## 5 Nonlinear Modeling and Dynamic Stiffness Comparison of the Newly Designed HEM and MRF Mount
### Comparison with the MR Fluid HEM
The results in Figure 12 show that if the magnetic field intensity is less than 20 A/mm, when the excitation frequency is less than the fixed-point \(P_{\rm 4}\) frequency, the dynamic stiffness increases with the increase in the magnetic intensity, while the dynamic stiffness decreases with the increase in the magnetic intensity when the excitation frequency is greater than the fixed-point \(P_{\rm 4}\) frequency; when magnetic field intensity is greater than 20 A/mm, the dynamic stiffness is affected slightly by the magnetic intensity. The change in the dynamic stiffness with a magnetic intensity greater than 20 A/mm means that the MR fluid flow in the track is impeded as the viscosity increases due to the increasing magnetic intensity, and this change is the same as the results in Ref. [31]. The dynamic stiffness of the newly designed HEM with inertia track length \(L_{\rm i}=1.0\ L_{0}\) is the same as that of the MRF mount at magnetic field strength \(H_{\rm m}=0\) A/mm. The dynamic stiffness of the newly designed HEM with an inertia track length less than 0.1 \(L_{0}\) is the same as that of the MRF mount at magnetic field intensity \(H_{\rm m}\) greater than 20 A/mm. In all, the newly designed HEM has a similar dynamic stiffness control effect to that of the MRF mount. On the other hand, the newly designed HEM does not require expensive MR/ER fluid or additional continuous external energy to maintain the magnetic intensity. The fluid properties for various MR fluids, distinct from those reported in the literature, are presented in Table 3. The fluids from Rang-Lin Fan [32] and Thanh Quoc Truong [14] have a lower fluid kinematic viscosity than the MRF in Quoc NV [33]. As shown in Figure 13, for the newly designed HEM with low-viscosity fluid, when the excitation frequency is less than the fixed-point \(P_{\rm 5}\) frequency, the lower control bound of the dynamic stiffness is lower; while the upper control bound of the dynamic stiffness is higher when the excitation frequency is greater than the fixed-point frequency. The overall dynamic stiffness control range for the newly designed HEM is wider than that of the MRF mount. The dynamic stiffness tunable range of both the newly designed HEM and MRF mount decreases as the viscosity of the fluid increases. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameters** & **Values** & **Units** \\ \hline \(M\) & 60 & kg \\ \(k_{\rm rm}\) & 102.9 & N\(\cdot\)mm\({}^{-1}\) \\ \(b_{\rm rm}\) & 0.045 & N\(\cdot\)s\(\cdot\)mm\({}^{-1}\) \\ \(A_{\rm pm}\) & 2047.8 & mm\({}^{2}\) \\ C\({}_{\rm 1m}\) & \(2.699\times 10^{4}\) & mm\({}^{5}\cdot\)N\({}^{-1}\) \\ \(C_{\rm 2m}\) & \(8.084\times 10^{6}\) & mm\({}^{5}\cdot\)N\({}^{-1}\) \\ \(L_{\rm 0m}\) & 138.95 & mm \\ \(A_{\rm im}\) & 139.4 & mm\({}^{2}\) \\ \(\rho_{\rm m}\) & \(1.15\times 10^{-6}\) & kg\(\cdot\)mm\({}^{-3}\) \\ \(\mu_{\rm m}\) & \(1.0\times 10^{-7}\) & N\(\cdot\)s\(\cdot\)mm\({}^{-2}\) \\ \(b_{\rm m}\) & 10 & mm \\ \(c\) & 2.57 & N\(/\)A \\ \(g\) & 11.6 & mm \\ \(k_{\rm\tau}\) & 0.58 & N\(/\)A \\ \(H_{\rm m}\) & \([0,200]\) & A/mm \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Detailed parameters of the MRF mount. Figure 12: Comparison of the dynamic stiffness of the MRF mount with different magnetic intensities and the newly designed mount with different inertia track lengths, \(P_{3}\), \(P_{4}\) are fixed frequency points. \begin{table}
\begin{tabular}{c c} \hline \hline
**Fluid Medium Source** & **Fluid Kinematic Viscosity \(\mu\)(N\(\cdot\)s-mm\({}^{-2}\))** \\ \hline Rang-Lin Fan [32] & \(2\times 10^{-7}\) \\ Thanh Quoc Truong [14] & \(3.67\times 10^{-7}\) \\ MRF-122-2 ED [33] & \(7.5\times 10^{-7}\) \\ MRF-132 DG [33] & \(1\times 10^{-6}\) \\ MRF-140 CG [33] & \(2.9\times 10^{-6}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Fluid properties for different fluids. Figure 13: Comparison of the dynamic stiffness between the newly designed HEM and MRF mount under different damping conditions, \(P_{5}\) is fixed frequency point.