## 3 Results and Discussion
### Numerical Results of Orbits  
The numerical analyses made here compare the orbits of shaft inside the bearings. Since the rotor is asymmetric to the presence of a decentralized disk, as seen in Figure 2, bearing 1 was chosen for viewing the orbits, since gyroscopic effects are more evident there. The first critical speed of the system is about 37 Hz. Thus, five rotating speeds were selected to evaluate the orbit behavior in conditions of high eccentricity (below first critical speed), low eccentricity (above first critical speed) and high amplitudes (around first critical speed) being 15, 30, 35, 45 and 60 Hz. Additionally, the Sommerfeld number range goes from 0.049 to 0.2. Figure 3: Pressure field distribution—(**a**) Elliptical bearing; (**b**) Three-lobe bearing. Figure 2: Sketch of the rotor used in the simulations (out of scale, for better visualization). Figure 5 shows the orbits results for the elliptical bearing with preload of 0.25. A total simulated time of 20 s is used, in which only the last second of the converged orbit is plotted. It is possible to visualize the \"self-centering\" effect, in which with increasing rotating speed the shaft tends to be centered in the radial clearance of the bearing. Additionally, the major discrepancies between linear and nonlinear models occur near the first critical speed of the system (35 Hz, green lines), where the vibration amplitude is higher. In this case, either shape or size can be different when considering the nonlinear model. For the other rotating frequencies, just slight differences can be observed between the models, maintaining similar orbit shapes and positions. By increasing the preload level to 0.8, the shapes of the orbits remained similar, except for 35 Hz, which has differences due to the proximity with the critical speed (Figure 6). However, the orbits are smaller than the previous case, since the higher the preload, the smaller the radial clearance in the vertical direction, which creates a stiffer oil film and, consequently, reduces the vibration amplitudes. Additionally, it is interesting to note that Figure 6 shows a zoom in the whole region containing the orbits. Figure 4: Dynamic Coefficients for elliptical and three-lobe bearings—(**a**) Stiffness; (**b**) Damping. Changing the bearing type, Figure 7 shows orbits for the three-lobed bearing with preload 0.25. In this case, differences in size can be observed at all rotating speeds, indicating a greater influence of nonlinearities in this bearing geometry. At 35 Hz, near the first critical speed, the discrepancies in size are ever higher, and it is possible to observe differences in shape and position as well. Figure 5: Orbits for all rotations simulated in elliptical bearing 1 (preload 0.25), with zoom in rotation 15 Hz. Figure 6: Orbits for all rotations simulated in elliptical bearing 1 (preload 0.8), with zoom in rotation 15 Hz. For the three-lobed bearing with preload 0.8, the central position of the orbits remained the same for all rotating speeds (comparing linear and nonlinear approaches), as shown in Figure 8. However, the orbits' size and shape change at all rotations in an evident way. In the proximity of the critical region (35 Hz), the amplitude of the orbit increases, but as stated, maintaining the same position. This shows that the nonlinearity substantially affects three-lobed bearings and has an even greater effect in the case of high preload. Figure 8: Orbits for all rotations simulated in three-lobed bearing 1 (preload 0.8), with zoom in rotation 15 Hz. Figure 7: Orbits for all rotations simulated in three-lobed bearing 1 (preload 0.25), with zoom in rotation 15 Hz. According to results obtained by the calculation of the orbits, the conclusion is that for elliptical bearings the nonlinearities are only influential in cases of higher orbit amplitudes, i.e., in the vicinity of the critical speed. This behavior is similar to that observed in cylindrical bearings, as shown in the work of reference [17].