Nonlinear Random VibrationThis page intentionally left blankThis
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Nonlinear Random Vibration Analytical Techniques and Applications
Second edition
Cho W.S. To Professor of Mechanical Engineering University of
Nebraska-Lincoln USA
CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW,
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Table of contents
Acknowledgements xv
2 Markovian and Non-Markovian Solutions of Stochastic Nonlinear
Differential Equations 3 2.1 Introduction 3
2.1.1 Classification based on regularity 3 2.1.2 Classification
based on memory 4 2.1.3 Kinetic equation of stochastic processes
4
2.2 Markovian Solution of Stochastic Nonlinear Differential
Equations 6 2.2.1 Markov and diffusion processes 6 2.2.2 Itô’s and
Stratonovich integrals 7 2.2.3 One-dimensional
Fokker-Planck-Kolmogorov equation 9 2.2.4 Systems with random
parametric excitations 9
2.3 Non-Markovian Solution of Stochastic Nonlinear Differential
Equations 13 2.3.1 One-dimensional problem 13 2.3.2
Multi-dimensional problem 15
3 Exact Solutions of Fokker-Planck-Kolmogorov Equations 19 3.1
Introduction 19 3.2 Solution of a General Single-Degree-of-Freedom
System 22 3.3 Applications to Engineering Systems 33
3.3.1 Systems with linear damping and nonlinear stiffness 33 3.3.2
Systems with nonlinear damping and linear stiffness 50 3.3.3
Systems with nonlinear damping and nonlinear stiffness 53
3.4 Solution of Multi-Degree-of-Freedom Systems 54 3.5
Stochastically Excited Hamiltonian Systems 62
4 Methods of Statistical Linearization 65 4.1 Introduction 65 4.2
Statistical Linearization for Single-Degree-of-Freedom Nonlinear
Systems 66
4.2.1 Stationary solutions of single-degree-of-freedom systems
under zero mean Gaussian white noise excitations 66
4.2.2 Non-Zero mean stationary solution of a
single-degree-of-freedom system 76
4.2.3 Stationary solution of a single-degree-of-freedom system
under narrow-band excitation 78
4.2.4 Stationary solution of a single-degree-of-freedom system
under parametric and external random excitations 81
4.2.5 Solutions of single-degree-of-freedom systems under
nonstationary random excitations 84
vi Table of contents
4.4.1 Single-degree-of-freedom systems 94 4.4.2
Multi-degree-of-freedom systems 100
4.5 Uniqueness and Accuracy of Solutions by Statistical
Linearization 112 4.5.1 Uniqueness of solutions 112 4.5.2 Accuracy
of solutions 113 4.5.3 Remarks 114
5 Statistical Nonlinearization Techniques 115 5.1 Introduction 115
5.2 Statistical Nonlinearization Technique Based on
Least Mean Square of Deficiency 117 5.2.1 Special case 117 5.2.2
General case 118 5.2.3 Examples 122
5.3 Statistical Nonlinearization Technique Based on Equivalent
Nonlinear Damping Coefficient 133 5.3.1 Derivation of equivalent
nonlinear damping coefficient 134 5.3.2 Solution of equivalent
nonlinear equation of
single-degree-of-freedom systems 135 5.3.3 Concluding remarks
143
5.4 Statistical Nonlinearization Technique for
Multi-Degree-of-Freedom Systems 143 5.4.1 Equivalent system
nonlinear damping coefficient and exact solution 144 5.4.2
Applications 146
5.5 Improved Statistical Nonlinearization Technique for
Multi-Degree-of-Freedom Systems 148 5.5.1 Exact solution of
multi-degree-of-freedom nonlinear systems 149 5.5.2 Improved
statistical nonlinearization technique 154 5.5.3 Application and
comparison 156 5.5.4 Concluding remarks 158
5.6 Accuracy of Statistical Nonlinearization Techniques 161
6 Methods of Stochastic Averaging 163 6.1 Introduction 163 6.2
Classical Stochastic Averaging Method 164
6.2.1 Stationary solution of a single-degree-of-freedom system
under broad band stationary random excitation 166
6.2.2 Stationary solutions of single-degree-of-freedom systems
under parametric and external random excitations 172
6.2.3 Nonstationary solutions of single-degree-of-freedom systems
178 6.2.4 Remarks 187
6.3 Stochastic Averaging Methods of Energy Envelope 188 6.3.1
General theory 190 6.3.2 Examples 194 6.3.3 Remarks 201
6.4 Other Stochastic Averaging Techniques 202 6.5 Accuracy of
Stochastic Averaging Techniques 227
6.5.1 Smooth stochastic averaging 227 6.5.2 Non-smooth stochastic
averaging 228 6.5.3 Remarks 229
Table of contents vii
7 Truncated Hierarchy and Other Techniques 231 7.1 Introduction 231
7.2 Truncated Hierarchy Techniques 231
7.2.1 Gaussian closure schemes 234 7.2.2 Non-Gaussian closure
schemes 235 7.2.3 Examples 237 7.2.4 Remarks 239
7.3 Perturbation Techniques 239 7.3.1 Nonlinear
single-degree-of-freedom systems 239 7.3.2 Nonlinear
multi-degree-of-freedom systems 240 7.3.3 Remarks 242
7.4 Functional Series Techniques 242 7.4.1 Volterra series
expansion techniques 242 7.4.2 Wiener-Hermite series expansion
techniques 251
Appendix Probability, Random Variables and Random Processes 255 A.1
Introduction 255 A.2 Probability Theory 255
A.2.1 Set theory and axioms of probability 255 A.2.2 Conditional
probability 256 A.2.3 Marginal probability and Bayes’ theorem
257
A.3 Random Variables 258 A.3.1 Probability description of single
random variable 258 A.3.2 Probability description of two random
variables 260 A.3.3 Expected values, moment generating and
characteristic functions 261
A.4 Random Processes 263 A.4.1 Ensemble and ensemble averages 263
A.4.2 Stationary, nonstationary and evolutionary random processes
264 A.4.3 Ergodic and Gaussian random processes 265 A.4.4 Poisson
processes 266
References 269 Chapter 1 269 Chapter 2 271 Chapter 3 273 Chapter 4
275 Chapter 5 281 Chapter 6 283 Chapter 7 287 Appendix 291
Index 293
To My Parents
Preface to the first edition
The framework of this book was first conceptualized in the late
nineteen eighties. However, the writing of this book began while
the author was on sabbatical, July 1991 through June 1992, at the
University of California, Berkeley, from the University of Western
Ontario, London, Ontario, Canada. Over half of the book was
completed before the author returned to Canada after his
sabbatical. With full-time teaching, research, the arrival of a
younger daughter, and the moving in 1996 from Canada to the
University of Nebraska, Lincoln the author has only completed the
project of writing this book very recently. Owing to the long span
of time for the writing, there is no doubt that many relevant
publications may have been omitted by the author.
The latter has to admit that a book of this nature is influenced,
without exception, by many authors and examples in the field of
random vibration. The original purpose of writing this book was to
provide an advanced graduate level textbook dealing, in a more
systematical way, with analytical techniques of nonlinear random
vibration. It was also aimed at providing a textbook for a second
course in the analytical techniques of random vibration for
graduate students and researchers.
In the introduction chapter reviews in the general areas of
nonlinear random vibration appeared in the literature are quoted.
Books exclusively dealing with and related to are listed in this
chapter. Chapter 2 begins with a brief introduction to Markovian
and non-Markovian solutions of stochastic nonlinear differential
equations. Chapter 3 is concerned with the exact solution of the
Fokker-Planck-Kolmogorov (FPK) equation. Chapter 4 presents the
methods of statistical linearization (SL). Uniqueness and accuracy
of solutions by the SL techniques are summarized. An introduction
to and discussion on the statistical nonlinearization (SNL)
techniques are provided in Chapter 5. Accuracy of the SNL
techniques is addressed. The methods of stochastic averaging are
introduced in Chapter 6. Various stochastic averaging techniques
are presented in details and their accuracies are discussed.
Chapter 7 provides briefly the truncated hierarchy, perturba- tion,
and functional series techniques.
C.W.S. To Lincoln, Nebraska 2000
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Preface to the second edition
Various theoretical developments in the field of nonlinear random
vibration have been made since the publication of the first
edition. Consequently, the latter has been expanded somewhat in the
present edition in which a number of errors and misprints has been
corrected.
The organization of the present edition remains essentially the
same as that of the first edition. Chapter 1 is an updated
introduction to the reviews in the general areas of nonlinear
random vibration. Books exclusively dealing with and related to
analytical techniques and applications are cited. Chapter 2 is
concerned with a brief introduction to Markovian and non-Markovian
solutions to stochastic nonlinear differential equations. Exact
solutions to the Fokker-Planck-Kolmogorov (FPK) equations are
included in Chapter 3. Methods of statistical linearization (SL)
with uniqueness and accuracy of solutions are presented in Chapter
4. Some captions and labels of figures in this chapter have been
changed to commonly used terminology. Chapter 5 deals with the
statistical nonlinearization (SNL) techniques. Section 5.5 is a new
addition introducing an improved SNL technique for approximating
multi-degree-of-freedom nonlinear systems. Methods of stochastic
averaging are presented in Chapter 6. In the present edition, more
detailed steps are added and some reorganization of steps are made.
Chapter 7 includes truncated hierarchy, perturbation, and
functional series techniques. In the present edition, more steps
have been incorporated in the Volterra series expansion techniques.
An appendix presenting a brief introduction to the basic concepts
and theory of probability, random variables, and random processes
has been added to the present edition. This new and brief addition
is aimed at those readers who need a rapid review of the
prerequisite materials.
