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August  2014, 7(4): 785-791. doi: 10.3934/dcdss.2014.7.785

Oscillations in suspension bridges, vertical and torsional

P. J. McKenna 1,

1. 

Department of Mathematics, University of Connecticut, Storrs, CT 06269, United States

Received  November 2013 Published  February 2014

We first review some history prior to the failure of the Tacoma Narrows suspension bridge. Then we consider some popular accounts of this in the popular physics literature, and the scientific and scholarly basis for these accounts and point out some failings. Later, we give a quick introduction to three different models, one single particle, one a continuum model, and two systems with two degrees of freedom.
Keywords: nonlinear oscillations torsional, Suspension bridge, vertical..
Mathematics Subject Classification: 35L75, 70K40, 35J6.
Citation: P. J. McKenna. Oscillations in suspension bridges, vertical and torsional. Discrete & Continuous Dynamical Systems - S, 2014, 7 (4) : 785-791. doi: 10.3934/dcdss.2014.7.785
References:
[1]

Ahmad M. Abdel-Ghaffar, Suspension bridge vibration: Continuum formulation, Journal of Eng. Mech., 108 (1982), 1215-1232. Google Scholar

[2]

A. A. Abdel-Ghaffar and R. H. Scanlan, Ambient vibration studies of Golden Gate Bridge. Suspended structure, Jour. Eng. Mech., 11 (1985), 463-481. Google Scholar

[3]

N. U. Ahmed and H. Harbi, Mathematical analysis of dynamic models of suspension bridges, SIAM J. Appl. Math., 58 (1998), 853-874 doi: 10.1137/S0036139996308698.  Google Scholar

[4]

O. H. Ammann, T. von Karman and G. B. Woodruff, The Failure of the Tacoma Narrows Bridge, Federal Works Agency, 1941. Google Scholar

[5]

K. Y. Billah and R. H. Scanlan, Resonance, Tacoma Narrows bridge failure, and undergraduate physics textbooks, Am. Jour. Physics, 59 (1991), 118-124. Google Scholar

[6]

F. Bleich, C. B. McCullough, R. Rosecrans and G. S. Vincent, The Mathematical Theory of Suspension Bridges, U.S. Dept. of Commerce, Bureau of Public Roads, 1950. Google Scholar

[7]

A. Castellani,, Safety margins of suspension bridges under seismic conditions, J. Structural Eng., 113 (1987), 1600-1616. Google Scholar

[8]

Y. S. Choi, K. C. Jen and P. J. McKenna, The structure of the solution set for periodic oscillations in a suspension bridge model, IMA J. Appl. Math, 47 (1991), 283-306. doi: 10.1093/imamat/47.3.283.  Google Scholar

[9]

M. Diaferio and V. Sepe, Smoothed "slack cable" models for large amplitude oscillations of suspension bridges, Mechanics Based Design of Structures and Machines, 32 (2004), 363-400. Google Scholar

[10]

P. Drábek, H. Leinfelder and G. Tajčová, Coupled string-beam equations as a model of suspension bridges, Appl. Math., 44 (1999), 97-142. doi: 10.1023/A:1022257304738.  Google Scholar

[11]

P. Drábek and P. Nečesal, Nonlinear scalar model of a suspension bridge: existence of multiple periodic solutions, Nonlinearity, 16 (2003), 1165-1183. doi: 10.1088/0951-7715/16/3/320.  Google Scholar

[12]

F. Gazzola and R. Pavani, Wide oscillation finite time blow up for solutions to nonlinear fourth order differential equations, Arch. Ration. Mech. Anal., 207 (2013), 717-752. doi: 10.1007/s00205-012-0569-5.  Google Scholar

[13]

J. Glover, A. C. Lazer and P. J. McKenna, Existence and stability of large-scale nonlinear oscillations in suspension bridges, Z.A.M.P., 40 (1989), 171-200. doi: 10.1007/BF00944997.  Google Scholar

[14]

D. Green and W. G. Unruh, The failure of the Tacoma Bridge: A physical model, American J. Phys., 74 (2006), 706-716. doi: 10.1119/1.2201854.  Google Scholar

[15]

