Abstract
A time-delayed proportional-derivative controller is proposed in this paper to reduce the horizontal vibration of a magnetically levitated system having quadratic and cubic nonlinearities to primary and parametric excitations. Applying multiple scales perturbation technique, a second-order approximate solution is sought to analyze the nonlinear behavior of the system. The effects of the time delay are studied to indicate the stable range of time delays for the best performance. The results are compared with the numerical simulations, and this shows a good verification for the approximate solution and for the control algorithm used in this paper. A comparison with previously published work is included at the end.
Similar content being viewed by others
Abbreviations
- \(y,\dot{y},\ddot{y}\) :
-
Displacement, velocity and acceleration
- \(\mu \) :
-
Linear damping coefficient
- \(\alpha _{2},\alpha _{3}\) :
-
Quadratic and cubic stiffness nonlinearity parameters
- \( f_{j},~f_{jt}\) :
-
External and modulated excitation force components
- \(\Omega \) :
-
External excitation frequency
- \(\Omega _{t},\,{\psi }\) :
-
Modulated excitation frequencies
- \(p,~d \) :
-
Proportional and derivative gains
- \(k_{1},~k_{2},~k_{3}\) :
-
Constants dependent on the applied magnetic forces
- \(\varepsilon \) :
-
Small perturbation parameter
- \(\sigma _{1},~\sigma _{2},\sigma _{t}\) :
-
Detuning parameters
- \(\tau _{1},~\tau _{2}\) :
-
Time delays
References
Jo, H., Yabuno, H.: Amplitude reduction of primary resonance of nonlinear oscillator by a dynamic vibration absorber using nonlinear coupling. Nonlinear Dyn. 55, 67–78 (2009)
Jo, H., Yabuno, H.: Amplitude reduction of parametric resonance by dynamic vibration absorber based on quadratic nonlinear coupling. J. Sound Vib. 329, 2205–2217 (2010)
Kamel, M., Kandil, A., El-Ganaini, W.A., Eissa, M.: Active vibration control of a nonlinear magnetic levitation system via Nonlinear Saturation Controller (NSC). Nonlinear Dyn. (2014). doi:10.1007/s11071-014-1323-3
Bauomy H. S.: Active vibration control of a dynamical system via negative linear velocity feedback. Nonlinear Dyn. doi:10.1007/s11071-014-1306-4.
El-Ganaini, W.A., Saeed, N.A., Eissa, M.: Positive position feedback controller (PPF) for suppression of nonlinear system vibration. Nonlinear Dyn. 72, 517–537 (2013)
Eissa, M., Sayed, M.: A comparison between active and passive vibration control of non-linear simple pendulum, Part I: transversally tuned absorber and negative \(G\,\dot{\varphi }^{n} \) feedback. Math. Comput. Appl. 11(2), 137–149 (2006)
Eissa, M., Sayed, M.: A comparison between active and passive vibration control of non-linear simple pendulum, Part II: Longitudinal tuned absorber and negative \(G \ddot{\varphi }\) and \({G\,\varphi }^{n}\) feedback. Math. Comput. Appl. 11(2), 151–162 (2006)
Eissa, M., Bauomy, H.S., Amer, Y.A.: Active control of an aircraft tail subject to harmonic excitation. Acta Mech. Sin. 23, 451–462 (2007)
Zhao, Y.Y., Xu, J.: Effects of delayed feedback control on nonlinear vibration absorber system. J. Sound Vib. 308, 212–230 (2007)
Xu, J., Chung, K.W., Zhao, Y.Y.: Delayed saturation controller for vibration suppression in a stainless-steel beam. Nonlinear Dyn. 62, 177–193 (2010)
Zhao, Y.Y., Xu, J.: Using the delayed feedback control and saturation control to suppress the vibration of the dynamical system. Nonlinear Dyn. 67, 735–753 (2012)
Nayfeh, A.H., Mook, D.T.: Nonlinear Oscillations. Wiley, New York (1995)
Author information
Authors and Affiliations
Corresponding author
Appendix
Appendix
Rights and permissions
About this article
Cite this article
Eissa, M., Kandil, A., El-Ganaini, W.A. et al. Analysis of a nonlinear magnetic levitation system vibrations controlled by a time-delayed proportional-derivative controller. Nonlinear Dyn 79, 1217–1233 (2015). https://doi.org/10.1007/s11071-014-1738-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11071-014-1738-x