Calculation of Eigensolution Derivatives for Nonviscously Damped Systems Using Nelson’s Method

A method to calculate the derivatives of the eigenvalues and eigenvectors of multiple-degree-of-freedom damped linear dynamic systems with respect to arbitrary design parameters is 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 given kernel functions. Because of the general nature of the damping, eigensolutions are generally complex valued, and eigenvectors do not satisfy the classical orthogonality relationship. The proposed method to calculate the eigenvector derivative depends only on the eigenvector concerned. Numerical examples are provided to illustrate the derived results. Nomenclature c j = constant associated with the derivative of u j D(s) = dynamic stiffness matrix d j = constant associated with the derivative of v j G(s) = Laplace transform of G(t) G(t) = nonviscous damping matrix K = stiffness matrix L [ ] = Laplace transform of [] M = mass matrix m = total number of eigenvectors N = number of degrees of freedom of the system n = number of relaxation parameters p = design parameter rank ( ) = rank of a matrix s = Laplace domain parameter u j = j th eigenvector (t) = displacement vector v j = j th adjoint (left) eigenvector x j = vector associated with the derivative of u j yj = vector associated with the derivative of v j θ j = normalization constant for the j th eigenvector λ j = j th eigenvalue μk = relaxation parameters Subscripts ()T = matrix transpose () ′ = derivative with respect to s I