Bifurcation analysis of the nonlinear vibration of an inextensible cantilever beam is analyzed by using the nonlinear normal mode concept. Two flexural modes of the cantilever beam, one in each transverse plane is considered. Two degrees-of-freedom nonlinear model for the vibration in the transverse direction is obtained by the discretization of the governing equation using Galerkins method based on the eigenmodes in each direction. The method of multiple scales is used to derive two first-order nonlinear ordinary differential equations governing the modulation of the amplitude and the phase of the dominant mode for the case of 1:1 internal resonance. The bifurcation diagrams are computed considering the frequency of excitation and the magnitude of the excitation as the control parameters. The stability of the fixed point is determined by examining the eigenvalues of the Jacobian matrix. The results show that a saddle-node-type bifurcation of the solution can occur under certain parameter conditions.

中文翻译：

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谐波激励下悬臂梁非线性模态的分叉

利用非线性法则模式概念分析了不可伸展悬臂梁非线性振动的分叉分析。考虑悬臂梁的两种弯曲模式，每个横向平面中的一种。通过基于各个方向的本征模的Galerkins方法对控制方程进行离散化，获得了横向振动的两个自由度非线性模型。使用多尺度方法可导出两个一阶非线性常微分方程，这些方程控制在内部共振为1：1的情况下主导模式的振幅和相位的调制。分岔图的计算以激励频率和激励幅度为控制参数。通过检查雅可比矩阵的特征值来确定不动点的稳定性。结果表明，在某些参数条件下，溶液的鞍形节点分叉会发生。