## 4 Conclusions  
The paper presents comparative results of linear and nonlinear models for elliptical and three-lobed hydrodynamic bearings, operating under severe loading conditions. The analysis included the comparison of the orbits (by the numerical integration of the system in time-domain) and signal amplitudes in the frequency domain (by DFT full spectrum).  
Regarding the presented orbits, the results showed that for elliptical bearings, the situations with the greatest presence of nonlinearities occur with high vibration amplitudes, near the critical speed region, whereas for three-lobed bearings, the nonlinearities have greater influence even in smaller vibration amplitude situations.  
An important point was seen in the analysis of the full spectrum, which showed that nonlinearities are not always evidenced by the appearance of higher order harmonic components but can also be presented through a significant increase in the \(-1\times\) component.  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Rot. Speed**} & \multicolumn{8}{c}{**Amplitudes (m)**} \\ \cline{2-13}  & \multicolumn{4}{c}{**Linear**} & \multicolumn{8}{c}{**Nonlinear**} \\ \cline{2-13}  & \(-3\times\) & \(-2\times\) & \(-1\times\) & \(1\times\) & \(2\times\) & \(3\times\) & \(-3\times\) & \(-2\times\) & \(-1\times\) & \(1\times\) & \(2\times\) & \(3\times\) \\ \hline
15 Hz & - & - & \(9.82\times 10^{-9}\) & \(3.08\times 10^{-8}\) & - & - & - & - & \(1.96\times 10^{-8}\) & \(3.84\times 10^{-8}\) & - & - \\
30 Hz & - & - & \(3.73\times 10^{-8}\) & \(1.95\times 10^{-7}\) & - & - & - & - & \(9.50\times 10^{-8}\) & \(2.24\times 10^{-7}\) & - & - \\
35 Hz & - & - & \(1.20\times 10^{-7}\) & \(7.04\times 10^{-7}\) & - & - & - & - & \(3.29\times 10^{-7}\) & \(8.06\times 10^{-7}\) & - & - \\
45 Hz & - & - & \(3.57\times 10^{-8}\) & \(2.71\times 10^{-7}\) & - & - & - & \(9.36\times 10^{-8}\) & \(2.52\times 10^{-7}\) & - & - \\
60 Hz & - & - & \(6.83\times 10^{-9}\) & \(6.99\times 10^{-8}\) & - & - & - & - & \(3.56\times 10^{-8}\) & \(1.02\times 10^{-7}\) & - & - \\ \hline \hline \end{tabular}
\end{table}
Table 4: Magnitudes of the most relevant peaks of Figure 12—three-lobed bearing (preload 0.8).  
Figure 12: DFT full spectrum of simulated results in three-lobed bearing 1 (preload 0.8): (**a**) linear, (**b**) nonlinear.  
Finally, the combined analyses presented here are of great importance for mapping the influence of nonlinearities in different types of multi-lobed hydrodynamic bearings with fixed geometry. As the number of lobes, and/or the preload value, increases, the vibration levels decrease, generally pointing to a more linear bearing. Consequently, the traditional linear analysis could be employed without major drawbacks. This study is important for monitoring and fault diagnosis, mainly those that are model based since representative mathematical models are mandatory for a reliable diagnosis.  
Conceptualization, C.A.A.V., D.S.A. and T.H.M.; methodology, C.A.A.V., D.S.A. and T.H.M.; software, C.A.A.V. and D.S.A.; validation, C.A.A.V., D.S.A. and T.H.M.; formal analysis, C.A.A.V., D.S.A. and T.H.M.; investigation, C.A.A.V., D.S.A. and T.H.M.; resources, T.H.M.; data curation C.A.A.V., D.S.A. and T.H.M.; writing--original draft preparation, C.A.A.V.; writing--review and editing, D.S.A. and T.H.M.; visualization, C.A.A.V. and D.S.A.; supervision, D.S.A. and T.H.M.; project administration, D.S.A. and T.H.M.; funding acquisition, C.A.A.V., T.H.M. All authors have read and agreed to the published version of the manuscript.  
Coordination of Superior Level Staff Improvement--CAPES, National Council for Scientific and Technological Development--CNPq, grant # 242899/2018-3, and grants # 2017/07454-8 and # 2018/21581-5 from the Sao Paulo Research Foundation (FAPESP).  
The authors declare no conflict of interest.