## 5 Nonlinear Modeling and Dynamic Stiffness Comparison of the Newly Designed HEM and MRF Mount
### Comparison with the MR Fluid HEM  
Because the MR/ER-based semiactive mount would tune the damping, we take the MR-based mount as an example to show the advantages of the newly designed HEM over an MR/ER-based mount. An MR-based HEM can be obtained by adopting the MRF as a  
Figure 11: Dynamic stiffness spectrum for different lengths under nonlinear model and linear model, \(P_{3}\), \(P_{1}\) are fixed frequency points. Figure 10: Comparison of the numerical calculation results of the dynamic stiffness at different frequencies with the experimental results [26]. fluid medium inside a common passive HEM and by placing a coil holder with a uniformly wound copper wire coil around the inertia track. The magnetic field generated by the energized coil changes the flow characteristics (viscosity) of the MR fluid and thus changes the dynamic performance of the HEM. We compare the dynamic performance of a typical MR mount in the literature [28] to the performance of the newly designed HEM. The pressure difference between the upper chamber pressure \(p_{1}(t)\) and the lower chamber pressure \(p_{2}(t)\) of an MRF mount [29] is  
\[p_{1}(t)-p_{2}(t)=\Delta P_{\rm I}+\Delta P_{\rm\mu}+\Delta P_{\rm\tau}, \tag{47}\]  
where \(\Delta P_{\rm I}\) is the pressure drop due to the inertia of the fluid flow; \(\Delta P_{\rm\mu}\) is the pressure drop caused by the fluid viscosity and the pressure drop caused by the energy loss when the fluid flows through the inertia track; and \(\Delta P_{\rm\tau}\) is the pressure drop due to the yield stress of the fluid. According to Equations (41)-(46), \(\Delta P_{\rm I}\) and \(\Delta P_{\rm\mu}\) are  
\[\Delta P_{\rm I}=I_{\rm im}\dot{Q}_{\rm im}(t)\] \[\Delta P_{\rm\mu}=\frac{128I_{\rm im}\dot{L}_{\rm im}}{\pi D_{\rm nm }}Q_{\rm im}+\frac{1}{2}\rho_{\rm m}(\zeta_{\rm Im}+\zeta_{\rm dm}+\zeta_{\rm bm })\;\frac{|Q_{\rm im}|}{A_{\rm im}^{2}}\;Q_{\rm im}\;\;. \tag{48}\]  
According to the literature [29], \(\Delta P_{\rm\tau}\) can be obtained as follows:  
\[\Delta P_{\rm\tau}=\frac{c\tau L_{\rm im}k_{\rm\tau}}{g}\frac{1}{|Q_{\rm im}|} \;Q_{\rm im}, \tag{49}\]  
where the coefficient \(c=[2.07-3.07]\) depends on the flow velocity profile [30], for convenience, \(c=2.57\); \(k_{\rm\tau}\) is the ratio of effective area caused by magnetic field; \(g\) is the gap of MRF; the parameters \(I_{\rm im}\), \(Q_{\rm im}\), \(\mu_{\rm m}\), \(L_{\rm im}\), \(D_{\rm mm}\), \(\zeta_{\rm lm}\), \(\zeta_{\rm dm}\), \(\zeta_{\rm bm}\), and \(A_{\rm im}\) have the same physical meanings as parameters such as \(I_{\rm i}\) without the subscript m, and the parameters with subscript m are the parameters of the MR mount; and the yield shear stress \(\tau\)[28] is  
\[\tau=C_{\rm\tau}\cdot 2.717\times 10^{5}\cdot\Phi^{1.5239}\cdot{\rm tanh}\; \left(6.33\times 10^{-6}\cdot H_{\rm m}\right), \tag{50}\]  
where \(H_{\rm m}\) is the magnetic field intensity, the tunable range is \(H_{\rm m}=[0-200]\;({\rm A/mm})\); the constant \(C_{\rm\tau}\) depends on the carrier oil type, \(C_{\rm\tau}=1.16\); and \(\Phi\) is the iron volume fraction, \(\Phi=0.22\). Equation (47) can be rewritten as follows:  
\[p_{1}(t)-p_{2}(t)=I_{\rm im}\dot{Q}_{\rm im}(t)+R_{\rm im}Q_{\rm im}(t), \tag{51}\]  
where the equivalent viscous resistance coefficient of MRF flow \(R_{\rm im}\) is  
\[R_{\rm im}=\frac{128\mu_{\rm m}L_{\rm im}}{\pi D_{\rm hm}{}^{4}}+\frac{1}{2} \rho_{\rm m}(\zeta_{\rm lim}+\zeta_{\rm dm}+\zeta_{\rm bm})\frac{|Q_{\rm im}| }{A_{\rm im}^{2}}+\frac{c\tau L_{\rm im}k_{\rm\tau}}{g}\frac{1}{|Q_{\rm im}|}. \tag{52}\]  
To better compare the performances of our newly designed HEM and MRF mount, the MRF HEM adopts the same parameter value set of the newly designed HEM, except for the field shear stress \(\tau\)-related parameters. Table 2 lists the values of all the key parameters of the MRF mount. \(\Delta P_{\rm\tau}\) in Equation (47) is added to the nonlinear model in Section 5.1 to determine the dynamic stiffness of the MRF mount when the magnetic field intensity \(H_{\rm m}\) is in the range of 0 to 200 A/mm.