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STABILITY OF VISCOELASTIC PLATES IN

SUPERSONIC FLOW UNDER STOCHASTIC AXIAL

THRUST

by

Noh M. Abdelrahman

Department of Mechanical and Materials Engineering

Faculty of Engineering Science

Submitted in partial fulfilment

of the requirement for the degree of

Master of Engineering Science

Faculty of Graduate Studies

The University of Western Ontario

London, Ontario, Canada

May 1997

Noh M. Abdelrahman 1997

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ABSTRACT

The objective of this study is to examine the almost-sure asymptotic stability of

aviscoelastic plate in a supenonic gas flow and subjected to stochastically fluctuating axial

thmst. The viscoelastic constitutive relations of the plate material are represented in

integral forms by using the Boltzman superposition principle. Piston theory is used to give

a quasi-steady first order approximation for the aerodynamic loading on the plate. The

linearized integro-partial differential equation of motion of the plate is derived using the

VoItera correspondence principle. By the use of the Bubnov-Galerkin method, the equation

of motion is discretid to a two -degree of freedom system. Through a non-dimensional

time, an appropriate transformation and a suitable CO-ordinate scaling, the discretized

integro-ordinary differential equations are transformed to those of more convenient

generalized CO-ordinate. By making use of the method of variation of parameters, the

transformed equations are converted to equations in amplitudes and phases. For small

excitation intensity, system damping and material relaxation mesure, the amplitude and

phase equations are then approximated to a system of Ito equations whose solution is a

Markov process, by making use of the Stratonovich - Khas'minskii stochastic averaging

method.Through a specific transformation and by making use of Ito's lemma and

Khas'minskii's technique, expressions for the largest Lyapunov exponents are obtained

analytically. The corresponding results for the elastic plate are deduced by setting the

relaxation term to zero. Numerical results are presented and suggestions for further study

to deal with nonlinear effects arising from stnictural and aerodynamic terms are also made.

DEDICATED

TO

MY WIFE MAZAHIB

The author wishes to express his sincere and deep gratitude to his supervisor, Professor

S .T. Ariaratnam , for his guidance, constructive instruction and con tinuous encouragement

throughout the course of this work. Thanks are extended to Professor C. W.S .To for his

guidance. The author gratefully acknowleùges the support of his CO-supervisor Professor

T.Base, and the assistance of Professor P.YU. Thanks are also extended to Professor

A.T.Olson and to feiiow graduate students Zhidong Chen and Eltayeb Mohamedelhassan.

Finally, the author wishes to thank his wife, Mazahib,for her patience, support and

inspiration throughout the years which Ied to this work.

TABLE OF CONTENTS

CERTIFICATE OF EXAMINATION

ABSTRACT

DEDICATION

ACKNOWLEDGEMENTS

TABLE OF CONTENTS

LIST OF FIGURES

CHAPTER 1 INTRODUCTXON

1.1 Introductory Remarks

1.2 Outline of The Thesis

CHAPTER 2 MATHEMATICAL REVIEW

2.1 Mathematical Review

2.1.1 Markov Process

2.1.2 Fokker - Planck Equation

2.2 Tools of Stochastic Andysis

2.2.1 Diffusion Process

2.2.2 Wiener Process

2.2.3 Ito Stochastic Differential Equations

Page

. . Il

... lu

2.2.4 Ito Differential Lemma 14

2.2.5 Stochastic Averaging Method 16

2.3 S tochastic Stability 17

2.3.1 Definitions 18

2.3.2 Infante's Lyapunov Function Method 20

2.3.3 Lyapunov Exponents of Continuous Stochastic Dynamical

S ystems 23

2.4 Concluding Remarks 28

CHAPTER 3 CONSTITUTIVE RELATIONS FOR AGING AND

NON-AGING VISCOELASTIC MATERIALS

3.1 Introduction

3.2 Piston Theory

3.3 Viscoelastic Materials

3.4 Concluding Remarks

CHAPTER 4 STABILITY OF A VISCOELASTIC PLATE UNDER

A STOCHASTIC AXIAL THRUST

4.1 Problem Formulation

4.2 Approximation to Markov Process

v i i

Lyapunov Exponent

4.3.1 Nonsingular Case

4.3.2 Singular Case

S tability Anal y sis

4.4.1 Narrow-Band Excitation

4.4-2 White Noise Excitation

Numencai Results

Concluding Remarks

CHAPTER 5 STABILITY OF AN ELASTIC PLATE UNDER

A STOCHASTIC AXIAL THRUST

5.1 Problem Formulation

5.2 Stability Anal ysis

5.2.1 Narrow-band Excitation

5.2.2 White Noise

5.3 Numerical Results

5.4 Computer Simulation


CHAPTER 6 SUMMARY AND CONCLUDDJG REMARKS

v i i i

APPENDIX A Larionov's Method For Averaging The Viscoelastic Integral

Term 132

APPENDIX B Fortran Program For Evaluating The Largest Lyapunov

Exponent 134

VITA 146

LIST OF FIGURES

Figure Figure Caption Page

4.1 Schematic Diagram of Panel in Supersonic Flow under Stochastic h i a i

Thmst. 39

4.2a Stability Boundaries under a Nmow-band Excitation for a Viscoeiastic

Plate.

4.2b Effect of S ( 2 o , ) on Stability Boundaries under a Narrow-band

Excitation for a Viscoelastic PIate.

4.3 Effect of S (2 o, ) on Stability Boundaries under a Narrow-band

Excitation for a Viscoelastic Plate. 94

4.4 Effect of S ( o , - o, ) on Stability Boundaries under a Narrow-band


4.5 Effect of S ( o , + o, ) on Lyapunov Exponent under a Narrow-band


4.6 Effect of S ( 2 o , ) on Stability Boundaries under a Broad-band Excitation

for a ViscoeIastic Plate. 97

4.7 Effect of S (2 o2 ) on Stability Boundraies under a Broad-band Excitation

for a Viscoelastic Plate. 98

4.8 Effect of S ( o , - y ) on Stability Boundaries under a Broad-band


Figure Figure Caption

Effect of S ( a l + y ) on Stability Boundaries under a Broad-band

Excitation for a Viscoelastic Plate.

Effect of Spectral Density S on Stability Boundaries under a White-Noise

Excitation for a Viscoelastic Plate.

Effect of the Non-Dimensional Paramater a , on Stability Boundaries under

a Broad-band Excitation with Different Values of S ( 2 o , ) for a

Viscoelastic Plate.

Effea of the Non-Dimensional Pararnater a , on Stability Boundaries

under a Broad-band Excitation with Different Values of S (2 o, ) for a

Viscoelastic Plate.

Effect of the Non-Dimensional Paramater or , on Stability Boundaries

under a Broad-band Excitation with Different Values of S ( w , - oz ) for a Viscoeiastic plate.

Effect of the Non-Dimensional Pararnater a, on Stability Boundaries

under a Broad-band Excitation with Different Values of S ( m l + m2 )

for a Viscoelastic Plate.

Effect of the Pararnater a , on Stability Boundaries under a White-noise

Excitation with Different Spectral Density S for aViscoelastic Plate.

Values of Stifiess Terms pl, / 8 , kL,/ 8 , k2 / 8 for Different

Values of the Non-dimensional Parameter a,.

Page

1 O0

101

102

Figure Figure Caption

Values of Frequencies a2, , o2 , for Different Values of the Non-

dimensional Paramater a ,.

Values of the One-sided Fourier Transform R , ( o , ) and R , ( o, ) of the

Relaxation Function R(t) for Different Values of Parameter a,.

Stability Boundaries under a Narrow-band Excitation for an Elastic Plate

Effect of S ( 2 o , ) on Stability Boundaries under a Narrow-band

Excitation for an Elastic Plate.

Effect of S ( 2 oz ) on Stability Boundaries under a Narrow-band


Effect of S ( w , - y ) on Stability Boundaries under a Narrow-band


Effect of S ( 2 o , ) on Stability Boundaries under a Broad-band


Effect of S ( 2 o ,) on Stability Boundaries under a Broad-band


EEect of S ( o , - oz ) on Stability Boundaries under a Broad-band


Effect of S ( a, + y ) on Stability Boundanes under a Broad-band


Effect of Spectral Density S on Stability Boundaries under a White-noise


Page

1 O8

xii

Figure Figure Caption Page

5.10 Largest Lyapunov Exponent for an Elastic Plate in a Supersonic Flow

under a White-noise Axial Thmst 127

... Xlll

Chapter 1

Introduction

1.1 Introductory Remarks

The needs of modem technology have pushed design into realms where random

inputs, random excitation, random environments, as well as randorniy varying system

components are encountered. It is gradually being recognized that a deterministic modelling

may not be adequate for certain types of extemal excitation, and a probabilistic system

modelling is needed. This probabilistic modelling absorbs the uncertainty in the response of

engineering structures owing to unpredictability of this excitation, and to imperfection or

lack of information in the modelling of physical problems [l].

Many engineering structures are subjected to forces of a random nature such as

those arising fkom wind, earîhquakes, ocean waves and jet noise which can be descnbed

satisfactorily only in probabilistic terms. When a excitation appears in the govemhg

equations of motion of a system as a parameter, the system is said to be parametncally

excited. This excitation causes the basic characteristics of the dynamical system to change

randorniy with time.

Initiated by the work of Andronov, Pontryagin and Witt [2], dynarnic stability which

qualitatively describes the dynarnical system behaviour is described in probabilistic terms.

Probabilistic modeiling with stochastic differential equations began in the 1930's. The

foundation for the great leap forward in the topics was laid by developments in the theories

of Markov processes, diffusion processes, and stochastic differential equations which were

initiated principaily by Kolmogrov [3] and Itô [4].

The classical stability snidy for linear Itô stochastic difTerentia.1 equations, whose

solutions are Markov diffision processes, was developed by KhaSminskii in 1967 [5].

The main concept is to n o m the solution process and study the properties of the normed

process on the surface of a unit sphere [6]. The objective of this method is to determine an

alrnoa - sure stochastic aability indicator without solving the basic differential equations. This

indicator is the maximum Lyapunov exponent which characterizes the exponential rate cf

growth of nearby system States, and is one of the most important characteristics in the study

of stochastic stability [7]. It has been s h o w by Arnold and Klieman [8] that the Lyapunov

exponents are analogous to the real part of the eigenvalues of detenninistic time - invariant

systems and the vanishing of the maximal Lyapunov exponent yields the almost-sure stability

boundary in the system parameter space. If the maximum exponent is positive, the system is

unstable with probability one and if it is negative, the system is stable with probability one.

Thus the vanishhg of the maximum Lyapunov exponent indicates the transition to an unstable

system or the onset of a stochastic bifurcation.

The direct use of Khaiminskii method for higher than two-dimensional systems has

not met with much success because of the d'ifliculty of studying diffusion processes occumng

on surfaces of unit hyperspheres in high dimensional Euclidean spaces 191.

When the excitation is non-white, the solution process is not a Markov diffision

process. Stratonovich (101 and KhaSminskii [Il] discovered that, when the excitation has a

small correlation time as compared to the relaxation time of the system, and when the limit

of the averaged physical system equations exists, a physical non-Markov process can

3

converge in the weak sense to a Markov difision process whose goveming Itô equations

are obtained by making use of the so- called stochastic averaging procedure. In the case of

linear stochastic syaems disturbed by a real noise process, stability conditions can be obtained

through the approximated Markov process.

