关键词: 概周期运动, 渐近摄动法, 内共振, 平均方程

Abstract: By considering the nonlinear strains-displacement relation and the effect of the shear deformation, the nonlinear governing equations of motion for the structure of plate and shell in the mechanical engineering are a two-degree-of-freedom nonlinear system including the quadratic and cubic nonlinear terms under external excitations. To consider the influence of the quadratic terms on the nonlinear dynamic characteristics of the nonlinear system, it is difficult for one to use the method of multiple scales to obtain the second-order approximate solution which includes the quadratic terms. By using the asymptotic perturbation method, based on the Fourier series and time rescaling, the quadratic terms can be included in the average equations. By introducing proper scale transformations, the asymptotic perturbation method is used to reduce the second-order non-autonomous nonlinear differential equations to autonomous nonlinear differential equations. The resonant case considered is 1:2, principal parametric resonance-1/2 subharmonic resonance. Then numerical analysis of the governing averaged equation is carried through by using Runge-Kutta method. It is found from numerical results that complex periodic, double-period and quasi-period motions exit intle four-degree-of-freedom system.

Key words: Asymptotic perturbation method, Averaged equations, Internal resonance, Quasi-period motions