Dynamic responses of axially moving viscoelastic beam under a randomly disordered periodic excitation

We investigate dynamic responses of axially moving viscoelastic beam subject to a randomly disordered periodic excitation. The method of multiple scales is used to derive the analytical expression of first-order uniform expansion of the solution. Based on the largest Lyapunov exponent, the almost sure stability of the trivial steady-state solution is examined. Meanwhile, we obtain the first-order and the second-order steady-state moments for the non-trivial steady-state solutions. Specially, we discuss the first mode theoretically and numerically. Results show that under the same conditions of the parameters, as the intensity of the random excitation increases, non-trivial steady-state solution fluctuation will become strenuous, which will result in the non-trivial steady-state solution lose stability and the trivial steady-state solution can be a possible. In the case of parametric principal resonance, the stochastic jump is observed for the first mode, which indicates that the stationary joint probability density concentrates at the non-trivial solution branch when the random excitation is small, but with the increase of intensity of the random excitation, the probability of the trivial steady-state solution will become larger. This phenomenon of stochastic jump can be defined as a stochastic bifurcation.