## 6 Conclusions  
The article investigates the frequency response and vibration stability of the rigid rotor with unbalance, coupled with SFDs. To analyse the rotor's response, three methods were used: the computation of the synchronous response with a circular centred orbit, the harmonic balance method, and the direct time integration method. All three methods accurately computed the frequency response curve, and the results were in good agreement. The stability of the induced vibrations was evaluated by means of the Floquet theorem. New computational procedures were developed to solve the equation of motion and assess vibration stability.  
The results of the computational simulations show that the nonlinear damping forces greatly impact the motion of the rotor, particularly in the resonance region for high values of the unbalance. It was verified that the frequency response curve is composed of four regions with varying numbers of solutions, where multiple branches of the frequency  
Figure 18: The effect of the dimensionless parameters on the presence of an isolated solution for the case without (**a**) and with (**b**) external damping.  
response are unstable. Techniques such as time series analysis, Fourier transformation, trajectory analysis in state space, Poincare sections, and bifurcation diagrams were used to study the nonlinear behaviour.  
A substantial amount of the newly presented work focuses on illustrating how initial conditions can affect the long-time behaviour of the response. The study presents basins of attraction for various values of the nondimensional speed. It has been proven that the basins of attractions are sensitive to the choice of initial velocity values. The regions of attraction can contain more than one solution, and the borders between them can be either smooth or without a clear structure.  
The simulation results indicate that the nonsynchronous response has a quasiperiodic character and occurs solely at rotational speeds that exceed the speeds where the isolated solution originates. The nonsynchronous response is generally considered to be an undesirable behaviour. Therefore, a new range of parameters for the dimensionless SFD constant, unbalance, and rotational speed was defined, which determines the conditions for the appearance of an isolated solution.  
The results of the carried-out research are highly valuable for contemporary investigations of nonlinear vibrations, particularly for high-speed rotating machines coupled with SFDs.  
Conceptualization, M.M. and P.F.; methodology, M.M., J.Z. and P.F.; software, M.M. and P.F.; validation, P.F. and J.Z.; formal analysis, M.M., P.F. and J.Z.; investigation, P.F., J.Z. and M.M.; resources, M.M., P.F. and J.Z.; data curation, P.F. and M.M.; writing--original draft preparation, P.F. and M.M.; writing--review and editing, P.F., M.M. and J.Z.; visualization, M.M.; supervision, P.F. and J.Z.; project administration, P.F. and M.M.; funding acquisition, M.M. and P.F. All authors have read and agreed to the published version of the manuscript.  
The authors are grateful for the support and funding provided by the Ministry of Education, Youth and Sports of the Czech Republic through the e-INFRA CZ (ID:90140), the Doctoral Grant Competition of Technical University of Ostrava (No. CZ/02.2.69/0.0./0./0.19_073/0016945) within the Operational Programme Research, Development and Education, under project DGS/TEAM/2020-033, the Support for Science and Research in the Moravian-Silesian Region (No. RRC/02/2020), and the Project of Specific University Research (No. SP2022/26).  
Not applicable.  
Data Avalability Statement: Data sharing does not apply to this article.  
The authors declare no conflict of interest.