the history of random vibrations through 1958

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Mechanical Systems and Signal Processing Mechanical Systems and Signal Processing 20 (2006) 1783–1818 The history of random vibrations through 1958 Thomas L. Paez Sandia National Laboratories, Albuquerque, NM, USA Received 6 March 2006; received in revised form 5 July 2006; accepted 5 July 2006 Abstract Interest in the analysis of random vibrations of mechanical systems started to grow about a half century ago in response to the need for a theory that could accurately predict structural response to jet engine noise and missile launch-induced environments. However, the work that enabled development of the theory of random vibrations started about a half century earlier. This paper discusses contributions to the theory of random vibrations from the time of Einstein to the time of an MIT workshop that was organized by Crandall in 1958. r 2006 Elsevier Ltd. All rights reserved. 1. Introduction Random vibrations are the oscillations of mechanical systems subjected to temporally, and perhaps spatially, randomly varying dynamic environments. Their study is particularly important because practically all real physical systems are subjected to random dynamic environments at some times during their lives, and many systems will fail due to the effects of these exposures. Mathematical and experimental studies of random vibrations have historically been pursued to explain observed phenomena, to predict the characteristics of system responses to as yet unrealised environments, to aid in the design of mechanical systems and systems that isolate them, and to demonstrate the survivability and response character of physical systems in the laboratory. Though random vibrations have been observed for millennia because of the effects on structures of earthquakes, wind, ocean waves, and other natural environments, they have only been studied in a mathematical framework since about the turn of the previous century. Einstein performed the first mathematical analysis that could be considered a random vibration analysis when he considered the Brownian movement of particles suspended in a liquid medium. The results of his study were published in 1905. (This is the same year in which his results on the photoelectric effect, for which he received the 1921 Nobel prize in physics, and his results on special relativity were published.) Numerous studies whose results would later be used to explain the random vibration of mechanical systems were carried out in the decades to follow, and in 1930 Norbert Weiner formally defined the spectral density of a stationary random process, i.e., a random process in a temporal steady state. Spectral density is the de facto fundamental quantitative descriptor of stationary random processes in use today. But it was not until the 1950s that the subject of random vibrations ARTICLE IN PRESS www.elsevier.com/locate/jnlabr/ymssp 0888-3270/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.ymssp.2006.07.001 E-mail address: [email protected].

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Page 1: The history of random vibrations through 1958

ARTICLE IN PRESS

Mechanical Systemsand

Signal Processing

0888-3270/$ - se

doi:10.1016/j.ym

E-mail addr

Mechanical Systems and Signal Processing 20 (2006) 1783–1818

www.elsevier.com/locate/jnlabr/ymssp

The history of random vibrations through 1958

Thomas L. Paez

Sandia National Laboratories, Albuquerque, NM, USA

Received 6 March 2006; received in revised form 5 July 2006; accepted 5 July 2006

Abstract

Interest in the analysis of random vibrations of mechanical systems started to grow about a half century ago in response

to the need for a theory that could accurately predict structural response to jet engine noise and missile launch-induced

environments. However, the work that enabled development of the theory of random vibrations started about a half

century earlier. This paper discusses contributions to the theory of random vibrations from the time of Einstein to the time

of an MIT workshop that was organized by Crandall in 1958.

r 2006 Elsevier Ltd. All rights reserved.

1. Introduction

Random vibrations are the oscillations of mechanical systems subjected to temporally, and perhapsspatially, randomly varying dynamic environments. Their study is particularly important because practicallyall real physical systems are subjected to random dynamic environments at some times during their lives, andmany systems will fail due to the effects of these exposures. Mathematical and experimental studies of randomvibrations have historically been pursued to explain observed phenomena, to predict the characteristics ofsystem responses to as yet unrealised environments, to aid in the design of mechanical systems and systemsthat isolate them, and to demonstrate the survivability and response character of physical systems in thelaboratory.

Though random vibrations have been observed for millennia because of the effects on structures ofearthquakes, wind, ocean waves, and other natural environments, they have only been studied in amathematical framework since about the turn of the previous century. Einstein performed the firstmathematical analysis that could be considered a random vibration analysis when he considered the Brownianmovement of particles suspended in a liquid medium. The results of his study were published in 1905. (This isthe same year in which his results on the photoelectric effect, for which he received the 1921 Nobel prize inphysics, and his results on special relativity were published.) Numerous studies whose results would later beused to explain the random vibration of mechanical systems were carried out in the decades to follow, and in1930 Norbert Weiner formally defined the spectral density of a stationary random process, i.e., a randomprocess in a temporal steady state. Spectral density is the de facto fundamental quantitative descriptor ofstationary random processes in use today. But it was not until the 1950s that the subject of random vibrations

e front matter r 2006 Elsevier Ltd. All rights reserved.

ssp.2006.07.001

ess: [email protected].

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of mechanical systems would be addressed directly. A comprehensive theory of random vibrations was neededto accurately predict structural response to jet engine noise and missile launch-induced environments. In 1958,Crandall organised a special summer programme at the Massachusetts Institute of Technology to addressproblems in the various areas of random vibrations of mechanical systems. Specifically, the papers coveredtopics such as analysis of random vibrations and design for random environments, random vibration testing,and the analysis of data from random environments. The history of many contributions to the theory andpractice of random vibrations from the time of Einstein to 1958, and the years immediately following, aredescribed.

The scheme used for the presentation of historical material is chronological and, to the limited extentpossible, graphical. The major developments from four eras, seen by the author as well defined, yet, necessarilyoverlapping, are covered, to the extent possible, in order. A few of the mathematical ideas are supported withgraphics. Some limited mathematics are included.

Many texts are available for those seeking a detailed introduction to, or even a more advanced presentationof the mathematics of random vibrations. Among these are the texts by Crandall [1,2], Crandall and Mark [3],Robson [4], Lin [5], Elishakoff [6], Nigam [7], Newland [8], Bolotin [9], Augusti et al. [10], Ibrahim [11], Yang[12], Schueller and Shinozuka [13], Roberts and Spanos [14], Ghanem and Spanos [15], Soong and Grigoriu[16], Wirsching et al. [17], and Bendat and Piersol [18]. Texts that contain discussions on historicaldevelopments in the theory of stochastic processes (The term ‘‘stochastic’’ is used interchangeably with‘‘random’’.) include those by Gnedenko [19], and Feller [20]. Many important topics in the theory of randomvibration of mechanical systems that are not discussed in this paper, are discussed in the texts listed, includingfirst passage and peak response of structures, numerical techniques in random vibration and signal analysis,random vibration of structures that are themselves random, stochastic fatigue, random vibration of structuresmodelled via finite elements, random vibration of non-linear structures, and non-Gaussian random vibrations.

The important ideas of a random vibrations framework can be described using a simple schematic diagram.Consider Fig. 1. Random vibrations occur in a mechanical system when it is subjected to a stochasticenvironment—one applied as forces or pressure on the system, or one applied at system boundaries through itssupport structure. Each mechanical system has its own characteristics and features—simple or complex, linear(quasilinear) or non-linear, time varying or not, etc. Ensembles of mechanical systems have their own rangesof characteristic random variations; however, classical random vibration studies consider excitationrandomness only. The effect of a random environment on a structural system is stochastic response motion.

The activities of random vibration analysis can be succinctly described with the following (scalar or vector)equation representation:

_X ¼ gðX ;Q; aÞ; X 0ð Þ ¼ X0; �1oto1. (1.1)

The quantity X represents system response (such as displacement), Q represents system excitation (such as aforce applied on a structural surface), a represents system parameters, the dot denotes differentiation withrespect to time, and g(.) is the deterministic functional form that relates the former quantities to the responsederivative. Bold type is used to denote vector and matrix quantities. The purpose of random vibration analysisis to specify the stochastic system response, X, in terms of the system characteristics gðX ; aÞ (deterministicmathematical form, g, and deterministic parameters, a), and the random excitation, Q. The response, X, canbe described in the framework of probability theory. The probabilistic character of the excitation and responsemay be specified completely, at one extreme, by higher-order probability distributions, or, as more commonlyoccurs, only partially specified by some of their average features. Random vibration analyses can be

Mechanical

System

Excitation Response

Fig. 1. Schematic of excitation/system/response.

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ARTICLE IN PRESST.L. Paez / Mechanical Systems and Signal Processing 20 (2006) 1783–1818 1785

performed by representing the mechanical system with a set of differential equations, then solving theequations, or they can be performed without writing the differential equations of dynamic response, by writingequations that govern the probability distribution of system response.

The purpose of design for random vibration is to specify the characteristics of a random excitation, Q, specifythe stochastic characteristics of the system response, X, or, perhaps, a deterministic limit on the randomresponse, then establish the characteristics of a system, gðX ; aÞ, that will yield the desired response.

Testing of mechanical systems in random environments may serve many purposes. It may be done simply toestablish the character of a particular system in a random environment. It may be used to show that therandom response satisfies certain criteria. Testing of physical systems may also be used to explore ourcapabilities to model the same physical systems. However, it is usually only out of indirect necessity that asystem model relating excitation to response is developed in the course of random vibration testing.

Random signal analysis uses measured data to estimate the measures critical for description of randomprocesses. Of course, this is fundamental to the pursuit of practical environment description and testspecification.

In spite of the importance of random vibration design, testing and signal analysis, this paper will focusmainly on the description of historical developments in random vibration analysis. Section 2 summarizes thefirst investigations into random vibrations, from Einstein’s description of Brownian motion as a diffusionprocess to description of mechanical system response in terms of averages. Section 3 summarizes thedevelopment of spectral density, the fundamental descriptor of stationary random processes, and traces somepreliminary thoughts on the subject back to 1889. Section 4 summarizes advances that were motivated byproblems in electrical and communications systems that arose prior to and during World War II. Analysis ofthe random vibrations of mechanical systems, as practised today, started in the 1950s, and the beginnings ofthe analytical developments are covered in Section 5. The summary provided here concludes with a descriptionof some of the works from Crandall’s 1958 workshop and a few others that followed immediately thereafter.

2. Einstein’s era

Around the turn of the previous century, Einstein [21] constructed a framework for analysing the Brownianmovement—the random oscillation of particles suspended in a fluid medium and caused by the molecularmotion associated with the kinetic theory of matter. Brownian movement had been recognised about a centuryearlier during observations of microscopic particles of pollen immersed in a liquid medium; it is characterizedby the erratic movement of the pollen particles. The particle motion characteristics depend on the mass andgeometry of the particle and the physical characteristics (such as viscosity and temperature) of the fluidmedium.

Because the problem Einstein solved yields the probabilistic description of the motion of a mass attached viaa viscous damper to a fixed boundary and excited with white noise (a random excitation with frequenciescovering a broad band), his development can be thought of as the first solution to a random vibration problemand the dawning of the era of random vibration analysis. Einstein, however, did not consider the solution ofthe random vibration problem as the most important breakthrough of the analysis. He stated, ‘‘If themovement discussed here can actually be observed (together with the laws relating to it that one would expectto find), then classical thermodynamics can no longer be looked upon as applicable with precision to bodieseven of dimensions distinguishable in a microscope: an exact determination of actual atomic dimensions isthen possible.’’

In his solution of the problem of Brownian movement Einstein did not use a direct formulation that writesand analyses the differential equation governing motion of the system. (The direct approach would eventuallybecome the one most commonly used for random vibration analysis.) However, for reference, the governingequation is

m €X þ c _X ¼W ðtÞ tX0; X ð0Þ ¼ 0; _X ð0Þ ¼ 0, (2.1)

where fX ðtÞ; tX0g is the one-dimensional particle displacement response random process, m is particle mass, c

is the damping that ties the mass to an inertial frame, fW ðtÞ;�1oto1g is the white noise excitation randomprocess, and dots denote differentiation (in a sense appropriate for a random process) with respect to time.

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The white noise excitation random process has the constant spectral density SWW ðoÞ ¼ SWW ; �1ooo1.(Spectral density defines the mean square signal content of a random source as a function of frequency. Thespectral density defined here is two-sided because it is defined for positive and negative frequencies. Negativefrequencies are to be interpreted in the sense that harmonic functions are defined for negative arguments. Thedefinition of spectral density, along with some examples, and some ideas underlying random processes andtheir notations are provided in the following section.) The system of Eq. (2.1) is shown schematically in Fig. 2.The white noise random process is a source with mean square signal content that is uniformly distributed overthe entire range of frequencies (up to infinity, in theory). This idea will be discussed in more detail in thefollowing section, and examples will be presented.

One way of thinking about a random process is to consider it as a sequence of random variables. In thisinterpretation, the excitation and all measures of the response are random processes. The random variables inthese random processes characterize the quantity under consideration (excitation or response) at a given time,t. For example, X ðtÞ is the random variable representing displacement response at time t. The random variablehas a formal definition which we will not explore, here, but the practical idea behind a random variable is thatwhen we perform a sequence of random experiments, the values that the random variable assumes (calledrealisations) are observed empirically to follow a probability distribution. One descriptor of a probabilitydistribution of a random variable X is the probability density function (PDF), f X ðxÞ;�1oxo1. The PDF isnon-negative, and has a unit integral on ð�1;1Þ. The integral of the PDF over an interval (a, b),

R b

af xðxÞdx,

where apb, is the relative chance—probability—that the realisation of the random variable X will occupy theinterval (a, b] when one random experiment is performed. The expected value, or mean, of a random variableis the average value of all possible random variable realisations. It is denoted E½X ðtÞ� ¼ mX ðtÞ. The variance ofa random variable is the average of the square of the deviations of the random variable realisations from themean. The variance is denoted V ½X ðtÞ� ¼ s2X ðtÞ. The standard deviation of a random variable is the squareroot of its variance. Every random process has PDF, mean, variance, standard deviation, and many othermeasures for each of its random variables. Beyond these things, a random process also has other measures thatcharacterize the simultaneous behaviour of pairs, triplets, etc., of its random variables. Some of these will beconsidered later in this section and in the following sections. (See [22], for more details.)

