## 4 Results and Discussion  
A modified rotor model based on the framework presented in [27] is implemented to investigate the dynamic response characteristics and stability behavior under base motion excitation. As shown in Figure 8, the rotor model consists of an offset disk rotor system. The system incorporates a SFD with centralized elastic support at the left bearing position, while the right bearing position utilizes linear elastic support. The primary structural parameters of the rotor system are as follows: the shaft has a total length of 0.5994 m, with an inner and outer radius of 20.65 mm and 30.15 mm, respectively. The offset disk is positioned at \(L/3\) distance from the left support, with an inner and outer radius of 30.15 mm and 101.6 mm, respectively, and a thickness of 50.8 mm. The rotor system is constructed from a single material with the density \(\rho=7810\) kg/m\({}^{3}\), elastic modulus \(E=211\) GPa, and shear modulus \(G=81.783\) GPa. In the finite element model, the shaft is uniformly divided into 20 elements with 21 nodes, where SFD is located \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Mass Unbalance** & \multicolumn{2}{c}{**Literature [27]**} & \multicolumn{2}{c}{**The Proposed Model**} \\ \((\mathbf{g\cdot mm})\) & \(\mathbf{\tilde{\Omega}}\) & \(\boldsymbol{u_{max}}\) & \(\mathbf{\tilde{\Omega}}\) & \(\boldsymbol{u_{max}}\) \\ \hline
144.02 & 1.0641 & 1.0014 & 1.0565 & 1.0048 \\  & 1.0578 & 1.0098 & 1.0556 & 1.0351 \\ \hline
360.04 & 1.3124 & 1.0755 & 1.2820 & 1.0047 \\  & 1.1281 & 1.0514 & 1.1245 & 1.0189 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Maximum Floquet multipliers at the bifurcation points. Figure 7: Dimensionless amplitude–frequency response curve at nodal point of: (**a**) SFD ([PERSON] et al. [27]); (**b**) disk ([PERSON] et al. [27]); (**c**) SFD (the proposed model); (**d**) disk (the proposed model). at node 1, and the disk element is positioned at node 8. The primary parameters of the SFD are listed in Table 2, where oil viscosity \(\mu\) and oil clearance \(C\) are fixed parameters, while the film length \(L_{sfd}\) is determined by the length-to-diameter ratio \(L_{sfd}/D\). The eccentricity of the unbalanced mass at the disk node is denoted by \(U\). Setting the harmonic order as H = 3, modal analysis reveals the first two critical speeds of the rotor system as \(\Omega_{1X}=811.3\) rad/s and \(\Omega_{2X}=811.3,2784.6\) rad/s. The system response is characterized through dimensionless parameters: the dimensionless speed is expressed as \(\widetilde{\Omega}=\Omega/\Omega_{1X}\), and the normalized displacement components in horizontal and vertical directions are denoted as \(\widetilde{y}=y/C,\widetilde{z}=z/C\), respectively. The dimensionless deflection is defined as \(\widetilde{r}=\sqrt{y^{2}+z^{2}}/C\). Given that base pitching and yawing motions induce speed-dependent static displacement at the SFD node, the analysis focuses exclusively on the first-mode response. The investigation examines the normalized displacement amplitudes at both SFD and disk nodes, with characteristic amplitudes determined from the half-peak values of each harmonic component, excluding the static (zero-order) term. Figure 9 illustrates the harmonic response curves in the horizontal direction at both SFD and disk nodes under fixed base conditions, with a clearance-to-radius ratio of \(C/R=0.485\%\) and unbalanced magnitude of \(U=360.04\) g \(\cdot\) mm. Unstable solutions are denoted by dashed lines. At bifurcation points, the system exhibits characteristic saddle-node bifurcation behavior: minor perturbations may trigger state transitions from unstable to other stable solutions. This jump phenomenon is attributed to the hardening spring characteristics of the SFD [8], where nonlinear oil film stiffness increases with amplitude.