# American Institute of Mathematical Sciences

2014, 3(3): 373-397. doi: 10.3934/eect.2014.3.373

## On the viscoelastic coupled suspension bridge

 1 Dipartimento di Matematica, Università degli studi di Salerno, Via Giovanni Paolo II 132, 84084 Fisciano (SA), Italy 2 DICATAM, Università degli studi di Brescia, Via D.Valotti 9, 25133 Brescia, Italy

Received  August 2013 Revised  February 2014 Published  August 2014

In this paper we discuss the asymptotic behavior of a doubly nonlinear problem describing the vibrations of a coupled suspension bridge. The single-span road-bed is modeled as an extensible viscoelastic beam which is simply supported at the ends. The main cable is modeled by a viscoelastic string and is connected to the road-bed by a distributed system of one-sided elastic springs. A constant axial force $p$ is applied at one end of the deck, and time-independent vertical loads are allowed to act both on the road-bed and on the suspension cable. For this general model we obtain original results, including the existence of a regular global attractor for all $p\in\mathbb{R}$.
Citation: 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
##### References:
 [1] A. M. Abdel-Ghaffar and L. I. Rubin, Non linear free vibrations of suspension bridges: Theory,, ASCE J. Eng. Mech., 109 (1983), 313. doi: 10.1061/(ASCE)0733-9399(1983)109:1(313). [2] A. M. Abdel-Ghaffar and L. I. Rubin, Non linear free vibrations of suspension bridges: Application,, ASCE J. Eng. Mech., 109 (1983), 330. doi: 10.1061/(ASCE)0733-9399(1983)109:1(330). [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] Y. An, Nonlinear perturbations of a coupled system of steady state suspension bridge equations,, Nonlinear Analysis, 51 (2002), 1285. doi: 10.1016/S0362-546X(01)00899-9. [5] A. V. Babin and M. I. Vishik, Attractors of Evolution Equations,, North-Holland, (1992). [6] J. M. Ball, Initial-boundary value problems for an extensible beam,, J. Math. Anal. Appl., 42 (1973), 61. doi: 10.1016/0022-247X(73)90121-2. [7] J. M. Ball, Stability theory for an extensible beam,, J. Differential Equations, 14 (1973), 399. doi: 10.1016/0022-0396(73)90056-9. [8] I. Bochicchio, C. Giorgi and E. Vuk, Steady states analysis and exponential stability of an extensible thermoelastic system,, Comunication to SIMAI Congress, 3 (2009), 232. doi: 10.1685/CSC09232. [9] I. Bochicchio, C. Giorgi and E. Vuk, Long-term damped dynamics of the extensible suspension bridge,, Int. J. Differ. Equ., 2010 (2010). doi: 10.1155/2010/383420. [10] I. Bochicchio and E. Vuk, Buckling and longterm dynamics of a nonlinear model for the extensible beam,, Math. Comput. Modelling, 51 (2010), 833. doi: 10.1016/j.mcm.2009.10.010. [11] I. Bochicchio and E. Vuk, Longtime behavior for oscillations of an extensible viscoelastic beam with elastic external supply,, Int. J. Pure Appl. Math., 58 (2010), 61. [12] I. Bochicchio, C. Giorgi and E. Vuk, On some nonlinear models for suspension bridges,, in Evolution Equations and Materials with Memory, (2012), 1. [13] I. Bochicchio, C. Giorgi and E. Vuk, Long-term dynamics of the coupled suspension bridge system,, Math. Models Methods Appl. Sci., 22 (2012). doi: 10.1142/S0218202512500212. [14] I. Bochicchio, C. Giorgi and E. Vuk, Asymptotic dynamics of nonlinear coupled suspension bridge equations,, J. Math. Anal. Appl., 402 (2013), 319. doi: 10.1016/j.jmaa.2013.01.036. [15] I. Bochicchio, C. Giorgi and E Vuk, Long-term dynamics of a viscoelatic suspension bridge,, Meccanica, 49 (2014), 2139. doi: 10.1007/s11012-014-9887-z. [16] Q. H. Choi and T. Jung, A nonlinear suspension bridge equation with nonconstant load,, Nonlinear Anal., 35 (1999), 