the beam. In this section the
experimental procedure and the results of three test cases
representing the stability due to transient and steady state motion
are presented and discussed. The significant increase in stability of
the test beam demonstrates the feasibility of employing active
parametric vibration control. The experimental results presented will

be compared with the simulation results in chapter 5.

In the following test cases the control force P(t) was set "on"
for negative end beam velocity and 'off" elsewhere (Figure 3.2) . The
control gain here is the amplitude of the control force or. The

locations of the sensors and actuators are shown in (Figure 4.8).

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Test Case 1: Transient lotion.

This case is similar to an initial value problem in which an
initial displacement, velocity or both were given to the beam. To
realize this, the beam was excited at the desired mode until reached a
steady state, then the excitation was turned "off" and the control at
a test gain or was turned ”on" until the vibration decayed. Data was
recorded from a time just before the excitation was turned "off”.
Figures 4.9-4.12 show the real time record of the transient motion
for control gain a: - 0, 10.63, 13.98 and 58.67 after being converted
into physical values using the calibration formulas given in the
appendices. Figure 4.9 shows the results of the transient response
for the uncontrolled beam. In figures 4.9a, 4.10b, 4.lla and 4.12a
the response curves are marked “2" while the excitation signals are
marked "1". The excitation signals where plotted with the response
curves just to give an idea when the excitation was turned'off". In
Figures 4.10b, 4.11b, and 4.12b the amplitudes of the control force
decrease with the decay of the response, that is because the control
law uses the end.beam velocity (which is more or less proportional to
the response) of the observed signal to apply the control action P(t).
Table 4.1 summarizes the experimental test results of the transient
case for the control action on the amplitude of the cycles number
1,5,10,15,20 and 48. By comparing figures 4.9 with 4.12a and from
table 4.1, the amplitudes were reduced with the increase of the
control gain 0:; e.g. for n - 48, the amplitude reduced from .40 mm to

.16 mm, for control gain a - 58.67 N.

99

 

 

 

 

 

 

 

 

 

 

RESPONSE (mm)

 

 

1.50
.1,50 . . ‘ ' ' ' ‘ l I 545°
0'00 TIME (59°) .

x - 367.2 mm for the uncontrolled beam

100

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Figure 4.10 Transient vibration, first-mode test results
for control gain a - 10.63 N.
a)the response at x - 367.2 mm
b)the control force P(t)

101

 

 

 

 

 

RESPONSE (mm)

 

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TIME (Sec)
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Figure 4.11 Transient vibration, first-mode test results
for control gain a - 13.98 N.
a)the response at x - 367.2 mm
b)the control force P(t)

102

1.50

RESPONSE (mm)

 

-1.50
0.00 5.00

TIME (Sec)

 

 

 

 

 

 

 

 

 

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Figure 4.12 Transient vibration, first-mode test results
for control gain at a - 58.67 N.
a)the response at x - 367.2 mm

b)the control force P(t)

103

Table 4.1 Effect of the control gain.on the transient motion

amplitudes test results.

 

Amplitude (mm) at n -

 

 

gain a

N 1 5 10 15 20 48
00.00 1.10 1.0 0.88 0.78 0.71 0.40
10.63 1.10 0.98 0.84 0.74 0.65 0.27
13.98 1.10 0.97 0.83 0.73 0.64 0.24
58.67 1.10 0.90 0.79 0.67 0.59 0.15

 

104

Test Case 2 Steady State Resonant Response

The most severe working conditions for a mechanical structure
is when it is excited with a harmonic force having one or more
frequencies equal to one or more of its eigenvalues , and it is the
role of the active control to limit the resonant amplitudes. In this
case the results of the effect of the control action on the steady-
state resonant amplitudes are presented and discussed. Subsections
(a) and (b) discuss the effect of the control action on the resonant
amplitude due to single mode and two-mode excitations respectively.
In both subsections the obtained results showed the efficiency of the

control law in limiting the steady state resonant amplitudes.
a) One-lode Excitation:

to obtain a steady-state resonant response, the wave generator
was adjusted to provide a magnitude and frequency (which corresponded
to the first or the second natural frequency of the beam) of the
sinewave signal to drive the excitingpower amplifier, after a while
the beam responded with the steady-state resonant amplitude for the
uncontrolled beam. To obtain the steady-state resonant amplitude for
the controlled beam; the previous step was done first while the
control was "off", then control action at certain gain a: was set "on”
until the steady-state amplitude for the controlled beam was obtained.
This procedure was repeated up to five times and the average of the
steady-state resonant amplitude was obtained. Table 4.2 shows the
experimental test results of the effect of the control gain on the

steady-state first-mode amplitude.

105

The control action could reduce the resonant amplitude from
1.26 n- to 1.0 m for a! - 12.55 N and to .73 mm for a: - 28.2 N. It
was expected to observe some increase in the resonant frequency caused
by the stiffening effect of the control force, but this did not

happen .

Figures 4.13a, b and c show the steady-state second-mode test
results of a complete record of the exciting force, the response and
the control force for control gain a: - 35.46 N for the steady-state
second mode. These Figures show the relationship between the
exciting force, the response and the control force at any instance
during the time record. The control force is applied during the
period while the beam tends to get shorter, and zero while it tends to
get longer which demonstrates the control action mechanism, which was
explained before in chapter 3, (test case 7). Table 4.3 shows the
experimental test results of the effect of the control gain on the
steady state second—mode response. It is clear from table 4.3 that
the steady state amplitude is reduced from 1.42 mm for a: - 0 to .72 m
for or - 60 N. A subset of the measured displacement, exciting force
and control force histories for the steady-state second mode test

results for a: - 35.46 N will be shown in appendix 8.
b) Two-lode Excitation:

As mentioned in the beginning of test case 2, that the most
severe working conditions for a mechanical structure is when it works
under the effect of an exciting force having a frequency equal to one
of the eigenvalues of the structure. A more dangerous situation is

when the exciting force has more than one frequencies that are equal

106

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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Figure 4.13 Steady state second-mode test results for a - 35.46 N.
a) the exciting force.
b) the response at x - 367.2 mm.

c) the control force

107

Table 4.2 Effect of the control gain.on the steady state first mode

response for test case 2.

