# Dynamics Analysis of a Variable Stiffness Tuned Mass Damper Enhanced by an Inerter
## 1 Introduction  
In mechanical engineering, civil engineering, and space engineering, vibrations often lead to a decrease in the efficiency of machines and electronic components [1]. To overcome this limitation, the tuned mass damper (TMD) consisting of a stiffness component, a damping component, and a mass was proposed to reduce the vibration from external excitations [2]. The TMD has the advantages of high stability, low cost, and large damping [3]. It has been successfully applied in a wide range of engineering fields [4, 5, 6, 7, 8, 9, 10, 11]. [PERSON] et al. [11] performed a parameter optimization design to improve the vibration reduction performance of steel structures in earthquakes. [PERSON] et al. [12] investigated the multiple TMD for tall buildings and conducted an optimization analysis on the vibration absorber. [PERSON] et al. [13] proposed a radial magnetic pendulum TMD for a rotor-bearing system of flywheel energy storage, which can improve low-frequency vibration reduction. [PERSON] et al. [14] designed a multiple bounding TMD to reduce the vibration of a bridge and performed numerical simulations of traffic. However, the traditional TMD has inherent limitations, with good damping effect only near the resonant frequency of the primary system. When external excitation is not in this range, the vibration reduction effect of the traditional TMD are limited. Therefore, broadening the frequency band of the traditional TMD and enabling it to adapt to more complex external excitation has become an important focus for researchers [15]. In recent years, many new types of TMD have been studied. [PERSON] et al. [16] designed a bio-inspired X-shaped TMD to enhance the vibration reduction performance of the traditional TMD. Vibration reduction performance can be improved by introducing nonlinearities into the structure [17; 18], especially variable nonlinear stiffness. Many scholars have proposed the passive variable stiffness TMD to improve the vibration reduction performance with different excitations [19; 20; 21]. [PERSON] et al. [22] investigated a magnetorheological elastomer variable stiffness TMD to reduce wind-induced vibration from a long-span bridge and building. [PERSON] et al. [23] proposed a variable stiffness TMD with a nonlinear system identification method and developed a variable stiffness control method based on the decay of the potential energy of the system. When the stiffness of the vibration absorber is adjustable in real time, it has better vibration reduction performance. Compared with the passive variable stiffness TMD, the active variable stiffness TMD can be used to reconfigure the device by external dynamic signals [24; 25; 26]. Inspired by [PERSON]. [27; 28], [PERSON] et al. [29] proposed a time-varying stiffness TMD and applied the vibration absorber to a lattice sandwich structure for vibration reduction, which can achieve an assembly capable of fast dynamic stiffness control. These recent investigations have indicated that the active variable stiffness TMD has become a promising vibration reduction device. When the additional mass of the vibration absorber is smaller, it can be conducive to the engineering applications of the vibration absorber. [PERSON] [30] investigated a new type of mechanical device called the inerter based on the mathematical model of the capacitance. The inerter connects two terminals. [PERSON] et al. [31] proposed the ballscrew inerter and applied it to reduce automobile vibration. Moreover, the inertance provided by the inerter can be much larger than its own mass. The ballscrew inerter can transform hundreds of kilograms of inertia mass into a flywheel with a mass of a few hundred grams through a force amplification mechanism. Many scholars have replaced the traditional mass block with the inerter to reduce the mass of the vibration absorber, and they have applied the vibration absorber in practical engineering [32; 33; 34; 35; 36; 37; 38]. [PERSON] et al. [39] investigated a tuned inerter damper (TID) for a primary system.