有限挠度Bernoulli-Euler梁中的非线性波与混沌行为

Nonlinear Wave and Chaos Property in Finite-Deflection Bernoulli-Euler Beam

基于有限挠度理论,导出了Bernoulli-Euler梁的非线性偏微分方程形式的弯曲波动方程,利用行波解法和积分技巧,将非线性偏微分方程转化为常微方程.定性分析表明,在一定条件下,动力系统有异宿轨道,对应冲击波解.利用Jacobi椭圆函数法,得到了波动方程的准确周期波解,当Jacobi函数的模数m→1时,得到系统的冲击波解.显然,阻尼和外载荷的摄动将使异宿轨道破裂,得到横截异宿点.通过Melnikov函数法得到了系统出现横截异宿点的阈值条件,这表明,系统存在Smale马蹄意义下的混沌行为.

Based on the finite-deflection beam theory, the nonlinear partial differential equations for flexural waves in a Bernoulli-Euler beam are derived. Using the traveling wave method and integration skills, the nonlinear partial differential equations can be converted into an ordinary differential equation. The qualitative analysis indicates that the corresponding dynamic system has a heteroclinic orbit under a certain condition. The exact periodic solution of nonlinear wave equation is obtained by means of Jacobi elliptic function expansion. The shock wave solution is given when the modulus of Jacobi elliptic function m→1 in the degenerate case. It is easily thought that the introduction of damping and external load can result in break of heteroclinic orbit and appearance of transverse heteroclinic point. The threshold condition of the existence of transverse heteroclinic point is given by help of Melnikov function. It shows that the system has chaos property under Smale horseshoe meaning.