Notch flexure hinges: An effective theory

Abstract

This article presents effective tractable equations for rotational compliance (stiffness) of a simple monolithic flexure hinge with circular (radius R and crosspiece thickness t), elliptical [at semiaxis ax, ay, elliptical ratio ε (=ax/ay)] and other cross sections. These equations and the method by inverse conformal mapping of circular approximating contour used to derive them are different from the known and widely used theoretical equations originally derived in 1965 by Paros and Weisbord for circular notch hinges. Later it was found that the circular hinge represents the worst case error between known theoretical and finite element models. The conformal mapping equations data presented in this article are likely to be much closer (within less than 10%) to the finite element analysis and experimental data than other theoretical equations. In particular this is the case for circular notch hinges at relative thickness β(=t/2R) in the range 0.01 to 0.3 and for elliptical hinges at the elliptical ratio ε=1 to 10. The derived general equation is common for all types of notch hinges whose profiles can be approximated by two shifted contiguous circles and includes material parameters with Young’s modulus and Poisson’s ratio. The latter is totally omitted in known theoretical solutions by other authors. New tractable equations are derived from the general equation on the basis of trigonometric functions’ simplified series expansion in certain ranges of hinge crosspiece relative thickness. The corresponding graphs are presented. Experimental data were received by holographic interference and autocollimator measurement.