A Symmetric Inverse Vibration Problem

Abstract

This paper considers the inverse eigenvalue problem for linear vibrating systems described by a vector differential equation with constant coefficient matrices. The inverse problem of interest here is that of determining real symmetric coefficient matrices, assumed to represent the mass, damping, and stiffness matrices, given the natural frequencies and damping ratios of the structure (i.e., the system eigenvalues). The approach presented here allows for repeated eigenvalues, whether simple or not, and for rigid body modes. The method is algorithmic and results in a computer code for determining mass normalized damping, and stiffness matrices for the case that each mode of the system is underdamped.

Symmetric Inverse Eigenvalue Vibration Problem and Its Application

“…Danek [4] has solved this problem for the case of real non-singular coe$cient matrices and he has de"ned the inverse formulas which determine the coe$cient matrices M, D and K of the abovementioned systems with given spectral and modal data. Starek and Inman [1] have solved the inverse problem in the state-space form and they have determined the inverse formulas which directly determine real coe$cient matrices M\K and M\D for the case that D and K are singular coe$cient matrices (i.e. there exist rigid-body modes).…”

“…Previously, inverse spectral problems in vibration of lumped non-conservative systems have been solved by Danek [4], Gladwell [5], Lancaster and Maroulas [6], and Starek and Inman [1,2,7,8]. Gladwell [5] has solved the inverse spectral problem for vibration of lumped conservative systems (D I "0) modelled by tridiagonal matrices.…”

“…In the asymmetric case, equation (2) may be written as qK (t)#H qR (t)#H q(t)"f (t) (2) where H "M\K, H "M\D and f (t)"M\ f (t). The matrices H and H are asymmetric and possibly singular to allow for rigid-body motion and the possibilities of semi-de"nite damping.…”

“…Here we consider linear lumped parameter system which can be modelled by a vector di!erential equation in the second-order form given by MqK (t)#DqR (t)#Kq(t)"f (t) (1) where q(t) is an n vector of time-varying elements representing the displacement of the masses in a lumped mass model of some structure or device. The vectors qR (t) and qK (t) represent the velocities and accelerations, respectively.…”

“…The inverse problem of interest here is that of determining real symmetric coe$cient matrices assumed to represent mass normalised velocity and position coe$cient matrices, given a set of speci"ed complex eigenvalues and eigenvectors. There are given two solutions of a symmetric inverse eigenvalue problem presented by Starek and Inman [1,2].The theory of inverse eigenvalue problem is applied to the model updating problem. The goal of this paper is to recognise that the model updating problem is a subset of the inverse eigenvalue problem.…”