Skip to main content
SpringerLink
Log in

A novel fractional-order model and controller for vibration suppression in flexible smart beam

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

Vibration suppression represents an important research topic due to the occurrence of this phenomenon in multiple domains of life. In airplane wings, vibration can cause discomfort and can even lead to system failure. One of the most frequently used means of studying vibrations in airplane wings is through the use of dedicated flexible beams, equipped with sensing and actuating mechanisms powered by suitable control algorithms. In order to optimally reject these vibrations by means of closed-loop control strategies, the availability of a model is required. So far, the modeling of these smart flexible beams has been limited to deliver integer order transfer functions models. This paper, however, describes the mathematical framework used to derive a fractional-order impedance lumped model for capturing frequency response of a flexible beam system exposed to a multisine excitation. The theoretical foundation stems from fractional calculus applied in combination with transmission line theory and wave equations. The simplified model reduces to a minimal number of parameters when converging to a limit value. It is shown that the fractional-order model outperforms an integer order model of the smart beam. Based on this novel fractional-order model, a fractional-order \(\hbox {PD}^\mu \) controller is then tuned. The controller design is based on shaping the frequency response of the closed-loop system such that the resonant peak is reduced in comparison to the uncompensated smart beam system and disturbances are rejected. Experimental results, considering a custom-built smart beam system, are provided, considering both passive and active control situations, showing that a significant improvement in the closed-loop behavior is obtained using the proposed controller. Comparisons with a fractional-order \(\hbox {PD}^\mu \) controller, tuned according to classical open-loop frequency domain design specifications, are provided. The experimental results show that the proposed tuning technique leads to similar results as the classical approach. Thus, the proposed method is a viable alternative, being based on closed-loop specifications, which is more intuitive for practitioners.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16

Similar content being viewed by others

References

  1. Ionescu, C.M.: The phase constancy in neural dynamics. IEEE Trans. Syst. Man Cybern. Part A Syst. Hum. 42, 1543–1551 (2012)

    Article  Google Scholar 

  2. Chen, L., Basu, B., McCabe, D.: Fractional order models for system identification of thermal dynamics of buildings. Energy Build. 133, 381–388 (2016)

    Article  Google Scholar 

  3. Zhu, L., Knospe, C.R.: Modeling of nonlaminated electromagnetic suspension systems. IEEE-ASME Trans. Mech. 15, 59–69 (2010)

    Article  Google Scholar 

  4. Sapora, A., Cornetti, P., Carpinteri, A., Baglieri, O., Santagata, E.: The use of fractional calculus to model the experimental creep-recovery behavior of modified bituminous binders. Mater. Struct. 49, 45–55 (2016)

    Article  Google Scholar 

  5. Monje, C.A., Vinagre, B.M., Santamara, G.E., Tejado, I.: Auto-tuning of fractional order PID controllers using a PLC. In: 14th IEEE ETFA Conference (2009)

  6. Muresan, C.I., Folea, S., Mois, G., Dulf, E.H.: Development and Implementation of an FPGA based fractional order controller for a DC motor. J. Mech. 23, 798–804 (2013)

    Google Scholar 

  7. Muresan, C.I., Ionescu, C., Folea, S., De Keyser, R.: Fractional order control of unstable processes: the magnetic levitation study case. J. Nonlinear Dyn. 80, 1761–1772 (2015). https://doi.org/10.1007/s11071-014-1335-z

    Article  Google Scholar 

  8. Oustaloup, A., Sabatier, J., Lanusse, P.: From fractional robustness to CRONE control. Fract. Calc. Appl. Anal. 2, 130 (1999)

