Dynamical Mechanical Systems Under Random Impulses

DYNAMICAL MECHANICAL SYSTEMS UNDER RANDOM IMPULSES

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Series on Advances in Mathematics for Applied Sciences - Vol. 36

DYNAMICAL MECHANICAL SVSTEMSUNDER RRNDOM IMPULSES

Radostaw Iwankiewicz Institute of Materials Science and Applied Mechanics Technical University of Wroclaw

World Scientific Singapore • New Jersey • London • Hong Kong

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Library of Congress Cataloging-in-Publication Data Iwankiewicz, Radoslaw. Dynamical mechanical systems under random impulses / Radoslaw Iwankiewicz. p. cm. — (Series on advances in mathematics for applied sciences ; vol. 36) Includes bibliographical references. ISBN 9810222815 1. Random vibration. 2. stochastic processes. I. Title. II. Series. QA935.I88 1995 53r.32'015192-dc20 95-22061 CIP

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Preface This monograph presents methods of analysis of a particular class of prob­ lems in stochastic dynamics, which is the dynamic behaviour of mechanical, in general, systems under random impulses. The methods and approximate an­ alytical techniques relevant to linear and non-linear dynamical systems under random trains of impulses and general pulses, which are covered herein, are the result of the author's research during the past years. In Chapter 1 random vibration problems in general and problems of stochas­ tic excitations which can be adequately idealized as random pulse trains are introduced. A brief review of the general methods of random vibration has been written from the point of view of their applicability in random impulses problems. Finally an overview of the literature on dynamical systems under random impulses is given. Chapter 2, which is a prerequisite for the subsequent chapters, is an intro­ duction into the basic methods of stochastic point processes. The stochastic point processes, which are a tool of a sound mathematical treatment of prob­ lems of random trains of impulses, are dealt with in many mathematical books. However, to the best of the author's knowledge, one could hardly find a book covering the collection of the methods of stochastic point processes made from the standpoint of their applicability in problems of response of dynamical sys­ tems to random pulse trains driven by different point processes. Therefore the relevant methods, scattered throughout the literature, have been collected and outlined in Chapter 2. That Chapter also provides some information about the asymptotic expansions of the probability density of the stochastic process. The methods for linear dynamical systems under external excitations in form of random pulse trains which are covered in Chapter 3 are, in general, moment methods. These methods allow the exact evaluation of response sta­ tistical moments for linear dynamical systems subjected to random pulse trains v

vi

Preface

driven by any stochastic point processes. Application of these methods to some problems of structural dynamics, such as the bridge response to a random stream of moving loads and vibration of shallow shells, is discussed in detail, and the numerical results are given. In addition the estimation, via moments, of the probability density function and of the mean upcrossings rate for the response of a linear system to a Poisson distributed pulse train is presented. In Chapter 4 the methods for non-linear dynamical systems under exter­ nal impulse process excitations are presented. The methods covered may be classified as those belonging to moment equations methods. For systems with polynomial non-linearities these are the methods of moment equations com­ bined with closure approximations and are valid for Poisson impulse process excitations. For hysteretic systems under Poisson impulses the equivalent lin­ earization and equivalent cubicization techniques are devised. Finally, the gen­ eralization of the moment equations technique to a class of renewal impulse process excitations is effectuated.

Radoslaw Iwankiewicz

Contents Preface

v

1

Introduction 1.1 Random vibration and random pulse trains problems 1.2 Most important general methods for random vibration 1.3 Methods for random pulse trains

2

Stochastic point processes — basic ideas and methods and other mathematical preliminaries 2.1 Characterization of stochastic point processes 2.1.1 General stochastic point process, counting process . . . . 2.1.2 Poisson process, compound Poisson process 2.1.3 Renewal processes 2.1.4 Markov renewal (semi-Markov) process 2.2 Filtered stochastic point processes 2.2.1 General case: stochastic integral with respect to a counting process, averaging of stochastic integrals . . . . 2.2.2 Filtered Poisson process 2.2.3 Filtered renewal process 2.2.4 Filtered Markov renewal process 2.3 Non-diffusive Markov process 2.4 Asymptotic expansions of the probability density function of the stochastic process - Gram-Charlier expansions

3

Linear dynamical systems under random pulse trains 3.1 Introductory remarks 3.2 Moments of the response process to a random pulse train . . . . vii

1 1 5 11

21 21 21 25 26 28 30 30 34 35 37 38 42 47 47 49

viii

CONTENTS 3.2.1 3.2.2 3.2.3

3.3

3.4

4

Correlated and Poissonian pulse train Pulse train driven by a delayed renewal process . . . . Pulse train driven by a Markov renewal (semi-Markov) process Examples of applications • ■• ... 3.3.1 Bridge structure under a random train of moving loads 3.3.2 Dynamic response of shallow shells to Poisson-distributed impulses • Estimation of higher-order characteristics of the response probability distribution . . 3.4.1 Probability density of the response of a linear oscillator to a Poisson-distributed pulse train 3.4.2 Mean upcrossings rate for a linear oscillator subjected to a Poisson-distributed pulse train

49 53 56 58 58 66 75 75 82

Non-linear dynamical systems under random pulse trains 91 4.1 Introductory remarks 91 4.2 Statement of the problem: stochastic equations and equations for moments for the systems subjected to a Poisson-distributed train of impulses 93 4.3 Moment equations technique for the systems with polynomial non-linearity, subjected to a Poisson-distributed train of impulses . . . . 96 4.3.1 Equations for moments and closure approximations . . . 96 4.3.2 Application of the usual cumulant-neglect closure approximations in the case of the system with cubic non-linearity: dynamic response of an elastic plate with moderately large deflections to a train of impulses . 103 4.3.3 Application of modified cumulant neglect closure approximations in the case of the system with cubic non-linearity: Duffing oscillator under a train of general pulses 112 4.4 Equivalent systems techniques for the system with non-analytical non-linearity, subjected to a Poisson-distributed train of impulses 116

CONTENTS 4.4.1 4.4.2 4.4.3

4.5

Statement of the problem Modified equivalent linearization technique Equivalent cubic form (equivalent cubicization) technique 4.4.4 Example: single-degree-of-freedom hysteretic system under a train of impulses 4.4.5 Example: single-degree-of-freedom hysteretic system under a train of general pulses Moment equations technique generalized for the problem of a pulse train driven by a class of renewal processes 4.5.1 Statement of the problem: stochastic equations and equations for moments 4.5.2 Correlation splitting equations

Bibliography

ix 116 120 125 128 136 140 . 140 145 149

Chapter 1 Introduction 1.1

Random vibration and random pulse trains problems

In many problems of vibration of mechanical, aerospace and civil engineering structures it is impossible to characterize the dynamic loading, or the excita­ tion, by a specified time function. Thus the statistical approach is used. The need of modelling the excitation by stochastic processes arises in the analysis of the dynamic behaviour of civil engineering structures, air-planes, vehicles, machines, etc. under the natural hazard or other irregular in nature dynamical loads. The most important problems of this kind are: - vibration of slender civil engineering structures (tall buildings, masts, industrial chimneys, suspension and cable-stayed bridges) under earth­ quakes and shock waves due to blasts, - vibration of slender civil engineering structures due to wind, - vibration of air-plane structural elements due to acoustic excitation from the jet engines and due to atmospheric turbulence, - ship motion on the sea, - behaviour of the ships and off-shore structures under the wave-induced loads, 1

2

CHAPTER

1.

INTRODUCTION

- response of a bridge to moving loads from the highway traffic, - vibration of vehicles (cars, taxying air-planes or railway vehicles) due to road or track surface unevenness, - behaviour of bucket-wheel excavators used in open mining engineering. The interest for random vibration problems, which arose from the needs for safe and rational design of mechanical and civil engineering structures should be mainly attributed to the demands of aerospace industry and to the needs of designing the structures withstanding the earthquakes and of designing off­ shore structures. The needs to predict the vibration response of different struc­ tures to random excitations stimulated the development of probabilistic theory of structural dynamics (methods for random vibration). This resulted in an extensive literature. The first textbooks giving the overview of the stochastic methods in dynamics were those by Crandall and Mark [24], Robson [109], Lin [70], Bolotin [9]. Complete solution of a random vibration problem means the characteriza­ tion, i.e. evaluation of the probability distribution, of the response process of a dynamical system to an excitation described by a random process with known probabilistic structure. However one must often be satisfied with determina­ tion of the response statistics (statistical moments) only. Probabilistic theory of structural dynamics was developed mainly for the excitations given by continuous, most often Gaussian-distributed, random pro­ cesses. Gaussian processes are important for two reasons in probabilistic struc­ tural dynamics. Firstly, many actual random excitations are in fact approx­ imately Gaussian-distributed. Secondly, even though the excitation distribu­ tion deviates from a Gaussian one, the response of a lightly damped linear system (being the output process of a linear filter) is more closely Gaussiandistributed, which follows from the central limit theorem [106]. If the Gaussian response assumption can be made, then the problem is very much simplified; it is sufficient to evaluate the first two response moments, i.e. the mean value and the covariance function, which uniquely determine the response probabil­ ity density. In case of non-linear dynamical systems subjected to Gaussian white noise excitations the Markov approach is used. This enables one to use for example the first-order ordinary differential equations governing the re­ sponse statistical moments or a parabolic partial differential equation (FokkerPlanck-Kolmogorov equation) governing the response probability density. The

1.1. RANDOM VIBRATION AND RANDOM PULSE TRAINS

3

Gaussianity assumption is also of crucial importance in the formulation of a classical equivalent linearization technique, which is so widely used for non­ linear problems, especially in connection with inelastic (hysteretic) structures. An important role is played in structural dynamics by the non-linear prob­ lems. This stems from the fact that the structural response, especially at large amplitudes levels, exhibits the non-linear behaviour. This is so, for exam­ ple, in the case of natural hazard loads, which are highly unpredictable and uncontrollable, and hence may inevitably result in large amplitude response levels, where the response exhibits the non-linear behaviour. Based on a linear model response predictions the more differ from the actual ones, the higher the excitation level is. Hence the non-linear model is certainly more adequate when high level loads or high-barrier crossings are considered. It is obvious that in the safe design those high-level loads should be taken into account and associated by the appropriate probability. In some problems the relaxation of the Gaussian excitation assumption, and more precise characterization of the excitation probability distribution become important. This is so, for example, when the response probability distribution is more adequately characterized, as it is the case when the response higher order statistics are required. These statistics are, as is well known, sensitive to the departures, even small, from the Gaussian behaviour [99, 108]. Finally, in some problems the excitation cannot be regarded, not even approximately, as a Gaussian process. For example in problems of response-depended loads re­ sulting from the fluid-structure or air-structure interaction, e.g. wave-induced forces, the forces acting on a structure are represented as a non-linear function (non-linear transformation) of a Gaussian process [106, 108], and hence are non-Gaussian processes. Another important class of non-Gaussian excitations are the discontinuous in time excitations in form of a train of randomly occurring short-duration loads. Such an excitation is called a random train of pulses (which can be Dirac delta impulses or general pulses with arbitrary pulse shape function) and is a sound model of a loading especially in these cases where the structure is subjected to a train of shocks or impacts. Such an impulsive excitation can occur individually or as an impulsive component of a complex discretecontinuous excitation. Typical problems in which the random pulse train is an adequate model of the loading process are: - dynamic response of a highway bridge to moving loads [133]

4

CHAPTER

1.

INTRODUCTION

- vibration under earthquake excitation regarded as a train of shocks (im­ pulsive changes of ground motion acceleration) [67, 18] - loading due to randomly occurring wind gusts associated with eddies [18, 78, 93], - loading due to atmospheric turbulence [81], - random "down-wash" exciting intermittently the tail of an airplane [66], - wave slamming loads acting on a ship hull structure or on some structural elements, i.e. horizontal members of the trussed towers, of off-shore structures [76], - dynamic behaviour of road vehicles travelling over a rough ground or of railway vehicles travelling over an imperfect track [73, 98], - impact loads acting on bucket-wheel excavators used in open mining engineering [26]. Often it appears that the filtered Poisson process applied as a model of the dynamic loading results in the linear system response probability density (being that of a filtered Poisson process as well) suiting much better to the experimental probability density histogram than the Gaussian density [81, 93]. In the case of dynamic behaviour of vehicles the shocks and impacts are pro­ duced by sudden jumps and bumps in the rough road surface. In the case of the bucket-wheel excavators the impacts on a structure are produced when the excavator bucket encounters the stony inclusions. It turns out that in the latter case the mean rate of the impacts is comparable with the fundamental natural frequency of the excavator boom, and consequently the resulting dy­ namic effects are severe. At present no adequate computational techniques are known for such dynamical problems [26]. The random impulsive excitation processes are described in terms of stochas­ tic point processes. Solution of arising dynamical problems requires on one hand the application of pertinent methods of point processes and, on the other hand, a suitable development of probabilistic theory of structural dynamics. It should be noted that the methods pertinent to dynamical systems, espe­ cially non-linear ones, subjected to random pulse trains have been much less developed than the methods for the continuous stochastic excitations.

1.2. GENERAL

1.2

METHODS FOR RANDOM VIBRATION

5

Most important general methods for ran­ dom vibration

Stochastic methods in dynamics aim at the evaluation of the probability den­ sity or different statistics of the response of a dynamical system to an excitation given by a stochastic process. One is usually interested in statistical response moments or the probability density function and also in the reliability char­ acteristics such as characteristics of the first passage time, statistics of the crossing process, or statistics of the local maxima. The pertinent methods are usually based on the specific assumptions concerning the excitation process. Presented herein an overview of the methods for random vibration, which is far from being complete, aims at presenting the possibilities and the range of versatility of the general methods in relation to their applicability to random pulses problems. The most general and versatile methods for linear systems, applicable to arbitrarily distributed random excitations are the correlation methods. As is well known these methods allow to evaluate the correlation function (or statistical moments) of the response process in terms of the respective corre­ lation function (or moments) of the excitation process. Statistical moments of arbitrary order are evaluated based on representation of the response as a Duhamel integral, i.e. the convolution of the excitation with the impulse response (or Green) function of the system, and via subsequent averaging of the pertinent multifold integrals. Consequently the response statistical mo­ ments are expressed in terms of the respective order moments of the excitation process. Hence in case of a linear system the response statistical moments are obtainable for arbitrarily distributed excitation process if only its correla­ tion structure is known. Exact solutions exist also for the response moments of linear systems to random pulse trains. However performing the averaging of pertinent integrals leads to the expressions for moments which are differ­ ent from those in the case of continuous stochastic excitations. For linear systems subjected to stationary random excitations a spectral method is of a widespread use. Simple relationships between the response and excitation spectral densities allow to evaluate the mean-square stationary response of the system. For the non-stationary excitations the generalized spectral method exists, where the bispectra, trispectra, etc. are introduced. A complete information about the response process is contained, of course,

6

CHAPTER

1.

INTRODUCTION

in its probability distribution. However exact determination of the response probability density, even that of a linear vibratory system, is only feasible when the random excitation is Gaussian distributed. Then, as is well known, the response process is also Gaussian distributed and its probability distribution is uniquely determined by the first two statistical moments, i.e. the mean value and the covariance (or variance) function. Otherwise, i.e. when the response probability distribution deviates from a Gaussian one, the probability density function can be at best approximately evaluated. Then it can be estimated by moments, by making use of the asymptotic Gram-Charlier expansion (or Edgeworth series) [21], in which the coefficients of expansion of the density function in terms of the Hermite polynomials are expressed by response mo­ ments. Another possibility lies in the application of the maximum entropy principle [119, 131]. Then the probability density function involves a num­ ber of free parameters which are determined from the maximization condition of the informational (Shannon's) entropy functional of a system considered. When the response process is Gaussian distributed the reliability character­ istics are most easily obtainable. For example the joint probability density of the response process and its first derivatives allow to exactly evaluate the average crossings rate and the distribution of local maxima [22]. For non-linear systems no explicit solutions for the response to a stochas­ tic excitation are known. The best developed are the methods for the systems driven by the Gaussian white noise process. These methods are relevant to the problems where the excitation is a white noise itself or a white noise filtered by an auxiliary linear system (filter), the latter being the case of random exci­ tations with rational spectral density. Then the state vector of the dynamical system is the multidimensional (multivariate) diffusive Markov process and the variety of Markov approach methods may be used. An extensive overview of such methods was given for example in the reference [103]. Joint probability density function of the state vector is then governed by a partial differential equation of a parabolic type - Fokker-Planck-Kolmogorov equation. The exact stationary solution of this equation, for an arbitrary non-linear function, exists however only for a univariate diffusive process [127]. For a vibratory system, described by at least bivariate diffusive process, the exact stationary solution is known for some specific forms of a non-linear function only, e.g. for a cubic function of the displacement (see e.g. [70]). Recently further progress has been made in this respect, by developing the methods of exact solution for FokkerPlanck-Kolmogorov equation belonging to the class of generalized stationary

1.2. GENERAL

METHODS FOR RANDOM

VIBRATION

7

potential [13, 72]. Otherwise, the approximate methods can be used, e.g. the generalized Galerkin method [144]. The numerical solution of the Fokker-Planck-Kolmogorov equation may also be performed using the so-called path integration method known also as a cell-to-cell mapping. This method consists in linearization of the equations of motion in small intervals, resulting in a Gaussian behaviour in these intervals. Consequently the evaluated conditional mean values and conditional covariances allow to determine the transitional probability density function of the Markov vector process. The method, which originates from physics (see e.g. [143]), has different applications which were found recently (e.g.[83, 129, 130]). Markov methods also allow to evaluate the characteristics of the first pas­ sage time (see e.g. [103, 105]). The probability of the first passage in a given time interval, which is the distribution function of the first passage time, is governed by the backward Kolmogorov equation and the same equation is sat­ isfied by the reliability function [151]. Either the solution to this equation can be attempted, e.g. with the help of a Petrov-Galerkin method [6], or based on this equation the differential equations governing the moments of the first passage time can be derived [103, 105]. Alternatively, a moment equations approach may be used. Obviously the response moments for a non-linear system can be evaluated neither by averag­ ing of explicit expressions (as it is the case for linear systems) nor by evaluating the integrals with respect to the probability density function (which is a priori unknown). If the dynamical system is excited by a Gaussian white noise or filtered white noise process then the equation of motion can be recast into stochastic Ito's differential equations, and the ordinary differential equations governing the response moments can be derived. The latter can be done either by suitable integration of the Fokker-Planck-Kolmogorov equation, or with the help of Ito's differential rule (e.g. [31]), which is the formula for evaluating the differential of a function of a stochastic process. The obtained equations for moments are not directly solvable, since they do not form a closed set, i.e. they involve additional unknowns - expectations of non-linear functions of a response process (of state variables). These functions have the form of non-linearities present in the equations of motion, hence in the case of poly­ nomial non-linearities (or power functions) the higher order moments appear, which are unknown, i.e. redundant for the considered set of equations. In the case of other non-linearities the appearing expectations of non-linear functions (which can be non-analytical) cannot be expressed in closed form in terms of

8

CHAPTER

1.

INTRODUCTION

moments. In order for the set of moment equations to be solvable first it must be closed, i.e. the unknown expectations of the non-linear functions of the response process must be evaluated, or rather expressed in terms of the same moments which are on the left-hand side of the equations. Such a closure of moment equations is performed with the help of closure approximations, which are formulated based on different hypotheses. For the systems with polyno­ mial non-linearity the most often used are the closure due to neglecting the higher-order central moments [96], the cumulant neglect closure [150] or the quasi-moment neglect closure [12]. The use of closure approximations is often very effective and leads to reliable results, if the closure is effectuated at the level by a few orders higher than the highest moment required. For example, if first and second order moments are required, the set of moment equations is usually truncated at fourth or sixth order moments [150]. However the ef­ fectiveness of the closure approximations may depend on the kind of problem and sometimes they may even yield the false results [128]. In general, the closure of moment equations requires evaluation of the ex­ pectation of certain (analytical or non-analytical) function of response process (of state variables), which is only feasible via inegration with respect to the response probability density function. Exact form of this density function is unknown, hence it can only be assumed in an approximate, tentative form. The assumed, approximate form of the density function should, first of all, render its non-Gaussianity. Most often a tentative, non-Gaussian density function is assumed in form of a Gram-Charlier expansion [2, 4, 5, 23, 25]. Such a closure technique is called a non-Gaussian closure; in a special case when the Gaussian density function is assumed it is a Gaussian closure. The quality of approxi­ mation depends, of course, on how close the approximate density function to the exact one is. Therefore in some problems the special modifications, moti­ vated by a physical sense of the problem, of the tentative density function are made (e.g. [80]). Application of closure techniques to multivariate problems becomes cumbersome, due to the need of evaluation of multifold integrals with respect to multivariate probability density functions. A special attention should be paid to the stochastic averaging method [127], the essence of which is the substitution of a given stochastic differential equation by a stochastic Ito's differential equation. Accordingly, the response process is approximately treated as a diffusive Markov process. From the phys­ ical point of view such an approach, i.e. approximating of an actual response process by a diffusive Markov process is justified, if the time intervals for which

1.2. GENERAL METHODS FOR RANDOM VIBRATION

9

the values of the process are practically independent (or time intervals lengths are larger than the correlation time) are at the same time sufficiently small compared to the relaxation time of the system, for the increments of the pro­ cess during these intervals to be small. Small increments of the process during small time intervals is just the property of the diffusive Markov process. This method is justified in Khasminski theorem [58], which deals with the process governed by the stochastic differential equation driven by certain function of this process and of the excitation process. This theorem states that the process governed by such a differential equation tends to a diffusive Markov process, i.e. to the solution of certain Ito's stochastic equation, if the following condi­ tions are fulfilled: the function on the right-hand side of the equation must be bounded, continuous and differentiable with respect to all its arguments and it must satisfy the so-called strong mixing condition [117, 118]. The method of stochastic averaging may be applied to those dynamical problems, where the excitation is a wide-band stationary process and the system is lightly damped. Then the response is a narrow-band process and can be represented in the form of a quasi harmonic process with slowly varying amplitude and phase. Stochastic averaging method is applied to non-linear equations governing such a response process. The technique of performing the averaging is similar to the one in a Bogoliubov-Mitropolski method known in deterministic theory of non-linear vibration [8]. An overview of applications of the stochastic averag­ ing method is given in the references [106, 107]. It should be noted that the requirement of small (infinitesimal) increments of the response process during small (infinitesimal) time intervals, hence of small velocities, is not fulfilled by the response to impulsive excitations (Dirac delta impulses), where finite jump changes in the velocity occur during infinitesimal time intervals (exactly - at the occurrence times). Hence the impulsive processes are excluded out of the range of applicability of the stochastic averaging method. A method, or a technique, which has a wide variety of applications to dif­ ferent non-linear problems, is the equivalent (or statistical) linearization tech­ nique. Its idea consists in considering, instead of the original system governed by a non-linear differential equation, an equivalent system governed by a linear differential equation. Such an approach was originally used in deterministic problems governed by non-linear differential equations. Next, this technique was adapted by Booton [10] and Kazakov [57] to stochastic problems of con­ trol theory, and by Caughey [16] to non-linear stochastic problems of structural dynamics. An extensive overview of various application of the equivalent lin-

10

CHAPTER

1.

INTRODUCTION

earization technique is given e.g. in the references [104, 108, 124]. An essential feature of this technique is that the coefficients of the equivalent linear equation are evaluated from the condition of minimization of the mean-square difference of equations and are expressed in terms of the moments and of the expecta­ tions of non-linear functions of the response process. If the non-linearities are of polynomial (power) type then, of course, the equivalent coefficients are ex­ pressed in terms of moments only. An important question which arises is with respect to what measure (probability density) these expectations, including moments, should be evaluated, because the exact probability density function of the response of the original system is unknown. It can be proved that if the exact density were assumed, the first and second order response moments evaluated from the equivalent linear system would be exact (of course, if the exact density were known, the linearization would be unnecessary). In prac­ tice, one would like the moments obtained from the equivalent linear system to be the most accurate, and therefore the response probability density func­ tion should be assumed in tentative form as close to the exact one as possible. Very often, however, this probability density function can be assumed, at best, as the probability density of the response of a linearized system. This is es­ pecially easy if the Gaussian excitation is considered, since the response of a linearized system is a Gaussian distributed process as well, and hence the tentative density function is assumed as the Gaussian one. The equations gov­ erning the first- and second-order moments derived from the equivalent linear system become non-linear, and can only be solved numerically; the equivalent coefficients are updated in each step of numerical integration. Unfortunately, if the tentative Gaussian density is assumed, the departure of the response process from Gaussianity, which is an important property of behaviour of the response of a non-linear system cannot be investigated. Despite this inherent shortcoming, the equivalent linearization technique yields in many cases the reliable estimates of the response mean value and variance, and therefore it is often applied, especially to multi-degree-of-freedom systems if the first- and second-order moments are required [108, 124]. As a generalization of the idea of the equivalent linearization the technique of an equivalent non-linear equation (or system) was proposed [17], in which in­ stead of an original non-linear system, another non-linear system is considered, for which the exact solution is known. Equivalent coefficients of a non-linear equation are evaluated from the mean-square optimization condition. This technique allows to investigate the departure of response probability distri-

1.3. METHODS FOR RANDOM PULSE TRAINS

11

bution from the Gaussian one. Its range of applicability is, however, rather restricted, mainly due to the fact that the class of non-linear systems for which the exact soluton exists, is narrow. A particular version of the technique of the equivalent non-linear equation is the technique of equivalent polynomial expansion (or polynomial form). In this technique an equivalent non-linear system, is considered, in which the original, often non-analytical, non-linear functions of the state variables are substituted by polynomials in theses variables, for example by a cubic form [84, 85]. Coefficients of the equivalent polynomial form are evaluated from the mean-square optimization condition, i.e. from the condition of minimization, in the mean-square sense, of the equation difference and are expressed in terms of expectations of non-linear functions of the state variables. Equivalent equa­ tions are solved with the help of known methods, for example with the help of the moment equations technique combined with non-Gaussian closure approx­ imation [84, 85]. Of course in order to perform the necessary expectations the unknown response probability density must be assumed in a tentative form. It can be proved that if an exact response probability density were assumed, an equivalent non-linear system with n-th order polynomial, would yield the exact moment up to and including n+1 order [84, 85]. This technique allows in many cases to obtain much more exact results than the equivalent linearization technique. Many of the classical stochastic methods of structural dynamics have their analogues, or generalizations, for random impulsive excitations. Most of the results can be obtained for the trains of independent impulses, described by a Poisson process. Then, for example, the methods of non-diffusive Markov processes can be used. The pertinent methods are overviewed in section 1.3.

1.3

Methods for random pulse trains

Random impulsive excitations, or random pulse trains, are adequately char­ acterized in terms of stochastic point processes. The name "point process" is due to Wold [149], and can be justified by the fact that these processes describe events, which can be interpreted as the points on the line. Theory of point processes, whose formulation should be attributed mainly to Moyal [82], is discussed, for example, in [64, 65, 116, 125]. Methods of determining the response of dynamical systems to random pulse trains are based on the

12

CHAPTER

1.

INTRODUCTION

methods of these processes. However the pertinent methods of stochastic point processes are rather disseminated in the literature and a full collection of such methods can hardly be found in a book. Therefore in the second chapter a brief account of the pertinent methods of stochastic point processes is given. The problem of the cumulative response of a linear dynamical system to a random train of impulses, or general pulses, can be converted, by virtue of a linear superposition principle, to the problem of a filtered process driven by the underlying stochastic point process. Each general pulse of the filtered process is the response of the system to a pulse of the excitation. The theory of filtered point processes provides some fundamental exact results, concerning mainly the moments, see e.g. [92, 95, 125]. For example for one of the best known processes of this kind, i.e. for a filtered Poisson process, the classical results are those for the mean value, the variance and the cumulants [95]. Solution of the detailed problems requires, however, suitable adaptations, extensions and generalizations of these methods. Also special techniques must be de­ veloped for performing the averaging of suitably constructed, often multifold, stochastic integrals with respect to a point (random counting) process. The existing methods for linear dynamical systems are the exact methods for the response moments and the approximate methods for estimating other response characteristics by the moments. First applications of the filtered point processes, e.g. of the filtered Poisson process, were concerned with problems of physics and electronics (shot noise effect). As the earliest and the most important contributions to the problem of mechanical vibration under random impulses one should certainly acknowledge those due to Lin [67, 68, 69, 70], Roberts [97, 98] and Srinivasan [126]. To the author's knowledge in Polish literature the first contribution to such a problem is the one due to Miadowicz and Tylikowski [79]. For a linear dynamical system, the exact correlation methods exist, which allow to evaluate the response moments (correlation functions) in terms of the product density functions describing the underlying point process. The corre­ lation methods are valid for the pulse train driven by any regular stochastic point process. These methods allow to solve, at the second order moments level, a variety of detailed problems, both for the Dirac delta impulses and for the general pulses, characterized by certain pulse shape function. For ex­ ample, many detailed results concerning the mean value and the variance of the response to a random train of general pulses, together with the technique of evaluating the pertinent integrals, have been given in reference [56]. There

1.3. METHODS FOR RANDOM PULSE TRAINS

13

have appeared a number of papers, in which the methods of filtered point processes have been applied to evaluate the mean value and the variance of the response of a linear system to impulsive excitations, often described by complicated models accounting for different possible features of the actual im­ pulsive excitations. The general analytical technique is in principle the same. The main problem being averaging of multifold stochastic integrals having a complicated structure. Thus, for example the mean value and the variance can be evaluated for the response to the train of impulses with a stochastic spatial shape [36, 37], or to the train of general pulses with random duration [48]. An interesting technique of performing the averaging of stochastic integrals for a non-Poissonian process constructed from a Poisson counting process is devised in reference [40]. The results have been obtained for different, practi­ cally oriented problems, such as the response of an elastic plate to a correlated train of impulses driven by a general counting process [50] or the response of shallow cylindrical [39, 51] and spherical shells [38] under Poisson-distributed impulses. For shallow shells an important problem has been investigated, of the contribution of modal responses and their cross-covariances in the variance of the shell response (radial displacement). An exact, closed-form solution has also been given for the mean value and the variance of the beam response (deflection) to a Poisson-distributed train of moving concentrated forces, with equal, constant speeds [49]. As is well known a Poisson process model is justified by the experimental evidence for rare, and hence independent, events only. In many other problems it is necessary to use other than Poissonian models for random train of events. One of the possibilities lies in assuming a general stochastic point process, which allows to take into account the correlation between events. However in different applications some more specialized models are needed. A class of point processes, which is especially useful are the renewal pro­ cesses, adequate in those situations where the interarrival times of events are independent random variables (as it is the case for a Poisson process), but have other than negative-exponential probability distribution. Within the frame­ work of renewal processes a realistic, adequate for an actual problem, form of the interarrival times probability density, can be assumed. Perhaps the most typical example of the applications of renewal point processes is the problem of a random stream of vehicles in a highway traffic [1, 34], which constitute the train of moving loads on the bridge structure. The theory of filtered renewal point processes provides some general formulae, which allow to evaluate the

14

CHAPTER

1.

