变温环境下粘弹性梁的混沌运动

Chaotic Motion of a Viscoelastic Beam in Time Dependent Temperature Field

根据Kelvin粘弹性材料本构关系、梁的运动方程及变形几何方程建立了同时具有温度扰动和横向分布力扰动的粘弹性梁非线性动力学模型.用Galerkin方法将系统简化为参数激励和强迫激励耦合的单模态Duffing振子,得到了系统的不动点和同宿轨道.用Melnikov函数法推导出系统混沌运动的临界条件,分析了系统通向混沌的途径.研究表明,非线性粘弹性梁在周期性横向激励及周期性温度联合作用下可能进入混沌运动,并且在发生Smale马蹄意义下的混沌前,将经历多次的次谐分岔.

A nonlinear dynamic model for a viscoelastic beam under a laterally distributed excitation in a time dependent temperature field was derived, which is based on the constitutive description of Kelvin viscoelastic materials, motion equations and strain-displacement relations of a beam with large deflections. One of multiple modes of the governing equation was obtained using Galerkin＇s method to obtain a parametrically excited （caused by temperature） Duffing＇s oscillator, and the fixed points and homoclinic orbits were obtained according to the Dufting＇s oscillator. The critical conditions for chaos of the viscoelastic beam to occur were determined by the Melnikov function method to discuss the path to chaos. The results show that it is possible for the nonlinear viscoelastic beam under the laterally distributed excitation and temperature to enter a chaotic motion in the Smale＇s horseshoe sense, and the system experiences many times of subharmonic bifurcations before the chaos happens.