## 4 Dynamic Performance of the New Semiactive HEM  
### Lumped Parameters  
The semiactive HEM would tune only the lumped damping of the inertia track \(R_{\rm i}\) or the equivalent damping \(b_{\rm ie}\) without the other parameters of the TMD system. Let radius \(r_{\rm i}\) of the inertia track's neutral line and the rotation angle \(\theta\) of the moving plate screw be as shown in Figure 3d. The length of the inertia track is \(L_{\rm i}\):  
\[L_{\rm i}=\pi r_{\rm i}\theta. \tag{28}\]  
Equation (28) becomes  
\[\theta=L_{\rm i}/r_{\rm i}\pi. \tag{29}\]  
The height and cross-section area of the inertia track are  
\[h(L_{\rm i})=L_{\rm i}\tan\ \alpha=\pi r_{\rm i}\theta\tan\ \alpha, \tag{30}\]  
\[A(L_{\rm i})=bL_{\rm i}\tan\ \alpha=\pi br_{\rm i}\theta\tan\ \alpha, \tag{31}\]  
where \(\alpha\) is the spiral lift angle of the moving plate, and \(b\) is the width of the inertia track. The inertia parameters of the inertia track are  
\[I_{\rm i}(L_{\rm i})=\rho L_{\rm i}/A_{\rm i}(L_{\rm i})=\rho/(b\tan\ \alpha), \tag{32}\]  
\[M_{\rm ie}(L_{\rm i})=A_{\rm p}^{2}I_{\rm i}(L_{\rm i})=\rho A_{\rm p}^{2}/(b \tan\ \alpha). \tag{33}\]  
Since \(\rho\), \(b\), and \(\tan\alpha\) are invariant, considering Equations (32) and (13), the inertia of the inertia track \(I_{\rm i}(L_{\rm i})\) and \(M_{\rm ie}\) do not change with \(L_{\rm i}\). The wetted perimeter of the inertia track is  
\[P(L_{\rm i})=2(h(L_{\rm i})+b)=2L_{\rm i}\tan\ \alpha+2b=2\pi r_{\rm i}\theta \tan\ \alpha+2b. \tag{34}\]  
\(R_{\rm i}(L_{\rm i})\) and \(b_{\rm ie}(L_{\rm i})\) are  
\[R_{\rm i}(L_{\rm i})=\ 128\mu L/\pi\bigg{(}\frac{2 bL_{\rm i}\tan\ \alpha}{L_{\rm i}\tan\ \alpha+b}\bigg{)}^{4}\ = \ 128\mu/\pi L_{\rm i}^{3}\bigg{(}\frac{2b\tan\ \alpha}{L_{\rm i}\tan\ \alpha+b}\bigg{)}^{4}, \tag{35}\]  
\[b_{\rm ie}(L_{\rm i})=A_{\rm p}^{2}R_{\rm i}(L_{\rm i})=\ 128\mu A_{\rm p}^{2}/ \pi L_{\rm i}^{3}\bigg{(}\frac{2b\tan\ \alpha}{L_{\rm i}\tan\ \alpha+b}\bigg{)}^{4}. \tag{36}\]  
When \(L_{\rm i}=0\), we have  
\[R_{\rm i}(L_{\rm i}=0)=+\infty \tag{37}\]  
\[\theta=L_{\rm i}/r_{\rm i}\pi. \tag{29}\]0  
Let \(R_{\rm i}=R_{\rm i}(L_{0})\), \(b_{\rm 1}=b_{\rm ie}(L_{0})\), and Equations (37) and (38) indicate the tunable range of lumped damping and equivalent damping \(R_{\rm i}\ \in\ [R_{\rm 1},+\infty)\), \(b_{\rm ie}\ \in\ [b_{\rm 1},+\infty)\). Because the damping of the anti-freeze liquid is weaker than that of the MR/ER fluid with the same geometry, the lower limit of the damping of the newly designed HEM is smaller than that of MR/ER fluid. Because the tunable damping of the MR/ER fluid-based semiactive mount is proportional to the magnetic/electric field intensity, which cannot be \(+\infty\), the upper bound is finite. However, the upper bound of the damping of the newly designed HEM is \(+\infty\). In other words, the damping of the newly designed HEM has a much wider tunable range than that of the MR/ER fluid-based semiactive mount. According to [PERSON] [12], the upper compliance \(C_{1}\) can be calculated as follows:  
\[\theta=L_{\rm i}/r_{\rm i}\pi. \tag{29}\]1  
where the change in the upper chamber, denoted by \(\Delta V_{1}\), is a result of the variation in pressure, represented by \(\Delta P_{1}\). By closing the inertial track in the hydraulic mount finite element model and applying pressure to the top of the upper chamber, we can calculate the volume change \(\Delta V_{1}\) and the pressure change \(\Delta P_{1}\) to determine the compliance of the upper chamber \(C_{1}\). The compliance in the lower chamber \(C_{2}\) can also be computed. Using the parameters in Table 1, the damping \(R_{\mathrm{i}}(L_{\mathrm{i}})\) and \(b_{\mathrm{ie}}(L_{\mathrm{i}})\) become smaller when the length of the inertia track becomes larger, as shown in \[\theta=L_{\rm i}/r_{\rm i}\pi. \tag{29}\]2 This shows that the new semiactive HEM can utilize the change in the geometry of the inertia track to tune the damping of the inertia track instead of the expensive MR/ER fluid.