弹性约束浅拱的内共振非线性模态

摘要

When a dynamic system is elastically constrained,its natural vibration features and nonlinear normal modes are different from those under ideal boundary conditions.The constrain stiffness values change parametric regions of various nonlinear dynamic behaviors.The nonlinear normal modes of a shallow arch with one end elastically constrained along vertical direction in a case of 2∶1 internal resonance were constructed by using the multi-scale method and the direct perturbation here.The research results showed that an internal resonant nonlinear system has both a singled-mode motion and coupled-mode motions;the existence and various values of elastic constraint lead to different asymmetric linear modes,shape functions and internal resonance exciting conditions;moreover,the space-time effects of nonlinear normal modes with different constraint stiffnesses display different dynamic configurations.%动力系统采用弹性约束时自振特性、非线性模态与采用理想边界时存在差异，约束刚度值将改变各种非线性动力行为产生的参数域。采用多尺度法进行直接摄动构造了一端竖向弹性约束浅拱在发生2∶1内共振时的非线性模态。结果表明：内共振非线性系统存在单模态运动和耦合模态运动；弹性约束的存在及不同取值将导致不同的非对称线性模态、形函数及内共振激发条件；不同约束刚度时非线性模态的时空效应呈现不同的动力形态。