Skip to main content
SpringerLink
Log in

Theoretical aspects of transverse galloping

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

Abstract

We investigate theoretical aspects of a reduced-order model for transverse galloping, which is derived directly from the full governing equations, i.e., the fluid mass and momentum balances, the fluid far-field boundary condition, the kinematic and dynamic conditions at the interface, and the bluff body equation of motion. We show that the presence of low-intensity turbulence and back-action from the slow transverse oscillations of the bluff body yield a correction to the hydrodynamic force of the quasi-steady theory in the form of additive random excitation. The reduced-order model consists of a pair of nonlinear Langevin equations for the amplitude and phase of the transverse motion of the bluff body. We show that while the phase dynamics is associated with a strongly diffusive random walk motion, the amplitude dynamics is associated with a relatively weak diffusion and can be mapped onto the motion of an overdamped particle trapped in a potential well. This mapping provides a highly useful tool for understanding both the deterministic (no random excitation) and the stochastic (weak random excitation) amplitude dynamics, and hence, for yielding theoretical insights on the overall system dynamics.

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.

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

Notes

  1. Under the assumption that \(U^*\gg St^{-1}\); otherwise, when \(U^*\sim \ St^{-1}\), vortex-induced vibration can occur [39].

References

  1. Abdelkefi, A., Hajj, M., Nayfeh, A.: Piezoelectric energy harvesting from transverse galloping of bluff bodies. Smart Mater. Struct. 22(1), 015,014 (2012)

    Article  Google Scholar 

  2. Alonso, G., Meseguer, J., Sanz-Andrés, A., Valero, E.: On the galloping instability of two-dimensional bodies having elliptical cross-sections. J. Wind Eng. Ind. Aerodyn. 98(8), 438–448 (2010)

    Article  Google Scholar 

  3. Alonso, G., Pérez-Grande, I., Meseguer, J.: Galloping instabilities of z-shaped shading louvers. Indoor Built Environ. 26(9), 1198–1213 (2017)

    Article  Google Scholar 

  4. Alonso, G., Sanz-Lobera, A., Meseguer, J.: Hysteresis phenomena in transverse galloping of triangular cross-section bodies. J. Fluids Struct. 33, 243–251 (2012)

    Article  Google Scholar 

  5. Alonso, G., Valero, E., Meseguer, J.: An analysis on the dependence on cross section geometry of galloping stability of two-dimensional bodies having either biconvex or rhomboidal cross sections. Eur. J. Mech. B Fluids 28(2), 328–334 (2009)

    Article  MATH  Google Scholar 

  6. Andrianne, T., Aryoputro, R.P., Laurent, P., Colson, G., Amandolèse, X., Hémon, P.: Energy harvesting from different aeroelastic instabilities of a square cylinder. J. Wind Eng. Ind. Aerodyn. 172, 164–169 (2018)

    Article  Google Scholar 

  7. Barrero-Gil, A., Alonso, G., Sanz-Andres, A.: Energy harvesting from transverse galloping. J. Sound Vib. 329(14), 2873–2883 (2010)

    Article  Google Scholar 

  8. Barrero-Gil, A., Sanz-Andrés, A., Alonso, G.: Hysteresis in transverse galloping: the role of the inflection points. J. Fluids Struct. 25(6), 1007–1020 (2009)

    Article  Google Scholar 

  9. Bearman, P., Gartshore, I., Maull, D., Parkinson, G.: Experiments on flow-induced vibration of a square-section cylinder. J. Fluids Struct. 1(1), 19–34 (1987)

    Article  Google Scholar 

  10. Bearman, P., Luo, S.: Investigation of the aerodynamic instability of a square-section cylinder by forced oscillation. J. Fluids Struct. 2(2), 161–176 (1988)

    Article  Google Scholar 

  11. Blevins, R.D.: Flow-induced Vibration, 377 pp. Van Nostrand Reinhold Co., New York (1977)

  12. Bokaian, A., Geoola, F.: Effects of vortex-resonance on nearby galloping instability. J. Eng. Mech. 111(5), 591–609 (1985)