Einstein developed the diffusion construct for analysing the random vibration of mechanical systems. Thisframework models diffusion of a particle (rigid mass) under the influence of applied impacts. In theapplication he considered, the impacts arose from the motions of the molecular constituents. The paperEinstein wrote has two parts. The first part uses the idea of osmotic pressure and equilibrium of a spheremoving in a fluid medium to derive the coefficient of diffusion of such a particle. He showed that thecoefficient of diffusion can be modelled as D ¼ ðRT=NÞð1=6pcf rÞ, where R is the universal gas constant, T isabsolute temperature, N is Avagadro’s number, cf is the coefficient of viscosity of the fluid, and r is the radiusof the sphere. The coefficient of diffusion (to be used, later) governs the rate at which particles will spreadthroughout a fluid in an equilibrium condition.

The second part of Einstein’s paper considered particle diffusion in one dimension in more detail. Hedefined a time interval t that is short compared to an interval of visual observation but long enough thatparticle movements executed in two consecutive intervals are independent. He assumed that during the time t

m

c

x

W (t )

Fig. 2. Schematic of the system considered by Einstein in his solution of the Brownian movement problem/random vibration analysis.

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the x coordinate of an individual particle will increase by a random amount D (positive or negative). Thequantity D is a random variable with symmetric PDF f DðaÞ; �1oao1. (Einstein neither used the term‘‘random variable’’ nor ‘‘PDF’’ because the terminology was not yet established, however, these are what hedescribed.) He then specified that the PDF of the x coordinate of a particle after time t, i.e., the PDF of arandom variable X(t) in the random process fX ðtÞ; tX0g, is f X ðtÞðxÞ; tX0; �1oxo1, and wrote therelation

dx � f X ðtþtÞðxÞ ¼ dx �

Z 1�1

f X ðtÞðxþ aÞf DðaÞda tX0; �1oxo1. (2.2)

This equation states that the probability of finding the x coordinate of a particle in the interval ðx;xþ dx� attime t+t (left side) equals the ‘‘sum’’ of probabilities that the particle starts in the interval ðxþ a; xþ aþ da�at time t, and moves a distance �a during the time increment t (right side). He expanded f X ðtþtÞðxÞ in aTaylor’s series in t, and f X tð Þðxþ aÞ in a Taylor’s series in a and substituted the series into Eq. (2.2). Hesimplified the integral on the right hand side and noted that because of the symmetry of f DðaÞ, only the oddnumbered terms are non-zero. The term f X ðtÞðxÞ occurs on both sides and the two terms cancel. He retainedonly the linear term in t on the left hand side, arguing the permissibility of this approximation because of thesmall magnitude of t. He set the integral in the third term on the right hand side (the first non-zero term) to thecoefficient of diffusion:

D ¼1

t

Z 1�1

a2

2f DðaÞda. (2.3)

And he eliminated the remainder of the terms on the right-hand side, arguing that they are small relative tothe term retained. The result he obtained is

qf X ðtÞðxÞ

qt¼ D

q2f X ðtÞðxÞ

qx2tX0; �1oxo1. (2.4)

He pointed out that this is the equation governing diffusion of a particle in a liquid medium with coefficientof diffusion D.

Einstein specified as the initial condition

f X ð0ÞðxÞ ¼ dðxÞ; �1oxo1, (2.5)

where dðxÞ; �1oxo1, is the Dirac delta function. (The Dirac delta function was not actually defined untillater, so Einstein described an initial condition with this character, but not using the terminology. The Diracdelta function is a ‘‘distribution,’’ a function described by its behaviour under an integral. Its integral is one,R1�1

dðxÞdx ¼ 1, and it has ‘‘small’’ values away from the origin, dðxÞ ¼ 0; xa0. In the application writtenabove, it indicates certainty that the system starts with zero displacement.) Under these conditions, thesolution to Eq. (2.4) is

f X tð Þ xð Þ ¼1ffiffiffiffiffiffiffiffiffi4pDp

1ffiffitp exp �

x2

4Dt

� �tX0;�1oxo1. (2.6)

The displacement response random process has a normal (Gaussian) distribution with mean zero andvariance 2Dt. (This also equals the mean square, or mean of the square of X(t), because the random processhas zero mean.) The displacement of a particle in Brownian motion has a variance that increases linearly withtime. Its root-mean-square (RMS) departure from the origin increases at a rate of

ffiffitp

. Increases in thecoefficient of diffusion, D, imply increased response variance at a given time. D increases linearly withtemperature of the medium, and decreases as the inverse of coefficient of viscosity and particle size. Einsteinpointed out, and it was confirmed later, by more accurate analyses, that this result is not accurate for timesthat are small compared to (m/c). Nevertheless, Eq. (2.6) stands as the first solution of a random vibrationproblem.

It is interesting to note that the diffusion coefficient is related to the parameters of Eq. (2.1) via D ¼

SWW=2c2 when SWW has units of lb2/(rad/s). Fig. 3 shows five marginal PDFs (i.e., PDFs of single randomvariables) of response at normalised times t ¼ 2Dt ¼ 0:1; 1; 4; 7; 10. The normalised time versus x plane also

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0

5

10 -10

0

100

0.5

1

1.5

f X (

τ) (

x)

Fig. 3. Marginal PDFs of some normal distributions of the displacement response of a particle in Brownian motion for normalised times

t ¼ 2Dt ¼ 0:1; 1; 4; 7; 10.

T.L. Paez / Mechanical Systems and Signal Processing 20 (2006) 1783–18181788

shows the plus and minus one times the standard deviation curves, �ffiffiffitp

. The plots clearly indicate the effectsof the uniformly increasing variance. As time increases the standard deviation—and, thus, the width—of eachnormal PDF increases, and the peak of the PDF decreases. We reiterate the important fact that Einstein’sanalysis, summarised here, was the first random vibration analysis of a discrete mechanical system.

One of many important results demonstrated in Einstein’s work is that the motion of a mechanical systemexcited by a large number of independent impacts is governed by a normal probability distribution. Heaugmented his initial study with further investigations (all reprinted in [23], and written during the period of1905–1908). He considered, among other things, extensions to his original development and the problem ofmolecular parameter identification.

A characteristic of random vibrations of mechanical systems that Einstein (and other early researchers inthis field) did not fully pursue is the average temporal characteristics of the excitation and particle responsemotions. A sample time history drawn from a random source (a random process) is known as a realisationof the random process, and it may bear a certain average relation to itself in time. For example, considerFigs. 4(a–c). The first of these shows a realisation of a band-limited white noise force. It has signal content upto a maximum frequency of 50Hz, a spectral density of 1 lb2=Hz, and, therefore, a mean square of 50 lb2. Thesecond shows the corresponding realisation of displacement response random process of a rigid particle withmass m ¼ 1 lb s2/in attached via a viscous damper with constant c ¼ 0.1 lb s/in to an inertial frame. The thirdshows the corresponding realisation of the velocity response random process. All the plots have a timeincrement of Dt ¼ 0:01 s. Clearly, the random variable realisations in the first figure bear little relation to oneanother, on average; the sequence of random variable realisations is quite erratic. On the other hand, therandom variable realisations in the displacement response are highly correlated; the displacement responserealisation is quite smooth. The random variable realisations in the velocity response bear an intermediatelevel of correlation to one another, so, the curve shown is not as erratic as the force realisation, but not assmooth as the displacement response. It may be useful and important to know the averages of the responses atspecific times t, their mean squares at times t, the average of the product of one of the responses at times t ands, and even the average of the cross-product between the displacement response at time t and the velocityresponse at time s. These quantities are particularly important for systems that execute oscillatory responses,and the reason will be discussed in the following section.

According to Uhlenbeck and Ornstein [24], Smoluchowski [88] and Furth [25] independently generalisedEinstein’s analysis and performed Brownian motion experiments to verify the predictions of the theory.Smoluchowski was first to write a form of the equation that would later be known as Fokker–Planck equationfor systems in which a displacement-related force restrains the mass. In other words, he wrote the diffusionequation governing the PDF of the response of a single-degree-of-freedom (SDF) system connected to a fixed

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m

c

x

W (t )k

Fig. 5. Schematic of the system considered by Smoluchowski in his random vibration analysis.

0 2 4 6 8 10-20

0

20

Time, sec

w (

t), l

b

0 2 4 6 8 10-10

0

10

Time, sec

x (t

), in

0 2 4 6 8 10

-2

0

2

Time, sec

v (t

), in

/sec

(a) (b)

(c)

Fig. 4. (a) Realisation of a band-limited white noise excitation. (b) realisation of displacement response of the system shown in Fig. 2 to

the excitation shown in (a). (c) Realisation of velocity response of the system shown in Fig. 2 to the excitation shown in (a).

T.L. Paez / Mechanical Systems and Signal Processing 20 (2006) 1783–1818 1789

boundary via a linear spring and viscous damper. The SDF system in which linear damping and stiffness arepresent was called the case of the ‘‘harmonically bound’’ particle, and a schematic of such a system is shown inFig. 5. But as Smoluchowski and, later, Uhlenbeck and Ornstein pointed out, the Fokker–Planck equationthat Smoluchowski developed, and its solution, govern only the response of an over-damped SDF system, i.e.,a system with relatively high damping—one that executes a non-oscillatory response in free vibrations causedby non-zero initial conditions. The over-damped solution is less interesting to structural dynamicists than thelightly damped one because most practical structural dynamic systems are lightly damped.

The limitation on the Fokker–Planck equation would later be overcome [26], but in the short term Ornstein[27] developed a method for assessing the displacement and velocity response PDFs for the linear SDF system.Ornstein’s solution was summarised in detail in the 1930 paper by Uhlenbeck and Ornstein. The methodinvolves direct consideration of the response based on its governing equation of motion

m €X þ c _X þ kX ¼W ðtÞ tX0; X ð0Þ ¼ 0; _X ð0Þ ¼ 0, (2.7)

where the parameters and variables have the same meanings as defined following Eq. (2.1), and k is the SDFsystem stiffness. Their first step was to mathematically characterize the white noise excitation. They requiredthat the random process have zero mean, E½W ðtÞ� ¼ 0; �1oto1, and a second-order average, known nowas the autocorrelation function, with the following behaviour:

E½W ðt1ÞW ðt2Þ� ¼ RWW ðt1; t2Þ ¼ 2pSWWdðt2 � t1Þ; �1ot1; t2o1, (2.8)

where dðtÞ; �1oto1, is Dirac’s unit delta function. This terminology was still not used in 1930, but it does,effectively, describe the characteristics Ornstein specified. The requirement of Eq. (2.8) indicates that theaverage of the product of the excitation at times t1 and t2 is zero for t1at2, and effectively, infinite for t1 ¼ t2.

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One interpretation of this will be given in the following section, but the important implication is that meansquare displacement response to such an excitation is finite. Uhlenbeck and Ornstein also defined some higherorder average characteristics of the excitation, but we will not summarise those here.

Any measure of the response of the system governed by Eq. (2.7) (or any linear system) can be expressedwith a convolution integral. The displacement response is

X ðtÞ ¼

Z t

0

hðt� tÞW ðtÞdt tX0, (2.9)

where hðtÞ; tX0, is the impulse response function relating the system excitation to the response measure ofinterest. In this case, the impulse response function is the displacement response following application of theunit delta function excitation, and is given by

hðtÞ ¼1

mod

e�zont sinðodtÞ tX0, (2.10)

where on ¼ffiffiffiffiffiffiffiffiffik=m

pis the natural frequency of the SDF system, the frequency in rad/s at which the system

responds in free vibration, z ¼ c=2ffiffiffiffiffiffiffikmp

is the (unitless) system damping factor (or, critical damping ratio), an

indicator of the rate at which energy is dissipated in the system, and od ¼ on

ffiffiffiffiffiffiffiffiffiffiffiffiffi1� z2

pis the damped natural

frequency, the actual frequency at which the damped SDF system responds. Uhlenbeck and Ornstein arguedthat because the expected value of excitation is zero, the expected value of the response is also zero. The idea isthat, based on Eq. (2.9), the mean of X(t) equals the mean of the integral, which equals the integral of themean of the integrand. The quantity W(t) in the integrand has a mean of zero, therefore, the integral is zero.(The formal conditions under which the mean of an integral equals the integral of the mean weremathematically clarified, later.) Next, they wrote the expression for the square of the convolution integral, Eq.(2.9), and computed the expectation to obtain the mean square displacement response:

E½X 2ðtÞ� ¼ s2X ðtÞ

¼pSWW

2zo3nm2

1� e�2zont 1þ2z2

ð1� z2Þsin2ðodtÞ þ

zffiffiffiffiffiffiffiffiffiffiffiffiffi1� z2

p sinð2odtÞ

" #( )tX0. ð2:11Þ

This equation establishes the growth of displacement response mean square with time for a linear SDFsystem. Uhlenbeck and Ornstein went on to show that certain conditions on the random excitation yield aresponse that is Gaussian distributed.

