The geometrical mode and frequency analyses of a vibrating system with planes of symmetry

The geometrical mode and frequency analyses of a vibrating system can be performed via the theory of screw. From a screw theoretical standpoint, a vibration mode can be geometrically interpreted as a pure rotation about the center of vibration in a plane and as a twisting motion on a screw in a three-dimensional space. This paper presents a method to diagonalize a spatial stiffness matrix by use of a parallel axis congruence transformation when the stiffness matrix satisfies some conditions. It also describes that the diagonalized stiffness matrix can have the planes of symmetry depending on the location of the center of elasticity. For a system with the planes of symmetry, the vibration modes can be expressed by the axes of vibration. Analytical solutions for the axes of vibrations have been derived. The set of axes of vibrations yields the modal matrix and the response at the mass center is expressed by the reciprocal product between the axes of vibration and the applied wrench. A numerical example of an application to the vibrational analysis of an optical disc drive has been presented.