Dynamics of Nonviscously Damped Linear Systems

This paper is aimed at extending classical modal analysis to treat lumped-parameter nonviscously damped linear dynamic systems. It is supposed that the damping forces depend on the past history of velocities via convolution integrals over some kernel functions. The traditional restriction of symmetry has not been imposed on the system matrices. The nature of the eigenvalues and eigenvectors is discussed under certain simplified but physically realistic assumptions concerning the system matrices and kernel functions. A numerical method for calculation of the right and left eigenvectors is suggested. The transfer function matrix of the system is derived in terms of the right and left eigenvectors of the second-order system. Exact closed-form expressions for the dynamic response due to general forces and initial conditions are presented. The proposed method uses neither the state-space approach nor additional dissipation coordinates. Suitable examples are given to illustrate the derived results.