1.2 Outline of the Thesis

The main objective of this thesis is to study the stochastic stability of a viscoelastic

plate in a supersonic flow of gas and subjected to a stochastically fluctuating axial t h s t .

Chapter 1 is devoted to the introduction and to the outline of the thesis. In Chapter

II, a review about Markov processes and the associated Fokker-Planck equation is given.

Definitions and presentations of the main tools needed for stochastic analysis, such as

diaision process, Wiener process, Itô's dzerential equation, Itô's difiierential lernma and the

stochastic averaging method are presented. Also, various stochastic stability definitions are

introduced, together with the formulation and methods of evaluation of the largest Lyapunov

exponent. In Chapter LU, the phenornenon of panel flutter and the importance of stochastic

stability of plates in supersonic gas flow are introduced. A quasi-steady first-order

approximation for the aerodynarnic loading, known as piston theory, is presented. Definitions

of creep kemel and relaxation measure, together with the constitutive equations for non-

aging viscoelastic materials are introduced.

In Chapter N, the integro-partial differentiai equation of motion of a viscoelastic

plate in a supersonic flow of gas and subjected to a stochastically varying axial thmst is

derived. With the help of the Bubnov-Galerkin method, the integro-pariial differential

4

equation is discretized to a two- degree -of fkedorn system and transformed to those in ternis

of more convenient generalized CO-ordinates. The transformed integro-ordinary differential

equations are convertecl to equations in amplitudes and phases by the method of variation of

parameters. For srnall damping and srnail intensity of the axial t h s t , these equations are then

approximated to a system of Itô equations, whose solution is a Markov process, by making

use of the stochastic averaging procedure. A pair of Itô equations goveming the naturd

logarithm of the nom of the averaged amplitude vector and the phase angle are obtained by

making use of Itô's lemma. This pair of Itô equations is used to obtain the stochastic stability

conditions of the original integro-ordinq differential equation of the plate in a first

approximation. Numerical results for various forms of the stochastic axial t hmst are presented

to give a qualitative picture of the effect of the excitation spectrarn on the almost-sure

stochastic stability.

In Chapter V, the case of an elastic plate is considered. Again expressions for the

largest Lyapunov exponent are denved and stochastic stability conditions for various forms

of the axial thmst are obtained. Cornputer simulation is presented, to compute the largest

Lyapunov exponent under white noise excitation. Numencal results are presented to show

stability regions for different excitation spectrarn and plate parameters.

Chapter VI, is devoted to a surnmary of this thesis and some suggestions for future

work are presented.

Chapter II

Ma thema tical Review

This section contains some definitions and background material for the purpose of

making this thesis self- contained.

2. 1.1 Markov Process

Roughly, a Markov process is a random process whose future state depends only

on its most recently known state and ali relevant predictions of the future can neglect the past

[12]. A random process X( t ) ; z .G T. is said to be Markov if the conditional probability

for any tz and for any t , , where t , < t , . . . < r , . Here P [ ] denotes the probability

of an event. The symbol x denotes the realization of the state of the process X( z ) if for any

t F P [x-il< X(I) c x-h ] > O for every h > 0- A suficient condition For X( t ) to be

a Markov process is that its increments be independent in nonoverlappinç time intervals. For

stochastic dynarnics applications, a Markov process X( t ) is generally continuously valued.

A Markov process is completely specified by its transition probability distribution function

P [ X( t ) c x ( X( t ,) = x, ] and its probability distribution at some initial time t , . For a

diffierentiable transition probability distribution function we deal with the transition probability

density defined as [13] :

The concept of a scalar Markov process is generalized to a vector Markov process.

T h u s X ( t ) = { X , ( t ) , X , ( 2 ) . . . . ,X, ( 1 ) ) isanm-dimensionalMarkovvector,ifit

has the property

P

where n denotes the joint occurence of the multiple events. A sufficient condition for a

vectonally valued stochastic process to be a Markov vector is that its vectorial increments be

independent in nonoverlapping time intervals. The transion probability density of a vector

Markov process is a generaiization of (2-2):

The higher-order probability densities, descnbing the behaviour of a Markov process at

several instants of time, can be constructed fiom the initial probability density P ( x , ) and

the transition probability density 1141, through the relation :

2. 1.2 Fokker-Planck Equation

Let X( t ) be an n-dimensional Markov vector ; then the transition density Q satisfies

the Chapman-Kolmogrov-Smoluchowski equation [ l ]

Here $y denotes an infinitesimal element in the n-dimensional state space. The Chapman-

Kolmogrov-Srnoluchowski equation is an integral equation goveming the transition

probability density of a Markov process implying that the integration of transition probability

is independent of the path. Consider the integral

where R (y) is an arbitrary scalar fùnction of y, , j = 1.2,. . . .n , such that

R ( y ) - O a r y,-*- andforany s = k + I + ...- r ,

It is also assumed that R(y) cm be expanded in a Taylor series about c

if the integral

converges uniformly in a neighbourhood of t . Using equation (2-6), substituting for R ( y )

from equation (2-9), integrating first on y and using the relation

for any x , t , and A t , one obtains

where

a , b ,, and Ci,, are cded the derivate moments and can be expressed more meaningfully

as follows :

10

Integrating equation (2-1 1) by parts, using the relations (2-8) and combi~ng the result with

equation (2-7), one can obtain, for arbitrary R ( X),

In equation (2-14), the arguments of q ( X , t 1 x0, fo ), aj (X, t ) ,bjk (X, t ) and q,, (X, t )

are omitted for brevity. Equation (2-14) is known as the Fokker-Planck equation. For a

difftsive Markov process, the denvate moments of orders higher than two C ,, , ,. . . are zero.

In this special case the F-P equation (2-14 ) reduces to :

with initial condition

2. 2 Tools Of Stochastic Analysis

2. 2. 1 Diffusion Process

A cifisive process is a Markov process for which the sarnple functions are continuous

with probability one. A sufficient condition for a Markov process to be dias ive is that the

11

derivate moments of order higher than two are zero and its Fokker-Planck equation has the

form of equation (2-1 5) [ 1 1.

2. 2.2 The Wiener Process

The Wiener process denoted by W( t ), is aiso known as the Brownian motion process

and it is the simplest form of a Markov process. The Wiener process is a mathematical

idealization, as a physical process cm be close to a W~ener process, but it can never be exactly

a Wiener process. Wiener process is a Gaussian process. Without loss of generality it is

considered to have the following properties [l]:

(a)W( t ) has zero value at initial zero time,W(O ) = 0;

(b) It has zero mean value, E[ W( t ) ] = 0;

(c) Its covariance function is given by;

where o' is a positive constant.

(d) W( t ) is not differentiable in the mean square sense.

where 6 is the Dirac delta fùnction.

(e) W( t ) is a process with orthogonal increments; that is to Say if t , c t < t 3 < t 4

(f) The expectation of the increment process of W( t ) for any t is zero.

(g ) The covariance function of the increments of the Wiener process is

Also dW( t ) = O( dt L ' 2 ) in mean square as well as with probability 1.

(h) The Wiener process is a diffusion process since ail the derivate moments of orders higher

than two are zero. The first derivate moment of the Wiener process vanishes ( a, = O )

and the second denvate moment is constant ( b, , = o ' ). (i) The Wiener process is of unbounded variation within any finite tirne interval.

(i) The Wiener process W( t ) grows as { t Zn ( In t ) 1''' as + which is much slower

than that of t.

2. 2.3 Itô Stochastic Differential Equations

A scalar Markov process X( t ) can be generated from the stochastic diEerential

equation [ 141

13

where m and u are caiied the drift and difision coefficients, respectively, and W( f ) is a unit

Wiener process, independent of X( t ). Equation (2-2 1) is equivalent to the integral equation

According to Itô [15], the second integrai of equation(2.22) which is of Stieltjes type is

interpreted as a fonvard (in the mean square sense) integral:

[ W( u , + , ) - W( u, ) ] is evaluated in a forward time interval foilowing the time instant at

which o (x(u) .u ) is evaluated. Equation (2-23) defines an Itô integral. There are direct

correspondences between an Itô stochastic differential equation and the associated Fokker-

Planck equation. For the first denvate moment a( x ,t ) and second derivate moment

b( x . t ) of the corresponding Fokker-Planck equation, the following relations apply [l];

where x is the state variable of X( t ).

An arbitrary II-dimensional Markov vector process can be generated from the Itô

stochastic differential equation

where m, are the drift coefficients , a,, are the diffusion coefficients , and W , ( t ) are mutudy independent unit Wiener processes. The derivate moments of the corres ponding

Fokker-Planck equation of the system of Itô dinerential equations (2-25) can be obtained

fio m

2.2 -4 Itô's Differential Lemma

Consider an n-dimensional Markov vector process

Let F ( X. t ) be an arbitrary scalar function of the Markov vector X ( t ) and t, assumed to

be differentiable once with respect to t and twice with respect to the components of X ( t ).

Expansion of d F ( X , t ) will give

Substituting for a, and ndc f?om equation (2-27) into equation (2-28) gives;

Since Wk ( t ) are independent Wiener processes and d W, ( t ) is of order ( d t )' "

with prabability one, w.p. l then,

dX,.dXr =O] , . o,dt wp. 1

Keeping terms of the order ( d t ) and d Wk ( t ) , one obtains

The dierential of an arbitrary fùnction of a Markov process can easily be derived by

making use of Itô's differential mle (2-30). This property of Itô's differential equations

facilitate the investigations of stochastic stability [l]. Anaratnam and Srikantaiah [16] have

used the Itô differential mle for the investigation of moment stability without the utilization

of the corresponding Fokker-Planck equation goveming the probability density. The

necessary and sdiicient condition of KhaSminiskii for the almost sure asymptotic stability of

linear stochastic differential equations was obtained by using the Itô differential rule to

16

formulate the Itô differential equations for the naturai loganthrn of the nom and the phase

of the original state variables.

2.2.5 Stochastic Averaging Method

The method of stochastic averaging was onginally formulated by Stratronovich [IO]

based on physical and mathematical arguments, and by Khirniniskii [11],[17] on more

ngorous mathematical arguments. It is an extension to random differential equations of the

well-known Bogoliubov-Mitropolski [18] technique of averaging for ordinary differential

equations containing a small parameter. Consider the system of equations :

The first and second terms on the nght hand side of equation (2-3 1) are assumed to be of

orders O( E ) and O( E ' I L ) respectively, and their contributions to the system response are

commensurable. In equation (2.3 1), X, ( t ) are the stochastic solution processes, 5 .( f ) are

zero mean stationary random processes, f, ( X, 1 ) and g, , ( X, t ) are fùnctions of their

arguments which are bounded together with their first and second order denvatives.

Let M, (. ) denote the averaging operator

where the integration is performed over explicitly appearing t in the integrand. Suppose the

following assumptions hold 1171:

(i) the following limits exist unifody over X and t

In the above, the argument X in f, and g,, has been suppressed.

(ii) The correlation fûnction 4[ ( t )E ( t + s ) ] of the stochastic process [ ( t ) decays

sufficiently fast to zero as s increases, Le. 6 ( t ) has a small correlation time as compared to

the relaxation time of the system.

