Path curvature of a geared seven-bar mechanism

Abstract

This paper presents a graphical technique to locate the center of curvature of the path traced by a coupler point of a planar, single-degree-of-freedom, geared seven-bar mechanism. Since this is an indeterminate mechanism then the pole for the instantaneous motion of the coupler link; i.e., the point coincident with the instantaneous center of zero velocity for this link, cannot be obtained from the Aronhold–Kennedy theorem. The graphical technique that is presented in the first part of the paper to locate the pole is believed to be an important contribution to the kinematics literature. The paper then focuses on the graphical technique to locate the center of curvature of the path traced by an arbitrary coupler point. The technique begins with replacing the seven-bar mechanism by a constrained five-bar linkage whose links are kinematically equivalent to the second-order properties of motion. Then three kinematic inversions are investigated and a four-bar linkage is obtained from each inversion. The motion of the coupler link of the final four-bar linkage is equivalent up to and including the second-order properties of motion of the coupler of the geared seven-bar. Then the center of curvature of the path traced by an arbitrary coupler point can be obtained from existing techniques, such as the Euler–Savary equation. An analytical method, referred to as the method of kinematic coefficients, is presented as an independent check of the graphical technique.