研究了压电位移放大机构的运动学和动力学建模问题。基于能量守恒原理和弹性梁弯曲理论推导了桥式位移放大机构的位移放大比等静力学解析模型; 在此基础上, 通过拉格朗日方程建立了桥式位移放大机构的固有频率解析模型。通过有限元计算验证和分析了提出的解析模型的可行性和优越性, 并与国内外典型的位移放大比数学模型进行了比较。结果表明: 由于本文提出的模型考虑了位移放大机构的拉伸和弯曲变形, 并且摒弃了国内外普遍采用近似几何关系进行数学推导的思路, 因此所建立的位移放大比解析模型精度更高; 固有频率解析计算结果与有限元模态分析结果的相对误差约为5%。得到的结果显示: 本文给出的建模方法以及位移放大比、固有频率等解析模型可为柔性机构的优化设计和研制提供依据和参考。

Kinematic and dynamic modeling of piezoelectric displacement amplifying mechanisms was researched. The static analytical models(such as displacement amplifying ratio) for a bridge type compliant displacement amplifying mechanism was derived based on the law of conservation of energy and elastic beam theory. Then, an analytical model of natural frequency was also built by employing the Lagrange equation. The finite element analysis was used for verification of the feasibility and superiority of proposed analytical models and for comparison with several typical mathematical models deduced by other authors. The results show that the proposed theoretical formula of the displacement amplification ratio has the highest accuracy, because it considers both the translational and rotational stiffnesses of the mechanism during modeling and abandons the approximate geometric relationship between input and output displacements of the bridge-type compliant mechanism. Moreover, the discrepancy between the theoretical formula of natural frequency in this paper and the finite element calculation results is kept within 5%. The modeling method and corresponding theoretical formulas of the displacement amplification ratio and natural frequency proposed in this paper provides a useful and accurate reference for optimal designing and manufacturing of satisfactory structures of bridge-type displacement amplification mechanisms.