Systematic use of velocity and acceleration coefficients in the kinematic analysis of single-DOF planar mechanisms

In single-degree-of-freedom (DOF)mechanisms, velocity coefficients (VCs)and their 1st derivative with respect to the generalized coordinate (acceleration coefficients (ACs))only depend on the mechanism configuration. In addition, if the mechanism is a planar linkage (i.e., a mechanism containing only lower pairs), complex numbers are an easy-to-use tool for writing linkage's loop equations that are formally differentiable and are the only constraint equations of linkages. Actually, the use of complex numbers for systematically writing the constraint equations of planar mechanisms is often limited to linkages. The possible presence of higher pairs in these mechanisms is usually managed through either equivalent linkages or apparent velocity/acceleration equations. Both these methods are simple to implement for a single mechanism configuration, but become cumbersome when continuous motion has to be analyzed. Other approaches use ad hoc auxiliary equations. Here, first, a general notation that brings to select particular auxiliary equations is proposed; then, such notation is adopted to propose an algorithm that systematically uses VCs and ACs for solving the kinematic-analysis problems of single-DOF planar mechanisms. The proposed notation and algorithm, over being efficient enough for constituting the kinematic block of any dynamic model of these mechanisms, are easy to present planar kinematics, with the complex-number method extended to higher pairs, in graduate and/or undergraduate courses.