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September  2019, 8(3): 489-502. doi: 10.3934/eect.2019024

A dynamic problem involving a coupled suspension bridge system: Numerical analysis and computational experiments

1. 

Departamento de Matemáticas, Universidade da Coruña, ETS de Ingenieros de Caminos, Canales y Puertos, Campus de Elviña, 15071 A Coruña, Spain

2. 

Departamento de Matemática Aplicada I, Universidade de Vigo, ETSI Telecomunicación, Campus As Lagoas Marcosende s/n, 36310 Vigo, Spain

3. 

Dipartimento di Ingegneria Civile, Architettura, Territorio, Ambiente e di Matematica, Università degli Studi di Brescia, Via Valotti 9, 25133 Brescia, Italy

* Corresponding author: José R. Fernández

Received  April 2018 Revised  November 2018 Published  May 2019

Fund Project: This work has been supported by Ministerio de Economía y Competitividad under the project MTM2015-66640-P (with the participation of FEDER)

In this paper we study, from the numerical point of view, a dynamic problem which models a suspension bridge system. This problem is written as a nonlinear system of hyperbolic partial differential equations in terms of the displacements of the bridge and of the cable. By using the respective velocities, its variational formulation leads to a coupled system of parabolic nonlinear variational equations. An existence and uniqueness result, and an exponential energy decay property, are recalled. Then, fully discrete approximations are introduced by using the classical finite element method and the implicit Euler scheme. A discrete stability property is shown and a priori error estimates are proved, from which the linear convergence of the algorithm is deduced under suitable additional regularity conditions. Finally, some numerical results are shown to demonstrate the accuracy of the approximation and the behaviour of the solution.

Citation: 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
References:
[1]

M. Aassila, Stability of dynamic models of suspension bridges, Math. Nachr., 235 (2002), 5-15.  doi: 10.1002/1522-2616(200202)235:1<5::AID-MANA5>3.0.CO;2-J.  Google Scholar

[2]

O. H. Amann, T. Von Karman and G. B. Wooddruff, The failure of the Tacoma narrows bridge, Federal Works Agency, Washington D.C., 1941. Google Scholar

[3]

A. Arena and W. Lacarbonara, Nonlinear parametric modeling of suspension bridges under aerolastic forces: torsional divergence and flutter, Nonlinear Dyn., 70 (2012), 2487-2510.  doi: 10.1007/s11071-012-0636-3.  Google Scholar

[4]

G. Arioli and F. Gazzola, A new mathematical explanation of what triggered the catastrophic torsional mode of the Tacoma Narrows Bridge, Appl. Math. Model., 39 (2015), 901-912.  doi: 10.1016/j.apm.2014.06.022.  Google Scholar

[5]

G. Arioli and F. Gazzola, Torsional instability in suspension bridges: The Tacoma Narrows Bridge case, Commun. Nonlinear Sci. Numer. Simul., 42 (2017), 342-357.  doi: 10.1016/j.cnsns.2016.05.028.  Google Scholar

[6]

J. M. Ball, Initial-boundary value problems for an extensible beam, J. Math. Anal. Appl., 42 (1973), 61-90.  doi: 10.1016/0022-247X(73)90121-2.  Google Scholar

[7]

I. Bonicchio, C. Giorgi and E. Vuk, Long-term dynamics of the coupled suspension bridge system, Math. Models Methods Appl. Sci., 22 (2012), 1250021, 22pp. doi: 10.1142/S0218202512500212.  Google Scholar

[8]

M. CampoJ. R. FernándezK. L. KuttlerM. Shillor and J. M. Viaño, Numerical analysis and simulations of a dynamic frictionless contact problem with damage, Comput. Methods Appl. Mech. Engrg., 196 (2006), 476-488.  doi: 10.1016/j.cma.2006.05.006.  Google Scholar

[9]

P. G. Ciarlet, Basic error estimates for elliptic problems., in Handbook of Numerical Analysis (eds. P.G. Ciarlet and J.L. Lions), Elsevier, (1993), 17–351.  Google Scholar

[10]

F. Dell'OroC. Giorgi and V. Pata, Asymptotic behaviour of coupled linear systems modeling suspension bridges, Z. Angew. Math. Phys., 66 (2015), 1095-1108.  doi: 10.1007/s00033-014-0414-9.  Google Scholar

[11]

Z. Ding, Traveling waves in a suspension bridge system, SIAM J. Math. Anal., 35 (2003), 160-171.  doi: 10.1137/S0036141002412690.  Google Scholar

[12]

P. DrábekH. HolubováA. Matas and P. Necesal, Nonlinear models of suspension bridges: discussion of the results, Appl. Math., 48 (2003), 497-514.  doi: 10.1023/B:APOM.0000024489.96314.7f.  Google Scholar

[13]

