Exact steady-state probability density functions for second order nonlinear systems under external stationary excitations

The main aim of this paper is to present mathematically rigorous foundations of a probabilistic analysis of vibrations in nonlinear dynamic systems driven by some stationary processes. A class of solvable Fokker-Planck equations is given and a new method is presented to obtain a probability density function of the response of a nonlinear oscillator to stationary excitations. A one-dimensional vibrating system with a nonlinear elastic force is considered.

We analyse the case when the excitation force is a stationary 2nd order stochastic process with a mean value equal to zero and a spectral density of the form S_z(omega)=S_0/(1+omega^2r^2), omega in R^1, where S_0>0 and tau are certain constants.

Thus, utilizing the Fokker-Planck equations, determination of the density of the three-dimensional Markov vector of the following components: displacement, velocity and acceleration of a nonlinear oscillator can be circumvented. It is presented that the density function has the following form w^(3)(x,dot x ,ddot x )=Phi(x,ddot x )=exp[Psi(x,dot x )] where Phi(.,.) and Psi(.,.) are analytically determined functions.