## 2 Materials and Methods
Oil flow is laminar for \(\mathrm{Re}_{i}=\frac{\rho\omega\cdot D_{i}\cdot\xi}{4\eta}\leq\mathrm{Re}_{i \mathrm{/CR}}\approx\frac{41.2}{\sqrt{\psi_{i}}}\)[48]; the velocity of the oil film near the surface is equal to the velocity of the surface rotating inertia ring or housing; the oil flow rate in the axial direction is much greater than the flow rate of oil flowing in the circumferential direction. 6. Fluid heat exchange takes place by convection [49; 50; 51]. 7. Ambient temperature \(T_{0}=\mathrm{const}\). 8. The heat is absorbed by conduction through the damper housing and discharged to the environment. 9. The quantity describing the heat transfer intensity is the heat transfer coefficient \(\alpha=20\) W\(\cdot\)m\({}^{-2}\cdot\)K\({}^{-1}\)[48]. \begin{table}
\begin{tabular}{l l} \hline \multicolumn{3}{c}{**Parameters of the torsional vibration viscous damper**} \\ \hline
1. The inner radius of the inertia ring & \(R_{I,1}=78.5\) mm \\
2. The outer radius of the housing interior & \(R_{H,2}=130\) mm \\
3. Inner radial clearance & \(\delta_{1}=R_{I,1}-R_{H,1}=0.14\) mm \\
4. Outer radial clearance & \(\delta_{2}=R_{H,2}-R_{I,2}=0.52\) mm \\
5. Width of the inertia ring & \(b_{1}=b_{2}=33\) mm \\
6. External surface of the damper & \(A_{B}=0.105154\) m\({}^{2}\) \\
7. Weight of the inertia ring & \(F=89.6\) N \\
8. The heat transfer coefficient & \(\alpha=20\) W\(\cdot\)m\({}^{-2}\)K\({}^{-1}\) \\ \hline \multicolumn{3}{c}{**The remaining input values for calculation of the damper operating parameters**} \\ \hline
9. Type of silicone oil & M30000 \\
10. Silicone oil density & \(\rho(T)=990-0.8\cdot T\) kg\(\cdot\)m\({}^{-3}\) (Figure 5b) \\
11. Silicone oil dynamic viscosity & \(\eta(T)=\rho(T)10(\frac{793.1}{273+1}-4.082)\) kg \(\cdot\)m\({}^{-1}\)s\({}^{-1}\) \\  & (Figure 5a) \\
12. Ambient temperature & \(T_{0}=70\)\({}^{\circ}\)C \\
13. Relative angular velocity & \(\omega\in[0.2;\,2.0]\) rad\(\cdot\)s\({}^{-1}\) \\ \hline \end{tabular}
\end{table} 
Table 2: Simulation parameters. Figure 5: Viscosity (a) and density (b) functions of silicone oil M30000. With such defined assumptions, the shape of the oil gaps can be approximated using the equation  
\[h_{i}(\varphi)=\delta_{i}(1+\varepsilon_{i}\cos(\varphi)), \tag{5}\]  
and the position of the line connecting the geometric center of the housing and the geometric center of the inertia ring is defined by the angle \(\beta\), described by the following relationship:  
\[\beta=\arctan\sqrt{\frac{\pi(1-\varepsilon^{2})}{4\varepsilon}}. \tag{6}\]  
The resultant relative eccentricity of the inertia ring and the housing is described by  
\[\varepsilon=\frac{2e}{\delta_{1}+\delta_{2}}. \tag{7}\]  
The ability of a viscous damper to attenuate torsional vibrations can be described using the so-called oil flow resistance. If we assume that two surfaces between which the oil is located rotate relative to each other at a certain angular velocity, then the oil flow resistance is defined as the ratio of the generated torque to this angular velocity. In Figure 3, the layers for which the oil flow resistances are defined are indicated. The symbol \(c_{0}\) denotes the flow resistance for a single end-face surface, \(c_{1}\) denotes the flow resistance for the inner cylindrical surface, and \(c_{2}\) denotes the flow resistance for the outer cylindrical surface. For the purposes of further considerations, a polar coordinate system is defined on the \((x,y)\) plane, which is given by the following relationships:  
\[x =-r\sin(\varphi), \tag{8}\] \[y =r\cos(\varphi), \tag{9}\]  
where \(\varphi\) is the angle denoted in Figure 3, and \(r\) is the leading radius.