American Institute of Mathematical Sciences

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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 and 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.

[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.

[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.

[4]

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

[5]

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

[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.

[7]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[16]

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

[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.

[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.

[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.

[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.

[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.

[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.

[23]

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

[24]

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

[25]

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

[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.

[27]

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

show all references

References:
[1]

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

[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.

[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.

[4]

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

[5]

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

[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.

[7]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[16]

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

[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.

[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.

[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.

[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.

[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.

[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.

[23]

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

[24]

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

[25]

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

[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.

[27]

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

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