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Stability, bifurcation, and vibration control of a discontinuous nonlinear rotor model under rub-impact effect

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Abstract

This work aims to investigate the stability, bifurcation, and vibration control of a discontinuous dynamical model simulating the nonlinear oscillation of a horizontally suspended nonlinear rotor system. A novel Proportional-Integral-Resonance-Control (PIRC) algorithm is introduced to dampen the rotor's vibrations. An 8-pole electromagnetic bearing is employed as an active actuator through which the PIRC control signals are applied in the form of eight electrical currents. These currents, in turn, generate controllable attractive forces that counteract the rotor's vibrations. Based on the presented control structure, the mathematical model of the closed-loop system, including the magneto-electromechanical coupling and rotor-actuator rub-impact force, is derived as a 2-DOF discontinuous dynamical system connected to two first-order filters. A closed-form solution using the multiple scales analysis (up to the second-order approximation) of the system model was obtained. Additionally, the autonomous dynamical system governing the evolution of oscillation amplitudes and modified phases was derived when the rub-impact was neglected. Based on this derived autonomous system, the bifurcation characteristics, vibration control, and system stability have been explored. The analytical results obtained are utilized as a heuristic tool to report the conditions under which the system will experience rub and/or impact forces. Then, the entire discontinuous model was numerically analyzed using bifurcation diagrams, zero–one chaotic test, poincaré maps, orbital plots, and instantaneous radial and lateral oscillations at the rub and/or impact conditions reported analytically. The principal findings demonstrate that the introduced analytical investigation can be used to predict accurately when the rotor-actuator will be subject to a rub-impact force. Furthermore, it is proven analytically that the system's damping coefficients are proportional to the cartesian product of the controller's feedback and control gains. Moreover, the numerical analysis also illustrates that the presence of a rotor-actuator rub-impact can lead to period-1, period-2, period-3, period-4, or quasiperiodic oscillations, depending on the rotation speed and/or disc eccentricity. Finally, it is proven that optimizing the controller gains can prevent the rotor-actuator rub-impact effect regardless of the rotor speed and disc eccentricity.

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Funding

The authors are grateful to the Raytheon Chair for Systems Engineering for funding. This work has been supported by the Polish National Science Centre, Poland under the Grant OPUS 18 No. 2019/35/B/ST8/00980. The authors are very grateful for the financial support from the National Natural Science Foundation of China (Grant No. 11972129).

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Authors and Affiliations

  1. Department of Physics and Engineering Mathematics, Faculty of Electronic Engineering, Menoufia University, Menouf, 32952, Egypt

    Nasser A. Saeed

  2. Department of Automation, Biomechanics, and Mechatronics, Faculty of Mechanical Engineering, Lodz University of Technology, 90924, Lodz, Poland

    Nasser A. Saeed & Jan Awrejcewicz

  3. Mathematics Department, Faculty of Science, Galala University, Galala City, 43511, Egypt

    Nasser A. Saeed

  4. Basic Science Department, Higher Technological Institute, 6th of October City, Giza, Egypt

    S. T. Hafez

  5. School of Astronautics, Harbin Institute of Technology, Harbin, 150001, China

    Lei Hou

  6. Raytheon Chair for Systems Engineering (RCSE), Advanced Manufacturing Institute, King Saud University, 11421, Riyadh, Saudi Arabia

    Mohamed K. Aboudaif

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  1. Nasser A. Saeed
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  2. Jan Awrejcewicz
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  3. S. T. Hafez
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Correspondence to Mohamed K. Aboudaif.

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Appendix

Appendix

$$ \zeta_{10} = 4 + 8\cos^{2} (\chi ) - 8G_{1} \cos^{2} (\chi ) - 4G_{1} , $$
$$ \zeta_{11} = 8G_{3} \cos^{2} (\chi ) + 4G_{3} , $$
$$ \zeta_{12} = 8G_{2}^{2} \cos^{4} (\chi ) + 48\cos^{4} (\chi ) + 16G_{1} G_{2} \cos^{4} (\chi ) - 48G_{2} \cos^{4} (\chi ) - 24G_{1} \cos^{4} (\chi ), $$
$$ \zeta_{13} = 8G_{3}^{2} \cos^{4} (\chi ) + 4G_{3}^{2} , $$
$$ \zeta_{14} = 8G_{4}^{2} \cos^{4} (\chi ), $$
$$ \zeta_{15} = 12G_{3} - 8G_{1} G_{3} - 16G_{1} G_{3} \cos^{4} (\chi ) + 24G_{3} \cos^{4} (\chi ), $$
$$ \zeta_{16} = 24G_{3} \cos^{4} (\chi ) - 16G_{2} G_{3} \cos^{4} (\chi ), $$
$$ \zeta_{17} = 48G_{4} \cos^{4} (\chi ) - 16G_{2} G_{4} \cos^{4} (\chi ) - 16G_{1} G_{4} \cos^{4} (\chi ), $$
$$ \zeta_{18} = 16G_{3} G_{4} \cos^{4} (\chi ), $$
$$ \zeta_{19} = 4G_{1}^{2} + 8 - 24G_{1} \cos^{4} (\chi ) + 16\cos^{4} (\chi ) - 12G_{1} + 8G_{1}^{2} \cos^{4} (\chi ), $$
$$ \zeta_{20} = 4 + 8\cos^{2} (\chi ) - 8G_{2} \cos^{2} (\chi ) - 4G_{2} , $$
$$ \zeta_{21} = 8G_{4} \cos^{2} (\chi ) + 4G_{4} , $$
$$ \zeta_{22} = 8G_{1}^{2} \cos^{4} (\chi ) + 48\cos^{4} (\chi ) + 16G_{1} G_{2} \cos^{4} (\chi ) - 48G_{1} \cos^{4} (\chi ) - 24G_{2} \cos^{4} (\chi ), $$
$$ \zeta_{23} = 8G_{4}^{2} \cos^{4} (\chi ) + 4G_{4}^{2} , $$
$$ \zeta_{24} = 8G_{3}^{2} \cos^{4} (\chi ), $$
$$ \zeta_{25} = 12G_{4} - 8G_{2} G_{4} - 16G_{2} G_{4} \cos^{4} (\chi ) + 24G_{4} \cos^{4} (\chi ), $$
$$ \zeta_{26} = 24G_{4} \cos^{4} (\chi ) - 16G_{1} G_{4} \cos^{4} (\chi ), $$
$$ \zeta_{27} = 48G_{3} \cos^{4} (\chi ) - 16G_{1} G_{3} \cos^{4} (\chi ) - 16G_{2} G_{3} \cos^{4} (\chi ), $$
$$ \zeta_{28} = 16G_{3} G_{4} \cos^{4} (\chi ), $$
$$ \zeta_{29} = 4G_{2}^{2} + 8 - 24G_{2} \cos^{4} (\chi ) + 16\cos^{4} (\chi ) - 12G_{2} + 8G_{2}^{2} \cos^{4} (\chi ). $$

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Saeed, N.A., Awrejcewicz, J., Hafez, S.T. et al. Stability, bifurcation, and vibration control of a discontinuous nonlinear rotor model under rub-impact effect. Nonlinear Dyn 111, 20661–20697 (2023). https://doi.org/10.1007/s11071-023-08934-5

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