Tuned mass damper on a damped structure

The classic design of the tuned mass damper on an undamped structure is based on the fixed-point theory introduced by Den Hartog. Because of the lack of fixed points in the dynamic amplification of damped structures, a generalization of previous methods is presented in this study to identify the optimal parameters of the damper. The optimal frequency is identified by analyzing complex natural frequencies. The resulting frequency leads to a confluence point in the complex frequency locus and an equal modal damping ratio for the two vibration modes, provided that the damping ratio of the damper is less than a critical value. This critical value can be used as an upper bound of the damping ratio value in practice that can effectively restrict the relative motion of the damper. The optimal damping ratio is identified by ensuring that the dynamic amplification reaches an extremum value at a reference frequency. It is demonstrated that the resulting damping ratio accounts for the reduction in the motion of the primary structure and relative motion of the damper. Furthermore, a modified design procedure is proposed to determine the optimal parameters of the damper for damped structures with multiple degrees of freedom. Several numerical experiments indicate the efficiency of the proposed parameters and design procedure.