## 2 Materials and Methods  
The vibrations generated in the crank-piston system of internal combustion engines are the result of periodically varying forces, primarily arising from the combustion of the air-fuel mixture. Figure 2 illustrates the distribution of forces stemming from the piston \begin{table}
\begin{tabular}{l l} \hline \multicolumn{2}{c}{**Geometrical quantities and the quantities dependent on them**} \\ \hline \(D_{H,1},R_{H,1}\) & Inner diameter, radius of the housing body \\ \(D_{H,2},R_{H,2}\) & Outer diameter, radius of the housing body \\ \(D_{I,1},R_{I,1}\) & Inner diameter, radius of the inertia ring \\ \(D_{I,2},R_{I,2}\) & Outer diameter, radius of the inertia ring \\ \(D_{1}=\) & Average inner diameter \\ \((D_{H,1}+D_{I,1})/2\) & Average inner diameter \\ \(R_{1}=\) & Average inner radius \\ \((R_{H,1}+R_{I,1})/2\) & Average outer diameter \\ \(D_{H,2}+D_{I,2})/2\) & Average outer diameter \\ \(R_{2}=\) & Average outer radius \\ \((R_{H,2}+R_{I,2})/2\) & Inner and outer width of inertia ring (\(b_{1}=b_{2}\)) \\ \(b_{1},b_{2}\) & Inner and outer width of inertia ring (\(b_{1}=b_{2}\)) \\ \(b_{1}=R_{I,1}-R_{H,1}\) & Inner radial clearance \\ \(\delta_{2}=R_{H,2}-R_{I,2}\) & Outer radial clearance \\ \(A_{B}\) & External area of the damper housing \\ \(e\) & The eccentricity of the housing and the inertia ring \\ \(\epsilon_{1}=e/\delta_{1}\) & Inner relative eccentricity \\ \(\epsilon_{2}=e/\delta_{2}\) & Outer relative eccentricity \\ \(\epsilon=2e/(\delta_{1}+\delta_{2})\) & Resultant relative eccentricity \\ \(\beta\) & The angle between the \(y\)-axis and the line connecting \\  & the center of the housing and the center of the inertia ring \\ \(h_{1}(\varphi),h_{2}(\varphi)\) & Functions of the inner and outer oil film thicknesses \\ \(h_{1,\min},h_{2,\min}\) & Minimum of the inner and outer oil film thicknesses \\ \hline \multicolumn{2}{c}{**Physical quantities**} \\ \hline \(\omega\) & The angular velocity of the inertia ring relative to the housing \\ \(T\) & Temperature \\ \(T_{0}\) & Ambient temperature \\ \(T_{B}\) & Temperature of the damper housing \\ \(T_{B,\lim}\) & The maximum allowable operating temperature of the damper \\ \(\varphi(T)\) & Silicone oil density \\ \(\eta(T)\) & Silicone oil dynamic viscosity \\ \(F\) & Weight of the inertia ring \\ \(F_{L1},F_{L2}\) & Inner and outer hydrodynamic buoyant forces \\ \(f_{1},f_{2}\) & The fluid friction coefficients of the inner and outer film layers \\ \(\alpha\) & The heat transfer coefficient between the damper housing and \\  & the surrounding environment \\ \(M_{0},M_{1},M_{2}\) & The frictional moment generated by the front, inner, and outer \\  & oil film layers \\ \(c_{0},c_{1},c_{2}\) & The damping coefficient of the front, inner, and outer \\  & oil film layers \\ \hline \end{tabular}
\end{table} 
Table 1: _Cont._pressure \(P\) acting on the connecting rod and crankshaft during the combustion stroke. The following trigonometric relationships hold among the indicated forces:  
\[N =P\tan(\gamma) \tag{1}\] \[S =P\cos(\gamma)^{-1}\] (2) \[F_{t} =S\cos(\alpha+\gamma)\] (3) \[F_{r} =S\sin(\alpha+\gamma) \tag{4}\]  
Analyzing the forces at point \(O_{2}\), which describes the instantaneous position of the crankshaft and connecting rod, it can be observed that the radial force \(F_{r}(\alpha,\gamma)\) will induce transverse vibrations, while the tangential force \(F_{s}(\alpha,\gamma)\) will induce torsional vibrations. The angles defining the position of the crankshaft and connecting rod depend on time \(\alpha(t),\gamma(t)\). The transverse vibrations are damped in hydrodynamically lubricated sliding bearings. Meanwhile, torsional vibrations are damped by appropriately selected damper constructions, examples of which are shown in Figure 1.  
In the case of viscous dampers, the free spaces between the housing and the inertia ring are filled with silicone oil, as indicated in Figure 3. Torsional vibration damping in this scenario is the result of internal friction forces acting in the oil.