Abstract
This paper establishes a systematic procedure to determine the instantaneous invariants for a member in a planar mechanism and the curvature ratios of the path of a point fixed in the member. Closed-form expressions are derived for the instantaneous invariants and the curvature ratios as continuous functions of the input variable of the mechanism. The methods that are proposed in this paper can be applied to the design of planar mechanisms in general. For a given mechanism, the configuration defined by the input variable at which the member achieves the optimal approximation of a prescribed rigid motion can be determined. Alternatively, a point fixed in a member can be selected such that it will generate a trajectory matching a given curve with high precision. For purposes of illustration, the paper details the analytical procedure for a simple epicyclic gear train and a four-bar mechanism. The instantaneous invariants and the curvature ratios are generated for the complete operating cycle of each mechanism.