Then over a tirne intervai of order O( 1/c ) , X ( Z) can be approximated unifody in the weak

sense by a Markov diffusion process which satisfies the Itô equations

If the functions f, ( X, t ) and g , ( X, t ) are explicitly dependent on t,

this dependence is 10s through tirne-averaging. Where certain time dependent properties of

a dynamical system are of primary importance, tirne-averaging should not be used [I l .

2.3 Stochastic Stability

One of the purposes of a stochastic adysis is to determine the qualitative properties

of the solution. This is equivalent to determining the boundedness and convergence of the

nom of the solution and consequently its approach to the trivial or reference solution as time

increases. Since convergence of the solution can be interpreted in more than one way [18],

different definitions for stochastic stability are available [ 19 1.

A brief discussion of dynamic stability of deterministic systems is appropriate in

order to explain the fundamentals of stochastic stability.

2.3. 1 Defnitions

For deterministic systems defined by di dt = f ( x . r ) the following definition is

attnbutable to Lyapunov [20].

Lyapunov stability : The trivial solution is said to be stable if, for every c > O, there exists

a s ( r , t , ) > O suchthat

provided 11 x,li i 6 where xo = x ( to ). The stability is said to be uniform if, equation

(2.36 ) holds for any to .

Lyapunov asymptotic stability : The trivial solution is said to be asymptotically stable, if it

is stable, and if there exists a 6 '( t, ) > O such that

lim 1 x ( t ; x o , t , ) II = 0 r - w

provided Il x, 11 s 6'. The trivial solution is said to be asymptotically stable in the large if,

equation (2.37) holds for any x,.

Extending the above definitions for the stochastic case, a new meaning is assigned for

the sense of convergence of the above inequalities.

Lyapunov stabiity with probability 1: The trivial solution is said to be stable in the Lyapunov

sense W. p. 1 if. for every pair of el, q > 0, there exists a 6 ( e l , E, , t , ) > O such that

where x, = X( t, ) is determiniaic.The stabiiity is said to be unifonn if equation (2.3 8) holds

for any ta. Since E, , E, are arbitrarily small, this is also known as almost -sure or sarnple

stability.

Lyapunov asyrnptotic stability with probability 1: The trivial solution is said to be

asymptotically stable in the Lyapunov sense w.p. 1, if(2.38) holds and if, for every E > O, there

exists a ô'(&, t, ) > O such that

provided Il x, II < 6'. The stability is said to be in the large if equation (2-39) holds for any

x,. Since c is arbitrarily small, (2-39) is also known as almost- sure asymptotic stability.

Stability in the mth moment: The trivial solution is said to be stable in the mth moment

for eves, E > O, there exists 6 ( E, t, ) > O such that

provided /lxo Il i 6 , where xo = X( t, ) is deterministic. The stability is uniform if equation

(2.40) holds for any t,.

Asymptotic stability in the rnth moment : The trivial solution is said to be asymptotically

stable in the mth moment Xequation (2.40) holds and if there exists a 6'( E, t,, ) > O such that

provided Il x, II s 6. The stability is in the large if equation (2.4 1) holds for any x,.

Almost- sure aability and asymptotic sample stability describe the qualitative behaviour

of stochastic systems, since they characterize the boundedness and convergence of the

greatest excursions for the entire time domain. According to Arnold [21], and Kozin and

Sugirnoto [22], the asyrnptotic moment stability is more stnngent than the almost sure

asymptotic sample stability for linear stochastic systems, since convergence in the mean

square sense impiies convergence in probability, and since stability in probability is equivdent

to stability with probability one for linear systerns [23].

2.3.2 Infante's Lyapunov Function Method

The use of a Lyapunov function for the stability investigation of a stochastic sytern

was first made by Bertrarn and Sarachik [24] in the sense of stability in the mean. Infante [25]

obtained a sufficient condition for asyrnptotic sample stability of linear systern under non

white ergodic random excitation. Consider an n-dimensional Iinear stochastic system

where X( t ) is an n-vector solution process. A is an n x n constant rnatnx and F ( r ) is a

matnx whose nonzero elements f , , ( t ) are ergodic processes with zero mean. Choose a

Lyapunov fùnction V( X ) defined as:

V ( X ) = X T ~ x (2.43)

such that V( X) is positive for any nontrivial X( t ) and is zero only for X = O. B is an n x n

real symrnetric, positive definite matrix. Along the trajectories of system (2.42), define

which gives the exponential rate of growth of the Lyapunov fûnction at time t.

Using the min-max theorem for positive definite matrices, namely

one obtains

Now from quation (2.44a)

Since F ( 1 ) is a matrix of ergodic elements, 1 ( t ) also tends to be ergodic as t increases

Thus

It follows from equation (2.47) that Y[ X ( t ) ] + 0 as t + - provided Ef A ( t ) ] s - c ,

E > O . Thus, a sufficient condition for asymptotic sample stability is

Stochastic stability of multi-dimensional systems is defined in terms of a nom, and thus there

can be different sufficient stability conditions, depending on the choice of the nom. The

works of Kozin 1261, Caughey and Gray [27],and Ariaratnarn [28] were targeted towards the

sharpness of the sufficient stability conditions. The work of Ly [29] and recent works of

Ariaratnarn and Xie [30], [3 11, [32] optimized the sufficient stability conditions by varying

the elements of matrix B.

Plaut and Infante [33], Ly 1341 and Ahmadi [35] used the Lyapunov function method

to treat distributed parameter and noniinear dynamical systems. Kusher[36] and

Khaiminiskii [37] developed the Lyapunov function method for Itô differentiai equations.

2. 3. 3 Lyapunov exponents of continuous stochastic dynamical systems

According to Oseledec's multiplicative ergodic theorem [38], Lyapunov exponents

exist and are deterministic numben even though the system is stochastic. A well known

procedure by Khaiminiskii [ I l ] is foiiowed to obtain an expression for the largest Lyapunov

exponent for linear Itô stochastic differential equations.

Consider a system of n-linear Itô stochastic differential equations

where b, and a , ' are real constants and W ( t ) = [ w , , w, , . . . , w , ] are d-mutually

independent unit Wiener processes. Equation (2.50) descnbes the behaviour of a dynamical

linear system whose parameters are subjected to wide-band random excitations. According

to Arnold 1391, the unique solution process to the stochastic system of equation (2.50) is a

Markov diffusion process with the generating differential operator L given by :

where ( . , .) denotes the inner product

From matrix theory, it follows that matnx A is symmetric because a is a real rnatrix .

Let s = X 1 11 XII , p = log II X I I , where Il X il is the euclidean nom and given by

II XII = ( X , ' + X ' + - . . + Xn ' )in . This transformation maps the solution of system

(2.50) ont0 the surface of an n-dimensional unit sphere II s II =1. Applying Itô's lemrna the

equation for s( t ) [39] , is

3 ds(i) = { ( B - A ) s ( ( B - - ) s ) 2 2

where tr denotes the trace of the matrix. In (2-5 1), the coefficients of dt and dW , ( i )

of the Itô equation for ds( t ) depend only on s (2). Hence s( t ) generates a Markov difision

process on S "-', the unit hypersphere lls[l= 1. Let v (ds ) denote the invariant measure of this

process on the hypersphere. Again, using Itô's lemma p ( t ) satisfies the equation

d

( ( B -A)s , s ) + (ds , s )dwr( t ) 2

For the iinear system (2.50), L[ p ( z ) ] is a hnction of s only. Let Q( s ) be given by

1 o ( s ) = L [ p ( r ) ] = ( ( B - A )s,s) + - t r ( A )

2

Integrating equation (2.52) one obtains

Using the martingale property of the Itô integral [39], it can be shown that the second integral

tends to zero as r - =J, provided that the process s ( t ) is ergodic. Then equation (2.53) yields,

when I - m,

Hence E [ Q( s ) ] characterizes the rate of exponential growth or decay of the Euclidean

nom of the response of system (2.50) for large t. The largest Lyapunov exponent of the linear

Itô stochastic differential equation is then given by

1 Ii = iim -log 11x1 = E [ Q ( s ) ] r - - 2

If h < O, then the trivial solution X( t ) = O of system (2.50) is asymptotically stable

w.p.1, and therefore P(1lXI - O as t - - ) =1; while ifA > O then forX(0) + 0 ,

P{IXII - - as t - = ) = 1 .

26

E[Q ( s )] can be evaiuated directly for first order linear Itô equations, but for second and

higher order linear Itô equations, the knowledge of the invariant measure of the s- process

with respect to which the expectation is defined is required. A suficient condition for s ( t )

to be ergodic is that there exists a positive constant C, such that for any 11-vector V, the

matrix A= [a ] , , satisfies the relation

If condition (2.56) is not satisfied, the singulanties of the s- process have to be determined

and classified since the Markov process s( t ) may not be ergodic throughout the entire

hypersphere and the invariant measure has to be studied separately for each ergodic

component of the process s ( t ). In the subsequent chapters, we only meet the case when

IZ = 2; then the s- process is defined on the boundary of the unit circle s,* + s 2Z 4. Using

the transformation,

A 1 s, = cos @ = - lt xll

the s - process can be snidied in tems of the 4- process. By employing Itô's Lemma, the Itô

equations for p = log 11 x [J and @ = tan' ' ( x2 / x , ) are obtained as

where

Integrating the first of equations (2.57) results in

Taking the limits of both sides of equation (2.58) as z - - leads to

1 1 t

1 A = lim-log Rxll = lim-p(t) = l i m - I ~ ( $ ) d t t - - t r - - t f -- l o

provided the @ ( t )- process is ergodic. The Iargest Lyapunov exponent is obtained as

where the function p(4) is the density of the invariant measure of the process ( t ) with

28

respect to the uniform measure on the unit circle and is the solution of the Fokker-Planck

equation

When Y?(@) + O, there is a unique solution for p(+ ) on [ 0 , 2n 1. ~f'fR(@) is not positive,

Khasrninskii [ I l ] showed that the process 4 ( r ) has at most four ergodic components.

Nishioka [40] has given a general classification of singulanties of the difision process

4 ( t ) on a unit circle and the corresponding density p(4) of the invariant measure.

Ariaratnarn [41] pointed out that the KhaSrniniskii technique can be extended to certain

homogenous nonlinear systems also.


In this chapter, the concept of a Markov process and the associated Fokker-Planck

equation are introduced in some detail. The main tools needed for stochastic analysis, such

as difision process, Wiener process, Itô's differential equation, Itô's differential lemrna and

the stochastic averaging rnethod are presented. Vanous stochastic stability definitions are

introduced, together with the formulation and methods of evaluation of the largest Lyapunov

exponent such as Infante's method and Khas'minskii's procedure.

Chapter III

Constitutive relations for aging and non-aging viscoelastic materials

3.1 Introduction

Panel flutter has been an important structurai problern since ai:cr& and space

vehides fint exceeded the speed of sound. Defked as a selfexcited oscillation of the extemal

surface of the vehicle, panel flutter is a form of dynarnic instability resulting fiom dynamic

interaction of aerodynarnic, inertial, and elastic forces of the structural system. One of the

difficulties in studying this phenomenon is that aerodynarnic forces cannot in general be

sufficiently simply expressed in terms of disturbances of the surfaces exposed to the flow

After the existence of panel flutter was verified, various approximate methods for

aerodynamic loading have been used. The application of a two-dimensionai static

approximation was used by Hedgepeth [42]. The two-dimensionai static approximation was

only valid for a smaii range of Mach numbers and panel geometry ; therefore a new detailed

solution of the panel flutter problem was initiated by including three-dimensional unsteady

aerodynamics. Expressions for the three-dimensional unsteady aerodynamics are complicated

and thus it is difficult to attain a wide range of convergence.

