## 5 Computational Analysis
### Response Analysis with SFD Nonlinear Oil Film Force and Support Nonlinear Restoring Force  
Building upon the analysis without nonlinear terms, the analysis now incorporates both the SFD nonlinear oil film force and the support nonlinear restoring force to examine their effects on the rotor's amplitude-frequency characteristics, time-domain response, and orbit at the three supports. The forms of the SFD nonlinear oil film force and the support nonlinear force are as indicated in Equations (53) and (55), respectively. Consistent with the scenario without the application of nonlinear forces, the amplitude-frequency characteristics are calculated within the 2000 to 5000 rpm range. An unbalanced force is similarly applied at the sixth concentrated mass. The influence of the support nonlinear restoring force on the amplitude-frequency characteristics at the supports and at the concentrated mass where the unbalance is applied is investigated after imposing the nonlinear forces at each of the three supports. Starting with the first support, based on the conclusions from the previous sections, the parameters \(C\) = 5.52 \(\times\) 10\({}^{-5}\) m and \(k_{bml}\) = 1 \(\times\) 10\({}^{8}\) are selected to assess the impact of the support restoring force on the rotor's response. The results indicate that the effects are additive when both nonlinear forces are applied individually. As shown in Figure 18, after incorporating both types of nonlinear forces, the amplitude near the first critical speed decreases while the critical speed increases, with the amplitude-frequency curve coinciding with the curve obtained when applying the nonlinear support force alone. At the third critical speed, the rotor exhibits an increase in amplitude and critical speed. Additionally, the bi-stable phenomenon induced by the addition of the SFD nonlinear force vanishes, and the amplitude increase caused by the addition of the support nonlinear force is mitigated by the SFD. Subsequently, both nonlinear forces were applied to the second support to observe their impact on the rotor's response. Based on the conclusions from the previous sections, the oil film gap was set to \(C\) = 1.4 \(\times\) 10\({}^{-5}\) m, and the support nonlinear force coefficient was \(k_{bml}\) = 8 \(\times\) 10\({}^{8}\), which was compared with the cases where \(C\) = 1.6 \(\times\) 10\({}^{-5}\) m and \(k_{bml}\) = 8 \(\times\) 10\({}^{8}\) were applied individually. As shown in Figure 19, the results were consistent with those from the first support, indicating that the combined effect of both nonlinear forces is the sum of their individual effects. After applying both nonlinear forces, the amplitude-frequency curves near the first and second critical speeds coincided with the curves obtained when applying the nonlinear support force alone. Near the third critical speed, by narrowing the oil film gap to \(C\) = 1.4 \(\times\) 10\({}^{-5}\) m, the amplitude decreased by 19%, and the peak frequency increased by 2 Hz compared to the scenario without the application of nonlinear forces. Figure 17: Orbit for supports 1, 2, and 3 with restoring force applied. Finally, the third support was subjected to the addition of a nonlinear restoring force to observe its impact on the rotor's response. As shown in Figure 20, near the first critical  
Figure 19: Amplitude–frequency characteristics for bearing 2 with both nonlinear forces applied. Figure 18: Amplitude–frequency characteristics for bearing 1 with both nonlinear forces applied. speed, the curve with the SFD oil film force added coincides with the curve without the application of nonlinear forces, and the curve with both nonlinear forces applied coincides with the curve applying only the support nonlinear force.