 

 

gain a amplitude frequency exciting amplitude
N mm Hz N
00.00 1.26 9.412 0.22
12.55 1.00 9.412 0.22
28.20 0.73 9.412 0.22

 

Table 4.3 Effect of'the control gain.on the steady state second.mnde

response for test case 2.

 

 

gain, amplitude frequency exciting amplitude
«N mm Hz N
00.00 1.42 37.65 1.95
35.46 1.04 37.65 1.95

60.0 0.72 37.65 1.95

 

108

to the eigenvalues of the structure. The two-mode excitation case is
considered to demonstrate the efficiency of the control law to limit
resonant amplitudes resulting from two-mode excitation. The exciting
force was due to two sinsoidal waves having frequencies of the first
and second modes and were added together then fed to the magnet drive
power amplifier. The time domain data for various control gains will
be given in appendix C. Since the time domain data due to mixed modes
is difficult to demonstrate the efficiency of the control law , the
FFT of the steady-state response for various control gains are
plotted in Figure 4.14. Table 4.4 summarizes the two—mode excitationL
test results for various control gains. In this test case, the
amplitude of the first-mode was reduced from .95 mm to .55 mm and that
of the second-mode was reduced from .8 mm to .65 mm for a: - 50 N,
which demonstrates the effectiveness of the control action in limiting

the resonant amplitudes resulting from the two-mode excitation.

109

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110

Table 4.4 Effect of the control gain.on the steady state two-mode

response for test case 2.

 

 

 

 

 

response excitation
gain «
amplitude frequency amplitude frequncy
N mm Hz N Hz
mode 1 0.95 09.39 0.288 09.393
00.0
mode 2 0.80 37.57 0.537 37.57
mode 1 0.68 09.39 0.288 09.3
41.71
mode 2 0.73 37.57 0.537 37.57
mode 1 0.55 09.39 0.288 09.393
50.0

mode 2 0.65 37.57 0.537 37.57

 

111

Test Case 3 Random Noise Excitation.

The behavior of the beam under one and two-mode excitation were
presented in test case 2, in which the excitation were pure sinewaves.
The more practical case is when the beam is exposed to a force that
carries all the possible frequencies(including the eigenvalues of the
beam) within a bandwidth. The resulting response in this case will
cover all the possible resonant amplitudes within that bandwidth.

The results presented in this test case were for a random excitation
with 200 Hz bandwidth. Since the excitation was random, the given
results were the average of 100 records. Figures 4.15s and 4.15b show
typical test results done by the HP structural analyzer for a: - 0 and
13 N. A replot of the test results done by the HP analyzer is given
in figure 4.16 for a - 0, 13, 48.4 and 88.5 N. Table 4.6 summarizes
the most important results in this test case. It can be shown by the
aid of figure 4.16 and table 4.6 that when the gain increased the
amplitude of the first mode was reduced from 8.25 V/V to 4.87 V/V for
a: - 88.52 while the amplitude of fourth mode was reduced from .66 V/V
to .56 for the same gain. Also the table shows that the resonant
frequencies of the controlled beam were increased slightly from those
of the uncontrolled beam; e.g. the first mode frequency was 9.38 for o:
- 0 was increased to 10.156 for a; - 13 N. This increase in frequency
was expected since the control action was applied in one direction

i.e. tension force only.

112

x. 9. me 7. I. 2451
mus M. 1m

0.-

 

‘ (a)

 

 

 

 

t. I T fl I fl I I I I

 

 

 

 

 

I.‘ "I a.
fill!” 700.73!
m 'b 1-
0.-..
IIAB " (b)
I.. vi *r fi 4L I A I
0.. la a.

Figure 4.15 Typical test results of the transfer function
of the beam. The displacement sensor located
at x - 367.2 mm and the exciting force applied
at x - 82.55 mm.

a) the transfer function for a -.0
b) the transfer function for a - 13 N

113

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114

Table 4.5 Effect of the control gain.on the transfer function at the
first four modal frequencies for the random noise excitation

test results.

 

transfer function, frequency at mode no.

 

 

 

 

 

gain a
N l 2 3 4
freq. Hz 9.375 37.50 84.375 148.48
00.0
TF. V/V 8.2451 2.0272 .9467 .6598
freq. Hz 10.156 37.50 84.375 148.48
13.007
TF V/V 6.7651 2.0272 .9466 .6597
freq. Hz 10.156 38.281 85.156 149.22
48.3836
TF V/V 6.0847 1.9039 .7974 .5857
freq. Hz 10.156 38.379 85.094 149.23
88.527

TF V/V 4.8709 1.3171 .7299 .5612

 

115

4.4 qurison of. the Uncontrolled Ideal Beam With the Hodelled
Be- Iith Respect To Natural Frequencies and Young's Modulus

To see how well the modelled beam agrees with the ideal beam,

the experimental modal frequencies f1 and the Young's modulus E of the
modelled beam were compared with those of the ideal beam. The ideal
natural frequencies f1 of the uncontrolled beam were calculated from

the relation:

2
in L1
I1 - 2 12 I p A Hz (4.1)

2 .
where E - 210 GN/m is Young's modulus of the ideal beam. Also
Young's modulus E of the modelled beam was evaluated at the first four

modes using the experimental modal frequencies f1 in the relation:

“($1114 {91

The difference between the ideal and experimental natural frequencies
and Yetmg's modulus are less than 5% and 10% respectively (table 4.6)

which shows good agreement between the ideal and the modelled beam.

In this chapter; the active control prototype, the test set up
and the experimental test results were given and discussed. Although
the control action actedin tension only, the significant increase in

stability of the test beam demonstrated the feasibility of employing

116

Table 4. 6 Comparison of the uncontrolled natural frequencies
and Young's modulus of the ideal beam with those of
the ndelled beam.

 

mode no., i

 

 

 

 

 

 

1 2 3 4
E1 Hz 9.73 38.93 87.58 155.71
f1 Hz 9.375 37.50 84.375 148.48
8 err 3.65 3.67 3.66 4.64
E GN/mz 195.00 194.9 194.9 191.0
% error 7.14 7.19 7.19 9.05

 

117

active parametric vibration to control the motion due to initial data
and dynamic external disturbances. The comparison of the uncontrolled
modal natural frequencies and the Young's modulus for the modelled
beam with those of the ideal beam showed good agreement. In chapter 5
comparison between the simulation test results and the experimental

test results will be presented and discussed.

