## 4 Conclusions  
This paper proposed a nonlinear hybrid control law combining active friction damping and active control force, and simulations were conducted on a 3-DOF system. The following conclusions can be drawn from this study:  
(1) The proposed hybrid control is stable according to the Lyapunov criterion;  
(2) It is difficult to deal with the system vibration under different excitation frequencies at the same time using regular friction damping methods, and pure active control methods at high frequency are not efficient in terms of response speed and input energy. The hybrid control law proposed in this study can provide quite a good compromise;  
(3) The proposed nonlinear hybrid control law shows better control effects in steady-state kinetic energy of the system under different frequency excitation;  
(4) Under high-frequency excitation, the proposed hybrid control strategy reveals the advantage of reducing the input energy of active control force, compared with the pure active control method. Figure 9: Time histories of friction force under 13 Hz and 25 Hz excitation. (**a**) Time response of friction force under 13 Hz excitation (**b**) Time response of friction force under 25 Hz excitation. **Author Contributions:** Conceptualization, X.H. and X.Y. ; methodology, X.H., D.Y. and X.Y. ; validation, D.Y. ; formal analysis, X.H., D.Y. and X.Y. ; investigation, X.H., D.Y. and X.Y. ; resources, D.Y. ; data curation, D.Y. ; writing--review and editing, X.H., D.Y. and X.Y. ; visualization, D.Y. ; supervision, X.H. and X.Y. ; project administration, X.H. and X.Y. All authors have read and agreed to the published version of the manuscript. **Funding:** This research was supported by the National Natural Science Foundation of China, Grant Number 52105083, the major project of aeroengines and gas turbines, Grant Number J2019-IV-023-0091, the Innovation Centre for Advanced Aviation Power, Grant Number HKCX2020-02-016, and the National Natural Science Foundation of China, Grant Number 11804015. **Institutional Review Board Statement:** Not applicable. **Informed Consent Statement:** Not applicable. **Data Availability Statement:** There is no data reported. **Conflicts of Interest:** The authors declare no conflict of interest. [PERSON]; [PERSON]; [PERSON]; [PERSON] [PERSON] A state-of-the-art review of structural control systems. _J. Vib. Control._**2015**, _21_, 919-937. [CrossRef]
* _[PERSON] et al. (1997)_ [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON]. Structural Control: Past, Present, and Future. _J. Eng. Mech._**1997**, _123_, 897-971. [CrossRef]
* _[PERSON] (2016)_ [PERSON] Dry friction damper for supercritical drive shaft. _J. KONES_**2016**, _23_, 389-396. [CrossRef]
* _[PERSON] et al. (2021)_ [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON] Review of Magnetorheological Damping Systems on a Seismic Building. _Appl. Sci._**2021**, _11_, 9339. [CrossRef]
* _[PERSON] et al. (2021)_ [PERSON]; [PERSON]; [PERSON] Comparison of Strengthening Solutions with Optimized Passive Energy Dissipation Systems in Symmetric Buildings. _Appl. Sci._**2021**, _11_, 10103. [CrossRef]
* _[PERSON] et al. (2020)_ [PERSON]; [PERSON]; [PERSON] [PERSON]; [PERSON]; [PERSON] Parametric Study and Uncertainty Quantification of the Nonlinear Modal Properties of Frictional Dampers. _J. Vib. Acoust._**2020**, _142_, 1-23. [CrossRef]
* _[PERSON] et al. (2017)_ [PERSON] [PERSON]; [PERSON] [PERSON]; [PERSON] [PERSON]; [PERSON] [PERSON] A substructure-based numerical technique and experimental analysis of turbine blades damping with underplatform friction dampers. _Procedia Eng._**2017**, _199_, 820-825. [CrossRef]
* _[PERSON] and [PERSON] (2021)_ [PERSON]; [PERSON] An Evaluation Method for Dry Friction Damping of Ring Damper in Gas Turbine Engines under Axial Vibration. _Aerospace_**2021**, \(8\), 302. [CrossRef]
* _[PERSON] et al. (2021)_ [PERSON]; [PERSON] [PERSON] A Method to Minimize the Effort for Damper-Blade Matching Demonstrated on Two Blade Sizes. _Appl.