Nonlinear Dyn (2013) 72:517–537 DOI 10.1007/s11071-012-0731-5 ORIGINAL PAPER Positive position feedback (PPF) controller for suppression of nonlinear system vibration W.A. El-Ganaini · N.A. Saeed · M. Eissa Received: 4 October 2012 / Accepted: 19 December 2012 / Published online: 16 January 2013 © Springer Science+Business Media Dordrecht 2013 Abstract In this paper, a study for positive position feedback controller is presented that is used to sup- press the vibration amplitude of a nonlinear dynamic model at primary resonance and the presence of 1:1 in- ternal resonance. We obtained an approximate solution by applying the multiple scales method. Then we con- ducted bifurcation analyses for open and closed loop systems. The stability of the system is investigated by applying the frequency-response equations. The ef- fects of the different controller parameters on the be- havior of the main system have been studied. Opti- mum working conditions of the system were extracted to be used in the design of such systems. Finally, nu- merical simulations are performed to demonstrate and validate the control law. We found that all predic- tions from analytical solutions are in good agreement with the numerical simulation. A comparison with the available published work is included at the end of the work. Keywords Vibration control · Positive position feedback · Stability · Primary resonance Nomenclature W.A. El-Ganaini · N.A. Saeed ( ) · M. Eissa Department of Physics and Engineering Mathematics, Faculty of Electronic Engineering, Menoufia University, Menouf 32952, Egypt e-mail: eng_saeed_2003@yahoo.com u, ˙ u, ¨ u Displacement, velocity and acceleration of main system, respectively v, ˙ v, ¨ v Displacement, velocity and acceleration of controller, respectively μ 1 ,μ 2 Linear damping parameters of main system and controller, respectively ω 1 ,ω 2 Linear natural frequencies of main system and controller, respectively α 1 ,α 2 Cubic nonlinearity parameters of main system and controller, respectively δ Main system nonlinear parameter f External excitation force amplitude Ω External excitation frequency F c Control signal F f Feedback signal γ Control signal gain λ Feedback signal gain ε Small perturbation parameter 1 Introduction Vibration, occurring in most machines, vehicles, struc- tures, building and dynamic systems is undesirable, not only because of the resulting unpleasant motions, the dynamic stresses which may lead to fatigue and failure of the structure or machine, the energy losses and reduction in performance which accompany vibra- tions, but also because of the produced noise. Noise is an undesired phenomenon, and since sound is pro- duced by some source of motion or vibration causing