Trajectory and envelope synthesis of planar mechanisms via curvature theory

By incorporating instantaneous invariants, this paper extends curvature theory into a unified approach for synthesizing planar mechanisms involving point trajectories, straight-line envelopes, and circle envelopes. A canonical system is defined from the kinematics of the instantaneous center, where the j -th order motion requires only 2 j − 3 instantaneous invariants for j ∈ N . These invariants yield key curvature features, including the inflection circle, Bresse circle, return circle, cubic of stationary curvature (CSC), Ball’s point, and Burmester point, which enable the synthesis of four-bar mechanisms for path generation. The straight-line envelopes are established through the envelope method. Two curvature characteristic values, λ 1 and λ 2 , define the generalized Burmester line, supporting the synthesis of four-bar mechanisms and the profile design of disk cams with flat-faced followers as well as involute gears. For circle envelopes, a simplified formulation based on vector analysis provides up to five solutions for the generalized Burmester circles. By adjusting the generating circle radius, the method constructs a straight line or a circle on the four-bar coupler to envelop target curves. The same formulation applies to the synthesis of disk cams with roller followers and pin gear mechanisms.