## 3 Computational Methods and Validation
### Method Verification  
To validate the reliability of the proposed base motion rotor model and its stability analysis method, a linear single-disk rotor system, originally established by [PERSON] [34] and subsequently analyzed for stability by [PERSON] and [PERSON] [35], is selected as a comparative case study. The comparative analysis is conducted under operating conditions with rotor speed \(\Omega=250\) rad/s and base pitching harmonic motion angular velocity \(\omega_{by}=-A_{0}\omega_{b}\text{sin}(\omega_{b}t)\text{rad/s}\), where \(\omega_{b}\) represents the base harmonic motion frequency. Figure 6 presents a comparison of stability diagrams obtained through both methods, where the blue-shaded regions denote zones of system instability. The results demonstrate that  
Figure 5: Three types of Floquet multipliers leaving the unit circle.  
the unstable regions predicted by the present method show excellent agreement with the literature results, both exhibiting three primary unstable zones. Minor discrepancies between the two methods primarily stem from different theoretical approaches in shaft modeling: the literature employs Euler beam theory, whereas the present study incorporates shear deformation through Timoshenko beam theory. The comparative validation confirms the reliability of the established base motion rotor dynamics model and its stability analysis method while additionally accounting for shear effects.  
To further validate the accuracy of the HB-AFT method, Newmark-based Floquet multiplier solution method, and flexible rotor model with SFD employed in this paper, a comparison is made with [PERSON]'s [27] research findings on the dynamic characteristics and stability of a single-disk rotor system with centralized SFD on a fixed foundation. Figure 7 presents the dimensionless frequency-response amplitudes and stability curves at the SFD node and disk node under three unbalanced excitation conditions, where solid and dashed lines represent stable and unstable periodic solutions, respectively. The computational results demonstrate excellent agreement with the frequency-response characteristics reported in the literature [27].  
As shown in Figure 7, the system response with a mass unbalance of 72.008 g\(\cdot\)mm demonstrates complete stability throughout the analyzed speed range, with no bifurcation points or stability transitions detected. For cases with mass unbalances of 144.02 g\(\cdot\)mm and 360.04 g\(\cdot\)mm, Table 1 presents a comparison between the bifurcation points and corresponding maximum Floquet multiplier values calculated using the present model and those reported in the literature [27]. According to Floquet theory, saddle-node bifurcation occurs when Floquet multipliers cross the unit circle at (1, 0). The comparative results demonstrate good agreement in both the rotational speeds at bifurcation points and the way maximum Floquet multipliers traverse the unit circle. Discrepancies between the computational results from the two methods may be attributed to differences in finite element nodal discretization and solution methodologies (IHB combined with the Hsu method employed in the literature, while HB-AFT combined with the Newmark method is utilized in the present paper). The comparative validation confirms the reliability of the nonlinear rotor model and stability analysis method adopted in this paper.  
Figure 6: Stability chart calculated from: (**a**) [PERSON] and Chu [35]; (**b**) the proposed model.