Dynamic Analysis of Compliant Mechanisms

A systematic method for dynamic analysis of compliant mechanisms is presented including basic formulations for natural frequencies, modes, dynamic response, and frequency characteristics. Methods for design sensitivity analysis are developed to investigate the effect of various design parameters on the dynamic performance of compliant mechanisms. A micro compliant stroke amplifier mechanism, for MEMS actuator application, is presented as a design example to demonstrate significant differences between static and dynamic performance of compliant mechanisms. INTRODUCTION Designing a compliant mechanism for a specific application can be a complex problem with many considerations. The basic trade-off is between the flexibility to achieve deformed motion and the rigidity to sustain external load. Many researchers have addressed proposed various synthesis techniques for creating a viable compliant mechanism [1-11] including topology and shape optimization methods. Many of these methods for topology and dimensional synthesis are based on kinetostatic design specifications and do not consider the dynamic effects in the design stage. Therefore, the resulting designs are valid for static or low frequency applications. The dynamic effects can be significant and for instance, a complaint mechanism may exhibit a very different mechanical advantage at high frequencies compared to what it was designed for static situations. The synthesis algorithms should, ideally, account for operating frequency before generating an appropriate topology and dimensions. Towards this goal, we first developed analysis formulations to investigate the dynamic behavior of compliant mechanisms. This paper presents a systematic method for the dynamic analysis of compliant mechanisms including differential equations of motion, formulae for natural frequencies and modes, dynamic responses, dynamic compliance, and sensitivity analysis. A design example demonstrates the significance and the effectiveness of these methods in improving the dynamic behavior. Although the formulations presented in this paper are well understood and reported in the published literature, the paper highlights the significance of dynamic response of compliant mechanisms and the need to account for the same during the design phase. Additionally, methods to carry out sensitivity analysis are developed to investigate the effect of various design parameters on the dynamic performance of compliant mechanisms. We first present the basic dynamic analysis formulations. DYNAMIC EQUATIONS OF COMPLIANT MECHANISMS The dynamic differential equations of a compliant mechanism can be derived using the finite element method and take the form of [ ] { } [ ] { } [ ] { } { } 1 1 1 1 × × × × × × × = + + n n n n n n n n n n R D K D C D M & & & (1) where [M], [C] and [K] are system mass, damping and stiffness matrix respectively; {D} is the set of generalized coordinates representing the translation and rotation deformations at each element node in global coordinate system; {R} is the set of generalized external forces corresponding to {D}; n is the number of the generalized coordinates (elastic degrees of freedom of the mechanism). If we use frame elements shown in Fig. 1 in the formulation, the element mass matrix and stiffness matrix can be written as Eq. (2) and Eq. (3).