0

20

40 -20

2

0

0.5

1

1.5

2

Normalized Time, τNormalized Displacement, ξ

Nor

mal

ized

PD

F� X

(∞)

f X (�/

�n)

(�X (∞

) �)

Fig. 6. Marginal PDFs of some normal distributions of the displacement response of a linear SDF system excited by white noise, for

normalised times t ¼ 1; 10; 20; 30; 40; 50, as a function of the normalised displacements x ¼ x=sX ð1Þ.

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Fig. 6 shows a sequence of Gaussian, displacement response PDFs in normalised coordinates for a dampingfactor of z ¼ 0:05. Normalised time is t ¼ ont. The normalised displacement is x ¼ x=sX ð1Þ, wheres2X ð1Þ ¼ pSWW=2zo3

nm2, i.e., the displacement divided by the RMS response as t!1. The normalisedPDFs governing the displacement response at normalised times t ¼ 1; 10; 20; 30; 40; 50, are plotted.

Uhlenbeck and Ornstein showed many other things in their paper. They showed, for example, how tocompute a cross-correlation between the displacement and velocity responses. This is the expected value of theproduct between the displacement response at time t and the velocity response at time t. They showed that, foran SDF system, this cross-correlation starts at zero, then oscillates, and finally decays to zero. The reason isthat these two response measures tend, on average, to be ninety degrees out of phase.

This work of Uhlenbeck and Ornstein is extraordinarily important in the history of random vibrations forseveral reasons. Among them, it is the first treatment of random vibration for the under-damped case pertinentto structural dynamics. Further, it is the first analysis of response characteristics that directly uses theconvolution integral representation of the response, which is the approach that is used most frequently todayfor random vibration analyses.

Soon after publication of the 1930 paper by Uhlenbeck and Ornstein, Van Lear and Uhlenbeck [28] wroteanother paper using Ornstein’s approach to solve some problems in mechanical system random vibrations.They sought to analyse the oscillatory Brownian motion response of strings and elastic rods. Their specificobjective was to use modal analysis (or normal vibration analysis, as they called it) to decompose continuoussystem motion into a collection of components, each of which is governed by the equation for an SDF system.The simpler equations were analysed to establish mean square component response to a white noise excitation,then modal components were combined to establish mean square responses of the systems—string and rod—at various locations and at all times. Of course, the mean square responses in the various modes are transientat the start of response, then reach a steady state as time increases, at a rate dependant on damping in thesystem modes.

It is understandable that the collection of papers summarised in this section on the Brownian motion ofparticles, including particles that respond to input in an oscillatory fashion, may not have spurred theimagination of structural dynamics researchers interested in the motions of large-scale structures. However,this paper, written more than two decades prior to the time in the 1950s when widespread interest grew inrandom vibration of mechanical systems, might have served as an impetus to practical random vibrationinvestigations. It appears, though, that the paper was not widely known by structural dynamics researchers,and some of the earliest and strongest motivations for random vibrations research, namely the effects causedby jet noise and missile launch environments, did not yet exist. It appears that among those who participatedin early random vibrations research, only Lyon [29] referred to the 1931 paper by Van Lear and Uhlenbeck.For their part, Van Lear and Uhlenbeck refer to even earlier investigations into the random vibration ofstructural systems. Ornstein [30] wrote a paper on the random vibration of an elastic string, and Houdijk [31]wrote a paper on the random vibration of an elastic rod. Still, these three papers cannot really be consideredpart of the literature that inspired the popular development of random vibrations of mechanical systems.

Planck [32] and Fokker [33] used a different approach to obtain the diffusion problem formulation andsolution for Brownian motion. They started with the discrete space/discrete time framework of the randomwalk to characterize the dynamic motion of a particle in Brownian motion. Fig. 7 shows a particle on the real

Particle location at time j�t�x

Probability of moving one spatial step, �x, to the right during onetime step, �t, is p=0.5

Probability of moving one spatial step, �x, to the left during onetime step, �t, is q=1-p= 0.5

Fig. 7. The schematic idea that forms the basis for the derivation of the diffusion model from a discrete time/discrete space foundation—

the random walk.

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line. During this simple random walk a particle starts at the origin at time t ¼ 0, and moves at each step withprobability p ¼ 1=2, one step to the right, and with probability q ¼ 1=2 one step to the left. They specified thetime duration of each step as Dt, and the length of each step as Dx, so that the times under consideration aret ¼ jDt; j ¼ 0; 1; 2; . . ., and the displacements under consideration are x ¼ kDx; k ¼ 0;�1; . . . ;�j.

Let the notation f X ðtÞðxÞ ¼ f X ðjDtÞðkDxÞ; j ¼ 0; 1; 2; . . . ; k ¼ 0;�1;�2; . . . ;�j, denote the probability

that at time t ¼ jDt, following j impacts, the particle lies at the location x ¼ kDx, k steps to the right of theorigin (k can be positive or negative). If i equals the total number of steps to the right and j � i the totalnumber of steps to the left, then it must be that k ¼ i � ðj � iÞ ¼ 2i � j because k represents the net number of

steps to the right. This can be solved for i, and because x ¼ kDx, i ¼ 12ðk þ jÞ ¼ 1

2ðx=Dxþ jÞ. Because each step

in the random process involves a random binary movement, the probability distribution of the discretelocation of the particle is binomial (see [22]), and Planck and Fokker wrote

PX ðtÞ

Dx¼ k

� �¼

j

12ðj þ kÞ

!1

2

� �ð1=2ÞðjþkÞ1

2

� �ð1=2Þðj�kÞ

¼

j

12ðj þ kÞ

!1

2

� �j

¼ PðX ðtÞ ¼ kDxÞ j ¼ 0; 1; 2; . . . ; k ¼ 0;�1; . . . ;�j, ð2:12Þ

where the notationj

r

� �¼ j!=r!ðj � rÞ! refers to the binomial coefficient. Based on this, the expected value of

X(t) is zero, E½X ðtÞ� ¼ 0, and its variance is V ½X ðtÞ� ¼ ðDxÞ2j.The law of total probability indicates that the chance that the particle lies at the point x ¼ kDx at time

tþ Dt, i.e., X ðtþ DtÞ ¼ kDx, can be decomposed into two parts: the chance that the particle lies at the pointk � 1ð ÞDx ¼ x� Dx at time t times the chance that it moves one step to the right between t and tþ Dt, plus thechance that the particle lies at the point ðk þ 1ÞDx ¼ xþ Dx at time t times the chance that it moves one stepto the left between t and tþ Dt. This is

f X ðtþDtÞðxÞ ¼1

2

� �f X ðtÞðx� DxÞ þ

1

2

� �f X ðtÞðxþ DxÞ. (2.13)

This difference equation governs motion of the particle, subject to the initial condition

f X ð0ÞðxÞ ¼1 x ¼ 0;

0 xa0:

((2.14)

Planck and Fokker subtracted f X ðtÞðxÞ from both sides of Eq. (2.13) then divided the left side of the result byDt and the right side by a scaled version of Dt to obtain

f X ðtþDtÞðxÞ � f X ðtÞðxÞ

Dt¼

1

2

ðDxÞ2

Dt

ð1=2Þf X ðtÞðx� DxÞ � f X ðtÞðxÞ þ 1=2� �

f X ðtÞðxþ DxÞ

ðDxÞ2

!. (2.15)

Next, they let ðDxÞ2=2Dt ¼ D, a diffusion coefficient, and took the limit at Dt! 0 to obtain a continuousequation governing the PDF of the displacement response:

qf X

qt¼ D

q2f X

qx2tX0; �1oxo1 (2.16)

with initial condition f X ð0ÞðxÞ ¼ dðxÞ; �1oxo1. This is, of course, identical to the equation developed byEinstein, except that the present development has its origins in a scenario remote from physical diffusion.

Planck and Fokker could have stopped at this point and simply solved Eq. (2.16) for the PDF governingresponse, but instead they noted that the probability distribution of the response is already given, in its discreteform, by Eq. (2.12). The DeMoivre–Laplace Theorem (see [34]) holds that as the number of trials, j inEq. (2.12), increases the binomial distribution approaches a normal distribution with the same mean andvariance. Note that the mean and variance of X(t) were shown to be E½X ðtÞ� ¼ 0 and V ½X ðtÞ� ¼ ðDxÞ2j,therefore, because of the definition given in the previous paragraph, ðDxÞ2 ¼ 2ðDtÞD, and the variance

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approaches 2Dt. The PDF of the response is the same as that given in Eq. (2.6), but the result was derivedfrom a completely different perspective. Among other things, it is interesting to note here that the randomresponse is the accumulated result of numerous transitions, none of which is governed by a normaldistribution.

This discrete formulation was extended to cover several other cases including the one in which the particleexperiences a drift caused by a constant force superimposed over the oscillatory force, and the one in whichthe particle is harmonically bound. A summary of much of the early work is given by Kac [35].

Many special cases of the Fokker–Planck equation (written in its fully general form in [5]) have beendeveloped over the years, and work in this area continues. Kolmogorov [36] is credited with having greatlygeneralised the ideas of the Fokker–Planck equation. (Gnedenko [19] states that in the 1931 paper cited above,Kolmogorov started the ‘‘construction of the general theory of stochastic processes.’’) Also, among manyother things, Wang and Uhlenbeck [26] systematised the development of a Fokker–Planck equation for asystem governed by specific equations.

3. The development of spectral density

It is unlikely that the mathematical theory of random vibrations of mechanical systems would have becomeas popular and practically useful as it is today, had it not been for the development of spectral density byWiener [37] and those who preceded him. Spectral density (also know as mean square spectral density, powerspectral density, and by other descriptive names) is the fundamental characteristic of a weakly stationaryrandom process. It describes the distribution in the frequency domain of the mean square signal content of astationary random process.

Here, is what the terms in the previous paragraph mean, qualitatively, and why spectral density is essentialto random vibration analysis. First, an alternate way of thinking about random processes (alternate to thesequence-of-random-variables description, provided in the previous section) is to consider a random processto be an infinite ensemble (collection) of signals. Frequently, almost all the signals in the ensemble look alike,in a general sense, but, in detail, they all differ. For example, finite duration segments of two signals from theensemble of a single random process are shown in Fig. 8. They look alike in their general random character,but they differ in their details. When practically all of the signals in the ensemble of a random process are in arandom steady state from the infinite past to the infinite future, the random process is stationary. (Preciselywhat is meant by the phrase ‘‘random steady state’’ will be described in a moment. Of course, no real signalmaintains a steady state from the infinite past to the infinite future. In a practical sense, if signals from arandom source maintain a steady state for a ‘‘long time,’’ then the source is considered stationary.) The

0 0.05 0.1 0.15 0.2-5

0

5

x 1 (t

)

0 0.05 0.1 0.15 0.2-5

0

5

x 2 (t

)

Time, sec

Fig. 8. Segments of two realisations from a single random process.

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notation fX ðtÞ; �1oto1g indicates a random process defined at times in ð�1;1Þ. With the currentinterpretation this might be thought of as a collection of signals xjðtÞ; j ¼ 1; 2; . . . ; �1oto1. Each signalxjðtÞ is a realisation of the random process fX ðtÞ; �1oto1g.

Second, the stock in trade of structural dynamic analysts and experimentalists is frequency domain analysisof structures and their characteristics via Fourier expansion. Frequency domain analysis often reveals themodal characteristics of structural behaviour. The modal frequencies of a mechanical system are thefrequencies at which motion is amplified in structural response, and the mode shapes of the system arethe shapes the mechanical system naturally assumes when it responds at specific modal frequencies. For anyexcitation, it is useful to know the frequencies of substantial signal content because this information indicatesthe frequencies where substantial response might be excited in a structure. For any response, it is useful toknow the frequencies of substantial signal content because these frequencies indicate where the excitation hasat least some signal content, and the response amplified the input signal content. When the input and responseare finite duration and deterministic, direct Fourier analysis can be used to answer these questions aboutexcitation and response signal content. But when a source is a random process the ensemble of the randomprocess contains an infinite collection of signals, and further, each one has infinite duration. Infinite durationsignals do not have Fourier transforms in a traditional sense. Spectral density considers signal content in amean square sense, and in so doing enables characterization of the signal content of an infinite collection ofrandom process realisations, each of which has infinite duration.

Wiener started the process of defining the spectral density (in strict terminology, the autospectral density,for reasons that will be seen below) by defining a related function—the autocorrelation function. It is

RXX ðtÞ ¼ E½X ðtÞX ðtþ tÞ� ¼ limT!1

1

2T

Z T

�T

xðtÞxðtþ tÞdt; �1oto1, (3.1)

where xðtÞ; �1oto1, is a realisation of the random process fX ðtÞ; �1oto1g. Eq. (3.1) defines theautocorrelation function in terms of a temporal average. With this definition, it is assumed that the randomprocess realisation chosen to perform the time averaging computation in Eq. (3.1) is representative ofpractically all the other realisations in the random process. (When one arbitrarily chosen stationary randomprocess realisation can be used in the above formula to define a valid autocorrelation function, the randomprocess is said to be ergodic. See [5].) Wiener could have defined the autocorrelation function in terms of anaverage over all j of products like xjðtÞxjðtþ tÞ across the ensemble of the random process, but did not do so.An assertion of the formula in Eq. (3.1) is that the average presented is a function of t only. When thisassertion is correct for the random process, and when the mean of the random process is constant over alltime, then the random process is weakly stationary, and maintains the random steady state mentioned above.(The term ‘‘weak’’ used to describe stationarity refers to a random steady state that is characterized exclusivelyin terms of the mean and autocorrelation functions.) The function defined in Eq. (3.1) is referred to as theautocorrelation function because is defines the correlation of one random variable in the process, X(t), toanother random variable in the same random process, X(t+t). Finally, the autocorrelation function is asymmetric function in t, and it can be shown to be non-negative definite.

