Dynamic response of some dissipative systems by means of functions of matrices

A certain class of linear, viscously damped dynamical systems, the matrix coefficients of which satisfy a certain commutativity condition, are known to exhibit the same normal modes as the ones that exist in the absence of damping. These systems are said to have classical normal modes. In the present study, such systems are shown to be particular cases of dynamical systems that possess a zero commutator, which form a subset of separable systems. The response of the latter is investigated by means of functions of matrices, and its connection with the classical eigenfunction expansion approach is established. The paper continues the author's previous work where functions of matrices were used to (a) develop an algorithm for the numerical integration of the equations of motion for such systems, and (b) to establish their conservation laws.