    Article  Google Scholar 

  13. Chandrasekhar, S.: Stochastic problems in physics and astronomy. Rev. Mod. Phys. 15(1), 1 (1943)

    Article  MathSciNet  MATH  Google Scholar 

  14. Chawla, K.K., Meyers, M.: Mechanical behavior of materials. Prentice Hall, New Jersey (1999)

    MATH  Google Scholar 

  15. Demir, A., Mehrotra, A., Roychowdhury, J.: Phase noise in oscillators: a unifying theory and numerical methods for characterization. IEEE Trans. Circuits Syst. I Fundam. Theory Appl. 47(5), 655–674 (2000)

    Article  Google Scholar 

  16. Den Hartog, J.: Transmission line vibration due to sleet. Trans. Am. Inst. Electr. Eng. 51(4), 1074–1076 (1932)

    Article  Google Scholar 

  17. Dowell, E.H., Curtiss, H.C., Scanlan, R.H., Sisto, F.: A modern course in aeroelasticity, vol. 3. Springer, Berlin (1989)

    Book  MATH  Google Scholar 

  18. Dykman, M., Golding, B., Ryvkine, D.: Critical exponent crossovers in escape near a bifurcation point. Phys. Rev. Lett. 92(8), 080602 (2004)

    Article  Google Scholar 

  19. Dykman, M., Krivoglaz, M.: Fluctuations in nonlinear systems near bifurcations corresponding to the appearance of new stable states. Physica A Stat. Mech. Appl. 104(3), 480–494 (1980)

    Article  MathSciNet  Google Scholar 

  20. Dykman, M., Krivoglaz, M.: Theory of nonlinear oscillator interacting with a medium. Sov. Phys. Rev. 5, 265–442 (1984)

    Google Scholar 

  21. Gandia, F., Meseguer, J., Sanz-Andrés, A.: Static and dynamic experimental analysis of the galloping stability of porous h-section beams. Sci. World J. 2014, 746826 (2014). https://doi.org/10.1155/2014/746826

    Article  Google Scholar 

  22. Gao, G., Zhu, L.: Measurement and verification of unsteady galloping force on a rectangular 2: 1 cylinder. J. Wind Eng. Ind. Aerodyn. 157, 76–94 (2016)

    Article  Google Scholar 

  23. Gao, Gz, Zhu, Ld: Nonlinear mathematical model of unsteady galloping force on a rectangular 2: 1 cylinder. J. Fluids Struct. 70, 47–71 (2017)

    Article  Google Scholar 

  24. Guckenheimer, J., Holmes, P.J.: Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, vol. 42. Springer, Berlin (2013)

    MATH  Google Scholar 

  25. Hänggi, P., Talkner, P., Borkovec, M.: Reaction-rate theory: fifty years after kramers. Rev. Mod. Phys. 62(2), 251 (1990)

    Article  MathSciNet  Google Scholar 

  26. Ibarra, D., Sorribes, F., Alonso, G., Meseguer, J.: Transverse galloping of two-dimensional bodies having a rhombic cross-section. J. Sound Vib. 333(13), 2855–2865 (2014)

    Article  Google Scholar 

  27. Joly, A., Etienne, S., Pelletier, D.: Galloping of square cylinders in cross-flow at low reynolds numbers. J. Fluids Struct. 28, 232–243 (2012)

    Article  Google Scholar 

  28. Jung, H.J., Lee, S.W.: The experimental validation of a new energy harvesting system based on the wake galloping phenomenon. Smart Mater. Struct. 20(5), 055022 (2011)

    Article  Google Scholar 

  29. Khinchin, A.Y.: Theory of correlation of stationary stochastic processes. Uspekhi matematicheskikh nauk 5, 42–51 (1938)

    Google Scholar 

  30. Kluger, J., Moon, F., Rand, R.: Shape optimization of a blunt body vibro-wind galloping oscillator. J. Fluids Struct. 40, 185–200 (2013)

    Article  Google Scholar 

  31. Kramers, H.A.: Brownian motion in a field of force and the diffusion model of chemical reactions. Physica 7(4), 284–304 (1940)

