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Mathematical Models for Suspension Bridges pp 149–176Cite as

Models with Interacting Oscillators

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Part of the book series: MS&A ((MS&A,volume 15))

Abstract

In this chapter we model a suspension bridge through a number of coupled oscillators. This generates second order Hamiltonian systems which can be tackled with ODE methods. We first analyse a single cross section of the bridge and we model it as a nonlinear double oscillator able to describe both vertical and torsional oscillations. By means of a suitable Poincaré map we show that its conserved internal energy may transfer from the vertical oscillation of the barycenter to the torsional oscillation of the cross section. This happens when enough energy is present in the system, as for the fish-bone model considered in Chap. 3 We name again flutter energy the critical energy threshold where this transfer may occur.

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References

  1. A.M. Abdel-Ghaffar, Suspension bridge vibration: continuum formulation. J. Eng. Mech. 108, 1215–1232 (1982)

    Google Scholar 

  2. O.H. Ammann, T. von Kármán, G.B. Woodruff, The Failure of the Tacoma Narrows Bridge (Federal Works Agency, Washington, DC, 1941)

    Google Scholar 

  3. G. Arioli, F. Gazzola, Existence and numerical approximation of periodic motions of an infinite lattice of particles. Zeit. Angew. Math. Phys. 46, 898–912 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  4. G. Arioli, F. Gazzola, Periodic motions of an infinite lattice of particles with nearest neighbor interaction. Nonlinear Anal. T.M.A. 26, 1103–1114 (1996)

    Google Scholar 

  5. G. Arioli, F. Gazzola, Numerical implementation of the model considered in the paper: a new mathematical explanation of what triggered the catastrophic torsional mode of the Tacoma narrows bridge collapse (2013), http://mox.polimi.it/~gianni/bridges.html

  6. G. Arioli, F. Gazzola, Old and new explanations of the Tacoma narrows bridge collapse, in Atti XXI Congresso AIMETA, Torino, p. 10 (2013)

    Google Scholar 

  7. G. Arioli, F. Gazzola, A new mathematical explanation of what triggered the catastrophic torsional mode of the Tacoma narrows bridge collapse. Appl. Math. Model. 39, 901–912 (2015)

    Article  MathSciNet  Google Scholar 

  8. G. Arioli, F. Gazzola, S. Terracini, Multibump periodic motions of an infinite lattice of particles. Math. Zeit. 223, 627–642 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  9. G. Bartoli, P. Spinelli, The stochastic differential calculus for the determination of structural response under wind. J. Wind Eng. Ind. Aerodyn. 48, 175–188 (1993)

    Article  Google Scholar 

  10. E. Berchio, F. Gazzola, A qualitative explanation of the origin of torsional instability in suspension bridges. Nonlinear Anal. T.M.A. (2015)

    Google Scholar 

  11. F. Bleich, C.B. McCullough, R. Rosecrans, G.S. Vincent, The Mathematical Theory of Vibration in Suspension Bridges (U.S. Dept. of Commerce, Bureau of Public Roads, Washington, DC, 1950)

    Google Scholar 

  12. J.M.W. Brownjohn, Observations on non-linear dynamic characteristics of suspension bridges. Earthq. Eng. Struct. Dyn. 23, 1351–1367 (1994)

    Article  Google Scholar 

  13. E.A. Butcher, S. Radkar, S.C. Sinha, Order reduction of nonlinear systems with time periodic coefficients using invariant manifolds. J. Sound Vib. 284, 985–1002 (2005)

    Article  MATH  Google Scholar 

  14. A. Castro, A.C. Lazer, Critical point theory and the number of solutions of a nonlinear Dirichlet problem. Ann. Mat. Pura Appl. 70, 113–137 (1979)

    Article  MathSciNet  Google Scholar 

  15. S.H. Doole, S.J. Hogan, Non-linear dynamics of the extended Lazer-McKenna bridge oscillation model. Dyn. Stab. Syst. 15, 43–58 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  16. F.B. Farquharson, Letter to the Editor. ENR, 3 July 1941, p. 37

    Google Scholar 

  17. E. Fermi, J. Pasta, S. Ulam, Studies of Nonlinear Problems. Los Alamos Rpt. LA, n.1940 (also in “Collected Works of E. Fermi” University of Chicago Press, Chicago. 1965, vol II, pp. 978–988), 1955. http://www.physics.utah.edu/~detar/phys6720/handouts/fpu/FermiCollectedPapers1965.pdf

  18. F. Gazzola, Periodic motions of a lattice of particles with singular forces. Differ. Integr. Equ. 10, 245–264 (1997)

    MATH  MathSciNet  Google Scholar 

  19. F. Gazzola, Nonlinearity in oscillating bridges. Electron. J. Differ. Equ. 211, 1–47 (2013)

    MathSciNet  Google Scholar 

  20. H. Goldstein, C. Poole, J. Safko, Classical Mechanics, 3rd edn. (Addison Wesley, San Francisco, 2002)

    Google Scholar 

  21. M. Haberland, S. Hass, U. Starossek, Robustness assessment of suspension bridges, in Sixth International Conference on Bridge Maintenance, Safety and Management, IABMAS12, Stresa, 2012, pp. 1617–1624

