## 1 Introduction  
Vibration is a physical phenomenon that is prevalent in nature and human social activities, and the harm it causes cannot be ignored [1]. In the field of civil engineering, the force state of building structures is very complex, and vibrations caused by unpredictable natural disasters and other phenomena can pose a serious threat to the structure, leading to huge casualties and property damage, such as building collapses caused by earthquakes [2] and bridge fractures caused by resonance due to wind loads [3]. Therefore, the effective control of vibrations is imperative [4, 5]. Structural control techniques are an effective means of reducing the adverse effects of vibrations, and control devices are divided into three main types: active, passive, and semi-active control devices [6]. The main difference is whether the device requires external energy input. The use of active control in engineering is greatly limited due to the additional controls required and the high cost of use. Passive control devices are simple and basic in design, with the advantages of low cost, good stability, and no external energy input. Among these, tuned mass dampers (TMDs) are the most representative and most widely used in practical applications. TMDs originated in 1909 when [PERSON] invented a dynamic vibration absorber (DVA) in 1911 [7]. However, the DVA was originally designed without damping, which resulted in poor control in some frequency bands. [PERSON] improved the DVA by introducing damping, which significantly improved the vibration reduction and established the concept of a TMD. After this, [PERSON] optimized the TMD and provided optimized formulas for the frequency ratio and damping ratio [8]. [PERSON] et al. determined the optimal damping of the TMD based on the integral modal method [9]. TMDs have been successfully applied in many projects and have proven their effectiveness. The Sydney Television Tower was the first building to use a TMD as a control device. In 1970, to resist wind loads, TMDs were installed at the John Hancock Center, resulting in a 40% reduction of the impact of wind loads on the structure. [PERSON] used a water tank as a mass block for TMD [10] and obtained the desired control results. The Taipei 101 building in China uses a single pendulum TMD with a limiting device to control the vibration caused by external loads [11]. Meanwhile, TMDs are also widely used in the field of bridges. In 1988, a TMD system was installed on the main tower of the Akashi Kaikyo Bridge, and test experiments showed that the TMD system could increase the damping ratio of the tower from 1% to 3% when it was built [12]. TMDs absorb energy through resonance but are limited by their linearity, and they can only control a single mode, while structures usually have multiple modes. To overcome this shortcoming of TMDs, [PERSON] proposed the multiple tuned mass damper (MTMD) in 1988 [13], which is composed of TMDs with several different dynamic characteristics. In addition, a bidirectional and homogeneous TMD (BH-TMD) was proposed by [PERSON]' an et al. [14]. [PERSON] pointed out that the bandwidth of the controllable frequency of the damper can be effectively increased by adding nonlinearity, and the concept of a nonlinear energy sink (NES) was born [15]. To introduce nonlinearity into dampers, the most widely used are mass dampers and nonlinear stiffness dampers. [PERSON] et al. described in detail information about plasmonic dampers [16], which provided a valuable reference for subsequent research in this field. [PERSON] et al. proposed a viscoelastic damper that uses a viscoelastic material to efficiently absorb the energy generated during a collision, calling it a pounding tuned mass damper (PTMD) [17]. Collision-tuned dampers [18,19] and liquid impact dampers (TLPD) [20] are also impact dampers. [PERSON] and [PERSON] studied the stressed system response (SMR) near the 1:1:1 resonance [21], investigated the conditions for the existence of the SMR, and verified its numerical results using analytical methods [22].