Derivative of eigensolutions of non-viscously damped linear systems

Derivatives of eigenvalues and eigenvectors of multiple-degree-of-freedom damped linear dynamic systems with respect to arbitrary design parameters are presented. In contrast to the traditional viscous damping model, a more general nonviscous damping model is considered. The nonviscous damping model is such that the damping forces depend on the past history of velocities via convolution integrals over some kernel functions. Because of the general nature of the damping, eigensolutions are generally complex valued, and eigenvectors do not satisfy any orthogonality relationship. It is shown that under such general conditions the derivative of eigensolutions can be expressed in a way similar to that of undamped or viscously damped systems. Numerical examples are provided to illustrate the derived results. Nomenclature C = viscous damping matrix C = space of complex numbers c = damping constant D.s / = dynamic stiffness matrix G.s / = damping function in the Laplace domain G.t / = damping function in the time domai