Explicit Representation of Classical Damping Matrices by Caughey Series with Rational Fractional Powers

An explicit expression for a rational fractional powers Caughey damping series consistent with a set of prescribed modal damping ratios is presented. The result is obtained by use of Adhikari’s approach combined with a Lagrange interpolation of the modal damping constants as a function of the natural frequencies and independently by a modal expansion of the stiffness matrix. The explicit expression for the resulting classical damping matrix circumvents the need for solving an ill-conditioned system of equations for the coefficients in the Caughey series. The general result is specialized to the case of “linear hysteretic damping” in which all the modal damping ratios are equal. Closed-form expressions for the elements of a classical damping matrix for a simple multistory structure are also presented.