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Positive position feedback (PPF) controller for suppression of nonlinear system vibration

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Abstract

In this paper, a study for positive position feedback controller is presented that is used to suppress the vibration amplitude of a nonlinear dynamic model at primary resonance and the presence of 1:1 internal resonance. We obtained an approximate solution by applying the multiple scales method. Then we conducted bifurcation analyses for open and closed loop systems. The stability of the system is investigated by applying the frequency-response equations. The effects of the different controller parameters on the behavior of the main system have been studied. Optimum working conditions of the system were extracted to be used in the design of such systems. Finally, numerical simulations are performed to demonstrate and validate the control law. We found that all predictions 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.

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Abbreviations

\(u,\dot{u},\ddot{u}\) :

Displacement, velocity and acceleration of main system, respectively

\(v,\dot{v},\ddot{v}\) :

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

References

  1. Warminski, J., Bochenski, M., Jarzyna, W., Filipek, P., Augustyinak, M.: Active suppression of nonlinear composite beam vibrations by selected control algorithms. Commun. Nonlinear Sci. Numer. Simul. 16, 2237–2248 (2011)

    Article  Google Scholar 

  2. Shan, J., Liu, H., Sun, D.: Slewing and vibration control of a single-link flexible manipulator by positive position feedback (PPF). Mechatronics 15, 487–503 (2005)

    Article  Google Scholar 

  3. Wang, K., Xiong, S., Zhang, J.: Active vibration control of a flexible cantilever beam using piezoelectric actuators. Energy Procedia 13, 4367–4374 (2011)

    Article  Google Scholar 

  4. Creasy, M.A., Leo, D.J., Farinholt, K.M.: Adaptive positive position feedback for actively absorbing energy in acoustic cavities. J. Sound Vib. 311, 461–472 (2008)

    Article  Google Scholar 

  5. Ahamed, B., Pota, H.R.: Dynamic compensation for control of a rotary wing UAV using positive position feedback. J. Intell. Robot. Syst. 61, 43–56 (2011)

    Article  Google Scholar 

  6. Baz, A., Poh, S.: Short communications optimal vibration control with modal positive position feedback. Optim. Control Appl. Methods 17, 141–149 (1996)

    Article  MATH  Google Scholar 

  7. Baz, A., Hong, J.: Adaptive control of flexible structures using modal positive position feedback. Int. J. Adapt. Control Signal Process. 11, 231–253 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  8. El-Bassiouny, A.F.: Single-mode control and chaos of cantilever beam under primary and principal parametric excitations. Chaos Solitons Fractals 30, 1098–1121 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  9. Oueini, S.S., Nayfeh, A.H.: Single-mode control of a cantilever beam under principal parametric excitation. J. Sound Vib. 224(1), 33–47 (1999)

    Article  Google Scholar 

  10. El-serafi, S.A., Eissa, M.H., El-Sherbiny, H.M., El-Ghareeb, T.H.: Comparison between passive and active control of a non-linear dynamical system. Jpn. J. Ind. Appl. Math. 23, 139–161 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  11. Maccari, A.: Vibration control for the primary resonance of cantilever beam by a time delay state feedback. J. Sound Vib. 257(2), 241–251 (2003)

    Article  MathSciNet  Google Scholar 

  12. Alhazza, K.A., Daqaq, M.F., Nayfeh, A.H., Inman, D.J.: Non-linear vibrations of parametrically excited cantilever beams subjected to non-linear delayed-feedback control. Int. J. Non-Linear Mech. 43, 801–812 (2008)

    Article  MATH  Google Scholar 

  13. Eissa, M., Sayed, M.: Vibration reduction of a three DOF non-linear spring pendulum. Commun. Nonlinear Sci. Numer. Simul. 13, 465–488 (2008)

    Article  MATH  Google Scholar 

  14. Eissa, M., Kamel, M., El-Sayed, A.T.: Vibration reduction of multi-parametric excited spring pendulum via transverse tuned absorber. Nonlinear Dyn. 61, 109–121 (2010)

    Article  MATH  Google Scholar 

  15. Eissa, M., Kamel, M., El-Sayed, A.T.: Vibration reduction of a nonlinear spring pendulum under multi external and parametric excitations via a longitudinal absorber. Meccanica 46, 325–340 (2011)

    Article  MathSciNet  Google Scholar 

  16. Eissa, M., Kamel, M., El-Sayed, A.T.: Vibration suppression of a four-degree-of-freedom nonlinear spring pendulum via a longitudinal and transverse absorber. J. Appl. Mech. 79(1), 011007 (2012)

    Article  Google Scholar 

  17. Eissa, M., EL-Ganaini, W.A.A., Hamed, Y.S.: Saturation, stability and resonance of non-linear systems. Physica A 356, 341–358 (2005)

    Article  MathSciNet  Google Scholar 

  18. Eissa, M.: Vibration and chaos control in I.C engines subjected to harmonic torque via non-linear absorbers. In: Second International Symposium on Mechanical Vibrations (ISMV-2000), Islamabad, Pakistan (2000)

    Google Scholar 

  19. Eissa, M., El-Ganaini, W.A.A.: Multi absorbers for vibration control of non-linear structures to harmonic excitations. Part I. In: ISMV Conference. Islamabad, Pakistan (2000)

    Google Scholar 

  20. Eissa, M., El-Ganaini, W.A.A.: Multi absorbers for vibration control of non-linear structures to harmonic excitations. Part II. In: ISMV Conference. Islamabad, Pakistan (2000)

    Google Scholar 

  21. Eissa, M.: Vibration control of non-linear mechanical system via a neutralizer. Electronic Bulletin, No. 16, Faculty of Electronic Engineering, Menouf, Egypt, 1999, July

  22. Nayfeh, A., Mook, D.: Nonlinear Oscillations. Wiley, New York (1995)

    Book  Google Scholar 

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Authors and Affiliations

  1. Department of Physics and Engineering Mathematics, Faculty of Electronic Engineering, Menoufia University, Menouf, 32952, Egypt

    W. A. El-Ganaini, N. A. Saeed & M. Eissa

Authors
  1. W. A. El-Ganaini
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  2. N. A. Saeed
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  3. M. Eissa
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Correspondence to N. A. Saeed.

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El-Ganaini, W.A., Saeed, N.A. & Eissa, M. Positive position feedback (PPF) controller for suppression of nonlinear system vibration. Nonlinear Dyn 72, 517–537 (2013). https://doi.org/10.1007/s11071-012-0731-5

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  • DOI: https://doi.org/10.1007/s11071-012-0731-5

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