American Institute of Mathematical Sciences

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

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

Elvise Berchio 1, and  Filippo Gazzola 2,

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.
Keywords: Hill equation., Suspension bridges, torsional stability.
Mathematics Subject Classification: 37C20, 35B35, 34C1.
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, Washington D.C., (1941).

[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), 901-912.

[3]

E. Berchio and F. Gazzola, A qualitative explanation of the origin of torsional instability in suspension bridges, Nonlinear Analysis TMA, 121 (2015) 54-72.

[4]

K.Y. Billah and R.H. Scanlan, Resonance, Tacoma Narrows Bridge failure, and undergraduate physics textbooks, Amer. J. Physics, 59 (1991), 118-124.

[5]

C. Chicone, Ordinary differential equations with applications, Texts in Applied Mathematics 34, 2nd Ed., Springer, New York 2006.

[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), 1-36.

[7]

G. Holubová and A. Matas, Initial-boundary value problem for nonlinear string-beam system, J. Math. Anal. Appl., 288 (2003), 784-802.

[8]

H.M. Irvine, Cable Structures, MIT Press Series in Structural Mechanics, Massachusetts, 1981.

[9]

A. Larsen, Aerodynamics of the Tacoma Narrows Bridge - 60 years later, Struct. Eng. Internat., 4 (2000), 243-248.

[10]

P.J. McKenna, Oscillations in suspension bridges, vertical and torsional, Disc. Cont. Dynam. System S, 7 (2014), 785-791.

[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), 1411-1429.

[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), 80-91.

[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), 894-905.

[14]

A. Pugsley, The theory of suspension bridges, Edward Arnold, London, 1968.

[15]

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

[16]

R. Scott, In the wake of Tacoma. Suspension bridges and the quest for aerodynamic stability, ASCE Press, 2001.

[17]

N.E. Zhukovskii, Finiteness conditions for integrals of the equation $d^2y/dx^2+py=0$ (Russian), Mat. Sb., 16 (1892), 582-591.

show all references

References:
[1]

O.H. Ammann, T. von Kármán and G.B. Woodruff, The failure of the Tacoma Narrows Bridge, Federal Works Agency, Washington D.C., (1941).

[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), 901-912.

[3]

E. Berchio and F. Gazzola, A qualitative explanation of the origin of torsional instability in suspension bridges, Nonlinear Analysis TMA, 121 (2015) 54-72.

[4]

K.Y. Billah and R.H. Scanlan, Resonance, Tacoma Narrows Bridge failure, and undergraduate physics textbooks, Amer. J. Physics, 59 (1991), 118-124.

[5]

C. Chicone, Ordinary differential equations with applications, Texts in Applied Mathematics 34, 2nd Ed., Springer, New York 2006.

[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), 1-36.

[7]

G. Holubová and A. Matas, Initial-boundary value problem for nonlinear string-beam system, J. Math. Anal. Appl., 288 (2003), 784-802.

[8]

H.M. Irvine, Cable Structures, MIT Press Series in Structural Mechanics, Massachusetts, 1981.

[9]

A. Larsen, Aerodynamics of the Tacoma Narrows Bridge - 60 years later, Struct. Eng. Internat., 4 (2000), 243-248.

[10]

P.J. McKenna, Oscillations in suspension bridges, vertical and torsional, Disc. Cont. Dynam. System S, 7 (2014), 785-791.

[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), 1411-1429.

[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), 80-91.

[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), 894-905.

[14]

A. Pugsley, The theory of suspension bridges, Edward Arnold, London, 1968.

[15]

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

[16]

R. Scott, In the wake of Tacoma. Suspension bridges and the quest for aerodynamic stability, ASCE Press, 2001.

[17]

N.E. Zhukovskii, Finiteness conditions for integrals of the equation $d^2y/dx^2+py=0$ (Russian), Mat. Sb., 16 (1892), 582-591.

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