## 2 Mathematical Modeling  
Figure 1 illustrates the schematic diagram of the ERS-TQZS VI. The ERS-TQZS VI system comprises a primary mass \(m_{1}\) suspended on an upper-stage QZS VI, which is connected to a secondary mass \(m_{2}\). The secondary mass is then grounded through the lower-stage QZS VI. The ERS-TQZS VI system is composed using two horizontal linear springs with stiffnesses \(k_{h1}\) and \(k_{h2}\), along with two vertical linear springs with stiffnesses \(k_{v1}\) and \(k_{v2}\). Additionally, two vertical linear dampers with damping coefficients \(c_{1}\) and \(c_{2}\) are incorporated. The subscripts 1 and 2 are used to denote the upper and lower stages, respectively. The horizontal springs have an initial length of \(L\), and when compressed in the horizontal position, they reach a static equilibrium position denoted by \(a\). Moreover, the viscous dissipative element between the upper-stage QZS VI and lower-stage QZS VI is supplied by an electromagnetic transducer of coil resistance \(R_{i}\) and inductance \(L_{i}\). Figure 1 illustrates the schematic diagram of the ERSD equivalent circuit. The ERSD can take the form of either a linear or rotational mechanism with motion transmission and typically consists of a permanent magnet and a coil. When the ERS-TQZS VI is subjected to a harmonic force excitation \(f_{e}\), the relative motion between the mass \(m_{1}\) and \(m_{2}\) generates an induced voltage \(V_{ERSD}\) and an induced current \(I_{e}\) proportional to the relative velocity. The analytical model of the ERS-TQZS VI system mainly consists of two parts: three-spring QZS VI and ERSD. Among them, the analytical model of the three-spring QZS VI has been experimentally verified in our previous work [47]. In addition, the three-spring QZS VI and ERSD analytical models have also been separately studied in the following classic literature [41; 42; 48; 49; 50; 51]. Considering the ERS-TQZS VI system being subjected to a harmonic force \(f_{e}=F_{e}\cos\,(\omega t)\) while positioned on a rigid base, the governing equation of the ERS-TQZS VI system at the static equilibrium position can be derived using [PERSON]'s second law and [PERSON]'s second law. \[\left\{ \begin{array}{l}m_{1}\dot{z}_{1}+c_{1}(\dot{z}_{1}-\dot{z}_{2})+C_{d} I_{e}+\bigg{[}k_{v1}+2k_{h1}(1-\frac{L}{\sqrt{a^{2}+\left(z_{1}-z_{2}\right)^{2}}} )\bigg{]}(z_{1}-z_{2})=F_{e}\cos\,(\omega t)\\ m_{2}\dot{z}_{2}+c_{1}(\dot{z}_{2}-\dot{z}_{1})+c_{2}\dot{z}_{2}-C_{d}I_{e}+ \bigg{[}k_{v1}+2k_{h1}(1-\frac{L}{\sqrt{a^{2}+\left(z_{2}-z_{1}\right)^{2}}} )\bigg{]}(z_{2}-z_{1})+\bigg{[}k_{v2}+2k_{h2}(1-\frac{L}{\sqrt{a^{2}+z_{2}^{2} }})\bigg{]}z_{2}=0\\ L_{i}\dot{I}_{e}+(R_{i}+R_{e})I_{e}+C_{e}(\dot{z}_{2}-\dot{z}_{1})+\frac{1}{ \mathcal{C}}\int I_{e}dt=0\end{array} \right. \tag{1}\]  
where \(z_{1}\) and \(z_{2}\) represent the displacements of the masses \(m_{1}\) and \(m_{2}\), respectively, from the static equilibrium position. \(C_{d}\) denotes the electromagnetic coupling coefficient, and \(C_{e}\) represents the electromagnetic constant. For small displacements of masses \(m_{1}\) and \(m_{2}\), the spring forces in Equation (1) can be approximated using a Taylor series expansion around the respective static equilibrium positions. Consequently, Equation (1) can be simplified into two coupled Duffing equations:  
Figure 1: Schematic diagram of the ERS-TQZS VI system. \[\left\{