Concise Analytic Solutions for Random Seismic Response of High-Rise Structure With Series-Parallel Inerter System and Tuned Mass Damper

Tuned mass damper (TMD) is a single-terminal damper, while inerter is a two-terminal one, which are effective control devices. So, a hybrid damper with a series-parallel inerter system and a TMD (SPIS-TMD) in series is proposed. The main work of the manuscript is as follows. Firstly, based on the mechanical diagram of SPIS-TMD and its equipment on the roof of a high-rise structure, the general form of seismic motion equation was derived using dynamic finite element technology. Secondly, based on the method of quadratic decomposition for power spectrum density function (QD-PSDF), the concise analytic solutions for zero-, first-, and second-order response spectral moments (ZFSO-RSMs) of the general response of structure with SPIS-TMD were deduced. Thirdly, in response to the numerous parameters that affect the safety of high-rise structures and the varying difficulty for obtaining SPIS-TMD’s parameters, an optimization analysis technique was proposed, which is constrained by dynamic reliability and SPIS-TMD’s parameter weights. Finally, three examples were given and results show the following. (1) The proposed analytical solutions for ZFSO-RSMs are correct and high efficiency and can be extended to analysis of stationary seismic responses of general linear structures. (2) Calculating results of ZFSO-RSMs of general responses with the number of vibration modes corresponding to a participation weight of 100% is exactly the same as those with all vibration modes, and the calculating time of the case of the participation weight of 100% is less than 1/12 that of the case of all vibration modes. (3) The failure probabilities of the structure with SPIS-TMD with optimal parameters using the proposed method (with a limit failure probability of 0.1587), the structure only equipped with TMD with the same TMD’s parameters as SPIS-TMD, and the structure without dampers are 0.1570, 0.7060, and 0.9778, respectively. It indicates that under the same conditions, the hybrid damper SPIS-TMD has a better damping effect than a single TMD.