C.W.S. To Lincoln, Nebraska 2011
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Acknowledgements
ACKNOWLEDGEMENTS FOR THE FIRST EDITION
The author began his studies in random vibration during his final
year of undergraduate program, between 1972 and 1973, at the
University of Southampton, United Kingdom. The six lectures given
by Professor B.L. Clarkson served as a stimulating beginning. After
two years of master degree studies at the University of Calgary,
Canada, in October 1975 the author returned to the University of
Southampton to work as a research fellow in the Institute of Sound
and Vibration Research for his doctoral degree under the
supervision of Professor Clarkson. The fellowship was sponsored by
the Admiralty Surface Weapons Establishment, Ministry of Defence,
United Kingdom. During this period of studies, the author was
fortunate enough to have attended lectures on random vibration
presented by Professor Y.K. Lin who was visiting Professor Clarkson
and the Institute in 1976. The year 1976 saw the gathering of many
experts and teachers in the field of random vibration at the
International Union of Theoretical and Applied Mechanics, Symposium
on Stochastic Problems in Dynamics. The author was, thus,
influenced and inspired by these experts and teachers.
The conducive atmosphere and the availability of many publications
in the libraries at the University of California, Berkeley and the
hospitality of emeritus Professors J.L. Sackman, J.M. Kelly, Leo
Kanowitz and other friends at Berkeley had made the writing
enjoyable, and life of the author and his loved ones
memorable.
Case (ii) in page 131 and the excitation processes in almost all
the examples in Chapter 6 have been re-written and changed as a
result of comments from one of the reviewers. Section 7.4 has been
expanded in response to the suggestion of another reviewer. The
author is grateful to them for their interest in reviewing this
book.
Thanks are due to the author’s two present graduate students, Ms.
Guang Chen and Mr. Wei Liu who prepared all the drawings in this
book.
Finally, the author would like to express his gratitude to his
friend, Professor Fai Ma for his encouragement, and wishes to thank
the Publisher, Mr. Martin Scrivener and his staff for their
publishing support.
ACKNOWLEDGEMENTS FOR THE SECOND EDITION
Since the publication of the first edition in 2000 various
theoretical developments in the field of nonlinear random vibration
have been made. It is therefore appropriate to publish the present
edition at this time. The author has taken the opportunity to make
a number of corrections.
The appendix on Probability, Random Variables and Random Processes
is the result of the suggestion of a reviewer of the proposal for
the present edition. The reviewer’s suggestion and comments are
highly appreciated.
Finally, the author wishes to thank Mr. Janjaap Blom, Senior
Publisher, Ms. Madeline Alper, Customer Service Supervisor, and
their staff for their publishing support.
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1 Introduction
For safety, reliability and economic reasons, the n onlinearities
of many dynamic engineering systems under environmental and other
forces that are treated as random disturbances must be taken into
account in the design procedures. This and the demand for precision
have motivated the research and development in nonlinear random
vibration. Loosely speaking, the field of nonlinear random
vibration can be subdivided into four categories. The latter
include analytical techniques, computational methods, Monte Carlo
simulation (MCS), and system identification with experimental
techniques. This book is mainly concerned with the first category
and therefore the publications quoted henceforth focus on this
category. The subject of computational nonlinear random vibration
is dealt with in a companion book that is published recently
[1.1].
It is believed that the first comprehensive review on nonlinear
random vibration was performed by Caughey [1.2]. Subsequently,
other reviews appeared in the literature [1.3-1.15], for example.
There are books exclusively concerned with and related to nonlinear
random vibration [1.16-1.24]. Many books [1.25-1.39] also contain
chapter(s) on nonlinear random vibration.
While it is agreed that there are many techniques available in the
literature for the analysis of nonlinear systems under random
excitations, the focus of the present book is, however, on those
frequently employed by engineers and applied scientists. It also
reflects the current interests in the analytical techniques of
nonlinear random vibration.
Chapter 2 begins with a brief introduction to Markovian and
non-Markovian solutions of stochastic nonlinear differential
equations. This serves as a foundation to subsequent chapters in
this book.
Chapter 3 presents the exact solutions of the
Fokker-Planck-Kolmogorov (FPK) equations. Solution of a general
single degree-of-freedom (dof) system and applications to
engineering systems are included. Solution of a multi-degrees-of-
freedom (mdof) system and s tochastically excited Hamiltonian
systems are also considered.
2 Nonlinear Random Vibration
Chapter 4 deals with the methods of statistical linearization (SL).
Solutions to single dof and mdof nonlinear systems with examples of
engineering applications are given. Uniqueness and accuracy of
solutions by the SL techniques are summarized.
Chapter 5 provides an introduction to and discussion on the
statistical nonlinearization (SNL) techniques. Single dof and mdof
nonlinear systems are considered. Accuracy of the SNL techniques is
addressed.
Chapter 6 treats the methods of stochastic averaging. The classical
stochastic averaging (CSA) method, stochastic averaging method of
energy envelope (SAMEE), and various other stochastic averaging
techniques are introduced and examples given. Accuracy of the
stochastic averaging techniques is discussed.
Chapter 7 introduces briefly several other techniques. The lattter
include truncated hierarchy, perturbation, and functional series
techniques. The truncated hierarchy techniques include Gaussian
closure schemes and non-Gaussian closure schemes, while the
functional series techniques encompass the Volterra series
expansion techniques, and Wiener-Hermite series expansion
techniques.
It is assumed that the readers have a first course in random
vibration or similar subject. Materials in Chapters 2 and 3 are
essential and serve as a foundation to a better understanding of
the techniques and applications in subsequent chapters.
An outline of the basic concepts and theory of probability, random
variables and random processes is included in the appendix for
those who need a rapid review of the essential background
materials.
2 Markovian and Non-Markovian Solutions of Stochastic Nonlinear
Differential Equations
2.1 Introduction
Within the field of nonlinear random vibration of structural and
mechanical systems the statistical complexity of a stochastic
process (s.p.) is determined by the properties of its distribution
functions. Two types of classifications are important in the
analysis. These are classification based on the statistical
regularity of a process and classification based on its
memory.
In this section the above two types of classifications are
introduced in Sub- sections 2.1.1 and 2.1.2. Then in Sub-section
2.1.3 the kinetic equation associated with the s.p. is derived.
This provides the basis for distribution and density functions that
are important to subsequent analysis. Section 2.2 contains the
basic material for Markovian solution of stochastic nonlinear
differential equations. Essential features and relevant information
for non-Markovian solution of stochastic nonlinear differential
equations are included in Section 2.3.
2.1.1 Classification based on regularity
In this type of classification, s.p. are divid ed into two
categories. They are the stationary stochastic processes (s.s.p.)
and nonstationary stochastic processes (n.s.p.). Assuming time t is
the parameter of the strict sense or strong s.s.p. X(t), its
statistical properties are all independent of time t or are all
independent of the absolute time origin. On the other hand, for a
n.s.p. all statistical properties of that process are dependent of
time t.
When the absolute value of the expectation of the s.s.p. X(t) is a
constant and less than infinity, the expectation of the square of
X(t) is less than infinity, and the
4 Nonlinear Random Vibration
(2.1)
(2.2)
covariance of X(t) is equal to the correlation function of X(t),
the s.p. is called a wide-sense or weak s.s.p. Of course, when such
a s.s.p. is Gaussian it is completely specified by its means and
covariance functions. 2.1.2 Classification based on memory
If s.p. are grouped in accordance with the manner in which the
present state of a s.p. depends on its past history, then such a
classification is called classification based on memory. This
classification is centered around the Markov processes.
In accordance with the memory properties, the s implest s.p. is on
e without memory or is purely stochastic. This is usually called a
zeroth order Markov process. Clearly, a continuous-parameter purely
s.p. is physically not realizable since it implies absolute
independence between the past and the present regardless of their
temporal closeness. The white noise process is a purely s.p. The
Markov process to be defined in Sub-section 2.2.1 is usually called
a simple Markov process. There are higher order Markov processes
that are not applied in this book and therefore are not defined
here.
It may be appropriate to note that the memory of a s.p. is not to
be confused with the memory of a nonlinear transformation. The
latter is said to have memory if it involves with inertia.
2.1.3 Kinetic equation of stochastic processes
A technique that can give explicit results of joint distributions
of the solution process is introduced in this sub-section. The
foundation of the following derivation was presented by Bartlett
[2.1] and Pawula [2.2], and subsequently by Soong [2.3].
A s.p. X(t) with its first probability density function being
denoted by p(x,t) satisfies the relation that
where p(x,t+)t *y,t) is the conditional probability density
function of X(t+)t) given that X(t) = y.
Let R(u,t+)t *y,t) be the conditional characteristic function of )X
= X(t+)t) - X(t) given that X(t) = y,
Markovian and Non-Markovian Solutions 5
(2.3)
(2.4)
(2.5)
where the angular brackets denote the mathematical expectation. By
taking the inverse Fourier transformation, one has
By expanding the conditional characteristic function R(u,t+)t *y,t)
in a Taylor series about u = 0, Eq. (2.3) becomes
where
These expectations are known as the incremental moments of X(t).
Substituting Eq. (2.4) into (2.1) and after integration, one
obtains
This equation can be expressed as
Upon dividing this equation by )t and in the limit as )t 6 0, it
leads to
where
(2.6)
(2.7)
kEquation (2.5) is known as the kinetic equation of the s.p. X(t)
and " (x,t) are the derivate moments. It is a deterministic
parabolic partial differential equation and has important use in
the solution of stochastic differential equations.
2.2 Markovian Solution of Stochastic Nonlinear Differential
Equations
There are many physical quantities, such as the response of a
nonlinear system under a random excitation that can be represented
by a white noise process, can be described as Markov processes.
