## 6 Conclusions  
Based on the Euler-Bernoulli beam theory, this paper employs the finite element method to establish a dynamic model of the rotor-support system. The vibration equations for the lumped mass element, the squeeze film damper (SFD) element, and the support element are derived, with the linear components incorporated into the stiffness matrix, mass matrix, and gyroscopic matrix. The nonlinear components are treated as loads and written into the load matrix along with the excitation, thus constructing the overall vibration equation of the model. Subsequently, the Campbell diagram and critical speeds of the rotor are solved using both custom code and ANSYS software, and a comparison between the two validates the correctness of the finite element model coded. Considering the impact of nonlinearity, the Newmark Beta method is selected to solve the time-domain response of the rotor system. Four different scenarios were designed: (1) without nonlinear forces, (2) with SFD nonlinear oil film forces, (3) with support nonlinear restoring forces, and (4) the combined effect of SFD nonlinear oil film forces and support nonlinear restoring forces on the rotor response. The latter three scenarios were compared with the first to investigate the influence of nonlinear forces on the rotor model response. From this, the following conclusions are drawn:  
1. [leftmargin=*]
2. Squeeze film dampers can effectively reduce the amplitude of the rotor, and the smaller the oil film clearance \(C\), the more significant the reduction in amplitude. However, it is crucial to select an appropriate range for the oil film clearance to prevent the occurrence of a bi-stable phenomenon, which can compromise the stability of rotor vibration. When the amplitudes of two critical speeds are close, the selection of SFD parameters must be made with caution to avoid an increase in amplitude. The addition of an SFD at the first support resulted in a bi-stable phenomenon, and it was observed that as the oil film clearance \(C\) increased, the frequency range affected by the bi-stable phenomenon gradually decreased until it ceased to be an issue. The introduction of an SFD at the third support led to an increase in amplitude. This was due to the proximity of the second-order critical speed amplitude to the third-order amplitude, under the influence of the SFD, the second-order amplitude decreased while the third-order amplitude increased. 3. Support nonlinear forces induce a frequency shift in critical speeds, with an observed increase in frequency across all three supports. The impact of support nonlinear forces on amplitude varies among the supports. At the first support, the amplitude on third-order critical speed increases with the augmentation of the support nonlinear force coefficient; when \(k_{ml}\) = \(1\times 10^{8}\), the amplitude increases by 4%. At the second support, the third-order critical speed amplitude decreases with the increase in the support nonlinear force coefficient; when \(k_{ml}\) = \(8\times 10^{8}\), the amplitude near the third-order critical speed decreases by 11%. At the third support, both the second and third-order critical speed amplitudes increase with the support nonlinear force coefficient; when \(k_{ml}\) = \(2\times 10^{9}\), the amplitude near the second-order critical speed increases by 17%, with a corresponding increase in critical speed of 3.5 Hz, and the amplitude near the third-order critical speed increases by 11%. 4. The combined effect of the two types of nonlinear forces on the rotor is found to be a simple superposition of the results obtained when each force is applied individually. When both nonlinear forces are added to the first support, the frequency changes are consistent with those observed when only the support nonlinear force is applied, with an increase in amplitude and the disappearance of the bi-stable phenomenon.