INTRODUCTION

moments of the response of linear systems to random pulse trains driven by an arbitrary, i.e. with arbitrarily distributed interarrival times, renewal process. A classical result is the one given by Takacs [132], i.e. the integral equation governing the characteristic function of a univariate, filtered process driven by an ordinary renewal process (ordinary means the one in which the initial event occurs at the zero time - at the initial instant of observation and hence the waiting time to the first event has the same probability distribution as all other interarrival times). Prom this integral equation the recursive expressions for the statistical moments are derived. Making use of the known form of the product densities of the renewal process Srinivasan [125] gave the expressions for the mean value and the mean square of the filtered delayed renewal pro­ cess. The Takacs integral equation approach [132] has been generalized by Iwankiewicz [42] to a delayed renewal process, i.e. for the multivariate filtered process driven by a delayed renewal process the integral equation has been derived which governs its joint characteristic function. Based on this equation the recursive expressions for the joint statistical moments have been obtained. Explicit expressions for the mean values and the cross-correlation functions of the component processes have also been given for the general pulses. In a more complicated situation a random pulse train may be modelled as a Markov renewal (semi-Markov) process. The existing theory of filtered Markov renewal processes provides the set of integral equations governing the conditional characteristic functions (conditional upon the initial state of the semi-Markov process), from which the recurrent expressions for the conditional moments are obtained [113]. The latter results concern also the ordinary pro­ cesses. The Markov renewal process models have found applications in analysis of a noise due to a highway traffic [77] and in analysis of a static behaviour of a bridge under random streams of multilane highway traffic [29]. The ap­ proach to impulsive Markov renewal processes has also been extended [42] to a delayed process, i.e. the one in which the initial event does not occur at zero instant, but the waiting time to the initial event depends on the corresponding state of the semi-Markov process. For the multivariate filtered process driven by a delayed Markov renewal process the set of integral equations governing the conditional characteristic functions is obtained. The recursive expressions for the conditional moments are obtained therefrom. However a complete characterization of a response process, for example by determining its probability density function, appears to be a much more dif­ ficult task to do. The most comprehensive characterization can be given for

1.3. METHODS FOR RANDOM PULSE TRAINS

15

the process of response of a linear dynamical system to a Poisson distributed pulse train, which is just the filtered Poisson process. In that case the explicit expressions for the cumulants [95] and for the characteristic function (e.g. [92]) are known, also for the multivariate process [99]. However the evaluation of the probability density function as the inverse Fourier transform of the character­ istic function, is unfortunately generally not feasible, because of poor effective­ ness of the direct numerical techniques (such as the Fast Fourier Transform) [54, 93]. In a special case of the filtered pulse shape function (exponential), the probability density of the filtered Poisson process can be obtained exactly from the characteristic function, via exact evaluation of the inverse Fourier transform integral (see e.g.[88]), but this is relevant to the response of a first order, unoscillatory, system (filter) only. In another paper [32] the integral equation governing the distribution function of a filtered Poisson process has been formulated, next based on this equation the probability distribution has been evaluated for some particular forms of the filtered pulse shape. Since the exact solutions for the probability density function are difficult to obtain, the approximate solutions are often seeked. One of the possibilities is to use the Gram-Charlier expansion. This expansion is, as is well known, equivalent to the inversion of the characteristic function by expanding the nat­ ural logarithm of this function (the cumulant generating function) in McLaurin series [74] and it allows to estimate the probability density via the cumulants. Such an approach was used in problems of the response to Dirac delta impulses [98, 99, 137]. The use of the generalized, bivariate Gram-Charlier expansions allows to determine approximately the joint probability distribution of the re­ sponse process and its first derivative, and hence to evaluate the mean crossings rate, as it has been done in the case of a linear oscillator subject to Poisson distributed impulses [99]. The characteristic function can also be inverted with the help of expanding its natural algorithm in Taylor series about the saddle point [100]. The advantage of such an approach lies in the fact that it allows to give better estimates to the probability density tails than the Gram-Charlier expansion. However this approach was used in the reference [100] to the Dirac delta impulses only. It is known that the process of response to a random train of impulses is characterized by a non-Gaussian probability distribution. Essential informa­ tion about such a process is provided by the measures of the departure from a Gaussian behaviour (or from Gaussianity) such as e.g. the skewness (or asymmetry) and the excess coefficients. These measures can be used in order

16

CHAPTER 1.

INTRODUCTION

to investigate the departure of the examined probability distribution from the Gaussian behaviour [60, 91]. The determination of the probability density was also attempted in the case of Poisson-distributed general pulses, assuming a rectangular pulse shape function [41, 52]. First the skewness and excess coefficients were evaluated and their dependence on the pulse duration was investigated. Next, based on the truncated Gram-Charlier expansion the probability density function of the re­ sponse process was approximately evaluated [41]. Likewise, the joint, bivariate probability density function of the response process and its first derivative as well as the mean upcrossings rate [52] were evaluated. The approximate results obtained reveal the characteristic features of the behaviour of the probability distribution, in particular its tendency for the departure from Gaussianity. A special attention should be given to the references dealing with the re­ sponse of bridge structures to random trains of moving loads due to highway traffic. For example in the references [133] and [135] the numerical technique is developed to evaluate the univariate probability density function of the response process and the joint, bivariate probability density function of the response process and its first derivative. The life expectancy of the bridge structure under a random train of moving loads was also analysed [134]. Ba­ sically, this technique is based on the fact that the cumulative response at a given time instant is actually due to a certain number of moving forces having arrived in a limited time interval. This technique is suited for random trains of impulses driven by the Poisson process and the renewal process, with gamma distributed interarrival times. The methods for renewal impulsive processes, suitable for arbitrarily dis­ tributed interarrival times can be, in a natural way, applied to the problems of response of bridges to highway traffic loads. The expressions for the mean value and the cross-correlations of the modal responses were given in the case of a random train of moving forces travelling across the bridge with unequal, random velocities [42]. In the case of a bidimensional (plate) bridge struc­ ture under a multilane highway traffic the application of a Markov renewal (semi-Markov) process was discussed in detail [42]. If the process at the input of the dynamical system is a Poisson-distributed train of Dirac delta impulses then the state vector of the dynamical system (composed of the generalized coordinates and velocities) is a multivariate nondiffusive Markov process [30, 31,116]. In that case the joint probability density of the state variables satisfies a partial integro-differential equation (see e.g.

1.3. METHODS FOR RANDOM PULSE TRAINS

17

[94]), which is the Kolmogorov-Feller equation for the non-diffusive Markov process. Upon performing the Fourier transform of this equation, the first order partial differential equation is obtained, which governs the characteristic function of the response process. However the closed-form solution for the density function can only be obtained in a univariate case, i.e. the first order filter, [94, 138, 139]. The methods of the non-diffusive Markov processes allow to derive the ordi­ nary differential equations governing the response statistical moments. These equations can be obtained either by differentiating the equation for the char­ acteristic function [94] or by making use of the differential rule - an analogue of the Ito's differential rule (see e.g. [30, 31, 116]). The moment equations approach allows to deal with the problems with parametric excitation, such as the problem of stochastic dynamic stability (in the sense of moment stability) of a bar (beam) subjected to a compressive force in form of a Poisson train of impulses [94]. An interesting example of the problem leading to a parametric excitation is given in reference [137]. The problem considered is the dynamic behaviour of an oscillator subjected to a train of impacts due to collisions with concentrated masses having random velocities. Hence, in accordance with the momentum conservation principle, the magnitudes of the impulses depend on the state (velocity) of the system, which leads to the appearance of the mul­ tiplicative (parametric) excitation term. Stochastic dynamic stability under random impulsive excitation is also considered in reference [148]. Much less attention has been given in the literature to non-linear systems subjected to random pulse trains. Dynamic behaviour of a non-linear oscillator under a Poisson-distributed train of impulses was considered by Roberts [99], who based the analysis on the generalized Fokker-Planck-Kolmogorov equa­ tion containing higher order derivatives and valid for a non-diffusive Markov process. Next the solution to this equation, for the case of a train of impulses with high mean arrival rate, was obtained with the help of a perturbation technique. Tylikowski and Marowski [139] considering a Duffing oscillator un­ der a Poisson impulsive noise excitation applied the technique of equivalent linearization. Since in the considered problem the non-linearities were of cu­ bic form, the required expectations were just the higher order moments, which were substituted by the moments of a linearized system. In that case if the lin­ earization is performed in the steady-state, the moments are constant and the equivalent system is time-invariant, hence its response moments can be eval­ uated exactly from the known formulae as the moments of a filtered Poisson

18

CHAPTER

1.

INTRODUCTION

process, without assuming any tentative form of the joint probability density function. However the range of applicability of such an approach is confined to systems with polynomial non-linearities only. Most of the existing probabilistic methods of structural dynamics dealt with Gaussian (usually white noise) process excitations and are for obvious reasons inadequate for the problems of random pulse trains. Only recently the analytical techniques were essentially developed for non-linear dynamical systems under random trains of impulses. Some of these methods are covered herein. These are: moment equations technique together with closure approx­ imations for non-linear systems with polynomial non-linearities under Poisson impulsive process excitation, equivalent linearization and equivalent cubicization (or equivalent cubic form technique) for non-linear inelastic (hysteretic) systems subjected to Poisson-distributed random pulse trains and moment equations technique for non-linear systems under renewal impulsive process excitation. For the systems with polynomial non-linearity, subjected to Poisson dis­ tributed pulse trains, the moment equations technique together with closure approximations has been developed [45, 47]. The approach covers both Dirac delta impulses and general pulses, the latter being regarded as the responses of an auxiliary linear filter to individual impulses. Based on the fact that the system state vector is a non-diffusive (Poisson-driven) Markov process the differential equations governing the response statistical moments (joint mo­ ments of the state vector) are obtained from an integro-differential analogue of the Ito's differential rule. First, the equations for moments obtained for the system with cubic non-linearity were truncated with the help of an ordi­ nary cumulant-neglect closure technique [45]. Comparison with the results of Monte-Carlo simulations revealed that if the average arrival rate of impulses was not too small, the truncation of moment equations at the fourth order moments and using the closure approximations resulting from neglecting the fifth and sixth order cumulants gave very good estimates of the mean value and variance of the response process. For the problems of Poissonian impulses a special closure technique, a modified cumulant-neglect closure technique has also been devised [47]. The closure approximations are based on the assumed tentative discrete-continuous form of the joint probability density of the state variables. The discrete part of the density function is the conditional density, given that no Poisson impulse (Poisson event) has occurred yet and is just a Dirac delta spike at zero. The continuous part of the density function is the

1.3. METHODS FOR RANDOM PULSE TRAINS

19

conditional density, given that at least one impulse has arrived. The modified cumulant-neglect closure technique has been applied to a problem of a Duffing oscillator under a Poisson-distributed pulse train (filtered impulses). Equa­ tions for moments have been truncated at the fourth order moments level. The modified closure technique yields more accurate results than the usual cumulant-neglect closure technique and has a wider range of applicability, i.e. it allows to solve the problem for lower values of the Poisson process intensity. A moment equations technique is also used for a linear and for a Duffing oscil­ lator under a Poisson impulse excitation in reference [35], where the equations for moments are obtained from an analogue of Ito's differential rule involving a series of partial derivatives instead of an integral term. Recently further progress has been done in analysing the dynamic response of non-linear oscillators to Poisson impulse process excitations. Based on pre­ viously obtained exact solutions for the case of the Gaussian impulsive noise [13, 72] Cai and Lin have devised a perturbation technique [14, 15] to solve the generalized Fokker-Planck-Kolmogorov equation governing the probabil­ ity density function for the response to non-Gaussian, Poisson impulsive noise excitation. This technique was used to evaluate the approximate stationary probability density function for the response of a non-linear oscillator sub­ jected to both additive (external) and multiplicative (parametric) Poisson im­ pulsive noise [14, 15]. In reference [61] the integro-differential equation, a backward form of the Kolomogorov-Feller equation (an analogue of the back­ ward Kolmogorov equation) is solved with the help of a Petrov-Galerkin ap­ proach. Thereby the reliability function and the probability density of the first passage time are evaluated for the response of a non-linear oscillator to Poisson-distributed train of impulses. A path integration (cell-to-cell mapping) technique has also been used to solve the integro-differential equation (an ana­ logue of the forward Fokker-Planck-Kolmogorov equation) [62]. This allowed to obtain the response probability density and the reliability functions for the response of a non-linear oscillator to Poisson impulse process excitation. An important class of non-linear systems, for which the techniques appro­ priate for Poisson pulse processes have been developed, are the systems with non-linearity given by a non-analytical function of state variables [43, 44]. These are the inelastic systems, in which the hysteresis of the restoring force is taken into account. Two approximate analytical techniques have been devised for these systems: equivalent linearization and equivalent "cubicization", tech­ niques. In equivalent linearization technique the original non-linear function

20

CHAPTER

1.

INTRODUCTION

of the state variables is substituted by a linear form of these variables. In the other technique the equivalent system has a cubic non-linearity, i.e. the original non-linear function of the state variables is substituted by a cubic (in general a polynomial) form of these variables. The coefficients of the equivalent forms are evaluated from the minimization, in the mean-square sense, of the equation difference (i.e. the difference between the original and the equivalent equations) and are updated in each step of the numerical integration of equa­ tions for moments. Calculating the expectations of non-linear non-analytical functions of state variables, which appear in the equations for equivalent coef­ ficients, requires the use of a joint probability density of these variables. These density functions can only be assumed in a tentative form, usually as GramCharlier expansions. The equations for moments up to fourth order have been derived for equivalent systems. Comparison of analytically obtained (com­ puted) response mean values and variances with simulated ones revealed good accuracy of the approximate analytical techniques developed, especially of the technique of equivalent cubicization. Technique of moment equations has also been extended to a class of re­ newal impulsive processes, with gamma-distributed, with k — 2, interarrival times [46]. Of crucial importance is, in this case, the construction of the re­ newal impulsive process with the help of a Poisson counting process. To do that the non-Poissonian impulsive process given in [40] is used, in which the magnitude of an impulse depends on the number of the past Poisson counts and is assumed as a random telegraph process, hence the impulses occurrence times are Poisson-distributed, but their magnitudes are alternatively: positive and negative. Combining such a train of impulses with a Poisson one yields a renewal impulsive process. Equations for moments are obtained from the generalized Ito's differential rule, however additional unknown expectations appear in these equations. These are expectations af a random telegraph pro­ cess multiplied by different powers of the state variables (joint moments of the random telegraph process and of the state variables). Differential equations governing these expectations have also been derived in form of the correlation splitting formulae. The interest in problems of response of dynamical systems to random im­ pulses has recently increased. Especially the techniques for non-linear dy­ namical systems are still being developed and some new contributions in this domain can be expected to appear in the near future.

Chapter 2 Stochastic point processes — basic ideas and methods and other mathematical preliminaries 2.1

Characterization of stochastic point pro­ cesses

2.1.1

General stochastic point process, counting pro­ cess

Theory of stochastic point processes is useful in studying the phenomena which are discontinuous in time and in space, and which are a set of objects or events, the co-ordinates of which are randomly chosen from a given state space. Methods of description (characterization) of these processes are connected with characterization of the distribution of random points on the line. There exists a number of books on the theory and applications of point processes; the most representative of which being perhaps [64, 65, 116, 125]. An example of the problem which is adequately characterized by the stochastic point process is the random train of events, i.e. the train of events which occur at random times. Such a characterization is especially useful in studying the cumulative response phenomena, where the response is due to a random series of pulses. 21

22

CHAPTER 2. STOCHASTIC

POINT

PROCESSES

One of the earliest examples of the phenomena of this kind is the shot noise, being the cumulative effect of the passage of electrons from cathode to anode. Stochastic point processes find applications in many different fields such as statistical physics, astrophysics, astronomy, biology, communication theory, management science and last but not least, mechanics. Typical problems in which the point process models are used are e.g. kinetic theory of fluids, linear and non-linear response problems, theory of cosmic ray showers, stochastic problem of population growth, theory of queues, renewal theory, reliability theory. There exist several definitions and methods of characterizing of the point process, which arose from studying different particular problems and are due to different authors. It is convenient and purposeful to characterize the point process through a counting measure (see e.g. [125]). Consider a set of events, each of which is described by the point on the real line X. A stochastic set of events (population) is characterized by assigning to the subintervals A of the real line X the numbers N(A) of events, each of whose coordinate U belongs to A (U £ A). If the sequence of points {U} on the real line is assigned to the set of events, then the counting measure is introduced as N(A) — number of elements of sequence {U : U € A}. This measure is non-negative, integral-valued, countably additive and a - fi­ nite. It is also assumed to be a Borel measure, from which it follows that the expected number of events in every finite interval is finite, and which implies that in any finite interval the number of events is, with probability 1, finite. Let us denote N(t) - random variable specifying the number of events in an interval (0, t], N(dt) - random variable specifying the number of events in an infinitesimal interval (t, t + dt], i.e. N(dt) = N(t, t + dt)N(t). Thus, N(t) is a random counting measure, or a random counting process. Alternatively, N(t) can be specified as the number of events in the interval [0,i) [116, 120]. In what follows we confine our attention to regular or orderly point pro­ cesses. The point process is regular or orderly, if the probability governing the

2.1. CHARACTERIZATION OF STOCHASTIC POINT PROCESSES

23

counting measure satisfies the following condition:

J2 Pr{N(dt) = k} = o(dt), o(dt),

(2.1)

k>2

which means that in the infinitesimal time interval there can only occur, with non-zero probability, one event or no event. Let us choose from the interval (0, t] the disjoint infinitesimal time intervals (ti,U +dU],i = 1,2,... , n. Product density functions are defined as follows [125]: n

ffn(ti,t , ...,t n)dt 1---dt n n n (ti,t22,...,t n)dt 1---dt

Y[N{dU) ==EE Yl N (dU)

(2.2)

or, equivalently, if the point process is regular, as fn(h,t2,,...,..., Qdti Qdh ■■■dtn = Pl{N(dU) Pi{N(dU) -= 1}, 1}, 1i=l,2,...,n, = 1,2, . . . , n , t!^t ti =/=t ^--^t ^ . ••• ^t . 2 2n n

. . ^>

The product density function of degree n /„ (tu ..., tn) represents the prob­ ability that an event occurs in each of the disjoint intervals (U,U + dti\, irre­ spective of other events in the interval (0, £]. In particular, Pv{N(dt) = 1} = h(t)dt, Pr{N(dt) h(t)dt, (2.4) where fi(t) is the product density of degree one. The regularity assumption (2.1) implies that

and

Px{N(dt) = 0} = 1 - h(t)dt + o(dt), Pi{N(dt)

(2.5)

E[N{dt)] f1(t)dt o(dt) E \N{dt)) = h (t)dt + o(dt)

(2.6)

E{{N(dt)} (t) + o(dt), o(dt). E[{N(dt)}n] = f1l(t)

(2.7)

for arbitrary n. Product density of degree one fi(t) represents the mean rate of occurrence of events (mean arrival rate). It should be noted that fi(t) is not a probability density; its integration over the whole time interval (0, t] yields an expected number of events in this interval, which is usually not equal to one t

t

t

J fi(r)dr = j E [N{d [N{dT)\ =EEJJ N(dr) N(dr) = =EE[N(t)\. [N(t)\. T)\ = 0

0

0

(2.8)

24

CHAPTER 2. STOCHASTIC POINT PROCESSES Product density of degree two which satisfying the relationship hf2(t(t1,tut22)dt ) 1dt2xdt2 == E[N(dt E[N(dh) N(dt 1)N(dt 2)}, 2)},

h^t tx ±2,t2,

(2.9)

specifies the correlation between arrival rates at two different time instants ti, t2 (or the correlation of increments of the counting measure N(t) on disjoint infinitesimal time intervals). If k out of n time instants are set equal, i.e. tjx = tj2 — ■ ■ ■ = tjk, or k out of n infinitesimal intervals all overlap, the product density of degree n degenerates to (n — k 4- l)-th order product density, thus n

n k k = E 'f[N(dt 'f[N(dt i){N{dt i){N{dt Ji)} Ji)}

E 'f[N(dti) _i=l

J t ■ =-■■=( ■

n

(2.10)

L*=l

ik

— /n-Jfc+H^li ' " ,tn, tj^dti

■ • ■ dtndt^,

where i ^ j r , r = 1,2,... ,k. For example = E [{A^x)} [{NidtJ}2]2 ] = h{t)dt. h{t)dt.

E [N(dh)N(dt22)]\ E[N{dh)N{dt ))\h=t2 ti=^

(2.11)

Joint density function, defined as jr„(*i, 7r ■■■dt ■ ■ dt PrPr {N(dU) {N(dU)= = 1, 1, i =i =1,2,..., 1,2,..., n; n;N(t) N(t)= = n}, n}, n(ii, t22,..., tnn)dtidt22 ■ n = n = (2.12) specifies the probability that an event occurs in each of the disjoint intervals (ti,U +dU] and there is no other events in the whole time interval (0,4], i.e. that there are exactly N(t) — n events. The following relationships between the product density and the joint den­ sity functions hold (cf. [125]) "

i

1

f

n=fc n=k

ff 0

f 0

(n-fc) -fold

(2.13) ~,

t

-k

t

C(—l)"~ — l ' ) " * f/"

Kkjtl, T T f c ^ l■, . ■ -.■ , ^ >tk) ) = =/ /w w n=fc n=k

f/"

, s ,y , / ■/ ■ '" / / / nfn(tl,---,tk,t ( * l i - - - » k4+l,...,t . * f cn+)dt l ,k+- i- - 1 4 n ) d t f c + l ' , - d■■■dt f n - n. 0

0

(n-fc) -fold

(2.14)

2.1: CHARACTERIZATION OF STOCHASTIC POINT PROCESSES

25

The probability that exactly n events occur in the time interval (0, t] is evaluated as [125] t

t

Pi{N(t) Pr{iV(t) = = n} = = ^i [J -• -■ [- /irnir(n,t ,..., tn)dhdt tn)dhdt ■ ■■ ■dtn. dtn. n(tu2tt,..., 2 2■ ■ 0o o0

(2.15)

n—fold

Moreover the correlation functions are defined in terms of product densities as [64, 127] 9i(t) ffiW 9*{ti,h) 9z (tut2,t3)

= = =

/i(0, (2.16) /i(«). /a(*i,ta)-A(*i)/x(*s). f3 {tut2,ta) - 3{A (t0 h («a,^3)}s + 2/i (t0 h (h) h (*s),

where {■ ■ -} s denotes the symmetrizing operation, i.e. the arithmetic mean of all terms similar to the one in brackets and obtained by all possible permuta­ tions of t!,t 2 ,t3, e.g.

{/i (t0 (<0 h («2,h)} (*2, h)}s = = \i (/i (/i (t0 (t0 h/ 2 («a,is) (t2, t3) + +A /i (t (t22)) /»(t h (h, + hh (h) (*s) h/ 2 (ti, (h,h)). *a)) • l51h) 3) + (2.17)

2.1.2

Poisson process, compound Poisson process

Poisson process is a special case of a point process, whose increments N(dt) defined on disjoint time intervals dt are independent. The non-homogeneous Poisson process is completely characterized by its first-order product density /i(t) = v(t), fi(t) = K0.

(2.18)

which is the intensity of the Poisson process. Higher-order correlation functions equal zero Sn(fi,...,t„) = 0 5n(tl,...,t„)

(2.19)

for n > 1. Due to the independence of events (2.2) becomes n

^ (/ (f fii))//„(ti,...,t JI' n) = n n (ti,...,t„) i=l

(2-20)

26

CHAPTER 2. STOCHASTIC

POINT

PROCESSES

Substituting (2.20) in (2.14) and in (2.15) one obtains, respectively

7Tnn (*i, (tl, •. ■ , tn) == J\v f[ V(U) (U)exp exp Ij -- J/ u{r)dr v{T)dT J , , T • •. ,tn)

Pr{iV(t) =n} Pi{N(t) = n} = - (

f u(r)dT v{T)dT j exp j - jJ u(r)d V{T)
(2.21)

(2.22)

For a homogeneous Poisson process (i>(t) = v = const.) one obtains the following expressions fn(h,...,t fn(t = vvn,n, (2.23) 1,...,tnn)) 7r„n (t ir <,...,t . . .n,)* „ ) = = i/"exp(-i/t), i/"exp(-vt),

(2.24)

nn

(ut) (i>t) = n} = ii—(Lexp(-i/t). - f - exp(-i/t). n! n! Compound Poisson process {X(t),t > 0} is denned as Pi{N(t)

(2.25)

N(t) N(i)

*(*) X(t) = £$*>5 ,,

(2.26)

i=l

where {./V(£); i > 0} is the Poisson counting process and Yi are the random vari­ ables assigned to the random points U on the time-axis (instants of occurrence of Poisson events). These random variables are assumed to be independent and identically distributed and also independent of the counting process.

2.1.3

Renewal processes

The renewal process can be defined as a sequence of random time points ti,t2,...,tn on the positive real line, such that ti

tj_i

t\ h

=

A^,

— = XX\, u

I = Z, a , . . . ,

(2.27)

where the time intervals {Xi,i = 2,3,...} between the successive points are positive, independent and identically distributed random variables. The point process is called an ordinary renewal process if the time Xi measured from the

2.1.

CHARACTERIZATION

OF STOCHASTIC

POINT PROCESSES

27

origin to the first event has the same distribution as other time intervals Xi. This means that the time origin is placed at the instant of the zeroth, or initial event (which is not counted because N(t) gives the number of points in the interval (0, t\). If X\ has a distribution different from that of the other time intervals Xi, the point process is called a general or delayed renewal process. In that case the time origin is placed arbitrarily. An ordinary renewal process can be defined equivalently as the sequence of positive, independent and identically distributed random variables {Xi,i = 1, 2,.-.}. Consider an interval (0, x) of the time-axis. An ordinary renewal density hB(x) [19, 20] represents the probability that a random point (not necessarily the first) occurs in (x, x + dx], given that a random point occurs at the origin. A modified renewal density hm(x) [19] represents the probability that a random point (not necessarily the first) occurs in (x,x + dx), with arbitrarily chosen time-origin. A modified renewal density is the first-order product density of the renewal point process hm(x)dx

= Pi{N(dx)

= 1} = fi(x)dx. fi(x)dx.

(2.28)

If this probability is irrespective of the position of the interval (0, x) on the time-axis, the renewal process is stationary. Product density functions are obtained by making use of the independence of the times Xi between successive events as [125] fn{x1,...,xn)dx1---dxn = hm(xi)h0(x2

=

E[N{dx1)---N(dxn)]

- xi)h Xi)h00(x (x33 - x2) ■ ■ ■ h„(x h0(xnn - xn_i)dxidx _1)dx1dx22 (XI<X (xi<x2<2<

...< ■■■<

• ■■dx ■■ dxn,n,

Xxn),

hence fn(xi,

...,xn)

= hm(xi)h0(x2

- xy)h0(x3 - x2) ■ ■ ■ h0(xn - i „ _- i ) .,

(Xi < 2< X2 < ....< .. < {xi<x

xX n). n).

(2.30)

Let us denote the probability density of the random variable Xi as g\{x) and the probability density of each of the variables {Xi, i = 2,3,...} as g(x). It can be shown that the renewal densities hm(x) and hB(x) satisfy, respectively, the integral equations, called renewal equations, [125]

(2.29

28

CHAPTER 2. STOCHASTIC

POINT

PROCESSES

X

hhm(x) gi{x) + / hm(x - u)g(u)du, u)g(u)du, m(x) = gi(x) 0o

(2.31)

x X

(2.32) (x) = g(x) + / h0(x — (2.32) h0{x) - u)g(u)du. u)g(u)du. 0o The renewal densities can be evaluated by taking the Laplace transforms of the equations (2.31) and (2.32), which finally yields [19, 20]

«*> =£-'{T^)}' ^'--'I^U _1

< 233 »

(2.33)

"•*•«--'{.%}• <*> = £ " { r % } '

234 (2.34)

( »

where £ { . . . } denotes an inverse Laplace transform. A class of the renewal processes which is important in applications are Erlang processes, where the time-intervals between events have gamma (or Pearson type III) probability distribution, with the density function [34] k k 1-1 g{x) = vi/*** x ~ exp(-vx)/(k exp(-i/i)/(jfc - 1)!,

x > 00,,

(2.35)

where k = 1,2,3, A Poisson process is a special case of such a process, specified by letting k = 1, in which case the time-intervals have the negative exponential distribution characterized by the density function g(x) = L>exp(-i/x), vexp(-vx),

x>0.

(2.36)

An important property of the gamma distribution with density function given by (2.35) is that it is the distribution of the sum of k independent random variables, each of whose distribution is negative-exponential. Hence the events driven by an Erlang process with parameter k can be regarded as every fcth Poisson events.

2.1.4

Markov renewal (semi-Markov) process

A Markov renewal process is the stochastic process with continuous time and discrete (countably finite) state space i = 1,2,..., M, At random instant tn

2.1.

CHARACTERIZATION

OF STOCHASTIC

POINT PROCESSES

29

there is a transition (or jump) from the state Sn to the state Sn+i which is attained (visited) at the instant £ n+1 . The duration t n + 1 — tn is an arbitrar­ ily distributed random variable. The state Sn of the process at an instant tn assumes value from the discrete state space (set of states). The process S(t) describing these transitions is called a Markov renewal, or semi-Markov, process and is characterized by the distribution functions Gy(t) defined as [113] Gij(t) {Snn+1 i}, Gij{t) = Pr Pr{S
<7yy(0) (0) = 0. G

(2.38)

The distribution function Gij(t) represents the probability that the sojourn time in the state i is shorter than or equal to t and ends with the jump to the state j , or, in other words, it is the probability that there is a transition to the state j after the sojourn time shorter than or equal to t, given that the previous state was i. If the distribution functions Gy(£) are absolutely continuous there exist the corresponding probability densities gij(t). The renewal density hij{x) is introduced through the relation hij(x)dx = E [Nj(dx) | VV^O) N^O) = = 1],

(2.39)

where Nj(dx) is the increment of the counting process during an infinitesimal time interval (0,dx), the count being the jump to the state j . Hence the renewal density hij(x) can be interpreted as the conditional expectation of this increment, given that previous state was i. The renewal densities hij(x) satisfy a matrix analogue of the renewal equa­ tion (cf. [125]) X

H(z) u)du, H ( i ) = G(x) + I G(«)H(* - u)du, o

(2.40)

where H(z) = f/iy(x)] and G(:r) = [&j(x)] are the matrices of the renewal densities and of the probability densities of the sojourn durations, respectively. After taking the Laplace transforms of the matrix renewal equation (2.40) the matrix of renewal densities is evaluated as 1 1 1 H(x) ==C-1cH(x) {[I-G*(s)}{[i-g*(s)r g*(s)}g*(s)}

(2.41)

30

CHAPTER 2. STOCHASTIC

POINT

PROCESSES

where Q*{s) = [g*k(s)] is the matrix of Laplace transforms of the probability densities gjk(x), [■ ■ - ] _ 1 denotes the inverse matrix, and I denotes the identity matrix, i.e. I = [6jk], where 6jk is the Kronecker delta.