Rigorous fundamental treatment on the subject was presented by
Kolmogorov [2.4]. The solution by the analytical techniques
considered in this monograph is generally based on the concepts of
Markov processes. Thus, it is essential to introduce these
concepts. To this end, Markov and diffusion processes are defined
in Sub-section 2.2.1 while the Stratonovich and Itô's integrals are
presented in Sub-section 2.2.2. Sub-section 2.2.3 is concerned with
the one- dimensional Fokker-Planck forward or
Fokker-Planck-Kolmogorov (FPK) equation. To further clarify the use
of Stratonovich and Itô's integrals a single-degree-of- freedom
(sdof) quasi-linear system is included in Sub-section 2.2.4.
2.2.1 Markov and diffusion processes
A stochastic process X(t) on an interval [0,T] is called a Markov
process if it has the following property:
where P[.] designates the probability of an event and the
conditional probability of 0 0the Markov process X(t), P[X(t) <
x * X(t ) = x ] is known as the transition
probability distribution function. Equation (2.6) means that the
process forgets the n - 1past if t is being regarded as the
present.
Applying the Markov property (2.6), one can show that
i i i-1 i-1where p(x ,t * x ,t ), i = 2,3, are the transition
probability densities. Equation (2.7) 1describes the flow or
transition probability densities from instant t to another
instant
3t . It is known as the Smoluchowski-Chapman-Kolmogorov (SCK)
equation.
Markovian and Non-Markovian Solutions 7
(2.8a)
(2.8b)
(2.8c)
(2.9)
(2.10)
(2.11a)
(2.11b)
A Markov process X(t) is called a diffusion process if its
transition probability density satisfies the following two
conditions for )t = t - s and g > 0,
and
where the drift and diffusion coefficients, f(x,s) and G(x,s),
respectively, are independent of the time t when X(t) is stationary
because in this case p(y,t * x,s) = p(y,t - s * x) only depends on
the time lag )t.
2.2.2 Itô's and Stratonovich integrals
Consider a characteristic function on the interval [a,b] for 0 # a
< b # T,
For 0 # a < b # T, one defines
where B(t) is the Brownian motion process which is a martingale
because
1 2 n 1 nand for all t < t < ... < t and a ,..., a ,
0 1 mIf f(t) is a step function on [0,T] and 0 = t < t < ...
< t = b, then
8 Nonlinear Random Vibration
(2.12)
(2.13)
(2.14)
(2.15)
k k+1where the subscript [t ,t ) denotes the semi-open interval.
Now, one can define
The function f(t) can be a random function of B(t) and the value of
f(t) is taken at the left end point of the partition interval. When
f(t) is a random function of B(t) it is independent of the
increments B(t+s) - B(t) for all s > 0. Such a function is
called a non-anticipating function.
In the limit when m approaches infinity Eq. (2.13) becomes the
Itô's integral. The properties of the stochastic or Itô's integral
[2.5] are very much different from those of the Riemann-Stieltjes
integral. Applying the Itô's integral for f(t) = B(t), one can show
that
Now, another type of stochastic integral, the so-called
Stratonovich integral
[2.6], is defined for explicit function of B(t) by
k + 1 kwhere )t = t - t . If f(B(t),t) = B(t) Eq. (2.15) gives B
(T)/2. This result is2
very different from that in Eq. (2.14). The Stratonovich integral
satisfies all the formal rules of classical calculus and
therefore it is a na tural choice for stochastic differential
equations on manifolds. However, Stratonovich integrals are not
martingales. In contrast, Itô's integrals are martingales and
therefore they have an important computational advantage.
Markovian and Non-Markovian Solutions 9
(2.16)
(2.17)
(2.18)
2.2.3 One-dimensional Fokker-Planck-Kolmogorov equation
With the kinetic equation for s.p. X(t) derived in Sub-section 2.1
and diffusion 0 0process defined in Sub-section 2.2.1, the
transition probability density p(x,t * x ,t )
or simply p for an one-dimensional problem satisfies the following
parabolic partial differential equation
1 2in which " and " are the first and second derivate
moments,
where D = BS and S is the spectral density of the Gaussian white
noise process. Equation (2.16) is known as the Fokker-Planck
forward or Fokker-Planck- Kolmogorov (FPK) equation with the
following initial conditions
where *(.) is the Dirac delta function. In passing, it is noted
that the reduced FPK equation is a boundary value
problem and the classification of boundaries in accordance with
that proposed by Feller [2.7] for one-dimensional diffusion process
has been summarized by Lin and Cai [2.8]. A whole set of new
classification criteria that is equivalent but simpler than that of
Feller has been developed by Zhang [2.9]. For high dimensional
problems the classification of boundaries is still open.
The new criteria for classification of singular boundaries may be
applied to the time averaging method in order to investigate the
stability problem of nonlinear stochastic differential equations.
In this way, it can be applied to decide the existence of the
stationary probability density of a nonlinear system that may be
reduced to an one-dimensional problem. However, the stability
obtained by using the classification of singular boundaries is a
weak one. That is, it is the stability in probability. Moreover,
the time averaging on the differential operator is limited to a
specific form.
Finally, the issues of stability and bifurcation of nonlinear
systems under stationary random excitations are not pursued here.
These will be considered in a separate monograph to be published in
due course.
2.2.4 Systems with random parametric excitations
There are many practical engineering systems whose dynamical
behaviors can be
10 Nonlinear Random Vibration
(2.19)
(2.20)
(2.21)
described by governing equations of motion containing random
parametric excitations. The controversy in this type of systems is
the addition of the so-called Wong and Zakai (WZ) or Stratonovich
(S) correction term to the Itô's stochastic differential equation.
This issue was discussed by Gray and Caughey [2.10], Mortensen
[2.11, 2.12], and others [2.13-2.21]. The usual reason given, for
instance, in Refs. [2.15, 2.20], is that when the random parametric
excitations in the governing equation of motion are independent
physical Gaussian white noises, to convert the equation to the
corresponding Itô's equation the WZ or S correction term is
required. In Ref. [2.21], an example was presented to demonstrate
that such a given reason is not adequate.
In the following, the stochastic differential equation, of a
quasi-linear system with random parametric excitations, and
relevant concluding remarks in Ref. [2.21] are included since they
are important in the understanding and solution of many nonlinear
systems in subsequent chapters.
(a) Statement of Problem
The stochastic differential equation of interest is given by
where W(t) is a vector of Gaussian white noises and Z(t) is a
vector of s.p.; f(Z,t) and G (Z,t) are known vector and matrix
quantities and are nonlinear functions of Z(t) and t, in
general.
With the arguments omitted, Eq. (2.19) can be written as
where dB = Wdt, with B being the vector of Brownian motion or
Wiener processes. Equation (2.20) is the so-called Itô's stochastic
differential equation. The solution of Eq. (2.20) is as the
following
Note that the second integral on the right-hand side (RHS) of Eq.
(2.21) cannot be interpreted as an ordinary Riemann or
Lebesque-Stieltjes integral, since the sample function of a
Brownian motion is, with probability 1, of unbounded variation
[2.22]. Two interpretations of this second integral have been
presented in the literature. The
Markovian and Non-Markovian Solutions 11
(2.22)
(2.23)
first [2.22] leads to replacing Eq. (2.20) by the following matrix
equation
in which the second term inside the parentheses on the RHS of the
above equation involves with the division by the vector MZ which,
strictly speaking, is not allowed in the matrix operation. Thus,
the partial differentiation term, MG/MZ should be used in
accordance with the rules of matrix operation. More explicitly the
above equation may be written as
This equation is to be solved in the sense of Itô's calculus [2.5]
so that the property of the martingale [2.23] is retained. The
second term inside the parentheses on the RHS of Eq. (2.23) is
known as the WZ or S correction term. The solution of Eq. (2.23) is
equal to that of Eq. (2.20), provided that Eq. (2.20) is solved in
accordance with the second interpretation in which the second
integral on the RHS of Eq. (2.21) is defined in the sense of
Stratonovich calculus [2.6]. With such a definition it can be
treated in the same way as with the ordinary integrals of smooth
functions.
As introduced in Sub-section 2.2.2, the rules governing the Itô
calculus and the Stratonovich calculus are entirely different and
therefore, the WZ or S correction term is required not as a
consequence of converting the physical Gaussian white noises into
the ideal white noises. It may be appropriate to recall that the
white noise process is just a mathematical idealization. In
applying a mathematical approach to describe a physical phenomenon,
such as the dynamical behavior or response, one has inherently
adopted some form of idealization. The following example of a
quasi- linear random differential equation with random parametric
excitations will illustrate the reason for the addition of the WZ
or S term. (b) An Example
Consider a single degree of freedom (sdof) system disturbed by both
parametric and external stationary Gaussian white noise
excitations. The governing equation of motion for the system
is
or simply as
(I-1)
(I-2)
(I-3)
(I-4)
where x is the stochastic displacement, the over-dot and double
over-dot designate first and second derivatives with respect to
time t; a and b are constants, while w is the Gaussian white noise
excitation. Equation (I-1) can be converted into two first
1 2 1 2order differential equations by writing z = x, z = x0 and Z
= (z z ) such that T
or in similar form to Eq. (2.20)
in which the superscript T denotes the transpose of. The diffusion
coefficient G in Eq. (I-2) is a function of Z. Therefore,
according
to Refs. [2.10, 2.11, 2.22] the WZ or S correction term is
required. Accordingly, Eq. (2.23) becomes
Applying this equation to the system described by Eq. (I-1) one
has
The second term inside the square brackets on the RHS of the second
equation of (I-4) is the WZ or S correction. This term will not be
zero as long as a is not
1 2equal to zero or bz + az is not equal to unity.
