Inverse problems for damped vibrating systems

Linear damped vibrating systems are defined by three real definite matrices, M>0,D⩾0, and K>0; the mass, damping, and stiffness matrices, respectively. It is assumed that all eigenvalues of the system are simple and nonreal so that the eigenvectors (columns of a matrix Xc∈Cn×n) are also complex. It is shown that, when properly defined, the eigenvectors have a special structure consistent with Xc=XR(I-iΘ) where XR,Θ∈Rn×n, XR is nonsingular and Θ is orthogonal. By taking advantage of this structure solutions of the inverse problem are obtained: i.e., given complete information on the eigenvalues and eigenvectors, it is shown how M,D, and K can be found. Three points of view are developed and compared (namely, using spectral theory, structure preserving similarities, and factorisation theory).