## 5 Computational Analysis
### Response Analysis with Support Nonlinear Restoring Force  
Building upon the analysis without nonlinear terms, the analysis now incorporates the support nonlinear force to examine its impact on the rotor's amplitude-frequency characteristics, time-domain response, and orbit at the three supports. The form of the SFD nonlinear oil film force is as indicated in Equation (55). An unbalanced force is also 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 restoring force at each of the three supports. Starting with the first support, based on equivalent stiffness calculations, the restoring force coefficient \(k_{bnl}\) becomes significantly pronounced at 1 \(\times\) 10\({}^{8}\). A comparison is made between the results with the restoring force coefficient ranging from 0 to 1e8 to assess the impact of the support restoring force on the rotor's response. As the coefficient of the support nonlinear force increases, both the amplitude and frequency at the support are affected. As shown in Figure 14, at the first critical speed near the support, the amplitude decreases while the critical speed increases. With \(k_{bnl}\) at 1 \(\times\) 10\({}^{8}\), the amplitude is reduced  
Figure 13: Time-domain response and orbit at bearing 3 with SFD. by 6%, and the critical speed is raised by 0.8 Hz. Conversely, at the third critical speed near the support, the amplitude increases while the critical speed also increases. With \(k_{bnl}\) at 1e8, the amplitude increases by 4%, and the critical speed is elevated by 2.5 Hz. Subsequently, a nonlinear restoring force was added to the second support to observe its impact on the rotor's response. Based on equivalent stiffness calculations, the range of the restoring force coefficient \(k_{bnl}\) was set from 1 \(\times\) 10\({}^{8}\) to 8 \(\times\) 10\({}^{8}\). As shown in Figure 15, with the restoring force coefficient increased, the amplitudes near the first three critical speeds all decreased, and the critical speeds also rose. When the restoring force coefficient \(k_{bnl}\) was 8 \(\times\) 10\({}^{8}\), the amplitude near the first critical speed dropped by 3%, and the critical speed increased by 0.5 Hz; the amplitude near the second critical speed decreased by 4%, with no change in the critical speed; and the amplitude near the third critical speed fell by 11%, with the critical speed rising by 1.4 Hz. Finally, a nonlinear restoring force was introduced to the third support to observe its impact on the rotor's response, with the result shown in Figure 16. Based on equivalent stiffness calculations, the range for the restoring force coefficient \(k_{bnl}\) was set between 5 \(\times\) 10\({}^{8}\) and 2 \(\times\) 10\({}^{9}\). As the restoring force coefficient increased, the amplitude near the first critical speed decreased, with no change in the critical speed. However, the amplitudes near the second and third critical speeds increased, with a rise in the second critical speed. When the restoring force coefficient \(k_{bnl}\) is set to 2 \(\times\) 10\({}^{9}\), the amplitude change near the first critical speed is minimal, decreasing by 0.5%, with no variation in the critical speed. Near the second critical speed, the amplitude increases by 17%, and the critical speed rises by 3.5 Hz. Near the third critical speed, the amplitude increases by 11%, while the critical speed remains unchanged. In addition, nonlinear support restoring forces with coefficients \(k_{bnl}\) = 1 \(\times\) 10\({}^{8}\), \(k_{bnl}\) = 8 \(\times\) 10\({}^{8}\), and \(k_{bnl}\) = 2 \(\times\) 10\({}^{9}\) were applied to the three supports, respectively.