A General Method of Determining and Eliminating Branching in Planar Multiloop Mechanisms

In most mechanism systems, there exists more than one mode of assembly or configuration, of the mechanism for a given set of input conditions. During the synthesis process, branching occurs when the mechanism changes from one configuration to another. Branching in mechanisms renders them unsuitable because they have to be disassembled and reassembled in order for them to correctly perform the design tasks. In this paper, a comprehensive method for the determination and the elimination of branching problems during synthesis in planar multiloop mechanisms with lower kinematic pairs and with multiple modes of assembly is presented. A general method is developed for the identification of all the branching configurations in a given mechanism. Sub-Jacobian matrices are formed for the sets of constraints associated with each potential branching configuration. The signs of the determinant values of the sub-Jacobian matrices are evaluated. By maintaining the signs of the determinant values to be the same, at all times, during the synthesis process, solutions without the effect of branching are achieved for general planar multiloop mechanisms. Implementation of the method in a general mechanism synthesis package is discussed and applications of the method are illustrated by means of examples. This method of determining branching in mechanisms also is ideally suited for implementation in an optimization process to select the appropriate mechanism configuration that best satisfies the design requirements and for maintaining that configuration throughout the entire cycle of operation.