## 3 Results and Discussion  
In this work, the computational simulations are presented, and discussed, essentially in two parts, namely, the numerical results of orbits, and the spectrum of the system response (by Discrete Fourier Transform--full spectrum). Two types of radial lobed bearings were used in the system for comparation, which are elliptical and three-lobed bearings. The preload parameter is varied, being 0.25 and 0.8, in order to simulate higher and lower eccentricity conditions, respectively. Thus, combining all cases, it was possible to verify the influence of four different bearing configurations on the simulated system response. Both three-lobe (Equation (6)) and elliptical (Equation (4)) bearing models were considered to calculate the hydrodynamic forces. The orbits were plotted simultaneously and compared for each case, facilitating the discussion.  
The rotating system used is shown in Figure 2. It consists of 18 beam elements (19 nodes) and 1 disk element. The SAE 1020 steel shaft has elastic modulus of 2 \(\times\) 10\({}^{11}\) N/m\({}^{2}\), poison coefficient of 0.3, 7800 kg/m\({}^{3}\) density, 700 mm length, 15 mm diameter. Moreover, it is supported by two hydrodynamic bearings, placed approximately at 580 mm from each other, positioned at nodes 3 and 17, respectively, having a length of 20 mm, an inner diameter of 31 mm with 90 \(\upmu\)m of radial clearance. The lubricant is the ISO VG 32 oil at a constant average temperature of 40 \({}^{\circ}\)C (\(\mu\) = 2.7622 \(\times\) 10\({}^{-2}\) Pa-s). The disk was placed at node 7, positioned 200 mm from bearing 1, with 94.7 mm diameter and 47 mm length. In this case, the unbalanced mass was \(m_{0}=3\) g, located 37 mm away from the center of the disk. In addition to the unbalance, a 200 N external force was applied to the disk node to simulate a severe loading condition. As shown in Equation (12), the structural damping can be proportional to the mass and stiffness matrices, but for steel shafts, the recommendation is to only use a matrix proportional to the stiffness. Therefore, the coefficients used here were \(\alpha=0\) and \(\beta=1.5\times 10^{-5}\) (this value was experimentally estimated in [17]).  
Is worth noting that the procedures used for the simulations--the finite element modelling and time integration scheme, as well as the solution of the Reynolds equation--have been experimentally validated in [17] for a rotor system supported by cylindrical journal bearings. Although the bearing geometries presented are different from the cylindrical one, the procedure used to solve the Reynolds equation is the same of [17].  
Figure 3 shows the pressure field distribution in each bearing (elliptical and trilobed) in order to exemplify its behavior in each analyzed geometry. Preload 0.8 was used, since the pressure peaks can be better visualized in this situation. When considering the elliptical bearing, it is possible to observe the presence of two pressure peaks due to the two lobes. Similarly, for the three-lobe bearing, three pressure peaks are seen due to the three lobes.  
Figure 4 depicts the behavior of the stiffness (Figure 4a) and damping (Figure 4b) coefficients for the elliptical and three-lobe bearings, which are a function of the rotating speed. These coefficients are used in the linear bearing approach. Analyzing the dynamic coefficients, it is possible to note that the cross-coupled coefficients (xy, yx) for the three-lobe case are smaller than for the elliptical case, confirming that three-lobe bearings present greater stability in comparison to the elliptical one, as expected.