An algorithm using co-ordinate transformation for free vibration analysis of fully incompressible materials

This paper presents an algorithm for conducting eigenvalue analysis of fully incompressible isotropic materials (a Poisson’s ratio of exactly 0.5). This algorithm is based on the concept of co-ordinate transformation de ned by eigenvectors with positive eigenvalues of the system sti ness matrix. The transformation to the new co-ordinates of orthogonal bases in the system potential energy can reduce the size of the system equations. The original eigenvalue problem is reduced to a smaller one that can be solved. This algorithm possesses two drawbacks. One is the increase of system degrees of freedom (DOF) due to the mixed method for incompressible problems; the other is due to additional eigenvalue calculation. However, the present algorithm is inherently able to compensate for the above two drawbacks by reducing the DOF. Numerical demonstrations indicate that the exact transformation yields accurate results that converge to analytical results. They also indicate that an approximated transformation retaining only several per cent of all the eigenvectors produces accurate results in the lower frequency range that is generally of interest in engineering analyses. Therefore, the proposed algorithm is judged to be an e cient procedure for eigenvalue problems of fully incompressible materials. This co-ordinate-transformation scheme, derived from the system sti ness matrix, will be applicable to static incompressible problems