Elliptical flexure hinges

TLDR

The study derives closed‑form equations for the compliance of monolithic elliptic‑cross‑section flexure hinges, extending Paros and Weisbord’s 1965 model by incorporating the major‑to‑minor axis ratio ε. The authors evaluate the equations through finite‑element analysis across typical hinge geometries, compute stress concentration factors, and validate the predictions experimentally on CNC‑milled hinges using dead‑weight loading and laser‑interferometer deflection measurements. The equations converge to known circular‑section and beam‑theory limits and predict hinge compliance within 12 % accuracy for βx between 0.06 and 0.2 and ε from 1 to 10.

Abstract

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.