A Note on Nonproportional Damping

This note deals with three aspects of nonproportional damping in linear damped vibrating systems in which the stiffness and damping matrices are not restricted to being symmetric and positive definite. First, we give results on approximating a general damping matrix by one that commutes with the stiffness matrix when the stiffness matrix is a general diagonalizable matrix, and the damping and stiffness matrices do not commute. The criterion we use for carrying out this approximation is closeness in Euclidean norm between the actual damping matrix and its approximant. When the eigenvalues of the stiffness matrix are all distinct, the best approximant provides justification for the usual practice in structural analysis of disregarding the off-diagonal terms in the transformed damping matrix. However, when the eigenvalues of the stiffness matrix are not distinct, the best approximant to a general damping matrix turns out to be related to a block diagonal matrix, and the aforementioned approximation cannot be justified on the basis of the criterion used here. In this case, even when the damping and stiffness matrices commute, decoupling of the modes is not guaranteed. We show that for general matrices, even for symmetric ones, the response of the approximate system and the actual system can be widely different, in fact qualitatively so. Examples illustrating our results are provided. Second, we present some results related to the difficulty in handling general, nonproportionally damped systems, in which the damping matrix may be indefinite, by considering a simple example of a two degrees-of-freedom system. Last, we use this example to point out the nonintuitive response behavior of general nonproportionally damped systems when the damping matrix is indefinite. Our results point to the need for great caution in approximating nonproportionally damped systems by damping matrices that commute with the stiffness matrix, especially when considering general damping matrices. Such approximations could lead to qualitatively differing responses between the actual system and its proportionally damped approximation.