Representation of damping matrix

This paper deals with the representation of the damping matrix in terms of mass and stiffness matrices. A relationship is proposed that can represent not only Rayleigh damping, but also the general nonproportional damping. This proposed relationship is mathematically based on a theorem of matrix representation that, under certain circumstance (which can be easily checked), a matrix C can be represented accurately by a polynomial of two given matrices M and K. In engineering applications, in the sense of least-squares approach or series solutions, the first few terms of the polynomial of the given matrices M and K are shown to be sufficiently accurate to represent the damping matrix. An attempt is made to interpret the physical meaning of some of the parameters of the proposed damping matrix. Numerical examples are given to illustrate the simplicity and accuracy of the proposed method as applied to nonproportional damping problems in structural dynamics. The proposed approach can also be applied in system identification problems.