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A generalized van der Pol nonlinear model of vortex-induced vibrations of bridge decks with multistability

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Abstract

The mathematical model of vortex-induced vibrations (VIV) on long-span bridges is important to predict nonlinear structural responses. Such models can be divided into two categories: wake-oscillator and single-degree-of-freedom (SDOF) models. The SDOF model is widely used for wind-induced vibration calculations. However, the traditional SDOF model based on the standard van der Pol oscillator cannot simulate VIVs with multistability. In this study, a newly generalized van der Pol model is proposed to incorporate the limit-cycle oscillation (LCO) with multiple amplitudes, and the nonlinear damping is expressed by polynomial expansion. Next, the multiple LCO amplitudes can be determined from the energy evolution formula derived from the averaging method. Similarly, the evolution of the vibration amplitude during the transient response is also derived by the same method. Subsequently, nonlinear parameter identification methods based on constraint optimization are derived according to both the LCO amplitude and transient responses. In the last part of this study, the “energy map” is proposed to present the energy extracted from the fluid–structure interaction with different wind speeds and vibration amplitudes, and it is constructed by the parameters identified in the lock-in range of VIV. The “energy map” can provide a complete picture of the evolution of the energy of VIVs on bridge decks.

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Data availibility

The datasets generated and analyzed during the current study are not publicly available due to security concerns of the owner of the bridge but are available from the corresponding author on reasonable request.

References

  1. Chen, X., Kareem, A.: Coupled dynamic analysis and equivalent static wind loads on buildings with three-dimensional modes. J. Struct. Eng. (ASCE) 131(7), 1071–1082 (2005)

    Google Scholar 

  2. Ehsan, F., Scanlan, R.H.: Vortex-induced vibrations of flexible bridges. J. Eng. Mech. 116(6), 1392–1411 (1990)

    Google Scholar 

  3. Frandsen, J.: Simultaneous pressures and accelerations measured full-scale on the Great Belt east suspension bridge. J. Wind Eng. Ind. Aerodyn. 89(1), 95–129 (2001)

    Google Scholar 

  4. Fujino, Y., Yoshida, Y.: Wind-induced vibration and control of Trans-Tokyo Bay crossing bridge. J. Struct. Eng. 128(8), 1012–1025 (2002)

    Google Scholar 

  5. Gao, G., Zhu, L., Li, J., et al.: Application of a new empirical model of nonlinear self-excited force to torsional vortex-induced vibration and nonlinear flutter of bluff bridge sections. J. Wind Eng. Ind. Aerodyn. 205(104), 313 (2020)

    Google Scholar 

  6. Ge, Y., Yang, Y., Cao, F.: VIV sectional model testing and field measurement of xihoumen suspension bridge with twin box girder. In: Proceedings of the 13th International Conference on Wind Engineering, Amsterdam, Netherlands, pp. 11–15 (2011)

  7. Ge, Y., Zhao, L., Cao, J.: Case study of vortex-induced vibration and mitigation mechanism for a long-span suspension bridge. J. Wind Eng. Ind. Aerodyn. 220(104), 866 (2022)

    Google Scholar 

  8. Grasman, J., Verhulst, F., Shih, S.D.: The lyapunov exponents of the van der pol oscillator. Math. Methods Appl. Sci. 28(10), 1131–1139 (2005)

    MathSciNet  Google Scholar 

  9. Hajj, M.R., Mehmood, A., Akhtar, I.: Single-degree-of-freedom model of displacement in vortex-induced vibrations. Nonlinear Dyn. 103(2), 1305–1320 (2021)

    Google Scholar 

  10. Hartlen, R.T., Currie, I.G.: Lift-oscillator model of vortex-induced vibration. J .Eng. Mech. Div. 96(5), 577–591 (1970)

    Google Scholar 

  11. Hu, C., Zhao, L., Ge, Y.: Time-frequency evolutionary characteristics of aerodynamic forces around a streamlined closed-box girder during vortex-induced vibration. J. Wind Eng. Ind. Aerodyn. 182, 330–343 (2018)

    Google Scholar 

  12. Hu, C., Zhao, L., Ge, Y.: Mechanism of suppression of vortex-induced vibrations of a streamlined closed-box girder using additional small-scale components. J. Wind Eng. Ind. Aerodyn. 189, 314–331 (2019)

    Google Scholar 

  13. Hu, C., Zhao, L., Ge, Y.: A simplified vortex model for the mechanism of vortex-induced vibrations in a streamlined closed-box girder. Wind Struct. 32(4), 309–319 (2021)

    Google Scholar 

  14. Hu, C., Zhao, L., Ge, Y.: Wind-induced instability mechanism of old Tacoma Narrows Bridge from aerodynamic work perspective. J. Bridge Eng. 27(5), 04022029 (2022)

