## 1 Introduction
[23] analyzed the stability of the structure of a mass damper system with three-time nonlinear stiffness using both experimental and approximate analysis methods, and they verified the accuracy of the approximate analysis method. [PERSON] et al. [24] used the approximate analysis method to investigate the stability of a mass damper system with the bifurcation of triple nonlinear stiffness and verified its validity through numerical solutions. Subsequently, [PERSON] et al. [25] analyzed a system of nonlinearly tuned mass dampers with stiffness due to geometric position and investigated the closed separated resonant section of the frequency-amplitude response curve. [PERSON] et al. [26] proposed that the control frequency range of a nonlinear energy sink (NES) is wider than that of linear TMDs. [PERSON] et al. [27] found that an NES can distribute the input energy of a structure from low to high modes. [PERSON] et al. [28] demonstrated the existence of jumps in nonlinear TMDs. [PERSON] and [PERSON] [29] argued that the only way to make the steady-state response of a system better than that under linear TMD control is to eliminate or reduce the high-amplitude periodic solutions generated by nonlinear TMDs. [PERSON] and [PERSON] [30] used an approximate analytical method to obtain an optimal expression for the TMD frequency of a cubic stiffness nonlinear mass-damped system and verified its applicability and effectiveness through numerical methods. [PERSON] et al. [31] derived a design method for solving the optimal mass ratio of the TMD using the complex variable averaging method and the multiscale method and demonstrated that the method possesses better performance. [PERSON] et al. [32] used the harmonic balance method (HBM) to determine the steady-state amplitudes of structures controlled by nonlinear TMDs, obtained the formula for the optimal frequency ratio of TMDs and compared the analytical results with those obtained from numerical calculations to prove the correctness and validity of the formula. [PERSON] et al. [33] used the complex variable averaging method to derive the approximate analytical solution of the system amplitude and obtained the theoretical optimized design parameters of the TMD, which proved that the optimized parameters of the TMD obtained by a nonlinear design have a good vibration reduction effect before and after the nonlinear characteristics of the TMD are generated. In recent years, there have been significant developments in engineering applications of tuned mass dampers, leading to the introduction and implementation of various new types of tuned mass dampers that have been improved and innovated from the original design. [PERSON] et al. proposed an adaptive-passive eddy current pendulum tuned mass damper (APEC-PTMD), which can retune the frequency by changing the length of the pendulum and adjusting the damping ratio. The paper [34] verifies the function of frequency and damping retuning. Subsequently, [PERSON] et al. applied the APEC-PTMD to a 40-story high-rise building under four different soil conditions. The APEC-PTMD was able to identify the optimal TMD frequency for different soil types and reset its damping ratio, resulting in better seismic performance [35]. Another development is the semi-active eddy current pendulum tuned mass damper (SAEC-PTMD) proposed by [PERSON] et al. This type of TMD can adjust the frequency and damping ratio in real time to ensure the stability of the controlled structure [36]. [PERSON] et al. also applied the adaptive tuned mass damper to the vibration control of pedestrian bridges, achieving excellent results in vibration control [37]. With their higher reliability, wider applicability and increased intelligence, these new types of tuned mass dampers provide more effective solutions for engineering applications. They will play an important role in mechanical and vibration-damping fields and make a positive contribution to enhancing structural safety. In reality, a linear tuned mass damper (TMD) represents an ideal case.