Homology Design for Eigenvector of Bending Vibration using Finite Element Method.

Abstract

This paper presents a method of homology design for problems of undamped linear vibration analyzed by means of the finite element method. The homology design aims at achievement of a homologous vibration mode ; namely, a mode shape in which a certain geometric property is conserved in both static state and during vibration. The basic eigenvalue problem is modified by introducing a constraint matrix which describes the homologous mode shape, and the governing equation to determine design variables is obtained on the basis of the first-order sensitivity analysis of the eigenvalue and eigenvector of the modified eigenvalue problem. The stiffness matrix and mass matrix of the modified eigenvalue problem are asymmetric. The governing equation of the design variables is solved with the use of the Moore-Penrose generalized inverse. Bending vibration of a planar lattice structure is dealt with to show the validity and versatility of the proposed method in the case where the diameter or length of beam elements is taken as a design variable.