弹性约束浅拱的非线性动力响应与分岔分析

浅拱采用竖向、转动方向弹性约束时,自振频率和模态与理想的铰支/固结边界存在差异,不同约束刚度将改变外激励下的非线性响应及各种分岔产生的参数域．由浅拱基本假定建立无量纲动力学方程, 采用在频率和模态中考虑约束刚度大小的方法,通过Galerkin全离散和多尺度摄动分析导出极坐标、直角坐标形式的平均方程, 其中方程系数与约束刚度一一对应．用数值方法分析了周期激励下竖向弹性约束系统最低两阶模态之间1∶2内共振时的动力行为, 所得结果与有限元的对比以及平均方程系数的收敛性证明了所采用方法是可行的．随着激励幅值、频率的变化存在若干分岔点,分岔发生时的参数分布与约束刚度值有关,在由分岔点连接的不稳定区或共振区附近,存在一系列稳态解、周期解、准周期解和混沌解窗口,且随参数的变化可观测到倍周期分岔．

Nonlinear Dynamic Response and Bifurcation Analysis of the Elastically Constrained Shallow Arch

When the ends are elastically constrained in vertical and rotation directions for the shallow arch, the natural frequencies and modes are quite different from those of the case of ideal hinged or fixed boundary condition, and the different constraint stiffness will change the nonlinear responses and the parameter fields of various bifurcations under external excitation. The dimensionless dynamic equation was established by introducing the fundamental assumptions of shallow arch, and the method that the effects by the boundary constraint stiffness were considered in the natural frequencies and modes solution was employed, then the full-basis Galerkin discretization and the multi-scale perturbation methods were used to obtain the polar- and Cartesian-type averaging equations, of which the coefficients have one-to-one correspondence with the values of constraint stiffness. With the application of numerical calculation, the dynamic behaviors of the vertical elastically constrained system in the case of one-to-two internal resonance between the lowest two modes under periodic excitation were studied. Both the comparison of calculated results with finite element results and the convergence of the coefficients in averaging equations proved the feasibility of the present method. Also, the numerical results show that there exist several bifurcation points with the variation of the amplitude and frequency of excitation, and the parameter distributions for the occurrence of bifurcations are associated with the values of constraint stiffness. Moreover, there are a series of steady-state solution, periodic solution, quasi-periodic solution and chaotic solution windows in the vicinity of the unstable areas or resonance regions which are connected by the bifurcation points, and the period-doubling bifurcation can be observed with the variation of parameters.