Solution of Fokker-Planck Equation with Applications in Nonlinear Random Vibration

Although the theory of stochastic processes has found wide applications in information and communication sciences for many years, only recent advances in rocket propulsion and aerospace industries have made random vibration problems subjects of growing importance in mechanical and civil engineering. These problems involve structural responses due to random loadings and are in general nonlinear resulting from large motions. 1 Such nonlinear random transformation problems often encountered in practice are generally memory-dependent; that is, the equations of motion are described by nonlinear differential equations. 2-4 Under the Markov and Smoluchowski assumptions, it has been shown that the probability density function 2031 2032 T H E BELL SYSTEM TECHNICAL JOURNAL, JULY-AUGUST 1969 of a large class of random processes satisfies equations of the FokkerPlanck (F-P) type. 5 - 0 Recently Pawula showed t h a t generalized Fokker-Planck equations can be derived for many cases with both these assumptions removed. 7 M a n y interesting problems with their governing equations of the Fokker-Planck type have been investigated by various researchers. Rosenbluth, and others, studied the FokkerPlanck equation for the distribution function for gases with an inversesquare particle interaction force; 10 van Kampen used an FokkerPlanck equations to describe the thermal fluctuations in linear and nonlinear systems; 1 1 Ariaratnam found the steady state response distribution for a class of nonlinear two-mode mechanical oscillators by applying certain constraints to decouple the governing Fokker-Planck equation; 1 2 and Hempstead and Lax used Fourier transform techniques to eliminate the phase variable from the Fokker-Planck equation in the polar coordinates for a rotating-wave Van der pol oscillator and found the phase and amplitude spectra.