The history of random vibrations through 1958

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<ul><li><p>Mechanical Systemsand</p><p>Signal ProcessingMechanical Systems and Signal Processing 20 (2006) 17831818</p><p>1. Introduction</p><p>Though random vibrations have been observed for millennia because of the effects on structures of</p><p>process 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</p><p>ARTICLE IN PRESS0888-3270/$ - see front matter r 2006 Elsevier Ltd. All rights reserved.</p><p>doi:10.1016/j.ymssp.2006.07.001</p><p>E-mail address: tlpaez@sandia.gov.earthquakes, 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 rstmathematical 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 dened the spectral density of a stationary random process, i.e., a randomRandom 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.The history of random vibrations through 1958</p><p>Thomas L. Paez</p><p>Sandia National Laboratories, Albuquerque, NM, USA</p><p>Received 6 March 2006; received in revised form 5 July 2006; accepted 5 July 2006</p><p>Abstract</p><p>Interest in the analysis of random vibrations of mechanical systems started to grow about a half century ago in response</p><p>to the need for a theory that could accurately predict structural response to jet engine noise and missile launch-induced</p><p>environments. However, the work that enabled development of the theory of random vibrations started about a half</p><p>century earlier. This paper discusses contributions to the theory of random vibrations from the time of Einstein to the time</p><p>of an MIT workshop that was organized by Crandall in 1958.</p><p>r 2006 Elsevier Ltd. All rights reserved.</p><p>www.elsevier.com/locate/jnlabr/ymssp</p></li><li><p>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. Specically, 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.</p><p>ARTICLE IN PRESST.L. Paez / Mechanical Systems and Signal Processing 20 (2006) 178318181784The 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 dened, 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 presentation</p><p>of 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 withrandom.) 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, includingrst 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 nite 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.</p><p>Consider Fig. 1. Random vibrations occur in a mechanical system when it is subjected to a stochasticenvironmentone 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 featuressimple 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)</p><p>equation representation:</p><p>_X gX ;Q; a; X 0 X0; 1oto1. (1.1)The quantity X represents system response (such as displacement), Q represents system excitation (such as a</p><p>force 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 gX ; a (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 specied completely, at one extreme, by higher-order probability distributions, or, as more commonlyoccurs, only partially specied by some of their average features. Random vibration analyses can be</p><p>MechanicalSystem</p><p>Excitation Response Fig. 1. Schematic of excitation/system/response.</p></li><li><p>ARTICLE IN PRESST.L. Paez / Mechanical Systems and Signal Processing 20 (2006) 17831818 1785performed 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, specify</p><p>the stochastic characteristics of the system response, X, or, perhaps, a deterministic limit on the randomresponse, then establish the characteristics of a system, gX ; a, that will yield the desired response.Testing of mechanical systems in random environments may serve many purposes. It may be done simply to</p><p>establish the character of a particular system in a random environment. It may be used to show that therandom response satises 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 random</p><p>processes. Of course, this is fundamental to the pursuit of practical environment description and testspecication.In spite of the importance of random vibration design, testing and signal analysis, this paper will focus</p><p>mainly on the description of historical developments in random vibration analysis. Section 2 summarizes therst investigations into random vibrations, from Einsteins 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 Crandalls 1958 workshop and a few others that followed immediately thereafter.</p><p>2. Einsteins era</p><p>Around the turn of the previous century, Einstein [21] constructed a framework for analysing the Brownianmovementthe random oscillation of particles suspended in a uid 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 uidmedium.Because the problem Einstein solved yields the probabilistic description of the motion of a mass attached via</p><p>a viscous damper to a xed boundary and excited with white noise (a random excitation with frequenciescovering a broad band), his development can be thought of as the rst 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 nd), 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 writes</p><p>and 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</p><p>m X c _X W t tX0; X 0 0; _X 0 0, (2.1)where fX t; tX0g is the one-dimensional particle displacement response random process, m is particle mass, cis the damping that ties the mass to an inertial frame, fW t;1oto1g is the white noise excitation random</p><p>process, and dots denote differentiation (in a sense appropriate for a random process) with respect to time.</p></li><li><p>The white noise excitation random process has the constant spectral density SWW o SWW ; 1ooo1.(Spectral density denes the mean square signal content of a random source as a function of frequency. Thespectral density dened here is two-sided because it is dened for positive and negative frequencies. Negativefrequencies are to be interpreted in the sense that harmonic functions are dened for negative arguments. Thedenition 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 innity, 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 this</p><p>interpretation, 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,</p><p>ARTICLE IN PRESST.L. Paez / Mechanical Systems and Signal Processing 20 (2006) 178318181786t. For example, X t is the random variable representing displacement response at time t. The random variablehas a formal denition 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 x;1oxo1. The PDF isnon-negative, and has a unit integral on 1;1. The integral of the PDF over an interval (a, b), R b</p><p>af xxdx,</p><p>where apb, is the relative chanceprobabilitythat 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 EX t mX t. 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 X t s2X t. 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 syst...</p></li></ul>

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