    Article  MathSciNet  MATH  Google Scholar 

  32. Luo, S., Chew, Y., Ng, Y.: Hysteresis phenomenon in the galloping oscillation of a square cylinder. J. Fluids Struct. 18(1), 103–118 (2003)

    Article  Google Scholar 

  33. Luongo, A., Di Fabio, F.: Multimodal galloping of dense spectra structures. J. Wind Eng. Ind. Aerodyn. 48(2–3), 163–174 (1993)

    Article  Google Scholar 

  34. Luongo, A., Paolone, A., Piccardo, G.: Postcritical behavior of cables undergoing two simultaneous galloping modes. Meccanica 33(3), 229–242 (1998)

    Article  MATH  Google Scholar 

  35. Luongo, A., Piccardo, G.: Non-linear galloping of sagged cables in 1: 2 internal resonance. J. Sound Vib. 214(5), 915–940 (1998)

    Article  Google Scholar 

  36. Luongo, A., Piccardo, G.: Linear instability mechanisms for coupled translational galloping. J. Sound Vib. 288(4–5), 1027–1047 (2005)

    Article  Google Scholar 

  37. Luongo, A., Zulli, D., Piccardo, G.: Analytical and numerical approaches to nonlinear galloping of internally resonant suspended cables. J. Sound Vib. 315(3), 375–393 (2008)

    Article  Google Scholar 

  38. Luongo, A., Zulli, D., Piccardo, G.: On the effect of twist angle on nonlinear galloping of suspended cables. Comput. Struct. 87(15–16), 1003–1014 (2009)

    Article  Google Scholar 

  39. Mannini, C., Marra, A., Bartoli, G.: Viv-galloping instability of rectangular cylinders: review and new experiments. J. Wind Eng. Ind. Aerodyn. 132, 109–124 (2014)

    Article  Google Scholar 

  40. Mannini, C., Marra, A.M., Bartoli, G.: Experimental investigation on viv-galloping interaction of a rectangular 3: 2 cylinder. Meccanica 50(3), 841–853 (2015)

    Article  Google Scholar 

  41. Mannini, C., Marra, A.M., Massai, T., Bartoli, G.: Interference of vortex-induced vibration and transverse galloping for a rectangular cylinder. J. Fluids Struct. 66, 403–423 (2016)

    Article  Google Scholar 

  42. Mannini, C., Massai, T., Marra, A.M.: Modeling the interference of vortex-induced vibration and galloping for a slender rectangular prism. J. Sound Vib. 419, 493–509 (2018)

    Article  Google Scholar 

  43. Mannini, C., Massai, T., Marra, A.M.: Unsteady galloping of a rectangular cylinder in turbulent flow. J. Wind Eng. Ind. Aerodyn. 173, 210–226 (2018)

    Article  Google Scholar 

  44. Mannini, C., Massai, T., Marra, A.M., Bartoli, G.: Interference of vortex-induced vibration and galloping: Experiments and mathematical modelling. Procedia Eng. 199, 3133–3138 (2017)

    Article  Google Scholar 

  45. Maxey, M.R., Riley, J.J.: Equation of motion for a small rigid sphere in a nonuniform flow. Phys. Fluids 26(4), 883–889 (1983)

    Article  MATH  Google Scholar 

  46. McComber, P., Paradis, A.: A cable galloping model for thin ice accretions. Atmos. Res. 46(1), 13–25 (1998)

    Article  Google Scholar 

  47. Nakamura, Y., Matsukawa, T.: Vortex excitation of rectangular cylinders with a long side normal to the flow. J. Fluid Mech. 180, 171–191 (1987)

    Article  Google Scholar 

  48. Ng, Y., Luo, S., Chew, Y.: On using high-order polynomial curve fits in the quasi-steady theory for square-cylinder galloping. J. Fluids Struct. 20(1), 141–146 (2005)

    Article  Google Scholar 

  49. Niu, H., Zhou, S., Chen, Z., Hua, X.: An empirical model for amplitude prediction on viv-galloping instability of rectangular cylinders. Wind Struct. 21(1), 85–103 (2015)

