International Journal of Engineering Inventions e-ISSN: 2278-7461, p-ISSN: 2319-6491 Volume 2, Issue 12 (August 2013) PP: 43-51 www.ijeijournal.com Page | 43 Study of Structural Analysis of Mechanisms - Structural Isomorphism Dr. Ali Hasan Mechanical Engineering Department, F/O-Engineering & Technology, Jamia Millia Islamia, New Delhi ABSTRACT: In this paper, the adjacency and incidence matrices to determine the topological structure of a mechanism up to structural isomorphism are studied and found that these methods satisfy the uniqueness and decidability conditions but are computationally inefficient. The other methods of identification of structural isomorphism’s including the characteristic polynomials, the MAX code, and the degree code, are also presented. The paper is extremely useful for P.G students, research scholar and designer of mechanisms at the conceptual stage of design. KEYWORDS: mechanisms, isomorphism’s, matrix, kinematic graphs, degrees of freedom I. INTRODUCTION Structural analysis means the study of the connection among the members of a mechanism kinematic chains and its mobility. Mainly, it is related with the fundamental relationships among the dof, number of links, number and the type of joints used in a mechanism. The structural analysis does not deal with the physical dimensions of the links. The structural analysis only deals with the general functional characteristics of a mechanism. Mostly, graph theory is used as a helping tool in the study of the kinematic structure of mechanisms. The on hand study focused only on mechanisms whose corresponding graphs are planar and contain no articulation points or bridges. A graph having a bridge means that the mechanism is a combination of two mechanisms connected in series with a common link but no common joint, or with a common joint but no common link. Such mechanisms are considered as two separate mechanisms. The topological structure of a mechanism kinematic chain is represented by a graph. An important step in structural synthesis of kinematic chains and mechanisms is the identification of isomorphic structures. Undetected isomorphic structures lead to duplicate solutions, while falsely identified isomorphisms reduce the number of feasible solutions for new designs. Several methods of identification have been proposed. Some are based on visual approaches while others are based on heuristic approaches. Each method has its own advantages and disadvantages. An ideal algorithm, however, should satisfy the following conditions [1-23]. (a) Uniqueness. There exists a one-to-one correspondence between the kinematic chain and its mathematical representation so that structural isomorphism can be uniquely identified. (b) Efficiency. The algorithm for identification of isomorphic mechanisms should be simple and computationally efficient. This is essential for automated identification of structural isomorphisms. (c) Decidability. The mathematical representation can be transformed into a unique kinematic chain. This makes it possible for a large number of graphs or mechanisms to be stored in a computer for use by designers. Both the adjacency and incidence matrices determine the topological structure of a mechanism up to structural isomorphism. They satisfy the uniqueness and decodability conditions. However, they are computationally inefficient. For this reason, in this paper, other methods of identification have been proposed. II. NOTATIONS USED The following notations have been used. ci : degrees of constraint on relative motion imposed by joint i. F or dof: degrees of freedom of a mechanism. fi : degrees of relative motion permitted by joint i. j : number of joints in a mechanism, assuming that all joints are binary. ji : number of joints with i dof; namely, j1 denotes the number of 1-dof joints, j2 denotes the number of 2-dof joints, and so on. L: number of independent loops in a mechanism. n: number of links in a mechanism, including the fixed link. λ: degrees of freedom of the space in which a mechanism is intended to function. S: spherical kinematic pair (dof =3). E: plane kinematic pair (dof =3). G: gear pair (dof =2)