  15. Hwang, Y.C., Kim, S., Kim, H.K.: Cause investigation of high-mode vortex-induced vibration in a long-span suspension bridge. Struct. Infrastruct. Eng. 16(1), 84–93 (2020)

    Google Scholar 

  16. Iwan, W.D., Blevins, R.D.: A model for vortex induced oscillation of structures. J. Appl. Mech. 41(3), 581–586 (1974)

    Google Scholar 

  17. Javed, U., Abdelkefi, A., Akhtar, I.: An improved stability characterization for aeroelastic energy harvesting applications. Commun. Nonlinear Sci. Numer. Simul. 36, 252–265 (2016)

    Google Scholar 

  18. Kovacic, I., Mickens, R.E.: A generalized van der pol type oscillator: investigation of the properties of its limit cycle. Math. Comput. Model. 55(3–4), 645–653 (2012)

    MathSciNet  Google Scholar 

  19. Kumarasena, T., Scanlan, R., Ehsan, F.: Wind-induced motions of Deer Isle bridge. J. Struct. Eng. 117(11), 3356–3374 (1991)

    Google Scholar 

  20. Larsen, A., Esdahl, S., Andersen, J.E., et al.: Storebælt suspension bridge-vortex shedding excitation and mitigation by guide vanes. J. Wind Eng. Ind. Aerodyn. 88(2–3), 283–296 (2000)

    Google Scholar 

  21. Li, H., Laima, S., Ou, J., et al.: Investigation of vortex-induced vibration of a suspension bridge with two separated steel box girders based on field measurements. Eng. Struct. 33(6), 1894–1907 (2011)

    Google Scholar 

  22. Li, H., Laima, S., Zhang, Q., et al.: Field monitoring and validation of vortex-induced vibrations of a long-span suspension bridge. J. Wind Eng. Ind. Aerodyn. 124, 54–67 (2014)

    Google Scholar 

  23. Li, K., Han, Y., Cai, C., et al.: Experimental investigation on post-flutter characteristics of a typical steel-truss suspension bridge deck. J. Wind Eng. Ind. Aerodyn. 216(104), 724 (2021)

    Google Scholar 

  24. Liu, S., Zhao, L., Fang, G., et al.: Nonlinear aerodynamic characteristics and modeling of a quasi-flat plate at torsional vibration: effects of angle of attack and vibration amplitude. Nonlinear Dyn. 1–25 (2022)

  25. Menon, K., Mittal, R.: Flow physics and dynamics of flow-induced pitch oscillations of an airfoil. J. Fluid Mech. 877, 582–613 (2019)

    Google Scholar 

  26. Menon, K., Mittal, R.: Aeroelastic response of an airfoil to gusts: prediction and control strategies from computed energy maps. J. Fluids Struct. 97(103), 078 (2020)

    Google Scholar 

  27. Menon, K., Mittal, R.: Quantitative analysis of the kinematics and induced aerodynamic loading of individual vortices in vortex-dominated flows: a computation and data-driven approach. J. Comput. Phys. 443(110), 515 (2021)

    MathSciNet  Google Scholar 

  28. Mittal, S., Singh, S.: Vortex-induced vibrations at subcritical Re. J. Fluid Mech. 534, 185–194 (2005)

    Google Scholar 

  29. Nayfeh, A.H., Mook, D.T.: Nonlinear Oscillations. Wiley, Hoboken (2008)

  30. Nguyen, T., Koide, M., Yamada, S., et al.: Influence of mass and damping ratios on vivs of a cylinder with a downstream counterpart in cruciform arrangement. J. Fluids Struct. 28, 40–55 (2012)

    Google Scholar 

  31. Pagnini, L., Piccardo, G., Solari, G.: Viv regimes and simplified solutions by the spectral model description. J. Wind Eng. Ind. Aerodyn. 198(104), 100 (2020)

    Google Scholar 

  32. Raghavan, K., Bernitsas, M.: Experimental investigation of reynolds number effect on vortex induced vibration of rigid circular cylinder on elastic supports. Ocean Eng. 38(5–6), 719–731 (2011)

    Google Scholar 

  33. Strogatz, S.H.: Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering. CRC Press (2018)

  34. Tamura, Y., Amano, A.: Mathematical model for vortex-induced oscillations of continuous systems with circular cross section. J. Wind Eng. Ind. Aerodyn. 14(1–3), 431–442 (1983)

    Google Scholar 

  35. van der Pol, B.: The nonlinear theory of electric oscillations. Proc. Inst. Radio Eng. 22(9), 1051–1086 (1934)

    Google Scholar 

  36. van der Pol, B., van der Mark, J.: Frequency demultiplication. Nature 120(3019), 363–364 (1927)