2.2 2.2.1

Filtered stochastic point processes General case: stochastic integral with respect to a counting process, averaging of stochastic inte­ grals

Filtered stochastic point process {X(t),t

> 0} is defined as

AT(t)

y

X(t) = £/w(t,ti,Yt),

(2.42)

i=l

where {N(t),t > 0} is a general counting process, Y; is a vector random variable attributed to a random point U. In general the random variables which are the components of the vector Y* do not have to be independent, they can be correlated, and can be characterized by different probability distributions. Furthermore the vector random variables Yt attributed to different points are not necessary to be mutually independent or identically distributed. Let us only assume that these random variables are statistically independent of the counting process N(t). Non-random function w(t,ti,Y{) (w = 0 for U > t) represents the effect at the time t of an event occurring at the random instant U, the event being characterized by a vector random variable Y^. Hence the process (2.42) represents the cumulative effect of a train of point events occurring at random instants U belonging to the interval (0, t], described by a general stochastic point process. The process X{t) denned by the formula (2.42) can be interpreted as a random train of general pulses, or signals, with origins at the random times U; the function w(t,ti,Yi) being the pulse shape function. After the division of the interval (0, t] onto disjoint, contiguous subintervals, X(t) can be written as the Riemann-Stieltjes sum. The limit, in the mean-square sense, of the sequence of such sums, is the mean-square RiemannStieltjes integral with respect to the counting process N(t), or the stochastic integral

2.2. FILTERED STOCHASTIC POINT PROCESSES

31

t

X{t) X(t) = fw(t,Jw(t,T,Y T, YTT))N(dT), N(dr), o 0

(2.43)

where YT is the vector random variable prescribed to the point occurring in the interval (T, T + dr\. The expected value of the process X(t) is obtained by just averaging the expression (2.43), which yields t

t

E[X(t)} =JE[w(t,T, o

YT)] h{r)dr = jjw(t,r,y)g jjw(t,r,y)g T{y)h{r)dydT, T{y)h{r)dydr, (2.44) oyT

where gT(y) is the joint probability density (which may be time-variant) of the vector random variable Y T and yr is the sample space of this random variable. Subsequent moments of the process X(t) are evaluated by averaging of the pertinent multi-fold integrals obtained based on (2.43). For example, the second-order moment (the mean square) is formulated as t

t

2

E [X\t)\ [X (t)] = JJE[W jJE[w oo

(t, n,Yn)

w (t, r2, YT2)] E [N(dn)N(dT [N^N^)}. 2)}.

(2.45)

In order to evaluate this integral, the degeneracy property of the seconddegree product density must be taken into account, which takes place within the integration domain, for T\ = r2. This yields t

2

E[X (t)]

= IE jE o

[w2(t,r,YT)]

h{r)dr

t t

+ I I E [w(t,n,Yn)w(t,T2, 0 0 y-h{Ti,T Xf2(ri,T2)dT 2)dT XdT2. 1dT2.

YT2)] (2.46)

32

CHAPTER 2. STOCHASTIC

POINT

PROCESSES

Likewise the correlation function of the process X(t) can be obtained as m\n{t-±,t 2) min(t 1 ,«2)

E[X{h)X{t2)} =

jI £>(*!, T)Yr )i«(t,,T )]/1(r)dr ^K* 1 ,T,Y )]/i(r)dT rM*2,7-,Y l Y rT o 0 «1«2

-IYn)w(t22,r + j/ j/ #E [w{h,r [t«(tl,Ti, ,T22, , u

Y T2 )]

00 Xf2{TuT2)dT Xf2(n,T 1dT 2)dTidT 2. 2.

(2.47)

The general expression for the nth order moment is 1

' * rn i rn E[X (t)}= [■■■ [E f[w(t,Tk,YTk) E f[N(drk) { { U=i J U=i L=i n

.

(2.48)

n—fold

Of course, as rx ^ r 2 7^ . . . ^ r n , we have n

£

J|AT(dTfc ) J]/V(dT

= / n ( r i , . ....))TT„„))ddTr ll " - d T B .

jt=i

In the multidimensional integration domain any possible equatings of the ar­ guments Tic take place. Therefore in order to evaluate the integral (2.48), all possible degeneracies of n-th degree product density /„ ( n , . . . ,r„) must be taken into account. Moreover, in general case the integration is performed with respect to the joint probability density of n vector random variables Yk,k= 1,2,..., n. In particular the expressions for third- and fourth-order moments are ob­ tained as

E[X3(t)\

= J E [w3 (t,T,Y T )] h(r)dT o t

+3JJ 0 0

t

E [w2(i,TX, Y T1 )w(t,r 2 , Y T2 )] h{TUT2)dTXdT2

2.2. FILTERED STOCHASTIC POINT PROCESSES

33

t t t

+ JJJ JJJ

E[w(t,r E[w(t,ruu

Y nn )w(t, )w(t, r2, YT2 )«;(*, rr33,, Y YT3 )] T2 )w(t, T3)]

0 0 0 Xf3(Tl,T2,T X / 3 ( T l , T23)dT , 1dT2dT3, T3)dTidT2dT3,

(2.49)

t(

4

E[X\t)} = JJEE [«,*(*,r,Yr)] h{r)dr £[X (t)] = o t( t

+ZJJ E [W2(t,n, +3JJ (t,n, YTl)w2(t,T2, YT2)] h{T f2(TUT2)d )dTlTlddT T 2 2

0 0 t t t t

3 )] / 2 (n, T2)dndT"22 W{t, TuYn)w (t, +4 J J £E [[w{t, TUYn)w3(t, Tr2,2,YYT2T2 )] f2(n, T2)dndi

+4 ^

00 0 0 t t t t t (

[w22(t,n,YTl)w(t,T2,YT2)w(t,T3,YT3] [w (t,n,YTl)w(t,T2,YT2)w(t,T3,YT3]

+ 6 JJJE + 6 JJJE 0 0 0 0 0 0

x f3(n, r 2 , r3)dTidT2dr3 X f3(n, T 2 , T3)dTidT 2 dr3 t t t t t t * (

+ J J j J E [w(t, n , Y T1 ) • ■ • w(t, r 4 , Y T J ]

+ /0 0/ 0/ 0/

E

N*. 1, Y n ) • ■ • w(t, r4, Y T J]

0 0 0 0

Xf4(Ti,...,T4)dTi---dT4.

x / 4 ( n , • • •, 7"4)rfTi • • • dr4.

(2.50)

In applications we often deal with the process of the form N(t)

X ^WH^ ^PPi i^^ -- if ii )) ,

(2.51)

i=l

where the impulses magnitudes Pi are independent, identically distributed ran­ dom variables having the distribution as a common random variable P. Then evaluation of the integrals (2.44-2.50) is considerably simplified. In general, however, the evaluation of the above integrals, especially in the case of higher-order moments becomes cumbersome. Hence, in the case of a Poisson process, it is much easier to handle the cumulants, which can be obtained directly from the characteristic function (or a cumulant generating

34

CHAPTER 2. STOCHASTIC

POINT

PROCESSES

function). In the case of a filtered renewal process the recursive expressions for the moments can be obtained from the integral equations governing the characteristic function.

2.2.2

F i l t e r e d Poisson process

A filtered Poisson process {X(t),t

> 0} is defined as

N(t) W) *(*) = ^w{t ; > >tU,Yi), (Mi,y;), X(t) =

(2-52)

where {N(t),t > 0} is the counting Poisson process and Y{ are indepen­ dent, identically distributed random variables having the distribution as a common random variable Y. The random variables Yi are also assumed to be independent of the counting process N(t). The characteristic function $(t,u) = E[exp{iu>X(t)}] of a filtered Poisson process (2.52) is expressed as [92]

if

$(t, w) = exp I IE [exp (iuw (t, T,YT))

- 1}

\\ .

V{T)
(2.53)

$(t,w) = exp \ I E [exp (kuw (t, r, YT)) - 1] V(T)<1T > .

The cumulants Kn(t) are obtained by differentiation of the cumulant gen­ erating function K(t,u>) = ln$(f,w) as KnW

l

"()

—T^— **

'

U=Q'

(2-54'

which yields t

(2.55) Kn(t) = If E K (t, (i, T, Y yT)] V{T)
(2.56)

2.2. FILTERED STOCHASTIC POINT PROCESSES

35

where H{...} is the characteristic function of the random variable Y. Then the expression (2.55) for the cumulants simplifies to the form of t

nn

n n Kn(t) = E[Y E [Y]] f wIw (t,(tT)V{T)<1T. tr)u(T)dr.

(2.57)

0

The expressions (2.53) and (2.55) or (2.56) and (2.57) can be easily gener­ alized to the case of a vector random variable YT.

2.2.3

Filtered renewal process

Consider a filtered process {X(t),t > 0}, driven (generated) by an ordinary renewal process, in the form of N(t)

x(t)= Y, f(*-'**$)=5>(*-*».*i). 0
j=l j=l

(2'58)

where the random times (random points at the time axis) tj are distributed according to an ordinary renewal process, i.e. at the time t0 = 0 the so-called zeroth (or initial) event occurs, which is not counted. The random variables Yj are assumed to be independent and to have identical probability distributions characterized by the common density function v(y). The probability distribu­ tions of the interarrival times are characterized by the distribution function G(x) and by the density g(x). The characteristic function <$(£, u) of the process X(t) defined by (2.58) satisfies the integral equation [132] t

$(t,u) - I $(tx,w)T(t - x,u)dG(x) + 1l-G(t), - G(t), *(i,ii/)= Q(t-x,w)r(t-x,w)dG(x) o

(2.59)

where oo

r(« -x,w) T(t - x, w) = = E [exp{iuw(t -x,Y)}]= - x,Y)}}=

/ exp[iww(t - x, y))v(y)dy, y))v(y)dy, (2.60) —oo

is the characteristic function of a single general pulse w (t — x, Y).

36

CHAPTER 2. STOCHASTIC

POINT

PROCESSES

After differentiating the equation (2.59) r times with respect to u and after substituting u> — 0, we arrived at the following integral equation governing the moment m°(t) = E [XT(t)):

x x m°(t) = 53 (/0 r-i(t / *-*(*- _x)m°{t X ( f- ~x)g(x)dx, )9(x)dx, m°(t) = J2 Q i=0

(2.61)

00

where oo

kk

Mt k(t-x) (f> -x) ==EE[w [w(t(t - - x,x,Y)] Y)} ==j f wwk(tk(t - - x,x,y)v(y)dy. y)v(y)dy.

(2.62)

—oo —oo

Taking the Laplace transforms on both sides of equation (2.61), solving the obtained algebraic equation and taking the inverse transform one obtains the following recursive expression [132]:

m°(t) = = 5J23 ((rr )J // ^<j> 0(x)dx, m°(t) - ^r_i{t ~ x-)mx)m°(f i(t ~ - x)h x)K(x)dx, i=0 i=o

(2.63)

0

where h0(x) is the ordinary renewal density (cf. (2.34)). The specific formulae for the mean value £'[X(f)] = m°(t) and the mean square E [X2(t)] = mj(t) are obtained as t

m°(t)= mj(i) = / 4>i(t 0i(t - —x)hx)h 0(x)dx, 0(x)dx, o t

t t

™°2(t) = / <M* ~ x)h0{x)dx + 2 j j y(t - x^it 0

(2.64)

- u)h0{u -

x)h0(x)dudx,

0 i

(2.65) where E\w(t-x,Y)],) Mt-x) 1(t-x) = E[Ht-x,Y)]

(f> 22{t (t --x)x) = E [w\t [w\t - x,Y)] x, Y)] .

(2.66) (2.67)

2.2. FILTERED

2.2.4

STOCHASTIC

POINT PROCESSES

37

Filtered Markov renewal process

Consider a filtered process {X(t), t > 0} of the form:

X(t)= 53 wit-t^Sj.Yj), w(t-tj,Sj,Yj), *(*)= ^

(2.68)

0
where the pulse shape function w(t — tj,Sj,Yj) of a general pulse (signal) depends on the state Sj of a Markov renewal process (semi-Markov process) at the time tj, The random variables Yj are independent and characterized by the probability distributions dependent on the associated state Sj of the semi-Markov process. The respective conditional probability density functions are denoted as vkk(y) v (y)=v{y = v(y\S\Sj = k). (2.69) j The semi-Markov process is described by the distribution functions Gij(t) (2.37); the respective density functions are gij{t). Assume that at the initial time to = 0 the state of the semi-Markov is S0—j. The conditional characteristic functions of the process X{t), conditional on the initial state, denoted as w) = $ u |I S0 = j), M, **,(*, * ({t, t,w j ) , j = 1,2,..., 1,2,... ,M, rf (t,w)

(2.70)

satisfy the set of integral equations [113] M

i

M

Qfau) = Y,\ " x,u)Tk{t - x,u)dGjk(x) +1-Y,G Qfau) =5 3 / **(* -*.w)r*(t-x,w)dG 53jkG{t), ;*W> iA (i) +1 Jt=0 {

(2.71)

*=0 oo

where rk(t — x, u>) — J exp [iww(t - x, k, y)] vk(y)dy is the conditional char—oo

acteristic function of the pulse shape w(t - x, Sj, Yj) given that Sj = k. The conditional moments m°{t,j) = E[XT(t) \ S0 — j] satisfy the set of integral equations m f f_ x x k m°{t, *-*<*"~xx'fc>)fc)m ^ ~' k)g ' )9jk(x)dx, <(*» j)j) ==5£3 £S ((r )0 / *-^* i"( jk(x)dx,

(2.72)

38

CHAPTER 2. STOCHASTIC

POINT

PROCESSES

where oo

&_;(f -x,k)

T {

= E [w - (t - x, Sh Yj) I S53s- = Jfc] =

/ wT-\t

- x,

k,y)vk(y)dy.

—oo

(2.73) Taking the Laplace transforms on both sides of equation (2.72), solving the obtained algebraic equations and taking the inverse transform one obtains the following set of recursive expressions: r—1 /

M

\

»

<(*>i) = E E 1 • / *-*('- x '*) m ^~~ *■*)M*)<&. *=o fc=0 i=o i=0 \^ '' {{

(2-74)

where the renewal densities hjk{x) are evaluated from the matrix renewal equa­ tion and are the elements of the matrix H(x) (cf. (2.41)).

2.3

Non-diffusive Markov process

Response of a dynamical system to an excitation in form of an impulsive point process is an effect of responses to individual point events (impulses). In general case, if the system is not linear, the cumulative effect of the impulses cannot be regarded as a superposition of the individual effects. However the increment in the response process during the infinitesimal time interval can be attributed to the increment of the excitation process in this interval. This allows to formulate the stochastic equations governing the response process to a random train of impulses. Consider the vector stochastic process {X(t),t > 0} satisfying the stochas­ tic integral equation [30, 116] t

t

X(t) = X(t) = X 0 + j a(r, X(r))dr X(r))dT + J j b(r, X ( r ) , p)U(dT, dp), dp), 0

(2.75)

0 V

where the vector X 0 = X(0) of initial conditions can be, in general, a random vector. In the above equation R(dt,dp) is a Poisson random measure [30, 116] for the compound Poisson process, and V is the sample space of the vector random mark variable P The functions a(r, X(r)) and b(r, X ( T ) ) satisfy some

2.3. NON-DIFFUSIVE MARKOV PROCESS

39

general conditions, e.g. that of a boundedness [116]. The measure U(dt, dp) can be interpreted as a random number of events (points) in the interval [r, r+dr), with the values of the associated random variables P in the interval (p, p+dp). The counting Poisson process N(t) is here alternatively defined as the number of events in the interval [0,t). An additional assumption must hold: Pr{./V(0) = 0} = 1. The Poisson random measure satifies the following relationship:

fu(dt,dp) = N(dt), fu(dt,dp) N(dt), v

(2.76)

where N(dt) is the increment of the counting process in the interval [t, t + dt) and E [U(dt, dp)]== v(t)fp(p)dpdt, E[n(dt,dp)] v(t)fP(p)dpdt, (2.77) where v{t) is the intensity of the Poisson process and /p(p) is the joint prob­ ability density of the vector random variable P. The double integral in (2.75) assumes the following values t

//b(r,X //b(aw,p)n(*,*)= M ,p W Mp)= 0 V

r

0,

ggMbX ( t i ) , P i ) ,

if N(t) = 0 a m i l i

(»8)

■,

where b^ (X (U), Pj) describes the jump in the state vector X(£) which occurs at the time U and which depends on both the state vector X(f) itself at the time U and the vector random variable Pj. Next assume that the initial conditions X<> are statistically independent of the underlying stochastic point process (the compound Poisson process) in the interval [0, t). This assumption is not quite obvious. Of course the initial con­ ditions cannot depend on the anticipating point process, but the point process, e.g. the compound Poisson process, could depend on the initial conditions, e.g. the impulses magnitudes could depend on the initial conditions. Since the increments of the Poisson process are independent and if the underlying compound Poisson process is independent of the initial conditions, the vector process X(£) governed by the equation (2.75) is a Markov process. It is a non-diffusive, Poisson-driven Markov process.

40

CHAPTER 2. STOCHASTIC

POINT

PROCESSES

The integral stochastic equation (2.75) can be rewritten in an equivalent integro-differential form as dX(t)

=

a(t,X(t))dt+

fb(t,X(t),p)n(dt,dp), v

X(0) =

X 00,

(2.79)

where dX(t) — X(t + dt) - X(t) is the increment in X(t) during [t,t + dt). Equation (2.79) is an analogue, for the non-diffusive Markov process, of a classical ltd stochastic differential equation. The above equation can be recast in the form of dX(t) = a (t, X(t)) dt + b (t, X{t), P(t)) N(dt), N(dt),

(2.80)

where N(dt) is the increment in N(t) during the time interval [t, t + dt) and P(t) denotes the vector random variable - associated with an impulse occurring in this interval. If the terms b (analogues of the diffusion terms), or the jumps are inde­ pendent of the state of the system, the joint probability density fx(t, x) of the non-diffusive Markov process governed by the equations (2.75) and (2.79) or (2.80) satisfies the following equation (e.g. [94]) a xx + + EE^:( ^ T ( a ii^ ^ ))/x(*,x)) / x ( * , x ) ) ++ ^)/x(t,x) ^)/x(t,x)

^^ TT ^^

i

-

v(t)ffx(t,x-b{t,p))fPP(p)dp v(t)Jfx(t,x-b(t,p))f = 0. (2.81) v V In a more general case, when the terms b depend on the state variables X(f), a similar equation is obtained, after some extra transformations [94]. Integro-differential Kolmogorov-Feller equation (2.81) is an analogue, for the non-diffusive Markov process, of the classical Fokker-Planck-Kolmogorov equa­ tion. This equation can be recast in the form of a partial differential equation (an analogue of the forward Fokker-Planck-Kolmogorov equation)

5J

^

+ T,£rM>,*)M,*))-*)£,^ (2 . 82 ) dTdT *^c*^c dXii ..dXir ^(t,x)/n^('.p)^(p)^j ^.x)/n^(^p)/p(pMp =°dXii ..dXir t

r=l

2.3. NON-DIFFUSIVE MARKOV PROCESS

41

The differential of the function V (t, X(<)) of the process governed by the equations (2.75, 2.79) can be evaluated from the following differential rule (an analogue, for the non-diffusive Markov process, of the classical Ito's differential rule) [30, 31, 116]

^((,x«)) - M . ^ ' M , , , , ^ * + j1 [V ((, ((, X(() X(() ++ bb ((, ((, X((), X((),p)) p)) -- VV (1, (1, X(f»] X(f»] V

xU(dt,dp),

(2.83)

where the function V (t, X(i)) must be bounded for t and X(t) finite and must be once continuously differentiable with respect to all its arguments. It is worth noting that the differential rule (2.83) is valid even if the jumps are state-dependent, i.e. for b = b (£,X(i),p). The form of the differential rule corresponding to the equation (2.80) is

dV(t,m)

=

^^^^^M

a i ( f

,x

W

)^

V (t, X(t))] X(t))} + [V (t, X(t) + b (t, X(0, P(t))) - V (2.84) xN(dt). The differential rules (2.83) and (2.84) can be recast, with the help of Taylor expansion, in the forms containing the partial derivatives, respectively, as

dV(tMt)) dv«,X(i))

.= «£KMW) 5i2|W)MiM(,x(t)), ,x( i )),ti K i | p ^ +d(+E E ^M(ffl

E ( i ) E / | ™ r W , m Pt,>) + E(i)E/S^nM,x«,, xU(dt,dp), xll(df, dp),

dV((,x(1))

(2.85)

= ^MW)* +E 2KMW) M( ,x W )* i

42

CHAPTER 2. STOCHASTIC

POINT

PROCESSES

+ Za)£|™lW.xM,r<< )) ( 2 - 86 )

xN(dt). xW(df).

If the function V is assumed as the product of different powers of the state variables X{, then taking the expectation of the differential rule leads to the differential equations governing the statistical moments of the response process X(<), or the joint moments of the state variables Xi. It should be noted that the stochastic equations (2.75, 2.79, 2.80) and the respective differential rules (2.83 2.86) are valid for any point process which is uniformly regular and has a finite number of points in a finite interval [116]. However, effective averaging of the last term of the differential rule is only feasible if the correlation between the function V of the state variables and the increment (U(dt, dp) or N(dt)) of the underlying point process can be directly split. This can only be done if the point process has independent increments (a Poisson process).

2.4

Asymptotic expansions of the probability density function of the stochastic process — Gram-Charlier expansions

Univariate probability density function f(t, x) of the stochastic process X(t) can be expressed as an inverse Fourier transform of a characteristic function *(*,w) OO

f(t,x) = — /(*!*) =* « - // *(t,w)exp(-ttjs)dw. *(£»u;)exp(—iw$)duj.

(2.87)

—OO

Inverting can be effectuated, for example, by substituting (cf. e.g. [74, 87])

*(t,W) = e x pj| ff ;;^^ « j^ ( t )) J!), *(*.«)=op

(2.88)

2.4. GRAM-CHARLIER EXPANSIONS

43

where Kj(t) are the cumulants, and by expanding the exponential in the series. This results in the following expansion for the density function:

/(u) = m {1+E (i+E ^(0+5 *$W)+5 E ^ ^ ^ * ( o 1 f. A^OA^AKt)H

+ 3? 2 ^

^ilm

' j,k,l=3

J- ■ ■

1 J+k+i{Z) + ■■■>,

(2-89)

)

where


(2.90)

is a standardized Gaussian density function, (x - mi (t))/a(t), €£ -= (1 mi(t))/
*(t) = V ^ W - V

(*(*)).

m E[X(t)), mi(0 = ^[A-(t)], x{t) =

AAj(t) = Kj(*)M*), ;W = «i(0M*).

{

'

and Hj(£) are the Hermite polynomials satisfying the relationship Hj{MZ)==((-iy^m, #i(04>(0 "1)i^^)-

Jj==0,1,2,.... 0, i, 2,....

(2.92)

If the terms of the expansion (2.89) are collected according to the orders of the Hermite polynomials, the Gram-Charlier expansion is obtained as

/(u) = m | i + E ^jrWo} • /(*,o=^)|i+E^i«)l

(2-93)

The coefficients bj(t) of this expansion are the so-called standardized quasi-moment functions [127] and are expressed in terms of the standardized cu­ mulants Aj(t) as 63 63 64 64 65 b6 67 bb88

= AA 3 ,3 , = A4, = A5, = Ag + lOA2., = A7 + 35A3A4, = A8 + 35A2 + 56A 56A3A5. 3A5.

bl = xl'+ioxl

(2 94)

-

44

CHAPTER 2. STOCHASTIC

POINT

PROCESSES

The expansion (2.93) is also called Edgeworth series. In different problems, its coefficients involve the powers of certain parameter and the expansion is usually arranged by grouping the terms according to these powers. Expanding the probability density function in terms of Hermite polynomi­ als

/(«) = ^ ) E # ^ ) J

3=0 ' j=o ■>■

(2-95)

and employing the weighted orthogonality condition OO

oo

j M)Hn(QHn{t)d£ = 6mnn\,

(2.96)

—oo

one obtains the expansion coefficients in (2.95) as oo

cCjj(t) =E[H = E[Hj{E{t)}) j{E(t)})== JJ

HjiOmOdt, HjiOmOdt,

(2.97)

—oo

where

_ X(t) -

mi(t)

m-m^i.

(,98)

The coefficients Cj(t) take the following values: Co = Co

11,,

= Cl ci = = Cj(t)

0,

; : l:0, bj(t),

<**> j>3,

hence the expansions (2.93) and (2.95) are identical. It can be seen from (2.89) and (2.93) that the first term of the GramCharlier expansion is a Gaussian density function and the subsequent terms describe the deviation (departure) of the examined probability distribution from a Gaussian one. Thus the Gram-Charlier expansion is often used as an approximate form of a probability density of non-Gaussian processes. The fundamental questions which arise in a natural way are concerned with the covergence of a Gram-Charlier expansion and the accuracy which can be attained by approximating an unknown density function f(t, £) with the help

2.4. GRAM-CHARLIER

EXPANSIONS

45

of a truncated expansion. The convergence condition given by Cramer (1946) [21] is the following: the expansion (2.93) is convergent to /(£,£) in every point where f(t,£) is continuous, if /(£,£) is a function of bounded variation in (—00,00) and if the integral f™ exp (£ 2 /4) f(t, £)d£ is convergent. This means that as |£| —* 00 the density function /(£,£) must approach zero (i.e. the tails of the density curve must die out) faster than exp (—£ 2 /4), which can be slower than in the case of a Gaussian density function. Unfortunately this condition is of little practical value as the rate of ap­ proaching zero for the examined actual density functon is unknown. Despite the fact that the question of convergence of a Gram-Charlier expansion is open, the use of this expansion was justified by Cramer [21] as follows: in practical applications it is most important to know whether the truncated expansion (2.93), consisting preferably of a small number of terms, provides a good ap­ proximation to a probability density function. From this point of view the question of convergence is not important. On one hand, it may happen that the Gram-Charlier expansion truncated at a low level gives a good approxi­ mation to the density function, though the expansion is not convergent. On the other hand, if a very large number of terms is required to give a good approximation, the practical value of such an expansion is little, even if the expansion were convergent. The questions of convergence of the Gram-Charlier expansion were discussed e.g. in [7, 142]. The joint, bivariate probability density function / (t, Xi,x2) of the processes Xi(t), X2(t) can be expressed as an inverse two-fold Fourier transform of a joint characteristic function $(t,tJi,w2) 00 0 0

f{t,xuxi)

=j—^

/ / $(t,wi,w )exp[-i(wiar 1 +2:E w22a:)]dwidw2$(t,wi,w2)exp[-i(wiXi+w 2)]dw1dw2. — OO —00 —00 —OO

(2.100) Upon expressing the joint characteristic function in terms of joint cumulants Kj(t) as [74]

*(i,^ 2)=exp( $(f,w 1 ,w 2) = exp^

± 2L

(J^t \, Tifci Kjk{t) M')f>

(2.101)

46

CHAPTER 2. STOCHASTIC POINT PROCESSES

the transform (2.100) can be inverted, which yields the expansion / ( * , 6 , 6 ) ==

tf*,6,6)ji + E^ffi*«i.6.p) J I

A*

1 £ ^gWt)

+ 1 jLE ^^H^Mrn&^P) 2! j,k,l,m ^> v.kw.nv. J , 1 3!

2j

^ Ajfcft)A|m(^)Anrnr(t) (t) V^ ^jk(t)Xlm(t)\ ^—' — y.k\l\m\n\r\ ?!KU!m!n!r!

( (2-102) , , c. (c

v.

,

[I

Z

ji + +fc fc= = 3,4,5,...; 3,4,5,...; ii + +m m= = 3,4,5,...; 3,4,5,...; rc n+ + rr = = 3,4,5,.. 3,4,5,..., where

X! -- mi(t) xi mx(t) e x%:r2 - mm22(t) (t) 77; 1 s?2 =_ 777 > ^ at{t) ' ' o2{t) '

(2.103) (4.1V6)

Si =

(2.104) and (j)(t, 6 , 6 ) is the joint, bidimensional Gaussian density function: P )} . V)},

2 rh(t f f \ = — ~ = e xrvn J ff2 2 - •?«< I <^ hL ffifriub) 2 p 6«6 + s1)W// (1 ¥H'> m,(,2j — /; r c xpP |S - i0 fo V»l "PS1S2 1 S2,l \ l 2-Ky/l - p V *■

2

where p = p(t) = An(£) and i/jfc (6,6>P) are bivariate analogues of Hermite polynomials, which satisfy the condition Qj+k fij+k

(2.105) - i yi+ f ceT-j7^<M6,6). r7J7^(6,6)(2105) ^%( 6 (, 66, ,p6) <, ^/ >( (66, ,66 ) = ((-i) "^sl**S2 S I "S2 Likewise the expansions for the joint multivariate probability densities can be obtained, for example for the trivariate density [75]. There also exists another form of the expansion for the joint bivariate probability density function, in which instead of the bivariate analogues of Hermite polynomials the products of ordinary (one-dimensional) polynomials are used [2]. Unfortunately, besides the bounds for the Gaussian approximation error [142], very little information can be found in the literature as regards to the validity and the convergence of multivariate Gram-Charlier expansions.