(c) Remarks
The above example clearly demonstrates that the WZ or S correction
term is required regardless of whether the Gaussian white noise
excitation w is ideal or physical. Indeed, when the parametric
stationary white noise excitation associated
Markovian and Non-Markovian Solutions 13
(2.24)
(2.25)
with the next to highest derivative of the governing equation of
motion is zero, the solution of the equation by the Itô's calculus
rules is identical to that given by using the ordinary or
Stratonovich calculus rules, and consequently no WZ or S correction
is necessary. In other words, in the above sdof system the WZ or S
correction term is required because there is a random parametric
excitation associated with the velocity term. If the random
parametric excitation associated with the velocity term is zero and
the random parametric excitation associated with the restoring
force is retained the WZ or S correction term is zero, meaning the
solution in this case is identical whether one employs the
Stratonovich or Itô's calculus.
2.3 Non-Markovian Solution of Stochastic Nonlinear
Differential
Equations
While in practice many physical phenomena occur in structural and
mechanical systems can adequately be represented by Markovian
processes, there are important cases in other fields that have to
be modeled by non-Markovian processes. For example, in the problem
of magnetic resonance in a fluctuating magnetic field [2.24, 2.25],
nematic liquid crystals [2.26], and the behavior of the intensity
of a single mode dye laser [2.27] the non-Markovian processes were
employed. In fact the Markovian processes in the foregoing sections
are special cases to the non- Markovian processes. Therefore, it
may be of interest to p resent the essential features and relevant
information for the non-Markovian solution of stochastic nonlinear
differential equations.
2.3.1 One-dimensional problem
Consider an one-dimensional system described by the following
stochastic differential equation [2.28]:
where f(q(t)) or simply f(t) or f and g(q(t)) or simply g(t) or g
are general nonlinear functions of q(t) or simply q, while >(t)
or simply > is the colored noise excitation which is also known
as the Ornstein-Uhlenbeck process. The latter is a Gaussian process
with zero mean and correlation function given by
14 Nonlinear Random Vibration
(2.26)
(2.27)
where J is the finite correlation time and D is the noise parameter
of the stochastic disturbance >(t). Since any solution of Eq.
(2.24) is non-Markovian and nonstationary, Eq. (2.24) defines a
class of non-Markovian and nonstationary random (NMNR) processes.
The latter differ from each other in the selection of initial
conditions. In the limit J v 0, Eq. (2.24) defines a stationary
Markovian process when the distribution of initial conditions are
also stationary. As pointed out in Ref. [2.28], in the class of
processes defined by Eq. (2.24) the effects of non-Markovian and
nonstationary properties cannot be disentangled. This is due to the
fact that both properties have the same origin J.
In general, exact solution for moments and correlation functions of
the process defined by Eq. (2.24) is not available and therefore
approximate solution, in which the zeroth-order approximation is
the Markovian limit J = 0, is derived.
By expanding in powers of J, the approximate solution for the first
moment < q(t) > or simply < q > is obtained following
averaging of Eq. (2.24) as [2.28]
in which, by assuming t >> J, the second term on the RHS of
Eq. (2.26) can be shown to be
where
For nonlinear one-dimensional problems, results obtained by
applying the above equations can be found in Ref. [2.27].
It may be appropriate to point out that for a linear
one-dimensional NMNR problem that has the following relations in
Eq. (2.24)
Markovian and Non-Markovian Solutions 15
(2.28)
(2.29)
(2.30)
(2.31)
rThe steady-state relaxation time J is given by
rIn the limit J v 0, the steady-state relaxation time J v 1/a which
is the steady- state relaxation time of a Markovian problem.
2.3.2 Multi-dimensional problem
The equations above can be generalized to multi-dimensional
problems. Thus, the equations corresponding to (2.24) and (2.25)
are, respectively [2.28]
ij ij ijwhere * is the Kronecka delta such that * = 1 if i = j
otherwise, * = 0. The differential equations of first moments are
given as
where
While the above first order approximate solutions have been
obtained for quasi-linear single and multi-dimensional problems
[2.28, 2.29], and nonlinear one-dimensional problems [2.27], the
solution of general multi-dimensional nonlinear NMNR problems
remains a formidable challenge.
Before leaving this sub-section, a sdof or two-dimensional problem
is included here to illustrate application of the foregoing
procedure. Consider the system having unit mass such that the
equation of motion is
or simply as
where w is the zero mean Gaussian white noise such that
<w(t)w(t')> = 2BS*(8) with 8 = t - t' and S being the
spectral density of the white noise process, > the
Ornstein-Uhlenbeck process whose correlation function has been
defined by Eq. (2.25), and the remaining symbols have their usual
meaning.
To proceed further one can express the quantities of interest of
the above oscillator as x = q, and dx/dt = p such that the equation
of motion can be re-
Markovian and Non-Markovian Solutions 17
(I-2)
(I-3)
(I-4)
(I-5)
written as two first order stochastic differential equations
The solution process in Eq. (I-2) is NMNR due to the fact that >
is not a white noise. By applying Eq. (2.31), writing q(t) = q and
p(t) = p, one obtains the approximate equations for the first
moments as
By means of the J expansion [2.28], one can show that
and the approximate equations, to first order in J, for the second
moments are
Equations (I-3) and (I-5) can be solved in closed form or by so me
numerical integration algorithm, such as the fourth order
Runge-Kutta (RK4) scheme. They are dependent of J which is a
measure of the non-Markovian property of the solution process. In
the limit when J approaches 0 the solutions in Eqs. (I-3) and (I-5)
are Markovian.
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3 Exact Solutions of Fokker-Planck-Kolmogorov Equations
3.1 Introduction
The response of general nonlinear oscillator under parametric
random excitations and external random excitations has been
extensively studied in the last three decades. The foundation of
the development has been installed earlier by Rayleigh [3.1],
Fokker [3.2], and Smoluchowski [3.3], for example. In general, no
exact solution can be found. When the excitations can be idealized
as Gaussian white noises, in which case, the response of the system
can be represented by a Markovian vector and the probability
density function of the response is described by the FPK equation,
exact stationary solution can be obtained. The solution of the FPK
equation has been reported in the literature [3.4-3.15]. The
following approach is that presented by To and Li [3.15]. It seems
that the latter approach gives the broadest class of solvable
reduced FPK equations. It is based on the systematic procedure of
Lin and associates [3.11-3.14], and the application of the theory
of elementary or integrating factor for first order ordinary
differential equations. In Ref. [3.14] the solution of the reduced
FPK equation is obtained by applying the theory of generalized
stationary potential which is less restrictive than that employing
the concept of detailed balance [3.12]. The latter is similar to
that of Graham and Haken [3.16]. The basic idea of the concept of
Graham and Haken is to separate each drift coefficient into
reversible and irreversible parts.
In this chapter after the introduction of the FPK equation for a
vector process, the solution of a general sdof nonlinear system is
presented in Section 3.2. Section 3.3 includes solutions to various
systems that are frequently encountered in the field of random
vibration. Sections 3.4 and 3.5 are concerned with the solution of
mdof nonlinear systems.
20 Nonlinear Random Vibration
(3.1)
(3.2)
(3.3)
(3.4a,b)
(3.5)
(3.6)
The one-dimensional FPK equation in the last chapter can be easily
extended to the multi-dimensional cases.
Consider the following Itô equation for a n-dof system
1 2 2nwhere X = (x x , ..., x ) ; f(X,t) and G(X,t) or simply f and
G are the drift vectorT
of 2n × 1 and diffusion matrix of 2n × 2n, respectively. Note that
B is the jBrownian motion vector process such that B are the
elements of the vector B. The
latter should not be confused with the matrix [B] of second
derivate moments. The ijelements of [B] are B .
The associated FPK equation is
in which p(X,t) is the joint transition probability density
function or simply referred to as transition probability density,
and D is the matrix of excitation intensities
ij ij ijwhose ij'th element is D = BS where S are the
cross-spectral densities of the white noise processes.
i ijIn terms of the first and second derivate moments, A and B
rsespectively, one has
such that
It is understood that the first and second derivate moments are
evaluated at X = x. The initial conditions for the FPK equation
are
The FPK equation is invariant under translations in time t. In
other words,
Exact Solutions of Fokker-Planck-Kolmogorov Equations 21
(3.7)
(3.8)
(3.9)
such that one can write the backward Kolmogorov or backward FPK
equation and forward FPK equation, (3.3), respectively, as
where L is the adjoint operator to L. *
0Writing s = t the backward FPK equation becomes
0in which the first and second derivate moments are functions of
X(s) or X(t ), and the initial conditions are defined in Eq. (3.5).
The backward FPK equation can be applied to derive partial
differential equations for the moments of the response of the
system, while the forward FPK equation is employed mainly to
evaluate the transition probability density.
Finally, the so-called Itô's differential rule for an arbitrary
function Y(X,t) or simply Y of a Markov vector process X(t) is
important and useful for subsequent application and therefore is
included at this stage. Starting from the classical chain
rule,
iSubstituting dX, and remembering that elements of the latter
vector are x , in Eq. (3.1) and adding the WZ or S correction terms
to (3.8), one can show that
Xwhere (.) is the generating differential operator of the Markov
process X and is defined as
Equation (3.9) is the Itô's differential rule which is also known
as Itô's lemma
or Itô's formula.
Consider the stochastic system
1 2where x = x, x = dx/dt, and the double over-dot denotes the
second derivative with 1 2 i 1 2 irespect to time t, h(x ,x ) or
simply h and f (x ,x ) or f are generally nonlinear
1 2 i ifunctions of x and x , and w ( t) or w are Gaussian white
noises with the delta type correlation functions
Applying the technique in Section 3.1, the FPK equation for the
system described by Eq. (3.10) becomes
In general, exact solution for the transition probability density
function p is not savailable and only the stationary probability
density function, p can be obtained
from the reduced FPK equation
i ijThe first derivate moment A and second derivate moment B of
Eq.(3.13) are
According to the method described in Ref. [3.14], the first and
second derivate moments are divided as in the following
Exact Solutions of Fokker-Planck-Kolmogorov Equations 23
(3.14)
(3.16)
(3.15)
(3.17)
(3.18)
(3.19)
(3.20)
Then Eq. (3.13) is solvable if the following equations are
satisfied
The stationary probability density function can be shown to
be
1 2where N is a function of x and x and C is a normalization
constant. It should be noted that Eqs. (3.14) through (3.16) are
similar to Eqs. (17) through (19) of Ref.
