基于磁化电流法的双稳压电悬臂梁磁力精确分析

Accurate analysis of magnetic force of bi-stable cantilever vibration energy harvesting system with the theory of magnetizing current

双稳压电悬臂梁结构常常用于振动能量采集系统,其中的非线性磁力与系统势函数和动力学方程的建立紧密相关,非线性磁力的正确分析和精确计算对系统振动响应和能量采集效果的准确预测至关重要.本文采用形状函数分析方法,通过悬臂梁弯曲斜率的整体积分计算,得到了悬臂梁末端的运动轨迹及其末端磁铁精确的位置与姿态,并由此根据磁化电流理论建立了双稳压电悬臂梁能量采集系统的磁力计算模型,给出了末端磁铁受到的水平轴向磁力和竖直纵向磁力及其合磁力的变化规律.数值模拟发现,随着末端磁铁竖直纵向位移逐渐增大,磁铁受到的水平轴向磁力和竖直纵向磁力都会依次由排斥力转变为吸引力,从而导致磁力合力的方向会随磁铁位移发生跨越两个象限的大幅度变化.实验验证表明,磁力计算结果与实验测量结果符合良好,其精确度优于现有文献方法的精度,因此本文的方法可以准确预测双稳压电悬臂梁振动过程的磁力变化规律.

In the study of piezoelectric cantilever energy harvesting system, a bi-stable nonlinear cantilever with magnets added to the structure has a wider frequency band response and a higher energy output efficiency. Hence, the calculation accuracy of the magnetic force on which the potential function and dynamics of the system depend is essential to predicting the output response and energy harvesting effect. In this work, we use a shape function to describe the relation between the deflections of an arbitrary point and the free-end point on the beam, and then calculate the trace and deflection angle of the beam＇s free-end by integrating the entire slope of the cantilever beam. The magnetic force is consequently derived from the magnets＇ real-time relative positions and postures by using the magnetizing current method. With comprehensively considering the axial magnetic force and the lateral magnetic force, the simulation results demonstrate that when the displacement of the magnet at the end of the beam is large enough, the directions of axial and lateral magnetic force change from repulsive to attractive, which leads to a large veer of the resultant magnetic force across two quadrants. So, it means that a smaller interval between magnets may not cause a larger deflection of the beam, and the magnetic force existing as attractive force could diminish the well space of potential function（that is,the distance between two equilibrium positions of the system）. The experimental data in this work are nicely consistent with the simulation results. And in this work, we also make a comparison of the simulation results with those from our method and existing method, showing that the accuracy of the proposed method is much higher than that from the existing calculation method, especially in the scenario where the magnet at the end of the beam is far from the horizontal axis.