processes, and traces some preliminary thoughts on the subject back to 1889. Analysis of the random vibrations of mechanical systems, as practiced today, started in the 1950’s, and the beginnings of the analytical developments are covered in Section 4. Some of the works from Crandall’s 1958 workshop are discussed. Examples are provided throughout the paper. 2.0 Einstein’s Era Around the turn of the previous century, Einstein (1905) constructed a framework for analyzing the Brownian movement - the random oscillation of particles suspended in a fluid medium and excited by the molecular motion associated with the kinetic theory of matter. Brownian movement had been recognized about a century earlier during observations of microscopic particles of pollen immersed in a liquid medium; it is characterized by the erratic movement of the pollen particles. The particle motion characteristics depend on the mass and geometry of the particle and the physical characteristics (such as viscosity and temperature) of the fluid medium. Because the problem Einstein solved yields the probabilistic description of the motion of a mass attached via a viscous damper to a fixed boundary and excited with white noise (a random excitation with frequencies covering a broad band), his development can be thought of as the first solution to a random vibration problem and the dawning of the era of random vibration analysis. In his solution of the problem of Brownian movement Einstein did not use a direct formulation that writes and analyzes the differential equation governing motion of the system. (The direct approach would eventually become the one most commonly used for random vibration analysis.) However, for reference, the governing equation is ( ) ( ) ( ) 0 0 0 0 0= = ≥ + = t ,X ,X mX cX W t & & && (1) where { ( ) }0≥ X t ,t 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, { ( ) } −∞< <∞ t W t , is the white noise excitation random process, and dots denote differentiation (in a sense appropriate for a random process) with respect to time. The white noise excitation random process has the constant spectral density ( ) = −∞< <∞ ω ω , S S WW WW . (Spectral density defines the mean square signal content of a random source as a function of frequency. The spectral density defined here is two-sided because it is defined for positive and negative frequencies. The definition of spectral density, along with some examples, and some ideas underlying random processes and their notations are provided in the following section.) The system of Eq. (1) is shown schematically in Figure 1. The white noise random process is a source with mean square signal content that is uniformly distributed over the entire range of frequencies (up to infinity, in theory). Figure 1. Schematic of the system considered by Einstein in his solution of the Brownian movement problem/random vibration analysis. With a very intuitive discussion, Einstein developed the diffusion construct for analyzing the random vibration of mechanical systems. This framework models diffusion of a particle (rigid mass) under the influence of applied impacts. In the application he considered, the impacts arose from the motions of the molecular constituents in the fluid surrounding the particle. The paper Einstein wrote has two parts. The first part uses the idea of osmotic pressure and equilibrium of a sphere moving in a fluid medium to derive the coefficient of diffusion of such a particle. He showed that the coefficient of diffusion can be modeled as ( )( ) c r D RT N f π 1 6 = , where R is the universal gas constant, T is absolute temperature, N is Avagadro’s number, m c x W(t) 106
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