When a random process is weakly stationary the spectral density exists; it was defined by Wiener as theFourier transform of the autocorrelation function:

SXX ðoÞ ¼1

2p

Z 1�1

RXX ðtÞe�iot dt; �1ooo1. (3.2)

Because of the characteristics of the autocorrelation function, mentioned above, the spectraldensity is symmetric and non-negative. Eq. (3.2) is called a two-sided spectral density because it is definedfor negative as well as positive frequencies. (As mentioned earlier, negative frequencies are to be interpreted inthe sense that harmonic functions are defined for negative arguments.) Among many other things, Wienershowed that the autocorrelation function can be recovered from the spectral density via inverse Fouriertransformation:

RXX ðtÞ ¼Z 1�1

SXX ðoÞeiot do; �1oto1. (3.3)

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Because, from Eq. (3.1), RXX ðtÞ ¼ E½X ðtÞX ðtþ tÞ�, the mean square of the random process is the constants2X ¼ RXX ð0Þ. Therefore, evaluation of Eq. (3.3) at t ¼ 0 gives the mean square of the random processfX ðtÞ; �1oto1g. This is

RXX ð0Þ ¼

Z 1�1

SXX ðoÞdo. (3.4)

As Wiener pointed out, Eq. (3.4) shows that the mean square of a random process is the area under thespectral density curve, and it is clear that the integral of the spectral density over a portion of the frequencyaxis yields the part of the mean square of the random process due to contributions over that frequency range.

The function defined in Eq. (3.2) is, in strict terms, an autospectral density because it is the Fouriertransform of the autocorrelation function, and because it is the frequency domain characterization of themean square behaviour of a random process.

In his 1930, paper defining spectral density, Wiener attributed the fundamental idea underlying spectraldensity to Schuster [38–40]. He wrote: ‘‘The germs of the generalised harmonic analysis of this paper arealready in the work of Schuster, but only the germs. To make the Schuster theory assume a form suitable forextension and generalisation, a radical recasting is necessary.’’

However, some of the ideas underlying spectral density preceded Schuster’s 1899 paper. Rayleigh [41]introduced the idea that a random process has an autocorrelation, and, in a footnote, he gave an excellentexample of how and why it exists. He did not define the autocorrelation via an equation, though. In the samepaper he showed how an energy spectrum can be defined as a frequency domain description of a stationaryrandom process. He recognised the difficulty in writing the Fourier transform of a signal that extends fromminus infinity to infinity, and he multiplied the time-domain signal by a symmetric, exponential lag weightingto make the Fourier transform well-defined. He did not hint at the possibility that the autocorrelation mightbe related to the spectral density, but he certainly introduced the idea that a spectral representation can bedeveloped for a stationary source.

Schuster wrote another paper [42] that discusses the idea of a spectral representation for random signals.His topic was the characterization of light, but, as Rayleigh had done earlier, he suggested that randomsources can be described in terms of their energy spectra, and that energy over limited ranges of frequencyrepresents the content of signal components. In his 1894 paper, Schuster appears not to have accommodatedthe problem of growth of the Fourier transform magnitude of a stationary signal as greater duration signalsare represented. But in a later paper Schuster [43] indicated that the modulus of the Fourier transform of asegment of stationary random signal taken on the interval ðt; tþ TÞ increases as the square root of T when thesignal has no periodic components. Still, he did not apply the correction in his spectral characterization in the1897 paper. It was not until later that Schuster [40] inserted the correction into his definition of theperiodogram. The expression he developed did not take the limit as T !1, but it is still the earliest definitionof a quantity that is very similar to the spectral density in use today.

During the time frame of the early 1900s and later, Rayleigh [44–47] wrote many other papers thatconsidered the spectral representation and probability distributions of random signals.

The first mathematical definition of the autocorrelation function appears to have been written by Taylor[48]. In a paper that considered the diffusion of heat in fluids in turbulent motion, he defined a random processthat proceeds in discrete, temporal steps. He defined the correlation between random variables in the processat adjacent steps and inferred a correlation structure for the process.

To understand in a practical and intuitive sense the meaning of spectral density, consider the schematic ofFig. 9. It indicates that every real signal, like the one shown at left in the figure, can be filtered into a finitenumber of components. A sequence of idealised filter gain functions is shown in the second column. The filtersare non-overlapping, band-pass, equal width, and vary from low to high frequency, top to bottom. Somecomponents of the signal at left are shown by the third column of signals. The signals vary from the lowestfrequency component at the top to the highest frequency component at the bottom. The mean square value ofeach signal can be estimated by squaring a long-duration segment of each component and computing theaverage. The mean square values of the signal components are plotted as a function of band-pass filter centrefrequencies in the fourth column of figures. Finally, the mean square values of the signal components aredivided by the effective filter bandwidths to obtain the estimate of spectral density of the random source from

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SignalMean

SquaresSpectralDensity

Time

TimeFreq

Freq Freq

SignalComponents

Bandpass Filters

Fig. 9. Schematic describing the meaning of spectral density.

T.L. Paez / Mechanical Systems and Signal Processing 20 (2006) 1783–18181796

which the measured signal arose. Of course, the units of the spectral density are the units of the measuredsignal squared per unit of frequency (usually Hz). This is how spectral density is related to random source. Theexplanation is meant only to be a rough description of the meaning of spectral density. Issues like the filtertype, shape and widths, and the measured signal duration are all important. Even so, a physicalimplementation of the process described here was used to estimate the spectral density of measured sourcesinto the 1960s and 1970s. Miles [49] mentions the need to obtain a spectral density estimate fromexperimentally measured signals, in order to perform random vibration analyses, and suggests a method likethe one outlined here. A paper by Rona [50] describes the process, shown graphically, here, for estimation ofspectral density of mechanical system motions.

Here are some examples of random process realisations, their autocorrelation functions, and spectraldensities. First, a zero-mean, wide-band random process fX ðtÞ; �1oto1g, is considered in Fig. 10a. It hasthe one-sided spectral density GXX ðf Þ; fX0, shown on the right. (The one-sided spectral density is defined onnon-negative frequencies and has twice the magnitude of the two-sided spectral density. The one-sided spectraldensity is traditionally used to describe the results of laboratory and field signal analyses, and it is used in mostplots of spectral density.) The random process is termed wide-band because the frequency band over which thespectral density has non-zero values is wide relative to the ‘‘centre’’ frequency. This particular random processis known as a band-limited white noise, by analogy to visible light, because the density of mean square signalcontent is evenly distributed over all the frequencies where it is non-zero. Such a random process hasrealisations most of which resemble the signal shown on the left. The autocorrelation function of a wide-bandrandom process, RXX ðtÞ; �1oto1, is narrow and sharp, as shown by the middle figure. It indicates thatthe random process loses correlation with itself over a very short time lag. Second, a zero-mean, narrowbandrandom process is considered in Fig. 10b. It has the one-sided spectral density shown on the right. The randomprocess is called narrowband because the frequency band over which the spectral density has non-zero valuesis narrow relative to the ‘‘centre’’ frequency. Such a random process has realisations most of which resemblethe signal shown on the left. The autocorrelation function of a narrowband random process is periodic anddecaying, as shown by the middle figure. It indicates that the random process has correlation with itself that isnearly periodic, and that diminishes slowly. The period of the autocorrelation function matches the averageperiod of the realisations, and these are the reciprocals of the ‘‘centre’’ frequency of the spectral densitymeasured in Hertz. Third, the response random process of a three degree-of-freedom (dof) system isconsidered in Fig. 10c. It has the one-sided spectral density shown on the right. There is a peak in the spectral

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-0.2 0 0.2

0

0.5

1

100 101 10210-4

10-2

0 0.5 1

-2

0

2

Time, sec Time Lag,τ Frequency, Hz

x (t )R

XX (�) S

XX (f )

-0.2 0 0.2

-1

0

1

100

100

101 10210-4

10-2

100

0 0.5 1-2

0

2

Time, sec

0 0.5 1Time, sec

Time Lag, τ

Time Lag, τ

Frequency, Hz

100 101 102

Frequency, Hz

x (t ) RXX

(τ) SXX

(f )

-0.5 0 0.5

-2000

200400

-50

0

50 x (t ) RXX

(τ) SXX

(f )

(a)

(b)

(c)

Fig. 10. (a) Band-limited white noise random process. realisation (left), autocorrelation function (centre), spectral density (right).

(b) Narrowband random process. realisation (left), autocorrelation function (centre), spectral density (right). (c) Random response of three

dof system. realisation (left), autocorrelation function (centre), spectral density (right).

T.L. Paez / Mechanical Systems and Signal Processing 20 (2006) 1783–1818 1797

density for each mode in the system that is substantially excited. The frequencies where the peaks occurindicate the modal frequencies of the system. Such a random process has realisations most of which resemblethe signal shown on the left. The realisations of this random process are superpositions of three narrowbandrandom processes. The autocorrelation function of the random response of the three dof system is thesuperposition of autocorrelation functions of three narrowband random processes with centre frequencies atthe modal frequencies of the system, as shown by the middle figure. In most situations it is difficult to interpretthe character of a stationary random process using response realisations or autocorrelation function, butrelatively easy to interpret stationary random process mean square signal content using spectral density.

A random process known as an ideal white noise is one whose two-sided spectral density is finite andconstant on ð�1;1Þ. Indeed, the excitation characteristics described mathematically by Uhlenbeck andOrnstein [24] imply a white noise excitation, though not in those words. Even prior to that description, theassumptions made by Einstein [21] in his early development of a Fokker–Planck equation imply anassumption of white noise-type excitation.

It was stated following Eq. (3.4) that the area under the spectral density curve is the mean square of a zero-mean random process, therefore, an ideal white noise has infinite mean square. In spite of this, the ideal whitenoise model is important because some measures of the mean square response of stable linear systems to whitenoise are finite. For example, the mean square displacement and velocity responses of force-excited, fixed-base,linear structures to a white noise excitation are finite. For this reason, the white noise excitation model is usedeven today to perform relatively simple, yet accurate, analyses. Part of modern analyses is the development ofthe relation between the spectral density of a random excitation and the spectral density of the response itexcites. In view of this, an understanding of the history of random vibrations requires knowledge of the originsof input/output spectral density relations.

Wiener pursued this topic in his 1930 paper. He did so in a far-ranging section entitled ‘‘Spectra andIntegration in Function Space.’’ Without writing the formula, he stated the relation that is most fundamentalto the modern practice of random vibrations of linear mechanical systems. He stated the result for a whitenoise excitation in three ways; here are two of them. First, he wrote, ‘‘the spectral density of [random linearsystem response] is half the square of the modulus of the Fourier transform of [the system impulse response

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function].’’ Later, ‘‘if a linear resonator is stimulated by a uniformly haphazard sequence of impulses, eachfrequency responds with an amplitude proportional to that which it would have if stimulated by an impulse ofthat frequency and of unit energy.’’ The excitation he is referring to is a type of white noise. (See discussions ofshot noise in [34].) The Fourier transform of the system impulse response function is the system frequencyresponse function (FRF). The FRF of a linear system is the coefficient of proportionality relating magnitudeand phase of a harmonic response component to magnitude and phase of a harmonic input component. Thatis, H(o) is the FRF in the relation

X ðoÞ ¼ HðoÞQðoÞ; �1ooo1, (3.5)

where QðoÞ ¼R1�1

qðtÞe�iot dt; �1ooo1, is the Fourier transform of a system excitation, and

X ðoÞ ¼R1�1

xðtÞe�iot dt; �1ooo1, is the Fourier transform of a linear system response.

When fW ðtÞ; �1oto1g is a zero-mean, ideal white noise random process with spectral densitySWW ðoÞ ¼ SWW ¼ constant; �1ooo1, Wiener’s statement is

SXX ðoÞ ¼ jHðoÞj2SWW ; �1ooo1, (3.6)

where SXX ðoÞ; �1ooo1, is the spectral density of the response random process fX ðtÞ; �1oto1g. Forexample, when a linear, SDF system with mass, m ¼ 1:0 lb 2=in, damping factor, z ¼ 0:05, and naturalfrequency on ¼ 2p rad=s, is excited by a zero-mean, ideal white noise with spectral density SWW ¼ 1 lb2=Hz,the displacement response spectral density is

SXX ðoÞ ¼1

m2ððo2n � o2Þ

2þ ð2zonoÞ

¼1

ððð2pÞ2 � o2Þ2þ ð0:1ð2pÞoÞ2Þ

; �1ooo1 ð3:7Þ

because

HðoÞ ¼1

mðo2n � o2 þ 2izonoÞ

; �1ooo1. (3.8)

This response spectral density is graphed in Fig. 11.It appears that Carson [51] had previously defined the input/output relation for spectral densities in work

Wiener was not aware of. Carson was motivated by a need to describe the effects of noise on an electricalcommunications system. He defined what he called the energy spectrum of random interference, as the modulussquared of the Fourier transform of a finite duration segment of a random process realisation, divided by theduration of the segment. This quantity is essentially the same as Schuster’s [40] definition of the periodogram,and, as Schuster, Carson failed to take the limit as T !1. But Carson wrote the expression for the meansquare response of a linear system to the random excitation in terms of the energy spectrum. Essentially, hewrote Eq. (3.6), but he wrote it in terms of the linear system impedance, the inverse of the system FRF. A fewyears later, Carson [52] modified the definition of what he now called the energy-frequency spectrum by takingthe limit as T !1. He re-wrote the input/output spectral density relations for a linear system in the 1931

10-1 100

10-5

100

Frequency, Hz

SX

X (

f), i

n2/H

z

Fig. 11. Spectral density of the response of an SDF system.

