Comparison Of Tests For Isomorphism In Planetary Gear Trains

Abstract

There are plenty of ways available for synthesis and analysis of Planetary Gear Trains (PGTs) of one DOF. However, every method has its own shortcomings. In this paper a comparison is made between Characteristic polynomial, Eigenvalues and Eigenvectors, Hamming Number Method and Modified Path Matrix (MPM) method. There are many methods available to test isomorphism in PGTs, out of which these four methods wasanalyzed in this paper. For a given PGT with a number of links and a single Degree of Freedom (DOF), adjacency matrix is enough to find out the Eigenvalues and Eigenvectors. Isomorphism of PGT is determined using the Eigenvalues. If Eigenvalues are similar then the PGTs are isomorphic. Similarly, if the characteristic polynomials of two PGTs are samethen it represents the isomorphic PGTs. Characteristic polynomials are determined from the Adjacency matrix. Hamming method also uses adjacency matrix to generate Hamming matrix. Hamming strings are developed from Hamming Matrix. Uniform hamming strings of two PGTsindicates isomorphism in PGTs. Along with isomorphism, Symmetry also known from the Hamming method. Modified path matrix method uses a connectivity matrix to generate MPM. If train values of two PGTs are same then they will be isomorphic otherwise non isomorphic. As per literature as the number of links increases the results may not be accurate with Characteristic coefficients and Eigenvalues methods, though all these methods are used to detect isomorphism among a group of PGTs. Whereas with the Hamming number approach, one can detect isomorphism, symmetry and number of possible level combinations of an PGT with a single hamming matrix.