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

In the course of analyzing the dynamic behavior of mechanical systems subjected to random excitations, we investigate the associated Fokker-Planck equation. We also discuss the relationship between the characteristics of the random excitation and the nonlinear intensity coefficients governed by the physical properties of the system. This relationship leads to some simplified methods for solving the response probability density of certain nonlinear systems. We present general solutions to a class of multidimensional problems with desirable constraints. The random motion of a single-mode mechanical oscillator with various nonlinear stiffnesses and a charged particle moving in an electromagnetic field are examples. Cosine-power and sech-power distributions are found to be associated with the steady state response of a tangent stiffness system and a hyperbolic tangent stiff ness system, respectively. When the influencing magnetic vector potential M is irrotational, the stationary probability for the moving particle in the 6-dimensional response phase-space is statistically independent.