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paper, and, in addition, proposed several shot noise-type models (see [5]) for the electrical noise. He venturedthat the energy-frequency spectrum of practical noise sources must be wide-band and slowly varying.

Wiener went much further in his 1930 paper and defined what is now known as the cross-spectraldensity. He called it coherency, and used it to define frequency domain relations among random processesin a quadratic mean sense. He developed the cross-spectral density in terms of cross-correlations betweenrandom processes. Consider a collection of M zero-mean, stationary, ergodic random processesfX mðtÞ; �1oto1g; m ¼ 1; . . . ;M. Each random process consists of an infinite collection of realisations,and one of the realisations in any of the random processes is representative of practically all the otherrealisations in the random process. The signals xmðtÞ; m ¼ 1; . . . ;M ; �1oto1, are realisations, onefrom each of the M random processes, respectively. He defined a function, one that we now call the cross-correlation function between a pair of random processes fX jðtÞ; �1oto1g; j 2 ð1; . . . ;MÞ, andfX k tð Þ; �1oto1g; k 2 ð1; . . . ;MÞ, as

RX jX kðtÞ ¼ lim

t!1

1

2T

Z T

T

xjðtÞxkðtþ tÞdt; j; k 2 ð1; 2; . . . ;MÞ; �1oto1. (3.9)

Normally, we consider jak. Wiener defined the cross-spectral density as the Fourier transform of the cross-correlation function:

SX jX kðoÞ ¼

1

2p

Z 1�1

RX jX kðtÞe�iot dt; j; k 2 ð1; 2; . . . ;MÞ; �1ooo1. (3.10)

He described many of the characteristics of the cross-spectral density, and noted that the matrix of all thecross-spectral densities of the random processes fX mðtÞ; �1oto1g; m ¼ 1; . . . ;M, namely,

SXX ðoÞ ¼

SX1X1ðoÞ SX1X2

ðoÞ � � � SX1XMðoÞ

SX2X1ðoÞ SX2X2

ðoÞ � � � SX2XMðoÞ

..

. ... . .

. ...

SXM X1ðoÞ SX M X2

ðoÞ � � � SXM XMðoÞ

2666664

3777775; �1ooo1 (3.11)

‘‘determines the spectra of all possible linear combinations of [the random processes].’’He noted that the cross-spectral density matrix is Hermetian (i.e., SXjXk

ðoÞ ¼ S�X kXjðoÞ), and that every

cross-spectral density matrix can be diagonalized. He pointed out that this makes it easy to generate correlatedrandom processes starting with completely uncorrelated random processes—a fact that has been rediscoveredmany times. This is important, for example, in the laboratory-experimental generation of coherent, stationaryrandom signals for multi-axis testing. (See [53,54].)

In terms of the cross-spectral density, Wiener defined what he called the coefficient of coherence as

gXjXkðoÞ ¼

SXjXkðoÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

SXjXjðoÞSX kXk

ðoÞp ; j; k 2 ð1; 2; . . . ;MÞ; �1ooo1, (3.12)

where SXjXjðoÞ and SXkX k

ðoÞ are the autospectral densities of the two random processes. He stated, ‘‘Themodulus of the coefficient of coherency represents the amount of linear coherency between [the two randomprocesses], and the argument [phase] the phase lag of this coherency.’’ Today, we frequently define a quantitythat is the modulus squared of the expression in Eq. (3.12) and call it the coherence. It is used frequently withexperimentally measured data to judge the extent of linear relation between a pair of random processes. (See,for example, [18].)

The use of the cross-spectral density to define the coefficient of coherency is useful and interesting, but themost important use for the cross-spectral density involves the estimation of the FRF. It appears that this is onevery important application that Wiener did not explicitly include in his 1930 paper. It happens that thefrequency domain input/output relation for a linear system, Eq. (3.5), above, can be used to develop a relationamong the excitation autospectral density, SQQðoÞ; �1ooo1, the cross-spectral density between theresponse and the excitation, SXQðoÞ; �1ooo1, and the FRF, HðoÞ; �1ooo1, of the linear system:

SXQðoÞ ¼ HðoÞSQQðoÞ; �1ooo1. (3.13)

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When the auto- and cross-spectral densities in the equation are estimated, the FRF can be inferred. Wienerdid suggest methods for estimating auto- and cross-spectral densities, though they are not the approaches usedin the current digital age. Details on this sort of estimation can be obtained in any text that discusses randomsignal analysis, for example, [18], or Wirsching et al. [17].

It is inappropriate to end this section without noting that the spectral density was independently defined in1934 by Khintchine [84]. He also defined the autospectral density as in Eq. (3.2), and in honour of his work,and, of course, that of Wiener, Eqs. (3.2) and (3.3) are usually known as the Wiener–Khintchine relations. Infact, many papers and texts, especially those written in Russia, acknowledge Khintchine as the person whodeveloped spectral density.

4. More foundations for random vibrations

The early and continued development of the Fokker–Planck equation, the development of direct analysistechniques for random vibration, and the definition of the spectral density all formed a foundation sufficientto develop a modern theory of random vibrations of mechanical systems. Yet many elements that would proveextraordinarily useful to the solution of real problems of mechanical systems were not present as of 1944. Forexample, the modern practice of random vibration testing of physical systems and the Monte-Carlo analysis ofcomplex, perhaps non-linear systems, require the generation of random process realisations. (Monte-Carloanalysis involves (1) the specification of a random process model for excitation, (2) the generation of excitationrandom process realisations, (3) the deterministic computation of the response of a system to the generatedexcitations, and (4) the statistical analysis of the family of generated responses.) The framework for doing sohad not been developed.

In a far ranging paper, Rice [55] developed techniques to perform these analyses, and many others,including a thorough investigation of the shot effect, an investigation into many measures of peak response,and an investigation of the averages and probability distributions of the outputs of non-linear devices excitedby noise inputs. (The shot effect is observed in phenomena which give rise to sequences of pulses that occur atrandom times, with potentially random amplitudes, and potentially random shapes. Any random process withsuch realisations is called a shot noise. A good summary of several types of shot noise is given in [52].)

We consider, first, representations for a stationary random process that can be used to generate stationaryrandom process realisations. Rice developed two methods for accomplishing this. He had preceded thisdevelopment with the analysis of a shot noise random process, and stated that ‘‘the Fourier seriesrepresentation of the shot effect current y suggests the representation’’

X ðtÞ ¼XN

k¼1

ðAk cos oktþ Bk sin oktÞ; �1oto1 (4.1)

for the random process fX ðtÞ; �1oto1g, where

ok ¼ 2pf k f k ¼ kDf k ¼ 1; . . . ;N (4.1a)

and the Ak; Bk; k ¼ 1; . . . ;N, are zero-mean, uncorrelated, normally distributed random variables withvariances DfGXX ðf kÞ; k ¼ 1; . . . ;N. The function GXX ðf Þ; 0pfpf max is the one sided spectral density of therandom process. Normally, f N is defined so that f N ¼ f max. He showed that if the ensemble of random processrealisations generated with Eq. (4.1) was to have the desired spectral density, then the random variables musthave the stated variance.

Realisations of the stationary random process can be generated by generating realisations of theAk;Bk; k ¼ 1; . . . ;N, and using them in place of the Ak;Bk; k ¼ 1; . . . ;N, in Eq. (4.1).

He also mentioned in a footnote that ‘‘this sort of representation was used by Einstein and Hopf forradiation.’’

The representation of Eqs. (4.1) and (4.1a) is usually associated with Rice because of his 1944 and 1945papers. For example, Wang and Uhlenbeck [26] include a section entitled ‘‘The Gaussian Random Process;Method of Rice.’’ But the idea of representing a random process and its realisations in a Fourier expansiongoes back at least as far as Rayleigh [41,44–47] and Schuster [40,42,43]. Both authors used the harmonic

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representation to motivate the idea of a spectral representation of a random process, but the harmonicrepresentation can be used for much more. In their 1945 paper, Wang and Uhlenbeck use it to discuss theprobability distribution of one random variable in a stationary, Gaussian random process; the jointprobability distribution of a stationary, Gaussian random process and its derivative at a fixed time, t; the jointprobability distribution of two separate random variables in a stationary, Gaussian random process; and thejoint probability distribution of any finite collection of random variables in a stationary, Gaussian randomprocess and/or its derivatives.

In what appears to be his only reference to the input/output relation for spectral density, he states ‘‘supposewe are interested in the output of a certain filter when a source of thermal noise is applied to the input. LetjHðf Þj be the absolute value of the ratio of the output current to the input current when a steady sinusoidalvoltage of frequency f is applied to the input. Then

SXX fð Þ ¼ C H fð Þ�� ��2:’’ (4.2)

He went on to explain that this formula can be used to establish C, the spectral density of a broadband inputwhen both jHðf Þj and SXX ðf Þ are known.

He also suggested an alternate representation for the stationary random process that is trigonometricallyequivalent to Eq. (4.1), yet different in its details:

X ðtÞ ¼XN

k¼1

Ck cosðokt� fkÞ; �1oto1, (4.3)

where fk; k ¼ 1; . . . ;N, are independent random variables, uniformly distributed on ð0; 2pÞ, theCk; k ¼ 1; . . . ;N, are defined as constants

Ck ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2DfGXX ðf kÞ

p; k ¼ 1; . . . ;N (4.3a)

and the ok and fk are defined as in Eq. (4.1a). This definition uses harmonic components with fixed amplitudes,and random phase angles. When N is sufficiently large, the random process in Eq. (4.3) has a normaldistribution by virtue of the Central Limit Theorem. (See [34].)

Realisations of the stationary random process can be obtained by generating realisations of thefk; k ¼ 1; . . . ;N, and using them in place of the fk; k ¼ 1; . . . ;N, in Eq. (4.3). This formula is used togenerate random process realisations for many applications, including the generation of random processrealisations for laboratory testing.

Much of the material in Rice’s paper deals with crossing rates of stationary random processes, and theprobability distribution of peaks values in stationary random processes and their envelopes. We will consideronly his work on level-crossings. But first, it is necessary to summarise another part of Rice’s work. Much ofthe work that Rice did depends on the joint distribution of the response and its derivatives. And the work wewill summarise below depends on the joint probability distribution of the random process and its firstderivative, i.e., the displacement and velocity responses. The zero-mean random process fX ðtÞ; �1oto1gwith spectral density SXX ðoÞ; �1ooo1, has the mean square

s2X ¼Z 1�1

SXX ðoÞdo (4.4)

as shown in Section 3. In addition, though, Rice showed that the derivative of the random processfX ðtÞ; �1oto1g has the means square

s2_X ¼Z 1�1

o2SXX ðoÞdo (4.5)

if the random process derivative exists in some probabilistic sense.Rice also showed (Uhlenbeck and Ornstein [24] had shown this previously) that the displacement and

velocity random processes are uncorrelated.The level crossing problem addresses the number of times that a stationary random processfX ðtÞ; �1oto1g crosses the level x ¼ a in the time interval ðt; tþ dtÞ. The problem is critical in structural

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Time, t

x (t)

a

Level x=a up-crossing Peak

Valley

Zero up-crossing

Zero down-crossing

Fig. 12. Schematic diagram indicating terminology with reference to level crossings and peaks.

x (t)

tdt

x (t ) dt•

a

Fig. 13. Schematic that shows signal conditions when an up-crossing of the level xðtÞ ¼ a occurs.

T.L. Paez / Mechanical Systems and Signal Processing 20 (2006) 1783–18181802

dynamics because of its association with failure analysis of certain system responses. With reference to Fig. 12,let Naþðt; tþ dtÞ denote the discrete, random number of up-crossings of the level x ¼ a by the random processin the time interval ðt; tþ dtÞ. Naþðt; tþ dtÞ is a discrete random variable with a probability mass function, i.e.,a probability that the random variable equals 0, a probability that the random variable equals 1, etc.

Rice assumed that the time interval dt can be defined such that either zero or one up-crossings occur inðt; tþ dtÞ. We denote the up-crossing rate of the level x ¼ a as naþ . Because the random process is stationary,the number of up-crossings depends directly on the interval length dt. These assumptions lead to

E½NaþðdtÞ� ¼ naþ dt ¼ Pðup�crossingÞ. (4.6)

Consider Fig. 13. An up-crossing occurs in ðt; tþ dtÞ when

X ðtÞoa; _X ðtÞ40; X ðtÞ þ _X ðtÞdt4a. (4.7)

The probability of up-crossing is

Pðða� _X ðtÞdtoX ðtÞoaÞ \ ð _X ðtÞ40ÞÞ. (4.8)

The region of integration implied by the probability of Eq. (4.8) is shown in Fig. 14. The probability isZ 10

dv

Z a

a�vdt

dx f X _X ðx; vÞ. (4.9)

Combining Eqs. (4.6) and (4.9) yields

naþ ¼

Z 10

dv

Z a

a�v dt

dx f X _X ðx; vÞ. (4.10)

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x

x.

a

a–x dt.

Fig. 14. Region of integration implied by the probability statement of Eq. (4.4).

