## 1 Introduction
On the other hand, in the inerter-based TMD, the inerter is primarily used in combination with the TMD to enhance its performance [36; 37; 38; 39]. Previous studies on the two typical inerter-based damping systems primarily focused on parameter optimization and performance evaluation of the structures subjected to harmonic loads and white-noise random loads [40; 41]. When the structure experiences ground motions, which are usually simulated with random loads, the reduction in the mean square of the structural response is usually selected as the optimization goal, which is also called H\({}^{2}\) optimization [42]. Generally, both algebraic and numerical methods were used to solve the optimization problem. Although this optimization method yields favorable results, it does not consider variations in the spectral characteristics of ground motion, which may limit the seismic reduction effect from reaching the desired level. In recent years, the combination of tuned inerter-based dampers or inerter-based TMD with base-isolation structure has been proposed to be extensively studied [43; 44; 45; 46]. Previous studies have primarily considered that the primary structure almost behaves almost like a rigid body, and it is typically simplified as a single-degree-of-freedom (SDOF) system, with its equivalent mass representing the total mass of the primary structure [47]. However, for multi-storey structures, the provided solutions may not be applicable, as the equivalent mass of the base-isolated structure is no longer a simple sum of the structural masses, and the impact of the damping in the isolated structure on parameter optimization must also be considered. More importantly, in the previous studies, the external excitation was often regarded as white-noise random loads, which cannot accurately describe the spectral differences between far-fault and near-fault ground motions [48; 49; 50; 51]. In general, the parameters of ground motion cause the difference in the optimal parameters of the TID to increase progressively as the mass ratio rises [52]. Additionally, the optimal parameters of TID are closely related to the main structural damping [53]. Most of these valuable results are derived from the analysis of SDOF structures, while research on multi-degree-of-freedom structures remains relatively scarce. Therefore, investigating how to select the optimal parameters of TID for multi-storey seismic isolation structures under various ground motions, along with assessing its vibration reduction control effectiveness, offers significant practical value for engineering applications [54]. This paper examines the combination of the TID and base isolation, evaluating the seismic reduction effects of the TID on simplified seismic isolation structure models of three different heights. A more general parameter optimization scheme is constructed, which can be directly applied to the base-isolated structure with different damping ratios subjected to far-fault and near-fault ground motions. The optimal frequency ratio and optimal damping ratio of the TID were determined through a numerical optimization method according to the H\({}^{2}\) optimization, when the ground motions are modelled as white-noise random process. Subsequently, influences of ground motion parameters on the optimal parameters of the TID are investigated, using the filtered Kanai-Tajimi spectrum to model far-fault and near-fault ground motions. The selection of optimal parameters for the TID system under various ground motions offers valuable guidance. Finally, the seismic performance of the TID with varying parameters for 5-storey, 10-storey, and 15-storey base-isolated structures was assessed, highlighting the impact of height on the damping effect of the TID.