Inverse problems for multidimensional vibrating systems

Abstract

Many vibrating systems involve interactions only between neighbouring parts of the system. Frequently, such systems are analysed by supposing that the mass distribution is lumped at the generalized coordinates. Undamped systems of this type involve a block tri-diagonal stiffness matrix K and a block diagonal inertia matrix M . The inverse problem is to reconstruct K and M from frequency response data. The known reconstruction of a Jacobi matrix using the Forsythe algorithm is generalized so that a block Jacobi matrix can be reconstructed from a certain spectral function. This analysis is used to construct the dynamic stiffness matrix A = L -1 KL -T ( M = LL T ) from the frequency response; A is determined to within a block diagonal orthogonal matrix. It is shown that K and M can be separated by using special information about their forms. A vibrating lattice composed of rods and masses is used as an example.