Catastrophic or rapid failure occurs, ifthe stress amplitude due to flutter exceeds the

yield stress of the plate material. On the other hand, if the stress due to flutter is relatively

small, then fatigue or long-tirne failure may occur [43].

A great deal of attention has been given to problems on the stability of plates in

supersonic gas flow. These problems are important in co~ec t ion with the vibration of the

30

skin of modem aircrafts. The ~tabiIity of plates in supersonic flow under deterministic loading

was considered in many works, e.g. Bolotin et al [44], Volmir 1451, and Dowel [46]. The

approximate aerodynamic load is obtained by a quasi-steady first order aerodynamic piston

theory 1471. The number of works concerned with the stability of long plates in supersonic

gas flow subjected to in- plane stochastic loading is much smaller, though such a problem is

of significant interest. Among those works are investigations by Plaut and Infante [48],

Kozin [49] and Ahmadi [50]. Elastic plates and ergodic stationary in- plane excitation were

considered. Sufficient conditions for almost -sure stability of the plates on the basis of the

Lyapunov method were obtained . V.D.Potapov [5 lland V.D.Potapov and Bonder [52],

treated the viscoelastic case and also obtained stability conditions in the mean and mean

square sense.

Represented by two modes of interaction, the stochastic flutter of elastic plates was

examined by Ibrahim et. al. [53] and Ibrahim and Orono [54]. In- plane excitation was

assumed as a Gaussian white noise process and the response moments equations were

generated by making use of the Fokker-Planck equation approach.

In the present investigation, the Iaw of plane sections or the so-called piston theory

[55] is used to approximate the aerodynamic forces due to supersonic flow. The partial

dinerential equation governing the motion of the plate in supersonic gas flow and subjected

to in-plane stochastic loading is discretized by making use of the Bobnov-Galerkin rnethod.

The stochastic averaging method described in Section (2.2.5) together with Larianov' s

method of averaging [56] is applied to the discretized equation of motion of the plate to

obtain equations goveming the averaged amplitudes. Then, KhaSminiskii's procedure,

3 1

described in Section (2.3.3 ), is employed to obtain expressions for the largest Lyapunov

exponents. Conditions for almost- sure asymptotic stability are obtained.

3-2 Piston Theory

A number of approximate methods have been suggested for determining aerodynamic

forces. The simplea variant is known as the "law of plane sections", or the piston theory

[SA. Piston theory provides a formula relating the local pressure on the body to the normal

component of the velocity at the point considered. By making use of piston theory, the

expression for the aerodynamic forces is considerably simplified, provided factors associated

with viscosity, dissociation and phase changes on the boundary between the body and the

fluid are not taken into account.

For the steady motion of a thin profile at a supersonic velocity u, the disturbances are

trammitted only in a down strearn direction. As the flow velocity increases, the disturbances

assume a more local character and in the limiting case of very high supersonic velocity, each

particle of gas moves oniy in a direction perpendicular to the flow velocity. It is as if the

profle cuts through the gas, particles of which move in narrow bands bounded by extremely

close vertical planes ; the greater the flow velocity, the more exactly does this "law of plane

sections" hold.

The local pressure Po on the surface of the body c m be calculated as that on a piston

in a one-dimensional tube moving with velocity q given by

where u is the flow velocity, w is the transverse deflection of the plate, x is the coordinate

measured in the direction of u and the local pressure is given by

where Pd . C, are the pressure and the velocity of sound in the undisturbed gas, x is the

polytrophy index.

From equations (3.1) and (3.2), the component of the aerodynamic pressure P( x . t ) caused

by the deviation of the plate from its undisturbed state is given by

3-3 Viscoelastic Materials

The term viscoelastic is reserved for materials which exhibit both viscous and elastic

properties in either shear or volumetric deformation, whether they be elastic liquids or

viscous solids. Viscoelastic materials possess both viscous and elastic properties in varying

degrees and may Vary from viscous solids such as rubber to elastic fluids such as molten

polymers. For viscoelastic materiais, a pronounced influence of the rate of loading is

observed, the strain being larger if the stress has grown more slowly to its final value [58].

Instantaneous stresses for viscoelastic materiais depend upon the instantaneous and the entire

past history of the deformation 1591. For real materials only the most recent history is

considered and are descnbed as having a fading memory. This influence of time upon the

relation between stress and strain can be described either by a differential equation involving

33

derivatives of stress and main with respect to time, or by a hereditary integral equation with

time as a variable. DEerences in the mechanical behaviour of viscous, elastic and viscoelastic

materials are most evident in unsteady or time-dependent situations.

To fornulate constitutive equations for uniaxial deformations of a linear viscoelastic

material with infinitesimal strains, a viscoelastic specimen in the form of a rectilinear rod is

considered, [60].

For a rod in its stress free state, unit tensile forces are applied at time r 2 O, to the

ends of the rod. The longitudinal strain is given by

where e , ( s ) =E-'(r ) is the instantaneous strain and e , ( t , r ) = c ( t ,s) is an

additional (creep) strain caused by the material viscosity. E ( r ) is called the current elastic

modulus, and c ( t ,s ) is called the creep measure and is assumed to be a sufficiently smooth

function of its arguments satisfing the condition c ( r ,r ) = O.

If we suppose that at instant r = O a time-varying longitudinal load is applied to the

rod , the longitudinal stress due to this load is denoted by o ( t ) and is considered to be

continuously differentiable satisfing the condition o ( O ) = O.

For linear viscoelastic materials the B O ~ ~ M superposition pnnciple is valid [6 11.

This pnnciple States that "the strain E ( r ) at instant t caused by a stress history

{O ( s ) ( O i s s t ) ) equals the surn of the strains caused by eiementary stresses da(r ) ",

that is

Integration of equation (3.5) by pans with the use of equation (3.4) and the conditions

c(s , s ) = O and a(O)=O yields

where

The function K ( t ,t ) is called the creep kernel. By rnaking use of equation (3.7), the

constitutive equation (3.6) can be presented in the form

where the first tenn of the nght hand side of equation (3 -8) determines the instantaneous

elastic drain, while the second term determines the creep strain. Equation (3.8) describes the

mechanical behaviour of an aging viscoelastic matenal with a time-dependent response. For

35

a non-aging viscoelastic material, the current elastic modulus E( t ) may be treated as a

constant, and the creep kemel K( t ,s ) rnay be considered as a fûnction of the time difference

( t -r ) only ; thus E( t ) = E and K( t , r ) = II( t- s). The constitutive equation (3.8) for

a non-aging viscoelastic material then takes the form

For a given strain history e ( t ) equations (3.8) and (3.9) may be treated as linear integral

equations for the stress o ( t ) mi, by solving these equations, we obtain relaxation equations

which express the stress u ( t ) as a function of the strain a ( t ). For an aging viscoelastic

materid, we have

a ( t ) = E ( t )

and for a non-aging viscoelastic material, since R ( t , t )=Re ( t - r ),the constitutive equation

(3.1 0) can be presented in the fom:

Several explicit expressions for creep and relaxation measures are suggested and

they are evaluated according to their agreement with experimental data. The simplest mode1

36

of regular relaxation measures corresponds to the standard viscoelastic solid 1621 and is given

by

where R' ( t ) is the relaxation kemel, X' is the material viscosity and T is the characteristic

time of relaxation. Although this mode1 has a simple mechanical interpretation, it allows the

creep kemel to be found in explicit form and describes qualitatively the material response

observed in experimens for both creep and relaxation, but it shows poor agreement with

experimental data. A more sophisticated expression for the relaxation kernel is provided by

the so-called Prony series

With M= 10 , a good agreement with experimental data is predicted for different materials

The Creep kemel K( t ) conespondhg to the above relaxation measure is presented

in a form similar to that of equation ( 3.13 )

where p. and TL are material parameters. The constants ri' are called the characteristic times

of relaxation.


In Chapter III, the phenornenon of panel flutter is introduced. In connedion with the vibration

of the skin of modem aircraft, a great deal of attention has been given to problems on the

stability of plates in supersonic flow. Piston theory as a quasi-steady first order approximation

for aerodynamic loading is presented. Definitions of kernel and relaxation rneasure for non-

aging and aging viscoelastic matenals are introduced. Constitutive relations for viscoelastic

materials are also presented together with some suggested forms for the relaxation measure

and the creep kemel.

Chapetr IV

Stability of a viscoelastic plate under a stochastic axial thmst

4-1 Problem Formulation

Consider a long viscoelastic plate, one side of which is exposed to a supersonic flow

of gas, performhg a small oscillation. The plate is assumed to be effectively infinitely long and

fieely supported dong the long edges, Figure (4.1). Let w(x . t ) be the transverse deflection,

where x is the longitudinal displacement fkorn one edge, D the cylindncal stifhess, since we

confine to the case of cylindrical bending. A uniform t h s t q( r ) is applied to the mobite

edges at the mid-plane.

Using the correspondence principle ( the Voltera pnnciple ), solutions for certain

viscoelastic problems can be obtained by the corresponding solutions to elastic problems [63]

This principle is based on the hypothesis of deriving constitutive relations for viscoelastic

materials from those of elastic materials by replacing the Young's modulus by an appropriate

Voltera operator. The relaxation operator E(l - Ra ) replaces Young's modulus for the stress

- strain relation, where R' is an operator given by;

It is customary to take R' ( z ) in the fom

z 1

1 , ./--.

I / ' - , w(x. t ) -\\, X

al- /- 1/> '\ /"A 90)

'v- -

Figure 4.1 Schematic Diagram of Panel in Supersonic Flow under Stochastic Axial Thnist

R* ( t ) = C y o i ~ , ' e

where yi = 1/ T, , T, being the characteristic time of relaxation and x i' is the characteristic

viscosity.

