Vortex-Induced Vibration of Bridge Decks: Volterra Series-Based Model
Publication: Journal of Engineering Mechanics
Volume 139, Issue 12
Abstract
A brief overview of vortex-induced vibration (VIV) of bridge decks is presented, highlighting special VIV features concerning bridge decks. A popular VIV model (Van der Pol–type model) for bridge decks is examined in detail. Alternatively, a truncated Volterra series–based nonlinear oscillator is introduced to model the VIV system. Typical features of VIV such as the limit cycle oscillation (LCO), frequency shift, hysteresis, and beat phenomenon are parsimoniously and accurately captured in the proposed nonlinear model. As a functional expansion of a nonlinear system, the Volterra series is convenient for estimating the linear and nonlinear contributions to VIV. It is demonstrated that the relative contribution of nonlinear effects in VIV is around 50% of the total response for a range of bridge cross sections. The efficacy of the Volterra series as a reduced-order model (ROM) in capturing aerodynamic nonlinearities eliminates the need for reliance on conventional phenomenological models as it promises to offer a unified framework for nonlinear wind effects on long-span bridges—for example, VIV, buffeting, and flutter.
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Acknowledgments
Support for this project, provided by NSF Grant No. CMMI 09-28282, is gratefully acknowledged.
References
Alper, P. (1965). “A consideration of the discrete Volterra series.” IEEE Trans. Automat. Contr., 10(3), 322–327.
Battista, R. C., and Pfeil, M. S. (2000). “Reduction of vortex-induced oscillations of Rio–Niterói bridge by dynamic control devices.” J. Wind Eng. Ind. Aerodyn., 84(3), 273–288.
Bearman, P. W. (1984). “Vortex shedding from oscillating bluff bodies.” Annu. Rev. Fluid Mech., 16(1), 195–222.
Billah, K. Y. R. (1989). “A study of vortex-induced vibration.” Ph.D. dissertation, Princeton Univ., Princeton, NJ.
Bishop, R. E. D., and Hassan, A. Y. (1964a). “The lift and drag forces on a circular cylinder in a flowing fluid.” Proc. R. Soc. Lond. A, 277(1368), 32–50.
Bishop, R. E. D., and Hassan, A. Y. (1964b). “The lift and drag forces on a circular cylinder oscillating in a flowing fluid.” Proc. R. Soc. Lond. A, 277(1368), 51–75.
Boyd, S. P., and Chua, L. O. (1985). “Fading memory and the problem of approximating nonlinear operators with Volterra series.” IEEE Trans. Circ. Syst., 32(11), 1150–1161.
Carassale, L., and Kareem, A. (2010). “Modeling nonlinear systems by Volterra series.” J. Eng. Mech., 136(6), 801–818.
Clancy, S. J., and Rugh, W. J. (1979). “A note on the identification of discrete-time polynomial systems.” IEEE Trans. Automat. Contr., 24(6), 975–978.
Currie, I. G., Hartlen, R. T., and Martin, W. W. (1974). “The response of circular cylinders to vortex shedding.” Flow-induced structural vibrations, E. Naudascher, ed., Springer, Berlin, 128–142.
Diana, G., Resta, F., Belloli, M., and Rocchi, D. (2006). “On the vortex shedding forcing on suspension bridge deck.” J. Wind Eng. Ind. Aerodyn., 94(5), 341–363.
Donley, M. G., and Spanos, P. D. (1990). Dynamic analysis of nonlinear structures by the method of statistical quadratization, Springer, New York.
Dowell, E. H. (1981). “Nonlinear oscillator models in bluff body aeroelasticity.” J. Sound Vibrat., 75(2), 251–264.
Dowell, E. H., and Hall, K. C. (2001). “Modeling of fluid-structure interaction.” Annu. Rev. Fluid Mech., 33(1), 445–490.
Doyle, F., Pearson, R., and Ogunnaike, T. (2001). Identification and control using Volterra models, Springer, Berlin.
Ehsan, F., and Scanlan, R. H. (1990). “Vortex-induced vibrations of flexible bridges.” J. Eng. Mech., 116(6), 1392–1411.
Ewen, E. J., and Weiner, D. D. (1980). “Identification of weakly non-linear systems using input and output measurements.” IEEE Trans. Circ. Syst., 27(12), 1255–1261.
