Design of Compliant Mechanisms Approximating Straight and Circular Paths

Compliant mechanisms are extensively employed in many fields due to their advantages over their rigid-body counterparts. They transmit force, motion, and energy through the elastic deformations of their flexible parts, that often consist of beams with uniform or variable cross-section, with straight or initially-curved axis. Although extensive efforts have been made in the analysis of the deflections, fewer studies dealt with the kinematic aspects associated to the deflection problem. Recently, some investigations focused on the behavior of the pole of the displacements, which characterizes the rigid displacements or on the inflection circle for developing pseudo-rigid-body models. In this paper, the instantaneous geometric invariants are applied to the description of the motion generated by straight flexures. The developed analytical formulation gives kinematic insights on the geometric characteristic of motion up to the fourth order. The invariants lead to the determination of fundamental geometric entities, that are the inflection circle, the cubic of stationary curvature and its derivative, the Ball's point, and the Burmester's points. In particular, Ball's and Burmester's points identify the special points on the plane that approximate straight paths to the third order, and circular paths to the fourth order, respectively. A possible implementation of the geometric invariants to the design of compliant mechanisms is briefly discussed.