Skip to main content
SpringerLink
Log in

Resonances of a forced van der Pol equation with parametric damping

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

Abstract

This work entails an analysis of secondary resonances in the parametrically damped van der Pol equation, with and without external excitation. A potential application of this system is a vertical-axis wind-turbine blade, which can have cyclic damping, aeroelastic self-excitation, and direct excitation. We analyze the system using the method of multiple scales and numerical solutions. For the case without external excitation, the analysis reveals nonresonant phase drift (quasiperiodic responses) and subharmonic resonance with possible phase drift or phase locking (periodic responses). The case of external excitation consists of a constant load and a harmonic load with the same frequency as the parametric term. Hard excitation is treated for nonresonant conditions and secondary resonances. Subharmonic and superharmonic resonances show possible phase drift and phase locking. Primary resonance is observed but not analyzed here.

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

Similar content being viewed by others

Data availability statement

The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.

References

  1. Afzali, F., Kapucu, O., and Feeny, B. F.: Vibrational analysis of vertical-axis wind-turbine blades. In: Proceedings of the ASME 2016 International Design Engineering Technical Conferences. Paper number IDETC2016-60374, Charlotte, North Carolina (2016)

  2. Afzali, F., Acar, G.D., Feeny, B.F.: A Floquet-based analysis of parametric excitation through the damping coefficient. J. Vib. Acoust. 143, 4 (2020)

    Google Scholar 

  3. Nayfeh, A.H., Mook, D.T.: Nonlinear Oscillations. Wiley, New York (2008)

    MATH  Google Scholar 

  4. Rand, R.H.: Lecture notes on nonlinear vibrations. https://ecommons.cornell.edu/handle/1813/28989 (2012)

  5. van der Pol, B.: The nonlinear theory of electrical oscillations. Proc. IRE 22(9), 1051–1086 (1934)

    Article  MATH  Google Scholar 

  6. Holmes, P.J., Rand, D.A.: Bifurcations of the forced van der Pol oscillator. Q. Appl. Math. 35(4), 495–509 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  7. Barbosa, R.S., Machado, J.T., Vinagre, B., Calderon, A.: Analysis of the van der Pol oscillator containing derivatives of fractional order. J. Vib. Control 13(9–10), 1291–1301 (2007)

    Article  MATH  Google Scholar 

  8. Náprstek, J., Fischer, C.: Super and sub-harmonic synchronization in generalized van der Pol oscillator. Comput. Struct. 224, 106103 (2019)

    Article  Google Scholar 

  9. Barrón, M.A., Sen, M.: Synchronization of four coupled van der Pol oscillators. Nonlinear Dyn. 56(4), 357–367 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  10. Ward, M.: Lecture Notes on Basic Floquet Theory. http://www.emba.uvm.edu/jxyang/teaching/ (2010)

  11. Hartono, Burgh, A.H.P.: An Equation Time-Periodic Damping Coefficient: Stability Diagram and an Application. Delft University of Technology, Delft (2002)

    MATH  Google Scholar 

  12. Acar, G., Feeny, B.F.: Floquet-based analysis of general responses of the Mathieu equation. J. Vib. Acoust. 138(4), 0410179 (2016)

    Article  Google Scholar 

  13. Afzali, F., Feeny, B.F.: Response characteristics of systems with parametric excitation through damping and stiffness. In: ASME International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, American Society of Mechanical Engineers. paper number DETC2020-22457 (2020)

  14. Month, L., Rand, R.H.: Bifurcation of 4–1 subharmonics in the nonlinear Mathieu equation. Mech. Res. Commun. 9(4), 233–240 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  15. Ng, L., Rand, R.H.: Bifurcations in a Mathieu equation with cubic nonlinearities. Chaos Solitons Fractals 14(2), 173–181 (2002)

  16. Tondl, A., Ecker, H.: On the problem of self-excited vibration quenching by means of parametric excitation. Appl. Mech. 72, 923–932 (2003)

  17. Veerman, F., Verhulst, F.: Quasiperiodic phenomena in the van der Pol–Mathieu equation. J. Sound Vib. 326(1–2), 314–320 (2009)

  18. Rugar, D., Grutter, P.: Mechanical parametric amplification and thermomechanical noise squeezing. Phys. Rev. Lett. 67(6), 699–702 (1991)

    Article  Google Scholar 

  19. Guennoun, K., Houssni, M., Belhaq, M.: Quasiperiodic solutions and stability for a weakly damped nonlinear quasiperiodic Mathieu equation. Nonlinear Dyn. 27(3), 211–236 (2002)

