Elliptical flexure hinges

This paper presents closed form equations based on a modification of those originally derived by Paros and Weisbord in 1965, for the mechanical compliance of a simple monolithic flexure hinge of elliptic cross section, the geometry of which is determined by the ratio ε of the major and minor axes. It is shown that these equations converge at ε=1 to the Paros and Weisbord equations for a hinge of circular section and at ε ⇒∞ to the equations predicted from simple beam bending theory for the compliance of a cantilever beam. These equations are then assessed by comparison with results from finite element analysis over a range of geometries typical of many hinge designs. Based on the finite element analysis, stress concentration factors for the elliptical hinge are also presented. As a further verification of these equations, a number of elliptical hinges were manufactured on a CNC milling machine. Experimental data were produced by applying a bending moment using dead weight loading and measuring subsequent angular deflections with a laser interferometer. In general, it was found that predictions for the compliance of elliptical hinges are likely to be within 12% for a range of geometries with the ratio βx (=t/2ax) between 0.06 and 0.2 and for values of ε between 1 and 10.