## 2 Mathematical Modeling
\begin{array}{l}m_{1}\tilde{z}_{1}+c_{1}(\hat{z}_{1}-\hat{z}_{2})+C_{d}I_{ e}+k_{\text{\tiny{c1}}}\alpha_{1}(z_{1}-z_{2})+k_{\text{\tiny{c1}}}\frac{\gamma_{1}}{ \delta^{2}}(z_{1}-z_{2})^{3}=F_{e}\cos\left(\omega t\right)\\ m_{2}\tilde{z}_{2}+c_{1}(\hat{z}_{2}-\hat{z}_{1})+c_{2}\hat{z}_{2}-C_{d}I_{e}+k_ {\text{\tiny{c1}}}\alpha_{1}(z_{2}-z_{1})+k_{\text{\tiny{c1}}}\alpha_{2}z_{2}+ k_{\text{\tiny{c1}}}\frac{\gamma_{1}}{\delta^{2}}(z_{2}-z_{1})^{3}+k_{\text{\tiny{c1}}} \frac{\gamma_{2}}{\delta^{2}}z_{2}^{3}=0\\ L_{i}\hat{I}_{e}+(R_{i}+R_{e})I_{e}+C_{e}(\hat{z}_{2}-\hat{z}_{1})+\frac{1}{C} \int I_{e}dt=0\\ \text{where }\alpha_{1}=1-2(\frac{1-l}{l})K_{h1};\;\alpha_{2}=K_{\text{\tiny{c2}}}-2( \frac{1-l}{l})K_{h2};\;\gamma_{1}=\left(\frac{1-l^{2}}{l^{3}}\right)K_{h1};\; \gamma_{2}=\left(\frac{1-l^{2}}{l^{3}}\right)K_{h2};\;\delta=\sqrt{L^{2}-a^{2}} ;K_{h1}=\frac{k_{\text{\tiny{h1}}}}{k_{\text{\tiny{c1}}}};\;K_{h2}=\frac{k_{ \text{\tiny{h2}}}}{k_{\text{\tiny{c1}}}};K_{\text{\tiny{c2}}}=\frac{k_{\text{ \tiny{c2}}}}{k_{\text{\tiny{c1}}}};l=\frac{a}{L}.\end{array} \right.\]  
To obtain the dimensionless form of Equation (2), the following non-dimensional parameters are introduced:  
\[ \begin{array}{l}\omega_{n}=\sqrt{\frac{k_{\text{\tiny{c1}}}}{m_{1}}};\;\mu= \frac{m_{2}}{m_{1}};\;\zeta_{1}=\frac{c_{1}}{2m_{1}\omega_{n}};\;\zeta_{2}= \frac{c_{2}}{2m_{2}\omega_{n}};\;\Omega=\frac{\omega}{\omega_{n}};\;\tau=\omega _{n}t;\;\widehat{F}_{e}=\frac{F_{e}}{k_{\text{\tiny{c1}}}\delta};\;\widehat{z}_ {1}=\frac{z_{1}}{\delta};\\ \widehat{z}_{2}=\frac{2a}{\delta};\;\widehat{T}_{e}=\frac{L}{l_{0}};\;\widehat{ R}=\frac{R_{i}+R_{e}}{\omega_{n}k_{i}};\;\widehat{C}=\omega_{n}^{2}I_{e}C;\; \widehat{C}_{d}=\frac{I_{\text{\tiny{cC}}}}{k_{\text{\tiny{c1}}}\delta},\; \widehat{C}_{e}=\frac{\delta C_{e}}{I_{0}L_{i}}\end{array} \]  
where \(\tau\) is the non-dimensional time; \(\mu\) is the mass ratio; \(\widehat{F}_{e}\) is the non-dimensional excitation force amplitude; \(\Omega\) is the frequency ratio; \(\zeta_{1}\) and \(\zeta_{2}\) are the non-dimensional resistances of the upper and lower linear system, respectively; \(\widehat{T}_{e}\) is the non-dimensional current; \(\widehat{R}\) is the non-dimensional resistance; \(\widehat{C}_{e}\) is the non-dimensional capacitance; \(\widehat{C}_{d}\) is the non-dimensional electromagnetic coupling coefficient; and \(\widehat{C}_{e}\) is the non-dimensional electromagnetic constant. By using these non-dimensional parameters, Equation (2) can be written in a non-dimensional forms  
\[\left\{ \begin{array}{l}\widehat{z}_{1}+2\zeta_{1}(\widehat{z}_{1}-\widehat{ z}_{2})+\widehat{C}_{d}\widehat{I}_{e}+\alpha_{1}(\widehat{z}_{1}-\widehat{z}_{2})+ \gamma_{1}(\widehat{z}_{1}-\widehat{z}_{2})^{3}=\widehat{F}_{e}\cos(\Omega\tau )\\ \widehat{\mu}\widehat{z}_{2}+2\zeta_{1}(\widehat{z}_{2}-\widehat{z}_{1})+2\mu \zeta_{2}\widehat{z}_{2}-\widehat{C}_{d}\widehat{I}_{e}+\alpha_{1}(\widehat{z }_{2}-\widehat{z}_{1})+\alpha_{2}\widehat{z}_{2}+\gamma_{1}(\widehat{z}_{2}- \widehat{z}_{1})^{3}+\gamma_{2}\widehat{z}_{2}^{3}=0\\ \widehat{T}_{e}+\widehat{R}\widehat{I}_{e}+\widehat{C}_{e}(\widehat{z}_{2}- \widehat{z}_{1})+\frac{1}{C}\int\widehat{T}_{d}dt=0\end{array} \right. \tag{3}\]  
Since the magnitude-frequency curve of the nonlinear system always has turning points in the solution process, analytical solutions cannot be obtained. To obtain the solution for the governing equation of the ERS-TQZS VI system, Equation (3), we use the HBM in combination with the PLM [4; 52]. It is the most efficient method for tracing out solution branches. The basic idea of the PLM is to regard the system excitation frequency \(\Omega\) as a new unknown variable, introduce the arc length \(s\) of the amplitude-frequency curve of the system as an auxiliary parameter, and build an independent constraint equation using the normalized tangent vector of the amplitude-frequency curve. The HBM is commonly employed alongside the assumption that the response amplitudes are dimensionless and follow a harmonic form.