A method to stochastic dynamical systems with strong nonlinearity and fractional damping

摘要

In this paper, a new technique is proposed to deal with strongly nonlinear stochastic systems with fractional derivative damping and random harmonic excitation. Combining the advantages of Linstedt-Poincar, (L-P) method and multiple scales method, introducing a different frequency expansion form and a time transformation, a series of perturbation equations is obtained according to the powers of parameter. Then, we eliminate the secular producing terms to solve the perturbation equations to derive the second-order approximate solution. Furthermore, the steady-state frequency-amplitude function in deterministic case is analyzed, and the first-order and second-order steady-state moments of the amplitude are also discussed in the presence of random harmonic excitation. In order to explore the effectiveness of the proposed approximate method, two classical examples were proposed to verify the theoretical results through numerical simulations. Especially, the method can be used to investigate some types of extremely strong odd nonlinear terms via the discussions of each example.