“PARIS“ 0? mm WITH snmunos RESULTS

The simulation results showed the effectiveness of the approach
in controlling the beam transverse vibration with respect to the
stability, the transient motions and the dynamic motions due to
external disturbances. Unfortunately, there are no published study on
the active parametric control beam transverse vibration; therefore it
was important to verify the effectiveness of the approach
experimentally. The experimental test results showed significant
increase in the stability and demonstrated the feasibility of
employing active parametric vibration to control the transverse motion
of the beam. In this chapter some considerations needed for the
comparison between the experimental and simulation results will be
presented, then two comparison test cases representing transient and
steady-state motions will be presented and discussed. The comparison
results showed good agreement between the experimental and numerical

test results.

5 . 1 Cowarison Considerations .

The experimental transient and steady-state cases given in
chapter 4 were used in the comparison with the numerical simulation.
In each comparison test case, the simulation was adjusted to agree
with the experimental test conditions. Therefore the following

factors were taken into account when adjusting the simulation:
118

119

l) The control forces in the experimental tests were applied in one
direction; i.e. tension forces only; as was mentioned in chapter 4.

2) Although the control forces in the experimental tests were a bang-
bang control; i.e. "on” and ”off"; they were proportional control;
i.e. , the control amplitudes decreased with the decrease of the

horizontal end velocity of the beam.

3) The damping of the uncontrolled beam which effected the response of
the beam. The damping sources were due to the structural damping,
friction in the bearings and the air resistance to the transverse

vibration .

Factors 1 and 2 reduced the efficiency of the experimental
control, and the simulation was modified to account for these
deficiencies. For factor 1 the simulation was modified to allow only

tension control forces. For factor 2 the simulation was modified by

choosing the modification control factor 6 (Figure 2.2) in which the
control forces of both the simulation and the experiment were close.
For factor 1 the following method was used to model the damping of the

uncontrolled beam .

5.1.1 inng nodal for the Uncontrolled Beam.

The damping model chosen for the simulation was the same one
given in chapter 3, equation (3.10). The reason of choosing that
damping model is that the amplitudes of normal modes of vibration are
attenuated at rates which are proportional to the oscillation

frequencies [39]. Equation (3.10) includes the damping coefficient C

120

which must be estimated. The procedure for the evaluation of C for
the test beam is to excite the test stand beam at its first natural
frequency until steady-state is reached. The excitation was then
turned 'off' and the displacement at x - 367.2 mm recorded and
converted using the calibration formula (A.l). A simulation using
equation (3.10) was given an initial displacement y(x,0) - 1.161 sin

in mm and the response (y(x,t)) for various trial values of C was

obtained. The C value which gave the closest simulation to the
experimental response was found to be .01558 Kg m/sec. This behavior
will be shown when comparing the experimental with results simulated
results for the transient case for no control force (or - 0) in

table 5. 2.

5.1.2 Open-loop Nodal Frequencies.

The first four open-loop modal frequencies of the test beam
were obtained using the HP dynamic analyzer. A random noise of
bandwidth of 200 Hz was used to excite the uncontrolled beam and the
transfer function obtained. The corresponding simulated results were

obtained by giving equation (3.10) initial displacement:
y(x,0) - sin i3 m

with mode number i - 1,2,3 and 4. The corresponding simulation
natural frequencies were obtained from these responses. Table 5.1
shows the first four open-loop natural frequencies of the experimental

and simulated results of the beam. The maximum error is 4.05 8.

121

Tihle 5.1 Open-loop natural frequencies of the experimental
and simulated results of the beam.

 

Frequency Hz

 

 

Mode No. Error %
Exp. Num.

1 9.375 9.6543 2.97

2 37.5 39.02 4.05

3 84.375 86.587 2.62

4 148.48 145.84 1.78

 

122

5.2 Coqarison Cases.

The above factors 1-3 accounted for through adjusting the
simulation conditions to agree with the experimental test conditions.

Two comparison cases will be presented and discussed.
Comparison Case 1: Tramient lotion

The details of the experimental test procedures and response
time history for the transient motion for the first mode for various
control gains were presented and discussed in chapter 4. A damping

coefficient C - .01558 Kg m/sec and an initial displacement y(x,0) -
1.161 sin f3 m were used in the simulation. The comparison between

the experimental and the corresponding numerical amplitudes, n for
various control gains and control modification factor, 8 will be given
in this subsection. The amplitude of control forces in the
experimental test results which were shown in Figure 4.10b, 4.11b and
4.12b decreased with the decrease of the amplitudes of the response,
therefore the simulation was modified to account for this effect by
changing 6. Table 5.2 shows the experimental and numerical results

for various control gains. The simulated values in table 5.2 were

.8
calculated for e - 2.4 x 10 m/sec. The difference between the

numerical and experimental test results is less than 12 % which shows
good agreement between the experimental and numerical results for

transient motion of the first beam mode.

Table 5.2 Experimental and.simu1ation results for transient

notion of the beam for e - 2.4x1o“3 m/sec

123

 

Amplitude (mm) at n -

 

 

 

 

 

gain a

N l 5 10 15 20 48
Exp. 00.00 1.10 1.0 0.88 0.78 0.71 .40
Num. 00.00 1.10 1.0 0.87 0.76 0.67 .38
a error 0.0 0.0 1.14 2.56 5.63 5.00
Exp. 10.63 1.10 0.98 0.84 0.74 0.65 .27
Num. 10.63 1.10 0.98 0.84 0.72 0.63 .26
8 error 0.0 0.62 0.00 2.70 3.07 3.70
Exp. 13.98 1.10 0.97 0.83 0.73 0.64 .24
Num. 13.98 1.10 0.98 0.83 0.71 0.62 .231
% Error 0.0 1.03 0.00 2.73 3.13 3.80
Exp. 58.67 1.10 0.90 0.79 0.67 0.59 .15
Num. 58.67 1.10 0.91 0.74 0.61 0.52 .132
% Error 0.0 1.11 6.33 8.95 11.86 12.00

 

Coqarison Case 2: Steady-State Notion.

