## 1 Introduction  
Structures may undergo excessive vibrations when subjected to various dynamic loads, including earthquakes, winds, traffic, and other service-related loads. To ensure the resident comfort and structural safety of the structures, structural control systems are employed to minimize unwanted structural vibrations [1]. In general, these systems can be broadly divided into four main groups, namely passive, active, semi-active, and hybrid [2]. The passive control system is commonly used in engineering practice due to its advantages of low cost and high reliability [3]. As a typical passive control system, base isolation has garnered significant attention from scholars in the areas of isolation structure design, the development of isolation bearings, and the creation of efficient numerical models [4; 5; 6; 7]. Base isolation successfully reduces the structural inter-storey drifts, but the structure itself undergoes unfavorable relatively large displacements [8]. One approach to reducing the displacement of base-isolated structures is to enhance the damping characteristics of the isolation layer, but it will increase the acceleration response of the primary structure, which brings a greater risk of damage to non-structural components [9; 10]. As another typical passive control device, tuned mass dampers (TMD) are extensively adopted to suppress the vibration of structures [11]. Therefore, to decrease the displacement of the based-isolated structures, a hybrid system consisting of both base isolation and TMD have been proposed [12]. It is commonly accepted that the control effectiveness of the TMD is strongly dependent on the mass ratio between the TMD and the primary structure. Research has shown that a larger mass ratio is more advantageous in reducing seismic response compared to a smaller mass ratio [13; 14; 15; 16; 17]. However, the large mass ratio TMD systems are only suitable in limited circumstances, such as inter-storey isolation structures, mega-substructure systems, industrial structures with large mass ratio equipment, etc. [18]. Therefore, a smaller auxiliary mass is typically used to achieve control effects similar to those of the above system. A type of mechanical component with two ends, introduced by [PERSON] in 2002 and referred to as an inerter component, has garnered significant attention from researchers in recent years [19]. A key characteristic of the inerter is its ability to generate a force that is directly proportional to the relative acceleration across its terminals. The proportionality constant between the force and the acceleration has the same units as the physical mass and is called the inertance, apparent mass, or inerter coefficient, etc. The rack-and-pinion and ball-screw mechanisms are typical physical implementation schemes of inerters and are easy to manufacture [20; 21; 22; 23; 24]. The apparent mass of the inerter far exceeds its physical mass, which means they have excellent mass amplification effect. For example, [PERSON], [PERSON], and [PERSON] have developed the inerters with mass amplification factors of 32.5, 175, and 3300, respectively [25; 26; 27]. Due to the special characteristic of the inerter, inerter-based damping systems have become a research hotspot recently in the field of controlling the dynamic response of civil structures [28; 29; 30]. Tuned inerter-based dampers and inerter-based TMD are two typical inerter-based damping systems [31]. The tuned inerter-based dampers are mainly composed of three elements: an inerter, a spring and a viscous damper, and these three elements are connected in series, parallel or hybrid [32; 33; 34]. Tuned viscous mass damper (TVMD), tuned inerter damper (TID), and tuned series viscous mass damper (TSVMD) are three basic types of the tuned inerter-based dampers. Among them, the inerter in the TID is installed in series with the spring and the viscous damper, similar to TMD [35].