Fail, R., Lawford, J. A. and Eyre, R. C. W. (1959). “Low-speed experiments on the wake characteristics of flat plates normal to an air stream.” R&M No. 3120, A.R.C. Technical Report, Ministry of Supply, London.
Feng, C. C. (1968). “The measurement of vortex-induced effects in flow past stationary and oscillating circular and D-section cylinders.” M.A.Sc. thesis, Univ. British Columbia, Vancouver, BC, Canada.
Frandsen, J. B. (2001). “Simultaneous pressures and accelerations measured full-scale on the Great Belt East suspension bridge.” J. Wind Eng. Ind. Aerodyn., 89(1), 95–129.
Fujino, Y., and Yoshida, Y. (2002). “Wind-induced vibration and control of Trans-Tokyo Bay Crossing Bridge.” J. Struct. Eng., 128(8), 1012–1025.
Fujiwara, A., Kataoka, H. and Ito, M. (1993). “Numerical simulation of flow field around an oscillating bridge using finite difference method.” J. Wind Eng. Ind. Aerodyn. 46–47, 567–575.
Gerrard, J. H. (1966). “The mechanics of the formation region of vortices behind bluff bodies.” J. Fluid Mech., 25(2), 401–413.
Goswami, I., Scanlan, R. H., and Jones, N. P. (1993a). “Vortex-induced vibration of circular cylinders—Part 1: Experimental data.” J. Eng. Mech., 119(11), 2270–2287.
Goswami, I., Scanlan, R. H., and Jones, N. P. (1993b). “Vortex-induced vibration of circular cylinders—Part 2: New model.” J. Eng. Mech., 119(11), 2288–2302.
Griffin, O. M., and Koopmann, G. H. (1977). “The vortex-excited lift and reaction forces on resonantly vibrating cylinders.” J. Sound Vibrat., 54(3), 435–448.
Griffin, O. M., and Ramberg, S. E. (1982). “Some recent studies of vortex shedding with application to marine tubulars and risers.” J. Energy Resour. Technol., 104(1), 2–13.
Gupta, H., Sarkar, P. P., and Mehta, K. C. (1996). “Identification of vortex-induced-response parameters in time domain.” J. Eng. Mech., 122(11), 1031–1037.
Gurley, K. R., Tognarelli, M. A., and Kareem, A. (1997). “Analysis and simulation tools for wind engineering.” Probab. Eng. Mech., 12(1), 9–31.
Hartlen, R. T., and Currie, I. G. (1970). “Lift-oscillator model of vortex-induced vibration.” J. Engrg. Mech. Div., 96(5), 577–591.
Iwan, W. D., and Blevins, R. D. (1974). “A model for vortex induced oscillation of structures.” J. Appl. Mech., 41(3), 581–586.
Kumarasena, T., Scanlan, R. H., and Ehsan, F. (1991). “Wind-induced motions of Deer Isle Bridge.” J. Struct. Eng., 117(11), 3356–3374.
Kumarasena, T., Scanlan, R. H., and Morris, G. R. (1989). “Deer Isle Bridge: Field and computed vibrations.” J. Struct. Eng., 115(9), 2313–2328.
Landl, R. (1975). “A mathematical model for vortex-excited vibrations of bluff bodies.” J. Sound Vibrat., 42(2), 219–234.
Larsen, A. (1995). “A generalized model for assessment of vortex-induced vibrations of flexible structures.” J. Wind Eng. Ind. Aerodyn., 57(2–3), 281–294.
Larsen, A., Esdahl, S., Andersen, J. E., and Vejrum, T. (2000). “Storebæl suspension bridge—vortex shedding excitation and mitigation by guide vanes.” J. Wind Eng. Ind. Aerodyn., 88(2–3), 283–296.
Larsen, S. V., Smitt, L. W., Gjelstrup, H. and Larsen, A. (2004). “Investigation of vortex-induced motion at high Reynolds numbers using large scale section model.” Proc., 5th Int. Colloquium on Bluff Body Aerodynamics and Applications, International Associations for Wind Engineering, Ottawa, 273–276.
Lee, S., Lee, J. S., and Kim, J. D. (1997). “Prediction of vortex-induced wind loading on long-span bridges.” J. Wind Eng. Ind. Aerodyn., 67–68, 267–278.
Li, H., et al. (2011). “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.
Li, Y., and Kareem, A. (1990). “Stochastic response of a tension leg platform to wind and wave fields.” J. Wind Eng. Ind. Aerodyn., 36(12), 915–920.
