12 Stochastic processes and random vibrations 12.1 Introduction A large number of phenomena in science and engineering either defy any attempt of a deterministic description or only lend themselves to a deterministic description at the price of enormous difficulties. Examples of such phenomena are not hard to find: the height of waves in a rough sea, the noise from a jet engine, the electrical noise of an electronic component or, if we remain within the field of vibrations, the vibrations of an aeroplane flying in a patch of atmospheric turbulence, the vibrations of a car travelling on a rough road or the response of a building to earthquake and wind loads. Without doubt, the question as to whether any of the above or similar phenomena is intrinsically deterministic and, because of their complexity, we are simply incapable of a deterministic description is legitimate, but the fact remains that we have no way to predict an exact value at a future instant of time, no matter how many records we take or observations we make. However, it is also a fact that repeated observations of these and similar phenomena show that they exhibit certain patterns and regularities that fit into a probabilistic description. This occurrence suggests taking a different and more pragmatic approach, which has turned out to be successful in a large number of practical situations: we simply leave open the question about the intrinsic nature of these phenomena and, for all practical purposes, tackle the problem by defining them as ‘random’ and adopting a description in terms of probabilistic statements and statistical averages. In other words, we base the decision of whether a certain phenomenon is deterministic or random on the ability to reproduce the data by controlled experiments. If repeated runs of the same experiment produce identical results (within the limits of experimental error), then we regard the phenomenon in question as deterministic; if, on the other hand, different runs of the same experiment do not produce identical results but show patterns and regularities which allow a satisfactory description (and satisfactory predictions) in terms of probability laws, then we speak of random phenomenon. Copyright © 2003 Taylor & Francis Group LLC 12.2 The concept of stochastic process First of all a note on terminology: although some authors distinguish between the terms, in what follows we will adopt the common usage in which ‘stochastic’ is synonymous with ‘random’ and the two terms can be used interchangeably. Now, if we refer back to the preceding chapter, it can be noted that the concepts of event and random variable can be conveniently considered as forming two levels of a hierarchy in order of increasing complexity: the information about an event is given by a single number (its probability), whereas the information about a random variable requires the knowledge of the probability of many events. If we take a step further up in the hierarchy we run into the concept of stochastic or random process. Broadly speaking, any process that develops in time or space and can be modelled according to probabilistic laws is a stochastic or random process. More specifically, a stochastic process X(z) consists of a family of random variables indexed by a parameter z which, in turn, can be either discrete or continuous and varies within an index set Z, i.e. In the former case one speaks of a discrete parameter process, while in the latter case we speak of a continuous parameter process. For our purposes, the interest will be focused on random processes X(t) that develop in time so that the index parameter will be time t varying within a time interval T; such processes can also be generally indicated with the symbol In general, the fact that the parameter t varies continuously does not imply that the set of possible values of X(t) is continuous, although this is often the case. A typical example of a random time record with zero mean (velocity in this specific example, although this is not important for our present purposes) looks like Fig. 12.1, which was created by using a set of software-generated random numbers. Also note that a random process can develop in both time and space: consider for example the vibration of a tall and slender structure under the action of wind during a windstorm. The effect of turbulence will be random not only in time but also with respect to the vertical space coordinate y along the structure. The basic idea of stochastic process is that for any given value of t e.g. is a random variable, meaning that we can consider its cumulative distribution function (cdf) (12.1a) or its probability density function (pdf) (12.1b) where we write and to point out the fact that, in general, these functions depend on the particular instant of time t 0 . Note, however, Copyright © 2003 Taylor & Francis Group LLC stochastic process, say X(t) and Y(t´), and follow the discussion of Chapter 11 to define their joint pdfs for various possible sets of the index parameters t and t’. Now, since we can characterize a random variable X by means of its moments and since, for a fixed instant of time the stochastic process X(t) defines a random variable, we can calculate its first moment (mean value) as (12.4) or its mth order moment (12.5) and the central moments as in eq (11.36). In the general case, all these quantities now obviously depend on t because they may vary for different instants of time; in other words if we fix for example two instants of time t 1 and t 2 , we have Similarly, for two instants of time we have the so-called autocorrelation function and the autocovariance (12.7) which are related (eq (11.67a)) by the equation (12.8) Particular cases of eqs (12.6) and (12.7) occur when so that we obtain, respectively, the mean squared value and the variance (12.9) When two processes are studied simultaneously the counterpart of eq (12.6) is the cross-correlation function (12.10) (12.6) Copyright © 2003 Taylor & Francis Group LLC which is related to the cross-covariance (12.11) by the equation (12.12) Consider now the idea of statistical sampling. With a random variable X we usually perform a series of independent observations and collect a number of samples, i.e. a set of possible values of X. Each observation x j is a number and by collecting a sufficient number of observations we can get an idea of the underlying probability distribution of the random variable X. In the case of a stochastic process X(t) each observation x j (t) is a time record similar to the one shown in Fig. 12.1 and our experiment consists of collecting a sufficient number of time records which can be used to estimate probabilities, expected values etc. A collection of a number—say n—of time records is the engineer’s representation of the process and is called an ensemble. A typical ensemble of four time histories is shown in Fig. 12.2. As an example, consider the vibrations of an aeroplane in a region of frequent atmospheric turbulence given the fact that the same plane flies through that region many times a year. During a specific flight we measure a vibration time history x 1 (t), during a second flight in similar conditions we measure x 2 (t) and so on, where, for instance, if the plane takes about 15 min Fig. 12.2 Ensemble of four time histories for the stochastic process X(t). Copyright © 2003 Taylor & Francis Group LLC to fly through that region, The statistical population for this random process is the infinite set of time histories that, in principle, could be recorded in similar conditions. We are thus led to a two-dimensional interpretation of the stochastic process which we can indicate, whenever convenient, with the symbol X(j, t): for a specific value of t, say is a random variable and are particular realizations, i.e. observed values, of X(j, t 0 ); on the other hand, for a fixed j, say is simply a function of time, i.e. a sample function x j0 (t). With the data at our disposal, the quantities of eqs (12.4)–(12.9) must be understood as ensemble expected values, that is expected values calculated across the ensemble. However, it is not always possible to collect an ensemble of time records and the question could be asked if we can gain some information on a random process just by recording a sufficiently long time history and by calculating temporal expected values, i.e. expected value calculated along the sample function at our disposal. An example of such a quantity can be the temporal mean <x> obtained from a time history x(t) as (12.13) The answer to the question is that this is indeed possible in a number of cases and depends on some specific assumptions that can often (reasonably) be made about the characteristics of many stochastic processes of interest. 12.2.1 Stationary and ergodic processes Strictly speaking, a stationary process is a process whose probabilistic structure does not change with time or, in more mathematical terms, is invariant under an arbitrary shift of the time axis. Stated this way, it is evident that no physically realizable process is stationary because all processes must begin and end at some time. Nevertheless the concept is very useful for sufficiently long time records, where by the expression ‘sufficiently long’ we mean here that the process has a duration which is long compared to the period of its lowest spectral components. There are many kinds of stationarity, depending on what aspect of the process remains unchanged under a shift of the time axis. For example, a process is said to be mean-value stationary if (12.14a) for any value of the shift r. Equation (12.14a) implies that the mean value is the same for all times so that for a mean-value stationary process (12.14b) Copyright © 2003 Taylor & Francis Group LLC Similarly, a process is second-moment stationary if (12.15a) for any value of the shift r. For eq (12.15a) to be true, it is not difficult to see that the autocorrelation and covariance functions must not depend on the individual values of t 1 and t 2 but only on their difference so that we can simply write (12.15b) By the same token, for two stochastic processes X(t) and Y(t) we can speak of joint second-moment stationarity when At this point it is easy to extend these concepts and define, for a given process, covariant stationarity and mth moment stationarity or, for two processes, joint covariant stationarity, etc. It must be noted that stationarity always reduces the number of necessary time arguments by one: i.e. in the general case the mean depends on one time argument, while for a stationary process it does not depend on time (zero time arguments); the autocorrelation depends on two time arguments in the general case and only on one time argument ( ) in the stationary case, and so on. Other forms of stationarity are defined in terms of probability distributions rather than in terms of moments. A process is first-order stationary if (12.16) for all values of x, t and r; second-order stationary if (12.17) for all values of and r. Similarly, the concept can be extended to mth-order stationarity, although the most important types in practical situations are first- and second-order stationarities. In general, a main distinction is made between strictly stationary processes and weakly stationary processes, strict stationarity meaning that the process is mth-order stationary for any value of m and weak stationarity meaning that the process is mean-value and covariant