649. doi: 10.1016/S0362-546X(97)00616-0. [17] M. Conti, S. Gatti and V. Pata, Uniform decay properties of linear Volterra integro-differential equations,, Math. Models Methods Appl. Sci., 18 (2008), 21. doi: 10.1142/S0218202508002590. [18] M. Conti and and V. Pata, Weakly dissipative semilinear equations of viscoelasticity,, Commun. Pure Appl. Anal., 4 (2005), 705. doi: 10.3934/cpaa.2005.4.705. [19] M. Conti, V. Pata and M. Squassina, Singular limit of differential systems with memory,, Indiana Univ. Math. J., 55 (2006), 169. doi: 10.1512/iumj.2006.55.2661. [20] M. Coti Zelati, Global and exponential attractors for the singularly perturbed extensible beam,, Discrete Contin. Dyn. Syst., 25 (2009), 1041. doi: 10.3934/dcds.2009.25.1041. [21] M. Coti Zelati, C. Giorgi and V. Pata, Steady states of the hinged extensible beam with external load,, Math. Models Methods Appl. Sci., 20 (2010), 43. doi: 10.1142/S0218202510004143. [22] C. M. Dafermos, Asymptotic stability in viscoelasticity,, Arch. Rational Mech. Anal., 37 (1970), 297. doi: 0.1007/BF00251609. [23] P. Drábek, G. Holubová, A. Matas and P. Nečesal, Nonlinear models of suspension bridges: Discussion of the results,, Applications of Mathematics, 48 (2003), 497. doi: 10.1023/B:APOM.0000024489.96314.7f. [24] A. D. Drozdov and V. B. Kolmanovskii, Stability in Viscoelasticity,, North-Holland, (1994). [25] C. Giorgi, V. Pata and E. Vuk, On the extensible viscoelastic beam,, Nonlinearity, 21 (2008), 713. doi: 10.1088/0951-7715/21/4/004. [26] C. Giorgi and E. Vuk, Steady-state solutions for a suspension bridge with intermediate supports,, Bound. Value Probl., 2013 (2013). doi: 10.1186/1687-2770-2013-204. [27] J. Glover, A. C. Lazer and P. J. McKenna, Existence and stability of a large-scale nonlinear oscillations in suspension bridge,, Z. Angew. Math. Phys., 40 (1989), 171. doi: 10.1007/BF00944997. [28] M. Grasselli and V. Pata, Uniform attractors of nonautonomous systems with memory,, in Evolution Equations, 50 (2002), 155. [29] J. K. Hale, Asymptotic Behavior of Dissipative Systems,, Amer. Math. Soc., (). [30] G. Holubová and A. Matas, Initial-boundary problem for the nonlinear string-beam system,, J. Math. Anal. Appl., 288 (2003), 784. doi: 10.1016/j.jmaa.2003.09.028. [31] W. Kanok-Nukulchai, P. K. A. Yiu and D. M. Brotton, Mathematical modelling of cable-stayed bridges,, Structural Engineering International, 2 (1992), 108. doi: 10.2749/101686692780616030. [32] 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. [33] Q. Ma and C. Zhong, Existence of strong solutions and global attractors for the coupled suspension bridge equations,, J. Differential Equations, 246 (2009), 3755. doi: 10.1016/j.jde.2009.02.022. [34] P. J. McKenna and W. Walter, Nonlinear oscillations in a suspension bridge,, Arch. Rational Mech. Anal., 98 (1987), 167. doi: 10.1007/BF00251232. [35] V. Pata, Exponential stability in linear viscoelasticity,, Quart. Applied Math., 64 (2006), 499. doi: 10.1007/s00032-009-0098-3. [36] V. Pata and A. Zucchi, Attractors for a damped hyperbolic equation with linear memory,, Adv. Math. Sci. Appl., 11 (2001), 505. [37] E. L. Reiss and B. J. Matkowsky, Nonlinear dynamic buckling of a compressed elastic column,, Quart. Appl. Math., 29 (1971), 245. [38] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics,, Springer, (). doi: 10.1007/978-1-4612-0645-3. [39] S. Woinowsky-Krieger, The effect of an axial force on the vibration of hinged bars,, J. Appl. Mech., 17 (1950), 35. [40] C. Zhong, Q. Ma and C. Sun, Existence of strong solutions and global attractors for the suspension bridge equations,, Nonlinear Analysis, 67 (2007), 442. doi: 10.1016/j.na.2006.05.018.

