The active car suspension system is presented here to suppress the vibration of the car by applying proportional derivative (PD) and positive position feedback (PPF) controllers with time delay. The control signal output of the controller is applied electrically to the magneto-rheological (MR) damper or Electrical-rheological (ER) damper which is attached parallel to the passive components to improve the suppression of the vibration. The electrical control signal is produced by electronic circuits or programmable logic controller and the two-position sensor feedback signal which are connected to the controller. The approximate solutions of PD and PPF suspension systems are obtained by applying multiple time scales perturbation method. The effects of parameters variation of both the system and the controllers are investigated to achieve the best performance. Simulation results show good performance of the designed controllers.

Within this work, the radial Proportional Derivative (PD-) controller along with the eight-poles electro-magnetic actuator are introduced as a novel control strategy to suppress the lateral oscillations of a non-linear Jeffcott-rotor system. The proposed control strategy has been designed such that each pole of the magnetic actuator generates an attractive magnetic force proportional to the radial displacement and radial velocity of the rotating shaft in the direction of that pole. According to the proposed control mechanism, the mathematical model that governs the non-linear interactions between the Jeffcott system and the magnetic actuator has been established. Then, an analytical solution for the obtained non-linear dynamic model has been derived using perturbation analysis. Based on the extracted analytical solution, the motion bifurcation of the Jeffcott system has been investigated before and after control via plotting the different response curves. The obtained results illustrate that the uncontrolled Jeffcott-rotor behaves like a hard-spring duffing oscillator and responds with bi-stable periodic oscillation when the rotor angular speed is higher than the system’s natural frequency. It is alsomfound that the system, before control, can exhibit stable symmetric motion with high vibration amplitudes in both the horizontal and vertical directions, regardless of the eccentricity magnitude. In addition, the acquired results demonstrate that the introduced control technique can eliminate catastrophic bifurcation behaviors and undesired vibration of the system when the control parameters are designed properly. However, it is reported that the improper design of the controller gains may destabilize the Jeffcott system and force it to perform either chaotic or quasi-periodic motions depending on the magnitudes of both the shaft eccentricity and the control parameters. Finally, to validate the accuracy of the obtained results, numerical simulations for all response curves have been introduced which have been in excellent agreement with the analytical investigations.

This article presents the Proportional Integral Resonant Controller (PIRC-controller) as a novel control strategy to suppress the lateral vibrations and eliminate nonlinear bifurcation characteristics of a vertically supported rotor system. The proposed control algorithm is incorporated into the rotor system via an eight-pole electromagnetic actuator. The control strategy is designed such that the control law (PIRC-controller) is employed to generate eight different control currents depending on the air-gap size between the rotor and the electromagnetic poles. Then, the generated electrical currents are utilized to energize the magnetic actuator to apply controllable electromagnetic attractive forces to suppress the undesired lateral vibrations of the considered rotor system. According to the suggested control strategy, the whole system can be represented as a mathematical model using classical mechanics' principle and electromagnetic theory, in which, the rub-impact force between the rotor and the stator is included in the derived model. Then, the obtained discrete dynamical model is analyzed using perturbation techniques and validated numerically through bifurcation diagrams, frequency spectrums, Poincare maps, time responses, and steady-state whirling orbit. The obtained results illustrate that the proposed control algorithm can mitigate the nonlinear vibration and eliminate the catastrophic bifurcations of the rotor system when the control gains are designed optimally. In addition, the system dynamics are analyzed when the rub-impact occurrence between the rotor and the pole housing is unavoidable. The acquired results revealed that the system may perform periodic-1, periodic-n, or quasiperiodic motion with one of two oscillation modes depending on both the impact stiffness coefficient and the dynamic friction coefficient. Article Highlights Nonlinearity dominates the uncontrolled rotor response, where it suffers from the jump phenomenon and multiple solutions. The proposed controller forces the Jeffcott rotor to respond as a linear system with small oscillation amplitudes. The rotor oscillates with full-annular-rub or partial-rub-impact mode when rub-impact occurs between the rotor and stator.

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