2 x y[3.14], except that A here is replaced by (- 8 /8 ) in Ref.
[3.14].(2)
By the characteristic function method, one has from Eq.
(3.16)
From the first two equations of Eq. (3.18), one has
In order to obtain the exact solution of Eq. (3.19) and to
incorporate a broader class 1 2of nonlinear systems, the
integrating factor method [3.15] is applied. Let M(x ,x )
or M be the integrating factor of Eq. (3.19), then
24 Nonlinear Random Vibration
Therefore,
Equation (3.22) gives
1 1where C (x ) is an arbitrary function and M is a general
function characteristic of a particular nonlinear system.
Substituting Eq. (3.23) into (3.19), one has
Integrating Eq. (3.24) leads to
The RHS of Eq. (3.25) is a constant. Equation (3.25) is the
implicit solution of Eq. 2(3.19) in which A is given by Eq.
(3.23).(2)
By the first and third equations of Eq. (3.18), one has
The above equation gives
(3.26)
(3.27)
(3.28)
(3.29)
(3.30)
Applying Eq. (3.19), Eq. (3.27) reduces to
This gives
0where N (r) is an arbitrary function. Substituting Eq. (3.28) into
Eq. (3.17), one has
Substituting Eqs. (3.28) and (3.23) into (3.15), and re-arranging
it leads to
26 Nonlinear Random Vibration
By Eqs. (3.14) and (3.28) one has
2 1 1where C (x ) is an arbitrary function of x . Substituting Eq.
(3.31) into the first two terms on the RHS of Eq. (3.30) one
can
show that
3 1 1where C (x ) is an arbitrary function of x . Substituting the
last equation into Eq. (3.30) gives
Note that Eqs. (3.29) and (3.33) constitute the broadest class of
solvable reduced FPK equations of nonlinear sdof systems. Previous
results in Refs. [3.4-3.14] are included in this class.
yIf the function M = 8 (x,y) which is the partial derivative of
8(x,y) with 2 1respect to y, where y = (x )/2, x = x and 8(x,y) or
simply 8 is an arbitrary2
function of x and y, then by Eq. (3.25)
1 1Without loss of generality, by setting C (x ) = 0 the above
equation becomes
By Eq. (3.28), one has
Exact Solutions of Fokker-Planck-Kolmogorov Equations 27
(3.36)
(3.37)
(I-1)
(I-2)
Substituting Eq. (3.35) into Eq. (3.17), one has
This is Eq. (7) obtained by Cai and Lin in Ref. [3.17]. By Eqs.
(3.33) and (3.34),
This is Eq. (21) in Ref. [3.14] and is Eq. (6) in Ref. [3.17]. From
the foregoing, it is apparent that Eq.(3.36) is a special case of
Eq. (3.29),
while Eq. (3.37) is a special case of Eq. (3.33). The stationary
probability density function for an energy dependent nonlinear sdof
system provided by Zhu and Yu in Eq. (3) of Ref. [3.18] is also a
special case of Eq. (3.29). The equation of motion associated with
Eq. (3) of Ref. [3.18] is one in which the coefficients of velocity
and random excitation are functions of total energy of the
oscillator. This case is included in the following as Example
V.
Several mathematical models are included in the following to
illustrate the application of the method presented above. These
mathematical models have previously been studied. As they are used
for illustration only the probability density function of every
case is considered.
Example I. Consider the model in Ref. [3.19]
where w(t) is a Gaussian white noise with a spectral density S,
g(x) is the nonlinear spring force, .(8) is an arbitrary function,
and 8 is the total energy
Applying the same symbols as in the method presented above, the two
Itô stochastic differential equations for Eq. (I-1) are
28 Nonlinear Random Vibration
(I-3)
(I-4)
(I-5)
(I-6)
(I-7)
(I-8)
(II-1)
(II-2)
where B(t) or B is a unit Wiener process. The corresponding reduced
FPK equation becomes
The first and second derivate moments are divided into two parts as
those in the 2 1procedure described above except that A = - g(x )
is chosen in accordance with(2)
3 1Eq. (3.23), and C (x ) = 0 is imposed in Eq. (3.32) such that
Eq. (3.14) is satisfied. 1Then, Eq. (3.15) for the reduced FPK Eq.
(I-5) in which f = 1 becomes
2 2From Eq. (I-2), one has M8 = x Mx . Thus, integrating Eq. (I-6)
leads to
2 1where C (x ) is an arbitrary constant and therefore may be set
to zero without loss of generality. Thus, Eq. (3.17) gives
Example II. Consider the Rayleigh or modified van der Pol
oscillator [3.19]
where $ is a positive constant. If one writes
Exact Solutions of Fokker-Planck-Kolmogorov Equations 29
(II-3)
(II-4)
(II-5)
(II-6)
(II-7)
(II-8)
(II-9)
(II-10)
then Eq. (II-1) becomes
This equation is of the type described by Eq. (I-1) in Example I
above. In the latter, 8 is replaced by H here. Therefore,
Substituting Eq. (II-4) into (I-8), it gives
where C is a normalization constant. One can re-write
Substituting for Eq. (II-2) and re-arranging,
By Eq. (II-7), Eq. (II-5) becomes
Equation (II-8) can also be written as
Equation (II-9) can be reduced to
1where C is a normalization constant. Thus, the response of the
system by Eq. (II-1) is not Gaussian.
30 Nonlinear Random Vibration
Example III. Consider a nonlinear oscillator with parametric and
external excitations
i ii where w (t) are independent Gaussian white noises with
spectral densities S ; ", $ and S are constant. This is the example
studied by Yong and Lin in Ref. [3.11].
2 1Applying Eq. (3.15) and setting A = - S x by using Eq. (3.23) as
well as(2) 2
3 1imposing C (x ) = 0 in Eq. (3.32), it results
1 1 2since f = - S x and f = 1. Integrating Eq. (III-2)
gives2
22 11In particular, if S /S = "/$, one has
Without loss of generality, one may choose
The stationary probability density function is therefore given
as
where C is the normalization constant. As noted by Yong and Lin
[3.11] that under a suitable combination of Gaussian parametric and
external random excitations, the response of the above nonlinear
system is Gaussian.
Example IV. The following system is the one considered by
Dimentberg [3.8], and Yong and Lin [3.11]
Exact Solutions of Fokker-Planck-Kolmogorov Equations 31
(IV-2)
(IV-3)
(IV-4)
(IV-5)
(IV-6)
(IV-7)
i iiwhere w (t) are independent Gaussian white noises with spectral
densities S , S is a constant, and
As the coefficient of velocity in Eq. (IV-1) has a parametric
random excitation the WZ correction term [3.20] is required. The
resulting Itô equations for Eq. (IV-1) are
The corresponding reduced FPK equation becomes
2 1Applying the procedure described above and specifying A = - S x
with reference(2) 2
3 1to Eq. (3.23), and imposing C (x ) = 0 in Eq. (3.32), then Eq.
(3.15) gives
1 2 2 1 3 11 22since f = - x , f = - S x and f = 1. If S = S S ,
then from Eq. (IV-6) one2 2
can show that
4 1Applying Eq. (3.16) results C (x ) = constant. Therefore,
integrating and substituting the result into Eq. (3.17) gives
32 Nonlinear Random Vibration
(IV-8)
(IV-9)
(V-1)
(V-2)
(V-3)
(V-4)
(V-5)
If one confines .(7) = $7 + " in which " and $ are constant, and
after integrating Eq. (IV-8) it leads to
where C is a normalization constant. Equation (IV-9) was
independently presented in Refs. [3.8] and [3.11], with different
notations.
Example V. The following equation of motion is for the so-called
energy dependent system considered by Zhu and Yu [3.18]
where w(t) is a Gaussian white noise with a spectral density S,
g(x) is the nonlinear spring force, .(8) and f(8) are arbitrary
functions, and 8 is the total energy
Note that Eq. (V-1) is similar to Eq. (I-1) above except for the
RHS. Applying the same symbols as in the method presented above,
the two Itô stochastic differential equations for Eq. (V-1)
are
and
where B(t) or written simply as B is a unit Wiener process. The
corresponding reduced FPK equation becomes
Exact Solutions of Fokker-Planck-Kolmogorov Equations 33
(V-6)
(V-7)
(V-8)
The first and second derivate moments are divided into two parts as
those in the 2 1procedure described above except that A = - g(x )
is chosen in accordance with(2)
3 1Eq. (3.23), and C (x ) = 0 is imposed in Eq. (3.32) such that
Eq. (3.14) is satisfied. 1Then, Eq. (3.15) for the reduced FPK Eq.
(V-5) in which f = f(8) becomes
2 2From Eq. (V-2), one has M8 = x Mx . Therefore, integrating Eq.
(V-6) gives
2 1where C (x ) is an arbitrary constant and therefore may be set
to zero without loss of generality. Thus, Eq. (3.17) gives
Equation (V-8) agrees with (3) of Ref. [3.18] except for different
notations.
3.3 Applications to Engineering Systems
In this section the extension to the theory and associated
procedure of generalized stationary potential described in the last
section is applied to various sdof systems frequently encountered
in engineering. They are grouped into three categories, namely, (a)
systems with linear damping and nonlinear stiffness, (b) systems
with nonlinear damping and linear stiffness, and (c) systems with
both nonlinear damping and nonlinear stiffness.
As far as possible the mean square or variance of response of every
system is included in addition to the stationary probability
density function. Unless stated otherwise it is assumed that the
stationary probability density function exists in the nonlinear
system.