show all references

##### References:
 [1] A. M. Abdel-Ghaffar and L. I. Rubin, Non linear free vibrations of suspension bridges: Theory,, ASCE J. Eng. Mech., 109 (1983), 313. doi: 10.1061/(ASCE)0733-9399(1983)109:1(313). [2] A. M. Abdel-Ghaffar and L. I. Rubin, Non linear free vibrations of suspension bridges: Application,, ASCE J. Eng. Mech., 109 (1983), 330. doi: 10.1061/(ASCE)0733-9399(1983)109:1(330). [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] Y. An, Nonlinear perturbations of a coupled system of steady state suspension bridge equations,, Nonlinear Analysis, 51 (2002), 1285. doi: 10.1016/S0362-546X(01)00899-9. [5] A. V. Babin and M. I. Vishik, Attractors of Evolution Equations,, North-Holland, (1992). [6] J. M. Ball, Initial-boundary value problems for an extensible beam,, J. Math. Anal. Appl., 42 (1973), 61. doi: 10.1016/0022-247X(73)90121-2. [7] J. M. Ball, Stability theory for an extensible beam,, J. Differential Equations, 14 (1973), 399. doi: 10.1016/0022-0396(73)90056-9. [8] I. Bochicchio, C. Giorgi and E. Vuk, Steady states analysis and exponential stability of an extensible thermoelastic system,, Comunication to SIMAI Congress, 3 (2009), 232. doi: 10.1685/CSC09232. [9] I. Bochicchio, C. Giorgi and E. Vuk, Long-term damped dynamics of the extensible suspension bridge,, Int. J. Differ. Equ., 2010 (2010). doi: 10.1155/2010/383420. [10] I. Bochicchio and E. Vuk, Buckling and longterm dynamics of a nonlinear model for the extensible beam,, Math. Comput. Modelling, 51 (2010), 833. doi: 10.1016/j.mcm.2009.10.010. [11] I. Bochicchio and E. Vuk, Longtime behavior for oscillations of an extensible viscoelastic beam with elastic external supply,, Int. J. Pure Appl. Math., 58 (2010), 61. [12] I. Bochicchio, C. Giorgi and E. Vuk, On some nonlinear models for suspension bridges,, in Evolution Equations and Materials with Memory, (2012), 1. [13] I. Bochicchio, C. Giorgi and E. Vuk, Long-term dynamics of the coupled suspension bridge system,, Math. Models Methods Appl. Sci., 22 (2012). doi: 10.1142/S0218202512500212. [14] I. Bochicchio, C. Giorgi and E. Vuk, Asymptotic dynamics of nonlinear coupled suspension bridge equations,, J. Math. Anal. Appl., 402 (2013), 319. doi: 10.1016/j.jmaa.2013.01.036. [15] I. Bochicchio, C. Giorgi and E Vuk, Long-term dynamics of a viscoelatic suspension bridge,, Meccanica, 49 (2014), 2139. doi: 10.1007/s11012-014-9887-z. [16] Q. H. Choi and T. Jung, A nonlinear suspension bridge equation with nonconstant load,, Nonlinear Anal., 35 (1999), 649. doi: 10.1016/S0362-546X(97)00616-0. [17] M. Conti, S. Gatti and V. Pata, Uniform decay properties of linear Volterra integro-differential equations,, Math. Models Methods Appl. Sci., 18 (2008), 21. doi: 10.1142/S0218202508002590. [18] M. Conti and and V. Pata, Weakly dissipative semilinear equations of viscoelasticity,, Commun. Pure Appl. Anal., 4 (2005), 705. doi: 10.3934/cpaa.2005.4.705. [19] M. Conti, V. Pata and M. Squassina, Singular limit of differential systems with memory,, Indiana Univ. Math. J., 55 (2006), 169. doi: 10.1512/iumj.2006.55.2661. [20] M. Coti Zelati, Global and