A. C. Lazer and P. J. McKenna, Large-amplitude periodic oscillations in suspension bridges: some new connections with nonlinear analysis, SIAM Rev., 32 (1990), 537-578. doi: 10.1137/1032120.  Google Scholar

[16]

A. Larsen, Aerodynamics of the Tacoma Narrows Bridge-60 years later, Structural Engineering International, 4 (2000), 243-248. Google Scholar

[17]

P. J. McKenna, Large torsional oscillations in suspension bridges revisited: Fixing an old approximation, Amer. Math. Monthly, 106 (1999), 1-18. doi: 10.2307/2589581.  Google Scholar

[18]

P. J. McKenna and C. ÓTuama, Large torsional oscillations in suspension bridges visited again: Vertical forcing creates torsional response, Amer. Math. Monthly, 108 (2001), 738-745. doi: 10.2307/2695617.  Google Scholar

[19]

P. J. McKenna, Large-amplitude periodic oscillations in simple and complex mechanical systems: outgrowths from nonlinear analysis, Milan J. Math, 74 (2006), 79-115. doi: 10.1007/s00032-006-0052-6.  Google Scholar

[20]

P. J. McKenna and W. Walter, Nonlinear oscillations in a suspension bridge, Arch. Rat. Mech. Anal., 98 (1987), 167-177. doi: 10.1007/BF00251232.  Google Scholar

[21]

P. J. McKenna and W. Walter, Travelling waves in a suspension bridge, SIAM J. of Applied Math, 50 (1990), 703-715. doi: 10.1137/0150041.  Google Scholar

[22]

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. doi: 10.1137/S0036141001388099.  Google Scholar

[23]

R. H. Scanlan, Developments in low-speed aeroelasticity in the civil engineering field, AIAA Journal, 20 (1982), 839-844. Google Scholar

[24]

R. H. Scanlan and J. J. Tomko, Air foil and bridge deck flutter derivatives, J. Eng. Mech. Division ASCE, 97 (1971), 1717-1737. Google Scholar

[25]

J. Turmo and J. E. Luco, Effect of hanger flexibility on dynamic response of suspension bridges, J. Eng. Mech., 136 (2010), 1444-1459. Google Scholar

[26]

Q. Wu, K. Takahashi and S. Nakamura, The effect of cable loosening on seismic response of a prestressed concrete cable-stayed bridge, J. of Sound and Vibration, 268 (2003), 71-84. Google Scholar

[27]

Q. Wu, K. Takahashi and S. Nakamura, Non-linear vibrations of cables considering loosening, J. of Sound and Vibration, 261 (2003), 385-402. Google Scholar

show all references

References:
[1]

Ahmad M. Abdel-Ghaffar, Suspension bridge vibration: Continuum formulation, Journal of Eng. Mech., 108 (1982), 1215-1232. Google Scholar

[2]

A. A. Abdel-Ghaffar and R. H. Scanlan, Ambient vibration studies of Golden Gate Bridge. Suspended structure, Jour. Eng. Mech., 11 (1985), 463-481. Google Scholar

[3]

N. U. Ahmed and H. Harbi, Mathematical analysis of dynamic models of suspension bridges, SIAM J. Appl. Math., 58 (1998), 853-874 doi: 10.1137/S0036139996308698.  Google Scholar

[4]

O. H. Ammann, T. von Karman and G. B. Woodruff, The Failure of the Tacoma Narrows Bridge, Federal Works Agency, 1941. Google Scholar

[5]

K. Y. Billah and R. H. Scanlan, Resonance, Tacoma Narrows bridge failure, and undergraduate physics textbooks, Am. Jour. Physics, 59 (1991), 118-124. Google Scholar

[6]

F. Bleich, C. B. McCullough, R. Rosecrans and G. S. Vincent, The Mathematical Theory of Suspension Bridges, U.S. Dept. of Commerce, Bureau of Public Roads, 1950. Google Scholar

[7]

A. Castellani,, Safety margins of suspension bridges under seismic conditions, J. Structural Eng., 113 (1987), 1600-1616. Google Scholar

[8]

Y. S. Choi, K. C. Jen and P. J. McKenna, The structure of the solution set for periodic oscillations in a suspension bridge model, IMA J. Appl. Math, 47 (1991), 283-306. doi: 10.1093/imamat/47.3.283.  Google Scholar

