L. Starek et Dj. Inman, A SYMMETRICAL INVERSE VIBRATION PROBLEM FOR NONPROPORTIONAL UNDERDAMPED SYSTEMS, Journal of applied mechanics, 64(3), 1997, pp. 601-605
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.