摘要
研究了功能梯度矩形薄板的非线性自由振动问题。采用幂律分布规律描述功能梯度材料沿厚度的梯度性质,基于von Kámán理论,建立了功能梯度薄板的非线性振动控制方程。应用Bubnov-Galerkin法得到了功能梯度矩形薄板的单模态非线性振动的时域常微分方程,借助其势能函数分析了系统的周期振动状态。采用Lindstedt-Poincaré法和Runge-Kutta法分别获得了功能梯度矩形薄板单模态非线性周期振动的摄动解和数值解。研究表明:功能梯度薄板非线性振动控制方程中包含表征拉弯耦合效应的控制项,这导致其常微分方程中出现二次项;系统振幅在板横向的正负两个方向上是不相等的,其振动存在关于板中面的不对称性。
Nonlinear free vibration of functionally graded rectangular thin plates was studied.The gradient properties of functionally graded material（FGM） were expressed as the volume fraction power-law distributions on the thickness direction.Based on von Kmn＇s theory,the nonlinear governing equations for FGM thin plates were established.Bubnov-Galerkin procedure was applied to derive the time-domain ordinary differential equation for single mode motion of FGM rectangular thin plates.The periodic vibration was analyzed according to potential function of system.The perturbation solution and numerical results of periodic motion of system were obtained,respectively,by using Lindstedt-Poincaré method and Runge-Kutta method.It shows that,there contain the control terms to characterize the stretching-bending coupling in the nonlinear governing equations of FGM thin plates,which leads to reveal the quadratic term in the ordinary equation of system;the nonlinear amplitude of FGM rectangular thin plates is unequal in the positive and negative direction of lateral orientation,i.e.the vibration of system is not symmetric relative to mid-plane of plates.
出处
《力学季刊》
CSCD
北大核心
2010年第2期250-255,共6页
Chinese Quarterly of Mechanics
基金
国家自然科学基金青年项目(10902092)