Consider an elastic plate having bending in the x-direction and no-bending in the

y-direction; then

Therefore

where v is Poisson's ratio. Using the correspondence pnnciple, Young's modulus in

equation (4.1) can be replaced by the Voltera operator E( 1- R' ). Therefore, for viscoelastic

matenals, with bending in the x-direction and no bending in the y-direction

E( 1 - R') E,(z) Ox =

( 1 -v2)

Using Kirchhoff s hypothesis :

where z is the distance fi-om the neutral axsis along the thichness, w is the transverse

deflection. Substituting for E, into equation (4.2) gives;

Defining Mx as the moment per unit length of the plate, we obtain

so that

Let D = E K / 1 2 ( 1 - v2),then

For a uniform in-plane tluust q( i ), the contribution to the downward forces per unit length

42

of the plate is equal to - q( z ) ( d2w( x , t ) / d x ' ). If the mass per unit cross-sectional area

of the plate is m = ph. then the inertial force is - ph( d2w( x ,f ) / d t ') and the damping

force is - prie ( d w ( x ,t ) /d f ), where c is the damping constant. Assuming the aerodynamic

component due to gas disturbance as P( x , t ), the equation of motion o f a viscoelastic plate

in a supersonic gas flow subjected to a stochasticaiiy fluctuating in-plane t h s t q( t ) is given

by

Substituting for M from equation (4.3) gives

By making use of the piston theory [ S I , it is possible to have an approximation to the

component of the aerodynaniic pressure caused by the deviation of the plate from its

undisturbed state. Substituting for P( x , t ) from the relation of equation (3.3):

and taking K as the surnrnation of the plate structural and aerodynamic damping, that is

the equation of motion (4.5) becomes

which satisfies the boundary conditions of simple support:

then, with the help o f the Bubnov-Galerkin method, equation (4.7) becomes

where

~ x P - u j " b,,' = CœL 1 / sin nx cos j xdx O

if(n*~)imdd

zf(n*j) iseven

It has been shown [64] that not only qualitative but, to some extent, quantitative results are

predicted rather reliabiy with the use of the first two modes. Hence considering ( m = 2 ),

the reduced equations with K= O , Ra = O , te = O are

Let the non-dimensional time i , be given by;

and the prime ( ') denote differentiation with respect to the new time t , . Equations (4.12)

now become

where

The frequency equation of (4.13 ) is

a i - l 7 w ' + ( a 2 + 1 6 ) = 0

The roots are real provided 17 - 4 ( ar ' + 16 ) > O which implies

a s 1 5 / 2 = a , , or x P , u / C - i ( ~ P . u / C . ) , , = 4 5 x ~ D / 1 6 L ~

Then the frequencies are

O ,',O,' = 17/2 F J[(l5 /2 ) ' - a']

Using the transformation T,

46

where c' is a parameter to be chosen for a suitabie CO-ordinate scaling. Then equation

( 4.10 ) with m = 2 transfoms to

~ q ' AT^ = -KL2 Tq' + n4R ( T q ) + n 2 B ( t ) Tq

7r2 JphD

where

Substituthg for matrices A, T and B( r ), from equations (4.13), (4.15) and (4.16) and letting

2P =KL2/[n2J(phD)],

equations (4.16) become

where

We now choose the constant c' such that

k12= -%,

which implies

Substituting for w , , o 2, into ( 4.18 b ) gives

(15 + a,)'" - (17 +

2 where a 0 = 2 ( a , , - a2)112.Finallysubstitutingfor o , ,a, and cm,into(4.18a)

yields

4-2 Approximation to Markov Process

Consider the system of equations (4.17) ;

Equations (4.17) admit the trivial solution q , = q , = 0. 5 ( t, ) is taken to be an ergodic

aochastic process with zero mean value and sufficiently small correlation time. If P , and

the spectral density of ( t , ) are of the sarne order of smallness as compared to unity, the

stochastic averaging method may be used to obtain approximate Itô equations by making

use of the following transformation:

Then by the method of variation of parameters, the following 4first-order equations in

a,. 8 , . a, . 8, are obtained:

a2 a, = k,,a, s in~2(t l )cosQl(t l ) f ( t , ) + k2,-sin2@2(t,)~(tl) - 2 ~ a , ~ i n ~ @ ~ ( t ~ ) 2

It is assumed that the damping constant P, the relaxation measure R ( 1, ) and the stochastic

perturbation are small and of the saine order, P = O( E ) , R = O( E ) , S(O ) = O( E ) ,

Y (o ) = O( E ) , O < 1 el (( 1 where S(w ) and Y (o ) are the cosine spectral density and

the sine spectral density of 5 ( z , ) respectively and are given by

Y ( 0 ) = 2 , f ~ [ f ( t J C ( t ~ + r ) ] s i n o s d r

with E [O ] denoting the expectation operation Therefore, the system of equations (4.21) falls

into the category of system (2 .3 1) and the stochastic averaging method of KhaSminskii

[IO] and Stratonovich [Il] can be applied. As e decreases, the solution of the system of

equations (4.21) converges in the weak sense and up to first order in e to a difisive

Markov process whose governing Itô equations are of the form

da, = m,dt + 2 olJdw Y

/ = l

5 1

where w , j , w , are mutually independent unit Wiener processes. The drift coefficients

m i , lli are determined by equation (2.33) and the difision coefficients O,, , p ,,

are determined by equation (2.34) The most imponant feature of the stochastic averaging

method is that the limiting averaged amplitude processes a, ( r ) are decoupled fiom the

lirniting averaged phase angle processes O, ( z ). By making use of this decoupling property

for the amplitudes a, ( t ) and the phases 0, ( ), the investigation fiom now on will only

consider the limiting averaged amplitudes a, ( z ). Performing somewhat lengthy algebraic

calculations by making use of equations (2.3 3) and (2.34), and Larianov's method [65] for

averaging the integral part of equation (4.2 1) -see Appendix (A)-, the following expressions

are obtained:

where the superscript T denoting matrix transpose. R ( o , ), R ( o , ) are the one-sided

Fourier sine transfonns of the relaxation fùnction R( t ) at the frequencies o , and o,

respectively, and are defined as follows:

Expressions S + and S ' are defined by

S' = S ( 0 , + w,) + S(0, - O,)

S - = S ( 0 , + 0,) - S(ol - O,)

4-3 Lyapunov Exponent

The averaged amplitude vector (a, , o ) is a two-dimensional diffusion process,

and the coefficients on the right-hand side of equations (4.23) are homogeneous in a , ,a, of degree one. Therefore KhaSrniniskiiys technique, descnbed in section (2.3.3), may be

employed to denve an expression for the largest Lyapunov exponent of the amplitude process

[66].Tothis end a further logarithmic polar transformation is applied:

By making use of Itô differential rule, descnbed in Section (2.2.4 ), the following pair of Itô

equations governing p, @ are obtained.

where

dp ecw = m, - +m2- + -

a2 P a2 P da2 2 [oaTll1 .i + [ < J ~ ~ I ~ ~ - +2raaT~12

a% '[ dal a da, da,

a@ @(O) = m, - aZ 0 +[oaT]222 + 2 [ o d 1 ~ ~ da , a% da, da2

a 2 p - az2-a, 2 2

-- -- d2p - a, -a; da,' (al2 + O~~

>

Sa,' (al2 + aZ2l2

M e r performing somewhat lengthy algebraic calculations, equations (4.28 ) can be

expressed as :

The constants A, and A, are defined by:

The second of equation (4.27) shows that the @ - process itself is a diffusion process on the

first quadrant of the unit circle. To examine the ergodic property of the 4-process in the

interval [O, d 2 ] , we need to know if the fùnction Y' ( 4 ) is positive in the interval [O, 7~/2].

From the fourth of equation (4.30),

Ci eariy

and therefore Y '( 4 ) z O in the interval [O, ir/ 21 so that the @- process is ergodic. From

equation (4.19)

k ,, = k , = - 1 .O43 at a, = J 207, therefore for Y? ( @ ) to vanish at

@ = @ = x / 4. one of the following sets of conditions must be satisfied:

( i ) k , , = O , S ( 2 a 2 ) = 0 , S ( a 1 + a 2 ) = 0

(zi) k,, = O , S ( 2 0 , ) = O , S ( o , + O,) = O

For 'fR ( ) to vanish at 4 = 4 , = O, x / 2, the following condition must be satisfied:

4.3.1 Nonsingular case

When the diision coefficient of the diffusion phase process 4 ( t ) does not vanish

57

at any point, the dfision process ( t ) is nonsingular. The density p ( 4 ) of the invariant

measure is govemed by the following Fokker - Planck equation:

the general solution of which is

where C, G are integration constants and

Substituting for @ (@ ) and Y * (@ ) from equations (4.30) into equation (4.37) and letting

we obtain

= exp

we finally obtain

Since the constant n is always positive, the integration in (4.38) depends on the sign o f the

constant b.

For no accumulation of probability mass at the boundaries, the stationary probability

59

flux represented by G is zero, and the 4 - process is ergodic throughout the interval

O i 4 s x /2 . The invariant density p (+ ) is then given by

where C is determined from the normalization condition

Performing the integration of equation (4.38 ) for b > O , b < 0 and b = O and using

equation (4.39), one obtains ;

C. sin24 -(Al - A 2 ) tan- l Y2u2(4) 2-

C. sin2G 'XP [ (A, - ri,) cos20

'u2(4) 2a

where A = a b. To obtain the normalizing constant C, equation (4.40) together with

equation (4.4 1) can be used.

( i ) b>O

Substituting for (@ ) from equation (4.30) and taking cos 2@ = t and

performing the integration in (4.42a) transforms it to

Letring

and carrying out the integration in (4.42b) leads to

so that

( i i ) b<O

Similarly, the normalking constant C for b c 0, can be obtained as

C = 0 1 - k2)

csch

( i i i ) b = O

By making use of equation (4.40) and substituting for p(4 ) from equation (4.4 l), one

obtains

Substituting for y' (4 ) = kL S' 1 8 = a , since b = 0 and taking cos 2 4 = t ,

the integration in (4.44 a ) changes to

which gives

In sumrnary, the normalking constant is obtained as:

1 -(A, - A,) C S C ~ 2

1 -(A, 2 - A,) csch

By making use of KhaSminiskii's procedure described in section (2.3.3), the largest

Lyapunov exponent for system (4.23) is given by

From equation (4.30) , Q(4 ) can be simplified as

63

( 1 ) b > O

Substituting for Q(@ ) and p(@ ) from equations (4.30) and (4.41) into equation

(4.46) gives

where

Let

so that

From equation (4.30),

Setting

and performing the Integration in (4.47b) transforms it to

After Integrating by parts one obtains

one Enally obtains

C.a ( A 2 - A l ) 26 - . coshy = C . coshy

Substituting the appropriate value of C from equation (4.45) gives

1 Il = -(A, - A,) coth(- y )

2

Therefore, the largest Lyapunov exponent for b > O is obtained as

( 2 ) b < O

Similarly expression for the largest Lyapunov exponent in this case can be

obtained as follows :

( 3 ) b = O

When b = O, a = k Z S ' 1 8 and hence 'fR ( ) = a. Substituting for Q(4 ) and

~ ( 4 ) fiom equations (4.30) and (4.4 1) into equation (4.46) gives

where

integration in (4.50b) transforms it to

= a.C

. 7 1 1 - y l ë Y d y = a. C (A2 -AL)

coshy' = C. coshy' ( A2 - Al (A2 -Al). a

Substituting the appropnate value of C fiom equation (4.45) gives

Therefore, for b = 0, the largest Lyapunov exponent is given by

In summary, the largest Lyapunov exponents for the system o f equations (4.23) c m be

obtained as

I 1 k2 1 -(A1 + A,) - -S- + -(A1 - A,) ~ 0 t h 2 8 2

(4.52)

The expressions in equation (4.52) can be simplified to another more convenient form .

For b > O : Let

A = '

so that

1 k2 1 - ( A l +A, ) - -S' + - ( A , -A,)coth b<O 2 8 2

1 kt 1 -(A1 + A,) - gS' + - (hl - A,) ~ 0 t h 2 2

¶

\

Substituting for a , b , and taking A , = 16 A , q ,, = 2 q gives ;

[ k l 1 2 S ( 2 q ) + k222S(202) + t k 2 S - ] cosh ri, =

Therefore, for b > 0,

For b < O, let

and t herefore

The largest Lyapunov exponent can be summarized as foliows:

4.3.2 Singular case

In this case the process @ ( t ) is not ergodic in the whole interval [ O , ir / 2 ] and

there are some points 4 , at which the diffusion coefficient Y($ , ) of the phase process

( r ) vanishes.

