Potential well escape in a galloping twin-well oscillator

  • Hussam AlhusseinEmail author
  • Mohammad F. Daqaq
Original paper


When a bi-stable oscillator undergoes a supercritical Hopf bifurcation due to a galloping instability, intra-well limit cycle oscillations of small amplitude are born. The amplitude of these oscillations grows as the flow speed is increased to a critical speed at which the dynamic trajectories escape the potential well. The goal of this paper is to obtain a simple yet accurate analytical expression to approximate the escape speed. To this end, three different analytical approaches are implemented: (i) the method of harmonic balance, (ii) the method of multiple scales using harmonic and elliptic basis functions, and (iii) the Melnikov criterion. All methods yielded an identical expression for the escape speed with only one key difference which lies in the value of a constant that changes among the different methods. A comparison between the approximate analytical solutions and a numerical integration of the equation of motion demonstrated that the escape speed obtained via the multiple scales method using the elliptic functions and the Melnikov criterion are in excellent agreement with the numerical simulations. On the other hand, the first-order harmonic balance technique and the multiple scales using harmonic functions provide analytical estimates that significantly underestimate the actual escape speed. Using the Melnikov criterion, the influence of parametric and additive noise on the escape speed was also studied.


Bi-stable Morphing Galloping Noise 


Compliance with ethical standards

Conflicts of interest

The authors declare that they have no conflict of interest.