Comparison of steady-state experimental and numerical first-
mode test results required a representative excitation force
distribution. The exciting force was simulated as a trapezoid
centered at x - 82.55 mm and having an area equal to the exciting

force amplitude, Fe' The base length of the trapezoid was equal to

the magnet length 88.9 mm (3.5 in) and the top length was equal to

76.2 mm. (3 in), therefore the height of the trapezoid w was:
_ 4.3.1992— _ _
w (88.9 + 76.2) Fe 2.698 N/m for Fe .2227 N. (table 4.2)

It was found that for low control gains the experiment and
-s
simulation control forces were close for 6 - 1.2x10 and for high

-6
control gains they were close for e - 3x10 . Figure 5.1 shows the
effect of the control gain on the steady-state amplitude simulation

results for damping coefficient C - .0155 Kg m/sec and for

6 - 1.2x10-3, 3x10J, 3.x10'5, and 3.x10'8. Also plotted in Figure
5.1 the experimental results for control various gains. Table 5.3
shows the experimental and numerical results for selected control
gains and e. The difference between the numerical and experimental .
test results is less than 11 % which shows good agreement between the

experimental and numerical results for the steady-state motion.

125

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126

Table 5.3 Experimental and analytical amplitude test results for the
steady-state motion of the beam.

 

 

 

Gain, N 0.0 12.56 28.21 44.48 c m/sec
Exp.(mm) 1.26 1.00 0.73 0.57
-3
Num.(mm) 1.24 1.03 0.85 0.753 1.2x10
-4
1.24 0.97 0.67 0.61 3.0x10
_6
1.24 0.73 0.56 0.483 3.0x10

-8
1.24 0.20 0.09 0.0001 3.0x10

 

127

The results for the transverse vibration of a beam under the
action of the active parametric control have not been previously
reported in the literature. In this chapter some considerations for
the comparison between the experimental and simulation were presented.
These considerations included the damping model and the nature of the
experimental control forces. Based on these considerations two
comparison.cases representing transient and steady-state motions were
presented and discussed. The simulation results agreed with the
experimental results with error up to 12-11 t for the transient and

the steady-state response.

SM AND WSIONS

The active parametric control theory was presented and applied
to control the transverse vibration of a modified, nonlinear, dynamic,
simply-supported Bernoully-Euler beam. The mathematical formulation of
the open-loop system equations of motion was derived using the extended
Hamilton's principle. More importantly, the functional found after
application of Hamilton's principle is a valid Liapunov functional.
Closed-loop stability was then investigated using the direct method of

Liapunov and the control algorithm for asymptotic stability was found.

The control law was tested analytically and experimentally. The
closed-loop system model derived from Hamilton's principle was reduced
to a nonhomogeneous wave equation for the longitudinal vibration u(x,t)
subject to nonhomogeneous boundary conditions and a parabolic equation
for the transverse vibration y(x,t) . The wave equation was solved
analytically using the finite Fourier transform. The nonlinear fourth
order parabolic equation for the transverse vibration y(x,t) was solved

approximately using the finite difference method.

A prototype control system was designed and constructed to
demonstrate and verify the approach and to evaluate its performance.
Both the simulation and the prototype control system were tested and
compared to evaluate stability of transient vibration and resonant

amplitude due to external disturbances. The methods used for measuring

128

129

the motion of the end of the beam, showed that an inductive, non-
contacting, proximity probe has satisfactory noise immunity for
observing displacement at frequencies up to 50 Hz and the accelerometer
is less noise sensitive at higher frequencies. Comparison of the
simulation with experiment results showed good agreement with errors
less than 12 8 and 11 t in transient and steady state tests

respectively .

The active parametric control approach was found to be an
efficient method to reduce vibration due to external disturbances. The
control algorithm is easy-to-implement. No truncation is required in
the control algorithm. It enables a single force to control all modes
of the been based on observation of one velocity and does not suffer

from the spillover problems.

Further work may be directed toward the design and
implementation of a double acting force actuator to increase the
efficiency of the approach, the digital realization of controller
feedback, the combination of observing both the displacement using
proximity probe and the acceleration of the end of the beam and the

analytical solution of the closed-loop system.

This work is the first time the direct method of Liapunov has
been used to derive a parametric active control law. This easy-to-
implement, single-input single-output control law was tested
analytically and experimentally and stabilized the beam in transient
and steady-state tests without suffering from spillover induced

instability.

APPENDICES

130

APPENDIX A
Displacement And Forces Measurements And Calibrations
APPENDIX A1
Displacement Sensor Calibration

The displacement sensor used was inductive, non-contacting
proximeter model Kd2400 manufactured by Kaman Sciences corporation.
One sensor consisting of a detector and detector driver was used in
tests. The sensor was fixed to the frame through a long threaded sleeve
which was tightened to the frame. The position of the sensor was chosen
to detect the first five modes. The sensor was calibrated in its mount
on the prototype controller test stand. At varying distances from the
stationary detector, the gap between the detector head and the test beam
was measured with feeler gages and the sensor output voltage measured.
These calibration measurements were then plotted and a least squared
error fit to a first order polynomial obtained over the measurement
range used during the prototype controller tests (Figure A1). The

deduced relationship between the displacement yd in mm and the probe
output xp in V is given by:

yd - 0 .4233 + 0.233 xp mm ‘ (A.l)

This polynomial fit was subsequently used in the test data conversions.

131

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Figure A1

MIX A2
Forces Heasurements and Calibrations
A2-l strain Gauge Orientation and Configuration:

The following circuits were used in measuring the axial force and the

exciting force.
1) Control Force measurement:

The axial control force applied by the control magnet was
measured by a four-arm strain gauge bridge which compensates for
temperature and bending effects. Figure A2 shows the strain gauge

orientation and the bridge configuration.
2) Exciting force moment:

The exciting force was applied by using two identical magnets one
of them pulls the beam for the positive part of the exciting signal and
the other for the negative part. The exciting force was measured by a
four-arm strain gauge bridge which compensates for temperature, axial
and torsional effects. Figure A3 shows the strain gauge orientation and

the bridge circuit .

