## 2 Materials and Methods
### Reynolds Equation  
The Reynolds equation used in this work considers a viscous fluid, and its formulation takes into account the momentum conservation and mass conservation equations. Its solution, in case of hydrodynamic bearings, provides the fluid pressure distribution, the information needed to analyze the vast majority of bearing problems. It comes from the Navier-Stokes equations, adopting some simplifying hypotheses for the model [4]. Equation (1) expresses the classical form of the isothermal Reynolds equation for a dynamically loaded journal bearing, where \(\theta\) is the circumferential coordinate, \(z\) is the axial coordinate, \(\mu\) is the fluid viscosity, \(h\) corresponds to the oil film thickness, \(P\) is the pressure, \(U\) the relative tangential velocity between journal and bearing surfaces and \(t\) is time. \[\frac{\partial}{\partial\theta}\left(h^{3}\frac{\partial P}{\partial\theta} \right)+\frac{\partial}{\partial z}\left(h^{3}\frac{\partial P}{\partial z} \right)=6\mu U\frac{\partial h}{\partial\theta}+12\mu\frac{\partial h}{ \partial t} \tag{1}\]  
The film thickness is shown in Equation (2), where \(C_{r}\) is the radial clearance and \(n\) is the parameter called \"eccentricity ratio\" given by \(n=e/C_{r}\). The parameter \(e\) is the \"eccentricity\" and represents the distance between the center of the bearing and the center of the shaft. Figure 1 shows two types of lobed radial bearing geometries, being (Figure 1a) an elliptical bearing and (Figure 1b) a three-lobed bearing. The angle \(\phi\), defined between the Y direction and the line of centers \(O_{b}\) and \(O_{j}\), in the same direction as the \(\theta\) coordinate, is called \"attitude angle\". \(W\) is the load referring to rotor weight. \[h(\theta)=C_{r}(1+n\cos\theta) \tag{2}\]  
For the solution of the Reynolds equation, the approach used in the work of [PERSON] and [PERSON] [28] is the basis for the numerical model for bearings applied in this paper. This approach utilizes the finite volume method, employing a uniform mesh. Regarding the boundary conditions, atmospheric pressure is considered at the axial edges of the bearing. Near the minimum film thickness location, the Swift-Steinber boundary condition (also known as Reynolds' condition) is applied along with a relaxation process that makes the negative pressures null after each iteration, assuming no cavitation inside the bearing, as showed in Equation (3). \[\left\{ \begin{array}{lcl}p=0&if&p_{iter}<0\\ p=p_{iter}&if&p_{iter}\geq 0\end{array} \right. \tag{3}\]  
Also, as previously seen, the oil film thickness \(h\) is a fundamental parameter to solve the Reynolds equation, and the whole analysis depends on its clearance. Its calculation differs for each bearing type. For obtaining the film thickness, it is necessary to introduce two new variables, which are called the \"eccentricity ratio\" and the \"equilibrium angle\" for each radial bearing lobe. Two circular arcs from the elliptical bearings, being their centers located on the same line and displaced from the center of the bearing by a distance known as \"ellipticity\" \(\varepsilon\). Defining \(m=\varepsilon/C_{r}\) as the \"ellipticity ratio\", the \"eccentricity ratios\" for the bottom (1) and top (2) lobes are given by Equation (4). With that, it is also possible to write the \"equilibrium angles\" for each lobe, as follows in Equation (5). \[n_{1}=\sqrt{n^{2}+m^{2}+2 nm\cos\phi};\quad n_{2}=\sqrt{n^{2}+m^{2}-2 nm\cos\phi} \tag{4}\]  
\[\alpha_{1}=\text{arcsen}\bigg{(}\frac{n\text{sen}\phi}{n_{1}}\bigg{)};\quad \alpha_{2}=\text{arcsen}\bigg{(}\frac{n\text{sen}\phi}{n_{2}}\bigg{)} \tag{5}\]  
Trilobed bearings consist in the intersection of three eccentric lobes, where the center of each lobe has the same distance from the center of the bearing (ellipticity). This configuration allows, in most cases, the formation of three hydrodynamic pressure wedges. The eccentricity ratios for the bottom (b), right (r) and left (l) lobes are given as Equation (6).