    Article  MATH  Google Scholar 

  20. Rhoads, J.F., Shaw, S.W.: The impact of nonlinearity on degenerate parametric amplifiers. Appl. Phys. Lett. 96(23), 234101 (2010)

    Article  Google Scholar 

  21. Ramakrishnan, V., Feeny, B.F.: Resonances of a forced Mathieu equation with reference to wind turbine blades. J. Vib. Acoust. 134(6), 064501 (2012)

  22. Inoue, T., Ishida, Y., Kiyohara, T.: Nonlinear vibration analysis of the wind turbine blade (occurrence of the superharmonic resonance in the out of plane vibration of the elastic blade). J. Vib. Acoust. 134(3), 031009 (2012)

    Article  Google Scholar 

  23. Sharma, A.: A re-examination of various resonances in parametrically excited systems. J. Vib. Acoust. 142(3), 03101011 (2020)

    Article  Google Scholar 

  24. Ramakrishnan, V., Feeny, B.F.: Primary parametric amplification in a weakly forced Mathieu equation. J. Vib. Acoust. 144(5), 051006 (2022)

    Article  Google Scholar 

  25. Goswami, I., Scanlan, R.H., Jones, N.P.: Vortex-induced vibration of circular cylinders. ii: New model. J. Eng. Mech-Asce. 119(11), 2288–2302 (1993)

    Article  Google Scholar 

  26. Pandey, M., Rand, R.H., Zehnder, A.T.: Frequency locking in a forced Mathieu–van-der-Pol–Duffing system. Nonlinear Dyn. 54(1–2), 3–12 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  27. Belhaq, M., Fahsi, A.: 2: 1 and 1: 1 frequency-locking in fast excited van der Pol-Mathieu-Duffing oscillator. Nonlinear Dyn. 53(1), 139–152 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  28. Szabelski, K., Warmiński, J.: Parametric self-excited non-linear system vibrations analysis with inertial excitation. Int. J. Non-Linear Mech. 30(2), 179–189 (1995)

    Article  MATH  Google Scholar 

  29. Warminski, J.: Nonlinear dynamics of self-, parametric, and externally excited oscillator with time delay: van der Pol versus Rayleigh models. Nonlinear Dyn. 99(1), 35–56 (2020)

    Article  MATH  Google Scholar 

  30. Chakraborty, S., Sarkar, A.: Parametrically excited non-linearity in van der Pol oscillator: resonance, anti-resonance and switch. Physica D 254, 24–28 (2013)

  31. Allen, M.S., Sracic, M.W., Chauhan, S., Hansen, M.H.: Output-only modal analysis of linear time-periodic systems with application to wind turbine simulation data. Mech. Syst. Signal Process. 25(4), 1174–1191 (2011)

    Article  Google Scholar 

  32. Acar, G.D., Acar, M.A., Feeny, B.F.: Parametric resonances of a three-blade-rotor system with reference to wind turbines. J. Vib. Acoust. 142(2), 0210139 (2020)

  33. Luongo, A., Zulli, D.: Parametric, external and self-excitation of a tower under turbulent wind flow. J. Sound Vib. 330(13), 3057–3069 (2011)

    Article  Google Scholar 

  34. Nayfeh, A.H.: Perturbation Methods. Wiley, New York (2008)

    Google Scholar 

  35. Guckenheimer, J., Holmes, P.: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer, New York (1983)

  36. Ramakrishnan, V.: Analysis of wind turbine blade vibration and drivetrain loads. PhD thesis, Michigan State University, East Lansing (2017)

Download references

Funding

This work is based on a project supported by the National Science Foundation, under grant CMMI-1435126. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.

Author information

Authors and Affiliations

  1. Michigan State University, East Lansing, MI, 48824, USA

    Fatemeh Afzali & Brian F. Feeny

  2. Brown University, Providence, RI, 02912, USA

    Ehsan Kharazmi

Authors
  1. Fatemeh Afzali
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ehsan Kharazmi
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Brian F. Feeny
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Fatemeh Afzali.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Afzali, F., Kharazmi, E. & Feeny, B.F. Resonances of a forced van der Pol equation with parametric damping. Nonlinear Dyn 111, 5269–5285 (2023). https://doi.org/10.1007/s11071-022-08026-w

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-022-08026-w

Keywords

Search

Navigation