Contents lists available at ScienceDirect International Journal of Non–Linear Mechanics journal homepage: www.elsevier.com/locate/nlm Nonlinear PD-controller to suppress the nonlinear oscillations of horizontally supported Jeffcott-rotor system Nasser Abdul-Fadeel Abdul-Hameed Saeed ⁎ , M. Kamel Department of Physics and Engineering Mathematics, Faculty of Electronic Engineering, Menoufia University, Menouf 32952, Egypt ARTICLE INFO Keywords: Primary resonance Bifurcation Multi-jump phenomenon Localized vibrations Non-localized vibrations Nonlinear PD-controller ABSTRACT This paper investigates the vibration control of a horizontally suspended Jeffcott-rotor system. A nonlinear restoring force and the rotor weight are considered in the system model. The system frequency (angular speed) -response curve is plotted at different values of the rotor eccentricity. The analysis illustrated that the system has a high oscillation amplitude and exhibits some nonlinear behaviors before control. A Proportional-Derivative (PD)-controller is integrated into the system via two pairs of electromagnetic magnetic poles. The nonlinearity due to the electromagnetic coupling is considered in the system model. A second-order approximate solution is obtained by utilizing multiple scales perturbation method. The bifurcation analyses of the controlled system are conducted. The results showed the high efficiency of the controller to mitigate the nonlinear vibrations of the considered system. Numerical simulations are carried out to validate the accuracy of the analytical results. The numerical results confirmed the excellent agreement with the analytical solutions. Then, the optimal working conditions of the system are concluded. Finally, a comparative study with previously published work is reported. 1. Introduction Rotating machinery has an important role in modern industry due to their numerous applications such as automobile engines, turbo- machinery, large-scale manufacturing, compressors, aerospace, gen- erators, and home appliances, etc. The existence of vibrations in rotating machinery is an unavoidable phenomenon, which ultimately can lead to machine failures and dangerous accidents. Such undesired motions are often caused by the dynamic interaction between the stator and rotating parts and the mass unbalance that occurs if a mismatch between the principal axis of the moment of inertia of the rotating shaft and its axis of rotation exists. Generally, nonlinear vibration is inherent phenomenon and has quite negative results in terms of reliability, durability and safe operation of machines. Therefore, vibration analysis and control have received major attention in the last few years. Great efforts have been made to mitigate vibrations in rotating machines. Passive control elements such as squeeze film damper bearings and active control such as magnetic bearings have been integrated into such systems to suppress and control their vibrations. Active magnetic bearings (AMB) system generates magnetic forces through magnetic fields to act against the undesired oscillations in the rotating parts. The main advantage of AMB system is no-contact between the bearing and the rotating parts and thus prevents the mechanical wear and the need for lubrication during the machine operation. Generating electromag- netic force of a controllable magnitude and direction via controlling the dynamics of the AMB system is the main principle. The sensors measure the rotor-required states (displacement, velocity, or accelera- tion) from their reference position. A controller drives the control action according to the control algorithm and the measured values. A power amplifier converts the control action to control current that produces a controllable magnetic force in a way such that the rotor does not oscillate away from its reference position. The control algorithm is responsible for the system stability and the vibration level as well as the damping and the stiffness of the controlled system. Many researchers studied different control methodology to mitigate or to suppress the nonlinear vibrations in the rotating machines. Ji et al. [1– 3] investigated two different nonlinear model simulating the vibration behaviors of a rotor system supported by active magnetic bearing. The authors studied the rotor system supported via eight poles baring at primary and super-harmonic resonance with 1:1 internal resonance [1,2]. They observed the existence of nonlinear phenomena that did not appear in the linear model such as static bifurcation, jump phenom- enon, sensitivity to the initial conditions, and existence of more than one stable solution. In additions, the authors showed that the rotor unbalance and the control gain have great influences on the system response. In Ref. [3], they studied four poles magnetic bearing supported rotor system. They derived two independent second-order differential equations that describe the motions in x and y directions. http://dx.doi.org/10.1016/j.ijnonlinmec.2016.10.003 Received 13 March 2016; Received in revised form 3 October 2016; Accepted 4 October 2016 ⁎ Corresponding author. E-mail addresses: Nasser.A.Saeed@el-eng.menofia.edu.eg, eng_saeed_2003@yahoo.com (N.A.-F. Abdul-Hameed Saeed). International Journal of Non–Linear Mechanics 87 (2016) 109–124 0020-7462/ © 2016 Elsevier Ltd. All rights reserved. Available online 05 October 2016 crossmark