Chapter 3 Linear dynamical systems under random pulse trains 3.1

Introductory remarks

Consider vibration of a continuous structure, governed by a general equation L [u(r, t)] + cu t (r, t) + mu«(r, t) = p(r, t) = J ]

Pi

(r, t, tit Yt),

(3.1)

0
where L[...] is a linear operator in a space TZ of geometrical variables r and m, c are the mass and damping per unit area, which can also be, in general, the functions of variables r. The excitation is a train of impulses occurring at random times U, driven by a general stochastic point process or by a counting process N(t). An impulse occurring at random instant U has a time-spatial distribution (shape) given by a function pt (r, t, U, Y*). The consecutive pulses can have different spatial shapes. The pulse shape depends, in general, on the vector random variable Yj, which describes the pulse magnitude, its duration, etc. The response u(r, t) can be expressed in a most general way in terms of a Green function G(r, s), or impulse influence function, which is the response at location r due to a unit impulse applied at location s and originating at time t — 0 (it may be a general pulse in the time domain). The Green function may be easily determined with the help of normal mode expansion. The normalmode approach can also be used as a direct approximate method. Otherwise, 47

48

CHAPTER 3. LINEAR DYNAMICAL

SYSTEMS

i.e. if exact normal modes are not known, other approximate methods can be used, e.g. Ritz or Galerkin method, or the finite elements method. Almost inevitably the problem is finally discretized, i.e. described by the set (possibly coupled) of ordinary differential equations. The methods for random pulse trains developed in this Chapter deal with the dynamical systems described by uncoupled equations. Such a description results from a normal-mode approach or can be obtained through a standard technique of a modal transformation of coupled ordinary differential equations of motion. Let us briefly describe the results of the technique of normal-mode expansion. The normal-mode expansion for the function u(r, t) is u(r,t) =='£q u(v,t) J2q j(t)W j(r)t j(r), j(t)W

(3.2)

i where qj(t) and WJ(T) denote the generalized coordinate (modal response) and the normal mode (eigenfunction), respectively. Upon performing the necessary operations the following uncoupled system of equations is obtained q)(t) + 2Qnjqj(t)

+ + n2jqj (t) = Fj(t), jQj(t)

j = 1,2,... = 1,2,...

(3.3)

where N(t)

Fj(t) BfatnYi)

=

Y,Bi(t,ti,Yi),

= J Pifrt^YjWjtfdT/ pifat^YJWMdr/ tf

=

j L[Wj(i)]Wj(T)dx/

Jf mW*(r)dr, mWj(r)dT, f

(3.4)

mWf{r)dr.

and fij, Q are the circular natural frequency and the damping ratio, respec­ tively. If the impulses have equal spatial distributions, i.e. N(t)

p(rIt)=p(r)£s(i,ti,Yi),

(3.5)

3.2. MOMENTS OF THE RESPONSE PROCESS

49

then N(t)

i=i(t) = / ? ; $ > ( M i , Y 4 ) ,

(3.6)

i=l

where 0j mW]{v)dt Pi = [f p(r)Wj(r)dT p(r)W,-(r)dr/I /f mW?(r)dr

(3.7)

is the excitation (or modal) participation factor. Response process qj(t) of a dynamical system, governed by the differen­ tial equation (3.3), to a random train of impulses is, by virtue of a linear superposition principle, a random train of responses to individual impulses of excitation, or a random train of filtered pulses. Hence the process to be ana­ lyzed is the multivariate filtered point process q(i) = {q\(t),...,
E[u(r,t)} =

J^m^W^r), i var(u{r,t)) = /i2(r,t) = £ M W ^ i W , i,* > i « ( < ) W i M W rfcf c(r)Wi(r), (rW(r)l /x3(r,f) = 553^H(*)Wi-(r)W

(3.8)

j,k,i

/i4(r,0 =

3.2 3.2.1

53

ftfcPiW^f^iW^fr).

Moments of the response process to a random pulse train Correlated and Poissonian pulse train

Consider a random train of impulses (i.e. with Dirac delta shapes) having equal, deterministic spatial shapes. The excitation function (3.6) has the form

50

CHAPTER 3. LINEAR DYNAMICAL

SYSTEMS

of N(t) N(t)

F j(t) = p& JY$ /P> i6(t-t *}(t) i * i), (t-*i),

(3.9)

j=i

where the times U of occurrence of impulses are given by the general stochastic point process, taking into account the correlations between these times, and the impulses magnitudes Pi are independent, identically distributed random variables, distributed as a common random variable P The modal response qj(t), as a solution to the equation (3.3) can be repre­ sented as a filtered process 9i(*) ^ P«
1,2,.. jj == 1,2,...

(3.10)

where a,- (i — ti) = fijhj (t — U) and hj (t — U) is the impulse response function of a system governed by equation (3.3), i.e. the response at time t to an impulse applied at time U. Mean value of the modal response is obtained by performing the averaging of the stochastic integral in equation (3.10) (cf. (2.44)) t

E \qjj(t)} E[q {t)\ = E[P] J E[P}Ja r)h{T)dT, aj(t - j(t-T)f 1(r)dT.

(3.11)

0

Averaging of the two-fold integral corresponding to the product of modal responses qj(t),qk(t) and taking into account (3.11) leads to an expression for the second-order central moment, or the cross-covariance function, of the modal responses, in the form of (cf. (2.45, 2.46)) t

(ijk(t)

2

= E[P } j o0

aj{t-T)ak{i-T)h{r)dT t t

+E2[P] jjf j aj{t - TlTi)ofc(* )ak(t - 7-2)52(71, T 2 )dr )dndT 1 dr 22,

(3.12)

0 0

where g% (TI,T 2 ) is the correlation function of the system of random points see (2.16) and [126].

3.2. MOMENTS OF THE RESPONSE PROCESS

51

Likewise, higher-order moments can be determined. When the underlying point process is a Poisson process, the modal re­ sponses qj(t) form a multivariate filtered Poisson process. The following gen­ eralization, the expression for the joint characteristic function of the processes {%•(*). 3 = 1,2,..., n}, is valid [99]

$(t,wi,w ) == exp< (t-T) $(*, wi, w2,,...,w ■ • •,n w„) exp I/ IH I I Hi^2w i^2u - T) - -1 \1u{r)dr \ u{r)drV i . kakka k(t Therefrom the expression for the joint r — ri -I multivariate process is obtained as

(3.13) V r„-order cumulants of the

t

rr

KTl...rn(t) = E E[P [P]] Ja?(t-T)---a' Ja?(t-r).-- n»(ta£{t - TT)v{r)dr. )v(r)dr. o

(3.14)

Consider now a random train of general pulses, characterized by a deter­ ministic shape function and a finite, deterministic pulse duration N(t)

Fj{t)=P F s(t-tiSi,T) (t-t j(t)=P i^Pii^P ) i,T),

(3.15)

i=i

where s (t — U, T) is the deterministic shape function of a pulse originating at time U, with a duration T. The pulse shape function is not equal to zero in the interval t € (ti,U + T) and it is equal to zero outside this interval , i.e. when t < U or t > U + T. The random variables Pi satisfy the same assumptions as before. The modal response qj(t) is represented as a filtered process f(t)

Qi(t) = E «=i

* PiZ

T

> (' ~ *•' ) = j z& ~ T' T)P(r)N(dT), T)P(r)N(dT),

(3.16)

o

where Zj (t — U, T) is the j-th modal response at the time t to a general pulse s(t —ti,T) originating at the time U, and with duration T. The response

52

CHAPTER 3. LINEAR DYNAMICAL

SYSTEMS

Zj (t — U, T) can be represented as a Duhamel integral min(t-ti,T) mm(t-tj,T)

zZjj(t-t (t -i,T) U, T)==/30j j

Ij

hhj (t-U6) s(9)dff, j{t-t i-0)s(6)d0,

(3.17)

o where the shape function s(8) plays the role of an excitation with a finite op­ eration time (duration) T, hence the integration interval length can be smaller or equal to T. After substitution of (3.17) into (2.44) and into (2.46), in order to perform the integration, the following two cases must be considered: - min(* -T,T)=t-r,

i.e. t-T

- min(i - r, T) = T, i.e.


t-r>T.

Consequently the function Zj(t — T,T) assumes two forms [56] t-T

zjti(t -T,T) zjti(t -T,T)

= 0j / hj(t -T= 0j / hj(t -T-

*,(t-r,T)J . - _ I Zj[t -T,l)=

6)s(6)d6, 6)s(6)d6,

t-°r
i

=

I zji2(t-T,T) I

=

T

^

Pjjhj{t-T-9)s{6)de,

° pjJhj{t-T-9)s{6)de, ° t-r>T, i.e. 0 < r < t - T. t-r>T, i.e.t\ 0 < rZj< t - T. Another technique for evaluating of functions Zj and <2 is devised in the references [33, 49]. Finally, after making use of (2.44) and of (2.46), one obtains t

t-T

E[tti(t)]=E[P\ E[qj(t)] = E[P]j JZjtlzjtl (t-T,T)h{T)dT (t-T,T)h{T)dT+E[P] + E[P] jj Zjt2 zjt2{t-T,T)h{T)dT, {t-T,T)h{T)dT, (3.19) t-T

0 t

fijk(t)

2

= covqjqk(t) = E [P ] J zu(t - T, T)zKl(t

-

r,T)h{r)dr

t-T t-T

+E [P2] J zj>2(t - T,T)zki2(t - T,T)h(T)dr 0

3.2. MOMENTS OF THE RESPONSE PROCESS t 2

+E [P\

53

t

j j «i,i(* - n, 2> M (t - r2,Dsafn, r2)drxdT2 t-Tt-T t

t-T

2

+2E [P] J J t-T t-T

{zjjl(t-T1,T)zkt2(t-T2,T)}sg2(T1,T2)dr1dT2

0 t-T

+E2[P] J J *zji2/ ,(t» ( -* -n,T)z r i . Tk}2 K(ta C t --T i . r jr^2,T)g C n 2.{T r aUJTd2n)ddTlTdT, ,2, (3.20) o o0 0 where {.. .}s denotes the symmetrizing operation, with respect to the indices 3, k, e.g. {zjtl(t - n , T)zk<2{t - r 2 , T)}s

=

\{zitl{t

- n , T)zkt2(t - T2, T)

+ZkAt-r1,T)zj<2(t-r2,T)). In the case of a Poisson-distributed train of pulses (3.15), the expression for the joint r = rx + • • • 4- r n -order cumulants takes the form of t

Kn...Tn(t)

r

= E [P ] J z\\{t - r , T ) • • ■ zZ(t - T, T)u{T)dr t-T t-T rT

+E [P ]] j z\)2{t - r, T) • • • z%(t zrn%t - r, T)u{r)dr. T)u{r)dr. (3.21) o

3.2.2

Pulse train driven by a delayed renewal process

In applications, an adequate model based on the renewal processes should take into account the fact that the origin of the time axis may be arbitrary, not necessarily placed at the time of an initial event. Then the delayed renewal process must be used. Random pulse train is represented as Fj(t) = Pj Y, 0
s(t-U,Yi),

(3.22)

54

CHAPTER 3. LINEAR DYNAMICAL

SYSTEMS

where the times U are distributed according to the delayed renewal process (sec. 2.1.3) and the random variables Yt satisfy the assumptions given in sec. 2.2.3. Modal response, as a solution to equation (3.3), is expressed as qj(t)=

J2 Wjit-tuYi),

(3.23)

0
where Wj (t — ti,Yi) is the response of the system to the pulse s(t — ti,Yi). Process qj(t), being the filtered delayed process, can be represented as the sum of an initial pulse occurring at time t = t\ = X% and of the filtered ordinary renewal process with time axis origin shifted by this time, i.e. in the form of wj (t - Xi, Xx, yj) YJ + + q° q] (t - Xj) Xx),, (3.24) qj(t) = Wj where Xi is the waiting time until the first (initial) pulse and q° (t — Xx) is the filtered ordinary renewal process (cf. (2.58)), with origin at time t = X\. It may be shown [42], that the joint characteristic function ty(t, « i , . . . , w„) of the process q(t) satisfies the following integral relationship t

^(t,w1,...,wn)

= / Y(t~x,wu«..,ujn)${t-x,Wi,...lun)dG1{x)

+

l-Gi{t),

n0

(3.25) where Gx(x) is the distribution function of the random variable Xi and the characteristic function $(£, a>i,... ,wn) of the multivariate filtered ordinary re­ newal process q°(t) satisfies the multivariate analogue of the integral equation (2.59). Joint moments mry..rn(t) — E [q^(t) ■ • ■ q^n{t)] are obtained by applying to both sides of the relationship (3.25) the operator gri+-+rn

j—(n+ •+rn)_r

dujl1 • • • duj^n In the resulting integral equation, similar to (2.61), the moments mTl...rn(t) of the filtered delayed renewal process are expressed in terms of the moments m°,.,.rn(t) of the filtered ordinary renewal process q°(t). After taking the Laplace transform of that equation and using for m° ...rn(t) the multivariate

3.2. MOMENTS OF THE RESPONSE PROCESS

55

analogue of the expression (2.61) (cf. [42]) the expression for the joint moment is arrived at in the form of [42] mnn...r ...rn(t) n(t)

= =

J2"YJ J2 "

) I ( (i / )) / / ~- x)m° ■ 5Z I E[ 0n-n...r„-i„(t x)m? ..in( t -- x x) n-iyrn-in(t y..inr(t

i1=0 in=0 > 1' i1=0 in=0 > * ' (*1 «n)#(r- 1 ,...,m) (*1 «n)#(r- 1 ,...,m) xhm(x)dx, x/im(x)dx,

{ £

(3.26)

where cf. (2.62) 1

V.r„(t)^K (t,y)...<((,y)]. 4> • • • tC(«,n] • n...Tn(t) = E [«,?(*,y)

(3-27)

For Dirac delta impulses, specifying Tj = l,rj. = 0 for j ^ 0,j,- = 0 and inserting
£E [qj(t)} fo(i)] ==TOi(i) mi(t) ==£[P] E[P] /I aj(t aj(t -- x)h x)hmm{x)dx (x)dx (3.28) o and for: Tj = 1, rk = 1, rm = 0 for m ^ j ^ k, ij = 0,1; tfc = 0,1; (ij, ik) ^ (1,1) inserting 0n(* -x) = E [P2] ctj(t - x)ak(t - x), one obtains t

[Pp22]

E s [qj{t)qk(t)} to(t)»(«)] == ££ [ ] / «j(t <**■(*- - *)<**(* *)<**(*- - x)h^m{x)dx ( ^ o 0 t t

t t

+2E22[P}j

j{ (t-x)ak(t-u)}s j{aj aj(t-x)ak(t-u)}s

+2E [P}j 0

x

xhm(x)h0(u -—x)dudx. x)dudx.

(3.29)

In the case of general pulses, substituting (3.18) in (3.16), one obtains [42] tt

E[qj(t)} = E[P] f zjtl jtl(t - T,T)hm m(T)dr t-T t-T

T)hmm(r)dT, +E[P] f Zj, Zj<22{t -~ r,T,T)h (T)dT, 0

(3.30)

56

CHAPTER 3. LINEAR DYNAMICAL

SYSTEMS

t 2

E [«,-(*)»(*)] = E [P ] J zjtl(t - T,T)zktl(t - T,T)hm(r)dT t-T t-T 2

+E [P ] j zji2(t - T, T)zkt2 K2(t - T, T)hm{r)dT 0

t-T t-T 2

+2E [P] J j 0

{zj<2(t - r,T)zkt2 (t k>2(t

u,T)}s

T

xh0(u — T)hmm(T)dTdu {r)dTdu t

t-T 2

+2E [P] J f

{zj>2(t - r, T)zKl(t

- u, T)}$s

t-T

0

xha0{u (u — T)hm{r)dTdu t

t

2

+2E [P] J J {%il(t - r, T)zktl(t - u, T)}s t-T T

xh0(u — T)h xh„(u T)hmm{r)dTdu. {r)dTdu. 3.2.3

(3.31)

P u l s e train driven by a Markov renewal (semi-Markov) process

Consider now a random pulse train driven by a Markov renewal process. Let us generalize the results given in the sec. 2.2.4. to the multivariate case, and moreover to the case of the process in which the first (initial) event occurs at an arbitrary instant t\ = X\, The modal response (t) qj(t)

= ^Y

W

Y tt ~ ~ ^ *» Si,5«Yi), J (* i) - j 3==1,2,..., 1,2,..., n,n,

(3.32)

0<*i
may be represented as the sum qj(t)

= Wj {t (t - Xu Si, F Yi) (t -- Xi), Xx), x ) + q° (t

(3.33)

3.2. MOMENTS OF THE RESPONSE PROCESS

57

where q° {t — X{) is the filtered Markov renewal process with the time origin at t — Xi (this is an ordinary process, i.e. its initial event occurs at the zero time t - Xi = 0). Assume that the probability distribution of the time Xi until the first event depends on the state Si of the process at the instant t\ — Xx; the conditional distribution function being Gj(t) G P r{X { Ax '< i <xa |; Si | 5 i== j) i} i (t) = Pr

(3.34)

and the conditional probability density being gj{x). The conditional joint characteristic functions of the process q(t) * ((t,wi,. * j (t,wi, ...,w B ), j = l,2,...,M 1,2,... , M < , w i , ...,u»\Si=j)<= . . , w , , | S 1 = j ) = *i(t,wi,.-.,w»). satisfy the set of integral equations

(3.35)

t

TTj(t-x,u ,...,uj„)$j(t-x,LUi,...,wnn)dG )dGjj(x) j(t-x,u11,...,uj„)$j(t-x,LUi,...,w

%(t,aii,...,w n ) =

o +1 - G&), Gj(t),

(3.36)

where $j (t — x,Ui,... ,w„) satisfy the multivariate analogue of the equations (2.71). After the application of the differential operator with respect to Uk, taking the Laplace transform and making use of the multivariate analogue of (2.72) one obtains the expression for the conditional moments mn...rn(t,j)

= E [tf (*)••■ I SiI =Sij]=,j] , [q?{t) ■ ■C■W CW

(3.37)

in the form of M

mV " (t,j) (*,.7) ri...r r rn„

-=

n

fn

n

/

\

i.

E E - - - E I I ( / ) fc=0ii=0 k=0ii=0

in=0q=l

n-rir N>ri-rirn-in(t-X,k)

V q

'' ' Q Q

xm

°n..4»(* ~ *. k)hjk{x)dx, ((»i,...,»n) »i,...,»n) # # (ri,...,r„), (ri,...,r„),

(3.38)

where the moments m° i n ( t - x , A;) are determined from the multivariate ana­ logue of equation (2.74), cf. [42], where h 0 (t - x, fc) = £ [to? (t - x, StSuitYi) 4> -in4B(t-x,k) E[v>?-*Ht-x Yd ri _ ri-iriv....rn rn_

■ w■?-■i"(t-x,S ... «£•-*»i,Y (ti)\S - x, = k= k] i Si, Yi) \Si

(3.39)

58

CHAPTER 3. LINEAR DYNAMICAL

SYSTEMS

and where hjk{x) are the elements of the matrix H(x) = [hjk(xj\

1 1 = C-1 {diag [g*(s)} [I - g^s)}' G*{s)T }} .

(3.40)

The unconditional moments are obtained by unconditioning as M

mn-r„(t) ...rn{t) = ^2m ^2m (t,j)P(j), n...rn rv.. rn(t,j)P(j),

(3.41)

i=i

provided that the probabilities of the individual states of the initial event are specified, i.e. P O ^=PP rr {^ S^1 j= j}},, j = l,...,M. (3.42) P(j) l,...,M.

3.3 3.3.1

Examples of applications Bridge structure under a random train of moving loads

One of the most important applications of presented above analytical technique is the problem of dynamic behaviour of the highway bridge under traffic loads. In highway traffic engineering it is generally agreed that the vehicular traffic flow should be adequately modelled as a random train of events [1, 34]. The vehicles arrive at the bridge at random times and the resulting loading of the bridge is the random train of moving loads. Since the mass of the road vehicle is usually small compared with the mass of the bridge span, the load on the bridge due to the vehicle may be idealized as the set of concentrated forces (axle pressures) or, if the length of a vehicle is very small compared with the length of the span, as a single concentrated force. The simplest model of occurrence of vehicles is the model of independent events. This is equivalent to the assumption that the events are Poisson-distributed and that the interarrival times (called also headways) are independent positive random variables with the negative exponential distribution and with the probability density function given by g(x) = ^exp(—vx), vexp(-vx), x > > 0. 0.

(3.43)

However such a model of a traffic flow has a substantial deficiency [34]; as the interarrival time tends to zero, the value of the probability density (3.43)

3.3. EXAMPLES

OF APPLICATIONS

59

increases and approaches the non-zero value, whereas under the actual traf­ fic conditions very small headways are associated with very low probabilities. Hence in the Poisson model the probabilities of short interarrival times are overestimated. Nevertheless, as the experimental studies indicate, the Pois­ son model is accurate enough under the light traffic conditions and when the vehicles can overtake freely [1, 110]. Thorough experimental analysis reveals that the interarrival times of ve­ hicles are indeed independent, but more often they are characterized by the distributions other than the negative exponential. Therefore a more adequate model of the vehicles occurrence is the renewal process [1, 34]. The reason for which the models other than a Poisson one are more suitable, is that traffic flow is not completely random, it is to some extent influenced by the regularity due to the interference between vehicles [1]. A class of renewal processes which is very often used in applications are Erlang processes, with interarrival times characterized by a gamma (or Pearson type III) probability distribution, with the density function given by (2.35). In that case the density function has the required property of passing through zero. By the suitable choice of parameters u and k this density function may fit different actual distributions (experimental histograms) [1, 34]. For exam­ ple in the reference [63] it is found that under the conditions of low velocities the gamma distribution with parameter k = 7 suits much better to the exper­ imental distribution (histogram) of headways than the negative exponential distribution. However under the actual traffic conditions the subsequent vehicle arrival is not at all possible during a certain time gap A, determined by the vehicle length augmented by a minimum spacing between the consecutive vehicles. The probability of the interarrival time being shorter than this time gap is equal to zero. It is found that in this case an appropriate probability distribution is the translated negative exponential distribution, with the density function [34] _ f 0, 0< x A ' x > A where A is of order of 1 sec. [1]. Under heavy traffic conditions the models of correlated interarrival times may be suitable [34]. Axle pressures or the vehicles weights should be regarded as random vari-

60

CHAPTER 3. LINEAR DYNAMICAL

SYSTEMS

ables. There exist some statistical studies on vehicles axle pressures [111]. Since the velocities of all the vehicles are not equal, the velocity of an arriving vehicle should be regarded as a positive random variable, i.e. the one assuming the values from the positive half-line, such as e.g. the gammadistributed random variable [1]. There can exist different relationships between the parameters describing the traffic flow. For example, as the velocity increases, the interarrival times decrease and at the same time the spacings between the vehicles increase. Consequently the mean arrival rate of vehicles is a function of their average velocity [110]. In the simplest model the stream of moving loads due to vehicles may be regarded as a random train of moving concentrated forces with random magnitudes, travelling with equal, constant speeds v. If the bridge structure is idealized as a beam with length /, bending stiffness EYI, mass and damping per unit length m and c, respectively, its behaviour is governed by the equation EYIutXXXX

+ 4- cutt + muitt -^2Pi6[x-v(t- ^2 PiS [x - v (t - ttt)], t)],

(3.45)

t=i

where u — u(x,t) is the beam deflection function, (.. .)iX = J | , (.. .),t = J^. In the equation (3.45) the force Pt is non-zero when 0 < v (t — U) < I and it is equal to zero when t < U or v (t — ti) > I. Normal mode expansion for the function u(x, t) (in the case of a simply sup­ ported beam with uniformly distributed mass and stiffness the normal modes are the sine functions: Wj(x) = sinjirx/l) leads to the equations for the modal responses qj(t) in the form of N(t)

qj(t) + 2Qnjqj(t)

+ tfjqj(t) = ft J2 PW

[v (t - ti)], U)],

(3.46)

»=i

where the excitation is a random train of general pulses with shape function Wj [v (t — t{)] and with finite, deterministic duration T — l/v. The pulse shape function is equal to zero when t < ti or t > U+l/v. In the simplest approach the forces magnitudes Pt can be regarded as independent, identically distributed random variables, distributed as a random variable P. Mean values and cross-covariances of modal responses can be evaluated based on the general formulae (3.19) and (3.20), and in the case of a Poisson

3.3. EXAMPLES OF APPLICATIONS

61

process the cross- (joint) cumulants of the modal responses can be evaluated based on the formula (3.21). Dynamic response of a beam to a Poisson-distributed train of moving con­ stant forces was considered in the paper [49]. In that paper the closed-form expressions are given for the beam deflection mean value and variance. The modal responses Zj(t) were not obtained from the formula (3.17), but directly from the differential equation of motion as its particular integral for the righthand side Wj [v (t — U)]. Taking into account the fact that the speeds of the vehicles can be different, assume that they are given by the random variables vt. Then the train of moving loads is expressed as N(t) x v p{x,t) p{x, t) = YlP*6 '£l\-Pi8[x-v ~ i(t~i(t-tU)}, i)],

(3.47)

i=l

where the random variables Vi assume values from the positive half-line. Upon the application of the normal mode expansion one obtains from (3.4) N{t) NW

( I \ Fj(t) F PJiW Y,PiW (t - U)J , j(t) = ^ £ (3 j [- j\^r(t-U)),

(3.48)

which is the random train of general pulses with random durations T, = l/v,, which are the times of passage of the loads over the bridge. Assume that the random variables Tj are mutually independent and have identical probability distributions as a common random variable T. Forces magnitudes are regarded, as before, as independent, identically distributed random variables. Assume also that the random variables T{ (hence the random speeds Vi) are statistically independent of the variables Pi. The latter assumption may not be obvious, seeing that the heavy cars (trucks) drive usually with different (lower) speeds than the light cars. This will be taken into account within a Markov renewal process model for a train of moving loads. Statistical moments of the response to random train of pulses with random durations may be obtained from the general formulae (2.44) and (2.46). How­ ever evaluation of the pertinent integrals requires the integration with respect to the probability density function griv) of the random variable TT (cf. e.g. '"* 1141). The following expectations must be evaluated

62

CHAPTER 3. LINEAR DYNAMICAL

SYSTEMS

T, IV)] TT)\ = = Jj Zj(t r, v)g E [Zj (t - r, v)9r(v)dv, Zj(t - T, T(v)dv,

(3.49)

E \ZJ [ZJ (t(t -- T, T, TTTT)) zzkk (t(t -- T, T, TTTT)\)\ == jj Zj(t Zj{t -- T, T, r))z rfizkk(t -- r,r, 77)^(77)^, r))gT{ri)dr),

(3.50)

where T is the sample space of the random variable T. Since the function Zj(t — r, r?) assumes two forms, depending on the relation between t — r and 77 (cf. (3.18)), i.e. it_TtV)

=

3

\^(t-r,r]) I Zj, zji22(t-T,ri), (t-T,ri),

t-Tr), t-T>T),

v

>

the integrals (3.49), (3.50) are split. In the case of a Poisson process, under the obvious assumption that the sample space of the variable T is the positive half-line, the result is t 00

E[qj(t)\

= E[P]J

J

zitl{t-T,rj)gT{ri)v{T)dr,dT

0 t-T t t-T

+E[P] J j zji2(t - r, V)g r))g T(r,)iy(T)d T(r,)v(T)dr,dT, VdT, 0

0

cov

«"W ) = E[qj(t)q E[qj(t)qkk(t)}-E[ ^ *«(*) (t)] -qj(t))E[q E fe(i)] E [qk{t)\ k(t)]

(3.53)

OO

t

=

(3.52)

E[P2} J J zjA{t - T, V)zK1(t - r, n)gT{ri)v{T)dr1dT 0 t-T t t-T

+E[P2} J j zjt2{t - r, r,)zk<2(t - r, r,)gT(r,)v(T)dr,dT. 0

0

For the delayed renewal process one obtains ([42]) E[P}E[z i{t - r= ) = E\P\E [Zj (tj(t-T,T - r, TT)), 1(t-T) T)\, 2 d> d>u(t -r) == E E[P [P2]] EE [[ZjZj (t(t -- r,r, TTTT)) zzkk (t(t -- r,r, TTTT)], )}, n(t -T)

(3.54) (3.55)

3.3. EXAMPLES OF APPLICATIONS

63

OO oo

t

E[ (t)} = = E[P]f E[P\J J E[qjqj(t)}

z^it-T^gTWh^dvdr zitl{t-T,r })gT{71)hm{T)d7]dr

0 tt-T -T t t-T

+E[P] J J zja(t - T,r,)g (r)dVVdT, dT, T, v)9T(v)h T(r,)hm m(r)d o0 n0 t

(3.56)

oo

2

E [qj{t)q [qj{t)qk(t)} k(t)} = E[P } J J 0

Zjil(t

- r, V)zk<1(t - r, 7i)g 77)ffrT(7 dr {ri)h (r)d 7)/im a!7V?dT m(r)

t-T t-T

t

2

+E[P } J J zji2(t - T, rj)zK2{t - r, T, V)gT(V)hm(T)dVdT 0

0 t

t

oo t—u

2

+2E [P] J J J j Ki(* - r, Vi)zk,2(t ~ «. %)h 0

T t-T

0

X9T(vi)9T(V2)hm(T)h0(u - r)dr] T)dr]22dr]idudT dr]idudT t

t

t-T

OO

2

T +2E [P] JJ JJ JJ JJ fei(* fei(* -- u, u, m)z V2)z T, T/ Vi)} 1 )}s5 k}2(t ~- > k,2(t

0

0 t-u t-U

T

dr]idudT X5T(?7i)flT(%)/im(T")/i0(w - T)dr)2dr)idudT t

t

oo

oo

2

+2E [P]J J J j {zjA(t - T,^)zk
0

X9T(Vi)9T(r)2)hm(T)h0(u t t-T

t

^d^d^dudr

t-U

+2E2[P]J J J j {zja{t - T,Vl)zki2(t 0

T

0

u,V2)}s

0

X9T(Vi)9T(V2)hm(T)h0(u - T)dr)2dr)idudT.

