Caughey Damping Series in Terms of Products of the Flexibility Matrix

Abstract

The paper focuses on the representation of a classical damping matrix in terms of a Caughey series including only negative or zero powers of ([m]−1[k]). An explicit expression for the series in terms of prescribed modal damping ratios at a set of natural frequencies is derived which avoids the need to solve an ill-conditioned problem for the coefficients of the series. In addition, optimal choices for the coefficients of the series are presented for cases in which the natural frequencies are not known or can change as a result of structural changes. Two optimization procedures are presented: (1) analytical application of a least-squares approach for an expansion of the damping ratio into a power series of the eigenvalues; and (2) expansion of the damping matrix into a series of Legendre polynomials of matrices. Finally, the particular case of uniform damping ratios is given special consideration.