T.L. Paez / Mechanical Systems and Signal Processing 20 (2006) 1783–1818 1803

When the random process is Gaussian distributed, the mean up-crossing rate of the level x ¼ a is

naþ ¼1

2ps _X

sX

exp �a2

2s2X

� �. (4.11)

The mean up-crossing rate of the level x ¼ 0 is

n0þ ¼1

2ps _X

sX

¼1

2p

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR1�1

o2SXX ðoÞdoR1�1

SXX ðoÞdo

s. (4.12)

Later, researchers in random vibrations and many other fields used this and other results developed by Riceto characterize system responses and their probabilities of barrier crossings. For example, see Siebert [56],Crandall [57], and Powell [58].

5. The popularization of random vibration of mechanical systems

The modern field of random vibrations of mechanical systems and probabilistic structural dynamics, ingeneral, has gained importance as the awareness that real mechanical environments are stochastic hasbroadened. Today random vibration analyses are performed frequently and in practical settings, usuallywithin the framework of a commercial finite element code. Commercial finite element codes include, ingeneral, rather limited capabilities to perform probabilistic structural dynamic analyses, at least without muchpre- and post-processing. The most common analyses are those wherein the auto- (and, perhaps, cross-)spectral densities of stationary excitation random processes are specified, and response auto- and cross-spectral densities are computed. Some of the developments that led to the formulas that have beenimplemented are summarised below.

Crandall is normally credited as the person (at least, in the United States) who made the topic of randomvibrations of mechanical systems accessible to practicing engineers. He organised a summer programme at theMassachusetts Institute of Technology devoted to presentations on the fundamental topics in randomvibrations. The presentations are published in Crandall [1], and cover analysis of and design for randomvibrations, testing, data analysis, spectral density estimation, and other topics. Chapters 1, 2, and 4, in theproceedings present an introduction to random vibrations on a most elementary level. Because of theirhistorical significance, and, in particular, the historical significance of Chapter 4, those three chapters will besummarised first. Next, a number of papers whose publication actually preceded the MIT summer programmewill be discussed. These papers contain some of the earliest modern efforts to develop methods for the analysis

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of random vibration of mechanical systems. Then we will return briefly to the proceedings of the summerprogramme to define the apparent extent of understanding of random vibrations as of 1958.

The first chapter in the programme proceedings by Crandall [59], entitled ‘‘Mechanical Vibrations withDeterministic Excitations,’’ develops, very briefly, the ideas of impulse response function and FRF, and it usesthese to write expressions for the responses of linear systems in terms of the convolution integral and itsFourier transform. The time-domain expression for scalar, transient response, xðtÞ; tX0, is

xðtÞ ¼

Z t

0

qðtÞhðt� tÞdt; tX0, (5.1)

where qðtÞ; tX0, is the scalar excitation to the system, and hðtÞ; tX0, is the system unit impulse responsefunction (also simply called the impulse response function). Eq. (5.1) is the same as the convolution integral ofEq. (2.9). When there is a single excitation under consideration and when response at a single point is ofinterest, then the functions xðtÞ; qðtÞ; hðtÞ are scalar. The equation can be written for any measure of responsedesired, for example, displacement, velocity or acceleration. The excitation can be any quantity of interest, forexample, force or imposed motion. The impulse response function is the response of the system at the locationwhere xðtÞ; tX0, is measured to a unit-impulse excitation (a unit delta function) applied at the location whereqðtÞ; tX0, is applied. Crandall’s development and explanation of Eq. (5.1) was very intuitive.

When the excitation to the system is applied starting at time less than zero, perhaps at t! ð�1Þ, then thelower limit on the integral in Eq. (5.1) can be changed to ð�1Þ and the upper limit can be changed to N. TheFourier transform of the resulting expression can be taken to obtain the frequency-domain equivalent to theconvolution integral. It is

X ðoÞ ¼ HðoÞQðoÞ; �1ooo1, (5.2)

where X ðoÞ; �1ooo1, HðoÞ; �1ooo1, and QðoÞ; �1ooo1, are, respectively, the Fouriertransforms of xðtÞ; �1oto1, hðtÞ; tX0, and qðtÞ; �1oto1, defined

X ðoÞ ¼Z 1�1

xðtÞe�iot dt; �1ooo1,

HðoÞ ¼Z 10

hðtÞe�iot dt; �1ooo1,

QðoÞ ¼Z 1�1

qðtÞe�iot dt; �1ooo1. ð5:3Þ

The function HðoÞ; �1ooo1, is called the FRF of the system, and it is the fundamentaldescriptor of linear system behaviour in the frequency domain. It is the factor by which aharmonic excitation can be multiplied to obtain the harmonic response of a linear system at a singlepoint. Its magnitude is the scale factor between input and response amplitudes, and its phase is the phasedifference between input and response. Crandall’s derivation of Eq. (5.2) was also very intuitive. He developedEqs. (5.1) and (5.2) because they form the basis for the fundamental input/output relations for linear randomvibrations.

The second chapter in the programme proceedings was written by Siebert [56], and is entitled ‘‘TheDescription of Random Processes.’’ It provides an introductory discussion of the ideas of probability, randomprocesses, moments, correlation functions and spectral density. It seeks to establish a motivation fordescribing structural dynamic excitations and responses as random processes.

Chapter 4 in Crandall [57], entitled ‘‘Statistical Properties of Response to Random Vibration,’’ starts withthe expression of linear system response in terms of the convolution integral and its Fourier transform, Eqs.(5.1) and (5.2). He proceeds to develop the single-input/single-output relations for randomly excited linearsystems in terms of integrals involving autocorrelation and impulse response functions, and autospectraldensities and FRFs. Specifically, he derived the expression for the response autocorrelation functionRXX ðtÞ; �1oto1, of the scalar, stationary random process fX ðtÞ; �1oto1g.

RXX ðtÞ ¼Z 1�1

dt1

Z 1�1

dt2hðt1Þhðt2ÞRQQðtþ t2 � t1Þ; �1oto1, (5.4)

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where RQQðtÞ; �1oto1, is the autocorrelation function of the stationary excitation random processfQðtÞ; �1oto1g. Intuitive interpretation of Eq. (5.4) is difficult, except, perhaps, to note thatRXX ð0Þ ¼ s2X , is the mean square of the response random process when the means of the excitation andresponse are zero.

In the same chapter, Crandall also derived the expression for response spectral densitySXX ðoÞ; �1ooo1, of the stationary random process fX ðtÞ; �1oto1g. It is obtained through Fouriertransformation of Eq. (5.4):

SXX ðoÞ ¼ jHðoÞj2SQQðoÞ; �1ooo1, (5.5)

where SQQðoÞ; �1ooo1, is the autospectral density of the stationary, excitation random process. Eq.(5.5) describes the distribution of the mean square of the response of a randomly excited system in thefrequency domain. It is much easier to visualise than Eq. (5.4), and is, Crandall said, the ‘‘central result’’ ofrandom vibrations. Using the inverse Fourier transform, Crandall next wrote the expression for the responseautocorrelation function as

RXX ðtÞ ¼Z 1�1

jHðoÞj2SQQðoÞeiot do; �1oto1. (5.6)

When evaluated at t ¼ 0, it shows that the response mean square is strongly influenced by structural systemmodes, the frequencies where the greatest amplifications occur in the FRF, HðoÞ; �1ooo1, and the meansquare distribution of signal content in the stationary excitation. The expression is

RXX ð0Þ ¼ s2X ¼Z 1�1

jHðoÞj2SQQðoÞdo. (5.7)

Following these developments, Crandall discussed the probability distribution of response in the Gaussianexcitation case. He developed expressions for the response of an SDF system to ideal white noise, and hediscussed how these results might be applied in the practical case where the excitation is not ideal white, butonly relatively constant in the vicinity of the natural frequency of a lightly damped structure. He went on toapply one of the techniques of Rice [55], the technique for estimating the number of zero-crossings (and peaks)in a narrowband random process, to the response of a lightly damped, SDF system.

The developments of researchers who investigated the random vibration responses of linear systems, severalyears before Crandall’s workshop, are, in some cases, much more detailed and complex. Some of thosedevelopments will be summarised in a moment. But this seems to indicate that Crandall sought to keep hisdevelopments and results simple and clear, and he certainly accomplished that goal.

The result in Eq. (5.5) is, in a practical sense, the most important result in linear random vibrations becauseit is the formula applied in the vast majority of practical analyses. Indeed, practitioners seeking to characterizethe random vibration response of systems which they are willing to approximate as linear use the formulas inthe following way. Consider a mechanical system that has been either experimentally or analytically evaluated,and whose FRF relating acceleration response at one location to force excitation at another location has themodulus shown in Fig. 15. The system motion may involve multiple modes, but the displayed FRF indicatesthat three of them are dominant—the ones at 821, 1381, and 1706Hz. Suppose that an analyst or designer

102 103

100

Frequency, Hz

|H (

f)|,

g/l

b

Fig. 15. Modulus of the FRF of acceleration response on a system to a force excitation.

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102 103100

101

102

Frequency, HzG

QQ

(f)

, lb2 /

Hz

Fig. 16. Spectral density of the system excitation.

102 103

100

Frequency, Hz

GX

X (f

), g

2/H

z

Fig. 17. Spectral density of the acceleration response of the system.

T.L. Paez / Mechanical Systems and Signal Processing 20 (2006) 1783–18181806

wishes to characterize the response motion via its spectral density to the excitation whose spectral density isspecified by the curve of Fig. 16. She or he would compute the product of the curves in Figs. 15 and 16 toobtain the approximate spectral density of the response shown in Fig. 17. Many features of the responsecharacter can be inferred from its spectral density, but perhaps most important, is the fact that its RMS valueis approximately 22 g. (This is the square root of the area under the response spectral density curve.) The RMSvalue of the excitation is approximately 181 lb. As Crandall explained, if the acceleration excitation isnormally distributed with zero mean, then the response at each time is a normally distributed random variablewith zero mean and standard deviation 22 g. Gaussian realisations of the excitation and response randomprocesses can be obtained using a process described, for example, by Wirsching et al. [17]. That was done here,and the excitation and response realisations are shown in Figs. 18 and 19.

Because Crandall’s is one of the earliest popular works on the subject of random vibrations of mechanicalsystems, it is important to note the references he provides as the sources for his work. He cites Davenport andRoot [60] for ideas in random processes, but more importantly, he cites Laning and Battin [61] for ideas in theresponse of linear systems to random excitation. Their book arose from a set of lecture notes on randomprocesses in the field of automatic control, first offered at MIT in 1951. They devote an entire chapter to thesubject of ‘‘Analysis of Effects of Time-Invariant Linear Systems on Stationary Random Processes.’’ In thatchapter, they express the autocorrelation function and autospectral density of the response of a linear systemto a stationary random excitation, and this is apparently the source of Crandall’s corresponding expressions.They also refer to the idea of filtering of random processes and mention the relation of that activity to theanalysis of system response. However, Lanning and Battin do not explicitly cite the source for their approachto the development of the input/output spectral density relation for randomly excited linear systems.

The reference to filtering by Lanning and Battin is interesting in view of the fact that Siebert [56], in thechapter mentioned above, also briefly develops the input/output spectral density relation for randomly excitedlinear systems, and he references an early source for that development as James, Nichols and Phillips [83].Their text is written on the subject of servomechanisms. The input/output spectral density relations aredeveloped in a chapter entitled ‘‘Statistical Properties of Time-Variable Data.’’ It was written by Phillips [62]with reference to the input/output relations for servomechanisms, and with much reference to the terminologyof signal filtering. Though the equations relating the spectral density of a stationary, white noise random input

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0.04 0.045 0.05 0.055 0.06-500

0

500

Time, secq

(t),

lb

Fig. 18. Realisation of the excitation random process.

0.04 0.045 0.05 0.055 0.06-100

0

100

Time, sec

x(t)

, g

Fig. 19. Realisation of the response random process.

T.L. Paez / Mechanical Systems and Signal Processing 20 (2006) 1783–1818 1807

to the spectral density of the output of a linear system had appeared several years earlier, no reference is madeby Phillips to any other derivation.

As mentioned in Section 3, an early development of the input/output spectral density relation for linearsystems is to be found in Wiener [37] in his paper defining spectral density. The 13th section of his paper isentitled ‘‘Spectra and Integration in Function Space,’’ and in that section he proves that ‘‘if a linear resonatoris stimulated by a uniformly haphazard sequence of impulses, each frequency responds with an amplitudeproportional to that which it would have if stimulated by an impulse of that frequency and of unit energy.’’This is a statement of Eq. (5.5) for the case of a white noise excitation—the case whereSQQðoÞ ¼ SQQ ¼ constant; �1ooo1. Still, if the importance of his development to practical applicationshad been recognised, it may have been applied more broadly, and earlier.

Another early development of the input/output spectral density relation for linear systems excited by idealwhite noise is the one given by Wang and Uhlenbeck [26]. That paper was not written with reference tomechanical systems in particular, but with reference to linear systems, in general. In it, the authors focus mostof their attention on finding the probability distributions of the solutions to white noise excited, first andsecond order linear differential equations, and systems of second-order linear equations using theFokker–Planck equation approach. However, they did write the input/output spectral density relation for afirst-order linear system and for a second-order linear system excited by white noise. The first order linearsystem they considered is a massless particle connected to a fixed boundary via a spring and damper, andexcited by white noise. They did not write the relations in a general way using FRF or impedance, and they didnot discuss the implications of the expressions for interpreting signal content of the response mean square.Nevertheless, the relations are presented, perhaps, for the first time since Weiner wrote them. Whether or notthey independently rediscovered the input/output spectral density relation is not clear, but their footnotes andcomments on other parts of the work of Weiner appear not to credit him with the idea.