(3.57)

The general formulae (2.44) and (2.46) also allow to evaluate the mean value and the covariance function of the modal responses in the case of correlated moving forces - described by the general stochastic point process. The problem of a random train of moving forces with random speeds was considered e.g. in [40, 42, 114]. If the random axle pressures (or vehicle

64

CHAPTER 3. LINEAR DYNAMICAL

SYSTEMS

weights) of vehicles and their speeds are assumed to be correlated, the pertinent expressions can be generalized, which requires the integration with respect to joint bivariate density functions of these variables [40]. In a more adequate model of the above mentioned problem it should be taken into account that the probability distributions of random weights (or axle presssures) and of speeds of vehicles depend on the type of vehicle (for example slow trucks and fast cars). Then a Markov renewal (semi-Markov) process model is suitable and mathematically sound. Within the framework of such a model the probability distributions of interarrival times of vehicles may also be dependent on the types of the two consecutive vehicles. The moving load is expressed as N(t)

p(x, t) = = Y, (Si) 6(x-Vi 6(x-Vi (Si) (Si) (t - U)), p(x, Y,Pi^ (Si) U)),

(3.58)

where Pi (Si), Vi(Si) denote independent random variables, the distributions of which are associated with the states 5, = k, (k = 1,2,..., M) of the semiMarkov process. The number M of the states is equal to the assumed, within the model, number of types of vehicles. Probability densities gu(x) of the interarrival times between the vehicles of the same type i may be assumed different from the probability densities gtj(x) of the interarrival times between the vehicles of two different types i and j . This allows to take into account the effect of clustering of the vehicles of the same type (cf. [77]). Upon the application of the normal mode expansion one obtains N(t)

T **>Td$i))d$i))-

(3-59)

Next, the response moments may be obtained from (3.38). The Markov renewal process model may also be applied in the problem of a dynamic behaviour of a bridge under the conditions of a multi-lane highway traffic. Then the different possible transverse arrangements of vehicles are regarded as the states of a semi-Markov process. These are either the single arrivals at different lanes or the possible simultaneous arrivals at two or few lanes. Different probability distributions may be prescribed to the interarrival times of vehicles arriving according to the same transverse arrangement (e.g. on the lane i) and according to different transverse arrangements (e.g. on the

3.3. EXAMPLES OF APPLICATIONS

65

lanes i and j). The respective density functions are denoted by gu(x), and Probability distributions of random forces Pi and speeds Vi are also depen­ dent on the state Si of the semi-Markov process, i.e. Pi — Pt (Si), Vi — vt (Si). This corresponds to the situation where different vehicles are moving along dif­ ferent lanes, i.e. slow and heavy trucks at the right lane and light and fast cars at the left lane. Hence the probability distributions of random weights Pi (St) and speeds Vi (Si) of vehicles moving at the lane k are characterized by the conditional density functions gP (rj | Si = k) and gv(u | 5, = k), respectively. In the case of a multilane traffic, the bridge structure must be regarded as a plate. The governing equation of motion is D (uiXXXX + 2u<xxyy + li.yyys/) + cu,t + mu,tt — p(x, y, t, S), S),

(3.60)

where D — EYh2/12 (l - v*) is the plate bending stiffness. In the case of a multilane unidirectional traffic the time-spatial distribution of the excitation, dependent on the semi-Markov process 5 may be represented as N(t) n„

p(x, y,t,S) = J2J2

Pu (Si) 6 (x - vu v« (Si) (St) (t - U)) 68 (y - yi), Vl),

(3.61)

i=l 1=1

where np is the number of lanes and Pu(Si) and vu(Si) are the random variables, describing, respectively, the magnitude and the speed of a moving force arriving at the time U at the lane I, the transverse arrangement of the loading being given by the state Si. Forces arriving at different times at the lane I, provided that the corresponding state is St = k, have identical probability distributions characterized by the conditional density function 9r,(p,k) = gp,(p\Si = k). 9p,(p,k) k).

(3.62)

Likewise, the speeds of the vehicles arriving at different times at the lane I, provided that the corresponding state is Si = k, have identical probability distributions characterized by the conditional density function (v\S| iSi = k). 9vl(v,k) = gvVl,(ri

(3.63)

Depending on the state Si = k of the semi-Markov process, i.e. on the trans­ verse arrangement k of the loading, the densities (3.62) and (3.63) associated

66

CHAPTER 3. LINEAR DYNAMICAL

SYSTEMS

with the lane I are equal, accordingly, to the densities of the random forces and speeds, or to zero. For example, in the case of a three-lane traffic, the possible states Si — k (k = 1,2,..., 7) can be represented as 1/k I 1 2 3 4 5 6 7 1 1 0 0 1 1 0 1 2 0 1 0 1 0 1 1 3 0 0 1 0 1 1 1 If the probability densities of random forces and speeds on the lane / are denoted by gp^p) and gVi{v)> respectively, then the densities (3.62) and (3.63) can be represented as a matrix, which in the case of a three-lane traffic has the form gPl(p) ' gP2(p) gPi{p) (3.64) With the help of usual methods of structural dynamics, the governing par­ tial differential equation (3.60) can be converted into the set of independent ordinary differential equations, and next into the multivariate filtered Markov renewal process. The response moments are obtained from (3.38). 9p,(p,k)=

3.3.2

' 9P1(P) 0 0 gp2(p) 0 0

0 0 gP3(p)

gPl(p) gP2(p) 0

0 0 gP3(p)

0 gp2(p) gP3(p)

Dynamic response of shallow shells to Poissondistributed impulses

Shallow shells (or panels) are the typical structural elements. Interest for the vibration of thin shallow shells under continuous stochastic excitations, also in form of time-spatial random fields, arose from the needs of analysis of structural elements under the acoustic pressure. This problem was the subject of many papers e.g. [27, 28, 89, 90, 112, 136]. If the dynamic loading is discontinuous in time (e.g. distinct pulses of the acoustic or atmospheric pressure - see e.g. [66]), the random pulse train becomes a more adequate model of the loading. Vibration of shells under ran­ dom pulse trains was considered e.g. in [38, 39, 51]. An important problem which usually occurs in the stochastic analysis of shells, is the one concerned

3.3. EXAMPLES OF APPLICATIONS

67

Fig. 1. Shallow cylindrical shell. with the cross-correlations between the modal responses (or generalized coor­ dinates) [27, 38, 51]. Consider vibration of a shallow cylindrical shell with simply supported edges (Fig. 1) under the train of randomly occurring impulses with determin­ istic spatial shapes (spatial distribution of a pressure pulse), assuming that the underlying point process is a Poisson process. In accordance with Vlasov simplified (technical) theory, the deflection (radial displacement) u(a, /?, t) and the Airy stress function (a, /?, t) satisfy the equation [86] L[u] + cu cu,iff + mu « = F(t), where T_ U

~

(3.65)

[ l ^ V 2 V 2 -R(...),aa] U(a,/?,*)' [ R(...),aa £>V2V2 ] ' [u(a,p,t) 4 c — diag (0, aR ) , m = diag (0, mR4),

^l^fl^-J' ^ ^i^K/^E^W-'*)]' 2

(3 66)

'

and V = (.. .)
^a,/M) = X>;*(W*(a,/?),

68

CHAPTER 3. LINEAR DYNAMICAL u(a,p,t) J2Qjk(t)W u(a, P,t) = J2 Qik(t)W /?), jk(a,p), jk{a,

SYSTEMS (3.67)

3,k

where Wjk(a,p) — sin XjOe sin fj,kp is the eigenfunction (normal mode), A^ = JTrR/l,Hk = kirf3//30, and eliminating the coordinates Pjk(t), one obtains the uncoupled equations N(t)

qjk(t) + + 2Qknjkqjk(t)

+ il%qjk(t) = Pjk £

P 6

Pit U), , i (* ~- **)

(3.68)

t=i

where Qk = c/2mQjk is the damping ratio corresponding to the normal mode Wjk. The natural frequency is evaluated as

a -

1

1 XlEYh

i D w i u>?

In order to perform the detailed analysis, the shells with aspect ratios a/l = 1, a/l = 2 and with height-to-span ratios f/a = 1/5, f/a — 1/20, in each case, are assumed. The impulses are assumed to have equal, deterministic, uniform spatial shapes p(a, P) = po = const, hence the excitation participation factor is 2 Pjk = 16PO/TT 16po/n2 jkm. jkm. (3.70) If the attention is confined to the mid-point deflection (radial displace­ ment), it suffices to take into account the symmetrical normal modes only. The natural frequencies corresponding to the first 25 symmetrical mode shapes are listed in Table 1. A specific feature of the presented frequency spectra of cylindrical panels is that the values of the frequencies do not increase uniformly as the indices j , k of modes increase. The frequencies first decrease and then increase. The lowest, i.e. fundamental, frequency is in most cases fiii3, which corresponds to the mode TI7

1

o\

> \

a

nR

.

„

0

.

TtX .

37T/3

,„ _ , .

VK1)3(a, P) — s m A j a s i n ^ / ? = sin —sin37r— = sin-— sin——. Wit3{a,p) =smAia:sin^3/? = sin —I sin37r— —£-. (3.71) On = sin—-sin I On I Po I Po Hence, it follows that the contribution of the modal responses qjk(t) (or mode shapes Wjk) is due to two initially (i.e. for small values of indices i, j)

3.3. EXAMPLES OF APPLICATIONS

T a b l e 1. Natural frequencies of shallow cylindrical shells.

j 1 1 1 1 1 1 11 1 1 3 3 33 33 3 3 33 5 5 5 5 55 5 5 5 5 7 7 7 7 7 7 77 77 9 9 9 9 9 9 99 99

A; 1 3 5 7 99 1 3 5 7 9 1 3 5 7 9 1 3 5 7 9 11 3 5 7 99

a/l == 11 f/a = 1/20 f/a = 1/5 f/a Qjk fijfc 11 945.422 514.237 514.237 530.778 530.778 392.692 392.692 842.761 985.761 842.761 985.761 11 585.570 11 893.828 2 592.675 592.675 2 33 105.400 105.400 33 267.241 987.925 987.925 22 043.404 854.017 854.017 11 564.921 11 319.844 11 997.861 22 206.363 22 929.618 33 413.848 11 392.492 33 569.298 11 498.593 33 002.554 11 972.485 22 314.033 22 835.529 22 847.972 44 033.388 33 645.644 22 158.549 33 973.370 22 379.835 33 761.869 22 905.417 33 653.348 33 938.232 33 769.825 44 961.903 44 681.223 33 300.248 44 706.231 33 566.754 44 694.183 44 127.705 44 812.541 55 003.297 55 219.577 66 195.694 55 986.069

a/l a/7 == 2 ff/a/ a = 1/5 //a 1/20 f/a = 1/20 Sljk fijfe 832.756 22 943.978 832.756 11 317.842 585.426 585.426 11 104.507 11 109.073 109.073 11 728.241 22 010.303 22 714.683 714.683 220.838 33 220.838 33 739.829 11 726.579 33 377.125 11 903.210 33 129.109 22 403.812 403.812 33 370.044 33 265.185 44 459.294 44 126.726 55 225.353 44 001.736 001.736 55 227.742 44 279.379 55 479.336 44 851.550 851.550 55 944.573 55 732.719 732.719 66 735.032 66 926.574 926.574 88 334.629 77 624.502 88 517.508 77 920.050 88 905.285 88 514.102 99 529.213 99 410.378 410.378 10 10 415.011 415.011 10 10 611.369 12 12 953.699 953.699 12 12 509.613 509.613 13 13 177.954 177.954 12 12 809.710 809.710 13 13 632.436 632.436 13 13 410.639 410.639 14 14 326.643 326.643 14 14 313.501 313.501 15 15 270.232 270.232 15 15 519.307 519.307

69

70

CHAPTER 3. LINEAR DYNAMICAL

SYSTEMS

contradictory influences (or tendencies): decrease of the contributions of modal responses with decreasing participation coefficient /3jk and initial increase of these contributions with decreasing frequencies £ljk- Contribution of modal responses, i.e. their variances and cross-covariances in the stationary variance a\ of the mid-point radial displacement, normalized as

is given in Table 2. T a b l e 2. Contribution of variances and cross-covariances of modal responses in the variance of the stationary response (radial displacement) in the midpoint of shallow cylindrical shells.

2 a < qil l ff|M o^' < <5 COl cov qiiqi3 Vl<7l.3 Zi,k=itj,i=i,3Cov £i,fc=l;j,I Iki = :l,3COVqi. qijqkl Zi,k=ij,i=i,3,5Cov £i,fc=l;j,Z= ■ 1,3,5C0V>Hjlkl qijqkl

4, E s

C0

i,j,M=l C 0%i<7H ?j,jy=i %i?M

E

ff

S?' «7='=I
a/l = a/l =1 a/l a/l = 2 1/5 1/20 f/a = 1/5 f/a = 1/20 f/a = 1/5 1/5 f/a = 1/20 f/a 25.00 25.00 25.00 25.00 37.32 4.76 13.86 5.62 5.33 0.27 7.10 0.56 0.02 0.08 0.02 0.05 38.82 62.29 29.61 30.52 44.55 67.23 29.88 31.10 772.36 2 36

70.68 70.68

330.14 014

449.23 923

31.16 31.16

50.39 50.39

3 1 31.99 " 32.07 32.07

It appears that the solution obtained for the expansion in terms of the first 16 symmetrical normal modes (j, k = 1,3,5,7), i.e. ^ljki=icovqijqki IS o n ly slightly different from that obtained for the first 25 modes (j, k = 1,3,5,7,9), i.e. Sfj kti=\COVqijqkl. Hence the normal mode expansion convergence is suf­ ficiently fast to regard the solution for the 25 normal modes as sufficiently accurate. The contribution of the response g1|3 in the normal mode Wi)3, which cor­ responds to the lowest natural frequency filj3, is high and may even be pre­ dominant in the case of shells with large height-to-span ratio. The use of the

3.3. EXAMPLES OF APPLICATIONS

71

normal modes W\t\ and Wit3 yields good approximation for the shells with low height-to-span ratio. In the case of larger height-to-span ratio (f/a = 1/5) it is also necessary to take at least Wi$ into account. The contribution of crosscovariances of modal responses, as compared to the contribution of variances, is small. Consider now the non-stationary train of impulses characterized by the intensity function u{t)E[P22{t)] v(t)E[P {tj\ =Aexp(-ct). =Aexp{-ct).

(3.72)

In order to evaluate the variance of the midpoint deflection the first 16 symmetrical normal modes have been taken into account (i,j = 1,3,5,7). Time histories of the non-stationary variance, for the cylindrical panel with aspect ratio a/l = 1 and with height-to-span ratios f/a = 1/5 and f/a = 1/20 are shown in Fig. 2 and in Fig. 3, respectively. The contribution of responses in the modes Wi,i, W\$ and Wlj5 in the non-stationary variance of the deflection is similar as before. In this case, however, the contribution of cross covariances (the difference between the curves nos. 4 and 5) is important. Consider now the shallow spherical shell (panel) with simply supported edges (Fig. 4). The radial displacement (deflection) u(x,y,t) and the Airy stress function (j>(x,y,t) satisfy the general equations (3.65) [86], where 2 2

__ [[ ^i^v V2V v2 ~[ iV2

L L L

2

--±v I V 2 1] DV 2 V 2

^"[^ESWM]' 2

where V = (.. .)<xx + (.. .)*vThe normal mode is expressed as . JTra; j^x .

Wjk(x, y) = sin XjX sin /j, i*ky — sin

kity kity

sin -r—.

The normal mode expansion of the functions (x, y, t) and u(x, y, t) (cf. (3.67)) yields the equations in the form of (3.68). The natural frequencies are obtained from the formula

n = * ^ylDW+l®i+i£<±-7gf<»+*^W-

(3 74)

-

72

CHAPTER 3. LINEAR DYNAMICAL

SYSTEMS

Fig. 2. Variance of the radial displacement at the midpoint of the shallow cylindrical shell with f/a = 1/5.

Fig. 3. Variance of the radial displacement at the midpoint of the shallow cylindrical shell with f/a = 1/20.

3.3. EXAMPLES OF APPLICATIONS

73

Fig. 4. Shallow spherical shell. The numerical analysis has been performed for the shells with aspect ratios b/a = 1, b/a = 2 and with height-to-span ratios f/a — 1/5, f/a = 1/20, in each case. The impulses are assumed to have uniform spatial distribution p(x, y) = Po = const. In order to evaluate the variance of the midpoint deflection of the shell, the first 25 symmetrical normal modes (i.e. for j , k = 1,3,5,7,9) are taken into account. The respective natural frequencies are listed in Table 3. It is seen that the frequencies spectrum is dense, especially in the case of the shells with higher height-to-span ratios f/a. The frequencies spectrum is more dense in the case of rectangular panel b/a = 2 than in the case of a square panel b/a = 1. Contribution of variances and cross-covariances of modal responses in the stationary variance of the midpoint deflection is given in Table 4. Since the frequencies spectrum is dense the solution obtained for first few normal modes only, e.g. for Wjtk,j,k = 1,3, is for the shell with f/a = 1/5 far from being accurate. The contribution of cross-covariances is, in the case of the shells with f/a = 1/5 very high. Especially important is the contribution of cov91 iqi 3 for (f/a = 1/5). Of course, the contribution of modal responses depends on the spatial dis­ tribution of the excitation (of an impulse) and is different in different points of the shell. In all points other than the midpoint the antisymmetrical normal modes also have to be taken into account. The contribution of modal responses and of their cross-covariances is dif­ ferent if the excitation has a stochastic spatial distribution. For example, if this distribution is given by a spatial white noise process (a delta-correlated spatial random function), the excitation participation factors /?,*, are equal for all the modes. Consequently the contribution of modal responses decreases, as the respective natural frequncies increase, much slower and much more modes

74

CHAPTER 3. LINEAR DYNAMICAL SYSTEMS

T a b l e 3. Natural frequencies of shallow spherical shells.

b/a == 22 b/a == 1 f/a = 1/20 f/a f/a = 1/5 f/a f/a = 1/20 f/a = 1/5 f/a j3 11 1 1 1 1 1 1 1 1 3 3 3 3 3 3 3 3 3 3 55 5 5 5 5 5 5 55 7 7 7 7 7 7 77 7 7 9 9 9 9 9 9 9 9 9 9

Jfc

k

1 3 5 7 9 1 3 5 7 9 1 3 5 7 9 1 3 5 7 9 11 33 5 7 99

Qjk tijk 1904.913 1941.652 2148.937 2702.406 3677.099 1941.652 2024.736 2307.494 2928.289 3942.922 2148.937 2307.494 2702.406 3418.172 4490.331 2702.406 2928.289 2928.289 3418.172 3418.172 4214.337 5338.625 3677.099 3677.099 3942.922 3942.922 4490.331 4490.331 5338.625 5338.625 6500.462 6500.462

Hjfc ^jfc

517.078 639.288 1120.984 1985.365 3187.433 639.288 859.322 1401.143 2283.320 3490.746 1120.984 1401.143 1985.365 2884.888 4098.993 1985.365 2283.320 2884.888 3794.643 5013.940 3187.433 3187.433 3490.746 3490.746 4098.993 4098.993 5013.940 5013.940 6236.560 6236.560

Qjk Cljk

798.077 806.335 843.803 945.023 1139.896 872.113 906.056 988.421 988.421 1139.896 1376.255 1254.260 1314.434 1439.515 1636.223 1910.183 2050.669 2121.387 2264.561 2481.374 2773.818 3217.552 3217.552 3291.953 3291.953 3441.062 3441.062 3665.402 3665.402 3966.567 3966.567

Qjk fijfc

210.669 240.062 345.626 548.200 840.724 409.922 477.903 620.025 840.724 1140.849 990.264 1065.453 1216.411 1443.842 1748.214 1900.710 1977.014 2129.716 2358.956 2664.868 3124.115 3124.115 3200.688 3200.688 3353.858 3353.858 3583.660 3583.660 3890.138 3890.138

3.4. RESPONSE

HIGHER-ORDER

CHARACTERISTICS

75

T a b l e 4. Contribution of variances and cross-covariances of modal responses in the variance of the stationary response (radial displacement) in the midpoint of shallow spherical shells. b/a == 1l 2 a <,2qil a < n3, a\\ <>fi cov cov qiiqi3 qi.iqi.3 sS COT , Ly,r=i l,fc,Z,r=l CO«,*?!r ' V«lr %k,l,r=icoviik
= 22 b/a = f/a = 1/20 25.00 25.00 2.72 2.14 0.37 0.89 6.51 0.15 116.67 6 67 227.64 7 64 28.20 18.75 18.75 28.20 32.28 28.48

= 1/5 f/a == 1/5 1/5 f/a == 1/20 1/20 f/a = 1/5 f/a 25.00 2.67 0.79 4.24 118.91 8 91 223.65 3 65 33.12

25.00 1.82 0.21 0.05 32.19 32.19 33.26 33.26 29.30

must be taken into account in the normal mode expansion. It is interesting that in such a case the cross-covariances vanish [39].

3.4

Estimation of higher-order characteristics of the response probability distribution

3.4.1

Probability density of the response of a linear oscillator to a Poisson-distributed pulse train

Probability density of the response of a harmonic oscillator will be analysed approximately, with the help of a Gram-Charlier expansion (2.89). Prom the central limit theorem it follows that, for E[P] = 0, if v —> oo and the impulses magnitudes tend to zero in such a way that vE[P2) — const, the Poissondistributed train of Dirac delta impulses approaches the Gaussian white noise with intensity I — i/E [P2]. Hence the response of a linear system to a Poissondistributed train of impulses approaches the Gaussian process as well. This can be shown e.g. with the help of a Gram-Charlier expansion. If v —> oo and vE [P2] = const, it means that E [P2] tends to zero as i/ _1 , hence all the mo­ ments E [PT] ,r>2, tend to zero faster than v~l. Consequently vE [P r ] -> 0

76

CHAPTER 3. LINEAR DYNAMICAL

SYSTEMS

for r > 2, which means that the cumulants of order r > 2 vanish (cf. (2.57) and (3.14) or (3.21)), and accordingly the coefficients Ar vanish as well (cf. (2.91)). Then the probability distribution becomes Gaussian. It is obvious that for finite v this distribution deviates from the Gaussian one. Consider a harmonic oscillator subjected to a Poisson-distributed train of square pulses. The governing equation of motion is N(t) 22

q(t) + 2Cnq(t) q(t) 2(Qq(t)++QQq(t) q(t)= =£ ] £P{Ps {s(t-titT),(t-titT),

(3.75)

»=i

where

s({ T); = i1' s(t-t-T)

T

^ < ^ + \ 0, ttifLVkV+r < U or t > U + T '-'"" ={J:

<3-76>

and N(t) is a counting process. The assumptions concerning the random variables are the same as in the sec. 2.2.1. Let us confine our attention to the stationary probability distribution, i.e. for t —> oo and for the homogeneous Poisson process (v = const). It should be noted that the normalized cumulants Xj are expressed in terms of the factors ( i ) j / 2 _ 1 , i = 3 , 4 , 5 , . . . , (cf. (2.91) and (2.57) or (3.21)), hence the GramCharlier expansion may be regarded as a power series of a parameter (-) 1 / 2 . If the terms of the expansion (2.89) are collected according to the powers of this parameter, the Edgeworth series is obtained. Results of some studies on the application of the Gram-Charlier expansion in the problem of response to random impulses are reported e.g. in [54]. It is found that if the mean arrival rate v decreases, the contribution of nonGaussian terms of the expansion (and the deviation of the probability dis­ tribution from the Gaussian behaviour) increases. Then the convergence of the expansion gets much worse. As a practical index of the deviation from Gaussianity the following quantity has been proposed

66 == (.2CQ/U)wn/v)-11..

(3.77)

The numerical investigations on the convergence of the Gram-Charlier ex­ pansion have revealed that 6 = 1 is certain limit value, i.e. if 6 < 1 the deviation is large, whereas if S > 1, it is small. Retaining in the expansion (2.89) the terms of order (£) and (-)1 one obtains the following truncated expansion for the stationary probability density

3.4. RESPONSE HIGHER-ORDER CHARACTERISTICS

2 /(€) = = tffl l + ^ H i ( 0 ++| §#4(o ^ 4 ( 0 ++~ ~ ( § (|f) ) ' H . «^(o| )| ■ ■ /(o m |ji+^Hi(fl

77

(3.78)

As a fundamental, rough measure of the departure of the examined prob­ ability distribution from the Gaussian behaviour, the skewness A3 and the excess A4 coefficients may be assumed. Let us investigate the behaviour of these coefficients depending on the square pulse duration T. Let us introduce the normalized coefficients K,3/a3 and K4/GT*, normalized with respect to the coefficients for the Dirac delta impulses, through = A33

4

«3 _ 16Cy/f K3 lecvc / nn E[I»] E[P3} kk3 3 2 l2 aa\ 3(l )V^{£[P'] 3/2<73' 3(1 ++ 8C 8C )V v{E\P2}\? ~°¥

^'^

4 _ /«c44 __ 3C 3C Q O £[P E[P4} ] A /cC4 2 2 2 4 ~ a\ ~ 2(l + 3C )^{E[P ]} d '

In the limit, as T — ► 0, the coefficients A3 and A4 for Dirac delta impulses are obtained. The normalized skewness R3/a3 and excess /c4/cr4 coefficients are shown in Figs. 5 and 6. Both coefficients are positive and attain maxima for QT = 2ri7r, n = 1,2,... and minima for fiT = (2n + l)7r, n = 1,2, If fiT = 2nn, the pulse time duration is equal to the multiple natural period of the oscillator, the system is only slightly excited, and the amplitudes of vibration are very small. Then the pulse effect is quasi-static. If, on the other hand, QT — (2n+l)7r, the vibration amplitudes are large, the dynamic effect is strong. The behaviour of skewness and excess coefficients allows to predict that in the case of a quasi-static effect of pulses the departure of the probability distribution of the cumulative response from the Gaussian behaviour will be larger than in the case of a strongly dynamic effect. In the numerical analysis the random variable P has been assumed to be Gaussian-distributed, with parameters mp,ap such that ap/mp — 1, E[P3]/(E[P2})3/2 = \/2, and E [P4] / (E [P2])2 = 2.5. If the pulses mag­ nitudes are assumed to be Rayleigh-distributed, similar values are obtained, i.e. E [P3] I (E [P2})3/2 = 0 . 7 5 ^ « 1.329 and E [P4] / (E [P2])2 = 2. The damping ratio has been assumed £ = 0.01.

78

CHAPTER 3. LINEAR DYNAMICAL

Fig. 5. Skewness coefficient of the oscillator stationary response to a Poissonian train of square pulses.

SYSTEMS

Fig. 6. Excess coefficient of the oscillator stationary response to a Poissonian train of square pulses.

It is especially interesting to investigate the influence of a mean arrival rate v of pulses and of the pulse duration T on the shape of probability density curves. Probability density curves obtained for different values Q./v and QT with the help of a truncated expansion (3.78) are shown in Figs. 7, 8 and 9. The contribution of consecutive (individual) terms of expansion is shown in Figs. 10 and 11. The density curves are characterized by positive skewness (asym­ metry), i.e. the probability of small negative displacements (i.e. negative as compared to the mean value) is larger than the probability of small posi­ tive displacements. Accordingly, probability of large positive displacements is larger than the probability of large negative displacements. In other words, the probability density tends to zero more slowly when £ —► oo than when £ —* —oo, i.e. the "right-hand tail" of the density curve decays more slowly and is "thicker". The positive skewness of the density curve increases as the pulse duration T increases from 0 to 2ir/Q (Figs. 7, 8) and as Sl/v increases i.e. as the mean arrival rate v decreases (Fig. 9). Contribution of each of the non-Gaussian terms of expansion (3.78) is im­ portant (Figs. 10,11). However the density curves obtained from the truncated expansion are negative in some intervals. By including into the truncated ex­ pansion the subsequent terms of (2.89) the values of the density functions in

3.4. RESPONSE

HIGHER-ORDER

CHARACTERISTICS

79

Fig. 7. Approximate probability density function for the oscillator re­ sponse to a Poissonian train of square pulses, for different pulse durations T, with 0,/v = 2. these intervals could again be positive, but the negative values could appear at larger values of £. Such a behaviour of the approximate probability density curves being a deficiency of the application of the Gram-Charlier expansion, has also been observed in [98]. Like in the case of Dirac delta impulses, also in the case of a train of general pulses, it can be shown by virtue of a central limit theorem that the response of a linear system is asymptotically, as v —> oo, the Gaussian-distributed process. Hence, as v decreases, the departure of the response probability density from a Gaussian behaviour, increases. This is supported by the results shown in the Fig. 9; the case Q./v = 0 corresponds to the asymptotic Gaussian case. The departure of the response probability density from a Gaussian be­ haviour is the largest when the duration of the square pulse is equal to the natural period of the oscillator T = 27r/fi. Since the effect of each pulse is then

80

CHAPTER 3. LINEAR DYNAMICAL

SYSTEMS

Fig. 8. Approximate probability density function for the oscillator re­ sponse to a Poissonian train of square pulses, for different pulse durations T, with 0,/v = 5. quasi-static, with very small natural vibration, hence the cumulative response at the time t is mainly due to the pulses which at that time are still acting on the system, i.e. which have initiated their operation in the time interval (t — T, t). In accordance with the Poisson distribution (2.25) the probability of occurrence of a large number of events (pulses) in a short time interval of a length T = 27r/fi is very small, especially if u is small. This means that the response is effectively due to a relatively small number of pulses, hence by virtue of a central limit theorem the departure of the probability distribution from a Gaussian one is large. The dynamic behaviour of the system is different when the pulse dura­ tion is short, e.g. T = 7r/fi, in which case the effect of a pulse is strongly

3.4. RESPONSE HIGHER-ORDER CHARACTERISTICS

81

Fig. 9. Approximate probability density function for the oscillator response to a Poissonian train of square pulses, for different v, with T = 2II/fi.

dynamic. The response is then due to the large number of past pulses, the length of the time interval in which these pulses occurred is, especially with the light damping, large - equal to the multiple natural period. The probabil­ ity of occurrence of a large number of pulses in such a time interval is large. Accordingly, probability distribution is close to a Gaussian one. Likewise it may be shown that the departure of the probability distribution from a Gaussian behaviour increases as the damping ratio £ increases.

82

CHAPTER 3. LINEAR DYNAMICAL

SYSTEMS

Fig. 10. Contribution of consecutive terms of the Gram-Charlier expan­ sion in the approximate probability density function, for Q./v = 2.

3.4.2

Mean upcrossings rate for a linear oscillator sub­ jected to a Poisson-distributed pulse train

Consider again a linear oscillator subjected to a Poisson-distributed train of square pulses, with the governing equation of motion (3.75). Let us make the same as in the sec. 3.2.1 assumptions about the random variables Pi. The state vector of the system, made up of the displacement response q(t) = Xi(t) and of the velocity response q(t) = X2(t), is a bivariate filtered Poisson process. In order to evaluate approximately the joint, bivariate probability density of the processes q(t) and q(t) the generalized, bivariate Gram-Charlier expansion (2.102) will be used. From the formulae (2.104) and (3.14) or (3.21) it follows that the coefficients Xjk of this expansion involve the factor ( I ) ^ + f c / 2 ) - 1 , j+fc = 3 , 4 , 5 , . . . , hence the expansion (2.102) is an expansion in the powers of a

3.4. RESPONSE

HIGHER-ORDER

CHARACTERISTICS

83

Fig. 11. Contribution of consecutive terms of the Gram-Charlier expan­ sion in the approximate probability density function, for Q/u = 5.

parameter (^) . Arranging the terms according to these powers yield the generalized Edgeworth series. If the attention is confined to the stationary distribution, as t —* oo, the response moments and cumulants are constants. Accordingly, the moments, which are expressed as the time derivatives of some other moments, vanish, hence K 0 I = «n — «2i = « 31 = 0. Since p — \ n = 0, the bivariate analogue of a Hermite polynomial splits into a product

#;*(&, 66,, <>)=#,■ #;*(&, <>)=#,■((6) 6 ) HkH fa).kfa).

