A SYMMETRICAL INVERSE VIBRATION PROBLEM FOR NONPROPORTIONAL UNDERDAMPED SYSTEMS

This paper considers a symmetric inverse vibration problem for linear vibrating systems described by a vector differential equation with con stant coefficient matrices and nonproportional damping. The inverse pr oblem of interest here is that of determining real symmetric, coeffici ent matrices assumed to represent the mass normalized velocity and pos ition coefficient matrices, given a set of specified complex eigen-val ues and eigenvectors. The approach presented here gives an alter-nativ e solution to a symmetric inverse vibration problem presented by Stare k and Inman (1992) and extends these results to include noncommuting ( or commuting) coefficient matrices which preserve eigenvalues, eigenve ctors, and definiteness. Furthermore, if the eigen-values are all comp lex conjugate pairs (underdamped case) with negative real parts, the i nverse procedure described here results in symmetric positive definite coefficient matrices. The new results give conditions which allow the construction of mass normalized damping and stiffness matrices based on given eigenvalues and eigenvectors for the case that each mode of t he system is underdamped. The result provides an algorithm for determi ning a nonproportional (or proportional) damped system which will have symmetric coefficient matrices and the specified spectral and modal d ata.