## 1 Introduction  
In practical engineering applications, harmful vibration under complex working conditions affects the efficiency, reliability, accuracy, and safety of the equipment structure. The dynamic vibration absorber (DVA) is one of the vibration control equipment, which is attached to the vibration control object, and the vibration state of the primary structure is changed by adjusting the mechanical parameters and structure of the DVA. DVA is widely used in engineering practices due to its superior properties of simple structure and low cost. The design and optimization of high-performance DVA has always been the focus of attention. Since [PERSON] [1] invented the first DVA, many scholars have optimized the structure of DVA to improve the effect of vibration suppression and formed three traditional models: Voigt-type DVA [2], three-element-type DVA [3], and grounded-type DVA [4]. At present, the fixed-point theory [5] has been widely used in the parameter optimization of DVA. With the gradual deepening of engineering practice research, the application of viscoelastic materials provides a greater space for development in various fields. Viscoelastic materials are very effective energy dissipation materials, and their force-displacement hysteresis curves are approximately elliptic. In many engineering fields, viscoelastic materials often have stiffness and damping characteristics at the same time. A three-element-typeDVA model was proposed, namely the Maxwell model, which could describe the characteristics of viscoelastic materials very well [3]. The Maxwell model has good vibration suppression effect, which greatly reduces the resonance amplitude value and broadens the range of vibration reduction frequency. [PERSON] et al. [6] proposed a modified fixed-point theory and applied to the derivation of the optimal parameters formulas of a viscoelastic DVA. [PERSON] et al. [7] presented the optimal design parameters of viscoelastic tuned mass dampers. [PERSON] et al. [8] proposed an extended Maxwell-type viscous dampers model and the state determination algorithm, which could be used for the simulation of various damped dynamic systems. [PERSON] et al. [9] introduced a Maxwell element into the tuned mass damper, and applied to reduce bridge vibration. Negative stiffness devices have the advantages of strong carrying capacity, little deformation, and great controllability, which are more and more used in vibration reduction. The force generated by the negative stiffness system or mechanism is in the same direction as the displacement, and the force decreases with the increase of the displacement. [PERSON] et al. [10; 11] designed different types of DVAs with negative stiffness and found that it could show good reduction effect. [PERSON] et al. [12] proposed a tri-stable nonlinear energy sink combining negative stiffness and found that it had good vibration suppression performance under transient and steady state excitation. [PERSON] et al. [13] discussed the nonlinear dynamic response of a vibration isolation system with superelastic hysteresis and negative stiffness. [PERSON] et al. [14] investigated a quasi-zero-stiffness DVA, which showed good vibration reduction performance under harmonic excitation. [PERSON] et al. [15] proposed a novel tuned inertial damper with double flywheel structure, and its optimal grounded stiffness ratios were negative, zero, and positive. The results showed that the proposed tuned inertial damper with positive grounded stiffness could provide better control performance. In recent years, many scholars have carried out a series of studies on nonlinear systems [16; 17; 18]. [PERSON] et al. [16] studied an optimization procedure involving nonlinear aeroelastic effect for self-excited vibration control, and the results showed that the nonlinear objective and optimization method was more reliable than the traditional optimization method. A typical force amplification mechanism, such as an inerter, has a good engineering application prospect.