High-order kinematics of uniform flexures

In the last decades, many efforts have been made to analyze and model small and large deflections of flexures, considering complex load cases and different solution techniques. However, few investigations focused on the kinematic aspects related to the deflection analysis of the flexible elements, and limited the study to the second-order kinematics. In this paper, an analytical formulation based on the instantaneous geometric invariants is developed to give deep kinematic insight, up to the fourth order, into the motion generated by the deflections. The problem is addressed from a geometrical point of view, defining the fundamental geometric entities that characterize the motion, that are inflection circle, cubic of stationary curvature and its derivative, Ball’s point, and Burmester’s points. Thanks to these special points on the plane, straight and circular paths can be approximated to the third and to the fourth order, respectively. The proposed formulation defines the geometric characteristics of flexures with different curvatures. An application regarding the definition of pseudo-rigid body models is discussed. Finite element simulations are performed to validate the results.