    MATH  Google Scholar 

  9. Oustaloup, A.: La Commande CRONE: Commande Robuste dOrdre Non Entiere. Hermes, Paris (1991)

    MATH  Google Scholar 

  10. Podlubny, I.: Fractional-order systems and PID controllers. IEEE Trans. Autom. Control 44, 208214 (1999)

    Article  Google Scholar 

  11. Folea, S., De Keyser, R., Birs, I.R., Muresan, C.I., Ionescu, C.: Discrete-time implementation and experimental validation of a fractional order PD controller for vibration suppression in airplane wings. Acta Hung. 14, 191–206 (2017)

    Google Scholar 

  12. Yoshitani, N., Kuroda, M.: Fractional-order controller design based on the Nyquist diagram for the vibration control of a flexible beam. In: 9th European Nonlinear Dynamics Conference (2017)

  13. Birs, I.R., Muresan, C.I., Folea, S., Prodan, O., Kovacs, L.: Vibration suppression with fractional-order PID controller. In: IEEE International Conference on Automation, Quality and Testing, Robotics AQTR, Cluj-Napoca, Romania, 19–21 (May 2016). https://doi.org/10.1109/AQTR.2016.7501365

  14. Onat, C., Sahin, M., Yaman, Y.: Performance analysis of a fractional controller developed for the vibration suppression of a smart beam. In: Proceedings of the 5th Thematic Conference on Smart Structures and Materials, 213–222, Saarland University, 6–8 July 2011, Saarbrcken (2011)

  15. Onat, C., Sahin, M., Yaman, Y.: Fractional controller design for suppressing smart beam vibrations. Aircr. Eng. Aerosp. Technol. 84, 203–212 (2012)

    Article  Google Scholar 

  16. Cao, J.-Y., Cao, B.-G.: Design of fractional order controller based on particle swarm optimization. Int. J. Control Autom. Syst. 4, 775–781 (2006)

    Google Scholar 

  17. Monje, C.A., Chen, Y.Q., Vinagre, B.M., Xue, D., Feliu, V.: Fractional Order Systems and Controls: Fundamentals and Applications. Springer, London (2010)

    Book  MATH  Google Scholar 

  18. De Keyser, R., Muresan, C.I., Ionescu, C.: A novel auto-tuning method for fractional order PI/PD controllers. ISA Trans. 62, 268–275 (2016)

    Article  Google Scholar 

  19. Fey, R.H.B., Wouters, R.M.T., Nijmeijer, H.: Proportional and derivative control for steady-state vibration mitigation in a piecewise linear beam system. J. Nonlinear Dyn. 60, 535549 (2010). https://doi.org/10.1007/s11071-009-9613-x

    Article  MATH  Google Scholar 

  20. Weldegiorgis, R., Krishna, P., Gangadharan, K.V.: Vibration control of a smart cantilever beam using strain rate feedback. Procedia Mater. Sci. 5, 113–122 (2014)

    Article  Google Scholar 

  21. Abdelhafez, H., Nassar, M.: Effects of time delay on an active vibration control of a forced and self-excited nonlinear beam. J. Nonlinear Dyn. 86, 137151 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  22. Bauomy, H.S.: Active vibration control of a dynamical system via negative linear velocity feedback. J. Nonlinear Dyn. 77, 413423 (2014)

    MathSciNet  MATH  Google Scholar 

  23. Zhang, S.Q., Schmidt, R.: LQR control for vibration suppression of piezoelectric integrated smart structures. Proc. Appl. Math. Mech. 12, 695 696 (2012)

    Google Scholar 

  24. Takcs, G., Polni, T., Rohal-Ilkiv, B.: Adaptive model predictive vibration control of a cantilever beam with real-time parameter estimation, shock and vibration, Article ID 741765 (2014). https://doi.org/10.1155/2014/741765

  25. Zori, N.D., Simonovi, A.M., Mitrovi, Z.S., Stupar, S.N., Obradovi, A.M., Luki, N.S.: Free vibration control of smart composite beams using particle swarm optimized self-tuning fuzzy logic controller. J. Sound Vib. 333, 52445268 (2014)

