## 5 Computational Analysis
### Response Analysis with SFD Nonlinear Oil Film Force  
Expanding upon the analysis conducted without considering nonlinear elements, the SFD nonlinear oil film force is integrated into the analysis to assess its impact on the rotor's amplitude-frequency characteristics, time-domain responses, and orbits at each of the three support locations. Referring to the amplitude-frequency characteristic diagram from the linear scenario, the SFD oil film gap is fine-tuned for each support to ensure that the oil film gap \(C\) exceeds the maximum displacement \(e\) observed at the rotor's supports. The formulation for the SFD nonlinear oil film force is depicted in Equation (53). Adhering to the same rotational speed range of 2000 to 5000 rpm as in the linear case, the amplitude-frequency characteristics are recalculated. An unbalanced force is similarly applied to the sixth concentrated mass. The specific parameters for SFD implementation are detailed in Table 1.  
The aforementioned parameters in Table 2 do not include the oil film gap \(C\). Based on the amplitude-frequency response diagram without nonlinear terms, the oil film gap for the SFD applied at the first support \(C_{1}\) ranges from 5.45 \(\times\) 10\({}^{-5}\) m to 6 \(\times\) 10\({}^{-5}\) m. For the second support, the oil film gap \(C_{2}\) ranges from 1.6 \(\times\) 10\({}^{-5}\) m to 1.8 \(\times\) 10\({}^{-5}\) m, and for the third support, the oil film gap \(C_{3}\) is 1 \(\times\) 10\({}^{-5}\) m.  
For the first support, calculations were performed based on the scenario without an SFD, assigning an oil film gap \(C\) greater than its maximum amplitude value. Within the range of \(C\) = 5.45 \(\times\) 10\({}^{-5}\) m to 5.5 \(\times\) 10\({}^{-5}\) m, a bi-stable phenomenon was observed, as illustrated in the figure below. As \(C\) increased, the bi-stable phenomenon gradually diminished. Further increasing \(C\) to 6e\(-\)5 m resulted in an insignificant effect from the SFD; hence, the range of \(C\) = 5.45 \(\times\) 10\({}^{-5}\) m to 6 \(\times\) 10\({}^{-5}\) m was selected for computation. The following figure displays the amplitude-frequency characteristic curve exhibiting the bi-stable phenomenon. Upon further increasing the oil film gap \(C\), the bi-stable phenomenon has ceased to exist once \(C\) reaches 6 \(\times\) 10\({}^{-5}\) m. Moreover, the SFD exerts a minimal influence on the response and amplitude of the first three critical speeds. When the oil film gap \(C\) varies between 5.5 \(\times\) 10\({}^{-5}\) m and 6 \(\times\) 10\({}^{-5}\) m, the amplitude changes are less than 0.3%, and the frequency changes are less than 0.5 Hz. Referring to Figure 8, at the data point where \(C\) = 5.5 \(\times\) 10\({}^{-5}\) m and the frequency is 77.1 Hz, the extracted time-domain response and orbit are illustrated in Figure 9. After the time \(t\) exceeds 1 s, the amplitude of the support begins to fluctuate between two values and stabilizes after approximately 3 s, indicative of the bi-stable phenomenon. The orbit of the support is characterized by a circular motion that oscillates between an outer and an inner loop. \begin{table}
\begin{tabular}{c c c} \hline
**Oil Film Dynamic Viscosity (Pa-s)** & **Oil Film Radius R (m)** & **Oil Film Width L (m)** \\ \hline
0.02976 & 0.025 & 0.005 \\ \hline \end{tabular}
\end{table} 
Table 2: SFD parameters. For the second support, calculations were initially performed without an SFD, assigning an oil film gap \(C\) larger than its maximum amplitude value. It was observed that the amplitude decreased; however, the effect of the SFD became negligible when the oil film gap exceeded 1.8 \(\times\) 10\({}^{-5}\) m. Subsequently, calculations were made with a smaller oil film gap \(C\) based on the amplitude when \(C\) was 1.8 \(\times\) 10\({}^{-5}\) m. It was found that when the oil film gap \(C\) was less than 1.6 \(\times\) 10\({}^{-5}\) m, the eccentricity distance became greater than the oil film gap, which is not practical.