竖向弹性支撑浅拱的非线性动力行为分析  

Research on the nonlinear dynamical behaviors of vertical elastic support shallow arch

作　　者：易壮鹏[1] 黎亮[1] 王连华[2] 

机构地区：[1]长沙理工大学土木与建筑学院,长沙410004 [2]湖南大学土木工程学院,长沙410082

出　　处：《应用力学学报》2013年第6期925-931,957-958,共7页Chinese Journal of Applied Mechanics

基　　金：国家自然科学基金(11002030;10972073;11032004);教育部新世纪人才项目(NCET-09-0335)

摘　　要：研究了两端竖向弹性支撑浅拱在周期激励作用下发生1:1内共振时的分岔与混沌等非线性动力行为。通过引入基本假定,得到了浅拱的基本动力学方程;采用Galerkin全离散并通过多尺度法进行摄动,得到了内共振的发生条件及平均方程;去掉阻尼、外荷载、非线性项后,在所得线性方程的自振频率和正交模态的基础上考虑竖向弹性支撑,推导得出了与弹性刚度值对应的平均方程系数。研究结果表明:不对称弹性边界使1:1内共振形式为模态转向,系统存在对刚度敏感的弹性支撑区域;激励幅值和频率发生变化时,在一定参数条件下存在稳态解、周期解、准周期解、混沌解窗口,并存在倍周期分岔现象。The bifurcation, chaos and other nonlinear dynamic behavior of shallow arches in case of one-to-one internal resonance under external periodic excitation when both ends are vertically elastic restrained are investigated. By introducing the fundamental assumptions of shallow arch, the basic dynamic equation can be determined. Then the occurrence condition for internal resonances and the averaging equations are achieved with the application of a full-basis Galerkin discretization and the multi-scale method in perturbation analysis. Further, the damper terms, external load and non-linear terms are removed in basic equation, and the vertical elastic support are taken into account in the natural frequencies and normal modes of the obtained linear system, which leads to the coefficients which has a one-to-one correspondence with the values of elastic support stiffness in the averaging equations. It is found that the type of one-to-one internal resonance is veering between modes due to the existence of unsymmetrical elastic boundary, and there has an elastic support region sensitive to stiffness in the system. Also, the nonlinear interaction between the involved internal resonance modes under external excitation is remarkable with variation of the amplitude and frequency of external excitation, and there exist a series of steady-state solution, periodic solution, quasi-periodic solution and chaotic solution windows for certain parameterconditions. Moreover, the dynamic system enters into chaos by period-doubling bifurcation can be observed as the variation of systematic parameters.

关 键 词：竖向弹性支撑浅拱 1 1内共振 倍周期分岔 混沌 多尺度法 

分 类 号：U311.2[交通运输工程]