133

134

 

(a)

 

 

 

 

 

fly rat-7.2] 6..

Figure A2 Four-arm strain gauge bridge for measuring the
control force.
a) bridge circuit.
b) strain gauge orientation

135

 

(a)

 

 

 

 

 

(b)

 

\
I
\
K
\
E
I

 

Figure A3 Four-arm strain gauge bridge for measuring the
exciting force.
a) bridge circuit.

b) strain gauge orientation.

l 36
A2-2 Force Calibration:

In this section; the calibration procedures are presented and
the calibration constants are deduced for both the exciting and the

control forces.
A2-2. 1 Control force calibration:

The beam was taken out and clamped firmly by a machine vise.
Weights were hung by a wire through holes at the end of the beam and the
corresponding strain gauge output signals were recorded. Figure A4
shows a typical results of that calibration. Then the beam was mounted
back on the test support and the control magnet was excited by the power
amplifier. Both signals from the strain gauge and the power amplifier
outputs were recorded and plotted using a least square approximation as
shown in Figure A5. The relationship between the strain gauge output in

volts; ya and the control force magnet drive power amplifier output; xc

in volts is given by ( using the least square approximation):
2
ys - .0061xc + 1.5056xc V (A.2)

and the relationship between the weight which were applied axially; yw

in pounds and the strain gauge output is ( using the least square
approximation):

yw - 3.413 ys LB (Ao3)

137

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from (A.2) and (A.3) the control force, yW can be calculated by knowing

the control magnet drive signal; xc:

2 s
y - 3.413 (1.2 x - 0.3241 x + 0.5119 x ) LB
w c c c

2 s
- 18.218 x - 4.92 x +7.77l x N (A.4)
c c c

A2-2.2 Exciting Force Calibration:

First the outer bracket which supports the outer exciting magnet
was taken out and the whole set up was turned 90 degree. Dead weights
were mounted at the point of application of the exciting force and the
corresponding strain gauge output signals were recorded. A typical plot
of the calibration results is shown in Figure A6. The set up was put
back again and one of the magnets was excited by the exciting power
amplifier. Both signals from the strain gauge and the power amplifier
outputs were recorded and plotted using the least square approximation
as shown in Figure A7. The relationship between the strain gauge output

in volts; ys and the exciting magnetic drive power amplifier output; xe
in volts is given by:

2
y - 0.02998x + 0.22966x V (A.5)
s e e

The relationship between the weights; yw in pounds and the strain gauge

output; ys is:

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e

0
0.

(8‘1) 1119130

yw - 2.97975ys LB (A.6)
substituting from (A.5) into (A.6) yields

2
yw - 2.97975(o.02988ze + 0.22966xe ) LB

2
- 0.397xe + 3.044xe N (A.7)

By knowing the exciting magnet drive output signal; xe the corresponding

exciting force can be calculated using equation (A.7).

APPHIDIXB

Real Time Record For The Steady-State

Second-lode Test Results For a! - 35.46 N

In chapter 4, the prototype control system was presented to
demonestrate and verify the approach and to evaluate its performance.
The basic measured quantities were the transverse vibration y(x,t) , the

control force and the exciting force. The measured quantities were fed

into DEC 1.81 /23+ for the data acquisition. A subset of the measured
quantities for the steady-state second-mode test results for control

gain - 35.46 N are given in table 8.1 as an example.

143

144

Table 8.1 A subset of the measured displacement, exciting
force and control force histories for the

steady state second mode test result for a -

35.46 N
Time Exciting Force bisplsceeent Control Force
0.. ' V V

0.0000 g 0.005 .2.395 0.049
0.1250E-02. 0.459 _2 121 0.044
OeZSOOE-OZ. 0027‘ -1.667 0.044
0.37502-02, 0.059 _1'051 0.03,
0.50008-02. -0.156 -0:323 0.034
0.10002-01, -0.730 2 204 -1.35,
0.12505-01, ”0.579 2.43‘ .0.073
0.16255'01, '0.181 1.173 0.039
0.18755-01, 0.259 -o.415 -1_354
0.20003-01 ' 0e445 _1 .09: -1.354
0 .2125E‘01 . 0 e 596 -1 .743 -1 .344
0.22502-01, 0.694 -2.234 -1.344
0.23752-01, 0.720 -2.345 -1.359
0.2625E-01. 0.616 -2.531 0,049
0 .2730E-01 . 0 e 469 -2.312 0 .04‘
0.28752-01, 0.288 -1.85, 0,944
0.30002-01. 0.070 _1.251 9,039
0 .3125E-01 . “'0 .142 -o .523 o .03,
0.32502-01. -0.352 0.23. -o.132
0.35002-01, -o.seo 1.61, -1,354
0 e 3625:”‘01 . -0 e 733 2 . 082 -1 . 364
0 e 375°E'01 . -0 .743 2.336 -1 .251
0 e387SE-01 . -0 e 68‘ 2e366 -0 .010
0.40002-01, -o.567 2_185 0,044
0.41258-01, -0.401 1.774 0,039
0.42506-01. -0-196 1,133 0.039
0.43755‘01 . 04029 0.469 -o.239
0.4625E‘01, 0.435 -1.022 -1.354
0 e 47508'01 . 0 e 587 -1 .637 -1.. 349
0 e 4875E-01 . 0 e 689 -2.13‘ -1 .349
0.50002-01, 0.733 -20434 -1,359
0.51252-01, 0.709 -2.545 0.054
0.52305-01. 0.621 -2.453 0.044
0.53752-01, 0.484 -2.135 0.044
0.55002-01, 0.298 -..725 0.039
0.56252-01, 0.093 -1.105 0.039
0.57505-01. '0 .127 ”0.367 0.039
0.58752-01. -0-337 0,425 0.029
0 e 6125E-01 . -0 s 650 1 . 823 -1 .373
0 e 62505-01 . -0 e 733 2.297 -1 . 369
0 e 6375:“01 . -0 e 7‘3 2. 556 ...o . 533
0.6500E-01, ”0.689 2.60: 0.044
0.6625E-01. -0.577 2.43, 0.039
0.6875E-01. -o.215 1.422 0.029
0.70002-01, 0.015 0.777 0.029
0.7125E-01, 0.230 0.020 -i.163
0.72505-01. 0.420 -o,719 -l.369
0.73755-01, 0.577 -i.369 -1.359
0.76255-01, 0.733 -2,195 -1.202
0.77502-01, 0.714 -2.326 0.044
0 . 7875E-01 . 0 e 530 -2 .263 O . 039
0.8000E-01, 0.494 -2.004 0.039