stationary (note that some authors define weak stationarity as stationarity up to order 2). If we consider the interrelationships among the various types of stationarity, for our purposes it suffices to say that mth order stationarity implies all stationarities of lower order, while the same does not apply for mth moment stationarity. Furthermore, mth-order stationarity also implies mth moment stationarity so that, necessarily, an mth-order stationary process is also stationary up to the mth moment. Note, however, that it is not always possible to establish Copyright © 2003 Taylor & Francis Group LLC a hierarchy among different types of stationarities: for example it is not possible to say which is stronger between second-moment stationarity and first-order stationarity because they simply correspond to different behaviours. First-order stationarity certainly implies that all moments E[X m (t)]—which are calculated by using p X (x, t)—are invariant under a time shift, but it gives us no information about the relationship between X(t 1 ) and X(t 2 ) when Before turning to the issue of ergodicity, it is interesting to investigate some properties of the functions we have introduced above. The first property is the symmetry of autocorrelation and autocovariance functions, i.e. (12.18) which, whenever the appropriate stationarity applies, become (12.19) meaning that autocorrelation and autocovariance are even functions of . Also, if we note that we get from which it follows that (12.20) for all . Similarly, for all (12.21) where the first equality is a direct consequence of the second of eqs (12.9) where stationarity applies. Moreover, it is not difficult to see that eq (12.8) now reads (12.22a) so that, as it often happens in vibrations, if the process is stationary with zero mean, then When from eq (12.22a) it follows that (12.22b) Two things should be noted at this point: first (Chapter 11), Gaussian random processes are completely characterized by the first two moments, Copyright © 2003 Taylor & Francis Group LLC i.e. by the mean value and the autocovariance or autocorrelation function. In particular, for a stationary Gaussian process all the information we need is the constant µ X and one of the two functions R XX ( ) or K XX ( ). Second, for most random processes the autocovariance function rapidly decays to zero with increasing values of (i.e. ) because, as can be intuitively expected, at increasingly larger values of there is an increasing loss of correlation between the values of X(t) and Broadly speaking, the rapidity with which K XX ( ) drops to zero as | | is increased can be interpreted as a measure of the ‘degree of randomness’ of the process. If two weakly stationary processes are also cross-covariant stationary, it can be easily shown that the cross-correlation functions R XY ( ) and R YX ( ) are neither odd nor even; in general but, owing to the property of invariance under a time shift, they satisfy the relations (12.23) while eq (12.12) becomes (12.24) The final property of cross-correlation and cross-covariance functions of stationary processes is the so-called cross-correlation inequalities, which we state without proof: (12.25) (We leave the proof to the reader; the starting point is the fact that where a is a real number.) Stated simply, a process is strictly ergodic if a single and sufficiently long time record can be assumed as representative of the whole process. In other words, if one assumes that a sample function x(t)—in the course of a sufficiently long time T—passes through all the values accessible to it, then the process can be reasonably classified as ergodic. In fact, since T is large, we can subdivide our time record into a number n of long sections of time length Θ so that the behaviour of x(t) in each section will be independent of its behaviour in any other section. These n sections then constitute as good a representative ensemble of the statistical behaviour of x(t) as any ensemble that we could possibly collect. It follows that time averages should then be equivalent to ensemble averages. Assuming that a process is ergodic simplifies both the data acquisition phase and the analysis phase. In fact, on one hand we do not need to collect an ensemble of time histories—which is often difficult in many practical Copyright © 2003 Taylor & Francis Group LLC situations—and, on the other hand, the single time history at our disposal can be used to calculate all the quantities of interest by replacing ensemble averages with time averages, i.e. by averaging along the sample rather than across the number of samples that form an ensemble. Ergodicity implies stationarity and hence, depending on the process characteristic we want to consider, we can define many types of ergodicity. For example, the process X(t) is ergodic in mean value if the expression (12.26) where x(t) is a realization of X(t), tends to E[X(t)] as Mean value stationarity is obviously implied (incidentally, note that the reverse is not necessarily true, i.e. a mean-value stationary process may or may not be mean-value ergodic, and the same applies for other types of stationarities) because the limit of (12.26) cannot depend on time and hence (eq (12.13)) (12.27) Similarly, the process is second-moment ergodic if it is second-moment stationary and (12.28) These ideas can be easily extended because, for any kind of stationarity, we can introduce a corresponding time average and an appropriate type of ergodicity. There exist theorems which give necessary and sufficient (or simply necessary) conditions for ergodicity. We will not consider such mathematical details, which can be found in specialized texts on random processes but only consider the fact that in