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

Oscillations in suspension bridges, vertical and torsional

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.
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.

[2]

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

[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.

[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.

[6]

F. Bleich, C. B. McCullough, R. Rosecrans and G. S. Vincent, The Mathematical Theory of Suspension Bridges,, U.S. Dept. of Commerce, (1950).

[7]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[16]

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

[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.

[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.

[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.

[20]

P. J. McKenna and W. Walter, Nonlinear oscillations in a suspension bridge,, Arch. Rat. Mech. Anal., 98 (1987), 167. 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. 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. doi: 10.1137/S0036141001388099.

[23]

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

[24]

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

[25]

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

[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.

[27]

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

show all references

References:
[1]

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

[2]

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

[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.

[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.

[6]

F. Bleich, C. B. McCullough, R. Rosecrans and G. S. Vincent, The Mathematical Theory of Suspension Bridges,, U.S. Dept. of Commerce, (1950).

[7]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[16]

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

[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.

[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.

[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.

[20]

P. J. McKenna and W. Walter, Nonlinear oscillations in a suspension bridge,, Arch. Rat. Mech. Anal., 98 (1987), 167. 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. 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. doi: 10.1137/S0036141001388099.

[23]

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

[24]

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

[25]

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

[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.

[27]

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

[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]

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

[3]

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

[4]

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

[5]

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

[6]

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

[7]

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

[8]

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

[9]

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

[10]

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

[11]

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

[12]

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

[13]

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

[14]

José M. Amigó, Isabelle Catto, Ángel Giménez, José Valero. Attractors for a non-linear parabolic equation modelling suspension flows. Discrete & Continuous Dynamical Systems - B, 2009, 11 (2) : 205-231. doi: 10.3934/dcdsb.2009.11.205

[15]

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

[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]

Chandrani Banerjee, Linda J. S. Allen, Jorge Salazar-Bravo. Models for an arenavirus infection in a rodent population: consequences of horizontal, vertical and sexual transmission. Mathematical Biosciences & Engineering, 2008, 5 (4) : 617-645. doi: 10.3934/mbe.2008.5.617

[20]

Moatlhodi Kgosimore, Edward M. Lungu. The Effects of Vertical Transmission on the Spread of HIV/AIDS in the Presence of Treatment. Mathematical Biosciences & Engineering, 2006, 3 (2) : 297-312. doi: 10.3934/mbe.2006.3.297

2016 Impact Factor: 0.781

Metrics

  • PDF downloads (2)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]