3.3.1 Systems with linear damping and nonlinear stiffness
This category includes nonlinear systems with (a) elastic force of
polynomial type, (b) elastic force of trigonometric function type,
(c) elastic force with acceleration
34 Nonlinear Random Vibration
(I-6, 7)
jumps, (d) double bi-linear restoring force, and (e) in-plane or
axial random excitation. The equation of motion for every system
will be solved by the direct approach whenever it is possible, and
identification of the equations of the method in Section 3.2 will
also be made.
Example I. Consider the Duffing oscillator. This is the one with
the simplest polynomial type elastic force. It can be used to model
system with large displacement [3.21] or the so-called system with
geometrical nonlinearity. The equation of motion for this
oscillator under Gaussian white noise excitation is
where $ is the positive damping coefficient, S is the natural
frequency of the corresponding linear oscillator and g is the
strength of nonlinearity. The latter is assumed to be positive
henceforth.
The two Itô differential equations corresponding to Eq. (I-1)
are
1where g(x ) = S x + gx . The reduced FPK equation for Eqs. (I-2)
and (I-3) is2 3
In order to solve for Eq. (I-4) it may be written as
Equation (I-5) is solvable if it is satisfied by the following two
equations
Exact Solutions of Fokker-Planck-Kolmogorov Equations 35
(I-8)
(I-9)
(I-10)
(I-11)
(I-12)
(I-13)
Combining Eqs. (I-8) and (I-9), one has
where C is a normalization constant. Now, before deriving the
variance of displacement the method presented in
Section 3.2 is applied. With reference to Eq. (I-8) of Section 3.2,
8 is identified as the total energy of the present oscillator
and .(u) in Eq. (I-8) of Section 3.2 is $ in the present system.
Consequently, by applying Eq. (I-8) of Section 3.2 one has result
identical to Eq. (I-10) above.
The mean square of displacement is
Before substituting E q. (I-10) into (I-11) one separates the joint
stationary s 2 2 1 1probability density function into two parts as
p = p (x ) p (x ) where
and
where
Therefore,
By Eqs. (I-13) through (I-16), Eq. (I-11) can be expressed as
Both Eqs. (I-15) and (I-17) can be evaluated by applying the
parabolic cylindrical and gamma functions. Making use of the
following identity
where U(a,z) is the parabolic cylindrical function. Writing
Exact Solutions of Fokker-Planck-Kolmogorov Equations 37
(I-19)
(I-20)
(I-21)
(I-22)
(II-1)
so that
Substituting Eq. (I-19) into (I-15) and making use of Eq. (I-18),
one has
Simplifying, it gives
Applying Eq. (I-19) and with similar procedure as for the
derivation of Eq. (I-21), Eq. (I-17) can be obtained as
xIf S = S = g = 1.0 and $ = 0.1 Eq. (I-22) gives F = 3.5343 , and
other values are2
plotted in Figure 3.1. With reference to the latter, it is clear
that the mean square of displacement decreases with increasing
strength of nonlinearity, but it increases with increasing spectral
density of the Gaussian white noise excitation.
Example II. An example of a system with elastic force of
trigonometric function type is the following
0 0where k is the initial spring rate, x is the maximum deflection
obtainable with
38 Nonlinear Random Vibration
(II-2)
(II-3)
(II-4)
0 0 infinite force such that - x < x < x , m is the mass of
the system, and the remaining symbols have their usual meaning.
This nonlinear elastic force is shown in Figure 3.2. Clearly, this
oscillator is similar to that described in Eq. (I-1) above, except
that the polynomial elastic force is replaced by a so-called
tangent elasticity
characteristic [3.22]. The elastic force described in Eq. (II-1)
represents a hardening spring with limiting finite deflection, even
when it is subjected to an infinite force. Equation (II-1) was
dealt with by Klein [3.23]. The following results are taken from
the latter with change in the notations. A possible example of
application is in the analysis of a vibration isolator that uses an
elastomer, such as neoprene, as the spring element. Isolators of
this type are used in protecting electronic equipment from
vibration in aircraft and missiles [3.23].
The joint stationary probability density function of equation
(II-1) can be 1 1 0 0 1 0obtained by replacing g(x ) = S x + gx
with g(x ) = [2k x /(Bm)] tan[Bx /(2x )]2 3
so that
0 where S = k /m and C is the normalization constant. 2
0Writing F = BS/($S ) and performing the integration in Eq. (II-2),
2 2
The above probability density function can be factored by marginal
distributions as indicated in the last example due to the solutions
given in Eqs. (I-8) and (I-9). In
1 2other words, x and x are statistically independent. Thus,
where
Figure 3.1 Mean square of displacement of Duffing oscillator.
Figure 3.2 Elastic force of a hardening spring with finite
deflection.
40 Nonlinear Random Vibration
(II-5)
(II-6)
(II-7)
(II-8)
(II-9)
The normalization constant, C can be evaluated by the following
equation
0 0 0where n = [2x / (BF )] $ 0. Writing y = Bx /(2x ) and with
appropriate change of2
integration limits, the last equation becomes
Evaluating Eq. (II-7) gives
where '(.) is the gamma function or the Euler integral of the
second kind. Substituting C into Eq. (II-5) results
The mean square of displacement of the oscillator is
Exact Solutions of Fokker-Planck-Kolmogorov Equations 41
(III-1)
In general the integral in Eq. (II-9) can not be evaluated
explicitly. However, when n is a positive integer it can be
determined explicitly. Typical results are shown in Figure
3.3.
Figure 3.3 Mean square of displacement.
Example III. Consider a nonlinear oscillator with a set-up spring.
Its equation of motion is given by
where sgn x = 1 for x > 0 and sgn x = -1 for x < 0. When the
oscillator mass traverses through x = 0 it undergoes a jump in
relative acceleration of magnitude
02F /m, m being the mass of the oscillator, whereas its relative
velocity is continuous. This oscillator, shown in Figure 3.4, was
analyzed by Crandall [3.24]. Figure 3.5 presents the restoring
force as a function of relative motion x of the oscillator.
42 Nonlinear Random Vibration
Figure 3.4 Oscillator with a set-up spring.
Figure 3.5 Restoring force of a nonlinear oscillator with a set-up
spring.
Exact Solutions of Fokker-Planck-Kolmogorov Equations 43
(III-2)
(III-3)
(III-4)
(III-5)
(III-6)
The reduced FPK equation for (III-1) is similar to (I-4),
therefore
Following similar procedure in Example I above one obtains the jpdf
as
0where F = BS/(2.S ) and C is the normalization constant which is
defined by2 3
Then, applying the following definitions
the last relation can be found as
1 2Since the double integral is an even function of x and x and
therefore one can simply consider the following term
Substituting Eq. (III-5) into (III-4) one can show that
Substituting Eq. (III-6) into (III-3), it gives
44 Nonlinear Random Vibration
(III-7)
(III-8)
(III-9)
(IV-1a)
(IV-1b)
(IV-2)
Similarly,
By Eq. (III-7) the mean square of displacement is obtained as
The results in Eq. (III-7) through (III-9) are identical to those
presented by Crandall [3.24] except that they are different in
notations.
Example IV. Consider a sdof nonlinear oscillator with a general
double bi-linear restoring force, as shown in Figure 3.6.
Oscillators of this type can be applied to model materials
undergoing elasto-plastic deformation or systems with energy
dissipation absorbers. The equation of motion can be expressed
as
1 1 1 0 1where S = k /m, T = k /m, g = (k - k )x /k , and m is the
mass of the system.2 2
Equation (IV-1a) is linear and Eq. (IV-1b) is similar to Eq.
(III-1) above. By the procedure in the last example, the
probability density functions are
Exact Solutions of Fokker-Planck-Kolmogorov Equations 45
(IV-3)
(IV-4)
(IV-5a)
(IV-5b)
where
(IV-6)
(IV-7)
(IV-8)
(IV-9)
(IV-10)
1 2 and there are similar relations for p and p . These continuity
conditions will be satisfied if one assumes the following
relation
1The constant C can be evaluated from the normalization
condition
Therefore,
where
where
By making use of Eqs. (IV-2), (IV-3), (IV-6), (IV-8), and (IV-9)
one has
Exact Solutions of Fokker-Planck-Kolmogorov Equations 47
(IV-11)
(IV-12)
(IV-13)
where
The mean squares of displacement for several special cases can be
evaluated by applying Eq. (IV-10).
0 1Case (i) x = 0 and k = k
This is a linear system. Thus, the mean square of displacement
is
1Case (ii) k = k ( 2 = ( ) This case is also a linear system. It is
easy to show that the mean square of
displacement is identical to Eq. (IV-11) above.
1Case (iii) 2 v B /2 ( k v 4 ) 2From Eq. (IV-6), C = 0 and the
joint stationary probability density function is
given by Eq. (IV-2). Applying Eqs. (IV-8) and (IV-10) one can show
that the mean square of displacement is
1Case (iv) 2 = 0 ( k = 0 ) This case can be employed to model
elastic perfectly plastic materials. For this
case Eq. (IV-1b) becomes
48 Nonlinear Random Vibration
(IV-14)
(IV-15)
(IV-16)
(IV-17)
(V-1)
Applying Eq. (III-3) above or Eq. (IV-3), one can show that the
probability density function is given by
and the continuity condition gives
From the normalization condition one has
The mean square of displacement, after some algebraic manipulation,
can then be expressed as
Example V. Consider an oscillator with a parametric random
excitation as a coefficient of the cubic displacement term. The
equation of motion is
Equation (V-1) can be used to model a single mode vibration of a
plate structure under a transversal random excitation and an
in-plane random excitation when the second and higher modes of
vibration are well beyond the frequency range of interest.