(3.81)

If the terms of order (-) and (£) are only retained, the bivariate GramCharlier expansion for the stationary probability density / ( £ i , 6 ) becomes

84

CHAPTER 3. LINEAR DYNAMICAL

/(6,6) = 0(6,6) (l+ (1+

£

SYSTEMS

7jt#i(6W6) |g^«i)ft(6)

j',Jfc,(,m;j+fc=3,/+m=3 j,M,m;j+fc=3,Z+m=3 ^

)'

The mean upcrossings rate /z+ of a level g = X\ — a by the process q(t) — Xi(t) is expressed by the formula [95] oo CO Ma Ma ==

/ X2f(,CL,X , 2, x2f(a,x 2)dx22)dx

(3.83)

0

where f(xi,x2) is the joint probability density of the process X(t) = X\(t) and its time derivative X(t) — X2(t). Upon using in (3.83) the truncated expansion (3.82), and upon introducing the standardized variables, the following expression for the mean upcrossings rate of the level y is arrived at

ti

=

2 + 3X12Hl(y)] ^^^exp(-y - t / ^ e/2)il x p (-1^/2) f 1 + + i\[\[A3offs(») + 3^2^(2/)] 3QH3(y)

+

— [\4oH [A40i/4t(y) (2/) + + 6X22H2{y) - A04]

+

\x\2H2{y) + + ^X ±X30 X12 Hi4(y) (y) ^^loH S o 6^(y) e t o ) - jA^ffaGO 30X 12H

where 2/=

+ ^^A + Agg33||,, (3.84)

a — m, aa — - ^«io= a-rn L ;— = , V«20

01

which means that y is a relative level (barrier), measured with respect to the mean value mi, in terms of standard deviations o\. It is interesting to investigate the dependence of the mean upcrossings rate on the mean arrival rate v of the pulses, as well as on the pulse duration T. The mean upcrossings rate of the levels y = 0, y = 1 and y — 3, is presented in Figs. 12, 13 and 14, respectively, against the pulse duration T

3.4. RESPONSE HIGHER-ORDER CHARACTERISTICS

85

Fig. 12. Mean upciossing rate of the level y = 0 for the response of the oscillator to a Poissonian train of square pulses. and for £l/v = 0 (solid line), 0,/u = 2 (dotted line), Q/v = 5 (broken line). The value Sl/v = 0 corresponds to an asymptotic Gaussian case, when v —+ oo. As it is seen in Fig. 12, the mean upcrossing rate ji+ of the level y — 0, which is the mean response level (a = mi), assumes very much pronounced minima (very deep troughs) for QT — 2irn, i.e. when the square pulses have the quasi-static effect with small, residual oscillations about the zero displacements level. In that case the response is characterized by relatively high mean value and by the small oscillations, which are not about the mean value. Therefore the mean upcrossings rate is small. The pulses with other durations, espe­ cially with durations close to T = ^(2n + 1), induce the strongly oscillatory response characterized by a small mean value. The oscillations are performed about the zero displacements level, but since the mean value is small the mean upcrossings rate of the level a = rti\ is high. This observation is valid for all the considered values Sl/u — 0,2,5. Similar behaviour is observed in the case of upcrossings of the level y = 1, i.e. a = mi + o\ (Fig. 13). In the case of a quasi-static response, i.e. when QT = 2irn, the response mean value mi is high enough, and due to

86

CHAPTER 3. LINEAR DYNAMICAL

SYSTEMS

Fig. 13. Mean upcrossing rate of the level y — 1.

an unoscillatory character of the response the dispersion (the scatter) of the possible states of the system and hence the variance (or standard deviation (jj) is small. The level y = 1 is close to the mean value, but the oscillations are not performed about the mean value and the crossings of the level are not frequent. However if the response is strongly oscillatory, the response mean value is very small and though the response standard deviation o\ is larger than in the quasi-static case (hence the level a = mi + &i is quite distant from the mean response level) the mean upcrossings rate is high. As £L/v increases the mean upcrossings rate fi+ of a low level decreases, which is in accordance with the positive skewness of the probability density curve. The behaviour of the mean upcrossings rate of a level y = 3, i.e. a = m\ + 3CT! (Fig. 14) is essentially different. In the asymptotic Gaussian case Q/v = 0 the mean upcrossings rate is in the case of a quasi-static response only slightly smaller than in the case of a strongly oscillatory response. In the former case the low rate is due to the fact of quasi-static response behaviour, whereas in the latter case it is due to the fact that the level a = mi + 3ax is distant from the mean value level. As Q/v increases the mean upcrossings rate increases as well, especially for the pulse durations close to ClT = 2isn. This may be attributed to the increase of the skewness of the probability density curve, i.e. higher values of the probability density for large £ = {x\ — TOI)/<TI.

3.4. RESPONSE

HIGHER-ORDER

CHARACTERISTICS

87

Fig. 14. Mean upcrossing rate of the level y = 3. Mean upcrossings rate of a positive level y = 1 is lower than of a symmet­ rical negative level y = — 1 (Fig. 15). On the other hand, if the levels are more distant (y = 3 and y = —3), as shown in Fig. 16, the rate is, in the case of a positive level, higher. This observation is in agreement with the positive skewness of the response probability distribution. Small positive values of the displacement response are associated with smaller probability than small neg­ ative values, on the other hand the large positive values are associated with greater probability than the large negative values. The appearance of the negative values of /i+ for y — — 3 (Fig. 16) means that the truncated expansion (3.82) is not a satisfactory approximation to the actual probability density function. This means also that the results for y — 3 presented in Figs. 14 and 16 may be influenced by a substantial error. Nevertheless the qualitative interpretation of the results should remain the same. Mean upcrossings rate may be used to estimate the reliability of the system. Let us define the reliability R(t) as the probability of no upcrossing in the interval (0,t\. If the upcrossings are regarded as the independent, Poisson events, N+(t) being the corresponding counting process with intensity /z+(£), the reliability may be evaluated as (cf. (2.22))

88

CHAPTER 3. LINEAR DYNAMICAL SYSTEMS

Fig. 15. Comparison of mean upcrossing rates for the levels y = 1 and y = — 1 with Q/i> = 5. Solid line for 0,/v = 0.

Fig. 16. Comparison of mean upcrossing rates for the levels y = 3 and y = - 3 with 9,/u - 5. Solid line for 0,/v = 0.

3.4. RESPONSE HIGHER-ORDER

CHARACTERISTICS

R(t) = Pr {N i - j f i^{r)dr p+{r)dr > \ . R{t) = {N+ + = 0} = exp I

89

(3.86)

However the upcrossings are only asymptotically independent, as y —» oo. For a highly reliable system, i.e. if R(t) is slightly less than 1, the following estimate is valid R(t) = l-E[Nl-E[N +(t)},+(t)},

(3.87)

where t

E[N J^(r)dr. E[N+(t)] jfi+(T)dT. +(t))= =

(3.88)

oo

More precisely, the general point process of upcrossings should be ade­ quately characterized, with the help of correlations of upcrossings rates at different times, i.e. by higher order product densities of the underlying point process. Then the probability of no upcrossings may be evaluated more exactly [71, 145]. Otherwise, the upcrossings by the response envelope process instead of the response process itself may be regarded as independent events. Then the probability of no upcrossing by the envelope process can be evaluated. It should be noted that if the Dirac delta impulses are considered, or the general pulses with shape which can be represented as the response of an auxiliary filter to a random train of impulses, there are other possibilities to evaluate the reliability function. One possibility is to evaluate it from the integro-differential equation, a backward form of the Kolmogorov-Feller equation (an analogue of the backward Kolmogorov equation), cf. [61]. Al­ ternatively, the reliability function can be obtained with the help of a path integration (cell-to-cell mapping) technique applied to the integro-differential equation (an analogue of the forward Fokker-Planck-Kolmogorov equation) [62].

Chapter 4 Non-linear dynamical systems under r a n d o m pulse trains 4.1

Introductory remarks

Linear models of dynamic behaviour of structures are usually adequate in the situations of small vibrations, where amplitudes of vibration, or more gener­ ally the displacements, are small. The dynamical system can then be regarded in the first approximation as a linear. As the loading and the resulting dis­ placement response increase, the non-linear effects also increase and the pre­ dictions based on a linear model become quantitatively inaccurate and even qualitatively inadequate. Hence the need of constructing non-linear models of structural behaviour arises. Herein the discrete non-linear models of dynamical systems are considered, i.e. the ones described by non-linear ordinary differential equations. Such models can be constructed by substituting the actual structure with the sys­ tem of concentrated masses or rigid bodies (lumps) connected by the springs and dampers (spring and damping constraints). For example the dynamic models of vehicles can be constructed in such a way: the mass of the vehicle is substituted with the concentrated mass or as a rigid body and the suspension system is represented by the springs and the dampers. The wheels can also be represented by sprung masses. The generalized coordinates of the system are then the displacements of the concentrated masses or rigid bodies. Otherwise the discretization can be effectuated with the help of approximate methods, 91

92

CHAPTER 4. NON-LINEAR

DYNAMICAL

SYSTEMS

such as Ritz or Galerkin method, or via finite elements or boundary elements methods. In the case of approximate methods the expansion coefficients play the role of the generalized coordinates. It should be noted that the question of the convergence of the expansion and of the accuracy of the approximate solution is much more difficult for non-linear systems than for the linear ones. Therefore the conclusions based on the discrete, especially one degree of free­ dom, models must be formulated with care. Finally the adequatness of the assumed model can be assessed via the experimental verification of the ana­ lytical predictions. Two types of non-linear systems are considered herein. The first one are the systems in which the restoring force is the non-linear function of the displace­ ment, as it is in the case of geometrical non-linearities (e.g. large deflections of an elastic system) or physical non-linearities (when due to the high stress level the non-linear stress-strain relationship must be taken into account). In these cases, especially in the case of geometrical non-linearities, the non-linear functions are of power (polynomial) form in terms of displacements. Therefore this form of the non-linearity is analyzed in more detail. The technique devel­ oped is, however, equally suitable if the damping is non-linear and is expressed as the polynomial form in the generalized velocities. The second type are the non-linear inelastic systems in which the plastic deformations occur and stress-strain relationship is of hysteretic nature. Hence the relationship between the restoring force and the displacement is hysteretic as well. Such a behaviour must usually be taken into account if the heavy dynamic loads are considered. The Bouc-Wen differential model of hysteresis is assumed [11, 145], in which the hysteresis loop is governed by a differential equation. This equation is coupled with the equation of motion of the system. In this case the non-linearity is given by the non-algebraic, non-analytical function of the state variables. Moreover it is assumed that the inertia forces do not depend explicitly on the displacements and velocities (the kinetic energy depends explicitly only on the generalized velocities and not on the generalized coordinates). Then the equations of motion have the form of Mii(t) + D (ii(t), u(*)) = QF(t), Mu(t) D(u(t),u(*)) QF(t),

(4.1)

where M is a constant matrix and D is the function of instantaneous values of the processes u(t) and (u(t) at the time t. Then with the help of standard

4.2. STOCHASTIC EQUATIONS FOR POISSON IMPULSES

93

techniques the equations (4.1) can be recast into the form q(*) + g(q(«).q(*)) = bF(t), q(t)

(4.2)

and next into the following state-vector form X(f) = a(X(f)) + cF(i),

(4.3)

where X(t) = {&(£), qi(t), i = 1,2,..., nd} is the vector of state variables and rid is the number of dynamical degrees of freedom.

4.2

Statement of the problem: stochastic equations and equations for moments for the systems subjected to a Poisson-distributed train of impulses

Consider a multi-degree-of-freedom non-linear dynamical system subjected to a Poisson-distributed train of impulses. The system state variables {Xi(t),i = 1,2,..., n s }, where ns is the number of state variables, are governed by the equation X(i) = a (X(t)) + « (-**-,*-,))■. + cc£] /£*W

(4.4)

i=l

The initial conditions can be, in general, non-zero and random, given by the vector random variable X(0) = X„. X0.

(4.5)

Assumptions about the train of impulses, i.e. about the counting process N(t) and about the random variables Pi are the same as in the sec. 2.2.2. It is assumed that the compound Poisson process characterized by the counting process N(t) and the random variables Pi is statistically independent of the initial conditions, i.e. of the random vector X 0 . Integration of the governing equation (4.4) over the time leads to the fol­ lowing equation:

94

CHAPTER 4. NON-LINEAR

ft

X(t) = = X 0++

DYNAMICAL

SYSTEMS

*<*)

/ a(X(T))dT a ( X ( T ) )
(4-6)

where the last term (the excitation term), is just the compound Poisson process [it exists if N(t) > 1, otherwise, i.e. if N(t) — 0 it vanishes], and l{t~tt) is the unit step (Heaviside) function. The last term can be substituted by a two-fold integral [cf. (2.78)], specifying b* (t, X (U), Pi) = cP; and b (r, X ( r ) , p) = cp. Finally we obtain an integral equation in the form of (2.75) X(t) = X(t) = X 0 + f a (X(r)) dr + c f f pU(dr, dp) Jo Jo Jo Jo Jv Jv or an equivalent integro-differential form (2.79)

(4.7)

dX(t) = a (X(«)) dt + + c /I Ppn(rfr, n(dr, dp). (4.8) Jv Jv Hence the state vector X(t) of the dynamical system subjected to a Poissondistributed train of impulses is, provided that the impulse process is statisti­ cally independent of the initial conditions, a non-diffusive Markov process, the so-called Poisson-driven Markov process. In what follows let us assume, for the sake of simplicity, the zero initial conditions. Averaging of the stochastic equations (4.8), taking into account the proper­ ties of the Poisson random measure (2.77) leads to the equations for the mean values E \Xi{t)] = m,i(t) in the form of mi(t) = jhiit) = E[a £ [oj (X(*))] + + v(t)E[P]c i/(t)^[P]ci, i(X(t))\ i,

, 2 , 3 , . . . , n, ns .s . *z == l1,2,3,...

(4.9)

Equations for the centralized state variables (zero-mean response processes) Yi(t) — Xi(t) — mi(t) are obtained as dY(t) = rfY(t) = {a 0 (Y(t) (Y(«) + m(t)) - v(t)E[P]c} i/(t)E[P]c} dt + + c f/ PpU(dt, n(
(4.10)

(4.11)

4.2. STOCHASTIC EQUATIONS FOR POISSON IMPULSES

95

Let us denote the n-th order joint central moments of the state variables as n

lHr..i {t) = E Y[Yiq{t) lk v..nin(t)

, i, = l,2,...,n,. l,2,...,n,.

(4.12)

-9=1

Equations for moments are obtained from the differential rule (2.83) for the function V(Y) = JIILi Yiq(t). After performing the expectation the result is

E[V(Y)]= ^={ g[ ^E(JY^ Y ffttE[V(Y)] + + mm ) -) -^ ^^) ]^ ^ ^ ]^ ^}^ } +!/(*) / E £ [V(Y [V(Y + pc) - V(Y)] V(Y)} J>(p)dp. fP(p)dp. (4.13) Jv Jv In particular the equations for the joint second-, third- and fourth-order moments, which are denoted, respectively, as fUj = E \YiYj\, Hijk = E [YiYjYk], fcjki = E [YYjYkYi], are obtained in the form of

iiijkl(t)

im(t) == 22{E iHj{t) {E [Y [Yiaia°(Y)] °(Y)] }}$s ++ v{t) v{t) [P [P22]]acj, CiCj,

(4.14)

fiijk{t) ji (t) = 3 {E [YiYjaKY)] } s + v{t) v(t) [P3] cciCj iCjcckk,,

(4.15)

= 4{E[YYiYka°(Y)]}s + 4 +v(t)E [P ] acjCkCt. +v(t)E CiCjCkci.

6{ciCjm}su(t)E[P2] (4.16)

The differential equations (4.9) and (4.14-4.16) are associated with the zero initial conditions. Equations for moments involve unknown expectations of non-linear func­ tions of the state variables. Forms of these functions depend on the forms of non-linear functions a;(X) in the governing equations of motion (4.4). Equa­ tions for moments only form a closed set if aj(X) is a linear form in the state variables X. When this is a polynomial of degree r > 1, the equations for moments form an infinite hierarchy, i.e. the equations for the moments of up to ra-th order involve the moments up to and including the n + r — 1 or­ der. If aj(X) is of form other than a polynomial in the state variables, then

96

CHAPTER 4. NON-LINEAR

DYNAMICAL

SYSTEMS

the pertinent expectations appearing in the equations for moments cannot be expressed explicitly in terms of moments. They must be evaluated by perform­ ing the integrals with respect to the probability density function. The exact density function is, however, unknown and the approximate, tentative, den­ sity function must be assumed. If the Gaussian density function is used such a technique is called a Gaussian closure. A non-Gaussian tentative density function is often assumed in form of an asymptotic, Gram-Charlier expansion. The accuracy of the results obtained, i.e. of approximate response moments depends on how close to the exact one the assumed tentative density function is. Therefore some modifications of the tentative density function have been proposed in various problems [47, 80],

4.3

Moment equations technique for the sys­ tems with polynomial non-linearity, sub­ jected to a Poisson-distributed train of impulses

4.3.1

E q u a t i o n s for m o m e n t s a n d c l o s u r e a p p r o x i m a ­ tions

In this section let us consider the systems in which the non-linear stiffness (restoring force) and/or the non-linear damping are given by polynomials in the state variables. Let us study in detail the case of the 3d-order polynomial i.e. the system with cubic stiffness a

flj(X) Xnn ++ dimnpX XnXXmpX, nXp, i(-"-) = — bi OimmXXmm ++ Ci Cimn dimmnp mnXmX

(4-17)

where the summation convention is applied to the repeating indices. After introducing the centralized state variables Yt = X{ - m-i the central­ ized drift term is obtained as a°(Y + m) = Ai + BimYm + CimnYmYn + Dimnp Dimnp YmYYmnYnpY , p, where J A-i CimnUmn **i =— ~Cimnf 'mn

(•'imnpl^mnp ~ •JUimnp "J"«mnj v M m n ^ p / j , ~ "•imnplJ'mnp

(4.18)

4.3. MOMENT EQUATIONS FOR POISSON IMPULSES

97

Him — Oim++22{ci {cimn mnn}}ss ++ 33 {d {dirnnp mnnm mpp}}aa,, mnm imnpm

Cimn==Ci Cimn Cimnmn++33{di {di mpm }gp}g , , mnp mnp

(4.19) (4.19)

Uimnp ~~ ("imnp' Dimnp

Equations for moments (4.9) and (4.14-4.16) take the form of ihi(t)

=

ihi(t)

= T C j m T l / / m n ~r d&imnpXftmn'W'pfs ~r ^imnpf^mnp T C j m T l / / m n -+" «Jttimnp\/^mn^pJs ~r ^imnpf^mnp

+v(t)E[P]ci, +v(t)E[P]*,

tHjV')

(4.20)

=

— & \**imfl'mji3 £\&imf1'mjJ "T" * i^jmnMmnj'/j T ^ai£\l-'imnplJ>mnpjj \DimnplJ>mnpj Jaa s~i~^\^imnf1'mnjf

+v{t)E acj, +v(t)E [P2] dCj,

P-ijk(t)

=

^{AifJLjk),

(4.21)

+ 3{Bimllmjk}s

+3{Dimnp +HD npMmn mnpjk}s Pjib}5 + imH

P'ijkiit) V-ijkl{t)

+

u(t)E[P v(t)E\P ]a^c \CiCjC k, k,

= + jki}s + = ^{Aifijki}, 4{AiHjki}s + 4:{B 4{BimfJ, l}s + imfj, mjkm +4{Dimnpflmnpjkl}, +^{Di mnpHmnpjkl}S

+v(t)E[P4i]c +u(t)E[P ]cicicjcjckci. kc,.

3{CimnfJ.mnjk}s 3 Z

(4.22)

4{Ci fj.fJ.mnmnj jki} 4{Ci mnmn kl}ss 22

+ 66j/(f).E[P i / ( f ) £ [ P ]]{^yCfcC.} {^yCfcC.}a

(4.23)

Usual cumu Ian t-neglect closure approximations. If we decide to trun­ cate the moment equation at a given level, all higher-order redundant moments have to be expressed approximately in terms of lower-order moments. Different closure approximations, based on different hypotheses, can be used to do that. The technique frequently used is the cumulant neglect closure, e.g. [150], in

98

CHAPTER 4. NON-LINEAR

DYNAMICAL

SYSTEMS

which the cumulant of orders higher than n are neglected, which results in the approximations for the moments above the n-th order. For example neglecting the third-, fourth, fifth- and sixth-order joint cumulants yields, respectively, the following closure approximations for the joint central moments fUjk = 0, Vijk 0,

(4.24)

Hijki = 3 {MyMfcijj {/J-ijm},, > Hijkl

(4-25)

Ikjkl-m IJ-ijklm = ~ 10 {Hijmm} {HijHklm} s s

fMjkbnn = 15 {^ijUklmn} Vijklmn {fJ.ijfJ.klmn} s s

,

+ + 10 {(J
(4-26)

(4.27)

Truncation of the set of moment equations (4.20-4.23) for the system with cubic non-linearity at the second-order moments level requires using the ap­ proximations (4.24) and (4.25) - which is equivalent to the Gaussian response assumption. This is just the Gaussian closure. If the set of moment equations is to be truncated at third- or fourth-order moments, the closure approxima­ tions (4.25) and (4.26), or (4.26) and (4.27), respectively must be used. Use of the above closure approximations is equivalent to assuming the ten­ tative joint probability density function of the state variables in the form of the truncated Gram-Charlier expansion. Then if the higher order moments are evaluated as the integrals with respect to the approximate, Gram-Charlier type tentative probability density function the same expressions are obtained as from neglecting the cumulants. For example if the set of moment equations is truncated at the fourth-order moments, accordingly the truncated fourthorder Gram-Charlier expansion (involving third- and fourth-order cumulants), can be constructed. If this expansion involves only the terms for i + j — 3,4 of the single sum of the expression (2.102), the joint fifth-order moment evaluated by performing the integral with respect to such a density function is obtained exactly as (4.26). The joint sixth-order moment is obtained in the form of (4.27) if in the expansion the terms for i + j = 3, k +1 = 3 of the double sum in (2.102) are included. If only the terms of the single sum are included, the following closure approximations are obtained

4.3. MOMENT EQUATIONS

FOR POISSON IMPULSES

Vijklmn ==15 l^ijklmn 15{HijVklmn} {l^ijfiklmn} 5 {MijMfc/Mmn} s -32--21-51 {MijMfc/Mmn} s. s.

99

(4.28) (4.28)

Obviously, the accuracy of the results obtained with the help of closure approximations depends on whether or not the pertinent Gram-Charlier prob­ ability density function is a good approximation to the actual density function. Modified cumulant-neglect closure approximations for Poisson im­ pulse process problems. If the system is subjected to a random train of impulses and to zero initial conditions, it is still at rest at the time t if in the time interval [0, t) no impulse occurred. If the train of impulses is driven by a homogeneous Poisson process, the probability P<) of no impulse occurrence in the time interval [0, i) is expressed as (cf. (2.25))

P0 = Px{N{t) Pi{N{t) = 0} = exp(-i/t). exp(-i^).

(4.29)

The probability PQ may be high, close to the unity, if the length t of the time interval is small, i.e. at the early transient stage, especially if the mean arrival rate v is also small. Let us introduce two expected values: unconditional one rriiq{t) = E \Xiq(t)\ and the conditional expected value, given that at least one impulse occurred, denoted as m°iq(t) = E [Xiq(t) \ N(t) > 0]. These expected values are related as

miqiq(t) (t)

= E[X E[Xiqiq(t)\N(t) (t)\N(t)

= 0]PT{N(t) 0]PT{N(t) = 0}

+E [Xiqiq(t) +E (t) | N(t) N(t) > 0] Pr{iV(i) > 0} = (l-P (1-P0)m° )m° (t). (t). 0 iq iq

(4.30)

Furthermore let us introduce the processes centralized with respect to both expected values

Yiqiq = Xiqiq-m -miqiq,,

yiqiq = xiqiq -m -miqiq,,

(4.31)

Y°=Xiqiq-m<> Y°=X -m<>q,q,

yl=x yl=xiqiq-m° -m°q.q.

(4.32)

Joint probability density function of the state variables Yiq(t) can be rep­ resented in form of the sum of the continuous and discrete parts as

100

CHAPTER 4. NON-LINEAR DYNAMICAL

M y ) = fYY(y\N(t) (y\N(t)

SYSTEMS

= o)Pi{N(t) o)Pi{N(t) = o}

+fv(y\N(t)>0)Pi{N(t)>0} = Pof[6(yiq+miq)

00 + (l-P0)f )ft(y Y(y ).

(4.33)

9=1

In the first, discrete, part the Dirac delta spike 6(yiq + miq) represents the finite probability of the system being at rest. This probability is, of course, concentrated at the zero displacement Xi — yiq+rriiq. The second, continuous, part in (4.33), i.e. fY (y°) is the conditional density, given that at least one impulse occurred. Let us define, accordingly, two types of the central moments: unconditional moments n

fM1...in

»

= E J[Yiq = / .,=1 .,=!

J

J ns

K

n

\JyiqfY(y)dy,

(4.34)

,=1

and the conditional ones

<•*= E ° f[ nI K ^ lI = Jnn / n
J

(4.35)

> 9=1 ,=i

In the situations when the first, discrete, term in (4.33) is predominant, it can be predicted that the probability density function / y ( y ) will be diffi­ cult to approximate by the truncated Gram-Charlier expansion. The reason is that the Dirac delta is difficult to approximate in terms of Hermite polynomi­ als. Accordingly the closure approximations (4.24-4.27) may not be valid for the unconditional central moments Mt^-in- Then the Gram-Charlier expansion can be tried for the continuous part in (4.33) only, i.e. for the conditional density / Y ( y 0 ) - This is equivalent to assuming the closure approximations for the conditional moments. In order to obtain the closure approximations for the unconditional moments they must be expressed in terms of the con­ ditional moments, the latter will be subjected to the closure approximations, and in turn the lower-order conditional moments will be expressed in terms of unconditional moments. Hence the relationships (identities) between the unconditional moments fHy-in a n d conditional moments fi° .... are required.

4.3.

MOMENT EQUATIONS FOR POISSON IMPULSES

101

The density function /y(y) given by (4.33) is inserted into (4.34), which yields

i p £;

n

(4-36)

[~ " nn

Mi1...in=Pon(-^)+( - o) o n ^ •

lkv..in = Pol[{-tMq) + (l~Po)E0 l[Yiq 9=1 1.9=1 9=1 L=l Substituting (cf. (4.31) and (4.32))

.

(4.36)

Y

iq = Yiq + ^p-QmH

(4-37)

one obtains

M»x-*» — Mir-tn —

l 1 - (Po (P„ -1)- l ) -11]] A pPotPo"A mi 0[pr 1

(1-Po)"(I-PQ)"-1

i LL l l m
+a- p o)E ( ; ) k , _ n »Hr} r=0

X

'

\

q=n-r+l

)

s

(4 38

"(i^s)'-

- '

The inverse identity is obtained from (4.35) by substituting the conditional density function / Y ( V ° ) evaluated from (4.33). The result is

4"U - = ir** [jw]" r^a (i3s)"n"v [jN " r^K (i3i)"n"v

<«-3«9> «

Upon inserting

K n, _- 1^ ^- " x^ , *S == *'

(44 0) (4.40)

+ «-fi.)(l^fr) W - f l . ) ( l ^ ) " ] nI n"V "'K,-

(«•«) (4-41)

one obtains

^

' q=X

102

CHAPTER 4. NON-LINEAR

DYNAMICAL

SYSTEMS

For example, if t h e set of moment equations (4.20-4.23) is t o be closed at the fourth-order moments, t h e identity (4.38) must be specified for t h e joint fifth-order moments fiijkim a n d sixth-order moments fajkimn a n d next the con­ ditional moments l4jum a n d /4jkimn a r e subject t o t h e closure approximations

^ J^L},4 ^m l =OioK'4},'

(4-42)

Ajklran ^ K {^llMmnL}s, „ } 5 + 10 ™ {/*>?»»}. { / * > ? „ » } . -2-15 " 2 - 1 5 K M{&&&„}. 4^n = = 15 °^US

(443)

Next, all t h e conditional moments must be expressed in terms of t h e uncondi­ tional moments with t h e help of identity (4.41). Finally we obtain t h e following closure approximations for t h e fifth- and sixth-order moments: 1 Mijfcfm fJ.ijklm

= =

- pp 11 _

p[i 1 100

{VijVklm}, ~ {MijMttm},

oo pp ~ ( l - p ) 2 11 55 { ^ J * } .

~(l_p)2 { ^ i * } . 2 P ( l + 4 P i 0 0 (-P02)inr 1)0i n r P ( l + 4 P ( P i 0 0 0 + / l _ p \3 +

Hijklmn Hijklmn

= =

/ l _ p \3

i

p]

f \ __ pp \2 1 ° { TT OO''mmjjM (" MM « m}g

10

+

+

p !_p

5

5

x _ p

{ m iMjfcIm}

{miMjfcim}ss

{mimimkVlm}, {mimimklJ'lm}s

, P 0 ( l + l l P p + llPp 2 + PP0 3) H 7; ^ mimjm mim + 7z 5—w ^-mimjm , m, kmlmkm (* ~ "a) W ~ "a) 15 f , 15P 0 f

7 5 " X^ij^klmnf 1 — 15XT) 10

f

s

,

.. ... (4.44)

~ Z JT 1.TOimj7J fcJmn ) 115P — xT) 0 f

2

r ( l +1.TOimj7J P o - 2 PfcJmn 7 5 " X^ij^klmnf 0 )) „ „ r s i ~ Z ^ oJT 20 1 — XT)5- {MiifcAUmn}, +1 — XT)Y +z m\ {™-iTnjmkHi7 mi j s

10 i ^ o ( l +^i P—o -r20)P 0 2 ) „ „ r i — xT)r 15 + 1 60Pp 5- {MijJfeWmn} Y 2 53T 20{m i m J m f c /Xi m n } s f s + . _

71i — xT) B 1 2 {mPjkHlnm}s ~ ~^ i — xT0- 2j {iMjHdlimn}, (.1 6OP0 - XT>; f ( l 2- 15 .To) . _ ,7 i 2 P 0 -B4152 { m i M j * W m r J s T - 2 {MiiMfeiMmn}, (.1 (1 - -To) ( ! -- -H); Po) , 2P 0 -45 2 3 + P 0 ( l + 7 P 0 - 5P 0 - 3 P )

7i 5^377 _ p{mvj {mimjmkTnifimn}s imjnkuimn}s 15 (! - Po) 2 3 3 4 P 0 ((ll + -1- 20P 40P 7P 0 0- - 5P )0 ) 0 -0 -3P5P 15 (l_p)4 {mimjmkTni[lmn} H 7T 5^7 3 m.imjmkmim mmns. P 0 ( l + 2 0 {I P Q -- P))° 40P50 - 5P<) + (1 - P ) m,mjmfcm,mm7nn.

