Nonlinear Theory of Random Vibrations

Random vibration analysis of mechanical systems has become an important subject in recent years, principally because of advances in high speed flight. To design structures and equipment that will survive the randomly fluctuating loads caused by the flow of turbulent air or the efflux of jet or rocket engines, it has become necessary to develop a theory capable of analyzing the effect of such fluctuating loads on structures and equipment. Many of the techniques developed for the analysis of random excitation of nonlinear control systems are applicable to the analysis of nonlinear random vibrations, and conversely many of the techniques developed in the theory of nonlinear random vibrations are equally applicable to problems in communication theory and electronics. The chapter presents modeling in nonlinear random vibrations by Markov processes. The chief reason for adopting the idealized model of a system of differential equations excited by white noise is that the computations are much simpler in this case. One of the difficulties involved in modeling nonlinear random vibrations by Markov processes is that—one is restricted to quasi-linear systems. In the subsequent development of the theory, no distinction has been made between the physical nonlinearity and the mathematical model of that nonlinearity. Further, the chapter also discusses the basic theory of stochastic processes and its applications and solution techniques.