Second-order approximation pseudo-rigid model of flexure hinge with parabolic variable thickness

The paper deals with a study to determine an accurate pseudo-rigid model of flexure hinges with parabolic variable thickness subjected to pure applied moment. The work can be considered an extension of previous investigations on constant-thickness flexure hinges. Starting from the observation on second-order kinematic invariants of the relative motion between two bodies connected by such a flexure hinge, a pseudo-rigid model comprised of two circles rolling without slipping (planetary arrangement) is conceived. Thanks to parametric investigations, closed formulas for the computation of the radii of the two rolling circles and for the assessment of the concentrated spring are deduced in terms of main geometrical and elastic parameters. The proposed pseudo-rigid model is able to reproduce the elasto-kinematic behaviour of conic flexure hinges with accuracy, using a single degree of freedom surrogate mechanism. Finite element simulations confirm the accuracy of the closed form expressions. An example of application of the proposed methodology dealing with a vibration problem is presented in details.