Lucia, D. J., Beran, P. S., and Silva, W. A. (2004). “Reduced-order modeling: New approaches for computational physics.” Prog. Aerosp. Sci., 40(1–2), 51–117.
Marra, A. M., Mannini, C., and Bartoli, G. (2011). “Van der Pol–type equation for modeling vortex-induced oscillations of bridge decks.” J. Wind Eng. Ind. Aerodyn., 99(6–7), 776–785.
Marzocca, P., Librescu, L., and Silva, W. A. (2000). “Aerodynamic indicial functions and their use in aeroelastic formulation of lifting surfaces.” 41st Structures, Structural Dynamics, and Materials Conf., American Institute of Aeronautics and Astronautics, Reston, VA.
Marzocca, P., Librescu, L., and Silva, W. A. (2001). “Volterra series approach for nonlinear aeroelastic response of 2-D lifting surfaces.” 42nd Structures, Structural Dynamics, and Materials Conf., American Institute of Aeronautics and Astronautics, Reston, VA.
Naess, A. (1985). “Statistical analysis of second-order response of marine structures.” J. Ship Res., 29(4), 270–284.
Nakamura, Y., and Mizota, T. (1975). “Unsteady lifts and wakes of oscillating rectangular prisms.” J. Engrg. Mech. Div. 101(6), 855–871.
Nomura, T. (1993). “Finite element analysis of vortex-induced vibrations of bluff cylinders.” J. Wind Eng. Ind. Aerodyn., 46–47, 587–594.
Owen, J. S., Vann, A. M., Davies, J. P., and Blakeborough, A. (1996). “The prototype testing of Kessock Bridge: Response to vortex shedding.” J. Wind Eng. Ind. Aerodyn., 60, 91–108.
Pearson, R. K. (1995). “Nonlinear input/output modeling.” J. Process Contr., 5(4), 197–211.
Pearson, R. K., Ogunnaike, B. A., and Doyle, F. J. (1996). “Identification of structurally constrained second-order Volterra models.” IEEE Trans. Signal Process., 44(11), 2837–2846.
Raveh, D. E. (2001). “Reduced-order models for nonlinear unsteady aerodynamics.” AIAA J., 39(8), 1417–1429.
Rayleigh, Lord. (1896). Theory of sound, 2nd Ed., Macmillan, London.
Rugh, W. J. (1981). Nonlinear system theory, John Hopkins University Press, Baltimore.
Sarpkaya, T. (1978). “Fluid forces on oscillating cylinders.” J. Wtrwy., Port, Coast., and Oc. Div., 104(3), 275–290.
Sarpkaya, T. (1979). “Vortex-induced oscillations.” J. Appl. Mech., 46(2), 241–258.
Sarpkaya, T. (2004). “A critical review of the intrinsic nature of VIV.” J. Fluids Structures, 19(4), 389–447.
Sarwar, M. W., and Ishihara, T. (2010). “Numerical study on suppression of vortex-induced vibrations of box girder bridge section by aerodynamic countermeasures.” J. Wind Eng. Ind. Aerodyn., 98(12), 701–711.
Scanlan, R. H. (1981). “State-of-the-art methods for calculating flutter, vortex-induced, and buffeting response of bridge structures.” Rep. No. FHWA/RD-80/50, Federal Highway Administration, Washington, DC.
Scanlan, R. H. (1998). “Bridge flutter derivatives at vortex lock-in.” J. Struct. Eng., 124(4), 450–458.
Scanlan, R. H., and Tomko, J. J. (1971). “Airfoil and bridge deck flutter derivatives.” J. Engrg. Mech. Div., 97(6), 1717–1737.
Schetzen, M. (1980). The Volterra and Wiener theories of nonlinear systems, Wiley, New York.
Shiraishi, N., and Matsumoto, M. (1983). “On classification of vortex-induced oscillation and its application for bridge structures.” J. Wind Eng. Ind. Aerodyn., 14(1–3), 419–430.
Silva, W. A. (1997). “Discrete-time linear and nonlinear aerodynamic impulse responses for efficient CFD analyses.” Ph.D. thesis, College of William & Mary, Williamsburg, VA.
Silva, W. A. (2005). “Identification of non-linear aeroelastic systems based on the Volterra theory: Progress and opportunities.” Nonlinear Dyn., 39(1–2), 25–62.
