## 2 Materials and Methods
### Thermal Analysis
The temperature drop for the considered 12 cases **V1**-**V12** along the axis AB in a steady state is about 0.12-0.15 K. Maximum temperatures \(\phi_{max}\) in the working region for all the 12 considered cases are shown in Figure 11.  
Figure 10: Temperature drop along axis AB in a steady state for selected representative variants: **V1**, **V4**, **V9** and **V12**. Figure 9: Spatial temperature distribution in steady state for selected representative variants: **V1**, **V4**, **V9** and **V12**. As predicted, we can conclude that the greater the number of discs, the higher the maximum temperature, which is directly evidenced by Figure 5b, showing a substantial increase in copper losses as the number of discs increases. It needs an explanation as to why, with the same number of discs, the higher maximum temperature is associated with the lower torque clutches and not with the higher torque clutches (related to higher copper losses). The explanation can be based on simplified approximate reasoning from the description of the process of heating a homogeneous body treated as a source of heat that transfers the accumulated heat by convection through the surface of \(S_{air}\) to the surrounding air:  
\[\Delta P_{Cu}dt=c\cdot m\cdot d(\Delta\vartheta)+C_{h}\cdot S_{air}\cdot \Delta\vartheta dt, \tag{9}\]  
where \(\lambda\)--thermal conductivity; \(C_{h}\)--heat convection coefficient; \(c\)--heat capacity; \(m\)--mass; \(S_{air}\)--heat transfer area. The steady-state maximum temperature of such a homogeneous body as \(\vartheta_{max}\) is described by the following formula:  
\[\vartheta_{max}=\frac{\Delta P_{Cu}}{C_{h}\cdot S_{air}}+\vartheta_{0}, \tag{10}\]  
where \(\vartheta_{0}=20\)\({}^{\circ}\)C (293.15 K) is the temperature of the surrounding air. In the case of the considered magnetorheological clutches, the area marked in Figure 12 can be considered as the surface of heat transfers to the environment:  
\[S_{air}=2\pi\cdot\left((r_{ce}+Y^{\prime})^{2}-r_{bc}^{2}\right)+2\pi\cdot(r_ {ce}+Y^{\prime})\cdot(L_{C}+2Y^{\prime}) \tag{11}\]  
where \(r_{ci}\)--internal radius of the coil; \(r_{ce}\)--external radius of the coil; \(r_{bc}\)--external radius of the bearings coupling; \(Y^{\prime}\)--cylindrical and cover yoke thickness. It is the outer area of the cylindrical yoke and 2 outer areas of the cover yokes. How they are changed with the increasing number of discs and increasing clutching torque is shown in Figure 13a,b. As observed in Figure 13a, the area \(A_{conv}\) decreases for all variants with the increase in the number of discs, but with a fixed number of discs (Figure 13b), the area for a higher torque clutch is larger than that for a lower torque clutch. This clearly explains the problem posed at the beginning of the discussion: why, with the same number of discs, the maximum temperature of the higher-powered clutches is lower. Figure 11: Maximum temperature \(\vartheta_{max}\) in the working region (the highest possible temperature for MR fluid) vs. the number of discs N for \(T_{c}=20\), 35 and 50 Nm). Figure 14: Comparison of function courses: (**a**) the course of ratio \(\Delta P_{Cu}/A_{conv}\) vs. the number of discs for \(T_{c}=20,35\) and \(50\) Nm; (**b**) maximum temperature \(\theta_{hmax}\) vs. number of discs N for \(T_{c}=20\), \(35\) and \(50\) Nm. Figure 8:0 The surface of heat transfer to the environment (surrounding air). Figure 8:1 Heat convection areas: (**a**) areas \(A_{conv}\) vs. number of discs N for \(T_{c}=20\), \(35\) and \(50\) Nm; (**b**) areas \(A_{conv}\) vs. clutching torque \(T_{c}\) for the number of discs N = 1, 2, 3, 4. Even better, the compliance of the curves describing the maximum temperature, T, with the curves presenting the ratio \(\Delta P_{Cu}/A_{conv}\) can be seen in the 3D charts (Figure 15).