This paper proposes a practical evaluation method of vibration control performance of a tuned mass damper (TMD) mounted on rooftop of buildings without time history analysis. The response quantity of interest has been taken to be the peak seismic response of the buildings with the TMD. In this study, the seismic effectiveness of the linear TMD is assessed in terms of an equivalent damping factor of the overall vibration system. The objective of this paper is to formulate the equivalent damping factor of the building subjected to strong ground motions.
Firstly, the vibration system consisting of both the multi-story building and the TMD on the seismic response is idealized as an equivalent two degrees of freedom (2-DOF) system. Based on this simplified model, parameters of interest are a mass ratio and a tuning ratio, which govern the seismic effectiveness. The mass ratio is the ratio of the TMD mass to the effective mass of the building. Using these parameters, the equivalent damping factor based on random vibration theory is employed in optimal conditions (perfect-tuning case). The remaining parameters of interest are effective duration and a fundamental period of ground motions. The equivalent damping factor is empirically modified with the help of a solution of the time-dependent mean squared response under non-stationary random disturbances.
Secondary, the response to base excitation having a single sinusoidal wave, which is degenerated from historical records of ground motion pulses, is investigated. It is confirmed that the maximum response obtained by the SDOF system with the proposed damping factor show good accordance with those calculated by the 2-DOF system in a wide variety of natural periods through time history analysis. The closed form equation of the equivalent damping factor is also verified under population of historical earthquakes. Subsequently, the relationship between the maximum displacement of the building and the TMD is examined. Time history analysis demonstrates that the ratio of the TMD deformation to the building displacement is almost constant over a wide range of periods. The ratios, however, tend to significantly decrease in long period regions. A closed form expression of this ratio derived from random vibration theory is empirically modified to take account of this inherent dynamic characteristic through parametric study.
Finally, the evaluation method of the maximum response is developed to cover non-tuning cases in the light of design practice. The design damping factor of the TMD is assumed to be larger than its optimal value. The equivalent damping factor and the deformation ratio are theoretically decomposed into its optimal value and correction factors. Using these results, a performance curve diagram is presented. This diagram consists of the response reduction factor of the building and the amplification factor of the TMD in terms of the mass ratio and the damping factor. It is shown that the proposed method to design parameters of the TMD based on the diagram approach is useful in engineering applications.