Table 8.1 (continued)

Tile lfi¢1ttfll 39:3. Diepleceoont

0.0375E“01.
0.0625E“01.
0.0075E“01,
0.90006“01.
0.93752-01.
0.95006“01.
0.1000
0.1012
0.1025
0.1037
0.1050
0.1062
0.1075
0.1007
0.1100
0.1112
0.1125
0.1137
0.1150
0.1162
0 o 1175
0.1107
0.1200
0.1212
0.1225
0.1237
0.1250
0.1262
0.1275
0.1207
0.1300
0.1312
0.1325
0.1337
0.1350
0.1362
0.1375
0.1307
0.1400
0.1412
0.1425
0.1437
0.1450
0.1462
0.1475
0.1407
0.1500
0.1512
0.1525
0.1537
0.1550
0.1562
0.1575
0.1507
0.1600
0.1612_

V

0.100
“0.112
“0.323
“0.645
“0.699
“0.507
“0.425
“0.225

0.215

0.411

0.572

0.679

0.720

0.714

0.635

0.503

0.327

0.122
“0.090
“0.313
“0.494
“0.723
“0.704
“0.596
“0.440
“0.024

0.200

0.396

0.557

0.674

0,720

0.710

0.645

0.510

0.342

0.137
“0.070
“0.293
“0.740
“0.709

0.106

0.301

0.547

0.665

0.720

0.710

0.650

0.520

0.357

0.152

145

V

“0.230
0.547
1.205
1.921
2.634
2.603
2.512
2.141
1.564
0.050
0.003

“0.679

“1.359

-1 o 887
0.310
1.051
1.601
2.155
2.424
2.403
2.312
1.965
1.390
0.694

“0.060

“0.026

“1.500

“2.033

“2.390

“2.556

“2.527

“1.329

“0.630
0.132
0.000
1.544
2.053
2.366
2.463
2.346
2.019
1.406
0.011
0.060

“0.674

“1.344

“2.395

“2.100

“1.704

-1 e212

Control Force

V

0.034
0.034
0.029
“0.665
0.039
0.034
0.029
0 o 029
0.024
“1.153
“1.364
“1.364
0.044
0.039
0.034
0.034
0.034
0.029
0.024
“0.371
“1.256
“0.415
0.034
0.029
0.029
0.024
0.024
“1.129
0.039
0.034
0.029
0.029
0.029
0.024
0.024
“0.142
“1.150
“1.150
“0.420
0.029
0.024
0.024
0.024
0.020
“1.300
“1.370
“1.373
0.034
0.029
0.029
0.029
0.024

Table 3.1 (continued)

In.
.—

0.1662
0.1675
0.1607
0.1700
0.1712
0.1725
0.1737
0.1750
0.1762
0.1775
0.1707
0.1000
0.1012
0.1025
0.1037
0.1050
0.1062
0.1075
0.1007
0.1900
0.1912
0.1925
0.1937
0.1950
0.1962
0.1975
0.1907
0.2000
0.2012
0.2025
0.2037
0.2050
0.2062
0.2075
0.2007
0.2100
0.2112
0.2125
0.2137
0.2150
0.2162
0.2175
0.2107
0.2200
0.2212
0.2225
0.2237
0.2250
0.2262
0.2275
0.2207
0.2300
0.2312
0.2325
0.2337
0.2350
0.2362
0.2375
0.2307
0.2400
0.2412
0.2425
0.2437
0.2450

h-flungfmno thhn-nut 0am:d.flhuo

'
‘0 0621
“0.714
“0.616
“0.464
“0.274
“0.054
0.171
0.371
0.530
0.660
0.723
0 .723
0.660
0.530
0.367
0.166
“0.049
“0.264
“0.611
“0.709
“0.740
-0 .710
“0 0‘2‘
-0407:
0.156
0.357
0.520
0.650
0.723
0.723
0.665
0.547
0.301
0.101
“0.034
“0.249
“0.601
“0.704
“0.740
“0.723
“0.409
0.137
0.342
0.510
0.645
0.710
0.720
0.670
0.557
0.396
0.196
“0.015
“0.235
“0.435
“0.591

146

V

2.229
2.571
2.674
2.501
.2.270
1.774
1.114
0.367
“0.306
“2.240
“2.263
-1 0716
“1.170
-004“
0.259
1.007
1.672
2.190
2.512
2.630
2.532
2.224
1.711

1.041.

0.274
-10212
'1 e793
-2021,
-20454
“2.490

0.635

1.344

1.040

2.195

2.331

2.250

1.965

1.457

0.006

0.064
“0.609
“2.605
-2.146
-0.968
-0.215

0.547

1.251

1.023

0

“0.300
0.034
0.029
0.029
0.024

“1.056

“1.373

“1.359

“0.665
0.039

0.034
. 34

0.034
0.029
0.024
“0.929
“1.370
“1.373
0.039
0.034
0 o 029
0.029
-1 o 373
“0.500
0.044
0.039
0.039
0.034
0.034
0.029
“0.327
“1.373
0.039
0.034
0.029
0.029
“0.474
“1.364
“0.507
0 o 04‘
0.039
0.034
0.034
0.029
0.029
“0.391
“1.370
“1.373

147

Tabla 8.1.(continu-d)