Supplementary material


  1. 1.
    Zhou, S., Cao, J., Erturk, A., Lin, J.: Enhanced broadband piezoelectric energy harvesting using rotatable magnets. Appl. Phys. Lett. 102(17), 173901 (2013)CrossRefGoogle Scholar
  2. 2.
    Cottone, F., Vocca, H., Gammaitoni, L.: Nonlinear energy harvesting. Phys. Rev. Lett. 102(8), 080601 (2009)CrossRefGoogle Scholar
  3. 3.
    Mann, B.P., Owens, B.A.: Investigations of a nonlinear energy harvester with a bistable potential well. J. Sound Vib. 329(9), 1215–1226 (2010)CrossRefGoogle Scholar
  4. 4.
    Arrieta, A.F., Hagedorn, P., Erturk, A., Inman, D.J.: A piezoelectric bistable plate for nonlinear broadband energy harvesting. Appl. Phys. Lett. 97(10), 104102 (2010)CrossRefGoogle Scholar
  5. 5.
    Arrieta, A.F., Neild, S.A., Wagg, D.J.: On the cross-well dynamics of a bi-stable composite plate. J Sound Vib. 330(14), 3424–3441 (2011)CrossRefGoogle Scholar
  6. 6.
    Thompson, J.M.T.: Chaotic phenomena triggering the escape from a potential well. Proc. R. Soc. Lond. A Math. Phys. Sci. 421(1861), 195–225 (1989)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Virgin, L.N., Plaut, R.H., Cheng, C.-C.: Prediction of escape from a potential well under harmonic excitation. Int. J. Nonlinear Mech. 27(3), 357–365 (1992)CrossRefGoogle Scholar
  8. 8.
    Udani, J.P., Arrieta, A.F.: Efficient potential well escape for bi-stable duffing oscillators. Nonlinear Dyn. 92(3), 1045–1059 (2018)CrossRefGoogle Scholar
  9. 9.
    Cammarano, A., Burrow, S.G., Barton, D.A.: Modelling and experimental characterization of an energy harvester with bi-stable compliance characteristics. Proc. Inst. Mech. Eng. Part I J. Syst. Control Eng. 225(4), 475–484 (2011)CrossRefGoogle Scholar
  10. 10.
    Nguyen, S.D., Halvorsen, E., Paprotny, I.: Bistable springs for wideband microelectromechanical energy harvesters. Appl. Phys. Lett. 102(2), 023904 (2013)CrossRefGoogle Scholar
  11. 11.
    Arnold, D.P.: Review of microscale magnetic power generation. IEEE Trans. Magn. 43(11), 3940–3951 (2007)CrossRefGoogle Scholar
  12. 12.
    Mitcheson, P.D., Miao, P., Stark, B.H., Yeatman, E.M., Holmes, A.S., Green, T.C.: MEMS electrostatic micropower generator for low frequency operation. Sens. Actuators A Phys. 115(2–3), 523–529 (2004) CrossRefGoogle Scholar
  13. 13.
    Zarepoor, M., Bilgen, O.: Constrained-energy cross-well actuation of bistable structures. AIAA J. 54, 2905–2908 (2016) CrossRefGoogle Scholar
  14. 14.
    Gendelman, O.: Escape of a harmonically forced particle from an infinite-range potential well: a transient resonance. Nonlinear Dyn. 93(1), 79–88 (2018)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Neiman, A., Schimansky-Geier, L.: Stochastic resonance in bistable systems driven by harmonic noise. Phys. Rev. Lett. 72(19), 2988 (1994)CrossRefGoogle Scholar
  16. 16.
    Harne, R.L., Wang, K.: On the fundamental and superharmonic effects in bistable energy harvesting. J. Intell. Mater. Syst. Struct. 25(8), 937–950 (2014)CrossRefGoogle Scholar
  17. 17.
    Bibo, A., Alhadidi, A.H., Daqaq, M.F.: Exploiting a nonlinear restoring force to improve the performance of flow energy harvesters. J. Appl. Phys. 117(4), 045103 (2015)CrossRefGoogle Scholar
  18. 18.
    Alhadidi, A.H., Daqaq, M.F.: A broadband bi-stable flow energy harvester based on the wake-galloping phenomenon. Appl. Phys. Lett. 109(3), 033904 (2016)CrossRefGoogle Scholar
  19. 19.
    Kuder, I.K., Arrieta, A.F., Rist, M., Ermanni, P.: Aeroelastic response of a selectively compliant morphing aerofoil featuring integrated variable stiffness bi-stable laminates. J. Intell. Mater. Syst. Struct. 27(14), 1949–1966 (2016)CrossRefGoogle Scholar
  20. 20.
    Arrieta, A.F., Kuder, I.K., Rist, M., Waeber, T., Ermanni, P.: Passive load alleviation aerofoil concept with variable stiffness multi-stable composites. Compos. Struct. 116, 235–242 (2014)CrossRefGoogle Scholar
  21. 21.
    Novak, M.: Galloping oscillations of prismatic structures. J. Eng. Mech. 98, 27–46 (1972) Google Scholar
  22. 22.
    Païdoussis, M.P., Price, S.J., De Langre, E.: Fluid-Structure Interactions: Cross-Flow-Induced Instabilities. Cambridge University Press, Cambridge (2010)CrossRefGoogle Scholar
  23. 23.
    Marsden, J.E., McCracken, M.: The Hopf Bifurcation and Its Applications, vol. 19. Springer, Berlin (2012)zbMATHGoogle Scholar
  24. 24.
    Thompson, J.M.T., Sieber, J.: Nonlinear dynamic interactions between flow-induced galloping and shell-like buckling. Int. J. Mech. Sci. 91, 91–98 (2015)CrossRefGoogle Scholar
  25. 25.
    Barrero-Gil, A., Alonso, G., Sanz-Andres, A.: Energy harvesting from transverse galloping. J. Sound Vib. 329(14), 2873–2883 (2010)CrossRefGoogle Scholar
  26. 26.
    Sirohi, J., Mahadik, R.: Harvesting wind energy using a galloping piezoelectric beam. J. Vib. Acoust. 134(1), 011009 (2012)CrossRefGoogle Scholar
  27. 27.
    Daqaq, M.F., Bibo, A., Akhtar, I., Alhadidi, A.H., Panyam, M., Caldwell, B., Noel, J.: Micropower generation using cross-flow instabilities: a review of the literature and its implications. J. Vib. Acoust. 141(3), 030801 (2019)CrossRefGoogle Scholar
  28. 28.
    Parkinson, G., Smith, J.D.: The square prism as an aeroelastic non-linear oscillator. Q. J. Mech. Appl. Math. 17(2), 225–239 (1964)CrossRefGoogle Scholar
  29. 29.
    Szemplinska-Stupnicka, W., Rudowski, J.: Local methods in predicting occurrence of chaos in two-well potential systems: superharmonic frequency region. J. Sound Vib. 152(1), 57–72 (1992)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Bowman, F.: Introduction to Elliptic Functions: With Applications. Dover Publications, New York (1961)zbMATHGoogle Scholar
  31. 31.
    Barkham, P., Soudack, A.: An extension to the method of Kryloff and Bogoliuboff. Int. J. Control 10(4), 377–392 (1969)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Lakrad, F., Belhaq, M.: Periodic solutions of strongly non-linear oscillators by the multiple scales method. J. Sound Vib. 258(4), 677–700 (2002)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Nayfeh, A.H., Balachandran, B.: Applied Nonlinear Dynamics: Analytical, Computational, and Experimental Methods. Wiley, Hoboken (2008)zbMATHGoogle Scholar
  34. 34.
    Kovacic, I., Cveticanin, L., Zukovic, M., Rakaric, Z.: Jacobi elliptic functions: a review of nonlinear oscillatory application problems. J. Sound Vib. 380, 1–36 (2016)CrossRefGoogle Scholar

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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Department of Mechanical Engineering, Tandon School of EngineeringNew York UniversityBrooklynUSA
  2. 2.Division of EngineeringNew York University Abu DhabiAbu DhabiUAE

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