有限挠度下Timoshenko梁中的非线性弯曲波及其混沌行为  

Nonlinear Flexural Waves and Chaos Behavior in Finite-Deflection Timoshenko Beam

作　　者：张善元[1] 刘志芳[1] 

机构地区：[1]太原理工大学应用力学与生物医学工程研究所,太原030024

出　　处：《应用数学和力学》2010年第11期1276-1286,共11页Applied Mathematics and Mechanics

基　　金：国家自然科学基金资助项目(10772129)

摘　　要：以Timoshenko梁理论为基础,引入了有限挠度和轴向惯性,建立了支配梁运动的非线性偏微分方程组,采用行波法求解,通过某些积分技巧,将其转化为一个非线性常微分方程.常微分方程的定性分析表明,在一定条件下,系统存在异宿轨道,预示着有冲击波解存在.借助Jacobi椭圆函数展开求解,得到了非线性波动方程的准确周期解及其当模数m→1退化情况下的冲击波解.进而考虑阻尼和外加横向载荷对系统的摄动,利用Melnikov函数给出了横截异宿点出现的阈值条件,从而表明系统具有Smale马蹄意义下的混沌性质.On the basis of the theory of Timoshenko beam,taking into account finite-deflection and axial inertia,the nonlinear partial differential equations governing flexural waves in a beam were derived.When employing the method of the traveling wave solution,the nonlinear partial differential equations can be transformed into an ordinary differential equation by using certain integral skills.The qualitative analysis indicates that the corresponding dynamic system has heteroclinic orbit under certain condition.The exact periodic solution of nonlinear wave equation was obtained by means of Jacobi elliptic function expansion.When the modulus of Jacobi elliptic function m → 1 in the degenerate case,the shock wave solution was given.Further,small perturbations arising from damping and external load to original Hamilton＇s system are introduced and the threshold condition of the existence of transverse heteroclinic point is obtained by Melnikov＇s method.It is proved from this that the perturbed system has chaotic property under Smale horseshoe transform.

关 键 词：TIMOSHENKO梁 有限挠度 冲击波 混沌运动 Jacobi椭圆函数展开 MELNIKOV函数 

分 类 号：O347.4[理学—固体力学]