两自由度参数激励摆旋转运动分析和实验

研究具有支撑参数激励摆系统的支撑结构振动对摆旋转的影响，其中支撑结构是受到扭簧约束的刚性悬臂梁，参数激励摆与刚性悬臂梁的悬臂段铰接．首先，通过拉格朗日方程建立了系统两自由度的动力学方程．其次，利用多尺度法对建立的模型进行理论分析，得到悬臂梁的振动与上摆不同运动形式的关系，从而得到上摆不同运动形式下的参数平面分类和悬臂梁在上摆转动时的振动频响．最后，通过建立实验装置，观察理论预测，实验结果验证了理论分析的正确性．实验与理论对照得到，当参数激励频率接近悬臂梁的一阶固有频率时，悬臂梁的振幅变大，会破坏摆的转动稳定性．

Analysis and Experiment for the Rotation Movement of a Two Degrees of Freedom Pendulum Subjected to Parametric Excitation

The present paper is focused on the influence of vibration of a supportive structure, which includes a suported pendulum system under parametric excitation. The suportive structure was modelled as a rigid cantilever beam under tortional spring constraint. The parametrically excited pendulum was hinged at the upper end of the cantilever beam. Firstly, the Lagrange principle was employed to propose the dynamical equations of a two degrees of freedom system. Secondly, the multi-scale method was adopted to analyze dynamics of the system. Subsequently obtained was the relation between the vibration of cantilever beam and various movement of the upper pendulum. The classification of the upper pendulum’s movements under diferent excited frequencies and amplitudes was performed. Moreover, the amplitude-frequency response of the cantilever beam was also demonstrated when the upper pendulum was rotating. Finally, the experimental device was set up to observe the analytical predictions. The experiment results are in good agreement with those from the theoretical analysis. By comparing experimental results to analytical solutions, the oscillation of the cantilever beam becomes strong and induces the instability of the rotation movement of the pendulum, while the parametric excitation frequency approaches the first order natural frequency of the cantilever beam.