( l ) @ , = n / 4

For the process @ ( t ) to have a singular point at @ , = n / 4, one of the condition

equation (4.33) must be satisfied . Substituting one of such conditions into equation (4.30)

gives the drift coefficient @(x/ 4 ) of the @ ( t ) -process at the singular point 4 , = sr / 4 .

The value of the drift coefficient at a singular point determines the nature of the singular

point [67] .

For such condition @(n 1 4 ) > O and therefore the singular point 4 , = x / 4 is a right

70

shunt, this means that even if an initial point @ is in the lefi haif interval ( O , KI 4 ), it will

eventually be shunted across to the right haKinterval ( x/ 4 , n/2 ) and remain there forever.

The probability density is concentrated in the nght half of the intervai O s 4 5 K I 2. The

density p(4 ) of the invariant measure is govemed by the Fokker-Planck equation (4.35)

whose solution now is of the form

where for one of the conditions of equation (4.33)

Substituting for @(a ) and Y(@ ) fiom equation (4.55) into equation ( 4.37 ) gives;

U(0 ) =

1 l R ) 1 6 R J q \ - k2 sin24 + -S(o, - y ) c o t 2 ~ c Q s 2 2 @

2 \ 20, e x p -2[

202 1 8 a

k2 - S ( o , - y)c0s22c#l 8

1 W O ) = - sin 2 4 e*P I - 202 ) sec24

k 2 s ( o , - o,)

The constant C in equation (4.54) is determined by the following normalization condition:

ubstituting for p(@ ) from (4.54), Y? ($ )fiorn equation (4.55) an( i U (@ )from (4.56 ) gives

Rs(q 16R, (02)

. exp k 2 S ( o l - 02)cos2@

Letting

The integral of (4.57a) transforms to

Substituting for Q($ ) and p(+ ) fiom equations (4.55) and equation (4.54) into equation

(4.46) results in

where

Let

so that

Integrating the Iast equation by parts, one obtains

Substituting for C fiom equation (4.57 b) gives

Finaily, substituting for 1, into equation (4.58 ), the largest Lyapunov exponent for the case

where there is a nght shunt singular point 4 , = x/ 4 ,which occurs when

R , ( o , ) I Z o , > 8 R , ( o J l a?, isgivenby

This gives @(x 14 ) < 0 and the singular point 4, = x 1 4 is a left tshtzriit. this means that

even if an initial point 4 is in the right half interval ( sr/ 4 , rcl 2 ), it will eventually be

shunted across to the lefi half interval ( 0 , x14 ) and remain there forever. The probability

density is concentrated in the lefi haif of the interval O s 4 s x 1 2. The density p(+ ) of the

invariant measure is govemed by the Fokker-Planck equation (4.35) whose solution now is

of the form;

Where U(@ ) is given by equation (4.56 ) and the constant of normalization C is determined

inasimilarwayasforthecase R , ( o , ) l Z o , > 8 R , ( w J l w , andisfound tobe

S d a r l y as for the right shunt case, the largest Lyapunov exponent corresponding to the lefl

shunt singular point @ , = nl4, which occurs when

R , ( o , ) / 2 o , < 8 R , ( o & l o z , isgivenby

Then a(@ ) = 0 and the singular point 4, = rr 14 is a trap. This means that regardless

ofwhere the initial point 41 is situated, it will eventually be attracted to the point 4, = sr 1 4

and remain there forever. The density p(@ ) of the invariant measure is the Dirac delta

furiction concentrated at x 1 4 and is given by

where 6 is the Dirac delta funetion. Substituting for Q(@ ) from equations (4.55) and

p(@ ) from equation (4.63), into equation (4.46), the largest Lyapunov exponent for the trap

singular point @, = x 14 when

R s ( o , ) / 2 o , = 8 R , ( o 3 / o, isfoundtobe;

For the 4 - process to have a singular point at O and n/ 2 , the condition

S( o , + w, ) = S( o ,- o, ) = O mus be satisfied. Since @(O ) = <P(x / 2 ) = O, both singular

points 4 , = O , x12 are trap points.

(2-1) For @ ,=O

If A, > A, The largest Lyapunov exponent is given by

(2-2 ) For @ , = sr1 2

If A, < A, The largest Lyapunov exponent is given by

If the largest Lyapunov exponent of the Ito system of equation (4.27 ) is taken as an

approximation to that of the system of equations (4.18 ), then the asymptotic approximation

for the largest Lyapunov exponent of system (4.18 ) can be summarized as follows:

( 1 ) Non-Singular Case

(1-1) b > O :

where

A , A , k and A , has the same expressions as for the case b > O

A, , A, , k and A , have the sarne expressions as for the case 6 > O .

( 2 ) Singular Case

(2-1) 4 ,= x / 4

qoa = cos-'.

r \

25 ( 9 - a 0 ) ' S ( 2 o , ) 25 ( 9 + ~ t , ) ~ S ( 2 u ~ ) 18 ( 1 5 ~ - ao2) + + S -

(17 - 0 ) (17 +a0) ( 1 7 ~ - u , ~ ) " ~

18(15~-a,2) 2 112

S' (17' -a , ) . 1

(4.68)

Substituting for o , , o , and k gives

Substituting for o , , o , and k gives

Substituting for 0 and k ,, gives

Substituting for o , and k , gives

4.4 S tability Analysis

The trivial solution of equation (4.18) is asymptotically stable w.p. 1 if 1 is negative

and unstable w.p. 1 if Ii is positive. The stable region of aimost-sure stability for the systern

of equation (4.18) is detemiined by L c 0, and the stability boundary may be defined by

setting A. = O, which gives a relation amongst P , R , ( o , ) , RI ( o , ) and S ( o ). P and

a, are fiindons of plate dimensions, plate material type, undisturbed gas condition and flow

velocity . R, ( o , ) , RI ( o , ) are the one-sided Fourier sine transfoms of the relaxation

f ic t ion R ( t ) evaluated at the resonance frequencies o , and o , , respectively. S(o , ) is

the spectral density of the in-plane stochastic excitation E(t , ) evaluated at the frequencies

o , = 2 0 , , 2 o , and o, * o , , sine only £ira order approximation is considered. If higher

order approximation is performed, values of excitation spectrum at other fiequencies may

enter the stability condition.

For this analysis a narrow-band excitation and a white noise excitation are

considered. For the narrow-band excitation, the spectrum is nonzero only in a neighbourhood

of some 6equency o , , where as for the white noise excitation, the spectrum has a constant

value for ali frequencies o. The narrow-band excitation case gives a good qualitative picture

of the effect of the excitation spectrum on the dmost-sure stability [68].

4.4. 1 Narrow-band Excitation

In this case only a narrow frequency bandwidth, o , - A o , 1 2 < o < w , + A o , 1 2

where A o , « a, is considered. The excitation spectrum S(o) vanishes outside this

narrow fiequency bandwidth. If the correlation time of the excitation process, which is of

order O(11 A o , ) , is small compared to the relaxation tirne of the system response, which

82

is of order O( 1/ E ), that is to Say A o , » E, then the Markovian approximation obtained

by making use of the stochastic averaging method [Il] is justified. Since only a first order

approximation is considered, the stability condition depends on the spectral density of the

stochastic in plane excitation at the fiequencies 2 0 , , 2 o , and o , & o , . Therefore it

is suficient to deal with the cases when o , is equal to 2 0 , , 2 o , and o , o , . (1) 0, = 2 0 ,

Inthiscase S ( 2 o , ) = S ( o 1 + o , ) = S ( o , - o , ) = O

25 ( 9 - a,) b = S ( 2 0 , ) > O

64a:(17 - a,)

For h, > h2 the largest Lyapunov exponent is given by

Substituting for a , , P and k,, , from equations (4. M), (4.16), and (4.19 b) the system is

always stable if

Otherwise the system is unstable.

(II) 0, = 2 0 2

Inthiscase S ( Z o , ) = S ( o , + o , ) = S ( o , - o , ) = O

For b, c A, , the largest Lyapunov exponent is given by

the system is dways stable, otherwise the system is unstable .

( W oo=S(o,+o,)

In this case S(2 o , ) = S(2 o , ) = S( o , - o , ) = O,

b = 0,

The largest Lyapunov exponent is given by

The system is stable if

9 ( 1 s 2 - a,') S ( 0 , + 0 2 )

2aO2( 1 7 ~ - a t ) 1 / 2 J Otherwise the system is unstable . Here o , and o , are given by

Al' = 1 - * A, k2S(o l + o,)

the stability boundary given by A , = O may be expressed in the form

( Z V ) o o = S ( ~ , - ~ * )

We now have S(2 a , ) = S(2 a, ) = S( o , + o ) = 0

Since in this case A, > A, for ail values o f a, the largest Lyapunov exponent is given by


Othenvise the system is unstable.

4.42 White noise excitation

For white noise excitation the spectral density of the excitation has a constant

value S for al1 values of the frequencies o , so that

S(20, ) = S( 2% ) = S ( 0 , - y ) = S(o, + y ) = S

S + = 2 S and s ' = O

Therefore

qom = cos-' [ ( k b 3 ]

By substituthg for the constant spectral density S into equation (4.32) it can easily be shown

87

that the drift coefficient of the diffusion phase process does not vanish and hence the white

noise excitation corresponds to the non-singular case. Putting A = O into equation (4.53) the

stability boundaries can be obtained and stability conditions can be deduced as follows:

For b > O ( or a , > 9.835 ), the system is stable if

Otherwise the system is not stable

For b < O ( or a , < 9.835 ), the system is stable if

For b = O ( or a , = 9.835 ), the stability condition is

88

4.5 Numerical results

As an example for performing some numencal results, polyurethane is considered

as the plate material which has density = 103 0 kg / m3 and modulus of easticity E= 3.1 x 1 o9

N lm'. To jus* using thin plates theory, L / h = 100 is taken for the length -thickness ratio

and aardingly the cyiindrical stifhess D = 283.883 L) N.m, To apply piston theory for the

aerodynamic loading approximation, Mach number must be M o > J 2 . Studying the stability

of a viscoelaçtic plate type structure at an altitude of 10 h , ,r pressure = 198.765 mm Hg

and the polytropy index is taken as 1.4 for air considered as a perfect gas [ 69 ] . Therefore

the value of the non-dimensional flow parameter at which supersonic flow can occur is a >

5.37 and hence maximum a , < 10.5 . For an appropriate fiequency difference 1 o , - o, 1 ,

a < 7.35 and therefore a , > 3.0 can be considered . A range of 5.37 c a < 7.35 and hence

10.5 > a , > 3 .O will be considered . If' the relaxation function is given by

then by using equations (4.25) and (4.85), the one-sided Fourier sine transforms at the

frequencies o , , o , of the relaxation function cm be given by

- m Y O ~ X , " ' ~

R,(ol) = 1 yol x,' e - Y O t r . s i n q tdt = O 1 = 1 2

' = l yor + O r 2

89

Considering polyurethane as the plate matenal and adopting data from Christensen [70],with

R '( t ) = R ( t , ) one can obtain for m = 1 , y,'= 18.52 sec -' and xi* =0.283,

First stability boundaries are plotted in Figure 4.2a for a narrow band excitation at

frequencies 0, = 2 0 , , 2 0 , , 1 a, - q 1 for a spectral density value of S(o, ) = 0.95 .

