The Stochastic Dynamics of a 1-DOF System under Random Discontinuous Piecewise Forcing

Albert Luo and Yang Wang, Mechanical and Industrial Engineering, University of Southern Illinois

In the paper, the discontinue phenomenon, non-periodic motions and stochastic responses of the 1-DOF dynamical system are investigated under the nonlinear, non-periodic and random, discontinuous, piecewise forcing. For periodic piecewise discontinuous forcing, the Fourier series approximate expansion is extensively adopted. However, the Fourier series expansion cannot accurately give the non-smooth responses of the dynamical system. In this paper, an alternative methodology is presented to obtain the exact solutions without the Fourier series expansion. The exact solution of the 1-DOF dynamical system can be obtained under not only periodic but non-periodic piecewise, discontinuous forcing. Further, this method is used to determine stochastic dynamics of the 1-DOF system under the random forcing, which will help us better understand random vibration in the dynamical system.