Evolutionary stochastic characteristics of nonlinear oscillator with one side barrier due to multiple modulated Gaussian white noise

Abstract

In this paper, the evolutionary exponential-polynomial-closure (EPC) method is extended to study the challenging problem of obtaining the evolutionary probability density function (EPDF) solutions of the nonlinear stochastic oscillators with one barrier under multiple modulated Gaussian white noise. Firstly, the original oscillator with the barrier is transformed into a new approximately equivalent nonlinear oscillator in the state space $R^2$ through the variable transformation. Subsequently, the presented procedure is employed to solve the non-stationary FPK equation in the transformed state space. After that, the EPDFs of the original oscillator can be acquired based on the variable transformation. To determine the accuracy of the presented procedure, three illustrative examples are investigated by considering distinct nonlinear terms, restitution factors and multiple random excitations. The accuracy of the EPDFs acquired by the presented procedure is compared with that of simulated results in each example. The outcomes illustrate that the presented procedure can provide accurate EPDF solutions and it is more efficient than numerical simulation. The EPDF solutions of the oscillator are greatly affected by the barrier, the correlation of the noise and the modulation function in the system. The asymmetry of EPDFs can also be influenced significantly by the barrier and the correlation of the modulated noise. In the evolving process, the statistical moments of responses also vary in a similar trend as the modulated function of the white noise due to the barrier.