[9]

M. Diaferio and V. Sepe, Smoothed "slack cable" models for large amplitude oscillations of suspension bridges, Mechanics Based Design of Structures and Machines, 32 (2004), 363-400. Google Scholar

[10]

P. Drábek, H. Leinfelder and G. Tajčová, Coupled string-beam equations as a model of suspension bridges, Appl. Math., 44 (1999), 97-142. doi: 10.1023/A:1022257304738.  Google Scholar

[11]

P. Drábek and P. Nečesal, Nonlinear scalar model of a suspension bridge: existence of multiple periodic solutions, Nonlinearity, 16 (2003), 1165-1183. doi: 10.1088/0951-7715/16/3/320.  Google Scholar

[12]

F. Gazzola and R. Pavani, Wide oscillation finite time blow up for solutions to nonlinear fourth order differential equations, Arch. Ration. Mech. Anal., 207 (2013), 717-752. doi: 10.1007/s00205-012-0569-5.  Google Scholar

[13]

J. Glover, A. C. Lazer and P. J. McKenna, Existence and stability of large-scale nonlinear oscillations in suspension bridges, Z.A.M.P., 40 (1989), 171-200. doi: 10.1007/BF00944997.  Google Scholar

[14]

D. Green and W. G. Unruh, The failure of the Tacoma Bridge: A physical model, American J. Phys., 74 (2006), 706-716. doi: 10.1119/1.2201854.  Google Scholar

[15]

A. C. Lazer and P. J. McKenna, Large-amplitude periodic oscillations in suspension bridges: some new connections with nonlinear analysis, SIAM Rev., 32 (1990), 537-578. doi: 10.1137/1032120.  Google Scholar

[16]

A. Larsen, Aerodynamics of the Tacoma Narrows Bridge-60 years later, Structural Engineering International, 4 (2000), 243-248. Google Scholar

[17]

P. J. McKenna, Large torsional oscillations in suspension bridges revisited: Fixing an old approximation, Amer. Math. Monthly, 106 (1999), 1-18. doi: 10.2307/2589581.  Google Scholar

[18]

P. J. McKenna and C. ÓTuama, Large torsional oscillations in suspension bridges visited again: Vertical forcing creates torsional response, Amer. Math. Monthly, 108 (2001), 738-745. doi: 10.2307/2695617.  Google Scholar

[19]

P. J. McKenna, Large-amplitude periodic oscillations in simple and complex mechanical systems: outgrowths from nonlinear analysis, Milan J. Math, 74 (2006), 79-115. doi: 10.1007/s00032-006-0052-6.  Google Scholar

[20]

P. J. McKenna and W. Walter, Nonlinear oscillations in a suspension bridge, Arch. Rat. Mech. Anal., 98 (1987), 167-177. doi: 10.1007/BF00251232.  Google Scholar

[21]

P. J. McKenna and W. Walter, Travelling waves in a suspension bridge, SIAM J. of Applied Math, 50 (1990), 703-715. doi: 10.1137/0150041.  Google Scholar

[22]

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. doi: 10.1137/S0036141001388099.  Google Scholar

[23]

R. H. Scanlan, Developments in low-speed aeroelasticity in the civil engineering field, AIAA Journal, 20 (1982), 839-844. Google Scholar

[24]

R. H. Scanlan and J. J. Tomko, Air foil and bridge deck flutter derivatives, J. Eng. Mech. Division ASCE, 97 (1971), 1717-1737. Google Scholar

[25]

J. Turmo and J. E. Luco, Effect of hanger flexibility on dynamic response of suspension bridges, J. Eng. Mech., 136 (2010), 1444-1459. Google Scholar

[26]

Q. Wu, K. Takahashi and S. Nakamura, The effect of cable loosening on seismic response of a prestressed concrete cable-stayed bridge, J. of Sound and Vibration, 268 (2003), 71-84. Google Scholar

[27]

Q. Wu, K. Takahashi and S. Nakamura, Non-linear vibrations of cables considering loosening, J. of Sound and Vibration, 261 (2003), 385-402. Google Scholar

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