Likewise, it can be used to model a single mode vibration of a beam
structure in bending and simultaneously subjected to an axial
random excitation. Of course, the random excitations considered
here are Gaussian white noise processes. This, in theory, should
have an infinite range of frequency and therefore would cover
Exact Solutions of Fokker-Planck-Kolmogorov Equations 49
(V-2)
(V-3)
(V-4)
(V-5)
(V-6)
all the modes in the plate or beam structures. However, in
practice, the single mode assumption is acceptable in that the
duration of excitation is finite rather than infinite.
Applying similar procedure as that in Example III of Section 3.2,
one can show that
11 1 1 2where " = g S S and in Eq. (3.15) f = - gS x , and f = 1. 2
4 2 3
Integrating Eq. (V-2) leads to
Without loss of generality, one can choose
so that
Strictly speaking, this equation satisfies only the solvability
conditions (3.14) and (3.15) in Section 3.2 above. To also satisfy
the solvability condition (3.16) one requires
1 2 22 1 2where " = 5S /3 and " = 2S /(3"). When x = 0 then x = 4
such that N = 4.2
This leads to a zero probability density function. By Eq. (V-4),
the joint stationary probability density function becomes
2 1 where C is the normalization constant. For x > Sx the
probability density given by Eq. (V-6) is stable but
non-Gaussian.
50 Nonlinear Random Vibration
3.3.2 Systems with nonlinear damping and linear stiffness
This class of problems includes self-excited oscillators, such as
the van der Pol oscillator and modified van der Pol or Rayleigh
oscillator. For small strength of nonlinearity the latter two
oscillators exhibit limit cycles and the responses of both
oscillators are essentially sinusoidal. As the value of the
strength of nonlinearity increases the limit cycles become
distorted and the responses non-sinusoidal.
Example I. The Rayleigh or modified van der Pol oscillator excited
by Gaussian white noise has been considered by Caughey and Payne
[3.7] and To [3.19]. This model can be applied to analyse flow
induced vibration of a slender cylinder if the excitation is small
and only the first mode of vibration for the cylinder is of
interest. The equation of motion for such an oscillator is
where $ is a positive constant. If one let
Then Eq. (I-1) becomes
This equation is of the type described by Eq. (I-1) in Section 3.2
above. In the latter, 8 is replaced by H here. Therefore,
Substituting Eq. (I-4) into Eq. (I-8) of Section 3.2, it
gives
where C is a normalization constant. One can re-write
Substituting for Eq. (I-2) and re-arranging,
Exact Solutions of Fokker-Planck-Kolmogorov Equations 51
(I-7)
(I-8)
(I-9)
(I-10)
(I-11)
(I-12)
Equation (I-8) can also be written as
Equation (I-9) can be reduced to
1where C is a normalization constant. Thus, the response of the
system described by Eq. (I-1) is not Gaussian. Expanding the square
term in the exponential function in Eq. (I-10), it can be expressed
as
or
2where C is the normalization constant. Therefore, the joint
stationary probability density function is given by
Equation (I-12) agrees with Eq. (41) of Ref. [3.7].
52 Nonlinear Random Vibration
Similarly, the mean square of velocity can be obtained as
1 2 As the probability density function in Eq. (I-12) is symmetric
in x and x , hence
Introducing the change of variables,
Therefore,
Performing the double integrations, one can show that [3.7]
Example II. Consider a nonlinear oscillator with a parametric
excitation. This model can be applied to the simplified response
analysis of a rotor blade. The equation of motion is
Exact Solutions of Fokker-Planck-Kolmogorov Equations 53
(II-1)
(II-2)
(II-3)
(II-4)
(II-5)
(I-1)
where w(t) is the Gaussian white noise with spectral density S; ",
$ and S are constant. This is Example III in Section 3.2 above
except that the external random
2 1 excitation is disregarded. Applying Eq. (3.15) and setting A =
- S x by using Eq.(2) 2
3 1(3.23) as well as imposing C (x ) = 0 in Eq. (3.32), it leads
to
1 1since f = - S x . Integrating Eq. (II-2) gives2
Without loss of generality, one may choose
The stationary probability density function can then be expressed
as
where C is the normalization constant. Clearly, the response of the
above oscillator is not Gaussian.
3.3.3 Systems with nonlinear damping and nonlinear stiffness
Many practical engineering systems belong to this category.
However, explicit solution is difficult to obtained if not
impossible. The following example is included to illustrate its
derivation rather than to present an analysis of a reasonably
practical problem.
Consider a nonlinear oscillator having the following equation of
motion
54 Nonlinear Random Vibration
(I-2)
(I-3a,b)
(I-4)
(I-5)
(I-6)
where w(t) or simply w is a Gaussian white noise with a spectral
density S and the total energy is
Note that in the foregoing the limits of integration are not
identified as the reference level of the potential energy may be
chosen arbitrarily. Equation (I-2) can be easily verified if one
applies the following co-ordinate transformation
Using Eq. (I-2), Eq. (I-1) can be written as
Equation (I-4) is similar to Eq. (I-1) of Example I in Section 3.2
and therefore the joint stationary probability density function can
be expressed as
Performing the integration in Eq. (I-5) and substituting Eq. (I-2),
it results
3.4 Solution of Multi-Degree-of-Freedom Systems
Generalization of Eq. (3.10) for lumped-parameter systems with
multi-degree of freedom (mdof) is straightforward though the amount
of algebraic manipulation is substantially increased. For example,
the scalar variables x and x0 in Section 3.2
1 2 3 n 1 2 3 n ibecome vectors. That is, X = (x x x ... x ) and Y
= (x0 x0 x0 ... x0 ) such that fT T
ir i rbecomes f and w changes to w , where i = 1,2, ..., n and r =
1,2,..., m. Accordingly, the equations of motion for a n dof system
may be written as
Exact Solutions of Fokker-Planck-Kolmogorov Equations 55
(3.38)
(3.39)
This set of equations can be expressed in matrix form as
where Y and h(X;Y) are vector functions of order n × 1. In order to
identify the first and second derivate moments with the above
1 1 2 3 n 2 1 2 3 nnonlinear mdof system, one writes Z = X = (x x x
... x ) , Z = Y = (x0 x0 x0 ... x0 ) ,T T
1 2 1 2 3 2nand Z = (Z Z ) = (z z z ... z ) such that the state
vector equation becomesT T
m × 1where (w) is the vector of delta correlated white noise
processes. The corresponding Itô's equation is
m × 1 m × 1where 6 is the WZ correction term and is a vector of
order n×1, (db) = (w) dt, r rin which db = w (t)dt is the Brownian
motion or Wiener processes, and
The latter equation may also be expressed as
where the subscripts r,s = 1,2,...,m. The first and second derivate
moments of the FPK equation associated with Eq.
(3.39) are
Splitting the first and second derivate moments into
Then applying Eq. (3.40), following similar steps between Eqs.
(3.14) through (3.36) in Section 3.2, and now suppose that the
elementary or integrating factor is
1 2 3 n 1 2 3 n 1 2 3 n 1 2 3 n i iM(x , x , x ,...,x ;y ,y ,y
,...,y ), r = 8(x , x , x ,..., x ; y , y , y ,..., y ), y = (x )
/2, and2
yiM = 8 , one can obtain the stationary probability density
as
To illustrate the application of the foregoing procedure, it
suffices to consider the following two dof system
ir 1 2 1 2where f , in general, are functions of x , x , x0 and x0
. To identify the first and second derivate moments of this two dof
system one re-
writes Eq. (I-1) into the following four first order differential
equations
Exact Solutions of Fokker-Planck-Kolmogorov Equations 57
With reference to the above equations, and recall the notation,
that the first and second derivate moments of the corresponding FPK
equations can be determined as
58 Nonlinear Random Vibration
(I-3)
(I-4)
The remaining second derivate moments are zero. The above first and
second derivate moments are split and making use of the sdof
Eq. (3.33) for mdof systems, one can show that
If a consistent function dN(8) /d8 can be found from Eqs. (I-3) and
(I-4) then the above problem is of the generalized stationary
potential type.
Consider the simple case of Scheurkogel and Elishakoff [3.25] in
which the two
Exact Solutions of Fokker-Planck-Kolmogorov Equations 59
(I-5)
(I-6)
(I-7)
(I-8)
(I-9)
(I-10a,b)
dof system with equations of motion similar to Eq. (I-1) above
but
and
where H is non-negative potential function. From Eq. (I-5) and
comparing with Eq. (I-3) in the foregoing, one has M = 1,
and
where ( is a constant and therefore, the system admits a stationary
solution
Note that Eq. (I-8) is independent of the choice of (. This result
was obtained by Cai and Lin [3.14], and was also obtained by
Scheurkogel and Elishakoff [3.25] applying a different procedure.
It was pointed out by Cai and Lin that the system is in detailed
balance when ( = 1/2.
To derive the statistical moments for this particular case, Ref.
[3.25] uses
1 2where " and " are positive, and g is a positive small parameter.
Recall that
Introducing the new variables
60 Nonlinear Random Vibration
and the marginal probability density functions
1 s where C is a normalization constant, the probability density
function p can then be written as
Equation (I-15) implies that the velocities and new displacement
variables defined by Eq. (I-10) are pairwise independent. From Eqs.