    Google Scholar 

  26. Stavroulakis, G.E., Foutsitzi, G., Hadjigeorgiou, E., Marinova, D., Baniotopoulos, C.C.: Design and robust optimal control of smart beams with application on vibrations suppression. Adv. Eng. Softw. 36, 806813 (2005)

    Article  Google Scholar 

  27. Mattice, M., Coleman, N., Craig, K.: Tip-position control of a flexible beam: modelling approaches and experimental verification. Technical report ARFSD-TR-90003 (1990)

  28. Rathi, V., Khan, A.H.: Vibration attenuation and shape control of surface mounted, embedded smart beam. Latin Am. J. Solids Struct. 1, 1 25 (2012)

    Google Scholar 

  29. Ionescu, C., Machado, J.A.T., De Keyser, R.: Fractional-order impulse response of the respiratory system. Comput. Math. Appl. 62, 845–854 (2011)

    Article  MATH  Google Scholar 

  30. Meral, F.C., Royston, T.J., Magin, R.: Fractional calculus in viscoelasticity: an experimental study. Commun. Nonlinear Sci. Numer. Simul. 15, 939945 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  31. Sasso, M., Palmieri, G., Amodio, D.: Application of fractional derivative models in linear viscoelastic problems. Mech. Time-Depend. Mater. 15, 367–387 (2011)

    Article  Google Scholar 

  32. Xue, D., Chen, Y.: A comparative introduction of four fractional order controllers. In: Proceedings of the 4th IEEE World Congress on Intelligent Control and Automation, pp. 3228–3235 (2002)

  33. Luo, Y., Chen, Y.Q., Wang, C.Y., Pi, Y.G.: Tuning fractional order proportional integral controllers for fractional order systems. J. Process Control 20, 823831 (2010)

    Article  Google Scholar 

  34. Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions: With Formulas, Graphs and Mathematical Tables. U.S. Dept. of Commerce, Washington, D.C., USA (1972)

    MATH  Google Scholar 

  35. Oustaloup, A: Diversity and Non-integer Differentiation for System Dynamics (E-book). Wiley (2014)

  36. Oustaloup, A.: La Derivation Non-entiere. Hermes, Paris (1999)

    MATH  Google Scholar 

  37. Ionescu, C.M.: The Human Respiratory System. Springer, Berlin (2013)

    Book  MATH  Google Scholar 

  38. Lewandowski, R., Pawlak, Z.: Dynamic analysis of frames with viscoelastic dampers modelled by rheological models with fractional derivatives. J. Sound Vib. 330, 923936 (2011)

    Article  Google Scholar 

  39. Hu, S., Chen, W., Gou, X.: Modal analysis of fractional derivative damping model of frequency-dependent viscoelastic soft matter. Adv. Vib. Eng. 10, 187–196 (2011)

    Google Scholar 

Download references

Acknowledgements

This work was supported by a grant of the Romanian National Authority for Scientific Research and Innovation, CNCS UEFISCDI, Project Number PN-II-RU-TE-2014-4-0598, TE 86/2015.

Author information

Authors and Affiliations

  1. Department of Automation, Technical University of Cluj-Napoca, Gh. Baritiu, No. 26-28, Cluj-Napoca, Romania

    Cristina I. Muresan, Silviu Folea & Isabela R. Birs

  2. Department of Electrical Energy, Metals, Mechanical Constructions and Systems, Ghent University, Technologiepark, 914, 9052, Ghent, Belgium

    Clara Ionescu

Authors
  1. Cristina I. Muresan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Silviu Folea
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Isabela R. Birs
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Clara Ionescu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Cristina I. Muresan.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Muresan, C.I., Folea, S., Birs, I.R. et al. A novel fractional-order model and controller for vibration suppression in flexible smart beam. Nonlinear Dyn 93, 525–541 (2018). https://doi.org/10.1007/s11071-018-4207-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-018-4207-0

Keywords

Search

Navigation