American Institute of Mathematical Sciences

Advanced
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. 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. Google Scholar

[3]

N. U. Ahmed and H. Harbi, Mathematical analysis of dynamic models of suspension bridges,, SIAM J. Appl. Math., 58 (1998), 853. 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. 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, (1950). Google Scholar

[7]

A. Castellani,, Safety margins of suspension bridges under seismic conditions,, J. Structural Eng., 113 (1987), 1600. 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. 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. 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. 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. 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. 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. 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. 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. doi: 10.1137/1032120. Google Scholar

[16]

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

[17]

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

show all references

References:
[1]

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

[3]

N. U. Ahmed and H. Harbi, Mathematical analysis of dynamic models of suspension bridges,, SIAM J. Appl. Math., 58 (1998), 853. 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. 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, (1950). Google Scholar

[7]

A. Castellani,, Safety margins of suspension bridges under seismic conditions,, J. Structural Eng., 113 (1987), 1600. 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. 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. 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. 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. 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. 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. 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. 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. doi: 10.1137/1032120. Google Scholar

[16]

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

[17]

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

[1]

Ivana Bochicchio, Claudio Giorgi, Elena Vuk. On the viscoelastic coupled suspension bridge. Evolution Equations & Control Theory, 2014, 3 (3) : 373-397. doi: 10.3934/eect.2014.3.373
[2]

Marco Campo, José R. Fernández, Maria Grazia Naso. A dynamic problem involving a coupled suspension bridge system: Numerical analysis and computational experiments. Evolution Equations & Control Theory, 2019, 8 (3) : 489-502. doi: 10.3934/eect.2019024
[3]

Ling Xu, Jianhua Huang, Qiaozhen Ma. Upper semicontinuity of random attractors for the stochastic non-autonomous suspension bridge equation with memory. Discrete & Continuous Dynamical Systems - B, 2019, 24 (11) : 5959-5979. doi: 10.3934/dcdsb.2019115
[4]

Suping Wang, Qiaozhen Ma. Existence of pullback attractors for the non-autonomous suspension bridge equation with time delay. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 0-0. doi: 10.3934/dcdsb.2019221
[5]

Jonathan E. Rubin, Justyna Signerska-Rynkowska, Jonathan D. Touboul, Alexandre Vidal. Wild oscillations in a nonlinear neuron model with resets: (Ⅱ) Mixed-mode oscillations. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 4003-4039. doi: 10.3934/dcdsb.2017205
[6]

Olena Naboka. On synchronization of oscillations of two coupled Berger plates with nonlinear interior damping. Communications on Pure & Applied Analysis, 2009, 8 (6) : 1933-1956. doi: 10.3934/cpaa.2009.8.1933
[7]

Jonathan E. Rubin, Justyna Signerska-Rynkowska, Jonathan D. Touboul, Alexandre Vidal. Wild oscillations in a nonlinear neuron model with resets: (Ⅰ) Bursting, spike-adding and chaos. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3967-4002. doi: 10.3934/dcdsb.2017204
[8]

N. Romero, A. Rovella, F. Vilamajó. Dynamics of vertical delay endomorphisms. Discrete & Continuous Dynamical Systems - B, 2003, 3 (3) : 409-422. doi: 10.3934/dcdsb.2003.3.409
[9]

Donghi Lee, Makoto Sakuma. Simple loops on 2-bridge spheres in 2-bridge link complements. Electronic Research Announcements, 2011, 18: 97-111. doi: 10.3934/era.2011.18.97
[10]

Donghi Lee, Makoto Sakuma. Simple loops on 2-bridge spheres in Heckoid orbifolds for 2-bridge links. Electronic Research Announcements, 2012, 19: 97-111. doi: 10.3934/era.2012.19.97
[11]

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
[12]

Alberto Ferrero, Filippo Gazzola. A partially hinged rectangular plate as a model for suspension bridges. Discrete & Continuous Dynamical Systems - A, 2015, 35 (12) : 5879-5908. doi: 10.3934/dcds.2015.35.5879
[13]

Jianlu Zhang. Suspension of the billiard maps in the Lazutkin's coordinate. Discrete & Continuous Dynamical Systems - A, 2017, 37 (4) : 2227-2242. doi: 10.3934/dcds.2017096
[14]

Jon Aaronson, Omri Sarig, Rita Solomyak. Tail-invariant measures for some suspension semiflows. Discrete & Continuous Dynamical Systems - A, 2002, 8 (3) : 725-735. doi: 10.3934/dcds.2002.8.725
[15]

Salvador Addas-Zanata. Stability for the vertical rotation interval of twist mappings. Discrete & Continuous Dynamical Systems - A, 2006, 14 (4) : 631-642. doi: 10.3934/dcds.2006.14.631
[16]

Peng Feng, Menaka Navaratna. Modelling periodic oscillations during somitogenesis. Mathematical Biosciences & Engineering, 2007, 4 (4) : 661-673. doi: 10.3934/mbe.2007.4.661
[17]

Freddy Dumortier, Robert Roussarie. Bifurcation of relaxation oscillations in dimension two. Discrete & Continuous Dynamical Systems - A, 2007, 19 (4) : 631-674. doi: 10.3934/dcds.2007.19.631
[18]

Maria Conceição A. Leite, Yunjiao Wang. Multistability, oscillations and bifurcations in feedback loops. Mathematical Biosciences & Engineering, 2010, 7 (1) : 83-97. doi: 10.3934/mbe.2010.7.83
[19]

Oleksiy V. Kapustyan, Pavlo O. Kasyanov, José Valero, Michael Z. Zgurovsky. Strong attractors for vanishing viscosity approximations of non-Newtonian suspension flows. Discrete & Continuous Dynamical Systems - B, 2018, 23 (3) : 1155-1176. doi: 10.3934/dcdsb.2018146
[20]

Anthony Quas, Terry Soo. Weak mixing suspension flows over shifts of finite type are universal. Journal of Modern Dynamics, 2012, 6 (4) : 427-449. doi: 10.3934/jmd.2012.6.427

2018 Impact Factor: 0.545

Tools

Metrics

  • PDF downloads (9)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences