# American Institute of Mathematical Sciences

2015, 2015(special): 112-121. doi: 10.3934/proc.2015.0112

## The role of aerodynamic forces in a mathematical model for suspension bridges

 1 Dipartimento di Scienze Matematiche, Politecnico di Torino, Corso Duca degli Abruzzi 24, Torino, 10129, Italy 2 Dipartimento di Matematica Politecnico di Milano, Piazza Leonardo da Vinci, 32, 20133 Milano

Received  September 2014 Revised  April 2015 Published  November 2015

In a fish-bone model for suspension bridges previously studied by us in [3] we introduce linear aerodynamic forces. We numerically analyze the role of these forces and we theoretically show that they do not influence the onset of torsional oscillations. This suggests a new explanation for the origin of instability in suspension bridges: it is a combined interaction between structural nonlinearity and aerodynamics and it follows a precise pattern.
Citation: Elvise Berchio, Filippo Gazzola. The role of aerodynamic forces in a mathematical model for suspension bridges. Conference Publications, 2015, 2015 (special) : 112-121. doi: 10.3934/proc.2015.0112
##### References:
 [1] O.H. Ammann, T. von Kármán and G.B. Woodruff, The failure of the Tacoma Narrows Bridge,, Federal Works Agency, (1941).   Google Scholar [2] G. Arioli and F. Gazzola, A new mathematical explanation of what triggered the catastrophic torsional mode of the Tacoma Narrows Bridge collapse,, Appl. Math. Modelling, 39 (2015), 901.   Google Scholar [3] E. Berchio and F. Gazzola, A qualitative explanation of the origin of torsional instability in suspension bridges,, Nonlinear Analysis TMA, 121 (2015), 54.   Google Scholar [4] K.Y. Billah and R.H. Scanlan, Resonance, Tacoma Narrows Bridge failure, and undergraduate physics textbooks,, Amer. J. Physics, 59 (1991), 118.   Google Scholar [5] C. Chicone, Ordinary differential equations with applications,, Texts in Applied Mathematics 34, (2006).   Google Scholar [6] G.W. Hill, On the part of the motion of the lunar perigee which is a function of the mean motions of the sun and the moon,, Acta Math., 8 (1886), 1.   Google Scholar [7] G. Holubová and A. Matas, Initial-boundary value problem for nonlinear string-beam system,, J. Math. Anal. Appl., 288 (2003), 784.   Google Scholar [8] H.M. Irvine, Cable Structures,, MIT Press Series in Structural Mechanics, (1981).   Google Scholar [9] A. Larsen, Aerodynamics of the Tacoma Narrows Bridge - 60 years later,, Struct. Eng. Internat., 4 (2000), 243.   Google Scholar [10] P.J. McKenna, Oscillations in suspension bridges, vertical and torsional,, Disc. Cont. Dynam. System S, 7 (2014), 785.   Google Scholar [11] K.S. Moore, Large torsional oscillations in a suspension bridge: multiple periodic solutions to a nonlinear wave equation,, SIAM J. Math. Anal., 33 (2002), 1411.   Google Scholar [12] B.G. Pittel and V.A. Yakubovich, A mathematical analysis of the stability of suspension bridges based on the example of the Tacoma Bridge (Russian),, Vestnik Leningrad Univ., 24 (1969), 80.   Google Scholar [13] R.H. Plaut and F.M. Davis, Sudden lateral asymmetry and torsional oscillations of section models of suspension bridges,, J. Sound and Vibration, 307 (2007), 894.   Google Scholar [14] A. Pugsley, The theory of suspension bridges,, Edward Arnold, (1968).   Google Scholar [15] R.H. Scanlan and J.J. Tomko, Airfoil and bridge deck flutter derivatives,, J. Eng. Mech., 97 (1971), 1717.   Google Scholar [16] R. Scott, In the wake of Tacoma. Suspension bridges and the quest for aerodynamic stability,, ASCE Press, (2001).   Google Scholar [17] N.E. Zhukovskii, Finiteness conditions for integrals of the equation $d^2y/dx^2+py=0$ (Russian),, Mat. Sb., 16 (1892), 582.   Google Scholar

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##### References:
 [1] O.H. Ammann, T. von Kármán and G.B. Woodruff, The failure of the Tacoma Narrows Bridge,, Federal Works Agency, (1941).   Google Scholar [2] G. Arioli and F. Gazzola, A new mathematical explanation of what triggered the catastrophic torsional mode of the Tacoma Narrows Bridge collapse,, Appl. Math. Modelling, 39 (2015), 901.   Google Scholar [3] E. Berchio and F. Gazzola, A qualitative explanation of the origin of torsional instability in suspension bridges,, Nonlinear Analysis TMA, 121 (2015), 54.   Google Scholar [4] K.Y. Billah and R.H. Scanlan, Resonance, Tacoma Narrows Bridge failure, and undergraduate physics textbooks,, Amer. J. Physics, 59 (1991), 118.   Google Scholar [5] C. Chicone, Ordinary differential equations with applications,, Texts in Applied Mathematics 34, (2006).   Google Scholar [6] G.W. Hill, On the part of the motion of the lunar perigee which is a function of the mean motions of the sun and the moon,, Acta Math., 8 (1886), 1.   Google Scholar [7] G. Holubová and A. Matas, Initial-boundary value problem for nonlinear string-beam system,, J. Math. Anal. Appl., 288 (2003), 784.   Google Scholar [8] H.M. Irvine, Cable Structures,, MIT Press Series in Structural Mechanics, (1981).   Google Scholar [9] A. Larsen, Aerodynamics of the Tacoma Narrows Bridge - 60 years later,, Struct. Eng. Internat., 4 (2000), 243.   Google Scholar [10] P.J. McKenna, Oscillations in suspension bridges, vertical and torsional,, Disc. Cont. Dynam. System S, 7 (2014), 785.   Google Scholar [11] K.S. Moore, Large torsional oscillations in a suspension bridge: multiple periodic solutions to a nonlinear wave equation,, SIAM J. Math. Anal., 33 (2002), 1411.   Google Scholar [12] B.G. Pittel and V.A. Yakubovich, A mathematical analysis of the stability of suspension bridges based on the example of the Tacoma Bridge (Russian),, Vestnik Leningrad Univ., 24 (1969), 80.   Google Scholar [13] R.H. Plaut and F.M. Davis, Sudden lateral asymmetry and torsional oscillations of section models of suspension bridges,, J. Sound and Vibration, 307 (2007), 894.   Google Scholar [14] A. Pugsley, The theory of suspension bridges,, Edward Arnold, (1968).   Google Scholar [15] R.H. Scanlan and J.J. Tomko, Airfoil and bridge deck flutter derivatives,, J. Eng. Mech., 97 (1971), 1717.   Google Scholar [16] R. Scott, In the wake of Tacoma. Suspension bridges and the quest for aerodynamic stability,, ASCE Press, (2001).   Google Scholar [17] N.E. Zhukovskii, Finiteness conditions for integrals of the equation $d^2y/dx^2+py=0$ (Russian),, Mat. Sb., 16 (1892), 582.   Google Scholar
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