Stochastic averaging on a nonlinear oscillator with coordinate-dependent mass excited by Gaussian white noises

In this paper, a new stochastic averaging procedure is presented in detail to illustrate the improvement which develops the former stochastic averaging methods which can only deal with randomly excited oscillators without the coordinate-dependent mass (also called as inertia nonlinearity in many literature) into a new method which can handle nonlinear oscillators with both coordinate-dependent mass and stiffness nonlinearity terms. The crucial issue of this method is balancing the input and the dissipation of the Hamiltonian energy of the oscillator. After the oscillator has been simplified to be an I$\hat{t}$o type stochastic differential equation, the stationary probability density function (PDF) of transient equivalent amplitude, as well as the joint PDF of the displacement and velocity is studied. Numerical simulations are carried out to verify the theoretical anticipations. The numerical results coincide with the theoretical results perfectly.