A unified synthesis method for timing-path generation of planar four-bar linkages

The exact and approximate path synthesis problems require different synthesis methods, and mastering multiple design methods increases the engineers' burden. A unified drawing method can solve the synthesis problem of four and five positions in the existed studies. In this article, a novel unified synthesis method is proposed for the exact and approximate timing-path synthesis. The approximate timing-path synthesis problem is recombining into many groups of the exact timing-path synthesis. The exact timing-path synthesis formulas are derived based on the classical dyad method, while the simplified synthesis formulas are obtained by the linear elimination method with a specific order of elimination. A distance error formula is proposed to find the optimal group from all groups of solutions, where each group of solutions is the possible solution of the pre-recombination synthesis problem. Exact timing-path synthesis usually obtains several groups (12 or 2 groups, theoretically may be 0) of solutions with different kinematic properties, then the error formula is used to find the group of solutions that has the smallest rounding error. Finally, three cases test the proposed method and prove the effectiveness of the method in solving the timing-path synthesis problem by comparing the errors with the literature results.