(I-11) through (I-13), one can conclude that the velocities and u
are normally distributed with zero mean
Applying the following identity [3.26]
the second moments of velocities and u can be shown to be
With reference to Eqs. (I-10) and (I-16), and the independence of u
and v one can obtain
and therefore
To evaluate Eq. (I-20) one requires < v >. Note that the
marginal probability density2
function of v is symmetrical about the origin and therefore all its
odd moments are zero. Furthermore, applying Eqs. (I-10) and (I-16)
one has
Exact Solutions of Fokker-Planck-Kolmogorov Equations 61
(I-21)
(I-22)
(I-23)
(I-24)
(I-25)
(I-26)
(I-27)
By definition the moments of even order of v are
By making use of the following substitution
equation (I-22) can be obtained as
mwhere the function Q [.] is given by
Setting m = 0 in Eq. (I-24), the normalization constatnt
Hence, Eq. (I-24) becomes
As g is small one can show that the second moment of v is given by
[3.25]
62 Nonlinear Random Vibration
(I-28)
(I-29)
(I-30)
(3.42)
1Applying Eqs. (I-10), (I-18), (I-19), (I-21) and (I-28) the second
moments of x and 2x can be shown to be
and
3.5 Stochastically Excited Hamiltonian Systems
Another general technique of dealing with a somewhat larger but
still restricted class of mdof nonlinear systems in terms of
Hamiltonian formulation has been provided in Refs. [3.27-3.29]. The
technique in Refs. [3.27, 3.29] is a generalization of that by
Soize [3.28]. The basic steps in the technique are included
here.
Consider the mdof nonlinear system governed by
Exact Solutions of Fokker-Planck-Kolmogorov Equations 63
(3.43)
(3.44)
(3.45)
(3.46)
(3.47)
(3.48)
where "(q) is an arbitrary function of q; H is the Hamiltonian with
continuous first r irorder derivatives; w (t) are Gaussian white
noises; $(H), ( (q;p), and f(H) are twice
ij 1 2 3 n 1 2 3 ndifferentiable; c (q;p) are differentiable; q =
(q q q ... q ) ; p = (p p p ... p ) ;T T
i iq and p are generalized displacements and momenta, respectively.
The system in Eq. (3.42) encompasses both additive and
multiplicative random
excitations. Following the procedure in Ref. [3.29], one has
i j ij ijwhere N is the probability potential; < w (t) w (t + J)
> = 2BS *(J); and B is(i)
related to the second derivate moments. Equation (3.43) may be
re-written as
which has a general solution
Therefore, the stationary probability density
ij ir ij ji ijSuppose $ is constant, c and ( depend on q only, and
c + c = :B , one obtains
ij ir ij ji ijIf $ is a function of H, c and ( depend on q only,
and c = c = :B /2,
64 Nonlinear Random Vibration
Consider another system whose Hamiltonian is given by [3.29]
i i ij jwhere q are the generalized displacements, p = m q are the
generalized momenta, and m(q) is a symmetric matrix. The system
corresponding to the above Hamiltonian has the following governing
equations of motion
1 2 3 n iwhere x = (x x x ... x ) , x being the displacement of the
i'th dof of the system,T
and the remaining symbols have their usual meaning. Thus, the
stationary probability density of x and x0 can be expressed in
terms of that for q and p by the following relation
where *J* is the Jacobian and is equal to the determinant of the
symmetric matrix m(x) in Eq. (3.51).
4 Methods of Statistical Linearization
4.1 Introduction
The systematic methods developed by Cai and Lin [4.1], and further
generalized by To and Li [4.2] give a broadest class of solvable
reduced Fokker- Planck-Kolmogorov (FPK) equations which contains
all solvable equations previously obtained and presented in the
literature. However, it is difficult to find a real mechanical or
structural system that corresponds to a solvable reduced FPK
equation other than those already reported in the literature and
representatively included in Chapter 3. Consequently, it is
necessary to apply approximation methods to deal with other real
mechanical or structural systems.
One popular class of methods for approximate solutions of nonlinear
systems is that of statistical linearization (SL) or equivalent
linearization (EL) techniques. These techniques are popular among
structural dynamicists and in the engineering mechanics community.
This is partially due to its simplicity and applicability to
systems with mdof, and systems under various types of random
excitations.
The SL technique was independently developed by Booton [4.3,4.4]
and Kazakov [4.5,4.6] in the field of control engineering. Further
developments in this field were presented and reviewed by Sawaragi
et al. [4.7], Kazakov [4.8,4.9], Gelb and Van Der Velde [4.10],
Atherton [4.11], Sinitsyn [4.12], and Beaman and Hedrick [4.13]. In
the control and electrical engineering communities the SL
techniques are also known as methods of describing functions. In
the field of structural dynamics Caughey [4.14] independently
presented the SL technique as an approximate method for solving
nonlinear systems under external random forces. Subsequently,
generalization of the SL technique in the field of structural
dynamics was made by Foster [4.15], Malhotra and Penzien [4.16],
Iwan and Yang [4.17], Atalik and Utku [4.18], Iwan and Mason
[4.19], Spanos [4.20], Brückner and Lin
66 Nonlinear Random Vibration
(4.1)
(4.2)
(4.3)
[4.21], and Chang and Young [4.22]. Many applications of the SL
technique have been made since its introduction in mid-1950 and
early 1960. Examples can be found in the survey articles of
Sinitsyn [4.12], Spanos [4.23], Socha and Soong [4.24], and the
books by Roberts and Spanos [4.25], and Socha [4.26]. The
underlying idea of the SL techniques is to replace the nonlinear
system by a linear one such that the behaviour of the equivalent
linear system approximates that of the original nonlinear
oscillator. In essence the techniques are generalizations of the
deterministic linearization method of Krylov and Bogoliubov [4.27]
in the sense that equivalent natural frequencies are
employed.
In this chapter representative SL techniques, in the field of
structural dynamics, the issues of existence and uniqueness,
accuracy, and various applications are presented and
discussed.
4.2 Statistical Linearization for Single-Degree-of-Freedom
Nonlinear Systems
In this section the methods of SL for sdof nonlinear systems with
stationary solutions, sdof systems with nonstationary random
response, non-zero mean stationary solution, stationary solution of
a nonlinear sdof system under narrow-band excitation, stationary
solution of a sdof system under parametric and external random
excitations are introduced.
4.2.1 Stationary solutions of single-degree-of-freedom systems
under zero
mean Gaussian white noise excitations Consider a sdof nonlinear
oscillator described by the equation of motion
where the symbols have their usual meaning. In particular,
in which S is the spectral density of the Gaussian white noise
process w(t). The underlying idea of the SL technique is to replace
Eq. (4.1) by the following equivalent linear equation of
motion
Methods of Statistical Linearization 67
(4.4)
(4.5)
(4.6)
(4.7)
(4.8)
(4.11)
(4.12a,b)
e ewhere $ and k are the equivalent damping and stiffness
coefficients that best approximate the original nonlinear equation
of motion, (4.1). To achieve this, one simply adds the equivalent
linear damping and restoring force terms to both sides of Eq. (4.1)
and re-arranges to give
where D is the deficiency or error term in the approximation. The
deficiency
In order to minimize the error, a common criterion is to minimize
the mean square e evalue of the error process D. Therefore, the
parameters $ and k have to be
chosen such that
and
For < D > to be a true minimum it requires that 2
68 Nonlinear Random Vibration
(I-1)
By virtue of Eqs. (4.7), (4.8) and (4.12) one has the following
pair of algebraic equations
e eClearly, the solution for $ and k requires the knowledge of the
unknown expectations. There are two possible approximations
[4.4,4.14]. The first approach is to replace the joint transition
probability density function by the joint stationary probability
density function. This, in turn, enables one to replace the
time-dependent mean square values of displacement and velocity by
their corresponding stationary mean square values. The other
approach is to replace the joint stationary probability density
function by the joint stationary probability density function of
the linearized equation. In this approach the expectations are now
implicit functions of the equivalent damping and stiffness
coefficients. The consequence of this is that Eqs.
e e(4.13) and (4.14) become nonlinear in $ and k . It should be
noted that the most general formulas for the determination of
the
equivalent linear damping and stiffness coefficients, which are
applicable to stationary and nonstationary Gaussian approximations
of the response are
Equations (4.15) and (4.16) are obtained from the corresponding
relations for mdof systems that were presented by Atalik and Utku
[4.18]. This SL technique for mdof systems is included in Section
4.3 and therefore is not dealt with here.
In the following several examples are included to illustrate the
application of the SL technique presented above. These are systems
with (a) nonlinear restoring forces and linear dampings, (b)
nonlinear dampings and linear restoring forces, and (c) nonlinear
dampings and nonlinear restoring forces.
Example I. Applying the method of SL, determine the stationary
variances of displacement and velocity of a Duffing oscillator
whose equation of motion is given by Eq. (4.1) in which
Methods of Statistical Linearization 69
(I-2)
(I-3a,b,c)
(I-3d,e)
(I-4a,b)
(I-5)
(I-6)
(I-7a,b)
(I-8)
where S is the natural frequency of the associated linear
oscillator, that is, when g = 0 in Eq. (I-1).
The equivalent linear equation is
The approximation adopted here is to assume that x and dx/dt are
stationary, independent, and with zero means. Consequently, for the
oscillator defined by Eqs. (4.1) and (I-1)
Applying Eqs. (4.13), (4.14) and (I-3) immediately leads to
The variance of x can be determined from the following
relation
where the power spectral density of x is given by
win which "(T) and S (T) are the frequency response function or
receptance of the equivalent linear system and the power spectral
density function of the excitation, respectively. Thus,
The stationary variance of x for the equivalent linear equation is
given by
Writing the stationary variance of x for the linear oscillator,
that is g = 0,
70 Nonlinear Random Vibration
(I-9)
(I-10)
(II-1)
(II-2)
(II-3)
(II-4)
(II-5)
and solving for the variance of x of the equivalent linear
oscillator by making use of Eqs. (I-8) and (I-4b), one has
Example II. Consider the nonlinear oscillator of Eq. (4.1)
where
Let the equivalent linear equation be given by Eq. (I-2) above.
Applying Eq. (4.15) one can show that
Similarly, applying Eq. (4.16) gives
From Ref. [4.28]
where the stationary probability density is assumed to be Gaussian
since the excitation is Gaussian, that is
Therefore,
(II-6)