    Google Scholar 

  22. L.D. Humphreys, P.J. McKenna, When a mechanical model goes nonlinear: unexpected responses to low-periodic shaking. Am. Math. Mon. 112, 861–875 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  23. L.D. Humphreys, R. Shammas, Finding unpredictable behavior in a simple ordinary differential equation. Coll. Math. J. 31, 338–346 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  24. H.M. Irvine, Cable Structures. MIT Press Series in Structural Mechanics (MIT Press, Cambridge, 1981)

    Google Scholar 

  25. D. Jacover, P.J. McKenna, Nonlinear torsional flexings in a periodically forced suspended beam. J. Comput. Appl. Math. 52, 241–265 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  26. W. Lacarbonara, Nonlinear Structural Mechanics (Springer, New York, 2013)

    Book  MATH  Google Scholar 

  27. A.C. Lazer, P.J. McKenna, Large-amplitude periodic oscillations in suspension bridges: some new connections with nonlinear analysis. SIAM Rev. 32, 537–578 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  28. M. Lepidi, V. Gattulli, Parametric interactions in the nonlinear sectional dynamics of suspended and cable-stayed bridges, in Atti XXI Congresso AIMETA, Torino, 2013, p. 100

    Google Scholar 

  29. P.J. McKenna, Torsional oscillations in suspension bridges revisited: fixing an old approximation. Am. Math. Mon. 106, 1–18 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  30. P.J. McKenna, Large-amplitude periodic oscillations in simple and complex mechanical systems: outgrowths from nonlinear analysis. Milan J. Math. 74, 79–115 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  31. P.J. McKenna, Oscillations in suspension bridges, vertical and torsional. Discrete Continuous Dyn. Syst. S 7, 785–791 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  32. P.J. McKenna, K.S. Moore, The global structure of periodic solutions to a suspension bridge mechanical model. IMA J. Appl. Math. 67, 459–478 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  33. P.J. McKenna, C.Ó. Tuama, Large torsional oscillations in suspension bridges visited again: vertical forcing creates torsional response. Am. Math. Mon. 108, 738–745 (2001)

    Article  MATH  Google Scholar 

  34. R.H. Plaut, F.M. Davis, Sudden lateral asymmetry and torsional oscillations of section models of suspension bridges. J. Sound Vib. 307, 894–905 (2007)

    Article  Google Scholar 

  35. H. Poincaré, Introduction to the collected mathematical works of George William Hill, vol. I (Carnegie Institution, Washington, 1905), pp. vii–xviii

    Google Scholar 

  36. H. Poincaré, Les méthodes nouvelles de la mécanique céleste (Dover Publications, New York, 1957)

    MATH  Google Scholar 

  37. Y. Rocard, Dynamic Instability: Automobiles, Aircraft, Suspension Bridges (Crosby Lockwood, London, 1957)

    Google Scholar 

  38. R.H. Scanlan, The action of flexible bridges under wind, I: flutter theory, II: buffeting theory. J. Sound Vib. 60, 187–199, 201–211 (1978)

    MATH  Google Scholar 

  39. R.H. Scanlan, J.J. Tomko, Airfoil and bridge deck flutter derivatives. J. Eng. Mech. 97, 1717–1737 (1971)

    Google Scholar 

  40. R. Scott, In the Wake of Tacoma: Suspension Bridges and the Quest for Aerodynamic Stability (ASCE Press, Reston, 2001)

    Book  Google Scholar 

  41. C. Simó, T.J. Stuchi, Central stable/unstable manifolds and the destruction of KAM tori in the planar Hill problem. Physica D 140, 1–32 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  42. F.C. Smith, G.S. Vincent, Aerodynamic Stability of Suspension Bridges: With Special Reference to the Tacoma Narrows Bridge, Part II: Mathematical Analysis. Investigation conducted by the Structural Research Laboratory, University of Washington (University of Washington Press, Seattle, 1950)

    Google Scholar 

  43. Tacoma Narrows Bridge Collapse (1940), http://www.youtube.com/watch?v=3mclp9qmcgs (Video)

  44. H. Wagner, Über die entstehung des dynamischen auftriebes von tragflügeln. Zeit. Angew. Mathematik und Mechanik 5, 17–35 (1925)

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. Dipartimento di Matematica, Politecnico di Milano, Milano, Italy

    Filippo Gazzola

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  1. Filippo Gazzola
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Gazzola, F. (2015). Models with Interacting Oscillators. In: Mathematical Models for Suspension Bridges. MS&A, vol 15. Springer, Cham. https://doi.org/10.1007/978-3-319-15434-3_4

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