## 4 Conclusions  
To assess reliability or durability, it is necessary to understand the procedures that allow for the calculation of the operational parameters of the device being studied. This issue can be considered for the components of the device that are in a state of static equilibrium or in a state of dynamic equilibrium.  
For the purpose of the quantitative evaluation of a damper's operating parameters, a function \(\varepsilon(e,\delta_{1},\delta_{2})\) was introduced to describe the position of the inertia ring relative to the housing. As a result of the conducted studies, it was found that during the operation of the viscous torsional vibration damper, there is a small displacement of the center of the inertia ring relative to the center of the housing (\(\varepsilon>0\)). As a result, the cylindrical surfaces of the inertia ring and the housing are separated by an oil film. The position of the center of the inertia ring is influenced by the geometry of the damper, the mass of the ring, and the properties of the oil. The damping of torsional vibrations generated by the crankshaft-piston system is also related to the relative velocity between the housing and the inertia ring, which is defined as the difference in angular velocities of both components. The energy of the torsional vibrations is damped through oil films separating both the cylindrical and the end surfaces. As a result of the conversion of torsional vibration energy into heat, the temperature of the oil filling the working spaces of the damper increases. The calculations clearly showed the dominant role of the end surfaces in heat dissipation and reductions in torsional vibration energy. For the assumed geometric and physical properties of the device, it was shown that the end surfaces were responsible for dissipating up to 82% of the vibration energy. For the chosen type of oil,  
Figure 10: The power dissipated by the damper as a function of the relative velocity \(\omega\).  
the maximum relative velocity was also determined, beyond which the oil temperature would exceed the permissible value.  
According to this study, the temperature of the oil generated by the oil friction forces should serve as a criterion for designers to determine the operational range of the damper.  
Increasing the relative velocity while simultaneously lowering the working temperature of the oil requires an expansion of the heat dissipation surface area. A possible solution to this problem could be the use of ribbing on the damper housing. It is also crucial to ensure proper air circulation in the engine compartment. In particular, special attention must be paid to avoid the cooling of the engine block causing the accumulation of air with excessively high temperatures in the immediate vicinity of the damper.  
It should also be noted that a near-concentric position of the damper housing and the inertia ring will favor the occurrence of transverse vibrations. The cause of these vibrations may be external forces or the variable radial force generated in the crankshaft-piston system of the internal combustion engine. This issue is an important topic that warrants further investigation.  
Conceptualization, A.M., A.C., and W.H.; methodology, A.M., A.C., and W.H.; software, A.M.; validation, A.C. and W.H.; formal analysis, A.M. and A.C.; investigation, A.M., A.C., and W.H.; resources, A.M., A.C., and W.H.; data curation, A.M. and A.C.; writing--original draft preparation, A.M., A.C., and W.H.; writing--review and editing, A.C.; visualization, A.M., A.C., and W.H.; supervision, A.M.; project administration, A.M.; funding acquisition, W.H. All authors have read and agreed to the published version of the manuscript.  
This research received no external funding.  
Not applicable.  
The original contributions presented in the study are included in the article, further inquiries can be directed to the corresponding author.  
The authors declare no conflicts of interest.