## 5 Nonlinear Modeling and Dynamic Stiffness Comparison of the Newly Designed HEM and MRF Mount  
Although the theoretical performance of the HEM can be clearly understood based on the linear model shown in Figure 2a, the simulation results in Figure 8 demonstrate that nonlinear factors affect the dynamic stiffness. Therefore, it is necessary to establish a nonlinear model of the newly designed HEM in this work and evaluate its dynamic performance. Additionally, we aim to compare its nonlinear dynamic performance with that of MR/ER fluid-based semiactive HEMs. While there are various nonlinear factors  
Figure 9: Vibration isolation performance. (**a**) Optimal length of the inertia track; (**b**) dynamic stiffness for different parameters; and (**c**) transmissibility for different parameters.  
involved, such as carbon-filled rubber, this paper only considers the turbulence-induced nonlinear damping while ignoring other nonlinear factors.  
Liquid flowing through the inertia track will produce energy losses for various reasons, and the energy losses are manifested as intuitive pressure losses. The \(R_{\rm i}\) in Equation (3) describes the pressure loss due to the inertia damping.  
The energy loss along the inertia track caused by the equivalent linear fluid resistance \(R_{\rm i}\) is described as follows [14]:  
\[\Delta P_{\rm I}=R_{\rm i}Q_{\rm i}=\frac{128\mu L_{\rm i}}{\pi{D_{\rm h}}^{4}} Q_{\rm i}. \tag{41}\]  
The local energy loss due to the sudden change in the cross-section of fluid flowing through the inlet and outlet can be expressed as follows [26]:  
\[\Delta P_{\rm d}=\frac{1}{2}\rho(\zeta_{\rm d1}+\zeta_{\rm d2})\frac{Q_{\rm i }}{A_{\rm i}}\bigg{|}\frac{Q_{\rm i}}{A_{\rm i}}\bigg{|}, \tag{42}\]  
where \(\zeta_{\rm d1}\) and \(\zeta_{\rm d2}\) are the local factor for the sudden contraction and expansion of the cross-section, respectively.  
According to the literature [27], the pressure loss due to the bending effect caused by the fluid flowing along the curved inertia track can be expressed as  
\[\Delta P_{\rm b}=\frac{1}{2}\rho\zeta_{\rm b}\frac{Q_{\rm i}}{A_{\rm i}} \bigg{|}\frac{Q_{\rm i}}{A_{\rm i}}\bigg{|}, \tag{43}\]  
where the empirical formula for the bending effect resistance factor \(\zeta_{\rm b}\) is  
\[\zeta_{\rm b}=\left(0.131+1.847\bigg{(}\frac{D_{\rm h}}{r_{\rm i}}\bigg{)}^{3.5}\right)\frac{\theta}{90^{\circ}}. \tag{44}\]  
As shown in Figure 3d, \(r_{\rm i}\) is the radius of curvature of the centerline of the inertia track, and \(\theta\) is the rotation angle of the inertia track.  
Considering the pressure loss due to the various reasons mentioned above, compared to Equation (3), the pressure change equation can be expressed as  
\[p_{1}(t)-p_{2}(t)=I_{\rm i}\dot{Q}_{\rm i}(t)+\left(R_{\rm i}+\frac{1}{2}\rho (\zeta_{\rm d}+\zeta_{\rm b})\frac{|Q_{\rm i}|}{A_{\rm i}^{2}}\right)Q_{\rm i} (t)=I_{\rm i}\dot{Q}_{\rm i}(t)+R_{\rm in}\;Q_{\rm i}(t), \tag{45}\]  
where  
\[R_{\rm in}=R_{\rm i}+\frac{1}{2}\rho(\zeta_{\rm d}+\zeta_{\rm b})\;\frac{|Q_{ \rm i}|}{A_{\rm i}^{2}}. \tag{46}\]  
The nonlinear dynamic model of an HEM can be defined by Equations (1), (2), (8), (45) and (46). Considering the existence of nonlinear terms in the system of equations, MATLAB was used to determine the solution. The dynamic stiffness of the dynamic characters in the frequency domain can be obtained by determining the relevant parameter values, selecting a suitable integration method, and giving the proper amplitude and frequency of the excitation signal.