In. lldshulhun Dhaka-um: annullMN-
nun V V V

o .2473 . -o . 740 2.214 ’1 ~373
0.2407 , “0.723 2.395 “0.259
‘002500 . “0.640 2.336 00039
0.2537 . “0.103 13025 0-029
0.2575 , 0.500 -1.124 “1.369
0 02587 g 0 o 640 ..1 . 701 -1 .364
0.2612 . 0.728 -2,351 -0.132
0.2623 . 0.679 -2,390 0.039
0.2637 . 0.567 ~2.239 0.034
0.2650 . 0.411 -1.ass 0.034
0.2662 . 0.210 -1.333 0.034
than . 0.000 -0.710 0.029
0.2607 . -0.220 0.033 0.029
0.2700 , “0.420 0,311 0.024
0.2725 , “0.694 2.102 “1.303
0.2750 . “0.720 2.674 “0.020
0.2762 , -0.650 2,549 0.029
0.2775 . “0.513 2,414 0.029
0.2707 , “0.327 1.375 0.029
0.2812 . 0.100 0,530 0.020
0 02°37 . 00494 -0 0846 -1 0373
0.2050 , 0.630 “1.457 “1.364
0.2862 g 0071‘ -1.911 -1036,
0.2925 . 0.230 -1.359 0.029
0.2937 , 0.015 -o.713 0.029
0.2975 . “0.572 1.457 -1.193
0 .3000 . “0 .743 2.405 -1 .378
0.3012 , “0.733 2.535 “0.425
0.3025 , “0.655 2,555 0.034
0.3037 , “0.523 2.322 0.029
0.3050 . “0.342 1.077 0.029
0.3062 . -0.132 1.246 0 .024
0.3075 . 0.093 0.513 -0.088
0.3007 . 0.303 -o.259 -1.383
0.3100 , 0.404 “1.002 “1.369
0.3125 , 0.709 -2.111 -1.364
0.3137 . 0 0733 -2.405 -10129
0.3150 , 0.609 -z,512 0.039
0.3162 . 0-597 -2.424 0.034
0.3175 , 0.430 -2.146 0.034
0.3187 . 0-239 -1.557 0.029

APPENDIX C

rm mm TEST RESULTS ma Tm-KJDB EXCITATIOK

As mentioned in test case 2 chapter 4, that the most severe
working conditions for a mechanical structure is when it works under the
effect of an exciting force having more than one frequencies that are
equal to the eigenvalues of the structure. The two mode excitation case
was considered to demonstrate the efficiency of the control law to limit
resonant amplitudes resulting from two-mode excitation. The exciting
force was due to two sinsoidal waves having frequencies of the first and
second modes and were added together then fed to the magnet drive power
amplifier. The time domain data for control gains c - 0, 41.71 and 50 N
are shown in Figures Cl, CZ and C3. Figures Cla, 02a and 03a show the
exciting force versus time for various control gains. Figures Clb, C2b
and 03b show the response under the action of the exciting force for
various control gains. The required control forces are shown in Figures
C2c and C3c. As discussed in case 21: chapter 4 and by the help of FFT
of the responses for various control gains, (Figure 4.14), the amplitude
of the first-mode. was reduced from .95 mm to .55 mm and that of the
second-mode was reduced from .8 mm to .65 mm for a: - 50 N which
demonstrates the effectiveness of the control action in limiting the

resonant amplitudes resulting from the two-mode excitation.

148

149

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

3 :MHMMMMMMMM
E WMMUMMMU
0.000 TIME (sec) 0.500
1.50 ?; n n
a: :: M N N N
°‘ M A A x
. 0.000 TIME (sec) 0.500

uncontrolled beam.
a) the exciting force.
b) the response at x - 367.2 mm

150

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1.0
- W n n M M
. H W H
0.
Q
§
'0.
Q
3 (a)
't
'0
M M M H L
-10,:::..-.fi.:
0.000 0.500
nuns (Soc)
1.50
.\
E
E.
in
u.
'2
Q
3.
t (b)
'1-50 ‘ e L : t : r 2* . . {
0.000 0.500
TIME (Sec)
-5000
.\
3.. (c)
'10
9
§
K ‘.
u I 3
O I .
EE .
2 I
Q
Q
50-01—5 : :“4. er .L : :4.
0.000 0.500
TIME (Soc)
Figure C2 Steady state two-mode test results for control

gain a - 41.7 N.
a) the exciting force.
b) the response at x - 367.2 mm

c) the control force

151

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1.0
” M M M
- W“
3, n H A
N
Q
6
h.
<3: J (8)
':
3 U H U U1
h. “
..UU W M H H
.1 o %, L : : :e:%
0.000 0.500
rmE(s.c)
1.50
.\
E
E
'0
0. ..
0 4 ’
3; . (b)
|“ .
k I.
'1-50' :. : : r7 : : : : : i
0.000 0.500
TIME($ec)
”70001-
-\ u
s. L
'3 .. (e)
'6
k ..
Q
‘Q
E .
it
<3 ..
c:
70.0 ::::‘r::-fi44.
0.000 0.500
IZUE'(Sec)
Figure C3 Steady state two-mode test results for control

gain a - 50 N

a) the exciting force.

b) the response at x - 367.2 mm
c) the control force

LIST 0PM

152

REFERENCES

1. Schaechter, B. 8., Optimal Local Control of Flexible Structures,
AIAA J. Guidance and Control, Vol. 4, No. 1, 1979, pp. 22-26.

2. Schaechter, B. 8., Hardware Demonstration of Flexible Beam
Control, AIAA Paper 80-1794, Aug. 1980.

3. Schafer B.E. and Holzach 1%., Experimental Research on Flexible
Beam Modal Control, AIAA, J. Guidance, Vol.8, No. 5, Sept. Oct. 1985,
pp. 597-604. ’

h. Meirovitch, L. , Analytical Methods in Vibrations, The Macmillan
Co. NY,1967.
5. Balas, M. J. , Trends in Large Space Structure Control Theory:

Fondest Hopes, Wildest Dreams, 1888 Transactions on Automatic Control,
Vol. AC-27, No. 3,June 1982, pp.522-535.

6. Balas, M. , J. , Modal Control of Certain Flexible Dynamic Systems,
SIAM J. Control and Optimization, Vol. 16, No. 3, May 1978, pp. 450-462.

7. Meirovitch, I... Baruh, 11., On the Problem of Observation
spillover in Self Adjoint Distributed Parameter Systems, Journal of
Optimization Theory and Applications, Vol. 39, No. 2, Feb. 1983,
pp.269-291.

8. Radcliffe, C. J. and Mote, C. D., Identification and Control of
Rotating disc vibration, Journal of Dynamics Systems, Measurement and
Control, Vol. 105, March 1983.

