## 2 Research Method
### Mathematical Modeling
#### 2.1.2 Viscous Damping Force Model  
To mathematically model \(F_{viscosity}\), which corresponds to the first term in Equation (4), the following assumptions based on hydrodynamic theory are applied to use [PERSON]'s equation. 1. The MR fluid is assumed to be an incompressible fluid due to its high density and minimal volume change even under operating input;
2. The flow within the MR damper is considered steady-state, assuming laminar flow in the low-velocity, low-Reynolds-number region. Therefore, it is treated as steady flow;
3. The flow in the additional flow path and upper and lower chambers is friction-free;
4. The flow in the additional flow path and upper and lower chambers is the flow along the streamline;
5. To simplify the energy conservation analysis, the control volume is divided into sections A and B, as shown in Figure 3;
6. It is assumed that the MR fluid conserves energy within each section, so the pressure drop between the inlet and outlet of each MR fluid section is the same;
7. To reflect that the kinetic energy of the piston is fully transferred to the fluid, it is assumed that the piston velocity \(V_{piston}\) and the MR fluid velocity \(V_{inlet}\) at the inlet are the same. Based on the above assumptions, the values in the lower and upper chambers of the MR fluid are expressed using the Bernoulli equation. \[\frac{P_{1}}{\rho g}+\frac{V_{1}^{2}}{2g}+z_{1}+h=\frac{P_{2}}{\rho g}+\frac{ V_{2}^{2}}{2g}+z_{2} \tag{5}\]  
Figure 3: Schematic of flow mode with regions A and B. In Equation (5), \(P_{1}\) and \(P\) are the pressures in the lower and upper chambers, respectively, \(V_{1}\) and \(V_{2}\) are the average flow rates in the lower and upper chambers, respectively, \(z_{1}\) and \(z_{2}\) are the gross pump heads in each chamber, and h is the total head. P and g are the density and gravitational acceleration of the MR fluid, respectively, and in incompressible flow, the following holds true:  
\[V_{1}\approx V_{2} \tag{6}\]  
Because Equation (6) is established, Equation (5) can be reorganized as follows:  
\[\Delta P=P_{2}-P_{1}{=\rho gh} \tag{7}\]  
Additionally, the following theory is applied to \(F_{viscosity}\) to formulate it in the same manner as \(F_{MR}\). In the proposed MR damper, the additional flow path can be interpreted as a fully developed laminar flow in a circular pipe covered by hydrodynamics. In the proposed additional flow-path-type MR damper, the flow in the non-magnetic orifice is shown in Figure 4. This can be interpreted as fully developed laminar flow within a circular pipe in fluid dynamics, which provides structural characteristics that may help alleviate the block-up phenomenon. The hydrodynamic theoretical equation used for a fully developed laminar flow is as follows:  
\[\Delta P_{\text{\emph{orifice}}}=\frac{L_{\text{\emph{orifice}}}}{D_{\text{ \emph{orifice}}}}\frac{\rho V_{\text{\emph{orifice}}}^{2}}{2}f_{\text{\emph{ orifice}}} \tag{8}\]  
In Equation (9), \(\Delta P_{\text{\emph{orifice}}}\) is the pressure drop in the additional passage, \(L_{\text{\emph{orifice}}}\) and \(D_{\text{\emph{orifice}}}\) are the length and hydraulic diameter of the additional passage, respectively, \(V_{\text{\emph{orifice}}}\) is the average flow velocity of the MR fluid while flowing in the additional passage, and \(f_{\text{\emph{orifice}}}\) is the friction in the additional passage and is a coefficient. The friction coefficient in laminar flow in a fully developed circular pipe can be expressed in terms of the Reynolds number as follows:  
\[f_{\text{\emph{orifice}}}=\frac{64}{Re}=\frac{64\mu}{\rho V_{\text{\emph{orifice }}}D_{\text{\emph{orifice}}}} \tag{9}\]  
Using this approach, the flow in the primary gap is shown in Figure 5, which can be considered as fully developed laminar flow between two parallel plates. Using this approach, the flow in the underlying gap can be regarded as a laminar flow between two fully developed plates.