微分求积法在悬臂梁结构非线性动力学中的应用研究

Application of Differential Quadrature Method to Nonlinear Dynamics of Cantilever Beam Structures

利用微分求积法对受横向载荷和轴向载荷联合作用的粘弹性悬臂梁的非线性动力学偏微分控制方程直接离散求解,并提出了一种新的边界条件施加方法处理悬臂梁的边界条件。在数值结果的基础上结合非线性动力学理论,利用分叉图,时间历程图,相图等对受横向载荷和轴向载荷联合作用的粘弹性悬臂梁的非线性动力学特性进行了分析。由以上图形得到的其非线性动力学性质是一样的,因而表明微分求积法以及施加边界条件的新方法能够有效地用来分析悬臂梁结构的非线性动力学性质。

In this paper, a differential quadrature method (DQM) is developed to study the nonlinear dy-namic behaviors of a viscoelastic cantilever beam subjected to transverse loads and axial loads. The partial differential nonlinear governing equation of the cantilever beam is discretized in space region using DQM. For the boundary conditions of the cantilever beam, a new method is proposed to deal with the boundary conditions. Based on the numerical results, the nonlinear dynamical behaviors, such as the bifurcations and chaotic motions of the viscoelastic cantilever beam, are investigated by using the bifurcation diagrams, Poincare maps and phase portraits. It is drawn the conclusion from numerical simulation results that the DQM and a new method of applying the boundary conditions can be effectively used to analyze the nonlinear dynamics properties of cantilever beam structures.