Silva, W. A. (2008). “Simultaneous excitation of multiple-input/multiple-output CFD-based unsteady aerodynamic systems.” J. Aircr., 45(4), 1267–1274.
Silva, W. A., et al. (2001). “Reduced-order modeling: Cooperative research and development at the NASA Langley Research Center.” Int. Forum on Aeroelasticity and Structural Dynamics, American Institute of Aeronautics and Astronautics, Reston, VA.
Skop, R. A., and Griffin, O. M. (1973). “A model for the vortex-excited resonant response of bluff cylinders.” J. Sound Vibrat., 27(2), 225–233.
Smith, I. J. (1980). “Wind induced dynamic response of the Wye bridge.” Eng. Struct., 2(4), 202–208.
Strouhal, V. (1878). “Ueber eine besondere Art der Tonerregung.” Annalen der Physik, 241(10), 216–251.
Tognarelli, M. A., Zhao, J., and Kareem, A. (1997). “Equivalent statistical cubicization: A frequency domain approach for nonlinearities in both system and forcing function.” J. Eng. Mech. 123(8), 890–894.
Van der Pol, B. (1920). “A theory of the amplitude of free and forced triode oscillation.” Radio Review, 1, 701–754.
Van der Pol, B., and Van der Mark, J. (1927). “Frequency demultiplication.” Nature, 120(3019), 363–364.
Volterra, V. (1959). Theory of functionals and of integral and integro-differential equations, Dover, New York.
von Kármán, T., and Rubach, H. (1912). “Uber den mechanismus des flussigkeits-und luftwiderstandes.” Phys. Z., 13, 49–59.
Weihs, D. (1972). “Semi-infinite vortex trails, and their relation to oscillation airfoils.” J. Fluid Mech., 54(4), 679–690.
Wiener, N. (1942). “Response of a non-linear device to noise.” Rep. V-16S, MIT Radiation Lab, Cambridge, MA.
Williamson, C. H. K. (1996). “Vortex dynamics in the cylinder wake.” Annu. Rev. Fluid Mech., 28(1), 477–539.
Williamson, C. H. K., and Govardhan, R. (2004). “Vortex-induced vibrations.” Annu. Rev. Fluid Mech., 36(1), 413–455.
Winterstein, S. R., Ude, T. C., and Marthinsen, T. (1994). “Volterra models of ocean structures: Extreme and fatigue reliability.” J. Eng. Mech., 120(6), 1369–1385.
Wood, K. N., and Parkinson, G. V. (1977). “A hysteresis problem in vortex-induced oscillation.” Proc., 6th Canadian Congress of Applied Mechanics (CANCAM), Univ. of British Columbia, Vancouver, BC, Canada, 697–698.
Wooton, L. R. (1969). “The oscillations of large circular stacks in wind.” ICE Proc., 43(1), 573–598.
Wu, T., and Kareem, A. (2011). “Nonlinear modeling of bridge aerodynamics.” Proc., 13th Int. Conf. on Wind Engineering (ICWE13), July, Amsterdam.
Wu, T., and Kareem, A. (2012a). “Modelling of nonlinear bridge aerodynamics and aeroelasticity: A convolution based approach.” Proc., Int. Conf. on Structural Nonlinear Dynamics and Diagnosis, Univ. Hassan II of Casablanca, Casablanca, Morocco, 1–4.
Wu, T., and Kareem, A. (2012b). “Nonlinear aerodynamic and aeroelastic analysis framework for cable-supported bridges.” 3rd American Association for Wind Engineering Workshop, AAWE, Hyannis, MA, 1–8.
Wu, T., and Kareem, A. (2012c). “Volterra series based nonlinear oscillator for vortex-induced vibration modeling.” Proc., 2012 Joint Conf. of the Engineering Mechanics Institute and the 11th ASCE Joint Specialty Conf. on Probabilistic Mechanics and Structural Reliability, ASCE, Reston, VA, 1–10.
Zdravkovich, M. (1982). “Scruton number: A proposal.” J. Wind Eng. Ind. Aerodyn., 10(3), 263–265.
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Published In
Journal of Engineering Mechanics
Volume 139 • Issue 12 • December 2013
Pages: 1831 - 1843
Copyright
© 2013 American Society of Civil Engineers.
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Received: Oct 11, 2012
Accepted: Mar 14, 2013
Published online: Mar 16, 2013
Published in print: Dec 1, 2013
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