Dynamical analysis of a stochastically excited nonlinear beam with viscoelastic constitution

In the present article, the dynamics of a nonlinear beam with viscoelastic constitutions under stochastic excitation are studied. The viscoelastic constitution is adopted to model the viscoelastic properties of the beam. First, the dynamical model of the viscoelastic beam is established, which contains integral terms both in linear and nonlinear terms due to the viscoelastic constitution. Then, the Galerkin discretization method is applied to obtain a set of nonlinear ordinary differential equations for each vibration mode. Through numerical calculation, it is found that the responses of the primary vibration mode are much greater than those of other vibration modes under constant and linearly distributed Gaussian white noise excitations. Then, the stochastic averaging method for the primary vibration mode with viscoelastic terms is developed. An equivalent system without viscoelastic terms is derived to replace the original system, where the modified damping and conservative force are derived to approximate the linear and nonlinear viscoelastic force. It is found that the equivalent conservative force is amplitude-dependent. Finally, the analytical responses can be obtained by solving the associated Fokker-Plank-Kolmogorov (FPK) equation. The effects of the excitation intensity, damping ratio, and viscoelastic parameters are analyzed. The theoretical results and the numerical simulation results are in good agreement, which shows the effectiveness of the proposed analytical method.