, Volume 50, Issue 1, pp 233–251 | Cite as

On controlling the response of primary and parametric resonances of a nonlinear magnetic levitation system



In this paper, a proportional-derivative controller is proposed to reduce the horizontal vibration of a magnetically levitated system having quadratic and cubic nonlinearities to primary and parametric excitations. A second order approximate solution is sought using the method of multiple scales perturbation technique to clarify the nonlinear behavior for both amplitude and phase of the system. The effect of feedback signal gain is studied to indicate the optimum values for best performance. Validation curves are included to compare the approximate solution and the numerical simulation. A comparison with previously published work is included.


Multiple scales perturbation technique Second order approximation Proportional derivative controller Controller effectiveness Jump phenomenon 

List of symbols

\( y,\,{\dot{y}},\,{\ddot{y}} \)

Displacement, velocity and acceleration

\( \mu \)

Linear damping coefficient

\( \alpha_{2} ,\,\alpha_{3} \)

Quadratic and cubic stiffness nonlinearity parameters

\( f \)

External excitation force amplitude

\( \Omega \)

External excitation frequency

\( p,\,d \)

Proportional and derivative gains

\( k_{1} ,\,k_{2} ,\,k_{3} \)

Constants dependent on the magnetic forces between magnets

\( \varepsilon \)

Small perturbation parameter

\( \sigma_{1} ,\,\sigma_{2} \)

Detuning parameters


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Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  • M. Eissa
    • 1
  • A. Kandil
    • 1
  • M. Kamel
    • 1
  • W. A. El-Ganaini
    • 1
  1. 1.Department of Physics and Engineering Mathematics, Faculty of Electronic EngineeringMenoufia UniversityMenoufEgypt

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