common practice—unless there are obvious physical reasons not to do so—ergodicity is often tacitly assumed whenever the process under study can be considered as stationary. Clearly, this is more an educated guess rather than a solid argument but we must always keep in mind that in real-world situations the data at our disposal are very seldom in the form of a numerous ensemble or in the form of an extremely long time history. Stationarity, in turn—besides the fact that we can rely on engineering common sense in many cases of interest—can be checked by hypothesis testing noting that, in general, it is seldom possible to test for more than mean- value and covariance stationarity. This can be done, for example, by subdividing our sample into shorter sections, calculating sample averages for each section and then examining how these section averages compare with each other and with the corresponding average for the whole sample. Copyright © 2003 Taylor & Francis Group LLC [...]... eqs (12. 91), (12. 93), (12. 95) and (12. 96) in terms of modal characteristics For example, eqs (12. 91) and (12. 93) become (12. 98) and (12. 99) where it should be noted that in eqs (12. 98) and (12. 99) we took into account the symmetries and the fact that P*=P because P is a matrix of real eigenvectors Note that, explicitly, the (jk)th element of the spectral density matrix of eq (12. 99) is written (12. 100)... transform and we get (12. 47) which is plotted in Figs 12. 4(a) and (b) for the values and S0=1 Figure 12. 4(b) shows a detail of Fig 12. 4(a) in the vicinity of Fig 12. 3 Spectral density of narrow-band process Copyright © 2003 Taylor & Francis Group LLC which decays to zero for increasing values of | | In the limit of very small values of ∆ω , the spectral density becomes a Dirac delta ‘function’ at and (12. 48)... eqs (12. 88) and (12. 90) to form the product and taking expectations on both sides we get the response correlation matrix as (12. 91) where RFF is the input correlation matrix Explicitly, the jth diagonal element of eq (12. 91) represents the autocorrelation of the jth response and reads (12. 92) Note that if we have two inputs and the jth response is the only output, then eq (12. 92) becomes eq (12. 77) At... obtain (12. 43) so that (12. 44) and also (12. 45) Moreover (12. 46) showing that, if x(t) is a displacement time history, we can calculate the mean square velocity and acceleration from knowledge of the spectral density SXX(ω ) Copyright © 2003 Taylor & Francis Group LLC The final topic we want to consider in this section is the distinction that is usually made between narrow-band and wide-band random processes,... point x=xm can be directly obtained from eqs (12. 58) and (12. 61b) as (12. 102) then, the mean squared displacement response at x=xm can be obtained (eq .12. 35)), for example, from (12. 103a) and the mean squared velocity response can be obtained from the first of eqs (12. 46), i.e (12. 103b) If now we extend our reasoning to the case of multiple (say n) inputs and multiple outputs, we can be interested, for... desired result is given by (eq (12. 94a)) (12. 104) where, for is the cross-spectral density between the two inputs applied at x=xr and x=xs while, on the other hand, for r=s the function is the autospectral density of the input applied at x=xr If, on the other hand, we are interested in the cross-spectral density between the outputs at points x=xm and x=xk, we get (eq (12. 94b)) (12. 105) At this point, provided... in eq (12. 55); in fact, since it follows that (12. 56) and for an SDOF system (e.g eq (4.42)) we have H(0)=1/k, which leads precisely to the result of eq (12. 55) More generally, eq (12. 54) can also be written as (12. 57) Note that here and in what follows we represent the input as a force signal and the output as a displacement signal because this is the representation that we used for the most part of... cross-relationships (12. 65) from which we can obtain another expression for H(ω) In fact, putting together eq (12. 61b) and the second of eqs (12. 65) we have from which it follows that (12. 66) thus justifying the H2 FRF estimate of eq (10.29) Example 12. 1 SDOF system subjected to broad-band excitation From preceding chapters we know that the FRF of an SDOF system with parameters m, k and c is given by (12. 67a) so... of bandwidth of a random process The interested reader is referred, for example, to Lutes and Sarkani [5], or Vanmarcke [6] 12. 4 Random excitation and response of linear systems We are now in a position to start the investigation of how linear vibrating systems respond to the action of one or more stochastic excitation inputs The situations we are going to consider are those in which a random (and. .. Fourier transformation of both sides of eq (12. 91) leads to (12. 93) where The diagonal elements of the matrix SXX(ω ) are the Copyright © 2003 Taylor & Francis Group LLC response autospectral densities (12. 94a) while the off-diagonal elements are the response cross-spectral densities (12. 94b) and again, in the case of n inputs and one output, eq (12. 94b) becomes eq (12. 78b) A word of caution is necessary . have (12. 40a) and also, since (12. 40b) so that eqs (12. 40a and b) imply (12. 40c) and only a little thought is needed to show that is an odd function of . The result of eq (12. 40c). Fourier transform and we get (12. 47) which is plotted in Figs. 12. 4(a) and (b) for the values and S 0 =1. Figure 12. 4(b) shows a detail of Fig. 12. 4(a) in the vicinity of Fig. 12. 3 Spectral. counterpart of eq (12. 6) is the cross-correlation function (12. 10) (12. 6) Copyright © 2003 Taylor & Francis Group LLC which is related to the cross-covariance (12. 11) by the equation (12. 12) Consider