    Google Scholar 

  37. Wang, C., Hua, X., Feng, Z., et al.: Experimental investigation on vortex-induced vibrations of a triple-box girder with web modification. J. Wind Eng. Ind. Aerodyn. 218(104), 783 (2021)

    Google Scholar 

  38. Wang, W., Wang, X., Hua, X., et al.: Vibration control of vortex-induced vibrations of a bridge deck by a single-side pounding tuned mass damper. Eng. Struct. 173, 61–75 (2018)

    Google Scholar 

  39. Williamson, C., Govardhan, R.: Vortex-induced vibrations. Annu. Rev. Fluid Mech. 36, 413–455 (2004)

    MathSciNet  Google Scholar 

  40. Wu, T., Kareem, A.: An overview of vortex-induced vibration (VIV) of bridge decks. Front. Struct. Civ. Eng. 6(4), 335–347 (2012)

    Google Scholar 

  41. Xu, F., Ying, X., Li, Y., et al.: Experimental explorations of the torsional vortex-induced vibrations of a bridge deck. J. Bridge Eng. 21(12), 04016093 (2016)

  42. Xu, K., Ge, Y., Zhao, L., et al.: Experimental and numerical study on the dynamic stability of vortex-induced vibration of bridge decks. Int. J. Struct. Stabil. Dyn. 18(03), 1850033 (2018)

  43. Xu, K., Bi, K., Han, Q., et al.: Using tuned mass damper inerter to mitigate vortex-induced vibration of long-span bridges: analytical study. Eng. Struct. 182, 101–111 (2019)

    Google Scholar 

  44. Xu, K., Ge, Y., Zhao, L.: Quantitative evaluation of empirical models of vortex-induced vibration of bridge decks through sectional model wind tunnel testing. Eng. Struct. 219(110), 860 (2020)

    Google Scholar 

  45. Xu, K., Dai, Q., Bi, K., et al.: Multi-mode vortex-induced vibration control of long-span bridges by using distributed tuned mass damper inerters (DTMDIs). J. Wind Eng. Ind. Aerodyn. 224(104), 970 (2022)

    Google Scholar 

  46. Xu, W.H., Zeng, X.H., Wu, Y.X.: High aspect ratio (L/D) riser viv prediction using wake oscillator model. Ocean Eng. 35(17–18), 1769–1774 (2008)

    Google Scholar 

  47. Zhang, B., Song, B., Mao, Z., et al.: Numerical investigation on viv energy harvesting of bluff bodies with different cross sections in tandem arrangement. Energy 133, 723–736 (2017)

    Google Scholar 

  48. Zhang, W., Li, X., Ye, Z., et al.: Mechanism of frequency lock-in in vortex-induced vibrations at low reynolds numbers. J. Fluid Mech. 783, 72–102 (2015)

    MathSciNet  Google Scholar 

  49. Zhao, L., Cui, W., Shen, X., et al.: A fast on-site measure-analyze-suppress response to control vortex-induced-vibration of a long-span bridge. Structures 35, 192–201 (2022)

    Google Scholar 

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Funding

The authors gratefully acknowledge the support of the National Key Research and Development Program of China (2022YFC3005302, 2021YFF0502200) and the National Natural Science Foundation of China (52008314, 52078383). Any opinions, findings, conclusions, or recommendations are those of the authors and do not necessarily reflect the views of the agencies mentioned above.

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Authors and Affiliations

  1. State Key Lab of Disaster Reduction in Civil Engineering, Tongji University, 1239 Siping Road, Shanghai, 200092, China

    Wei Cui, Lin Zhao & Yaojun Ge

  2. Key Laboratory of Transport Industry of Wind Resistant Technology for Bridge Structures, Tongji University, Shanghai, China

    Wei Cui, Lin Zhao & Yaojun Ge

  3. Key Laboratory of Urban Security and Disaster Engineering of Ministry of Education, Beijing University of Technology, 100 Pingleyuan, Beijing, 02115, China

    Kun Xu

Authors
  1. Wei Cui
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  2. Lin Zhao
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  3. Yaojun Ge
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  4. Kun Xu
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Contributions

All authors contributed to the conception and design of the study. Conception, modeling and computation were performed by Wei Cui. The wind tunnels were conducted by Kun Xu. The first draft of the manuscript was written by Wei Cui, and Lin Zhao supervised the whole study. Lin Zhao and Yaojun Ge provided financial support for this study. All authors read and approved the final manuscript.

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Correspondence to Lin Zhao.

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Cui, W., Zhao, L., Ge, Y. et al. A generalized van der Pol nonlinear model of vortex-induced vibrations of bridge decks with multistability. Nonlinear Dyn 112, 259–272 (2024). https://doi.org/10.1007/s11071-023-09047-9

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  • DOI: https://doi.org/10.1007/s11071-023-09047-9

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