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
A.M. Abdel-Ghaffar, Suspension bridge vibration: continuum formulation. J. Eng. Mech. 108, 1215–1232 (1982)
O.H. Ammann, T. von Kármán, G.B. Woodruff, The Failure of the Tacoma Narrows Bridge (Federal Works Agency, Washington, DC, 1941)
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)
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)
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
G. Arioli, F. Gazzola, Old and new explanations of the Tacoma narrows bridge collapse, in Atti XXI Congresso AIMETA, Torino, p. 10 (2013)
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)
G. Arioli, F. Gazzola, S. Terracini, Multibump periodic motions of an infinite lattice of particles. Math. Zeit. 223, 627–642 (1996)
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)
E. Berchio, F. Gazzola, A qualitative explanation of the origin of torsional instability in suspension bridges. Nonlinear Anal. T.M.A. (2015)
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)
J.M.W. Brownjohn, Observations on non-linear dynamic characteristics of suspension bridges. Earthq. Eng. Struct. Dyn. 23, 1351–1367 (1994)
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)
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)
S.H. Doole, S.J. Hogan, Non-linear dynamics of the extended Lazer-McKenna bridge oscillation model. Dyn. Stab. Syst. 15, 43–58 (2000)
F.B. Farquharson, Letter to the Editor. ENR, 3 July 1941, p. 37
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
F. Gazzola, Periodic motions of a lattice of particles with singular forces. Differ. Integr. Equ. 10, 245–264 (1997)
F. Gazzola, Nonlinearity in oscillating bridges. Electron. J. Differ. Equ. 211, 1–47 (2013)
H. Goldstein, C. Poole, J. Safko, Classical Mechanics, 3rd edn. (Addison Wesley, San Francisco, 2002)
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
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)
L.D. Humphreys, R. Shammas, Finding unpredictable behavior in a simple ordinary differential equation. Coll. Math. J. 31, 338–346 (2000)
H.M. Irvine, Cable Structures. MIT Press Series in Structural Mechanics (MIT Press, Cambridge, 1981)
D. Jacover, P.J. McKenna, Nonlinear torsional flexings in a periodically forced suspended beam. J. Comput. Appl. Math. 52, 241–265 (1994)
W. Lacarbonara, Nonlinear Structural Mechanics (Springer, New York, 2013)
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)
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
P.J. McKenna, Torsional oscillations in suspension bridges revisited: fixing an old approximation. Am. Math. Mon. 106, 1–18 (1999)
P.J. McKenna, Large-amplitude periodic oscillations in simple and complex mechanical systems: outgrowths from nonlinear analysis. Milan J. Math. 74, 79–115 (2006)
P.J. McKenna, Oscillations in suspension bridges, vertical and torsional. Discrete Continuous Dyn. Syst. S 7, 785–791 (2014)
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)
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)
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)
H. Poincaré, Introduction to the collected mathematical works of George William Hill, vol. I (Carnegie Institution, Washington, 1905), pp. vii–xviii
H. Poincaré, Les méthodes nouvelles de la mécanique céleste (Dover Publications, New York, 1957)
Y. Rocard, Dynamic Instability: Automobiles, Aircraft, Suspension Bridges (Crosby Lockwood, London, 1957)
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)
R.H. Scanlan, J.J. Tomko, Airfoil and bridge deck flutter derivatives. J. Eng. Mech. 97, 1717–1737 (1971)
R. Scott, In the Wake of Tacoma: Suspension Bridges and the Quest for Aerodynamic Stability (ASCE Press, Reston, 2001)
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)
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)
Tacoma Narrows Bridge Collapse (1940), http://www.youtube.com/watch?v=3mclp9qmcgs (Video)
H. Wagner, Über die entstehung des dynamischen auftriebes von tragflügeln. Zeit. Angew. Mathematik und Mechanik 5, 17–35 (1925)
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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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DOI: https://doi.org/10.1007/978-3-319-15434-3_4
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