## 2 Theory and Experimental Method  
### MR Fluid Power Distribution System  
#### 2.1.1 MR Fluid Properties  
MR fluid experiences an MR effect where the fluid flow's resistance increases under a magnetic field load, and paramagnetic particles are dispersed in a solvent with low permeability. In the absence of a magnetic field load, the MR fluid behaves like a Newtonian fluid, where the particles move freely; however, when a magnetic load is applied, the particles become charged and form a chain structure, exhibiting the behavior of a Bingham fluid that has yield stress. Generally, the rheological properties of an MR fluid change according to the strength of the magnetic field. Accordingly, \(\tau\) denotes the shear stress of the MR fluid, and it can be predicted based on the MRF's Bingham model, as shown in the equation below [20].  
\[\tau=\tau_{y}+\eta\dot{\gamma}. \tag{1}\]  
Here, \(\tau_{y}\) and \(\eta\) denote the MR fluid's yield stress and viscosity coefficient after yielding, and \(\dot{\gamma}\) denotes the shear rate of the MR fluid at the interface. Since the gap between the disc and the plate is very small, it can be assumed that the MR fluid's shear rate at the interface is distributed linearly, and it can be determined as shown below.  
\[\dot{\gamma}=\frac{r\omega}{d}. \tag{2}\]  
Here, \(r\) and \(d\) denote the radius of the disc's smallest area and the space between the disc and the plate, respectively, and \(\omega\) symbolizes the angular velocity difference between the disc and the plate. The shear stress of the MR fluid indicates the shear stress of the fluid when a magnetic field is applied from the outside. The fluid's flow begins shortly and behaves in a complex manner; therefore, it is difficult to accurately measure the yield shear stress. Accordingly, linear equations for the shear rate and the shear stress are found, and the shear stress when the shear rate is zero is defined as the dynamic yield stress. Generally, the dynamic yield stress is treated as the MR fluid's yield shear stress, and it is expressed as a function of the magnetic field's strength.  
The dynamic yield stress is found, as shown in Equation (3), using a model in which viscosity is affected by the magnetic field strength. The approximate formula constants found here include characteristic eigenvalues that are determined by the solvent used in the MR fluid, the type of particles, the composition ratio, and the composition environment. The MRF-132 DG fluid's intrinsic property values are listed in Table 1[17; 18].  
\[Y=Y_{\infty}+(Y_{0}-Y_{\infty})\Big{(}2e^{-B\sigma_{SY}}-e^{-2B\sigma_{SY}} \Big{)}. \tag{3}\]  
Here, \(Y_{\infty}\), \(Y_{0}\), \(B\), and \(\sigma_{SY}\) correspond to the MRF's magnetic field saturation state, the magnetic field removal state, the magnetic density applied to the MRF, and the saturated moment, respectively.  
For MR fluids, the viscosity coefficient and yield stress must be calculated separately. This can be obtained through the parameters listed in Table 1 and Equation (3). The viscosity coefficient can be obtained through the saturated viscosity coefficient \(\eta_{\infty}\), the natural viscosity coefficient \(\eta_{0}\), and the viscosity coefficient saturation moment \(\sigma_{5y_{l}}\) in the Y term of Equation (3). The yield stress can be obtained through the saturated yield stress \(\tau_{y0}\), the natural yield stress \(\tau_{y0}\), and the yield saturation in the Y term can be obtained with the saturation moment \(\sigma_{5y_{y}}\)[19].