The Geometrical Mode Analysis of a Vibrating System With the Planes of Symmetry

Abstract

Vibration modes obtained from a modal analysis can be better explained from a screw theoretical standpoint. A vibration mode can be geometrically interpreted as a pure rotation about the vibration center 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 vibration have been derived. A numerical example of an application to the vibration analysis of an optical disc drive has been presented.

Vibration analysis of single rigid-body systems having planes of symmetry

“…2. The directions of eigenwrenches and eigentwists are parallel to the principal axes of inertia [4,8].…”

“…N ow, the centre of elasticity, which is expressed by the vector h from G, can be computed using the relation h 6 ¡A ¡1 B [4,8]…”

“…R elocation of the axes of vibration can be done by changing the values of the design parameters. In order to investigate the in uence of any design parameter on the locations of the axes of vibration, the loci of the axes of vibration for both plane-normal and coplanar modes can be plotted using the analytical expressions for the axes [8].…”

“…The plane can be identi ed by inspection of the stiffness matrix at the origin of the coordinate frame expressed in terms of the principal stiffnesses at the centre of elasticity and the vector to the centre of elasticity from the origin, which will be discussed in detail later. D an and Choi [8] showed that the planes of symmetry depend on the location of the centre of elasticity and derived an analytical expression for the axes of vibration for a system having planes of symmetry.…”