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 ( 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)(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(*-*».*i). 0i(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) + 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(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 ( 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),...,