It cm be seen that S(202 ) has the largest destabilizing effect because it is associated with

the term 1 8 which is dominant over the other terms l?,, 1 8 and k? 1 8 , Figure ( 4.16).

It can also be seen that, as the non-dimensional parameter a, increases, the destablizing

efféct for the different excitation values decreases. S(2o , ) has the least destabilizing effect.

Figures 4.2 4.3 and 4.4 show the effect of the variation of the excitation spectral densities

S(20 , ), S(20 , ) and S(o , - ci5 ) for three different values 0.30,0.60 and 0.95 under a

narrow band excitation. The Lyapunov exponent as a stability indicator is plotted in Figure

4.5 for the narrow band excitation at 0 , = (a, + w , ), fiom which it may be infemed that the

spectral density S(o , + o, ) at this fiequency stabiiizes the system. In order to show

the interaction among dierent spearal densities S ( o , ), stability boundaries for broad-band

excitations are plotted in Figures 4.6 , 4.7,4.8 and 4.9 . The spectral density S(2a , ), has

an extremely srnaii destabiiizing effect while S(2w2 ) has the largest destabilizing effect as for

the case of narrow- band excitation. Figure 4.9 shows the stabilizing effect of S(w , + w , ) and it cm be observed that the system is more stabilized with an increase in the spectral

90

density S(w, + y ). Figure 4.10 shows the stability boundaries for the case of white noise

excitation Figures 4.1 1,4.12, 4.13, 4.14 and 4.15 show the stability boundaries for different

flow conditions and it can be concluded that more damping is required for stabilizing the

system as the flow becomes more supersonic.


In this chapter the Voltera correspondence principle is used to denve the integro-partial

differential equation of motion of a viscoelastic plate in a supersonic flow of gas under a

stochastically varying axial thrust. The Bubnov - Galerkin method is used to discretize the

integro-partial differential equations of motion to a two -degree of freedom system. A non-

dimensional tirne, together with a suitable transformation and a CO-ordinate scaling, is used

to transform the integro-ordinary differential equations of motion to a more convenient form.

A method of variation of parameters is used to convert the transformed integro-ordinary

differential equations to equations in amplitudes and phases. By making use of the stochastic

averaging method, together with Larionov's method for averaging the viscoelastic integral

term, the amplitude and phase equations are then approximated to a system of Itô equations

whose solution is a d f i s ive Markov process for damping and axial thrust of the same order

of srnallness . Khasminskii's technique is used to obtain asymptotic expressions for the largest

Lyapunov exponent . Both the non-singular case, when the drift coefficient of the phase

process does not vanish at any point, and the singular case are considered. Based on the

analytical renilts, a stability analysis is perforrned for narrow-band and white noise excitations

Numerical results are presented which show that the almost -sure stability depends on the

value of the spectral derisities at the four frequencies w, = 2 0 , , 2 0 , , o , - o, and

9 1

a, + o, and on the gas flow condition, which is detennined by the non-dimensional parameter

a ,. The individual effècts of the spectral density of the excitation at each of these fiequencies

are discussed. Also the effect of the variation of the spectral density for the white noise

excitation is presented.

Figure 4.1 1 Effect of the non-dimensional parameter a, on stability boundaries under a broad-band excitation with different values of S(20, )

Figure 4.13 Effect of the non-dimensional parameter u, on stability boundaries under a broad-band excitation with different vaues of S ( o, - a, ) for a viscoelastic plate

I

CHAPTER V

Stability of an elastic plate in a gas flow under a stochastic axial thrust

5.1 Problem formulation

An elastic plate, effectively infinitely long with one side exposed to a

supersonic flow of gas , performing a smail oscillation is considered. A unifonn stochastic

axial thnist is applied to the mobile edges at the mid-plane. The partial differential equation

of motion can be obtained fiom equation (4.5) by putting relaxation measure R' ( t ) = 0:

Using the same procedure as described in Chapter IV, expressions for the largest

Lyapunov exponent c m be obtained as

where

k,, ' , & * , kz and b have the sarne expressions as in Chapter IV .

5.2 Stability analysis

The system is stable if ri c O and the stability boundanes can be defined by setting

A = O. Since only a Brst order approximation is considered, the spectral density of the

in-plane axial t h s t is evaluated only at the fiequencies 2o , , 2 0 , and o , * o , . To give

a good qualitative p i a r e of the effect of the variation of the non-dimensional flow parameter

a, as well as that of the spectral density of the excitation, narrow band and white noise

excitations, as for the case of the viscoelastic plate, are considered.

5.2.1 Narrow-band Excitation

( i ) 0 , = 2 0 ,

S (2y )=S(01+o , )=S(o , -o,)=O

25(9 - b = S ( 2 0 , ) > O

64E(,Z( 17 - a,)

25(9 - AIg = - P + S(20,) , 1,' = - p

16aO2(17 - a,)

The largest Lyapunov exponent is given by A = A,' and the system is stable if

25(9 - a,)' P'

16aO2(17 - a,) W o , )

25 (9 + a,)' b = S(20,) > O

64a:(17 + a,)

25(9 + a,)' 4 * = - p , A , = = + + 16a02( 17 + a,)

S(20,)+

The largest Lyapunov exponent is given by A = A,' , the system is stable if

25 (9 + aJ2 P'

16aO2(17 + a,) W u 2 )


9 (225 - a,') P'

2 1/2 S ( 0 , -a,) 16a:(289 - a, )

The system is always stable

5.2.2 White noise Excitation

White noise excitation corresponds to the non-singular case.

S(2o,)=S(20 , ) = S ( o , - o , ) = S ( o , + o , ) = S , S + = 2 S , S = O

For b > 0 , the system is stable if

For 6 < 0 , the system is stable if

For b = O , the system is stable if

5.3 Numerical Results

To compare with the results for the viscoelastic case, the range of the non-dimensional

flow parameter 5.37 < a < 7.3 5 , or 10.5 > a, > 3 .O as in Chapter TV is considered. In

Figure 5.1 stability boundaries for a narrow-band excitation at fiequencies 2 0 , , 2 0 and

o , - o , , for a spectral density value of S(o, ) = 0.95 are plotted. The effect of spectral

densities at the fiequencies 0, = 2 0 , , 20, and o , - w on the stability boundaries under

a narrow-band excitation for three diEerent values 0.30 ,0.60, and 0.95 are plotted in Figures

5 -2 , 5 -3 and 5.4 . As for the viscoelastic plate , the largest destabilizing effect is due to the

excitation at o, = 2w, . S(w , - o, ) has a small destabilizing effect while S(2o , ) has the

least effect . Stability boundaries for the elastic plate show stight diference from those for

the viscoelastic one, especially at the frequencies 2 0 , , and o , - o , because of the small

terms k,, / 8 and k / 8 . A considerable sensitivity in the stability boundaries is observed

115

for changes in the spectral density S(2w , ). As the non-dimensional flow parameter q,

increases the effect of the variation of the spectral densities becomes insignificant. More

damping is required for stabilizing the elastic plate as the fiow becomes more supersonic than

that required for the viscoelastic plate. Stability boundaries for broad-band excitation for

different values of spectral densities at o , = 2 0 , , 2 0 , and a, * o , are plotted in

Figures 5.5,5.6, 5.7 and 5.8. Again an appreciable sensitivity of stability boundaries is

observed for changes h the spectrai density S(2o, ). As for the narrow-band excitation, the

variation in the excitation has small effect on the stability boundaries at low flow conditions.

The e f k t of the white noise excitation is shown in Figure 5.9, and it may be observed that

more damping is required for stabilizing the system as the excitation spectral density

increases.

5.4 Computer Simulation

To determine the largest Lyapunov exponent, the invariant measure of the Markov process

has to be evduated. Since it is difficult to solve the integral equation for the invarient measure

even in two-dimensionai cases, a numerical solution was pursued. An efficient algorithm was

devolped by Wedig ( 1990 ) which is based on Khas'minskii's concept [ 711. Computer

simulation is important to confimi the range of validity the obtained analytical results and the

applicability of the stochastic averaging method. Considenng the system of equations (4.17)

with R = O

and using the transformation

four first-order equations are obtained:

For simplicity, suppose that the random excitation ( t , ) cm be appromimated by a white

noise process with constant spectral density S = constant for al1 o. Therefore,

6 ( t , ) d t , = d S dw ( t 1 ) where w ( t 1 ) is a unit Wiener process, so that the Ito stochastic

differential equations corresponding to (5.14) are of the form

dx = F x d t + GxdW ,

where

117

As an example, the following values of the parameters are chosen: P = 0.1 and a , = 7.5 .

Therefore,q=2.18, o, =3.5, kl1=0.2294, k,,=0.9407 , k,,= -0.9407

and k, = - 1 S71. Talcing the step size for numencal integration as A t = 0.00 1 and the

number of iterations N, = 5 x 10 . 6 = - 0.03 18 S < O , the analytical result for the largest

Lyapunov exponent is given by

which gives A = - 0.1 + 0.359 S. The result of digital simulation and those obtained from

equation (5.16) are shown in Figure 5.10 for comparison.

It may be seen that for small values of S , the analytical results are consistent with the

results &om cornputer simulation . nie case S - O , gives the correct tangent line of the actual

curve of the largest Lyapunov exponent, which confirms that the stochastic averaging

method is a valid first approximation method for finding Lyapunov exponent.


The elastic plate under a narrow-band, broad-band or white noise excitation requires more

darnping for stabilization than the viscoelastic plate under the same excitation. As the flow

becornes more supenonic, the destabilizing effect of the spectral densities at each fiequency

increases. The spectral density S(2o, ) has the largest destabilizing effect, because of the

dominant stifhess tem b2 / 8, S(o , - o, ) has small destabilizing effect and S ( P o , ) has

an extremely small destabilizing effect.

V) c O .- Cr,

f!

SUMMARY AND CONCLUDING REMARKS

The main objective of the present thesis is to obtain an almost-sure stability condition for

a viscoelastic plate in a supersonic flow under a stochastically fluctuating axial thrust.

This was done by deriving anaiytical expressions for the largest Lyapunov exponent.

ModeMg based on probabilistic terms as a need of modem technology, together with the

concept of Lyapunov exponent as an almost-sure stability indicator, were introduced in

Chapter 1. Chapter II is devoted to a mathematical review of Markov processes and the

associated Fokker-Planck equation. Definitions of the main tmls needed for stochastic

analysis, such as diffusion process, Wiener process, Ito differential equation, Ito

di fferential lemma and the stochatic averaging rnethod were introduced. Different

stochastic stability definitions and Khas'minskii's procedure for the evaluation of the

largest Lyapunov exponent were presented. In Chapter III, flutter phenornenon, piston

theory as a quasi-steady first order aeorodynamic loading approximation, together with

constitutive relations for aging and non-aging viscoelastic matenais were presented.

