## 4 Discussion  
S Scilab [52] was used for solving the Newton equations.  
Figures 8-10 show the trajectories of the journal from the geometrical center of the bearing (x = 0, y = 0) to the equilibrium position for different external load and operating conditions. For the smallest rotational speed (i.e., 328 rad/s), additional masses of 0.25, 0.5, and 1 kg were considered. For the intermediate velocity (i.e., 1500 rad  
Figure 8: Trajectory of the journal (**b**,**d**,**f**) and damping (**a**–**c**) @328 rad/s for different masses: top 0.25 kg–middle 0.5 kg–bottom 1 kg.  
Figure 9: _Cont._  
Figure 10: Cont.  
Figure 9: Trajectory of the journal (**b**,**d**,**f**) and damping (**a**–**c**) @1500 rad/s for different masses: top 1 kg–middle 2.5 kg–bottom 5 kg.  
The right diagrams show the trajectory of the journal from the nominal point (0,0) to the equilibrium position. It can be clearly observed that the equilibrium position does not lie on the vertical axis but is slightly shifted also tangentially. This effect is well known and occurs due to the cavitation that produces an unbalance of the pressures between the convergent and the divergent gaps.  
The diagrams on the left show the amplitude of the oscillations (variation of the eccentricity).  
The equilibrium position and the trajectory calculated with the half-Sommerfeld and the CFD approaches differ. The differences can be significant depending on the rotational speed and the load of the bearing. Figure 11 shows the percentage error in terms of predicted positions. While for the intermediate rotational speed the differences are negligible, for the low and the high velocities the differences can rise to about 30%. This result is aligned with the differences in terms of pressure distributions observed with the two approaches (Figure 3).  
The eccentricity at equilibrium depends both on the load and the rotational speed. An increase of the rotational speed promotes an increase of the drag and lift forces. Consequently, for the same load, the higher the speed, the lower the eccentricity. On the contrary, for a given speed the eccentricity increases together with the mass. This phenomenon is predicted by both approaches (CFD vs. half-Sommerfeld), even if the latter overestimate the eccentricity for a given force.  
Figure 11: Error in the predicted position (**a**) and predicted position (**b**) (eccentricity at equilibrium)–Half-Sommerfeld vs. CFD-Kunz.  
Figure 10: Trajectory of the journal (**b**,**d**,**f**) and damping (**a**–**c**) @3000 rad/s for different masses: top 2.5 kg–middle 5 kg–bottom 10 kg.  
Figure 12 shows the damping times and the related excited frequencies during the transient startup. It can be appreciated that the damping time reduces for higher values of the mass. A higher mass implies a higher acceleration and an offset from the nominal position (0,0). This is in turn related to higher viscous forces that promote the damping. Even for what concerns the excited frequencies the CFD and the Half-Sommerfeld predictions differ. In particular, the latter predicts lower frequencies. This is mainly related to the lower drag forces.  
While the damping time is useful to estimate the wear of the journal related to each stop, the knowledge of the frequencies has the same importance and is fundamental for a preliminary evaluation of the NVH (noise vibration harshness) behavior of the system. The acceleration from the steady state to the operating conditions introduces energy in the system and excites different frequencies \(f\) that, in turn, interact with the eigenfrequencies of the system causing noise and undesired vibrations. For the studied configurations, the excited frequencies results in the range 4-24 Hz.