Studies on Coupler Curves of a 4-Bar Mechanism with One Rolling Pair Adjacent to the Ground

Abstract

This paper presents an expository study of the shapes of the coupler curve of a four-bar mechanism with one of its fixed pivots replaced with a rolling pair. Such rolling pairs provide the advantage of being friction and clearance free. In this paper, circle-on-circle and circle-on-line type of rolling pairs have been explored. This arrangement introduces three additional design variables to study the continuous change in the shape of the coupler curve of a conventional 4-bar mechanism. For a given input rotation at the crank, the mechanism does not have a closed-form solution for its configuration. However, providing input rotation at the rolling link allows easy derivation of a closed-form solution for both branches of the configuration. The tracing of the coupler curves is done for arbitrary radius ratios for the rolling link and choice of coupler point on the coupler link. A computer program has been written to study and visualize the coupler curves and its properties; the program not only finds the closure configurations but also identifies the situations of non-closure. Illustrative examples show variety and complexity of coupler curves which are not achieved in linkages. Although these coupler curves of mechanisms with rolling pair are transcendental in nature, the velocity states of the systems are easily derivable. Since the point of contact of the rolling pair is the instantaneous centre for the rolling link with respect to the ground, the velocity of the coupler point is determined using Kennedy's theorem. This helps in characterizing the occurrence of cusps in the coupler curves. The paper presents the geometric conditions for components, crunodes and cusps in the coupler curve with illustrative examples.