    Article  Google Scholar 

  50. Noel, J., Yadav, R., Li, G., Daqaq, M.: Improving the performance of galloping micro-power generators by passively manipulating the trailing edge. Appl. Phys. Lett. 112(8), 083503 (2018)

    Article  Google Scholar 

  51. Obasaju, E.: An investigation of the effects of incidence on the flow around a square section cylinder. Aeronaut. Q. 34(4), 243–259 (1983)

    Article  Google Scholar 

  52. Okajima, A.: Strouhal numbers of rectangular cylinders. J. Fluid Mech. 123, 379–398 (1982)

    Article  Google Scholar 

  53. Parkinson, G., Brooks, N.: On the aeroelastic instability of bluff cylinders. J. Appl. Mech. 28(2), 252–258 (1961)

    Article  MATH  Google Scholar 

  54. Parkinson, G., Smith, J.: The square prism as an aeroelastic non-linear oscillator. Q. J. Mech. Appl. Math. 17(2), 225–239 (1964)

    Article  MATH  Google Scholar 

  55. Parkinson, G., Wawzonek, M.: Some considerations of combined effects of galloping and vortex resonance. J. Wind Eng. Ind. Aerodyn. 8(1–2), 135–143 (1981)

    Article  Google Scholar 

  56. Pulipaka, N., Sarkar, P.P., McDonald, J.R.: On galloping vibration of traffic signal structures. J. Wind Eng. Ind. Aerodyn. 77, 327–336 (1998)

    Article  Google Scholar 

  57. Risken, H.: Fokker-planck equation. In: The Fokker–Planck Equation, pp. 63–95. Springer, Berlin (1996)

    Chapter  MATH  Google Scholar 

  58. Robertson, I., Li, L., Sherwin, S., Bearman, P.: A numerical study of rotational and transverse galloping rectangular bodies. J. Fluids Struct. 17(5), 681–699 (2003)

    Article  Google Scholar 

  59. Shoshani, O., Shaw, S.W.: Phase noise reduction and optimal operating conditions for a pair of synchronized oscillators. IEEE Trans. Circuits Syst. I Regul. Pap. 63(1), 1–11 (2016)

    Article  MathSciNet  Google Scholar 

  60. Simiu, E., Scanlan, R.H.: Wind Effects on Structures. Wiley, New Jersey (1996)

    Google Scholar 

  61. Sorribes-Palmer, F., Sanz-Andres, A.: Optimization of energy extraction in transverse galloping. J. Fluids Struct. 43, 124–144 (2013)

    Article  Google Scholar 

  62. Stratonovich, R.L.: Topics in the Theory of Random Noise, vol. 2. CRC Press, Boca Raton (1967)

    MATH  Google Scholar 

  63. Tennekes, H., Lumley, J.L.: A First Course in Turbulence. MIT Press, Cambridge (1972)

    MATH  Google Scholar 

  64. Wiener, N.: Generalized harmonic analysis. Acta Mathematica 55(1), 117–258 (1930)

    Article  MathSciNet  MATH  Google Scholar 

  65. Yang, Y., Zhao, L., Tang, L.: Comparative study of tip cross-sections for efficient galloping energy harvesting. Appl. Phys. Lett. 102(6), 064105 (2013)

    Article  Google Scholar 

  66. Zulli, D., Piccardo, G., Luongo, A.: Analysis of dry galloping on inclined cables under stationary wind. In: ENOC 2017, pp. 25–30. Budapest, Hungary (2017)

Download references

Author information

Authors and Affiliations

  1. Mechanical Engineering, Ben-Gurion University of the Negev, Beer-Sheva, 84105, Israel

    Oriel Shoshani 

Authors
  1. Oriel Shoshani 
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Oriel Shoshani .

Ethics declarations

Conflict of interest

The author declares that he has no conflict of interest.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Shoshani , O. Theoretical aspects of transverse galloping. Nonlinear Dyn 94, 2685–2696 (2018). https://doi.org/10.1007/s11071-018-4518-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-018-4518-1

Keywords

Search

Navigation