Finally, as mentioned in Section 3, the earliest reference (found by the author) to an input/output spectraldensity relation for linear systems is the one given by Carson [51] in his analysis of electrical systems. Hisexpression of the input/output relation has its limitations—namely, it is not based on expressions for spectraldensity defined as T !1, and it is not written explicitly—but it does convey the correct concept for theinput/output spectral density relation, and it does so for a fully general (non-white noise) input randomprocess.

Many papers on the subject of random vibrations of mechanical systems were written prior to thepublication of the proceedings of Crandall’s MIT summer programme. Among those was a paper by Fung

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[63], though it was not referenced by Crandall. It is a development that appears in the mechanical systemsliterature of the input/output spectral density relation for randomly excited linear systems. His paper on thestructural dynamics of aeronautical systems used the relation between the spectral density of an excitationforce and the spectral density of a system response to express the mean square value of the response. (Insteadof using the FRF, he used system impedance.) The equation is

s2X ¼Z 1�1

SQQðoÞjZðoÞj2

do, (5.8)

where ZðoÞ ¼ 1=HðoÞ; �1ooo1, is the impedance of a linear system. The complex impedance of a linearsystem at a given frequency is the quantity by which a harmonic component of the input must be divided inorder to obtain the harmonic component of the response. Fung did not separately express the spectral densityinput/output relation, as in Eq. (5.5), but he clearly opened to door to future developments in the area.Further, though he only wrote the relation for a simple system, he noted ‘‘The preceding relation holds for amuch wider class of dynamic systems than that represented by’’ the equation for an SDF system. ‘‘It holds forhigher order linear differential equations, linear integral equations, or linear integral-differential equationsunder mild restrictions.’’ Eq. (5.8) appears to be the first occurrence of the formula in a paper on structuraldynamics. In the paper Fung referenced Liepmann [64] who applied random process concepts to the study ofaeronautical buffeting.

Fung’s paper considers many aspects of stochastic structural dynamic response, and in addition to writingthe input/output relations for linear systems, he also considered the distribution of extremes in the response ofa system. He based his analysis of extremes on developments presented by Cramer [65], and showed that theapproximate probability distribution of response extreme values can be written. He also noted that formulasdeveloped by Gumbel [66] can be used with data to estimate the parameters of the response extreme valuedistribution. This is a practical issue, because engineers seeking to analyse the random vibrations of linearsystems were and are interested in characterizing the extremes of the response.

Further, Fung used the work of Housner [67] to express an envelope on the response of a structural dynamicsystem. Housner’s work is interesting in its own right because he considers earthquakes to be sequences ofrandom pulses that are time-varying. In his paper, Housner developed a means for analysing earthquakesbased on averages of their Fourier transform magnitudes. Specifically, he showed that the sample mean of thesquare of the Fourier transform modulus (the estimate of spectral density, when averaged) of ten earthquakesis nearly constant in the frequency range [0.5,5]Hz. However, he did not generalise the idea, or speculate on itsmeaning relative to limiting arguments on the number of signals averaged. Nor did Housner refer to thequantity he was estimating as a spectral density.

Fung’s [63] paper led to another contribution [68]. In the latter paper he developed—in an extraordinarilydirect and intuitive manner—results far beyond any that had previously been presented, and that were notused until much later. The objective of his main development was to obtain the formula for the expected valueof the nth power of the non-stationary random response of a linear structure given information on thecharacter of a non-stationary random excitation. He accomplished this in four steps. First, he specified thatnon-stationary random processes that are a function of time can be modelled as the sum of a time-varyingmean function plus a non-stationary deviation from the mean. (Here, the term non-stationary refers torandom processes that are not stationary, i.e., random processes that are not in a random steady-state.) Suchan excitation random process is fQ tð Þ; tX0g, and its mean and nth order autocorrelation functions aremQðtÞ ¼ E½QðtÞ�; tX0, and RQ;...;Qðt1; . . . ; tnÞ ¼ E½Qðt1Þ � � �QðtnÞ�; tiX0. Second, starting with a determinis-tic convolution integral similar to Eq. (5.1), he expressed the mean and mean square responses at location r ofa continuous linear system in the time domain as (integral) functions of the non-stationary excitation randomprocess mean and autocorrelation functions. (The location on the structure, r, is written in bold because it maydenote a vector location on a multidimensional structure.) His expression for the response mean and meansquare are

E½X ðr; tÞ� ¼

Z t

0

hðr; t� tÞmQðtÞdt r 2 V ; tX0, (5.9)

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E½X 2ðr; tÞ� ¼

Z t

0

dt1

Z t

0

dt2hðr; t� t1Þhðx; t� t2ÞRQQðt1; t2Þ r 2 V ; tX0, (5.10)

where hðr; tÞ; r 2 V ; tX0, is the impulse response function of a linear structure, i.e., the response of astructure at location r to an impulsive load at a specific, but unspecified location, at time zero. The second ofthese equations differs from Eq. (5.4), written 3 years later by Crandall, in that this is simply the expression forthe response mean square. But, in addition, both these expressions are for non-stationary excitation andresponse, and further, they represent the response moments for a continuous structure.

His third step was to write the general, higher-order, response correlation functions of linear systemresponse in the time domain in terms of the multivariate autocovariance function of the random excitation.That is,

E½X nðr; tÞ� ¼

Z t

0

dt1 � � �Z t

0

dtn

Yn

i¼1

hðr; t� tiÞRQ...Qðt1; . . . ; tnÞ r 2 V ; tX0. (5.11)

Fourth, he wrote the general, higher-order, response autocorrelation functions of linear system response interms of frequency domain integrals. These final expressions were obtained by, first, writing the expression forlinear system response in terms of Fourier transforms:

X ðr; tÞ ¼

Z 1�1

QðoÞZðr;oÞ

eiot do r 2 V ; tX0, (5.12)

where QðoÞ; �1ooo1, is the Fourier transform of the excitation QðtÞ; tX0, andZðr;oÞ; r 2 V ; �1ooo1, is the impedance of the linear structure at location r to an applied input. Aswell, the impedance is the reciprocal of the Fourier transform of the impulse response functionhðr; tÞ; r 2 V ; tX0. Next, Fung formed general, higher-order products of the expressions at timest1; . . . ; tn, then took the expected value of the result. The expression he obtained is

RX ;...;X ðr; t1; . . . ; tnÞ ¼

Z 1�1

do1 � � �

Z 1�1

donQðo1Þ � � �QðonÞ

Zðr;o1Þ � � �Zðr;onÞei o1t1þ���þontnð Þ r 2 V ; tiX0, (5.13)

The quantity Qðo1Þ � � �QðonÞ refers to the mean of the product of the function QðoÞ; �1ooo1,evaluated at its frequencies ok; k ¼ 1; :::; n. If Eq. (5.5) is referred to as the ‘‘central result’’ of linear randomvibrations, then Eq. (5.13) completely generalises linear random vibrations to continuous structures with non-stationary excitations and responses. Of course, Eq. (5.13) reduces to the important second order case forautocorrelation function of the linear system response when n ¼ 2.

A byproduct of his development is the indirect definition of the spectral density of a non-stationary randomprocess. It is

SQQðo1;o2Þ ¼ Qðo1ÞQðo2Þ; �1oo1;o2o1. (5.14)

Fung did not refer to the non-stationary random process spectral density by any special name. In fact, hedid not write the autospectral density of the linear system response. He wrote only the moments and thehigher-order correlation functions of the linear, time-domain response. (Note that Fung’s definition of thespectral density of a non-stationary random process does not correspond exactly to the definition used inmuch modern modelling and analysis. See, for example [69].)

Fung’s [68] paper covered many other subjects useful in random vibrations including the simplification ofrandom vibration problems using normal mode analysis, analysis of bending moments, and applications tostructural design, including extreme value considerations.

Here, is an example of the use of the formulas from Fung’s [68] paper. Consider, again, the system whoseFRF magnitude is shown in Fig. 16. It is excited by a non-stationary, Gaussian random process with the(estimated) spectral density magnitude (of Eq. (5.14)) shown in Fig. 20. The system has a random responsewhose spectral density magnitude (of Eq. (5.14)) is shown in Fig. 21. realisations of both the randomexcitation and response can be generated as shown in Figs. 22 and 23.

The subject of structural fatigue was and remains one of great importance in structural dynamics. Amongmany other sources, it is strongly motivated by the response of structural components in the presence of a

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0

1000

2000

3000

0

1000

2000

3000

-8

-6

-4

-2

Frequency, f 1

,HzFrequency, f

2 , Hz

log 10

|SX

X (

f 1,f 2)

|

Fig. 21. Magnitude of the response spectral density for the system with FRF of Fig. 5.1 to the excitation with spectral density shown in

Fig. 5.6.

0

1000

2000

3000

0

10002000

3000

-4

-3

-2

Frequency, f 1

, Hz

Frequency, f2 , Hz

log 10

|SQ

Q (f

1,f 2)

|

Fig. 20. Magnitude of the spectral density of the excitation.

T.L. Paez / Mechanical Systems and Signal Processing 20 (2006) 1783–18181810

rapidly varying pressure field of the sort that arises in connection with aerodynamic turbulence. Of course,developments were starting to take place in the field of jet propulsion during the early to mid 1950s. Miles [49]wrote perhaps the first paper that considered the subject of random fatigue of structural components inmechanical systems. His paper started with the assumption that many lightly damped structures can beapproximated as SDF systems (an assumption essential to his mode of analysis). He expressed the responsespectral density and noted that, as far as displacement and stress responses are concerned, the spectral densityapproximately equals the one that would be excited in a system by a white noise excitation with spectraldensity equal to the actual input spectral density at the natural frequency of the SDF system. Specifically, ifthe excitation random process is zero mean and stationary, with spectral density SQQðoÞ; �1ooo1, and ifthe SDF system has FRF HðoÞ; �1ooo1, then the response spectral density is given by Eq. (5.5). Miles’

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0 0.002 0.004 0.006 0.008 0.01-200

0

200

Time, secq

(t),

g

Fig. 22. Realisation of the non-stationary random process with spectral density shown in Fig. 5.6.

0 0.002 0.004 0.006 0.008 0.01

-10

0

10

Time, sec

x (t

), g

Fig. 23. Realisation of the non-stationary response random process with spectral density shown in Fig. 5.7.

T.L. Paez / Mechanical Systems and Signal Processing 20 (2006) 1783–1818 1811

approximation states that the response spectral density can be approximated by

SXX ðoÞ ¼ jHðoÞj2SQQðonÞ; �1ooo1, (5.15)

where on is the natural frequency of the SDF system. For example, when an SDF system with naturalfrequency on ¼ 2p rad=s, and damping factor z ¼ 0:05, is excited by an input with the spectral density shownin Fig. 24, its displacement response is a random process with the spectral density shown in Fig. 24. Theresponse spectral density is the product of the modulus squared of the FRF (also shown) and the inputspectral density. Miles’ approximation to the response spectral density is the product of the white noisespectral density with magnitude SQQðonÞ and the modulus squared of the FRF, both shown in Fig. 24. TheRMS value of the response is the square root of the area under the response spectral density curve. In this casethe exact value is 0.312 in, and the value that comes from the approximation is 0.297 in. For this example, theapproximation yields accuracy of about five percent. (Of course, in individual cases, the approximation maybe better or worse.) Miles’ paper may be better known for this intermediate result than for its final result, tofollow.

Miles continued his analysis by using his white noise approximation and noting (with reference to [55])that the probability distribution of the peaks in a zero-mean, stationary, narrowband random processfollow a Rayleigh distribution [86]. Further, he assumed that the number of cycles to a high-cycle fatiguefailure is related to the operation stress level via a power law. He adopted Miner’s rule for fatigue damageaccumulation of a specimen loaded at a varying levels. And he used these facts to show that the single stressthat is equivalent, on average, to the varying stress amplitudes of an SDF system in narrowband randomresponse is

sr ffias2s

e

� �1=2

; ab1, (5.16)

where �a�1 is the slope on a log–log plot of the stress amplitude that corresponds to number of cycles tofailure, s2s is the mean square value of stress at the point of interest, and e is Euler’s constant. Finally, using thewhite noise approximation, Miles showed that the amplification ratio of this equivalent stress over the stress s0

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10-1 100 101

10-4

10-2

100

Frequency, Hz

SQQ (f )

|H (f )|2

SXX (f )

SQQ (fn)

|H (f )|2SQQ (fn)

Fig. 24. Graphic showing the meaning of Miles’ approximation to the response spectral density.

T.L. Paez / Mechanical Systems and Signal Processing 20 (2006) 1783–18181812

caused by a static force F 0, is

sr

s0¼

pa4ez

onSQQðonÞ

F20

" #1=2, (5.17)

where z is the SDF system damping factor. Given knowledge of the system’s natural frequency, the formulapermits estimation of the expected time to failure.

The response spectral density approximation that Miles defined is so attractive that it has gained widespreadacceptance, and is in use in many applications. For example, NASA uses it in combination with finite elementmodels and a load combination scheme to establish an approximation for the overall load on a system. (See,for example, [70–72].)

