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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  September 2019 Early access  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 and 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. [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. [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. [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. [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. [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. [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. [8] M. Campo, J. R. Fernández, K. L. Kuttler, M. 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. [9] P. G. Ciarlet, Basic error estimates for elliptic problems., in Handbook of Numerical Analysis (eds. P.G. Ciarlet and J.L. Lions), Elsevier, Ⅱ (1993), 17–351. [10] F. Dell'Oro, C. 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. [11] Z. Ding, Traveling waves in a suspension bridge system, SIAM J. Math. Anal., 35 (2003), 160-171.  doi: 10.1137/S0036141002412690. [12] P. Drábek, H. Holubová, A. 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. [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. [14] J. Glover, A. 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. [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. [16] G. Holubová-Tajcová, Mathematical modeling of suspension bridges, Math. Comput. Simul., 50 (1999), 183-197.  doi: 10.1016/S0378-4754(99)00071-3. [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. [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. [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. [20] J. Malík, Nonlinear models of suspension bridges, J. Math. Anal. Appl., 321 (2006), 828-850.  doi: 10.1016/j.jmaa.2005.08.080. [21] J. Malík, Sudden lateral asymmetry and torsional oscillations in the original Tacoma suspension bridge, J. Sound Vib., 332 (2013), 3772-3789. [22] J. Malík, Spectral analysis connected with suspension bridge systems, IMA J. Appl. Math., 81 (2016), 42-75.  doi: 10.1093/imamat/hxv027. [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. [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. [25] C. Zhong, Q. 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.

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. [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. [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. [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. [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. [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. [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. [8] M. Campo, J. R. Fernández, K. L. Kuttler, M. 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. [9] P. G. Ciarlet, Basic error estimates for elliptic problems., in Handbook of Numerical Analysis (eds. P.G. Ciarlet and J.L. Lions), Elsevier, Ⅱ (1993), 17–351. [10] F. Dell'Oro, C. 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. [11] Z. Ding, Traveling waves in a suspension bridge system, SIAM J. Math. Anal., 35 (2003), 160-171.  doi: 10.1137/S0036141002412690. [12] P. Drábek, H. Holubová, A. 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. [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. [14] J. Glover, A. 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. [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. [16] G. Holubová-Tajcová, Mathematical modeling of suspension bridges, Math. Comput. Simul., 50 (1999), 183-197.  doi: 10.1016/S0378-4754(99)00071-3. [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. [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. [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. [20] J. Malík, Nonlinear models of suspension bridges, J. Math. Anal. Appl., 321 (2006), 828-850.  doi: 10.1016/j.jmaa.2005.08.080. [21] J. Malík, Sudden lateral asymmetry and torsional oscillations in the original Tacoma suspension bridge, J. Sound Vib., 332 (2013), 3772-3789. [22] J. Malík, Spectral analysis connected with suspension bridge systems, IMA J. Appl. Math., 81 (2016), 42-75.  doi: 10.1093/imamat/hxv027. [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. [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. [25] C. Zhong, Q. 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.
Example 1: Asymptotic behaviour of the numerical scheme
Example 2: Oscillations of the bridge for different values of p
Example 3: Bridge and cable deformed configurations at final time for different values of $k_*^2.$
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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