## 1 Introduction  
Building structures are often subjected to complex and unpredictable forces due to natural disasters such as earthquakes [1]. In order to mitigate the adverse effects of these vibrations on structures, various techniques for reducing structural vibrations have been developed [2]. Among the vibration control devices, smart and passive control devices are the most widely used [3; 4; 5]: smart dampers are further divided into two types of active and semi-active control systems. In passive control devices, the tuned mass damper (TMD), as a common damping device, is widely used in building, machinery and vehicle systems [6]. It is usually installed on the top layer of the main structure and forms a resonance mechanism when the TMD is tuned to the frequency of the main structure. When the main structure is subjected to external forces or excitations, the mass of the TMD moves in synchronization with the vibration of the structure and undergoes tuned motion due to the action of the spring. This results in the dissipation of energy through the TMD, effectively reducing the amplitude of the main structure's vibration. By improving the vibration characteristics of the building structure or other mechanical systems, the TMD enhances the stability and safety of the structure or system [7; 8]. The TMD originated in 1911 when [PERSON] [9] invented the undamped dynamic vibration absorber (DVA). However, due to its lack of damping, the control effect of the DVA in frequency bands other than tuning was even inferior to that of the uncontrolled structure. In 1956, [PERSON] et al. [10] improved the DVA by introducing damping, thus enhancing its vibration-damping effect and gradually forming the concept of a TMD. [PERSON] further optimized the frequency ratio and damping ratio of the TMD and proposed a specific optimization formula, which served as the basis for the optimal parameter design method of the TMD [11]. [PERSON] [12] used the integral modal method they discovered to obtain the optimal damping of the TMD. Over the past decades, researchers have conducted numerous studies on TMDs. [PERSON] et al. [13] conducted a systematic theoretical and experimental study on the optimal values of structural parameters for TMDs. [PERSON] et al. [14] investigated the control effects of TMDs when different methods were used to determine the TMD design parameters for seismic control. Additionally, researchers have made efforts to develop more general and effective TMDs. [PERSON] extended the single-degree-of-freedom TMD to a multi-degree-of-freedom TMD and proposed the concept of a multi-tuned mass damper (MTMD) [15]. [PERSON] proposed a bi-directional homogeneous TMD (BH-TMD) [16]. [PERSON] et al. [17] utilized a distributed TMD with configuration and parameter optimization to reduce vibrations in multimodal cable-stayed bridges. Theoretical studies of the TMD were carried out based on a linear TMD at an early stage of development [18; 19; 20]. However, linear systems are special cases, and the vast majority of vibrations are nonlinear vibrations. The vibrations of linear systems are only ideal cases obtained by neglecting some parameters. The study of NTMDs, as a practical direction of nonlinear vibrations, has been widely and deeply investigated by more and more researchers. In 1952, [PERSON] [21] introduced the concept of bandwidth and defined it as the region between two peaks of the amplitude-frequency curve. He found that the bandwidth of an NTMD is longer than that of a linear TMD and considered that the NTMD has more advantages than the linear version. He verified this conclusion by studying the structure of mass dampers with the introduction of three-time hardening stiffness. In 1992, [PERSON] [22] obtained the approximate steady-state solution of the three-time stiffness structure using the averaging method and proved the validity of the approximation method. [PERSON] et al.