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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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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Canada
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
5.5 Concluding Remarks
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
Excitation for a Viscoelastic Plate. 95
4.5 Effect of S ( o , + o, ) on Lyapunov Exponent under a Narrow-band
Excitation for a Viscoelastic Plate. 96
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
Excitation for a Viscoelastic Plate. 99
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
Excitation for an Elastic Plate.
Effect of S ( w , - y ) on Stability Boundaries under a Narrow-band
Excitation for an Elastic Plate.
Effect of S ( 2 o , ) on Stability Boundaries under a Broad-band
Excitation for an Elastic Plate.
Effect of S ( 2 o ,) on Stability Boundaries under a Broad-band
Excitation for an Elastic Plate.
EEect of S ( o , - oz ) on Stability Boundaries under a Broad-band
Excitation for an Elastic Plate.
Effect of S ( a, + y ) on Stability Boundanes under a Broad-band
Excitation for an Elastic Plate.
Effect of Spectral Density S on Stability Boundaries under a White-noise
Excitation for an Elastic Plate.
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.
2.4 Concluding Remarks
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.
3.4 Concluding Remarks
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
The system is stable if
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.
4.6 Concluding Remarks
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 )
The system is stable if
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.
5.5 Concluding Remarks
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
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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