## 3 Analysis of Torque Performance  
### Analysis of MRF Flow  
Figure 2 shows the shear model between disks and MRF. Consider two discs with radius \(r\) and distance \(h\) (\(h<<r\)), where the working gap between the discs is filled with MRF. The right disc rotates along the \(z\)-axis with angular velocity \(\omega_{1}\) and the MRF between the discs is sheared, which drives the output disc to rotate with angular velocity \(\omega_{2}\). As the working disk in the sudden change of speed, the impact of the working disk rotation on the fluid with time changes shows a certain hysteresis. The inertial force and vibration of the MRF during the drive process have some influence on the flow of the MRF, which increases the uncertainty of the MRF transient analysis [15]. Therefore, to negate the effect of response time on MRF flow, the steady-state assumption is used to analyze the parallel flow of the MRF in the working gap between the two disks. The following conditions are assumed: (1) the MRF is considered incompressible; (2) the flow of the MRF is steady-state; (3) there is no flow of the MRF in the radial direction; (4) the flow of the MRF is only a function of the radius; (5) ignore the volume force; (6) the magnetic field strength is uniformly distributed in the working gap; (7) the pressure in the MRF is along the thickness direction invariant; and (8) ignore the inertial force. According to the above assumptions, in the cylindrical coordinate system (\(r\), \(\theta\), \(z\)), the distribution of the flow velocity is assumed to be  
\[v_{r}=0,\ v_{\theta}=r\omega(z),\ v_{z}=0 \tag{1}\]  
where \(v_{r}\), \(v_{\theta}\) and \(v_{z}\) are the velocities along the radius direction, rotation direction and thickness direction, respectively, and \(\omega(z)\) is the angular velocity of the MRF in the thickness direction, which is a function of \(z\). Figure 1: Structure diagram of an MR transmission. (**a**) Schematic diagram, (**b**) Structure diagram. Figure 2: Circumferential flow of magnetorheological fluid (MRF) between two discs. Assume that the pressure gradient along the rotation direction is \(\mathrm{d}p/\mathrm{d}\theta\). For constant flow, \(\partial v_{\theta}/\partial t=0\), there is no change in the \(r\) direction. The flow equation along the \(\theta\) direction can be expressed as [16]  
\[\frac{\mathrm{d}^{2}\omega(z)}{\mathrm{d}z^{2}}=\frac{1}{\eta}\frac{\mathrm{d }p}{\mathrm{d}\theta} \tag{2}\]  
where \(\eta\) is the apparent viscosity of the MRF, which is related to the magnetic field strength \(H\), and can be expressed as:  
\[\eta=\frac{\tau_{z\theta}}{\dot{\gamma}} \tag{3}\]  
where \(\tau_{z\theta}\) is the dynamic shear yield stress of the MRF and \(\dot{\gamma}\) is the shear strain rate. Integrating Equation (2):  
\[\omega(z)=\frac{z^{2}}{2\eta}\frac{\mathrm{d}p}{\mathrm{d}\theta}+c_{1}z+c_{2} \tag{4}\]  
The flow boundary condition is:  
\[\left\{ \begin{array}{ll}\omega(z)=\omega_{1}&z=0\\ \omega(z)=\omega_{2}&z=h\end{array} \right. \tag{5}\]  
After determining the integration constant according to the boundary conditions, the distribution of the flow velocity along the thickness direction of the MRF can be expressed as:  
\[\omega(z)=\omega_{1}+\frac{(\omega_{2}-\omega_{1})z}{h}+\frac{1}{2\eta}\frac {\mathrm{d}p}{\mathrm{d}\theta}\Big{(}z^{2}-zh\Big{)} \tag{6}\]  
As can be seen from Equation (6), the flow of magnetorheological fluid can be divided into three forms according to different values of pressure gradient: (1) unyielding, the rigid-plastic flow of the MRF; (2) the shear-yielding flow of the MRF caused by the relative rotational speed of the two discs, with a linear distribution of flow velocity along the thickness direction of the MRF; and (3) the pressure flow of the MRF caused by the squeezing pressure, with a parabolic flow velocity along the thickness direction of the MRF distribution. The case where the MRF is used for continuous flow is discussed below.