传递矩阵法分析平面柔顺机构的振动特性

为了分析柔顺机构的动态性能和力学传递关系,提出一种新型的基于传递矩阵法的振动力学模型。柔顺机构由若干个柔性铰链与杆件顺序联接而成,采用传递矩阵法描述其力学状态量的传递关系。将柔性铰链视为拉伸和弯曲变形的弹性梁,应用材料力学理论建立反映其自振性能的传递矩阵。将杆件视为刚体,采用动量矩定理建立其动力学模型,得到描述振动刚体的传递矩阵。按照柔顺机构的联接形式,将每个元件的传递矩阵进行拼装,得到系统总传递矩阵。总传递方程以柔顺机构的边界点状态矢量为未知变量,矩阵中的元素为机构结构参数和频率的函数。应用边界条件,可得柔顺机构的特征方程,通过求解方程可得系统的固有频率和振型。将外力纳入传递矩阵,建立反映外力激励与系统的关系的扩展传递矩阵以求解系统的频率响应。通过2种常用平面柔顺机构的动态性能分析,结果表明所建立模型的正确性,能精确描述柔顺机构力学传递关系。

Mechanical Vibration Modelling and Analysis of Planar Compliant Mechanisms Based on Transfer Matrix Method

In order to analyze dynamic characteristics and the mechanical transfer relationship, a novel mechanics model is proposed based on the transfer matrix method for a complicated rigid-flexure compliant mechanism. Because the compliant mechanism consists of several flexure hinges and bars in sequence, the transfer matrix method is applied to describe its transform relationship between mechanical states. The flexure hinge is treated as an elastic beam with tensile and bending deformation, and the material mechanics theory is applied to build the transfer matrix, which reflects the vibration performance. The bar is regarded as a rigid body, and the dynamic model and the transfer matrix of the vibration of the rigid body is obtained using the theorem of moment of momentum. According to the connection of the compliant mechanism, the total transfer matrix of the system can be obtained by assembling the transfer matrix of each part. The unknown variables of the total transfer equations are the boundary state vector of the compliant mechanism, and the elements of the total transfer matrix are the function with respect to the structural parameters and frequency. The characteristic equation of the compliant mechanism can be obtained by applying the boundary conditions, and the natural frequency and vibration mode can be obtained by solving the equation. The external force is added to the transfer matrix. The extended transfer matrix describing the relationship between external excitation and the system can be built and solved in order to acquire the frequency response. The dynamic characteristic of two kinds of planar compliant mechanisms are analyzed, and the results show that the proposed model is correct, which can accurately describe the mechanical transmission of the compliant mechanism.