9. Butkovskiy, A. , Distributed Control Systems, American Elsavier,
New York, 1969.

10. Kohne, M. , The Control of Vibrating Elastic Systems, Distributed
parameter systems Identification, Estimation, and Control, Marcel

Decker; Inc, New York and Basel, 1978, pp.387-457.

153

154

11. Baily, T. and Hubbard Jr., J.E., Distributed Piezoelectric-
Polymer Active Vibration Control of a Cantilever Beam, AIAA J. Guidance,
Vol. 8, No. 5, Sept-Oct. 1985.

12. Lubkin, S. and Stoker J. J. , Stability of Columns and Strings
under Periodically Varying Forces, Quart. Appl. Math. Vol. 1, No. 3,
1943, pp.215-236.

l3. Woodall, S. 51., On the Large Amplitude oscillations of a Thin
Elastic Beam, Int. J. Nonlinear Mechanics, vol. 1, 1966, pp.217-238.

14. Zauderer, E., Partial Differential Equations of Applied
Mathematics, Viley-Interscience pub., Wiley & Sons, New York, NY, 1983,
pp.332-340.

15. Iaipholz,‘H., Application of Liapunov's Direct Method to the
Stability Problem of Rods Subject to Follower Forces ,Instability of
Continuous Systems , Symposium Herrenlab (Germany) Sept 8-12 , 1969.

16. Crandall, S. H. , Numerical Treatment of Fourth Order Parabolic
Partial Differential Equation, Journal Association for Computing
Machinary, 1., 1954, pp.lll-118.

l7. Ames,‘w. F., Numerical Methods for Partial Differential
Equations, pp.279-281, 2nd Edition, Academic Press, New York,NY,
pp.279-28l

18. Lin, C. C. and Segel, L. A., Mathematics Applied to
Deterministic Problems in the Natural Sciences, Macmillan Co., NY,1974,
pp.401-403

19. Chen, G. and russell, D., A Mathematical Model For Linear Elastic
Systems with Structural Damping, Quarterly of Applied Mathematics,
Jan. 1982, pp.433-454.

20. Ball” .1. and Slemrod, M. , Feedback Stabilization of Distributed
Semilinear Control Systems, Appl. Math. Optim. 5, 1979, pp. 169-179.

155

21. Ball, J., Marsden, J. and Slemrod, M., Controllability for
Distributed Bilinear Systems, SIAM J. Control and Optimization, Vol. 20,
No. 4, July 1982, pp. 575-597.

22. Courant,R. and Hilbert, D., Methods of Mathematical Physics,

Vol.1, Interscience,NY, 1453, pp. 164-274.

23. Weinstock, R., Calculus of Variations with Applications to
Physics and Engineering, Dover, 1974,pp.72-90.

24. Timoshinko, 8., Young, D and Weaver,W. , John Wiley, 1974,
chapter 5.

25. Meirovitch, L., Analytical Methods in Vibrations, Macmilian,
1967.

26. Bolotin, V. V., The Dynamic Stability of Elastic System, Holden-

Day, inc., Sanfrancisco, 1964, pp. 111-112.

27] RayLigh, J. W., The Theory of Sound, Vol. 1, Second Edition,
Dover Publications, NY, pp. 244-246 and pp. 296-297.

28. Hoff,lL J, The Dynamics of The Buckling of Elastic Columns,
Journal of Applied Mechanics, Vol. 18, Tran. AS'ME, Vol. 73, 1951,
pp. 68-74.

29. Sevin. , E. , On The Elastic Bending of Columns Due to Dynamic
Axial Forces Including Effects of Axial Inertia, Journal of Applied
Mechanics, March, 1960, pp. 125-131.

30. Davison, J. , Buckling of Structures Under Dynamics Loading,
Journal of The Mechanics and Physics of Solids, Vol. 2, 1953, pp. 54-66.

31. Reiss, E., and Matkowsky, E. , Nonlinear Dynamic Buckling of A
Compressed Elastic Column, Quarterly of Applied Mathematics, July, 1971,
pp. 245-259.

156

32. Zubov, V. I. ,Methods of A. M. Liapunov and their Application,
Leningard, 1957; English Translation, P. Noordhoff Ltd., Gromingen,
Holland, 1964.

33. Wang, P. E., Stability Analysis of Elastic and Aeroelastic
Systems Via Liapunov's Direct Method, Journal of Franklin Institute,
Vol. 281, No. 1, 1966, pp. 51-72.

34. Wang, P.l(. , Stability Analysis of A Sipmlified Flexible Vehicle
Via Liapunov's Direct Method, AIAA Journal, Vol. 3, No. 9, 1967, pp.
1764-1766.

35. Lapidus, L. and Berger, A., An Introduction to the Stability of
Distributed Systems Via A Liapunov Functional, Aiche Journal, July,
1968, pp. 558-568.

36. Leipholz, H. , Application of Liabunov's Direct Method to the
Stability Problem of Rods Subject to Follower Forces,Instability of
continuous Systems, Symposium Herrenlab (Germang) Sept. 8-12, 1969,
pp. 1-23.

37. LEipholz, H. , Some Remarks on Liapunov Stability of Elastic
Dynamical Systems, Buckling of Structures, IUTAM Symposium, Cambridge,
1974, Springer-Verlag, 1976, pp. 208-217.

38. Dym, C. L., Stability Theory and Its Applications to Structural
Mechanics, Noordhoff International Publishing, Leyden, the Netherlands,
1974, pp. 52-75.

39. Friedman, A. , Foundations of Modern Analysis, Dover, 1982,
pp. 90-102.
40. Richtmyer, R. and Morton, R., Difference Methods For Intial-Value

Problems, Second Edition, Wiley, NY, 1967 Chapters 3 and 11.

41. Lapidus, L. and Pinder, G.F., Numerical Solution of Partial
Differential Equations in Science and Engineering, Wiley, NY, 1982,
pp. 186-187.

157

42. Yakubovich, V. A. and Starzhinskii, V. M. , Linear Differential
Equations With Periodic Coefficients, Wiley, 1975, Chapter 2.

43. Radcliffe, C. J., Active Control of Vibration in Rotating
Circular Discs, Ph. D. Dissertation, University of California, Berkeley,
1980.

"1111111111“