A. Ferrero and F. Gazzola, A partially hinged rectangular plate as a model for suspension bridges, Discrete Contin. Dyn. Syst. Ser. A, 35 (2015), 5879-5908.  doi: 10.3934/dcds.2015.35.5879.  Google Scholar

[14]

J. GloverA. C. Lazer and P. J. McKenna, Existence and stability of large scale nonlinear oscillations in suspension bridges, Z. Angew. Math. Phys., 40 (1989), 172-200.  doi: 10.1007/BF00944997.  Google Scholar

[15]

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

[16]

G. Holubová-Tajcová, Mathematical modeling of suspension bridges, Math. Comput. Simul., 50 (1999), 183-197.  doi: 10.1016/S0378-4754(99)00071-3.  Google Scholar

[17]

G. Holubová and A. Matas, Initial-boundary value problem for the nonlinear string-beam system, J. Math. Anal. Appl., 288 (2003), 784-802.  doi: 10.1016/j.jmaa.2003.09.028.  Google Scholar

[18]

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

[19]

H. Leiva, Exact controllability of the suspension bridge model proposed by Lazer and McKenna, J. Math. Anal. Appl., 309 (2005), 404-419.  doi: 10.1016/j.jmaa.2004.07.025.  Google Scholar

[20]

J. Malík, Nonlinear models of suspension bridges, J. Math. Anal. Appl., 321 (2006), 828-850.  doi: 10.1016/j.jmaa.2005.08.080.  Google Scholar

[21]

J. Malík, Sudden lateral asymmetry and torsional oscillations in the original Tacoma suspension bridge, J. Sound Vib., 332 (2013), 3772-3789.   Google Scholar

[22]

J. Malík, Spectral analysis connected with suspension bridge systems, IMA J. Appl. Math., 81 (2016), 42-75.  doi: 10.1093/imamat/hxv027.  Google Scholar

[23]

C. Marchionna and S. Panizzi, An instability result in the theory of suspension bridges, Nonlinear Anal., 140 (2016), 12-28.  doi: 10.1016/j.na.2016.03.003.  Google Scholar

[24]

P. J. McKenna, Oscillations in suspension bridges, vertical and torsional, Discrete Contin. Dyn. Syst. Ser. S, 7 (2014), 785-791.  doi: 10.3934/dcdss.2014.7.785.  Google Scholar

[25]

C. ZhongQ. Ma and C. Sun, Existence of strong solutions and global attractors for the suspension bridge equations, Nonlinear Anal., 67 (2007), 442-454.  doi: 10.1016/j.na.2006.05.018.  Google Scholar

show all references

References:
[1]

M. Aassila, Stability of dynamic models of suspension bridges, Math. Nachr., 235 (2002), 5-15.  doi: 10.1002/1522-2616(200202)235:1<5::AID-MANA5>3.0.CO;2-J.  Google Scholar

[2]

O. H. Amann, T. Von Karman and G. B. Wooddruff, The failure of the Tacoma narrows bridge, Federal Works Agency, Washington D.C., 1941. Google Scholar

[3]

A. Arena and W. Lacarbonara, Nonlinear parametric modeling of suspension bridges under aerolastic forces: torsional divergence and flutter, Nonlinear Dyn., 70 (2012), 2487-2510.  doi: 10.1007/s11071-012-0636-3.  Google Scholar

[4]

G. Arioli and F. Gazzola, A new mathematical explanation of what triggered the catastrophic torsional mode of the Tacoma Narrows Bridge, Appl. Math. Model., 39 (2015), 901-912.  doi: 10.1016/j.apm.2014.06.022.  Google Scholar

[5]

G. Arioli and F. Gazzola, Torsional instability in suspension bridges: The Tacoma Narrows Bridge case, Commun. Nonlinear Sci. Numer. Simul., 42 (2017), 342-357.  doi: 10.1016/j.cnsns.2016.05.028.  Google Scholar

[6]

J. M. Ball, Initial-boundary value problems for an extensible beam, J. Math. Anal. Appl., 42 (1973), 61-90.  doi: 10.1016/0022-247X(73)90121-2.  Google Scholar

[7]

I. Bonicchio, C. Giorgi and E. Vuk, Long-term dynamics of the coupled suspension bridge system, Math. Models Methods Appl. Sci., 22 (2012), 1250021, 22pp. doi: 10.1142/S0218202512500212.  Google Scholar

[8]

M. CampoJ. R. FernándezK. L. KuttlerM. Shillor and J. M. Viaño, Numerical analysis and simulations of a dynamic frictionless contact problem with damage, Comput. Methods Appl. Mech. Engrg., 196 (2006), 476-488.  doi: 10.1016/j.cma.2006.05.006.  Google Scholar

[9]

