An Algebraic-Geometrical Method for the Identification and Elimination of Circuit Defect

Abstract

Abstract In this paper, an algebraic-geometrical method is proposed which unifies the resolution of the circuit defect, and full or partial input link rotatability in mechanism synthesis. The circuit defect is a gross motion defect. Starting from the follower link, the method develops algebraic curves of 20th order (H20 = 0), in a body reference system, which define the boundaries of the regions where the choice of the floating pin of the driving (input) link either would yield a mechanism free of branch and circuit defects, or most likely would yield potential-circuit-defective-mechanisms (PCDM). The point (B) representing the floating pin of the follower (BA) is a quadruple point of this curve. Therefore, polar rays passing through this point (B) yield from H20, effectively, a nontrivial polynomial in one variable of 16th order, thus simplifying the root finding problem. This analytical method also enables the unification of the PCDM regions with the branch defect regions (due to Filemon) and those for the design of mechanisms with transmission angle within prescribed limits (due to Gupta). The main conclusion is that to avoid branch and circuit defects, avoid the Filemon-defective regions as well as the portions of the PCDM-regions which lie in the Filemon-feasible regions. Interestingly, large portions of the PCDM-region are within the Filemon-defective region and are therefore easily excluded. It has been our experience that the circuit defect, which is a gross motion defect, does not occur frequently; in many examples, the entire PCDM-region was contained in the Filemon-defective region. However, the Filemon-feasible region may contain portions of the PCDM-region and this sub-region must be examined for potentially circuit defective mechanisms. The method is developed and illustrated with examples for the planar four-bar mechanisms, some of the numerical aspects of the solution are also discussed here.