双铰抛物线弹性拱的混沌行为

The Chaotic Behavior of Parabolic Elastic Arch with Two Hinge Supports

要设计出具有好的非线性动力学特性的拱结构,需要了解拱在外激励下的长期非线性动力学行为,对两铰抛物线弹性拱在横向周期荷载下的混沌运动行为进行了研究。基于变形体的几何方程及拱的单元平衡方程建立拱的非线性动力学模型,然后利用Galerkin原理得到控制拱横向振动的二阶三次非线性微分动力系统,并由此得无扰动系统的不动点与同宿轨道;使用Melnikov方法得到了拱混沌振动的临界条件;最后通过数值仿真得到该微分动力系统Lyapunov指数谱、Lyapunov维数、平面相轨线、Poincare映射等混沌特性,并以此判定系统的振动是定常运动还是混沌运动。研究表明:双铰抛物线弹性拱在横向周期荷载作用下的可能出现定常运动也可能出现混沌运动;当横向外激励振幅较小时系统运动为定常运动,激励振幅较大时系统运动为混沌运动。

In order to design an arch structure with good nonlinear dynamic characteristics, the nonlinear dynamic behaviors under a long time external force have to be investigated. The chaotic behaviors of the parabolic elastic arch with two hinge supports subjected to a transverse distributed varying periodic excitation are investigated in this paper. Based on the geometric equation of deformable body and the equilibrium equations of an arch element, the nonlinear dynamic model which dominates the transverse vibration of the elastic arch is established first, and then the nonlinear differential dynamic system is obtained by using Galerkin＇ s method, thus the fixed points and the ho- moclinic orbits are found out. The critical condition of chaotic vibration of the elastic arch is obtained through the Melnikovg method. Finally the dynamic characteristics （such as Lyapunov exponents and Lyapunov dimension and the phase trajectories and also the Poincare map etc. ） which can be used to explain the dynamic behaviors of the differential dynamic system of the elastic arch are calculated by using numerical simulation for different parameters. It is found that the motion of the parabolic elastic arch with two hinge supports subjected to a transverse periodic excitation may be stationary or chaotic motion. The stationary motion occurs if the amplitude of the transverse periodic excitation is small, but the chaotic motion occurs if the amplitude is large.