Customarily used fonns for relaxation functions and creep kemel were introduced. In

Chapter IV, the Voltera correspondence principle was used to derive the integro-partid

differential equation of motion for an effectively infinitely long, viscoelastic plate in a

supersonic gas flow and subjected to a uniform stochastically fluctuating axial thrust. The

integro-partial differential equation was then discretid to a two degree of freedom

system by the use of the Bubnov -Galerkin method. By making use of a non-dimensional

129

time t, , an appropnate Iinear transformation T and a suitable CO-ordinate scaling c*, the

equations of motion were transformed to those in terms of a more convenient generalized

CO-ordinate. The transformed integro-ordinary differential equations were converted to

equations in amplitudes and phases by the method of variation of parameters. Assuming

that the damping and the axial thrust intensity are of the same order of smailness, the

amplitude and phase equations were approximated to a system of Ito equations, whose

solution is a Markov diffusion process, by making use of the stochastic averaging rnethod.

Ushg the Ito differential rule, a pair of Ito equations goveming the natural logarithm of

the nom of the averaged amplitudes vector and the phase angle were obtained. By making

use of Khas'minskii's technique [Il], the pair of Ito equations were used to obtain the

stochastic stability conditions for the original integmordinary di fferential equations of the

plate in a first approximation. Numerical results were presented to show the effect of the

non-dimensional flow parameter a, and those of the spectral densities at different

frequencies. In Chapter VI, the special case of an elastic plate was presented. Numencal

results for narrow band and white noise excitations were obtained. The evaluation of the

invariant measure of the Markov diffusion process defined on a unit hypershere is

necessary for the determination of the largest Lyapunov exponent. Although it is easy to

set up an integral equation for this invariant measure, it is difficult or even impossible to

solve the equation for more than two-dimensional systems. Therefore, numerical analysis

is needed and efficient digital simulation schemes have to be developed. Wedig 1721 has

recently developed a numerical algorithm for evaluating the largest Lyapunov exponent of

linear continuous systems by following Khas' minskii' s concept. By applying the Wedig

130

simulation scherne, a numerical analysis was performed to confirm the range of validity

of the analytical results and the appiicability of the stochastic averaging method.

In the present study, only the Linearized form of the equation of motion was

considered to obtain the almost sure ( a.s.) stability conditions for the trivial solution. In

reality there will also be noniinearities arising from both stnictural and aeroelastic forces.

When these effects are considered, the trivial solution wiil become unstable and undergo

a dynamical ( or D- ) bifurcation to a new non-trivial solution when the maximal

Lyapunov exponent vanishes, Ariaratmm [73]. A sudden change in the qualitative

behaviour of a dynamical system as some parameter is varied is b o w n as bifurcation. It

is then necessary to examine the probability distribution and the stochastic stability of this

new solution. The probability distribution can in principle, be determined as the solution

of the associated Fokker-Planck equation. In order to examine the stability of the

bifurcating solution, one must linearize the equation of motion around the non-trivial

stationary solution of the original non-linear system and determine the corresponding

maximal Lyapunov exponent. In general, there wiil be a further bifurcation charac terized

by a qualitative change in the probability distribution of the non-trivial solution from a

unimodal to a bimodal distribution. This latter bifurcation is refend to as a

phenomenological ( or P-) bifurcation and is the one commonly studied by physicists.

Recently, it has been shown by Baxendaie [74], using large deviation theory, that the

critical parameter value at which a P-bifurcation occurs can also be determined by

calculating the so-caiied moment Lyapunov exponent of the l inhzed systern about the

trivial solution. The pth-moment Lyapunov exponent A( p ) p E W, is defined by

where

If d denotes the dimension of the system, which in the present case is 4, a stochastic P-

bifurcation will occur when p' = - d where 'p t O is the second zero of A( p )

therefore,A( p' )=O. The number p' is referred to as the " stability index ". It is aiso

hown Arnold [75],that the maximal Lyapunov exponen t 1, whose vanishing indicates the

onset of a D-bifurcation, is related to A( p ) by the relation A'( p ) = A.

Future work in the nonlinear aspects of the problem considered in this thesis is

needed to examine and confm the validity of the aforementioned statements. The method

of analysis presented in this thesis can be also applied to plates with other end conditions.

The only difference will be in the modal shape functions used in the expansion of the

deflection w ( x. r ). In the case when the plate cannot be regarded as being 'long' in one

direction, a double series modal expansion of w ( x, y, r ) has to be used to denve the

equations of motion for the modal amplitudes. For sufficient accuracy, at least a four

degree of freedorn system has to be considered. The stability of the plate has to be then

determined using numerical methods since the analytical approach used in this thesis

cannot be readily extended to more than two degree of freedom systems.

APPENDDC A

Larionov's Method for Averaging the viscoelastic Integral Term

The one-sided Fourier cosine and sine transforrns of the relaxation function are

respectively defined as:

R J o ) = R ( S ) coswsds O

R J o ) = [ ~ ( s ) sinosds

For averaging an integral term in an integral differential equation, Larianov method can

be used. The averaging operator M, { F( a , @ , t )} is defined by

When the quantities a , , ir. the integral are regarded as constants, the integral term for

the first of equation (4.2 1) cm be averaged as

1 T r

- 1 - - - lim - [ IR(? -r)alsinO,(t) cos8,rdsdt o, T-- T

O O

Letting s = t - r t h e n , 0 , t = o , t + @ , , O , t = o , t + @ , , T = t + s

and using the relations [65],

A ( f ) = 1 o(s) cospsds ; B ( t ) = / o(s) s i n p d s O O

Similarly the averaged of the integral term of the amplitude a , can be obtained as

Fortran program for evaluation the largest Lyapunov exponent for a linear system of ito stochastic differential equations VARIABLES DT - Time step dt NO - Number of equations NP - Number of points to be simulated NS - Number of iterations for each point simulated S - Vector monng the values of S SQ - Vector storing the new values of S VO - Vector storing the sign of S F - Matrix of drift coefficients G - Matrix of diffiision coefficients N - Interger, the seed for generating random numbers FLAG - Flag for availability of random number O - NO, 1 - YES VLA - the largest Lyapunov exponent IMPLICIT REAL.8 (A - Y O - 2) REAL.8 S(4), SQ(4), VO(4) REAL'8 F(4,4), G(4,4) COMMON BLKU SIGMA input parameters from data file 'simu-in' OPEN (1 ,FILE ='simu. in') output is written on the file 'simu.res' OPEN (2,FILE ='simu.rest) READ (1, *) DT,NO,NP,NS SQDT = DSQRTPT) DO 100 LOOP = 1, NP READ (1, *)SIGMA analytical result ANA = -0.1 + 0.359*SIGMA obtaining the elements of matrix F and G CALL COE(F, G, NO) initializing the veaor Su) S m = O.ODO DO 20 I = 1, NO-1 S(1) = OSDO SUM = SUM + S(I)*S(I)

CONTINUE S(N0) = DSQRT(1 .ODO-SUM) initializing the vector VO(I) DO 10 1 = 1,NO

VO(1) = 1 . ODO CONTINUE N = 2 3 IFLAG = O K A = 0-ODO starting simulation DO 1 I = 1,NS searching out the largest Se) WMAX = O.ODO D Q 2 J = l , N O VO(J) = DSIGN(VO(J),S(J)) IF @ABS(S(J)).GT.WMAX) THEN L = J WMAX = DABS(S(J)) END IF CONTINUE obtaining the random number and evaluating dw IF (IFLAG.EQ.0) THEN CALL RANDOM(N,RNl,RN2) RN=RNl XFL,AG= 1 ELSE RN = RN2 IFLAG = O END IF DW = DSQRT@T)*RN evaluating ALPHA, BETA AND GAMMA ALPHA = O.ODO BETA = O.ODO GAMMA = 0-ODO DO3 K = l,NO D O 3 M = l,NO ALPHA = ALPHA+F(K,M)* S(K)* S(M) BETA = BETA+G(K,M)*S(K)*S(M) D O 3 L L = l , N O GAMMA = GAMMA+G(LL,M)*G(LL,K)* S(M) * S (K) CONTINUE applying the iteration formula to find new SQ(J) SU = 0-ODO D O 4 J = l , N O IF (J.EQ.L) GO TO 4 SQ(9 = S(J) + ((1.SDO*BETA**2-ALPHA-O.SDO*GAMMA) *DT-BETA*DW)* S(J)

D O S K = l , N O SQ(J) = SQ(J) +((F(J,K)-BETA*G(J,K))*DT + G(J,K)*DW)* S(K) CONTINUE SU = SU + SQ(J)*SQ(J) CONTINUE IF SSU is LESS than O, then time step is too large 'error'

a smaller time step should be taken SSU = 1.ODO - SU IF (SSU.LT.O.OD0) THEN WRITE (2,*) 'ERROR' GO TO 1000 END IF SQ(L) = VO(L)*DSQRT(SSU) SQ(I) is ASSIGNED to S(J), ITERATION IS REPEATED D O 6 J = l , N O S(J) = SQ(J) CONTINUE VLA = VLA +ALPHA - BETA**2 + O.SDO*GAMMA CONTINUE the largest L yapunov exponent is obtained VLA = VLA /NS W T E (2,*) SIGMA, V L 4 ANA CONTINUE STOP END elements of the matrices F and G SUBROUTINE COE(F,G,NO) IMPLICIT REAL* 8(A-YO-2) COMMON /BLK l/SIGMA REAL*8 F(NO,NO),G(NO,NO) F(1,l) = O.ODO F(2,l) = -2.18DO F(3,l) = O.ODO F(4,l) = O.ODO F(1,2) = 2.18DO F(2,2) = -0.20DO F(3,2) = 0-ODO F(4,2) = 0-ODO F(1,3) = O.ODO F(2,3) = 0-ODO F(3,3) = 0-ODO F(4,3) = -3.50DO

F(1,4) = 0-ODO F(2,4) = O.ODO F(3,4) =3 SODO F(4,4) = -0.2ODO G(1,1) = 0-ODO G(2,l) = -0.2294DO *Dsqrt@SIGMA) G(3,l) = 0.0DO G(4,l) = 0.9407DO*Dsqn@SIGMA) G(1,2) = 0.0DO G(2,2) = O.ODO G(3,2) = 0.0DO G(4,2) = O.ODO G(1,3) = O.ODO G(2,3) = -0.9407DO*Dsqrt(DSIGMA) G(3,3) = O.ODO G(4,3) = 1.57 1DO*Dsqn@SIGMA) G(I,4) = O.ODO G(2,4) = O.ODO G(3,4) = O.ODO G(4,4) = O.ODO RETURN END

c generating N(0,l)-nomdly distnbuted random variables SUBROUTINE RANDOM@T,RNl ,W) IMPLXCIT REAL* 8(A-H,O-2) REALt8 SR(2)

c generating 2 independent (O, 1) uniforrniy distributed r.v.'s 2 DO 1 1 = 1,2

N = NS65539 IF (N.LT.0) N = (N+2 l47483647)+ 1 SR(1) = FLOAT (N)*0.4656613D-9

1 CONTIMiTE c applying transformation to obtain 2 independent N(0,l) rx ' s

W 1 = 2.0DO*SR(l)- 1 .ODO W2 = 2.0DOeSR(2)- 1 .ODO W = Wlt*2+W2**2 IF (W-GT. 1 .ODO) GOTO 2 VAL = DSQRT(-2.0DO*DLOG(W)/W) RN1 = W l * V L RN2 = W2*VAL RETURN END

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