Lyon [29,73] wrote two papers in 1956 that consider, first, in a very general sense, then in a specific sense,how random pressure fields fQðr; tÞ; r 2 V ; �1oto1g propagate and excite mechanical system responses.The random field specification shows that it is a function of a spatial parameter as well as the time parameter;the spatial parameter, r, is bold because it can describe the locations where the random process is to beconsidered in one, two, or three dimensions, and this parameter is limited to the volume, V. The randomresponse Lyon considered is also a random field; it varies as a function of space and time. As Miles did, he alsobased his analysis on the work of Rice [55], but in addition to considering processes that are random in time,he also considered spatial randomness. In the first (and more general) paper he expressed the excitation field asa ‘‘random superposition of elementary events (eddies)’’ that propagate in space and time. The spatialrandomness was included by permitting the eddies to originate at random locations. He defined a probabilitylaw to govern the random field, permitting the time and location of origination of eddies to be dependant, andrequiring amplitudes of the eddies to be independent. He used the joint probability distribution with the formof the random field to establish a space–time correlation function for the noise field. It isE½Qðr1; t1ÞQðr2; t2Þ�; r1; r2 2 V ; t1; t2X0. He then ‘‘propagated’’ the noise field through a linear system toobtain the correlation function of the response. To do so, he noted that the response of a linear, continuoussystem to a deterministic excitation qðr0; t0Þ; r0 2 V ; �1ot0pt is

xðr; tÞ ¼

Z t

�1

dt0

ZV

dr0gðr; t; r0; t0Þqðr0; t0Þ r 2 V ; �1oto1, (5.18)

where gðr; t; r0; t0Þ; r; r0 2 V ; �1ot0; to1, is the Green’s function of the system, i.e., the response of thestructure at time t and locations r, to a temporally impulsive excitation at time t0 and locations r0. By writingthis expression using the random field excitation, Qðr0; t0Þ, in place of the deterministic excitation, qðr0; t0Þ, inEq. (5.18), evaluating the result at ðra; tÞ, multiplying the result by the expression for the response xðrb; sÞ, and

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averaging the result over the random excitation products, Lyon obtained

RXX ðra; t; rb; sÞ ¼

Z t

�1

dt1

Z s

�1

dt2

ZV

dr1

ZV

dr2

gðra; t; r1; t1Þgðrb; s; r2; t2ÞRQQðr1; t1; r2; t2Þ ra; rb 2 V ; �1ot; so1, ð5:19Þ

where RQQðr1; t1; r2; t2Þ ¼ E½Qðr1; t1ÞQðr2; t2Þ�, the expected value of the product Qðr1; t1ÞQðr2; t2Þ, is theautocorrelation function of the random field fQðr; tÞ; r 2 V ; �1oto1g, and RXX ðra; t; rb; sÞ ¼E½Qðra; tÞQðrb; sÞ� is the autocorrelation function of the response field. Lyon concluded the first paper withoutsolving a numerical example for a specific physical system. He also pointed out that an alternate means toobtain this result had been developed by Eckart [74] who propagated individual excitation realisations toresponse realisations, then averaged over the response realisations to obtain the autocorrelation function ofEq. (5.19).

Lyon’s second (and more specific) paper applied the theory of the first paper (i.e., Eq. (5.19)) to severaldifferent types of strings. He considered both finite and infinite strings, and experimented with some of thefinite strings. He compared his model-predicted results to experimental measurements to show theapplicability of the mathematical formulation.

Thomson and Barton [75] wrote a paper that extended the approximation of Miles [49]. They pointed outthat through modal analysis the equations of motion of a complex linear structure can be reduced to a set ofequations that have the form of the equation governing motion of an SDF system. These simple equations canbe evaluated individually, then the mean square responses in the modes can be synthesised to approximate themean square response at a point on the system. Their modal assumption is that the response at a location r ona complex system xðr; tÞ; r 2 V ; �1oto1, can be expressed as a series

xðr; tÞ ¼X

k

xkðtÞfkðrÞ r 2 V ; �1oto1, (5.20)

where fkðrÞ; k ¼ 1; 2; . . . ; r 2 V , are the mode shapes of the system, i.e., the shapes the system assumes whenresponse is harmonic and occurs at the modal frequencies ok; k ¼ 1; 2; . . ., andxkðtÞ; k ¼ 1; 2; . . . ; �1oto1, are the modal coordinates of the system, i.e., the amplitudes of theresponses in the individual modes. When a system is excited by a single excitation qðtÞ; �1oto1, thenresponse in the kth mode is governed by

€xk þ 2zkok_xk þ o2

kxk ¼ fkðrinÞqðtÞ; k ¼ 1; 2; :::; �1oto1, (5.21)

where zk; k ¼ 1; 2; . . ., are the modal damping factors, and rin is the location on the continuousstructure where the excitation is applied. Based on this equation, when the actual input is a stationaryrandom process, an approximation to the mean square response in each mode can be established usingMiles’ approximation. This is the approximation that Thomson and Barton made to obtain the overallmean square response at a point on the complex linear structure. When the excitation random processapplied at the point rin has zero mean, and is stationary with spectral density SQQðoÞ; �1ooo1, then theapproximate mean square response of the system at location r is (when the response achieves a stationary

102 103

10-6

10-4

Frequency, Hz

|H (

f)|,

in/l

b

Fig. 25. Magintude of the force in/displacement out FRF for a three-degree-of-freedom linear system.

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state) given by

s2X ðrÞ ¼X

k

pf2kðrinÞf

2kðrÞSQQðokÞ

2zko3k

; r 2 V : (5.22)

Development of this approximation includes the assumption that the correlation between pairs of modalresponses is zero.

Here, is an example that tests the accuracy of Thomson and Barton’s approximation in a specific case.Consider the system whose force input/displacement output FRF magnitude is given in Fig. 25. (The system isthe one considered in Fig. 15, except that here, displacement response is of interest.) The modal frequencieswere previously given as 821, 1384, and 1706Hz. It is now necessary to specify that the system has modaldampings of 0.020, 0.035, and 0.015. The system is a three dof system with masses of 0.025, 0.010 and0.005 lb s2/in. The mode shape amplitudes (normalised with respect to mass) at the input location are �2.66,�5.68 and �0.79, for the first, second and third modes, respectively. The mode shape amplitudes at theresponse location are 5.25, �1.30 and �8.41. The system is excited with a zero-mean stationary randomprocess with the spectral density shown in Fig. 26. The spectral density values at the modal frequencies are3.72, 2.84, and 4.78 lb2/Hz. The exact spectral density of the system response is shown in Fig. 27. The meansquare response is the area under the response spectral density curve. It is 3.45� 10�8 in2 with correspondingRMS response 1.86� 10�4 in. Thomson and Barton’s approximation can be used to estimate the samequantities using Eq. (5.22). The approximate results are a mean square response of 3.52� 10�8 in2 and anRMS response of 1.88� 10�4. In this case, the accuracy of RMS prediction is about 1%.

The final paper (published before 1958) to be summarised here was written by Eringen [76]. He considered,specifically, the random vibrations of linear beams and plates. The approach he took was to write thedeterministic solution for the response in terms of Fourier expansion. The response expressions so obtainedare series in the eigenfrequencies and eigenshapes of the beam or plate, and the excitation componentamplitudes. Eringen assumed that the excitation is the product of a temporal white noise and a shape functiondefined over the beam or plate. For these systems he developed expressions for the response auto- and cross-correlation functions, and the response auto- and cross-spectral densities. The autocorrelation functions andthe autospectral densities are the quantities that other researchers of the time were developing for general and

102 10310-1

100

101

Frequency, Hz

SQ

Q (

f),lb

2 /H

z

Fig. 26. Spectral density of force applied to the system.

102 103

10-10

Frequency, Hz

SX

X (

f), i

n2 /H

z

Fig. 27. Spectral density of system response.

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specific structural response random vibrations. The cross-correlation functions and cross-spectral densitieswere not commonly developed at the time, but are required to establish exact expressions for random responsespectral densities, mean squares, etc., when the response is the superposition of multiple components likemodes, or responses excited by multiple inputs simultaneously.

Eringen solved some specific random vibrations problems for beams and plates, and in those he consideredthe case of spatially concentrated loads. His discussion covers practical issues of linear system responseanalysis using the framework of Fourier analysis.

Several other noteworthy papers appeared in the proceedings of Crandall’s [58] workshop. Among these isone by Powell [58], entitled ‘‘On the Response of Structures to Random Pressures and to Jet Noise inParticular.’’ He also used modal analysis to express the response of continuous linear systems, and exploredthe importance of spatial distribution of loads. Further, he expressed the correlations between modal responsecomponents. In many respects the paper is similar to Fung’s [68] paper, and is, in fact, very advanced for theera. A paper by Dyer [77] in the MIT workshop, entitled ‘‘Estimation of Sound-Induced Missile Vibrations,’’is a very practical, empirical presentation of the load field, followed by vibration response characterization formissiles subjected to sound-induced noise.

Another paper in the 1958 Random Vibration volume by Rona [50] entitled ‘‘Instrumentation for RandomVibration,’’ considers both issues of experimental instrumentation and the estimation of spectral density. Hediscussed several types of accelerometer, and described how analog-to-digital conversion of measured signalsis performed. He also discussed the possibility of digital signal analysis, but pointed out that it was not ingeneral use at the time. He described what was, at the time of that writing, the technique used for estimation ofspectral density. In essence, his description is the one given by Fig. 9 in Section 3. He referenced Blackman andTukey [78] for the spectral density estimation technique.

Hardware implementations of the approach he described were used to estimate spectral densities ofstationary random sources in the laboratory and the field, through at least the mid-1960s. The input signalswere, of course, analog. The filters were band-pass filters, and were used in banks of 40–80. Spectral densityestimates were frequently obtained up to 1000–2000Hz. Mean square values of signal components wereobtained in two ways. First, the filtered signals might be run through a full-wave rectifier. The means of theresulting signals could be computed, and from this, the RMS values of the filtered signal components inferred.These quantities could be squared to obtain estimates of the component mean squares. A second methodinvolved the squaring of the filtered signal components using an analog squarer, then the averaging of theresulting signals to obtain the component mean squares. Finally, the mean square of each signal componentwas scaled by the factor 1/(BW) (the inverse of the effective bandwidth of the band-pass filters) to obtain theanalog estimate of spectral density.

Rona’s paper presented an early discussion (one of many) on the subject of test instrumentation, randomsignal analysis, and random vibration test control. Through the years, as much time, effort and funding hasgone to the performance of laboratory and field random vibration testing and analysis as has gone into thedevelopment of analytical techniques. The reason is simple; those burdened with the requirement to assurethat actual systems that are capable of being tested are robust and reliable have tended to trust the results ofphysical experiments over analytical predictions, or in addition to analytical predictions. That attitude ischanging, but a discussion of the reasons why requires another paper. For that matter, a complete discussionof the history of random vibration and mechanical shock testing requires a separate discussion.

The papers discussed in this section form a part of the legacy of the modern development of the theory ofrandom vibrations. During the 1960s and beyond the study of random vibrations of mechanical systemsflourished, and many papers dealing with the subject were published. Even the briefest summary of the topicscovered in random vibration studies during the past 45 years would require a substantial discussion and admitan enormous bibliography. For those reasons, the papers summarised here stand out, forming the startingpoint of the modern theory of random vibrations.

6. Conclusion

The history of the mathematical theory of random vibrations of mechanical systems spans the previouscentury, starting with the work of Einstein, and continues to the present. This paper briefly presents some

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Einstein 1905

Wiener 1930

Rayleigh 1880,1889

Smoluchowski, Fokker, Planck

1916

Rice 1944

Crandall 1958

Fung, et al. 1953

Schuster 1905

Uhlenbeck, Ornstein

1930

Wang, Uhlenbeck

1945

Fig. 28. Time line of contributors to the mathematical theory of random vibrations of mechanical systems.

T.L. Paez / Mechanical Systems and Signal Processing 20 (2006) 1783–18181816

developments in that history through the year 1958. And though the groundwork for random vibrationanalysis was laid over that entire period, the work dealing with application to structural and mechanicalsystems started in earnest in the 1950s. While numerous papers contributing to the development of randomvibrations have been summarised, there are some early texts that rate special mention. These are some of thetexts that form the foundations for the rich, formative contributions to come after 1958:

The text by Crandall and Mark [3] provides a particularly accessible development of the formulas andtechniques of random vibration of linear structures. It covers in great detail the random vibration of two-dof structures, and provides tables of formulas useful in the computation of random vibration integrals. � Lin [5] provided the first text with a detailed mathematical presentation of random vibration of mechanical

systems. He included a concise, yet thorough summary of probability and random process theory. Hecovered SDF and multi-dof linear systems, as well as non-linear systems. He covered direct moment-basedapproaches to random vibration as well as Markov vector-based approaches.

� The text by Robson [4] provides a very concise summary of deterministic structural dynamics and random

vibration of linear systems. It includes several developments fundamental to random vibrations andrandom processes that are omitted from other texts.

� The introduction of random vibrations as a theory useful for the solution of engineering problems

necessitated the development of techniques for estimation of the quantities required to apply the theory.The text by Bendat [79] satisfied many of these needs. It provides methods for the estimation of the spectraldensity and autocorrelation functions.

Finally, for the convenience of the reader, we present a time line of the major contributors to the field ofrandom vibrations. Fig. 28 shows many of the major contributors, the years of their contributions, and theirconnections to earlier and later contributors.

Acknowledgement

The author owes a great debt of gratitude to the individuals who reviewed the manuscript. They correctednumerous errors. They demanded clarity of wording and presentation, and suggested how to achieve it. Whenthe sourcing of historical developments appeared inadequate, they encouraged more research. They are AllanPiersol, Harry Himelblau, David Smallwood, Tim Hasselman, George Lloyd, Bill Hughes and MarkMcNellis. Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed MartinCompany, for the United States Department of Energy under Contract DE-AC04-94AL85000.

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