## 6 Conclusions  
The twelve-pole AMBS system vibrational behaviors are extensively analyzed within this article. The classical mechanics principle is applied to derive the system's equations of motion. Then, the derived mathematical model is solved analytically by utilizing the perturbation analysis to obtain the amplitude-phase modulating equations. Based on the extracted amplitude-phase equations, the steady-state oscillations are explored in terms of all system parameters. The stability charts are established in two-parameter space to investigate the stability margin of the different system parameters. In addition, the obtained results are validated numerically via plotting the system temporal oscillations, orbital motion, Poincare-map, frequency spectrum, and the bifurcation diagram. According to the presented study, we can conclude with the following important remarks:  
1. The proportional gain has a great influence on the twelve-pole system's dynamical behaviors, solution bifurcations, and stability conditions;
2. At the small values of the proportional gain (i.e., \(1.1<p<1.34\)), the rotor system responds with a small oscillation amplitude with complex bifurcation behaviors at the small eccentricity magnitude \(f\). However, the system may lose its stability to perform a quasiperiodic or chaotic oscillation when increasing the rotor eccentricity beyond a critical value;
3. At the large values of the proportional gain (i.e., when \(1.34<p<1.65\)), the twelve-pole rotor exhibits simple bifurcation behaviors and relatively large vibration amplitudes at the small disc eccentricities. In addition, the system responds periodically in the case of the strong values of the eccentricity \(f\) without losing its stability;
4. Regardless of the proportional gain magnitude, the rotor system vibrations amplitudes are a monotonic decreasing function of the derivative gain, where increasing \(d\) decreases the oscillation amplitudes and eliminate the motion bifurcations;
5. The stability margin of the rotor eccentricity \(f\) depends on the system angular speed \(\Omega\) at the small values of the proportional gain (i.e., when \(1.1<p<1.34\)). However, the rotor system performs a stable periodic motion regardless of the eccentricity magnitude and rotor angular speed at the large values of the proportional gain (i.e., when \(1.34<p<1.65\));
6. The optimal design of the control variables (\(p\) and \(d\)) is a compromising process that depends on the system parameters and the control objectives;
7. It is recommended to investigate different advanced control methodologies for the twelve-pole system in future works.  
Conceptualization, N.A.S. and J.A.; Data curation, S.M.E.-S.; Formal analysis, S.M.E.-S. and N.A.S.; Funding acquisition, M.K.A. and J.A.; Investigation, S.M.E.-S., N.A.S. and J.A.; Methodology, N.A.S. and J.A.; Project administration, N.A.S., M.K., K.R.R., M.K.A. and J.A.; Resources, N.A.S., M.K. and K.R.R.; Software, N.A.S.; Supervision, N.A.S. and J.A.; Validation, S.M.E.-S., N.A.S. and J.A.; Visualization, N.A.S.; Writing--original draft, S.M.E.-S., and N.A.S.; Writing--review and editing, N.A.S., M.K., K.R.R., M.K.A. and J.A. All authors have read and agreed to the published version of the manuscript.  
The authors are grateful to the Raytheon Chair for Systems Engineering for funding. This work is supported by the National Science Centre, Poland, under the grant OPUS 14 No. 2017/27/B/ST8/01330.  
Not applicable.  
The authors declare no conflict of interest.