## 2 Research Method
### Mathematical Modeling
#### 2.1.3 Magnetic Damping Force Models  
The formula for calculating the \(F_{MR}\) corresponding to the last term in Equation (4) is expressed as follows:  
\[F_{MR}= \tag{15}\]  
In Equation (16), \(A_{piston}\) is the cross-sectional area of the silver piston, and \(\Delta P_{MR}\) is the magnitude of the internal pressure drop due to the characteristics of the MR fluid. \(\Delta P_{MR}\) can be expressed as follows:  
\[\Delta P_{MR}=\frac{L_{gap}}{D_{gap}}C\tau_{y} \tag{16}\]  
In Equation (17), \(L_{gap}\) is the length of the basic gap, \(D_{gap}\) is the width of the basic gap, C is the flow rate coefficient of the MR fluid, and \(\tau_{y}\) is the yield stress of the MR fluid generated by the application of current which can be calculated using Equation (18):  
\[\tau_{y}=144+138708B+158790B^{2}-176510B^{3}+52962B^{4} \tag{17}\]  
In Equation (18), the yield stress of the MR fluid \(\tau_{y}\) is determined by the magnetic flux density B generated in the magnetic circuit of the MR damper. The magnetic flux density B is generated by the magnetic field that occurs when current flows in the solenoid coil according to [PERSON]'s law. The relationship is shown in Equation (19):  
\[\oint B\cdot dl=\mu 0\int J\cdot da \tag{18}\]  
Figure 5: Flow in a basic gap.  
To calculate the magnetic flux density B according to Equation (19), the magnetic circuit of the MR damper is expressed mathematically, and the permeance method is used to calculate the magnetic flux density. Permeability is the opposite of magnetoresistance R, indicates the degree to which the magnetic flux flows well in a magnetic material, and can be expressed by Equation (20):  
\[\int\frac{\mu_{0}uda}{l}=\frac{1}{R} \tag{19}\]  
The magnetic circuit of a general MR damper is composed of a solenoid coil, core, flux ring, and an MR fluid that fills the flow path. Each permeance is obtained from the shape parameters of the magnetic circuit components using Equation (20) and is calculated from the magnetic circuit. Because electric circuits are similar, we can calculate the magnetic flux of the desired part of the circuit. The similarities between the magnetic circuits and electric circuits are listed in Table 1.  
Based on the above calculations for the proposed MR damper with an additional flow path, an orifice hole is formed inside the coil such that the fluid flow is not affected when  
For the proposed MR damper with an additional flow path, the total damping force is shown in Figure 7. When the piston speed is high in this damper shape, the viscous damping force caused by the flow through the orifice is greater than the damping force of the basic MR damper with yield stress. The damper operates based on the smaller of the two damping forces, and this small value determines its overall damping force. This allows the damper system to balance the damping force to provide more stable damping performance (25). Accordingly, the total damping force of the proposed MR damper with an additional flow path, \(F_{damper}\), is defined as follows:  
\[F_{damper}= \text{min}\left(F_{viscosity_{\text{\emph{write}}}},\,F_{viscosity_{ \text{\emph{gap}}}}+F_{MR}\,\right) \tag{20}\]  
This confirms that Equation (4) can be expressed as Equation (20) for the MR damper with an additional flow path proposed in this study, and it was used to examine the total damping force based on the piston speed and applied current.