## 3 Results  
The operating temperature calculations for the viscous torsional vibration damper were conducted for the specified parameters presented in Table 2. For the study, it was assumed that the damper was filled with silicone oil M30000, and the weight of the inertia ring was \(F=89.6\) N. The partial differential equations were solved numerically using the finite difference method implemented in the Matlab software (Version R2024a). The convergence criterion adopted was the relative error, which should not exceed 1% between iterations. As a result of solving the system of Equations (5) and (6), which constitute the mathematical model of the damper, parameters describing its operational properties were obtained, including the relative position of the inertia ring center \(O_{i}\) and the housing center \(O_{H}\), expressed by \(\varepsilon_{i}(\omega)\); the minimum heights of the oil film \(h_{i}(\omega,\varepsilon)\); the oil flow resistances \(c_{0}(\omega),c_{1}(\omega,\varepsilon)\), and \(c_{2}(\omega,\varepsilon)\); the friction torques in the oil films \(M_{0}(\omega),M_{1}(\omega,\varepsilon)\), and \(M_{2}(\omega,\varepsilon)\); the housing temperature \(T_{B}(\omega,\varepsilon)\); and the heat generated by friction forces in the oil films \(P_{l}(\omega,\varepsilon)\). The resulting parameters described by these functions are presented in the graphs. The calculated values of the relative eccentricities for the internal oil film \(\varepsilon_{1}\) and the external oil film \(\varepsilon_{2}\) as a function of the relative velocity \(\omega\) are presented in Figure 6. Theseeccentricities take the following values: \(\varepsilon_{1}\in[0.081;0.260]\) and \(\varepsilon_{2}\in[0.021;0.070]\). An increase in the relative velocity \(\omega\) causes a decrease in the values of \(\varepsilon_{i}\). As the velocity \(\omega\) increases, the center of the inertia ring also shifts toward the center of the housing. For relative velocity values in the range \(\omega\in[0.2;2.0]\) s\({}^{-1}\), the values of the minimum oil film heights fall within the following ranges: \(h_{1,\min}\in[0.104;0.124]\) mm and \(h_{2,\min}\in[0.483;0.504]\) mm. Figure 7 shows the effect of the resultant relative eccentricity \(\varepsilon\) on the angle \(\beta\), defining the position of the centerline between the inertia ring and the housing. Their calculated values for the given range of relative velocities are as follows: \(\varepsilon\in[0.11;0.03]\) and \(\beta\in[81.9^{\circ};87.5^{\circ}]\). Analyzing the behavior of the function \(T_{B}(\omega)\) presented in Figure 8a, it can be observed that an increase in the relative velocity \(\omega\) results in a rise in the housing temperature \(T_{B}\) of the torsional vibration damper. Conversely, an increase in the resultant relative eccentricity  
Figure 6: Relative eccentricities of the internal and external oil films as a function of the relative velocity of the inertia ring. Figure 7: The total relative eccentricity \(\varepsilon\) as a function of the angle of the centerline position \(\beta\). \(\varepsilon\) contributes to a decrease in the housing temperature, as shown in Figure 8b. Assuming the permissible operating temperature of the oil to be \(T_{B,\text{lim}}=90\)\({}^{\circ}\)C, the maximum value of the relative velocity and the minimum resultant relative eccentricity were determined to be \(\omega=0.7\) s\({}^{-1}\) and \(\varepsilon=0.0452\), respectively. From the developed model, it is therefore possible to infer the maximum allowable relative angular velocity \(\omega\) and the minimum allowable value of the resultant relative eccentricity \(\varepsilon\) for a working damper. The use of the numerical model allows for the creation of graphs analogous to those shown in Figure 8 for specific damper dimensions and oil viscosity. This provides the designer with clear information regarding the permissible operating range of the damper.