(4.45)

4.3.

MOMENT EQUATIONS FOR POISSON IMPULSES

103

In the steady state, i.e. as t —► oo, it implies that P0 —* 0, hence the discrete part of the density function (4.33) vanishes, and the usual (classical) cumulant-neglect closure approximations (4.26) and (4.27) are obtained from (4.44) and (4.45), respectively. 4.3.2

Application of the usual cumulant-neglect clo­ sure approximations in the case of the system with cubic non-linearity: dynamic response of an elastic plate with moderately large deflections to a train of impulses

As an example illustrating the geometrically non-linear problem leading to the cubic non-linearity, let us consider the transverse vibration of the elastic, thin rectangular plate, with moderately large deflections (finite deflections and in­ finitesimal rotations). Equations of motion are obtained from the von Karman equations in the form of (cf. [141]) D V 2 V 2 U - h (U,xx<j),yy + U,yy4>tXX ~ 2U 2ll <xy(f> ,xy) + CU,f + TflU>tt

= p(x p(xty)Y,Pi6(t-ti), ty)Y,PiHt-ti),

(4.46)

i=l

—V 22V -!-V V22 - u% u2xyy + uuiXX u,
(4.47)

Hiy

where u — u(x, y, t) - the plate deflection, m,c - mass and damping per unit area, respectively, h - plate thickness, D = EYh3/12 (l - v$), (x, y, t) - Airy stress function. Assume that the plate edges are simply supported and free from stress. I.e. for x — ±a/2 U<XX + VpU^yy = 0 ,

,yy =— 4>, ,xy Xy ==00 and for y — ±6/2

(4.48)

104

CHAPTER 4. NON-LINEAR

DYNAMICAL

SYSTEMS

U
4„ = 0, 4>,xx = ,xy — 0,

(4.49)

where a, 6 - lengths of plate edges, and the origin of the coordinate system is assumed at the plate mid-point. The solution of the problem will be obtained with the help of the Galerkin method, using one-term approximation 7TX

TTIi

u(x, y, t) = hq(t) hq{t) cos — cos — , a b o

(4.50) (4.51)

As a result of employing the orthogonality condition the following equation is obtained: N(t) 2 3 q(t) + 2Cn0q(t) + n2q(t) 0q(t) + en q (t) = /? £

P
(4.52)

i=i

where the restoring force is a non-linear, cubic form of the displacement - this is a Duffing equation. The denotations are as follows:

n0 = (i + x*)r-) x2)r-)2J°, v

£ =

3A2 (l — v2) f 4(1 r 4(1 +A')' +A')'

+

,.,_

' \a/ A = a/6,

V m

1 2 2 A* _ A a(l/A)V A S^ 'W W iI W W -~ 2A 2AV ( l / A ) ^2((A)j A)j , sinh27rA 7rA + sinh 7rA 7TA cosh 7rA'

*
4.3. MOMENT EQUATIONS FOR POISSON IMPULSES

MX) = (1 + A2)2 K

^ '

yV

'

105

^ ( l + A 2 ) 2 -4a(A) 2

7T (1 + A 2 ) 4 - 1 6 A V ((A A))C( TT((11//A ) '

^

'

where the excitation participation factor /? is given by (3.7). It should be noted that the approximating functions assumed (4.50, 4.51) do not satisfy one of the boundary conditions in (4.48, 4.49), i.e. the condition <j>,xy — 0 for x — ±a/2 and y — ±6/2. This means that some shear stresses are imposed at the edges, i.e. iXy = —T. Hence it should be verified if the full variational equation (cf. [140]), including the work of the edge stresses at virtual displacements, is satisfied. The thorough examination of the pertinent variational equation reveals that this equation is satisfied. The stochastic equations for the state variables Xi — q,X2 = q, corre­ sponding to the equation of motion (4.52), are dXx = b12X2dt, dXi

(4.54)

3 dX2 = +M bnX dnnXf)dt-c pll(dt,dp), = (b (&2i*i z 2 ++ ckniXj ) dt2v(t)E[P]dt - c2v{t)E[P]dt ++ cc22 I[ pU(dt, dp), 21X1 + Jv where

b12 = 1, n =

- n 2o.., 621 = -ft

b22 = -2Cn -2Cfto, 0,

c

2 = 0,

= -&1-efi2,. d2U1 =

(4.55)

Equations for the mean values and for the centralized state variables are, respectively, mi = 6i2m 612^2, (4.56) 2, m2 — = himi b2\m\ + b22m2 + d2in (^iii + 3/inmi + m\) + c2v(t)E[P}.

dYi = dY2 =

B12Y2dt, 3 (A2 + B C2nY12 ++DDnu2niY BnYt Y*)l )dt* 21Y1 + B22Y2 + Can*? -c2v{t)E[P] + c2 I pU(dt, dp), -C2i/{t)E[P] Jv

where

A 3MH"H) , ,422 -= etlKum e^o (Mm + + 3Mnmi),

(4.57) (4-57)

106

CHAPTER 4. NON-LINEAR

B21 = -n20 (1 + Zem\), C7an = Cm = 3enjm 3 e ^ m i1 ,

DYNAMICAL

SYSTEMS

-2£n00, B22 = -2Cfi

D21U -= --eQ e f i 2j 0. . mi

(4.58)

It is interesting to examine the stationary solution of equations for the mean values mi,m2 in the case of zero-mean impulses: E[P] = 0. As t —*• oo, the left-hand sides of equations are equal to zero (4.56), hence m2 = 0 and the second equation becomes 0 = -tllmi eCll (fi (/ini + 3/inmi Sfinmi + ml). (4.59) -Cllmi - eQ,l mf). Ui + If the random variable P denoting the impulse magnitude has a zero mean value and asymmetrical probability distribution, i.e. the non-zero third-order central moment, then E [P3] ^ 0 and the probability distribution of the re­ sponse of linear system is asymmetrical as well (cf. (3.79)). Seeing that the type of non-linearity is of symmetrical type, i.e. g(—q) = —g(
f 21.2921s- 1 ], for i / = 10JJ0, \ 212.920 [s- 1 ], for v = 0.1fi 0.m0.

4.3. MOMENT EQUATIONS

FOR POISSON IMPULSES

107

2. Rayleigh-distributed random variable with parameters: ap

f 10.6461s- 1 ], for v = lOflo, lOfio, \ 106.460 [s- 1 ], for v = 0.lO 0 -

The use of the excitation process non-Gaussianity index (3.77) yields for £ = 0.05 the following condition of the large departure from Gaussianity: v < 0.1f20- Hence, for v — 10fi0 the departure should be considered small, whereas for v = 0.1fi0 it is moderately large. The parameter aP of the Rayleigh distribution is chosen in such a way as to keep the same, for both cases v = 10fi0 and v = 0.1fi0, the mean-square excitation level, i.e. to keep the same variance of the stationary response of a corresponding linear system (s = 0), given by

a

°-~w

( }

The set of moment equations has been truncated, for comparison, at three levels: using a Gaussian closure (formulae (4.24) and (4.25)), at third-order moments (formulae (4.25) and (4.26)), at fourth-order moments (formulae (4.26) and (4.27)). The obtained set of non-linear ordinary differential equations has been inte­ grated numerically with the help of a fourth-order Runge-Kutta method. The computed results are compared with the simulated ones, obtained by averaging over the ensemble of 50 000 independently generated response samples. In Figures 17-20 the mean value mi(t) = E[q(t)] and the variance fin(t) — E[q2(t)] — m\{t) of the displacement response q(t) are shown, evaluated in the time interval t € (0,12T 0 ), where TQ = 27r/fi0 is the natural period of the linear oscillator. After that time the oscillator reaches the stationary state. For comparison, the results for the corresponding linear oscillator (e = 0) are shown. Response mean values obtained from the set of moment equations truncated at third- and fourth-order moments are practically the same; the pertinent curves in Figures 17 and 19 coincide. As has been predicted the stationary

108

CHAPTER 4. NON-LINEAR

DYNAMICAL

SYSTEMS

Fig. 17. Mean response of a non-linear oscillator to a Poissonian train of Dirac delta zero-mean impulses: analytical, simu­ lated, a) v = ion 0 , b) v = o.in 0 -

4.3. MOMENT EQUATIONS

FOR POISSON IMPULSES

109

Fig. 18. Variance of the response of a non-linear oscillator to a Poissonian train of Dirac delta zero-mean impulses, — — — linear response, analytical for /C5 = «6 = 0, v = lOfio. analytical for K.5 = us = 0,v = O.lfioj analytical for K4 = K5 = 0 and for Gaussian closure for, v = lOflo and v = O.ino, • • • • • simulated for v = 10no simulated for v = O.lOo. response to zero-mean impulses has a non-zero mean value. This non-zeromean effect increases as v decreases, i.e. as the asymmetry of the probability distribution of the response process increases (the departure from the Gaussian behaviour increases). Simulated curves for the mean response presented in Figure 17 (and espe­ cially in Figure 17 a) exhibit a large scatter of the sample mean X. In other words in this case the simulated results (the sample mean) are associated by a large error. However it is known that sample mean is an unbiased estima­ tor, i.e. E{X] = mi. Therefore the simulated results may be interpreted in a mean sense. Simplifying the problem, we can use the time average of the stationary (nearly so) part of the simulated curve as an estimate of the mean of the stationary sample mean. Time averages of the stationary parts of the curves in Figures 17 a and b are indeed slightly negative. It is certainly quite

110

CHAPTER 4. NON-LINEAR DYNAMICAL SYSTEMS

Fig. 19. Mean response of a non-linear oscillator to a Poissonian train of Dirac delta non-zero mean impulses, — — — linear response, a) v = lOfio analytical and simulated, b) v = O.lfio analytical for K5 = KQ = 0, Gaussian closure, simu­ lated.

4.3. MOMENT EQUATIONS FOR POISSON IMPULSES

111

Fig. 20. Variance of the response of a non-linear oscillator to a Poissonian train of Dirac delta non-zero mean impulses: — — — linear response, analytical for K4 = K5 = 0, K5 = K& = 0, v = lOJlo, — analytical for Gaussian closure, v = lOfloi • • • • • simulated, v = 10f2o analytical for K5 = «6 = 0, v = O.lflo, an­ alytical for K4 = K5 = 0 and for Gaussian closure, v = O.lHoi simulated, v = O.lfioclearly visible in the Figure 17 b. Hence the simulated results support, at least qualitatively, the analytical predictions. In the case of non-zero mean impulses the simulated results for the mean values practically coincide with the analytical ones. The response mean values obtained from the set of moment equations trun­ cated at the third- or fourth-order moments are accurate enough. On the other hand the Gaussian closure is not appropriate: the predictions of the transient mean response (Fig. 19b) are much worse and in the case of zero-mean im­ pulses the predicted stationary mean response is always zero. Truncating the set of moment equations by neglecting the cumulants above

112

CHAPTER 4. NON-LINEAR

DYNAMICAL

SYSTEMS

the fourth order (fourth-order cumulant neglect closure) yields very good pre­ dictions of the response variance both in the case of zero-mean (Figure 18) and non-zero mean impulses (Figure 20). The Gaussian closure and the third -order cumulant neglect closure do not give satisfactory results. Difference between the results for a non-linear and for the linear oscillator, (mean values in Figure 19 and variance in Figures 18 and 20) reveals the essential non-linearity of the considered non-linear oscillator. It can be stated that the closure approximations obtained from neglecting the cumulants above the fourth order provide the good predictions for the mean values and variances of the response to Poisson distributed impulses, if the departure of the excitation process from Gaussianity is not very large (if the mean arrival rate of the train of impulses is high enough: v/Qo > 0.1). The effect of the non-zero mean response to zero-mean excitation is essential in the case of the systems with asymmetrical non-linearity, given by an even function: g{—X) = g(X). In the considered problem this effect is due to the fact that the excitation process has an asymmetrical probability distribution (asymmetrically distributed random magnitudes of the impulses), hence the response of the system with non-linearity given by an odd function g(—X) = —g{X) is asymmetrically distributed as well.

4.3.3

Application of modified cumulant neglect closure approximations in the case of the system with cubic non-linearity: Duffing oscillator under a train of general pulses

Consider the dynamic response of a Duffing oscillator to a Poisson train of general pulses, governed by the equation N(t) 2 2 2 32 3 q(t) ++2{n q{t) 2{n0q(t) q(t) q(t) + eQ + eQ q (t)q = (t)£= £PiW P(tiW- (tU), - U), 0q(t)++QQ

(4.62)

t=i

where q(t) is a non-dimensionalized generalized displacement and w (t - U) is a shape function of a general pulse. The response of a dynamical system is a non-diffusive Markov vector process if the input process is Poisson distributed train of impulses (cf. sec. 2.3). In order to use the Markov approach let us regard the pulse shape function w (t - U) as the response of an auxiliary

4.3. MOMENT EQUATIONS FOR POISSON IMPULSES

113

linear filter to a single Dirac delta impulse 6 (t — U). Then by virtue of a linear superposition principle, the cumulative effect of these responses u(t) = ^f=i PiW (t — U) is just the filter response to the train of Dirac delta impulses. In the case of a second-order filter the response u(t) is governed by the equation N(t)

u(t) + 2C/0/tt(t) 2C/fi/«(t) ++ n* 0/(i) (t) = J2 Pi6 P{6 (t(t-- u), U),

(4.63)

t=i

where Q and fi/ are the filter damping ratio and natural frequency, respec­ tively. If the damping ratio £/ is close to unity, the impulse response function of the filter a/ (t — U) decays very fast and becomes practically a single pulse (damped sine half-wave) with duration A = T//2, where 7/ = 2-K/Qf.il — (I is the natural period. The state vector of an augmented system, consisting of a non-linear oscil­ lator and of a linear filter, has four elements: X(i) = {q(t), q(t), u(t), u{t)}. It may be shown that the centralized state variables are governed by the following stochastic equations dYx = B12Y2dt, dY2 = (A2 + BnYt + B22Y2 + B23Y3 + C211Y2 + Dnu Y?) Y13) dt, 2111 dY3 = B3iY Yidt, 4dt, dYi = (B43Y3 + B44 Y44 - c2v{t)E[P])dt v(t)E[P])dt + c2 fI pU(dt, pll(dt, dp), i4Y where

(4.64)

B = 1, A2 = = eOg B12 efig (//in (//in + + 3/inmi), 3/inmi), 12 = 1, A2 2 = -fig (1 + Semi) -2Cfi00, 22 B21 = 3em ), B ,, „ . 21 22 = -2Cfi D0 B3i34 = l, Bai3 = -n}, -Q2f, B4i -2CfQf, 4i = -2QSl , ^ '' { 22 Can = -eQ . C2n = -3engmi, D2in 0 2in 0 In order to perform the example computations the following data has been assumed e = 0.5;C = 005;v = 0.1fi0 and v — 0.05fi0. Impulses magnitudes have been assumed as non-zero mean, Rayleigh-distributed random variables. The parameter op of a Rayleigh distribution is chosen in such a way that in both cases v — O.lfio aQd v = 0.05fi0 the variance of the stationary response of a corresponding linear oscillator (e = 0) has a unit value (cf. (4.61)). The

114

CHAPTER 4. NON-LINEAR

DYNAMICAL

SYSTEMS

Fig. 21. Mean response of a Duffing oscillator to a Poissonian train of general pulses with v = O.lfio: — — — ordinary closure technique, modified closure technique, simulated. following data has been assumed for the filter: fi/ = 2.4fi0.C/ = 0.95. Then A = ir/£lfJ\ — (j = §T0, hence the pulse duration is comparable with the natural period of a linear oscillator. The set of moment equations has been truncated at fourth-order moments level. In the case v = 0.1Q0 the closure approximations (4.26) and (4.27) resulting from the ordinary cumulant neglect closure have been applied, and for comparison - the modified closure approximations (4.44) and (4.45), both based on neglecting the fifth- and sixth-order cumulants. The response mean value (Fig. 21) and variance (Fig. 22) evaluated with the help of ordinary closure approximations are accurate enough, yet the use of modified closure

4.3. MOMENT EQUATIONS

FOR POISSON IMPULSES

115

Fig. 22. Variance of the response of a Duffing oscillator to a Poissonian train of general pulses with v = O.lfio: — — — ordinary closure technique, modified closure technique, simulated.

approximations gives even better predictions. In the case v = 0.05Q0 the ordinary cumulant neglect closure approximations yield erroneous results -the predicted response variance becomes negative. However the modified closure approximations lead to very good estimates of both the mean response (Fig. 23) and the response variance (Fig. 24).

116

CHAPTER 4. NON-LINEAR DYNAMICAL SYSTEMS

Fig. 23. Mean response of a Duffing oscillator to a Poissonian train of general pulses with v = 0.05fio: modified closure technique, simulated.

4.4

4.4.1

Equivalent systems techniques for the sys­ tem with non-analytical non-linearity, sub­ jected to a Poisson-distributed train of impulses S t a t e m e n t of t h e p r o b l e m

The problem of stochastic vibration of a non-linear inelastic system, with due account to plastic deformations and to the associated hysteresis of the stressstrain relationship, has been the subject of interest and studied for more than

4.4. EQUIVALENT

TECHNIQUES FOR POISSON IMPULSES

117

Fig. 24. Variance of the response of a Duffing oscillator to a Poissonian train of general pulses with v = 0.05fio: — — — ordinary closure technique, modified closure technique, simulated.

20 years. The main reason for this interest has been the need to predict the behaviour of structures under heavy loads resulting from the severe natural hazards such as earthquakes or strong sea waves. Non-linear inelastic (hysteretic) systems were usually analyzed under Gaussian stochastic excitations, idealized as a white noise or a filtered white noise. Different methods were used to study the stochastic response of hysteretic systems, for example the methods based on the Fokker-Planck-Kolmogorov equation, such as the direct linearization of this equation [55] or a generalized Galerkin solution to this equation [145]. In a series of papers the stochastic averaging method was used to study the dynamic response of hysteretic systems [101,102,122]. The Gaus-

118

CHAPTER 4. NON-LINEAR

DYNAMICAL

SYSTEMS

sian closure of moment equations was also applied to this problem [53]. The technique which is often used to analyze the response of hysteretic systems is the equivalent linearization technique adapted to this kind of problems by Wen [146]. Recently, an equivalent non-linear system (equivalent polynomial expansion) technique was developed for hysteretic systems under Gaussian stochastic excitation [84, 85]. The problem of a hysteretic system under a random pulse train requires another approach; such a problem was treated in [43, 44]. Here it is essential to formulate the analytical technique and to examine its effectiveness. Therefore we confine the attention to a single-degree-of-freedom dynamical system only. The governing equations are N(t)

2Cn0q + q + 2COo9 + atllq an20q + (1 - a)fijjZ a)n20Z = = J^ P{iWw (t - U), U),

(4.66)

where q(t) is the non-dimensionalized general displacement and the inelas­ tic component Z of the restoring force (which is the sum of the hysteretic component of the restoring force and non-viscotic damping force) satisfies the non-linear differential equation Z = aq - 0 I| qq | | ZZ | | ZZ pp""11 --7<7 7 ? | \ZZ \\nn=g(q,Z). = 9 (q, Z).

(4.67)

The equation (4.67) governing the hysteresis loop has been given by Bouc [11], and generalized by Wen [145]. By proper adjusting of parameters a, 0,7, n this equation allows to describe different hysteresis loops. There exist extensive studies on hysteresis loops pertinent to different materials, e.g. [121]. Example hysteresis loops governed by equation (4.67), for different values of parameters are shown in Figure 25, [145]. If the excitation is a random train of impulses then w (t — U) = 6 (t — U). The general pulses will be idealized, like in the sec. 4.3.3, as filtered impulses, i.e. for u(t) = Yli=l Piw (* — U) the filter equation (4.63) will be used. The plastic deformations are associated by the material deterioration or damage development. As a damage measure (or index) the hysteretic energy D(t) can be assumed [147], which is just the hysteretic loop area. The time evolution of this damage index is governed by the equation D = qZ = h(q,Z). h{q,Z).

(4.68)

4.4. EQUIVALENT TECHNIQUES FOR POISSON IMPULSES

119

Fig. 25. Example hysteresis loops governed by equation (4.67), [145]. All the governing equations are recast into the stochastic equations dX(t) = dX(t) - a(X(f)) a (X(t)) dt + c fI pU(dp, pU(dp, dt), dt), Jv

(4.69)

Jv

where in the case of impulses

"
~

r

9

Z(t) Z(t) ' a ( X j _ [ ./?(*) L mD(t)\ J

g(q,Z) 5(
h(q,Z) (9,Z)

"" 0 0 ''

cc ==

J1 . o • 0

(4.70)

120

CHAPTER 4. NON-LINEAR

DYNAMICAL

SYSTEMS

In the considered case of general pulses:

" ?(*) i

xmx m --

X ( i )

q(t) q(t) z{t) Z W D(t) u(t) u(t) u{t) u(t)

r annn n~-

, aa ( X )

a

-aQ220q - 2
c= c =

o o o

•

•

. (4-71) •

0 _ 1 Assume also the zero initial conditions X(0) - X0 = 0. In principle the equations for the mean values (4.9) and for the joint central moments (4.14-4.16) of the state vector X.(t) could be derived. Such equations would, however, involve the expectations of a non-linear non-analytical func­ tion g (q, Z) multiplied by different products of the state variables. These expectations would have to be evaluated by performing the multifold integrals with respect to the joint probability density functions of all the state vari­ ables. The tentative joint density function should, of course, be non-Gaussian, assumed e.g. as the truncated multivariate Gram-Charlier expansion. Per­ forming such multifold integrals, both analytical and numerical, is however a cumbersome task, especially if the number of state variables is large. In what follows the two analytical techniques are developed, in which this inconvenience is overcome.

4.4.2

Modified equivalent linearization technique

It has been found (sec. 4.3) that the systems with symmetrical (given by odd function) non-linearity exhibit the effect of non-zero mean stationary response to a zero-mean random excitation, if the excitation is asymmetrically dis­ tributed. Since the non-linearity in the equation (4.67) is symmetrical, i.e.

4.4. EQUIVALENT TECHNIQUES FOR POISSON IMPULSES

121

g (—q, —Z) = —g (q, Z) the similar effect should be expected. Hence the equa­ tions (4.69) cannot be linearized, because the linearized system would give the zero-mean (false!) prediction of the stationary response to a zero-mean exci­ tation. Therefore the approach used herein is similar to the one in problem with asymmetrical non-linearity [123]. First, by the averaging of the original stochastic equations (4.69) with (4.70) or with (4.71), the equations for the mean responses are obtained. Next, the equations for the centralized state variables (zero-mean responses) are linearized. Equations for the mean responses, in the case of impulses, become (cf. (4.9)) rh^t) m2(t) {t) m ihs(t) 3{t) m rrn(t) 4(i)

= m2, = -afigmi -anlm1-2Cn - 02CCl m2-(l-a)nlTn v(t)E[P], 0m2 - (13 - a)ftgm3++v(t)E[P], = E[g(X E[g(X ,X )), = ,X )}, 2 23 3 = = E[h(X2,X3)]

.

. ^U)

while in the case of general pulses rhi(t) ^(t) m3(t) m4(t) (f) m5(t) melt) m 6{t)

— = = = = =

m2, m — -aQ,lmi 2(Q0m a)Q.lm+ -aflo i —2(£l (1(l——a)fijjm3 mm 0m 3+ 22 - — 5, 5, E[g(X ,X )}, E[g(X ,X )}, 2 23 3 E[h(X E[h(X )},3)], 2,X23,X m6, -Q. -Q22fm mfm + u(t)E[P], v(t)E[P}, fTn 5-2C 5-2C fnffn 66

, V-'*'

where E [g (X2, X3)\ = am2 - $EX - 7yE £ 22,, E1 = E[\ X2 I X3 | X3 I""1] , E2 =

E[X2\X3\n],

E[h(X2,X3)]=E[X2X3}.

(4.74)

Equations for the centralized state variables Yi(t) = Xi(t) - rrii(t) have the general form of (4.10). Instead of the original system, governed by the equations (4.10), let us consider a linear system governed by the equations

122

CHAPTER 4. NON-LINEAR

DYNAMICAL

SYSTEMS

dY(t) = {BY(t) - v(t)E[P]c} dt + c f pU(dt, dp),

(4.75)

obtained by substituting the centralized drift term a°i (Y++ m) m)== aia{(Y (Y++ m) m)-- E[(n E[(n(Y (Y++m)] m)] a° (Y by an equivalent linear form e$„ (Y + e$„(Y + m) == B BimfmYYmm. .

(4.76)

The equivalent linear system governed by the equation (4.75) should op­ timally substitute for an original non-linear system governed by (4.10). It is required that the difference between equations (4.10) and (4.75) e (Y + m) = a 0 (Y + m) - B Y

(4.77)

be minimized in the mean-square sense. The necessary conditions of the min­ imum are T -^J-£[e | ~ £ [eT(Y (Y + (Y ++ m)] = 0. + m) m)te(Y m)]=0.

(4.78)

It follows that the equivalent coefficients Bim of the linear form satisfy the algebraic equations fjL \imjmj BB imim = E [Yja° (Y + m)] ,

(4.79)

where

E [y>? [Yrf (Y + m)] = E [Y [YjOi (Y++ mm))]], ,ii== 3,4; 3,4;jj ==2,3; 2,3;mm -=2,3. 2,3. jai (Y

(4.80)

In both considered cases of Dirac delta impulses (equations (4.70)) and of general pulses (equations (4.71)) only the third and the fourth equations are linearized, i.e. al B3333YY33,, aSJ,„ eq (Y + m) = B32Y2 + B

(4.81)

oJiB, (Y + al^(Y + m) m) = = B42 + BBi343YY3,3, 43Y2 t +

(4.82)

4.4. EQUIVALENT

TECHNIQUES

FOR POISSON IMPULSES

123

and the equation (4.79) assumes the form of ' M22 M32 | I- B32 Bi2 i 2 ]_ E \ Y2g . M23 M33 J L #33 Bi3\~ J [ Y3g

YY22hh ] Ygh Y 3h j '

V

V

where g = 3 (V2 + m2, Y3 + m3),h = (Y2 + m2) (Y3 + m3). Note that the equations for the mean values (4.72) and (4.73) involve the expectation of the non-linear function E[g(Y2 + m2, Y3 + m3)] and the second order moment E[h(Y2 + m2,Y3 + m3)) = E[(Y2 + m2)(Y3 + m3)}. Moreover, in the equations for the equivalent coefficients the expectations E[Yjg(Y2 + m2, Y3 + m3)] and third-order moments E[Yj(Y2 + m2)(Y3 + m 3 )], for j = 2,3, appear. The necessary expectations of the non-linear, non-analytical functions of the state variables Y2,Y3 must be evaluated by performing the integrals with respect to joint, bivariate probability density function of these processes. All the expectations which appear in (4.83), including the moments /xmj, are the expectations with respect to the same probability density function as the ex­ pectation in (4.78). Since the exact form of the probability density is unknown, this density function can only be assumed in a tentative, approximate form. Sometimes as the tentative density function the probability density of the response of a linearized system can be assumed, which is, of course, a rough estimate. For example in the classical equivalent linearization technique the excitation is a Gaussian process and the exact form of the probability density of the response of a linearized system is known. In the present problem the probability density of the response of a lin­ ear system is, however, unknown. The tentative probability density function should be assumed in a non-Gaussian form. To do that it is expedient to use the truncated Gram-Charlier expansion (cf. (2.102))

/ftn(t,&,6) = M 6 , 6 ) | i + I

E ij;«+J=3,4

^ur^(6,6,p)|, '•'■

where the coefficients (standardized cumulants) hj(t) brated from the moments

(4.84)

J

[cf. (2.104)] are cali­

124

CHAPTER 4. NON-LINEAR

im = IHi =E E [YiYi] [YJYJ]==EE Yy2^YV2Y3^_^Y V 3>V »>

3i

DYNAMICAL

^ == M ^ ^ 11

ii

SYSTEMS

(4-85)

00

The expectations E[g(Y2 + m2, Y3 + m3)\ and E[Yjg(Y2 + m2,Y3 + m3)],j = 2,3, regarded as integrals with respect to the density function (4.84), can be evaluated analytically or numerically. In order to construct the truncated expansion (4.84) the moments up to the fourth-order are required. Equations for moments of the response of the linearized system, derived based on the stochastic equation (4.75), become Mij(t) = fiij(t) = 22{B {Bimf imtixmj mj}}ss AO'fcW =

(iijki(t) (iijkt(t)

33

+ u{t)E [P2] acj, ciCj,

3 {Bimllmjk}, + v(t)E v{t)E [P [P3]] CiCjCk, CiCjCk, {BimfJ. mjk}s +

v{t)E [P2] 6 = 4 {BirnHmjki}s + v(t)E +u{t)E [P4] CiCjCkCi. +v{t)E

(4.86) (4.87)

{cicjnkl}s (4.88)

These equations form a closed set, however due to the coupling with equa­ tions (4.83) through the equivalent coefficients and due to the coupling with the equations (4.72) or (4.73) they are non-linear. Let us investigate the stationary solution of the equations for the mean re­ sponses (4.72). Assume that the impulses have zero-mean magnitudes, E[P] — 0. Then from the equations (4.72) it follows immediately that m2 = E\X2] — 0, /in = E[Y2Y3] = 0 and /( 0 = --aQlmi Q f i g m i _— (1 _- a )figm a)Q,lm33,, \\ 00 = -PEi jE . = ~m - JE22.

v-sy>

It can be shown that in general m 3 = E[X3] = 0 is not a solution of the equations (4.89). The expectations E\ and E2 become, for n = 1 [cf. (4.74)] EX1(Y {Y22,, Y3) = E[\Y2 + m2\ (Y3 + mm3)}, 3)},

(4.90)

E = E\{Y £2(12,13) E [(Y2 2 + m2) \\YY33 + + m33\]|] = = E E^X{Y , 3,Y Y22).). 2(Y2,Y3)

(4.91)

4.4. EQUIVALENT TECHNIQUES FOR POISSON IMPULSES

125

The expectation Ei evaluated as the integral with respect to the density function (4.84) becomes

Ei

= V? e x p ( 4^™ 3 (ff2 ~ ^ ) + ^ [ A 3 0 p ( / ? 2 22 - l ) + 3A21] + ^ [ A 3 o p ( / ? 2 - l ) + 3A21] +^[m 3 CT 2 A 4 o(/ ? 2 2

_ +^[m3(T2^o(Pt - *) 1)

-m2a3\i0p(/32

- 3) - 4ro 4ra2

+(Mn + m22m m33)[l )[l - 2erf(-ft)], 2erf(-ft)],

(4.92)

where ft = m2/a2, °2 a2 = =v,/fI^, =y S . aa 33 — vW, Substituting m2 = 0, m3 = 0 and fin = 0 yields [3E1 ++ 1jE (3E E2t~^=(^ = -}=(f3^

++ 1

V27T V \I2-K \ aO-i 2

^ 7)^). .

