Damping matrix of a lightly damped dynamic system

Two methods for constructing damping matrix of a lightly damped linear system are proposed. In the first method,  a matrix polynomial is employed to generalize Rayleigh damping so that damping of as many modes as desired can be matched. The classical Rayleigh damping is a special case two-term expansion of the generalized Rayleigh damping. In the second method,  a closed-form formula of the damping matrix,  using modal frequencies,  modal damping ratios,  and modal matrix,  is derived based on the equation of motion,  which avoids the presupposition of a form for the damping matrix. It is proved that for a system with only flexible modes, a unique closed-form damping matrix exists. Two numerical examples are presented to demonstrate the simplicity and efficiency of the proposed methods. Applications of damping matrices in systems with all flexible modes and with both flexible and rigid modes are discussed.