复合材料层合悬臂板的非线性动力学研究

 以飞机机翼的颤振为实际工程背景，考虑高阶横向剪切效应、几何大变形和横向阻尼的影响，基于Reddy的高阶剪切变形理论和von Karman的大变形理论，利用Hamilton原理对纤维增强复合材料层合悬臂板的非线性动力学问题进行了研究。建立了复合材料悬臂板在面内激励和横向外激励联合作用下悬臂板广义位移形式的偏微分运动控制方程。利用Galerkin方法，选取二阶模态对复合材料层合悬臂板偏微分形式运动控制方程进行二阶模态截断，得到了具有三次非线性项、参数激励项和横向激励项的常微分形式二自由度非线性动力学方程。在考虑主参数共振和1:2内共振的情况下，用多尺度法获得了复合材料层合悬臂板四维直角坐标形式的平均方程。在平均方程的基础上，利用数值方法分析面内激励和横向激励幅值对系统非线性动力学特性的影响，得到了1:2内共振时复合材料层合悬臂板动力学方程的平面相图、波形图、三维相图和频谱图。结果表明，随着外激励的变化，系统会出现单倍周期运动、多倍周期运动、概周期运动和混沌运动。

Nonlinear Dynamics of a Composite Laminated Cantilever Plate

The nonlinear dynamics of composite laminated cantilever plates subjected to in-plane and transversal excitations is investigated in this paper. With the wing flutter of a flying aircraft as the background, based on the Reddy's high-order shear deformation theory and von Kármán type equations for the geometric nonlinearity, the nonlinear governing partial differential equations of motion are derived for the composite laminated cantilever plate subjected to in-plane and transverse excitations by using the Hamilton's principle with considerations of the effects of higher-order transverse shear deformation, nonlinear geometry deformation and transversal damping. The Galerkin approach is utilized to transform the nonlinear partial differential governing equations of motion for the composite laminated cantilever plate to a two-degree-of-freedom nonlinear system. The principal parametric resonance, the 1/2 subharmonic resonance and the 1:2 internal resonance are considered. The method of multiple scales is employed to obtain the four-dimensional averaged equations of the composite laminated cantilever plate subjected to in-plane and transverse excitations. The dynamical behaviors of the cantilever plate under combined in-plane and transversal excitations are investigated with numerical simulations of the averaged equations. The two-dimensional phase portrait, waveform and three-dimensional phase portrait are obtained as results of the nonlinear dynamic behaviors of the composite laminated cantilever plate. The results of numerical simulations demonstrate that there exist one-periodic, multi-periodic and chaotic motions of the composite laminated cantilever plate.