(4.93)

03 cr3 Jj

Under the assumption that the probability distribution of the linearized system is unsymmetrical and that both the third-order central moments, i.e. /i2i, fii2, do not vanish, the expression (4.93) is not equal to zero. This means that the stationary solution is some m3 = E[Z\ ^ 0, hence mi = E[q] ^ 0. Moreover, it is obvious that the stationary values of E[Z] and E[q] have opposite signs (see (4.89)).

4.4.3

Equivalent cubic form (equivalent cubicization) technique

Equations for the mean responses remain the same, i.e. (4.72) or (4.73). Here, instead of the original non-linear system (4.10) an equivalent non-linear system is considered, governed by the equation dYi(t) = {Ai + B dYi(t) B^Ym CimnY^n imYm + C imnYmYn + +d f pU(dt,dp). +d pU(dt,dp).

DimnpYmYnYp-U^ElP^dt DimnpYmYnYp-v(t)E[P]ci}dt

(4.94)

126

CHAPTER 4. NON-LINEAR

DYNAMICAL

SYSTEMS

In this case the original centralized drift term a?(Y 4- m) has been substi­ tuted by a cubic form in the state variables a

i e o ( Y ~^~ + mm) ) ——AiAi ++BimYm YY i,eq\* BimYm ++CiCi mnmn mY nn mY

++ DimnpYmL}i Ynmnv Yp.YmYnYv.

(4.95)

Equations for the moments of the response of an equivalent system with cubic non-linearity have the form of (4.21-4.23). The difference between the equation (4.10) for an original system and the equation (4.94) for an equivalent system is m ) -- a a°,(Y ° , ( Y + m). e(Y + m) = a°(Y + m)

(4.96)

The equivalent coefficients of a cubic form (4.95) are evaluated based on the condition that the equation difference (4.96) be minimized, i.e.

A E [ e r e ] = „, J L . [ A ] - * -§-E[f<]=0,

j±-E[?t]=0.

These conditions lead to the following algebraic equations: A-i ^imf^mj ^im^mj

—

^imnl^mn

"

*-* *->L^J^tJ l/j^tj

^imnf^mnjk ^imn/J'mnjk DimnplJ-mnpjkl Dimnpl*>mnpjkl

& L | / j i^ f c&%J ^tj =

^imnpl^mnpt ^imnftmnj ^imnftmnj ^ifijk ^i^-jk

E [YjYkYi^] [3^Ffc>|a,-] — - Ai(lj A{fljkk[l

■L'imnpftmnpj) J^imnp^mnpj> ■^imf^mjk ^iml^mjk —B Bimfanjkl im(lmjkl

^4.yoJ ^4.yoJ

^imnpf^mnpjki Dimnpfl'mnpjki ~

Cimn\J,mnjkl. CimnlhnnjM*

The joint probability density function of the state variables Y2,Y3 which is needed to perform the expectations in the equations (4.72) (or (4.73)) and (4.98) could be assumed tentatively as a probability density of the response of an equivalent non-linear system. Since this density function is unknown, it will be assumed approximately in the form of a truncated Gram-Charlier expansion constructed from the moments of the order up to the fourth. It should be noted that all the expectations in the equations (4.98) are performed with respect to the same density function as in (4.97), hence the fifth- and sixth-order moments in (4.98), must also be evaluated by integration with respect to the density function (4.84). This yields the expression for the fifth-order moments (4.26),

4.4.

EQUIVALENT

TECHNIQUES

FOR POISSON

IMPULSES

127

and for t h e sixth-order moments (4.28). Obviously t h e moments of the order up t o t h e fourth, which appear in the equations (4.21)-(4.23) and (4.98) are the same. In order t o construct the expansion (4.84) t h e m o m e n t s of u p t o the fourth-order are required. All t h e r e d u n d a n t fifth- and sixth-order m o m e n t s which appear in t h e equations for moments (4.21)-(4.23) should be expressed with the help of some closure approximations. To do t h a t any closure technique can be used, for example the cumulant-neglect closure (4.26, 4.27). In the considered problem the equivalent cubic form must be used only in the t h i r d equation, because the non-linear fourth equation involves the quadratic form, which is a lower order polynomial and therefore it does not have to be s u b s t i t u t e d by an equivalent form. T h e equivalent cubic form for the third equation is

a°3,eq(Y2 + m2,Y3 + m3)

= =

A3 + B32Y2 + B33Y3 + C322Y? +C323Y2Y3

+ C D32222Y2 + C"333^3 + D322 333Y3 4-

+^3223^2 Y3 + 3 +D 3333Y3 . ■ +•03333^3

D3233Y2Y3 (4.99)

T h e algebraic equations for the equivalent coefficients can be presented in a convenient m a t r i x form of (where Hij — E \Y2lY33} - cf. (4.85))

' 1 0 /X22o0 0 /x M20 Mil M20 Mil M30 M30 M02 M21 M40

Mil Mil M21 M21 M12 M31 M22

M02 M02 M12 M12 M03 M22 Ml3 M13 M04

M30 M30 M40 M40 M3i M50 M41 M32 M60

M21 M21 M3i M3i M22 M41 M32 M23 M51 M42 M42

Ml2 M12 M22 M22 Mi3 M32 M23 Ml4 M14 M42 M33 M33 M24

M03 M03 1 [ ^ 33 Mi3 Mi3 -832 -832 M04 B33 M23 C322 Ml4 C323 _ ^ M14 M05 C333 M33 -D3222 M24 M24 D£3223 3223 Mis -D3233 M06 L £3333 #3333 JJ L Moe JJ L L

0 Y2g Y3g *2 9 Y2Y3g Y3 g *2 9 Y?Y33g° g° Y?Y Y22Y3322g° yYfg° 3 3 ff° -. (4.100)

and where g° = gg (Y22 + m22, Y33 + m33)-E ) - E [g {Y (Y2 + m 22, Y33 + m3m)}. 3)}.

(4.101)

128

CHAPTER 4. NON-LINEAR

DYNAMICAL

SYSTEMS

The expectations on the right-hand side of equation (4.100) as the integrals with respect to the density function (4.84) can be performed either analytically or numerically.

4.4.4

Example: single-degree-of-freedom hysteretic sys­ tem under a train of impulses

In order to assess the effectiveness and accuracy of the analytical techniques developed in sees. 4.4.2 and 4.4.3 these techniques will be used to analyze the dynamic response of a single-degree-of-freedom hysteretic system to a Poisson train of impulses, for an example set of data. The data for the system is: a = 0.05; C = 0.01;

fi0 = Ifs" Ifs-1], and the data for the hysteretic loop n = = 1; a = 1;

0 = 7 = 0.5. Hence the equation for the hysteretic component is Z = q-0.5\q\

Z - 0.5q

\Z\.

Random magnitudes P of the impulses have been assumed as: a) non-zero mean Rayleigh distributed random variable with <jP = 0.492161s-1]. b) zero-mean random variable P = R - E[R], where R is Rayleigh dis­ tributed random variable with aR = 1.0624[s-1]. As is well known, the Poisson train of Dirac delta impulses tends asymptot­ ically, as v —> oo, to the Gaussian white noise. Therefore the question arises, in a natural way, about the quality of the results obtained by substituting the train of impulses with finite mean arrival rate v by a white noise. For the sake of an additional comparison of the results with those of other authors

4.4. EQUIVALENT

TECHNIQUES FOR POISSON IMPULSES

129

the quantities v and o> have been assumed in the case a) in such a way as to obtain the same mean and mean-square excitation level as in the reference [3]. In that reference the mean excitation level is fip = 0.8 and the mean-square excitation level is characterized by the constant spectral density Go = 0.1. In the problem considered herein it has been assumed vE[P] — 0.8 and vE [P2] = 2TTG0 = 0.27T, hence in the case a): v = 1.297Cl0,aP = 0.49216[s -1 ]. In the case b) the data is chosen in such a way that the mean-square excitation level remains the same, thus v = 1.297fio>0'fl = 1.0624[s-1]. The mean values E[q], E[Z\, E[D] and the variances var(q), var(Z), var(D) of the response variables in the case of non-zero mean impulses are plotted in Figures 26-31 against the non-dimensional time (relative to the natural pe­ riod T 0 ). The analytical results for the train of impulses have been obtained from the proposed modified equivalent linearization technique and from the equivalent cubicization technique. The analytical results for the equivalent white noise have been obtained with the help of a classical equivalent lin­ earization technique. The response moments have been computed by numer­ ical integration of the equations for moments, with the help of a fourth-order Runge-Kutta technique. To verify the validity of the analytical techniques developed the approximate analytical results have been compared with those from the Monte-Carlo simulations. The simulated response moments have been evaluated based on the averaging over the ensemble of 50, 000 independently generated response sample functions. The results for the non-zero-mean impulses are shown in Figs. 32-37. The modified equivalent linearization technique provides good estimates of the mean responses in the case of non-zero mean impulses (Figs. 26-28) and at least qualitatively correct predictions in the case of zero-mean impulses (Figs. 32-34). The estimates of the variances var(q) and var(Z) (Figs. 28, 29, 35 and 36) are not accurate enough, they are rather of a qualitative nature, though in the case shown in Fig. 29 the transient maximum of var(q) is correctly predicted. The prediction of var(D) (Figs. 31 and 37) is totally incorrect, in the latter it is even equal to zero. Substitution of the Poisson train of impulses by the equivalent white noise and application of the classical equivalent linearization technique yields, in the case of non-zero mean impulses, sufficiently accurate results (Figs. 26-28). However in the case of zero-mean impulses the prediction of mean responses is incorrect, E[q] and E[Z] are zero. The accuracy of predicted variances is mostly similar as in the case of the modified equivalent linearization technique

130

CHAPTER 4. NON-LINEAR

DYNAMICAL

SYSTEMS

Fig. 26. Mean displacement response of a hysteretic system to a Poissonian train of Dirac delta non-zero mean impulses: classical and modified equivalent linearization, — — — equivalent cubic form, simulated.

Fig. 27. Mean hysteretic restoring force in a system subjected to a Poissonian train of Dirac delta non-zero mean impulses: clas­ sical equivalent linearization, modified equivalent linearization, — — — equivalent cubic form, simulated.

4.4. EQUIVALENT TECHNIQUES FOR POISSON IMPULSES

Fig. 28. Mean damage indicator in a hyst eretic system subjected to a Poissonian train of Dirac delta non-zero mean impulses: see Fig. 27.

Fig. 29. Displacement response variance in a hysteretic system sub­ jected to a Poissonian train of Dirac delta non-zero mean impulses: see Fig. 27.

131

132

CHAPTER 4. NON-LINEAR

DYNAMICAL

SYSTEMS

Fig. 30. Hysteretic restoring force variance in a system subjected to a Poissonian train of Dirac delta non-zero mean impulses: see Fig. 27.

Fig. 31. Variance of the damage indicator in a hysteretic system sub­ jected to a Poissonian train of Dirac delta non-zero mean impulses: see Fig. 27.

4.4. EQUIVALENT TECHNIQUES FOR POISSON IMPULSES

Fig. 32. Mean displacement response of a hysteretic system to a Poissonian train of Dirac delta zero-mean impulses: see Fig. 27.

Fig. 33. Mean hysteretic restoring force in a system subjected to a Poissonian train of Dirac delta zero-mean impulses: see Fig. 27.

133

134

CHAPTER 4. NON-LINEAR

DYNAMICAL

SYSTEMS

Fig. 34. Mean damage indicator in a hysteretic system subjected to a Poissonian train of Dirac delta zero-mean impulses: see Fig. 27.

Fig. 35. Displacement response variance in a hysteretic system sub­ jected to a Poissonian train of Dirac delta zero-mean impulses: see Fig. 27.

4.4. EQUIVALENT

TECHNIQUES FOR POISSON IMPULSES

Fig. 36. Hysteretic restoring force variance in a system subjected to a Poissonian train of Dirac delta zero-mean impulses: see Fig. 27.

Fig. 37. Variance of the damage indicator in a hysteretic system sub­ jected to a Poissonian train of Dirac delta zero-mean impulses: equivalent cubic form, simulated.

135

136

CHAPTER 4. NON-LINEAR

DYNAMICAL

SYSTEMS

(Figs. 30, 31 and 35-37), and in one case (Fig. 29) it is even worse. The equivalent cubicization technique yields, in all cases considered, suffi­ ciently accurate predictions of mean values and variances of response variables. In that case a very good accuracy of the estimate of variance of damage vari­ able var(D) (Figs. 31, 37) is due to the fact that the equation for the damage variable D involves the quadratic form, hence this equation is not, within the framework of the equivalent cubicization technique, substituted by any equiv­ alent form - it is taken into account as it is. In the case of zero-mean impulses the mean responses E[q] and E[Z], as it has been predicted in the sec. 4.4.2, tend to non-zero stationary values with opposite signs. The results obtained for the Gaussian white noise, presented in Figs. 26, 27, 29 and 30, are in complete agreement with those given in [3].

4.4.5

Example: single-degree-of-freedom hysteretic sys­ tem under a train of general pulses

In Figs. 38-43 further results of verification of the validity of the techniques developed are presented. A more complex problem of the hysteretic system under a random train of general pulses (idealized as filtered impulses - cf. eq. (4.63)) is analyzed. Pulses with non-zero mean magnitudes are assumed. The data is chosen in such a way that the mean value and the variance of the response of a corresponding linear oscillator, governed by the equations (4.66) and (4.63) with a = 1, remains the same as in the case of Dirac delta impulses (sec. 4.4.4) The data assumed for the hysteretic system is: n = l; 0o = 1; l; 0 = 7 = 0.5; a = 0.05; < =0.01; fto = l[s- x ]; and for the auxiliary linear filter: Qf = 2.401921s-1]; C/ = 0.95.

4.4. EQUIVALENT

TECHNIQUES FOR POISSON IMPULSES

Fig. 38. Mean displacement response of a hysteretic system to a Poissonian train of general pulses: — • modified equivalent linearization, — — — equivalent cubic form, simulated.

Fig. 39. Mean hysteretic restoring force in a system subjected to a Poissonian train of general pulses: see Fig. 38.

137

138

CHAPTER 4. NON-LINEAR

DYNAMICAL

SYSTEMS

Fig. 40. Mean damage indicator in a hysteretic system subjected to a Poissonian train of general pulses: see Fig. 38.

Fig. 41. Displacement response variance in a hysteretic system sub­ jected to a Poissonian train of general pulses: see Fig. 38.

4.4. EQUIVALENT

TECHNIQUES FOR POISSON IMPULSES

Fig. 42. Hysteretic restoring force variance in a system subjected to a Poissonian train of general pulses: see Fig. 38.

Fig. 43. Variance of the damage indicator in a hysteretic system sub­ jected to a Poissonian train of Dirac delta zero-mean impulses: see Fig. 38.

139

140

CHAPTER 4. NON-LINEAR

DYNAMICAL

SYSTEMS

The driving train of Dirac delta non-zero-mean Rayleigh distributed ran­ dom impulses is characterized by ^ = l[s- 1 ]; aP = 3.70834. Analytical results for the mean values and variances of the response processes have been verified against Monte-Carlo simulations based on the averaging over the ensemble of 50, 000 independent response sample functions. As before, the modified equivalent linearization technique yields good esti­ mates of the mean responses (Figs. 38-40), but the estimates of the variances are not satisfactory. Only the transient maximum of var(q) is well predicted (Fig. 41). The prediction of var(Z) (Fig. 42) is merely of a qualitative nature and the prediction of var(D) (Fig. 43) is quite incorrect. Here again, the equivalent cubic expansion (equivalent cubicization) tech­ nique provides better accuracy of the results than the modified equivalent linearization technique; the predictions of the variances are accurate enough.

4.5

4.5.1

M o m e n t equations technique generalized for t h e problem of a pulse t r a i n driven by a class of renewal processes Statement of the problem: stochastic equations and equations for moments

Consider a general multi-degree-of-freedom non-linear dynamical system under a random train of impulses, driven by an ordinary renewal process. The state vector is governed by the set of equations

X(t) = a(X(t)) a (X(t)) + c J2 Pi6PiHt-tj). (* - *i) •

(4-102)

0
If the occurrence times to an ordinary renewal impulses in the interval occurring at the initial

i, of Dirac delta impulses are distributed according process, the counting process giving the number of (0, t] does not take into account the event (impulse) instant t0 = 0. However this initial impulse with

4.5. MOMENT EQUATIONS

FOR RENEWAL

IMPULSES

141

random magnitude P 0 produces the dynamical effect, which must be taken into account by imposing the non-zero random initial conditions X(0) = X 0 = cP cP„. 0.

(4.103)

Let us now assume that the renewal process under consideration has gam­ ma-distributed, with k = 2, interarrival times, with the probability density function given by (2.35). As is well known, in that case the interarrival time is the sum of two independent, negative exponential distributed random vari­ ables, hence the renewals (renewal events) are every second Poisson events. Consequently the train of impulses in equation (4.102) may be regarded as a Poisson-distributed one with non-zero marks (impulses magnitudes) corre­ sponding to even Poisson events (counts), the impulses magnitudes correspond­ ing to odd Poisson events being equal to zero. Such a train of impulses may be obtained as a sum of an impulsive process with Poisson-distributed arrival times and with alternating signs of magnitudes (as constructed in [40]), and of the usual Poisson impulsive process (cf. [115]). In the latter reference such a formulation of the renewal impulsive process was used to evaluate, via linear combination of the results given in [40] and of those for the Poisson impulses, the mean value and the variance of the response of a linear system to a renewal distributed train of impulses. Equations of motion with the recast excitation term have the form 1

ff(t)

X(t)

=

a (X(*)) + - c £ ( l - ( - l ) ^ - > ) P 5 ( * - * , ) ,

X(0)

=

X Xoo--ccPPoo,,

(4.104)

where {N(t),t > 0;Pr{N(0) = 0} = 1} is a Poisson counting process giving the number of impulses in the interval [0,t). Thorough analysis reveals that the state vector governed by equations (4.102) and (4.104) differs by the jump change of momentum cP due to a possible impulse occurrence at time t, with probability h0(t)dt. It follows that the statistical moments of the state variables only differ by an infinitesimal quantity of order dt. In the recast governing equation (4.104) the time points U are Poisson distributed and the impulses magnitudes are expressed in terms of a Poisson counting process.

142

CHAPTER 4. NON-LINEAR

DYNAMICAL

SYSTEMS

Integration of equations (4.104) over the time yields the integral equations = X 0 + /f a ( X ( r ) ) dr+ \c (t / (l - ( - l ) ^ r ) ) PU(dr, U(dT, dp), X(t) = dp) Jo * Jo ./o ./p Jv

(4.105)

where f m tt / / (1 - ( - 1 ) *" « ) PpU(dr, U(dr, dp) = ££ (1 (1 -- (-1)"<*>) (-l)**)) P P<% Jo ii = •/o Jv JP = 11

(4.106)

for JV(t) > 1 (cf. (3.76)). It can be shown that even if the compound Poisson process, determined by the counting process N(t) and the random variables Pi associated with the points U, is statistically independent of the initial conditions Xo = cPo, the state vector governed by equation (4.105) is not a Markov process. Increments of the compound point process (renewal process) during the time interval [r, r + dr) are expressed in terms of the increments of the compound Poisson process (Poisson random measure) and in terms of the Poisson counting process defined on the whole interval [0, T), and hence they are not independent. The state vector X(t) depends explicitly on the whole past history of the point process during the interval [0, t), and does not satisfy the Markov property. It can also be shown by proceeding similarly as in [116, 120]. The stochastic equations (4.105) are recast in the integro-differential form as

dX(t)

=

a(X(t))dt

X (0) X(0)

-

cP00.. cP

^(l-(-l)N^)cjpU(dt,dp),

+

(4.107)

Let us now derive the equations governing the statistical moments of the state variables X(t). Equations for the mean values are obtained by direct averaging of the governing stochastic equations (4.107), making use of the relationship

E [(-!)*»] = B - l ) ^ fc=o fc=o

6 K

* ^ -

= «p(-2*).

(4.108)

4.5. MOMENT EQUATIONS FOR RENEWAL IMPULSES

143

The result is m(t) = m(t) = E[a(X(t))] £7 [a (X(t))] ++ cho(t)E[P], ch0(t)E[P],

(4.109)

h(t) = (11 -- exp(-2i4)) exp(-2i4)) V*) = |I C

(4.110)

where is the ordinary renewal density in the case of gamma-distributed, with k = 2, interarrival times. Equations for zero-mean response processes are 0 dY{t) = aa°(Y (Y + + m) m)dtdt + 7j(N(t)) (N(t)) f PpU(dt, U(dt,dp), dp), Jv

(4.111)

where a 0 (Y + m) = a (Y + m) - E [a (Y + m)] - ch ch00(t)E[P], (t)E[P],

(iV(t)) 7(JV(t)) T

= ii cc (( l - ( - l )) ""W « ))..

(4.112)

(4.113)

Equations for the joint moments of the state variables Yi(t) are obtained by making use of the differential rule (2.83) applied to the process Y(t) governed by the equations (4.111), i.e.

dV(Y(t)) = £ ^ I p a ° (Y + m)
+ [f [V (Y + ~f(N(t))p) j(N(t))p) - V (Y)] U(dt, dp). Jv

(4.114)

Effective averaging of the integral term in (4.114) is feasible owing to the recasting of the excitation process in terms of a process with independent increments - the Poisson process. The quantities appearing in the integral are the functional of the counting process, i.e. V (Y) = V [N(t)\ and the function of this process: 7 (N(t)), and hence these quantities are statistically independent of the increment of the Poisson counting process defined on the anticipating interval [t, t + dt), and consequently are independent of U(dt, dp) as well.

144

CHAPTER 4. NON-LINEAR DYNAMICAL SYSTEMS

For example the equations for the second-, third-, and fourth-order joint central moments become fait) = 2 {E [Y Mt) [YaPjiY)] vE[P){c {clVjlVj}}s s ++acjhoWE acjhaWE [P [P22]], , ia°(Y)] }}s s -- vE[P]

(iijk(t)

[YiYjaKY)} } s + \uE[P}3 = 3 {E [YiYACY)] \vE[P\Z {c, (fijk -

Vjk)}s

~uE [P2] 3 {c -^uE {ciC ciCjckhkh0(t)E [P33]], , icjrjVk ]k}ss + dCjC Q{t)E [P

ftijkl(t)

= 4 {E [YJTjYrfOO] }s + \"E[P\4 i* Oh** ~

(4.115)

(4.116)

m))s

l 3 +-vE + -vE [P2] 6 {dcj {acj {fi (m U - T] r)ki)} {cicjckrll}s kl)}s - -vE [P ] 4 {cjCjCfcift]

+cicjckc,h clh00(t)E [P [P441] ,

((4.117)

where N rVili(t) (t) = E [(-l)wE[(-l) <»ty]^,,

rVijlij(t) (t)

(4.118)

N N = E[(-l)E[(-l) ^YiYj^Y ), iYj],

(4.119)

rr,hjk E[(-l)N^YYiYk].. (t) = = E [(-lftoYiYtYt] (4.120) ijk(t) These expectations, which are the joint moments of the random telegraph process T(t) = ( - 1 ) ^ and of the state variables Yi(t), are unknown and cannot be directly split, seeing that Y = Y [iV(<)]. Because of the non-zero initial conditions (4.103) the non-zero initial con­ ditions corresponding to the equations for moments (4.115-4.117) are 2

cicjE[(P-E[P]) iij{0) == ciciE[{P-E[P\f\, f(Mi(0)

],

M ° ) ==WkE - -E\P]f] IHjkiP) CiCjCkE[{P [{P E\P]f], , IMJU(0) Wi«(0)

= = CiCjCkciE [(P - E[P]Y] .

(4.121) (4.122) (4.123)

4.5. MOMENT EQUATIONS FOR RENEWAL IMPULSES 4.5.2

145

Correlation splitting equations

Equations for moments (4.115-4.117) must be completed by equations for the unknown expectations (4.118-4.120). The problem is similar to the one of joint moments of the excitation and response processes, where the moments cannot be split because of the functional dependence of the response process on the excitation process. In some cases the differential equations governing such moments can be formulated, which are known as the correlation splitting formulae. Such a formula is known, for example, for the excitation in form of a random telegraph process [59]. Let us now derive the differential equation for the expectation 77 = E [(-1)N^V (Y(t),t)], where the function V(Y(t),t) satisfies the same assumption as in the case of a generalized Ito's differential rule (see sec. 2.3). The derivative dt]/dt may be represented as N fftt = ±E[(-l) ±E[(-l)N^V(Y,t)}=E^(-l)^V(Y,t)j ^V(Y,t)}=E^(-l)^V(Y,t)j

(4.124)

+dt = E [(-l)"<<+dt >>V(Y, V(Y, t + dt)dt) - (-1) W «V(Y, *)] t)] /dt f w = E [(-l)"< >(-l) <*> (V(Y, t) + dV(Y, t)) - (-1)*«V(Y, (-!)*«V(Y, 0] t)] /dt,

where N(dt), dV(Y, t) denote the increments in N(t) and V(Y, t), respectively, in the time interval [t,t + dt). After employing the independence of N(t) and N(dt) and the relationship (cf. (2.1, 2.4, 2.5)) E [[(-l)^**] (-l)ww]

= (-1)° Pi{N(dt) Pr{N{dt) = 0} + (-1) 1 Pi{N{dt) = 1} = \-2vdt, (4.125)

one obtains ^ = -2vn + E [{-l)N®(-l)NWdV(Y,

t)] /dt.

(4.126)

Prom the property (2.76) of the Poisson random measure the following impli­ cations are obtained N(dt) = I U(dt, dp) dP) = 0=^ U(dt, dp) = 0, Jv

146

CHAPTER 4. NON-LINEAR DYNAMICAL SYSTEMS

N(dt) = I N(dt) ( U(dt, dp) = 1 = => > U{dt, dp) = fP{p)dp. {p)dp.

(4.127)

After making use of the differential rule in the form of (4.114) one obtains E[(-l)N^(-l)N^dV(Y,t)]

= E ( - l ) N ( f ) ^ dt

+EE[(-vmwAdt + (I E [[(-l) ( - l ) w « {V(Y + 7P, t) - V(Y, t)}] Jv xE [(-l)N^dt)U(dt, dp)] . (4.128) With the help of (4.127) the following expectation can be evaluated E[(-i)NN^U(dt, E[(-l) ^n(dt,

dp)} = E[{-i) E[{-l)NN^^dt)dt)U(dt, u(dt, dp) 1 | N(dt) N(dt)==0]0]Pi{N{dt) Pv{N{dt)==0}0} N(dt) N(dt) +E[(-l) U(dt,dp) +E[(-l) U(dt,dp) |I N(dt) = 1] PT{N(dt) Pv{N(dt) = 1} 1) ll = (-1)° (-1)° • 0 • (1 - vdt) + {-l) {-l)fPfP{p)dpvdt {p)dpvdt - -vfp{p)dpdt. -vfP(p)dpdt. (4.129 (4.129)

Finally, the differential equation for 77, or the correlation splitting formula, is arrived at in the form ft

= -2,V

+

E[(-ir^+^E[(-lf^a^

(4.130)

-v f E [(-1)"« {V(Y + j(N(t))p, j(N(t))p, t) - V(Y, t)}] ffPP(p)dp. (p)dp. Jv Based on the above differential equation the equations for rji, r/y, rjijk, i.e. the correlation splitting equations, are obtained as N r,i(t) = -2UTH -2VTH + +E E [(-l) [(-!)»& Wa»(Y)] a^Y)] ++ Cih dh0(t)E[P], 0(t)E[P],

(4.131) (4.131)

NN Tfo(t) = -2vVi -2mJ 3 + HE[(-l) 2{E[(-l) ^Y^Y ia° ia° j(Y)]} j(Y)]} s-uE[P}{c s-vE[P}{c iriVj 1]}s 2 (4.132) +ciCj7i0(i)£ [P ] ,,

4.5. MOMENT EQUATIONS FOR RENEWAL IMPULSES

147

ViM*) = - 2 " % * + 3 {E [(-i)"W [(-ifVY-Yjal] y i y. a °] }ss + \vE\P\Z {Ci{njk - r,jk)}s --uE[P2}3{cicjr)k}s

+ ciCjckh0{t)E [P3] .

(4.133)

The initial conditions corresponding to equations (4.131-4.133) are

m(0)

77,(0) = E [(-1)°1$(0)] [(-l)°y;(0)] = 0, ^(0)

(4.134)

= E [(-1)^(0)^(0)] = Mij ^ ((0) 0 ) = ciCj E[P})2] ,, iCjE [(P - E\P]f)

(4.135)

ifc (o) Vijkt(o)

= E [(-i)°y;-(o)is.(o)n(o)] [(-i)°y»(o)y,(o)n(o)] = Mijfc ^ ((o) o) 3 = dCjCkE Q C ^ E [(P [(P-- E[P]) E[P}f] ] ..

(4.136)

If the system under consideration is non-linear, i.e. a°(Y) is a non-linear function of the state variables Y, the system of equations (4.115-4.117) to­ gether with (4.131-4.133) is not closed, since the unknown expectations of the expressions involving a°(Y) appear in it. The set of moment equations is only closed when a°(Y) is a linear form in the state variables Y{. Hence in the case of a non-linear system the equations for moments cannot be solved di­ rectly. However they can be used in two ways. Firstly, the moment equations can be combined with suitable closure approximations. Then the unknown expectations are evaluated approximately and the resulting closed set can be solved. Such an approach has been used for the problem of a Duffing oscilla­ tor in reference [46]. Secondly, the moment equations can be combined with the maximum entropy principle [118, 119, 131]. In that case, even though the moment equations do not form a closed set, they provide the information suf­ ficient for the approximate evaluation of the response probability density for the system considered.

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