Optimal Caughey series representation of classical damping matrices

A least squares approach to determine the coefficients in a Caughey series representation of a classical damping matrix is presented. Instead of solving an ill-conditioned Vandermonde system of equations involving the modal damping ratios at a set of pivots, the procedure uses a least squares fit between the polynomial representing the damping ratios and the assumed known frequency dependence of the damping ratios. The proposed approach eliminates the large fluctuations of the damping ratios associated with the standard approach. Three alternative procedures are presented: (i) analytical application of the least squares approach for an expansion into power series, (ii) numerical solution of the least squares equations for a dense set of pivots, and (iii) expansion of the damping matrix into series of Legendre polynomials of matrices. In each alternative, the cases of series including or excluding the mass-proportional term are considered separately.