## 5 Computational Analysis
### Response Analysis with SFD Nonlinear Oil Film Force
Therefore, the range for the SFD oil film gap \(C_{2}\) was set between 1.6 \(\times\) 10\({}^{-5}\) m and 1.8 \(\times\) 10\({}^{-5}\) m. As shown in the figure below, as the oil film gap \(C\) increases, the SFD's restrictive effect on the amplitude diminishes. Compared to the scenario without an SFD, when \(C\) is 1.6 \(\times\) 10\({}^{-5}\) m, the amplitude is reduced by 8.2%, and when \(C\) is greater than 1.8 \(\times\) 10\({}^{-5}\) m, the amplitude change is less than 1%. Referring to Figure 10, at the data point where \(C\) = 1.6 \(\times\) 10\({}^{-5}\) m and the frequency is 76.5 Hz, the time-domain response and orbit are extracted. As shown in Figure 11, after the time \(t\) exceeds 0.1 s, the amplitude of the support stabilizes. The orbit of the support is characterized by a circular trajectory. For the third support, employing the same method as described above, the solution range is selected to be \(C\) = 9.5 \(\times\) 10\({}^{-6}\) m to 1 \(\times\) 10\({}^{-5}\) m. When the oil film width \(L\) is 0.005 m, it is not feasible to solve for cases where \(C\) is less than 9.8 \(\times\) 10\({}^{-6}\) m; hence, the oil film width \(L\) is reduced to 0.003 m. When the SFD is applied to the third support, the second-order critical speeds are all increased, but the amplitude changes vary. At \(C\) = 9.5 \(\times\) 10\({}^{-6}\) m, the SFD effect is most optimal, with the critical speed increasing by approximately 1 Hz and the amplitude decreasing by 0.6%. As the oil film gap \(C\) increases, the increase in critical  
Figure 8: Amplitude–frequency characteristic curve with bi-stable phenomenon for bearing 1 with SFD. Figure 9: Time-domain response and orbit at bearing 1 with SFD. speed diminishes, and the amplitude actually rises, which is a situation to be avoided in the design of the SFD. Referring to Figure 12, at the data point where \(C\) = 9.5 \(\times\) 10\({}^{-6}\) m and the frequency is 66.4 Hz, the time-domain response and orbit are extracted. As shown in Figure 13, after the time \(t\) exceeds 0.2 s, the amplitude of the support stabilizes. The orbit of the support manifests as a circular trajectory. Figure 11: Time-domain response and orbit at bearing 2 with SFD. Figure 10: Amplitude–frequency characteristics for bearing 2 with SFD. Below is a summary of our findings thus far:  
1. Under ideal conditions, the addition of an SFD at the support can serve to reduce amplitude and increase critical speeds. However, improper design of SFD parameters may not only fail to reduce the amplitude but could also lead to an increase in amplitude and even induce a bi-stable phenomenon, thereby enhancing the overall vibration and destabilizing the rotor's operating state. 2. After the addition of an SFD to the first support, a bi-stable phenomenon occurred near the third critical speed. It was observed that as the oil film gap \(C\) increased, which corresponds to a decrease in eccentricity, the bi-stable phenomenon gradually diminished until it disappeared. Thus, a bi-stable phenomenon is likely to occur when the length-to-diameter ratio is 0.2 and the eccentricity is greater than 0.95, and it ceases as the eccentricity decreases. 3. Following the addition of an SFD to the second support, the amplitude was somewhat reduced. As the SFD oil film gap \(C\) increased, which implies a decrease in eccentricity, the change in amplitude became less pronounced. Compared to the scenario without an SFD, when \(C\) exceeds 1.8 \(\times\) 10\({}^{-5}\) m, the amplitude variation is less than 1%. 4. After the addition of an SFD to the third support, the amplitude near the second critical speed was somewhat reduced, and the critical speed itself increased. When the length-to-diameter ratio is 0.12 and the eccentricity is 0.99, the amplitude decreased.