Winkler地基上材料非线性矩形薄板主参数共振研究

研究Winkler地基上材料非线性矩形薄板受参数激励的参数共振动问题。按照弹性力学理论建立Winkler地基上材料非线性矩形薄板受参数激励的动力学方程。利用Galerkin方法将其转化为非线性振动方程。应用非线性振动的多尺度法求得系统满足主参数共振条件的一次近似解，并进行数值计算，分析定常解的稳定性。给出主参数共振系统参数平面的分岔集和幅频响应方程的分岔图。分析激励、调谐值、阻尼系数、非线性参数、几何参数对共振响应曲线的影响。

STUDY ON PRIMARY RESONANCE OF A PARAMETRICALLY EXCITED THIN RECTANGULAR PLATE ON WINKLER FOUNDATION CONSIDERING MATERIAL NONLINEARITY

Primary resonance of a parametrically excited thin rectangular plate on Winkler foundation considering material nonlinearity is studied. Applying the elastic theory, its nonlinear dynamic equations are established. From them its nonlinear oscillation equation is obtained based on Galerkin's method. By means of the method of multiple scales for nonlinear oscillations, the first approximate primary parametric resonance of the system is acquired. The stability of the steady state solutions of the system is investigated. The bifurcation diagrams of the parameter plane and the response equation are obtained. Numerical analysis for influences of excitation, detuning, damping, nonlinear parameter and geometric parameters on the system is carried out.