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The IUP Journal of Mechanical Engineering
Distinct Mechanisms of 6 Links, 7 Joints and 1F Kinematic Chains
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The paper develops a new, easy, reliable and efficient method to detect isomorphism and prepares a catalogue of fixed link and its corresponding equivalent links in the distinct mechanisms in kinematic chains of 6 links, 7 joints and 1F kinematic chains. It helps the new researchers/designers to select the best kinematic chain mechanism and to perform the desired task at the conceptual stage of design. The proposed method is presented by comparing the structural invariants ‘Sum of the absolute values of the Characteristic Polynomial Coefficients’ (SCPC) and ‘Maximum absolute value of the Characteristic Polynomial Coefficients’ (MCPC) of Joint-Joint [JJ] matrices. These invariants may be used to detect isomorphism in the kinematic chain mechanism having simple joints. The method is explained with the help of examples of planar kinematic chain having simple joints.

 
 

Over the past several years, much work has been reported in the literature on the structural synthesis of kinematic chains and mechanisms. Undetected isomorphism results in duplicate solutions and unnecessary effort. Therefore, the need for a reliable and efficient algebraic method is necessary. Identifying isomorphism among kinematic chains using characteristic polynomials of adjacency matrices of corresponding kinematic chain is a simple method (Raicu, 1974; Mruthyunjaya and Raghavan, 1979; and Yan and Hall, 1982). But the reliability of these methods is a question, as several counter examples were found by Mruthyunjaya and Balasubramanium (1987). The test proposed by Mruthyunjaya and Balasubramanium (1987) was based on the characteristic coefficients of the ‘Degree matrix’ of the graph of the kinematic chains. The elements of the degree matrix were sum of the degree of vertices (degree or type of links) or unity in a link-link adjacency matrix. Later, this test was also found unreliable. Mruthyunjaya (1984a) proposed the representation polynomial for detecting isomorphism between two kinematic chains.

 
 

Kinematic chain, Fixed link, Equivalent link, SCPC, MCPC