Eigenrelations for Nonviscously Damped Systems

Multiple-degree-of-freedom linear dynamic systems with nonviscous damping is considered. The nonviscous damping is such that the damping forces depend on the past history of the velocities via convolution integrals over some kernel functions. For the sake of generality, the system matrices are allowed to be asymmetric. The mode-orthogonality relationships, known for undamped or viscously damped systems, have been generalized to such nonviscously damped systems. Some expressions are suggested for the normalization of the eigenvectors. A number of useful results relating the system matrices and the eigensolutions have been established. Numerical examples are provided to illustrate the derived results. Nomenclature C = viscous damping matrix C = space of complex numbers D.s / = dynamic stiffness matrix G.s / = damping function in the Laplace domain G.t / = damping function in the time domain IN = identity matrix of size N i = unit imaginary number, i Dp¡1