## 2 Materials and Methods
This phenomenon occurs both in the external and internal oil films, as well as in the oil located between the end faces of the inertia ring and the housing. As the frictional energy is converted into thermal energy, an increase in the oil temperature inside the damper can be observed. Heat is dissipated into the surroundings through the housing components. Under the influence of the periodically varying tangential force \(F_{s}(\alpha,\gamma)\), the center of inertia \(O_{I}\) moves in a pendulum-like manner along a specific trajectory, as shown in Figure 4. The limitation for proper damper operation is the eccentricity at which the housing temperature reaches the permissible value determined by the thermophysical properties of the oil. The pendulum motion occurs at a variable angular velocity \(\omega\), resulting in a change in the position of the inertia ring center \(O_{I}\) and a corresponding change in the generated heat energy. Consequently, for each point on the trajectory of the inertia ring center, we can assign its position, velocity, the value of the hydrodynamic lift force, and the amount of thermal energy generated in the oil films. After reviewing the available literature, we  
Figure 2: Distribution of forces in the crank–piston system of internal combustion engines, description of the quantities shown in the figure: \(P\)—the resultant force acting on the piston, \(S\)—the component force acting along the connecting rod axis, \(N\)—the component force perpendicular to the longitudinal axis of the cylinder sleeve, \(S\)—the tangential force relative to the connecting rod axis, \(F_{t}\)—the component of the force \(S\) perpendicular to the crank radius, \(F_{r}\)—the component of the force \(S\) parallel to the crank radius, \(|O_{1}O_{2}|=L\)—the length of the connecting rod, \(|OO_{2}|=R\)—the crank radius. observed that when assessing the operational parameters of such dampers, the changes in the position of the inertia ring center relative to the housing are often disregarded, treating their mutual position as concentric. The aim of this article is to investigate the impact of changes in the position of the inertia ring center \(O_{I}\) relative to the housing center \(O_{H}\) and the oil flow velocity on the operational properties of the oil films, taking into account the thermal energy generated by the oil motion in the annular torsional vibration damper. Additionally, it was decided to determine the contributions of the cylindrical and end-face surfaces of the damper housing in dissipating thermal energy. To achieve the aforementioned objectives, suitable mathematical models of the physical phenomena occurring during damper operation were developed. For the numerical modeling of the operation of a viscous torsional vibration damper, assumptions were made concerning the geometric parameters and physical properties of the housing, the inertia ring, and the silicone oil. The necessary quantities and their descriptions are presented in Table 2.  
\[N =P\tan(\gamma) \tag{1}\]0 Trajectory of the displacements of the inertia ring center \(O_{I}\) relative to the housing center \(O_{H}\). \[N =P\tan(\gamma) \tag{1}\]1 The geometry of the damper with designated oil layers and their corresponding resistance coefficients. Based on [43; 47], the following assumptions were also made:  
1. Housing temperature, \(T_{\rm B}\). 2. Minimum height of inner and outer oil film, \(h_{\rm min,1}\) and \(h_{\rm min,2}\). 3. The damper's construction elements are non-deformable, perfectly smooth, axial parallel, cylindrical surfaces, and their material is homogeneous. 4. Oil is an incompressible Newtonian fluid; its viscosity is a function of temperature \(\eta=\eta(T)\); the pressure generated in the oil film may take values greater than or equal to zero; the pressure in the direction of the radial variable \(y\) is constant and the ambient pressure is constant. 5.