Two general solutions of torsional compliance for variable rectangular cross-section hinges in compliant mechanisms

Several approaches exist for calculating the torsional compliance of rectangular cross-section beams, but most depend on the relative magnitude of the cross-section thickness and width, which might be changing during the design phase (especially for design optimization) or is variant for variable cross-section beams such as circular flexure hinges and tapered bars. After summarizing current equations and analyzing their computational accuracy, two new equations are proposed, which are thickness-to-width ratio independent, and suitable for variable cross-section beams and optimization design of torsional elements in compliant mechanisms. The closed-form equations for the torsional compliance of elliptical and circular flexure hinges are derived by using the new equations.