直线运动柔性梁非线性动力学--组合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内共振及其组合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，系统的平凡响应与双模态非平凡响应共存，由内共振所产生的非平凡响应皆为不稳定的鞍点，平凡及非平凡解分支都存在Hopf分岔现象，一些稳定的极限环随参数变化最终经倍周期分岔后产生混沌运动。

Nonlinear Dynamics of Flexible Beams Undergoing a Large Linear Motion of Basement:Combinational 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 to solve directly the nonlinear differential equations and derive a set of nonlinear modulation equations for the combinational parametric and 3:1 internal resonances between the first two modes. Then, the modulation equations are numerically solved to obtain the steady-state response and the stability of the beam. Results show that the trivial response and the two-mode non-trivial response of the system coexist and the responses arising from the 3:1 internal resonance belong to unstable saddles. Supercritical and sub-critical Hopf bifurcations in both trivial and non-trivial branches and the saddle-node bifurcations in non-trivial branches of the response curves are found. As usual, for the Hopf bifurcations, a limit cycle is found undergoing a series of period-doubling bifurcations and become a chaotic oscillation at last.