直线运动柔性梁非线性动力学--主参数共振与内共振联合激励

针对基础直线运动柔性梁，基于Kane方程建立了相应的非线性动力学方程。采用多尺度法并结合笛卡尔坐标变换，导出了系统受前两阶模态间3：1内共振及第二阶模态主参激共振时的非线性调制方程组．数值求解了该方程组的定常解及相应的稳定性问题。研究表明，系统的平凡、单模态、双模态稳态解共存，超临界及亚临界叉形分岔只发生在单模态状态下，相反，鞍结分岔及Hopf分岔只在双模态状态下产生，一些稳定的极限环随参数变化经一系列倍周期分岔后导致运动的突然跳跃。

Nonlinear Dynamics of Flexible Beams Undergoing a Large Linear Motion of Basement:Principal Parametric and Internal Resonances

A set of nonlinear differential equations is established for the planar oscillation of flexible beams undergoing a large linear motion of the basement. The method of multiple scales combined with the Cartesian transformation is used solve to the nonlinear differential equations and derive a set of nonlinear modulation equations for the principal parametric resonance of the first mode and 3:1 internal resonance between the first two modes. Then, the modulation equations are numerically solved to obtain the steady-state response and the stability assessment of the beam. Results show that the trivial, single, and two-mode solutions are possible. Supercritical and sub-critical fork bifurcation only occurs in single-mode equilibrium and the saddle-node bifurcation and Hopf bifurcation can be found in two-mode equilibrium. For a Hopf bifurcation, a limit cycle is found undergoing a series of period-doubling bifurcations and finally results in a jump of the response to either a single-mode or a two-mode stable equilibrium solution after a blue-sky catastrophe.