# American Institute of Mathematical Sciences

September  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 and 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), 313-329. 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), 330-345. 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), 853-874. doi: 10.1137/S0036139996308698. [4] Y. An, Nonlinear perturbations of a coupled system of steady state suspension bridge equations, Nonlinear Analysis, 51 (2002), 1285-1292. doi: 10.1016/S0362-546X(01)00899-9. [5] A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, North-Holland, Amsterdam, 1992. [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] J. M. Ball, Stability theory for an extensible beam, J. Differential Equations, 14 (1973), 399-418. 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), 232-243. 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), 19 pp. 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), 833-846. 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), 61-76. http://ijpam.eu/contents/2010-58-1/8/8.pdf. [12] I. Bochicchio, C. Giorgi and E. Vuk, On some nonlinear models for suspension bridges, in Evolution Equations and Materials with Memory, (eds. D.Andreucci, S.Carillo, M. Fabrizio, P.Loreti and D.Sforza), Sapienza Università, (2012), 1-18. [13] I. Bochicchio, C. Giorgi and E. Vuk, Long-term dynamics of the coupled suspension bridge system, Math. Models Methods Appl. Sci., 22 (2012), (22 pages). 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), 319-333. 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), 2139-2151. 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), 649-668. 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), 21-45. doi: 10.1142/S0218202508002590. [18] M. Conti and and V. Pata, Weakly dissipative semilinear equations of viscoelasticity, Commun. Pure Appl. Anal., 4 (2005), 705-720. 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), 169-216. 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), 1041-1060. 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), 43-58. doi: 10.1142/S0218202510004143. [22] C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Rational Mech. Anal., 37 (1970), 297-308. 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), 497-514. doi: 10.1023/B:APOM.0000024489.96314.7f. [24] A. D. Drozdov and V. B. Kolmanovskii, Stability in Viscoelasticity, North-Holland, Amsterdam, 1994. [25] C. Giorgi, V. Pata and E. Vuk, On the extensible viscoelastic beam, Nonlinearity, 21 (2008), 713-733. 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), 204. 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), 171-200. doi: 10.1007/BF00944997. [28] M. Grasselli and V. Pata, Uniform attractors of nonautonomous systems with memory, in Evolution Equations, Semigroups and Functional Analysis, Progr. Nonlinear Differential Equations Appl., Birkhauser, Boston, (eds. A. Lorenzi and B. Ruf), , 50 (2002), 155-178. [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), 784-802. 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), 108-113. 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), 537-578. 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), 3755-3775. 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), 167-177. doi: 10.1007/BF00251232. [35] V. Pata, Exponential stability in linear viscoelasticity, Quart. Applied Math., 64 (2006), 499-513. 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), 505-529. [37] E. L. Reiss and B. J. Matkowsky, Nonlinear dynamic buckling of a compressed elastic column, Quart. Appl. Math., 29 (1971), 245-260. [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), 35-36. [40] C. Zhong, Q. Ma and C. Sun, Existence of strong solutions and global attractors for the suspension bridge equations, Nonlinear Analysis, 67 (2007), 442-454. 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), 313-329. 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), 330-345. 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), 853-874. doi: 10.1137/S0036139996308698. [4] Y. An, Nonlinear perturbations of a coupled system of steady state suspension bridge equations, Nonlinear Analysis, 51 (2002), 1285-1292. doi: 10.1016/S0362-546X(01)00899-9. [5] A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, North-Holland, Amsterdam, 1992. [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] J. M. Ball, Stability theory for an extensible beam, J. Differential Equations, 14 (1973), 399-418. 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), 232-243. 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), 19 pp. 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), 833-846. 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), 61-76. http://ijpam.eu/contents/2010-58-1/8/8.pdf. [12] I. Bochicchio, C. Giorgi and E. Vuk, On some nonlinear models for suspension bridges, in Evolution Equations and Materials with Memory, (eds. D.Andreucci, S.Carillo, M. Fabrizio, P.Loreti and D.Sforza), Sapienza Università, (2012), 1-18. [13] I. Bochicchio, C. Giorgi and E. Vuk, Long-term dynamics of the coupled suspension bridge system, Math. Models Methods Appl. Sci., 22 (2012), (22 pages). 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), 319-333. 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), 2139-2151. 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), 649-668. 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), 21-45. doi: 10.1142/S0218202508002590. [18] M. Conti and and V. Pata, Weakly dissipative semilinear equations of viscoelasticity, Commun. Pure Appl. Anal., 4 (2005), 705-720. 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), 169-216. 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), 1041-1060. 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), 43-58. doi: 10.1142/S0218202510004143. [22] C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Rational Mech. Anal., 37 (1970), 297-308. 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), 497-514. doi: 10.1023/B:APOM.0000024489.96314.7f. [24] A. D. Drozdov and V. B. Kolmanovskii, Stability in Viscoelasticity, North-Holland, Amsterdam, 1994. [25] C. Giorgi, V. Pata and E. Vuk, On the extensible viscoelastic beam, Nonlinearity, 21 (2008), 713-733. 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), 204. 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), 171-200. doi: 10.1007/BF00944997. [28] M. Grasselli and V. Pata, Uniform attractors of nonautonomous systems with memory, in Evolution Equations, Semigroups and Functional Analysis, Progr. Nonlinear Differential Equations Appl., Birkhauser, Boston, (eds. A. Lorenzi and B. Ruf), , 50 (2002), 155-178. [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), 784-802. 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), 108-113. 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), 537-578. 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), 3755-3775. 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), 167-177. doi: 10.1007/BF00251232. [35] V. Pata, Exponential stability in linear viscoelasticity, Quart. Applied Math., 64 (2006), 499-513. 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), 505-529. [37] E. L. Reiss and B. J. Matkowsky, Nonlinear dynamic buckling of a compressed elastic column, Quart. Appl. Math., 29 (1971), 245-260. [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), 35-36. [40] C. Zhong, Q. Ma and C. Sun, Existence of strong solutions and global attractors for the suspension bridge equations, Nonlinear Analysis, 67 (2007), 442-454. doi: 10.1016/j.na.2006.05.018.
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