Analytical Compliance Analysis and Synthesis of Flexure Mechanisms

Abstract

This paper presents a symbolic formulation for analytical compliance analysis and synthesis of flexure mechanisms with arbitrary topologies. Compliance analysis or mapping is to determine the relationship between the deformation of a mechanism and the external loading applied. It is a crucial step for the control and design of flexure mechanisms. Most of the current work relies on physical experiments or numerical simulations for studying the compliance or stiffness of flexure mechanisms. There is a lack of formal tools for the compliance synthesis whose goal is to determine the geometry of flexures or assembly of multiple flexures for a prescribed compliance in the motion direction of interest. In this work, we first derive a symbolic formulation of the compliance and stiffness matrices for commonly-used flexure elements, flexure joints and simple chains. Elements of these matrices are all explicit functions of flexure parameters. To analyze a complex flexure mechanism, we subdivide the mechanism into multiple structural modules which we identify as serial, parallel or hybrid chains. We then analyze each module with the known flexure structures in the library. At last we use a bottom-up approach to obtain the compliance matrix for the overall mechanism. Our symbolic formulation enables subsequent compliance synthesis or sensitivity analysis which is to determine how each flexure parameter affects the overall compliance of the mechanism. Four practical examples are provided to demonstrate the approach. The result is a robust design method for the compliance analysis and synthesis of flexure mechanisms.