IDENTIFICATION OF ISOMORPHISM AND STRUCTURAL PROPERTIES OF KINEMATIC CHAINS WITH THE HELP OF ADJACENCY MATRICES

Abstract

Structural synthesis and analysis of mechanism are very important for the invention and innovation of mechanisms. Structural synthesis of kinematic chains usually involves the creation of a complete list of kinematic chains, followed by an Isomorphism test to discard duplicate chains. A significant unsolved problem in structural synthesis is the guaranteed precise elimination of all Isomorphs. Undetected Isomorphism result in duplicate solutions and an unnecessary effort and falsely identified Isomorphism eliminates possible candidates for new mechanisms, Thus it is absolutely essential to identify isomorphism from the point of view of time saving and correct synthesis of mechanism kinematic chain. Many methods are available to the kinematician to detect isomorphism among chains and inversions but each has its own shortcomings. The work presented in this dissertation deals with a new approach of Adjacency Matrices of planar kinematic chains which has been applied firstly for identification of Isomorphism among kinematic chains and among Inversions of given kinematic chain and secondly for determination of structural properties of kinematic chains. Prior to identification of Isomorphism with the help of proposed method, work include comparison of this proposed method with two important methods of Isomorphism identification, first Hamming Number Technique and second Loop Based Detection method with some common examples. iii The presented work has been divided into two parts, in the first part the proposed method of Adjacency Matrices of kinematic chains is applied for identification of Isomorphism among kinematic chains and among Inversions of a given kinematic chain. Firstly kinematic chains are represented by Adjacency Matrices of digits 1 and 0 and of dimensions n x n, where n is number of links of kinematic chain. Then use of 'MATLAB SOFTWARE' gives Eigenvalues and Eigenvectors of this n x n matrix. By comparing the Eigenvalues and corresponding Eigenvectors of these Adjacency Matrices, the Isomorphism among kinematic chains can easily be identified. Identification of structural similarity among Inversions of a given kinematic chain can be done by comparing its Eigenvectors row wise and similarity of Eigenvectors in these rows shows structural similarity of given kinematic chain with respect to links corresponding to these rows. This dissertation work includes problems related with one, two and three degree-of-freedom kinematic chains for Identification of isomorphism. In the second part of presented work, the proposed method is applied for identification of various structural properties as Degeneration identification, Identification of Type of freedom i.e. Total, Partial and Fractionated degree-of-freedom, in planar kinematic chains of one, two and three degree-of-freedom. iv