磁耦合悬臂梁双稳动力学分析与实验

Analysis and Experiment on the Cantilever Beam with Magnetic Interactions

研究了一个自由端附加小磁铁的悬臂梁在磁力作用下的双稳态动力学行为.首先,利用Hamilton原理和Euler-Bernoulli梁的基本方程建立了系统在非零平衡点处做微幅振动的动力学方程.其次,利用多尺度法对建立的模型进行理论分析,得到悬臂梁在非零平衡点处振动的幅频方程和位移解,并对解进行了稳定性分析.最后,通过建立实验装置,得到悬臂梁不同运动形式下的参数平面分类和悬臂梁在非零平衡点处振动的幅频关系,通过观察系统在非零平衡点处振动的理论预测,实验结果验证了非零平衡点处振动的理论分析的正确性.对照理论、实验和数值结果得到:在不同的外激励幅值和频率作用下,悬臂梁有三种不同的运动形式:在非零平衡点处的微幅振动;大范围往返运动;在两个非零平衡点之间的无规律运动.

The bistable phenomenon of cantilever beam with a magnet at the free end of the beam was studied. Firstly, the dynamic equations of the system was derived by employing Hamilton principles and basic equations for Euler-Bernoulli beam. Then, the amplitude-frequency equation of local solutions for the cantilever beam was obtained by the theoretical analysis of the model using the multi-scale method. The stability analysis of the local solutions was also conducted. Finally, the experimental device was established to observe the analytical predictions. The experiment results are in good agreement with those from the theoretical analysis. Comparison of the theoretical, experimental and numerical results shows that, under different frequencies and amplitudes of the etxternal loading, the beam undergoes three different motions： micro-amplitude vibration at non-zero balance point, wide-range periodic vibration and the vibration switched between two non-zero balance points.