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 X c ∈ C n × n ) are also complex. It is shown that, when properly defined, the eigenvectors have a special structure consistent with X c = X R ( I - i Θ ) where X R , Θ ∈ R n × n , X R 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).