Limiting Behaviors of Randomly Excited Hyperbolic Tangent Systems

In recent years, random vibrations of nonlinear systems have attracted considerable attention among engineers. 1 In this paper we investigate the Fokker-Planck equations 2 " 1 associated with a class of random processes whose steady-state probability density distributions, of the Liapunov potential function type. The random response statistics of a nonlinear single-degree-of-freedom model having a hyperbolic tangent stiffness function can be described as a softening spring whose force-deflection relationship is asymptotic to some maximum force level. Such a model can be used to represent an elastic-perfect-plastic system, material often encountered in classical mechanics. Limiting situations for a class of probability density functions such as those obtained in this study are examined. We show that the limiting behavior of the steady-state output probability density function of a system having a generalized hyperbolic tangent stiffness function, F(u) = (k0/ba~*) tanh bu, is closely related to the range of the parameter a. At the limit b --» 00, the probability density function becomes a Dirac delta (impulse) function or an exponential distribution, or identically approaches zero for all u, depending upon whether a 543