## 1 Introduction  
Since the concept of a structural control was proposed [1; 2; 3], it has been widely used to improve the performance of buildings under various excitations, especially for those in earthquake-prone areas. In recent years, the introduction of inerters, a novel mechanical element [4], further developed the control strategies. A large number of inerter-based vibration control systems have emerged [5; 6; 7; 8; 9]. An inerter is a two-terminal element, generating a resisting force proportional to the acceleration difference between its two ends, which is a significant difference with the traditional one-node mass whose inertial force is dependent on absolute acceleration. The liquid mass pump developed by [PERSON] [10] in the 1970s is regarded as the bud of the two-terminal inerter element and was initially utilized as an auxiliary mass to modify the natural frequency of the structure for its high inertial resistance. In 1999, a ball screw viscous damper proposed by [PERSON] [11; 12], called the rotary damping system, utilized the ball screw mechanism to enhance the damping effect, while the benefits of its mass amplification effect were not used intentionally. Subsequently, researchers found that combing the inerter in series or parallel with springs and damping elements could achieve a better energy dissipation capability. [PERSON] and his coworkers [5; 13; 14] proposed a tuned viscous mass damper (TVMD), which is the first to make full use of its damping enhancement effect and mass amplification effect. These devices, similar to the TVMD, combing the inerters, damping elements, and springs together for vibration control, are called the inerter system [15; 16] or an inerter-based damper [17]. Various studies on the topological combinations of the inerter system [18] and their composition parameter optimizations [9; 16; 19] have been published. The inerter element with the mass magnification effect can connect the mass of a traditional tuned mass damper to the structure, making the tuned mass damper lightweight and so its performance improves [20; 21; 22]. Meanwhile, due to its excellent damping capacity, the inerter system can serve as a damper element substituting the traditional damper in vibration control systems, such as a traditional tuned mass damper [23], and base isolation systems [24; 25; 26]. The performance improvements introduced by the inerter element and inerter system have been verified. Indeed, without being combined with other vibration control systems, taking advantage of the damping enhancement and mass magnification effects, the inerter system still prevails over traditional control strategies in terms of achieving a lightweight and efficient structural control performance. In 2012, [PERSON] et al. [5] provided a tuning design formula for a single-degree-of-freedom (SDOF) structure equipped with the TVMD based on the fixed-point method and verified the TVMD seismic control effect through shaking table tests. In [27; 28], [PERSON] et al., using sequential quadratic programming, studied the distribution and parameter optimizations of the TVMDs installed on each story of a multiple-degree-of-freedom (MDOF) structure, and further simplified the design by assuming the distribution of the masses of the TVMDs were proportional to the stiffness of each story. In 2014, through a modal analysis and a numerical example, [PERSON] et al. [6] studied the tuned inerter damper (TID), one of the typical inerter systems, and illustrated that the best installation location is the bottom layer. [PERSON] et al. [29] studied the distributions and parameter optimizations of the TVMD and the TID considering multi-modal seismic control and verified their effectiveness. [PERSON] and [PERSON] [15; 16] studied the closed-form response of SDOF structures equipped with three typical inerter systems, respectively, and provided a demand-based parametric design method. In [19], they found the damping enhancement equation and clarified the damping enhancement effects' fundamental principle.