非线性弹性地基上矩形薄板的非线性振动与奇异性分析

NONLINEAR VIBRATION AND SINGULARITIES ANALYSIS OF A THIN RECTANGULAR PLATE ON NONLINEAR ELASTIC FOUNDATION

研究非线性弹性地基上小挠度矩形薄板的非线性振动,应用弹性力学理论建立非线性弹性地基上小挠度矩形薄板受简谐激励作用的动力学方程,利用Galerkin方法将其转化为非线性振动方程。根据非线性振动的多尺度法求得系统主参数共振-主共振情况的一次近似解,并进行数值计算。分析了阻尼系数、地基系数、激励参数等对系统主参数共振-主共振的影响。系统主参数共振-主共振曲线均具有跳跃现象。随着阻尼、地基系数的改变,系统响应曲线具有“类软刚度特征”。随着参数激励幅值的改变,系统响应曲线具有“类硬刚度特征”。应用奇异性理论得到系统主参数共振-主共振稳态响应的转迁集和分岔图。

Nonlinear vibration of a thin rectangular plate with small deflection on the nonlinear elastic foundation is studied. By means of theory of elasticity, nonlinear dynamical equation of the rectangular thin plate on the nonlinear elastic foundation subject to harmonic excitation is established and furthermore, a nonlinear vibration equation is obtained by using Galerkin's method. Applying the method of multiple scales for nonlinear vibration, the first approximation solution of primary parametric resonance and fundamental resonance of the system is acquired. Numerical analysis on the influences of damping, foundation coefficient, and excitation parameter on the system is carried out. On the response curves of primary parametric resonance and fundamental resonance of the system exsit jump phenomena. The characteristic due to hard or soft stiffness may be altered as the parameters of damping, foundation, and excitation on the system are varied. The transition sets and bifurcation diagrams on the unfolding parametric plane and illustrated.