exponential attractors for the singularly perturbed extensible beam,, Discrete Contin. Dyn. Syst., 25 (2009), 1041. doi: 10.3934/dcds.2009.25.1041. [21] M. Coti Zelati, C. Giorgi and V. Pata, Steady states of the hinged extensible beam with external load,, Math. Models Methods Appl. Sci., 20 (2010), 43. doi: 10.1142/S0218202510004143. [22] C. M. Dafermos, Asymptotic stability in viscoelasticity,, Arch. Rational Mech. Anal., 37 (1970), 297. doi: 0.1007/BF00251609. [23] P. Drábek, G. Holubová, A. Matas and P. Nečesal, Nonlinear models of suspension bridges: Discussion of the results,, Applications of Mathematics, 48 (2003), 497. doi: 10.1023/B:APOM.0000024489.96314.7f. [24] A. D. Drozdov and V. B. Kolmanovskii, Stability in Viscoelasticity,, North-Holland, (1994). [25] C. Giorgi, V. Pata and E. Vuk, On the extensible viscoelastic beam,, Nonlinearity, 21 (2008), 713. doi: 10.1088/0951-7715/21/4/004. [26] C. Giorgi and E. Vuk, Steady-state solutions for a suspension bridge with intermediate supports,, Bound. Value Probl., 2013 (2013). doi: 10.1186/1687-2770-2013-204. [27] J. Glover, A. C. Lazer and P. J. McKenna, Existence and stability of a large-scale nonlinear oscillations in suspension bridge,, Z. Angew. Math. Phys., 40 (1989), 171. doi: 10.1007/BF00944997. [28] M. Grasselli and V. Pata, Uniform attractors of nonautonomous systems with memory,, in Evolution Equations, 50 (2002), 155. [29] J. K. Hale, Asymptotic Behavior of Dissipative Systems,, Amer. Math. Soc., (). [30] G. Holubová and A. Matas, Initial-boundary problem for the nonlinear string-beam system,, J. Math. Anal. Appl., 288 (2003), 784. doi: 10.1016/j.jmaa.2003.09.028. [31] W. Kanok-Nukulchai, P. K. A. Yiu and D. M. Brotton, Mathematical modelling of cable-stayed bridges,, Structural Engineering International, 2 (1992), 108. doi: 10.2749/101686692780616030. [32] 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. [33] Q. Ma and C. Zhong, Existence of strong solutions and global attractors for the coupled suspension bridge equations,, J. Differential Equations, 246 (2009), 3755. doi: 10.1016/j.jde.2009.02.022. [34] P. J. McKenna and W. Walter, Nonlinear oscillations in a suspension bridge,, Arch. Rational Mech. Anal., 98 (1987), 167. doi: 10.1007/BF00251232. [35] V. Pata, Exponential stability in linear viscoelasticity,, Quart. Applied Math., 64 (2006), 499. doi: 10.1007/s00032-009-0098-3. [36] V. Pata and A. Zucchi, Attractors for a damped hyperbolic equation with linear memory,, Adv. Math. Sci. Appl., 11 (2001), 505. [37] E. L. Reiss and B. J. Matkowsky, Nonlinear dynamic buckling of a compressed elastic column,, Quart. Appl. Math., 29 (1971), 245. [38] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics,, Springer, (). doi: 10.1007/978-1-4612-0645-3. [39] S. Woinowsky-Krieger, The effect of an axial force on the vibration of hinged bars,, J. Appl. Mech., 17 (1950), 35. [40] C. Zhong, Q. Ma and C. Sun, Existence of strong solutions and global attractors for the suspension bridge equations,, Nonlinear Analysis, 67 (2007), 442. doi: 10.1016/j.na.2006.05.018.
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