## 5 Conclusions  
This study establishes equations of motion for a single-rotor system with a centralized squeeze film damper under base motion excitation using the finite element method. The periodic solutions were obtained using the HB-AFT method with arc-length continuation. Bifurcation patterns were identified through Floquet multiplier analysis combined with the Newmark method, while non-periodic solutions were verified using Newmark numerical integration, ensuring result reliability. The following nonlinear dynamic phenomena were observed and analyzed:  
1. Under unbalanced excitation, the system exhibits saddle-node bifurcations accompanied by jump phenomena, with response spectra confined to synchronous components. Under base rolling motion, the system maintains hardening characteristics, with frequency-response curves shifting toward lower frequencies as rolling angular velocity increases due to additional stiffness effects. Base pitching motion significantly enhances nonlinear effects, manifesting as higher-order harmonic components \(2\Omega,3\Omega\) in the response. The inertial moment introduced by pitching motion induces horizontal static eccentricity. In the absence of gravitational effects, base yawing motion produces vertical static eccentricity with effects comparable to pitching motion. 2. Increasing unbalanced magnitude significantly expands the unstable speed range, with more pronounced centering phenomena near critical speeds. Conversely, increasing pitching angular velocity progressively reduces the unstable speed range. At lower unbalanced magnitudes, strategic implementation of pitching motion demonstrates the capacity for the complete elimination of instability regions, validating the stabilizing influence of base pitching excitation. Figure 24: Frequency–response curves at the SFD node under base pitching motion (\(C/R=0.728\%\), \(\omega_{\text{ly}}=10\) rad/s) (**a**) horizontal displacement; (**b**) vertical displacement. Under specific combinations of unbalanced magnitude and base pitching angular velocity, variations in the clearance-to-radius ratio of the SFD induce complex nonlinear phenomena: secondary Hopf bifurcations manifest within specific speed ranges, resulting in destabilization of periodic solutions and transition to quasi-periodic motion, with potential progression to chaotic motion. These complex dynamic behaviors demonstrate strong coupling characteristics between system parameters and motion states. This research presents a methodology for analyzing periodic stability in nonlinear rotor systems under base motion excitation. The findings enhance the understanding of nonlinear dynamic characteristics and motion stability in rotor-SFD-support systems under base excitation, providing theoretical foundations for dynamic parameter optimization and vibration control. Conceptualization, B.Z. and X.C. ; Data curation, X.C. ; Formal analysis, B.Z. ; Funding acquisition, X.C. ; Investigation, B.Z., X.C. and F.X. ; Methodology, B.Z., X.C. and F.X. ; Project administration, X.C., G.R. and X.G. ; Resources, X.C., G.R. and X.G. ; Supervision, X.C., G.R. and X.G. ; Validation, B.Z., X.C. and F.X. ; Visualization, B.Z. and F.X. ; Writing--original draft, B.Z. ; Writing--review and editing, X.C. All authors have read and agreed to the published version of the manuscript. This research was funded by [the National Natural Science Foundation of China] grant number 12302066, [the National Science and Technology Major Project] grant number [J2022-IV-0004-0021], [the Basic and Applied Basic Research Foundation of Guangdong Province] grant number [2022A1515110077], and [the Shenzhen Stability Support Program for Higher Education] grant number [20220815113550002] And The APC was funded by [20220815113550002]. The data presented in this study are available on request from the corresponding author. The authors are grateful for the support from the Young Elite Scientists Sponsorship Program (YESS) by the China Association for Science and Technology (CAST).