P. G. Ciarlet, Basic error estimates for elliptic problems., in Handbook of Numerical Analysis (eds. P.G. Ciarlet and J.L. Lions), Elsevier, (1993), 17–351.  Google Scholar

[10]

F. Dell'OroC. Giorgi and V. Pata, Asymptotic behaviour of coupled linear systems modeling suspension bridges, Z. Angew. Math. Phys., 66 (2015), 1095-1108.  doi: 10.1007/s00033-014-0414-9.  Google Scholar

[11]

Z. Ding, Traveling waves in a suspension bridge system, SIAM J. Math. Anal., 35 (2003), 160-171.  doi: 10.1137/S0036141002412690.  Google Scholar

[12]

P. DrábekH. HolubováA. Matas and P. Necesal, Nonlinear models of suspension bridges: discussion of the results, Appl. Math., 48 (2003), 497-514.  doi: 10.1023/B:APOM.0000024489.96314.7f.  Google Scholar

[13]

A. Ferrero and F. Gazzola, A partially hinged rectangular plate as a model for suspension bridges, Discrete Contin. Dyn. Syst. Ser. A, 35 (2015), 5879-5908.  doi: 10.3934/dcds.2015.35.5879.  Google Scholar

[14]

J. GloverA. C. Lazer and P. J. McKenna, Existence and stability of large scale nonlinear oscillations in suspension bridges, Z. Angew. Math. Phys., 40 (1989), 172-200.  doi: 10.1007/BF00944997.  Google Scholar

[15]

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

[16]

G. Holubová-Tajcová, Mathematical modeling of suspension bridges, Math. Comput. Simul., 50 (1999), 183-197.  doi: 10.1016/S0378-4754(99)00071-3.  Google Scholar

[17]

G. Holubová and A. Matas, Initial-boundary value problem for the nonlinear string-beam system, J. Math. Anal. Appl., 288 (2003), 784-802.  doi: 10.1016/j.jmaa.2003.09.028.  Google Scholar

[18]

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

[19]

H. Leiva, Exact controllability of the suspension bridge model proposed by Lazer and McKenna, J. Math. Anal. Appl., 309 (2005), 404-419.  doi: 10.1016/j.jmaa.2004.07.025.  Google Scholar

[20]

J. Malík, Nonlinear models of suspension bridges, J. Math. Anal. Appl., 321 (2006), 828-850.  doi: 10.1016/j.jmaa.2005.08.080.  Google Scholar

[21]

J. Malík, Sudden lateral asymmetry and torsional oscillations in the original Tacoma suspension bridge, J. Sound Vib., 332 (2013), 3772-3789.   Google Scholar

[22]

J. Malík, Spectral analysis connected with suspension bridge systems, IMA J. Appl. Math., 81 (2016), 42-75.  doi: 10.1093/imamat/hxv027.  Google Scholar

[23]

C. Marchionna and S. Panizzi, An instability result in the theory of suspension bridges, Nonlinear Anal., 140 (2016), 12-28.  doi: 10.1016/j.na.2016.03.003.  Google Scholar

[24]

P. J. McKenna, Oscillations in suspension bridges, vertical and torsional, Discrete Contin. Dyn. Syst. Ser. S, 7 (2014), 785-791.  doi: 10.3934/dcdss.2014.7.785.  Google Scholar

[25]

C. ZhongQ. Ma and C. Sun, Existence of strong solutions and global attractors for the suspension bridge equations, Nonlinear Anal., 67 (2007), 442-454.  doi: 10.1016/j.na.2006.05.018.  Google Scholar

Figure 1.  Example 1: Asymptotic behaviour of the numerical scheme
Figure 2.  Example 2: Oscillations of the bridge for different values of p
Figure 3.  Example 3: Bridge and cable deformed configurations at final time for different values of $ k_*^2. $
Table 1.  Example 1: Numerical errors for some discretization parameters
$ nd \downarrow k \to $ $ 10^{-1} $ $ 10^{-2} $ $ 10^{-3} $ $ 10^{-4} $
$ 10 $ 0.1435116 0.0868544 0.0923330 0.0940042
$ 10^2 $ 0.1639226 0.0174114 0.0070235 0.0069553
$ 10^3 $ 0.1641941 0.0161108 0.0017435 0.0007232
$ 10^4 $ 0.1646557 0.0163375 0.0015935 0.0001722
$ nd \downarrow k \to $ $ 10^{-1} $ $ 10^{-2} $ $ 10^{-3} $ $ 10^{-4} $
$ 10 $ 0.1435116 0.0868544 0.0923330 0.0940042
$ 10^2 $ 0.1639226 0.0174114 0.0070235 0.0069553
$ 10^3 $ 0.1641941 0.0161108 0.0017435 0.0007232
$ 10^4 $ 0.1646557 0.0163375 0.0015935 0.0001722
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