The active magnetic bearings system plays a vital role in high-speed rotors technology, where many research articles have discussed the nonlinear dynamics of different categories of this system such as the four-pole, six-pole, eight-pole, and sixteen-pole systems. Although the twelve-pole system has many advantages over the eight-pole one (such as a negligible cross-coupling effect, low power consumption, better suspension behaviors, and high dynamic stiffness), the twelve-pole system oscillatory behaviors have not been studied before. Therefore, this article is assigned to explore the effect of the magneto-electro-mechanical nonlinearities on the oscillatory motion of the twelve-pole system controlled via a proportional derivative controller for the first time. The normalized equations of motion that govern the system vibrations are established by means of classical mechanics. Then, the averaging equations are extracted utilizing the asymptotic analysis. The influence of all system parameters on the steady-state oscillation amplitudes is explored. Stability charts in a two-dimensional space are constructed. The stable margin of both the system and control parameters is determined. The obtained investigations reveal that proportional gain plays a dominant role in reshaping the dynamics and motion bifurcation of the twelve-pole systems. In addition, it is found that stability charts of the system can be controlled by simply utilizing both the proportional and derivative gains. Moreover, the numerical simulations showed that the twelve-poles system can exhibit both quasiperiodic and chaotic oscillations besides the periodic motion depending on the control parameters’ magnitude.

The performance of a nonlinear position-velocity controller in stabilising the lateral vibrations of a rotor-active magnetic-bearings system (RAMBS) is investigated. Cubic nonlinear position-velocity and linear position-velocity controllers are introduced to stabilise RAMBS lateral oscillations. According to the proposed control law, the nonlinear system model is established and then investigated with perturbation analysis. Nonlinear algebraic equations that govern the steady-state oscillation amplitudes and the corresponding phases are derived. Depending on the obtained algebraic equations, the different frequency response curves and bifurcation diagrams are plotted for the studied model. Sensitivity analysis for the linear and nonlinear controllers’ gains is explored. Obtained analytical results demonstrated that the studied model had symmetric bifurcation behaviours in both the horizontal and vertical directions. In addition, the integration of the cubic position controller made the control algorithm more flexible to reshape system dynamical behaviours from the hardening spring characteristic to the softening spring characteristic (or vice versa) to avoid resonance conditions. Moreover, the optimal design of the cubic position gain and/or cubic velocity gain could stabilise the unstable motion and eliminate the nonlinear effects of the system even at large disc eccentricities. Lastly, numerical validations for all acquired results are performed, where the presented simulations show accurate correspondence between numerical and analytical investigations.

In this study, the Integral Resonant Controller (IRC) is presented along with the Proportional-Derivative (PD) controller as a novel control technique to control the dynamical behaviors of the 12-pole rotor active magnetic bearing system. According to the proposed control strategy, the system model has been derived as a two-degree-of-freedom nonlinear dynamical system coupled with two first-order filters. The obtained mathematical model has been analyzed utilizing the asymptotic analysis. The nonlinear algebraic system that governs the steady-state vibration amplitudes and corresponding phase angles of the considered system has been extracted. The influence of the IRC control parameters on the rotor dynamics has been explored by plotting the different bifurcation diagrams. The analytical investigations demonstrated that the vibration suppression efficiency of the combined controller (i.e., PD and IRC) is proportional to the product of control and feedback gains of the IRC as well as the derivative gain of the PD-controller. In addition, it is found that the controller efficiency is inversely proportional to both the square of the internal loop feedback gain of the IRC and the position gain of the PD-controller. Accordingly, an objective function has been derived to design the best control gains of the proposed controller strategy. The analytical and numerical simulations confirmed that the suggested control method can suppress the system vibration and eliminate the catastrophic bifurcation behaviors if the control gains are selected according to the proposed objective function. Finally, the effect of failing one of the coupled integral resonant controllers on the rotor dynamics has been explored as a precautionary procedure. It is found that the failure of one of the coupled integral resonant controllers may breakdown the bifurcation symmetry of the rotor system, but the system does not lose its stability.

Nonlinear vibration control of the twelve-poles electro-magnetic suspension system was tackled in this study, using a novel control strategy. The introduced control algorithm was a combination of three controllers: the proportional-derivative controller, the integral resonant controller and the positive position feedback controller. According to the presented control algorithm, the mathematical model of the controlled twelve-poles rotor was established as a nonlinear four-degree-of-freedom dynamical system coupled to two first-order filters. Then, the derived nonlinear dynamical system was analyzed using perturbation analysis to extract the averaging equations of motion. Based on the extracted averaging equations of motion, the efficiency of different control strategies for mitigating the rotor’s undesired vibrations and improving its catastrophic bifurcation was investigated. The acquired analytical results demonstrated that both the and controllers can force the rotor to respond as a linear system; however, the controlled system may exhibit the maximum oscillation amplitude at the perfect resonance condition. In addition, the obtained results demonstrated that the controller can eliminate the rotor nonlinear oscillation at the perfect resonance, but the system may suffer from high oscillation amplitudes when the resonance condition is lost. Moreover, we report that the combined control algorithm () has all the advantages of the individual control algorithms (i.e., ), while avoiding their drawbacks. Finally, the numerical simulations showed that the controller can eliminate the twelve-poles system vibrations regardless of both the excitation force magnitude and the resonant conditions at a short transient time.