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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 nonlinear dynamic behaviors such as bifurcation and chaos in the case of 1: 1 internal resonance under cyclic excitation are studied. By introducing the basic assumptions, the basic dynamic equations of the shallow arch are obtained. After the Galerkin is fully dispersed and perturbed by the multi-scale method, the internal resonance conditions and the average equation are obtained. After the damping, the external load and the nonlinear terms are removed, Based on the natural frequency and the orthogonal modal of the linear equation obtained, the vertical elastic support is considered, and the average equation coefficient corresponding to the elastic stiffness value is deduced. The results show that the asymmetric elastic boundary modifies the 1: 1 internal resonance mode, and the system has elastic support zone which is sensitive to stiffness. When the amplitude and frequency of excitation vary, the steady state solution exists under certain parameters, and the period Solution, quasi-periodic solution, chaos solution window, and the existence of doubling period bifurcation phenomenon.