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 & 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). Google Scholar

[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). Google Scholar

[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. Google Scholar

[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. Google Scholar

[5]

A. V. Babin and M. I. Vishik, Attractors of Evolution Equations,, North-Holland, (1992). Google Scholar

[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. Google Scholar

[7]

J. M. Ball, Stability theory for an extensible beam,, J. Differential Equations, 14 (1973), 399. doi: 10.1016/0022-0396(73)90056-9. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[12]

I. Bochicchio, C. Giorgi and E. Vuk, On some nonlinear models for suspension bridges,, in Evolution Equations and Materials with Memory, (2012), 1. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[22]

C. M. Dafermos, Asymptotic stability in viscoelasticity,, Arch. Rational Mech. Anal., 37 (1970), 297. doi: 0.1007/BF00251609. Google Scholar

[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. Google Scholar

[24]

A. D. Drozdov and V. B. Kolmanovskii, Stability in Viscoelasticity,, North-Holland, (1994). Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[28]

M. Grasselli and V. Pata, Uniform attractors of nonautonomous systems with memory,, in Evolution Equations, 50 (2002), 155. Google Scholar

[29]

J. K. Hale, Asymptotic Behavior of Dissipative Systems,, Amer. Math. Soc., (). Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[34]

P. J. McKenna and W. Walter, Nonlinear oscillations in a suspension bridge,, Arch. Rational Mech. Anal., 98 (1987), 167. doi: 10.1007/BF00251232. Google Scholar

[35]

V. Pata, Exponential stability in linear viscoelasticity,, Quart. Applied Math., 64 (2006), 499. doi: 10.1007/s00032-009-0098-3. Google Scholar

[36]

V. Pata and A. Zucchi, Attractors for a damped hyperbolic equation with linear memory,, Adv. Math. Sci. Appl., 11 (2001), 505. Google Scholar

[37]

E. L. Reiss and B. J. Matkowsky, Nonlinear dynamic buckling of a compressed elastic column,, Quart. Appl. Math., 29 (1971), 245. Google Scholar

[38]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics,, Springer, (). doi: 10.1007/978-1-4612-0645-3. Google Scholar

[39]

S. Woinowsky-Krieger, The effect of an axial force on the vibration of hinged bars,, J. Appl. Mech., 17 (1950), 35. Google Scholar

[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. Google Scholar

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). Google Scholar

[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). Google Scholar

[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. Google Scholar

[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. Google Scholar

[5]

A. V. Babin and M. I. Vishik, Attractors of Evolution Equations,, North-Holland, (1992). Google Scholar

[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. Google Scholar

[7]

J. M. Ball, Stability theory for an extensible beam,, J. Differential Equations, 14 (1973), 399. doi: 10.1016/0022-0396(73)90056-9. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[12]

I. Bochicchio, C. Giorgi and E. Vuk, On some nonlinear models for suspension bridges,, in Evolution Equations and Materials with Memory, (2012), 1. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[22]

C. M. Dafermos, Asymptotic stability in viscoelasticity,, Arch. Rational Mech. Anal., 37 (1970), 297. doi: 0.1007/BF00251609. Google Scholar

[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. Google Scholar

[24]

A. D. Drozdov and V. B. Kolmanovskii, Stability in Viscoelasticity,, North-Holland, (1994). Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[28]

M. Grasselli and V. Pata, Uniform attractors of nonautonomous systems with memory,, in Evolution Equations, 50 (2002), 155. Google Scholar

[29]

J. K. Hale, Asymptotic Behavior of Dissipative Systems,, Amer. Math. Soc., (). Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[34]

P. J. McKenna and W. Walter, Nonlinear oscillations in a suspension bridge,, Arch. Rational Mech. Anal., 98 (1987), 167. doi: 10.1007/BF00251232. Google Scholar

[35]

V. Pata, Exponential stability in linear viscoelasticity,, Quart. Applied Math., 64 (2006), 499. doi: 10.1007/s00032-009-0098-3. Google Scholar

[36]

V. Pata and A. Zucchi, Attractors for a damped hyperbolic equation with linear memory,, Adv. Math. Sci. Appl., 11 (2001), 505. Google Scholar

[37]

E. L. Reiss and B. J. Matkowsky, Nonlinear dynamic buckling of a compressed elastic column,, Quart. Appl. Math., 29 (1971), 245. Google Scholar

[38]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics,, Springer, (). doi: 10.1007/978-1-4612-0645-3. Google Scholar

[39]

S. Woinowsky-Krieger, The effect of an axial force on the vibration of hinged bars,, J. Appl. Mech., 17 (1950), 35. Google Scholar

[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. Google Scholar

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