# American Institute of Mathematical Sciences

March  2013, 2(1): 35-54. doi: 10.3934/eect.2013.2.35

## Vibrations of a damped extensible beam between two stops

 1 Facoltà di Ingegneria, Università e-Campus Italia, Via Isimbardi 10, Novedrate (CO), 22060, Italy 2 Dipartimento di Matematica, Università degli Studi di Brescia, Via Valotti 9, Brescia, 25133, Italy

Received  July 2012 Revised  July 2012 Published  January 2013

A PDE system modeling the dynamics of an extensible beam having one of its ends constrained between two stops is considered. The existence of a weak global-in-time solution is established by a penalization method. In addition, the asymptotic behavior of such a solution is analyzed and the exponential decay rate for the related energy is shown.
Citation: Alessia Berti, Maria Grazia Naso. Vibrations of a damped extensible beam between two stops. Evolution Equations & Control Theory, 2013, 2 (1) : 35-54. doi: 10.3934/eect.2013.2.35
##### References:
 [1] K. T. Andrews, M. Shillor and S. Wright, On the dynamic vibrations of an elastic beam in frictional contact with a rigid obstacle, J. Elasticity, 42 (1996), 1-30. doi: 10.1007/BF00041221.  Google Scholar [2] J. M. Ball, Initial-boundary value problems for an extensible beam, J. Math. Anal. Appl., 42 (1973), 61-90.  Google Scholar [3] A. Berti and M. G. Naso, Unilateral dynamic contact of two viscoelastic beams, Quart. Appl. Math., 69 (2011), 477-507.  Google Scholar [4] E. Bonetti, G. Bonfanti and R. Rossi, Global existence for a contact problem with adhesion, Math. Methods Appl. Sci., 31 (2008), 1029-1064. doi: 10.1002/mma.957.  Google Scholar [5] G. Bonfanti, M. Fabrizio, J. E. Muñoz Rivera and M. G. Naso, On the energy decay for a thermoelastic contact problem involving heat transfer, J. Thermal Stresses, 33 (2010), 1049-1065. doi: 10.1080/01495739.2010.511903.  Google Scholar [6] G. Bonfanti, J. E. Muñoz Rivera and M. G. Naso, Global existence and exponential stability for a contact problem between two thermoelastic beams, J. Math. Anal. Appl., 345 (2008), 186-202. doi: 10.1016/j.jmaa.2008.04.003.  Google Scholar [7] G. Bonfanti and M. G. Naso, A dynamic contact problem between two thermoelastic beams, in "Applied and Industrial Mathematics in Italy III," Ser. Adv. Math. Appl. Sci., 82, World Sci. Publ., Hackensack, NJ, (2010), 123-133. doi: 10.1142/9789814280303_0011.  Google Scholar [8] M. I. M. Copetti and D. A. French, Numerical approximation and error control for a thermoelastic contact problem, Appl. Numer. Math., 55 (2005), 439-457. doi: 10.1016/j.apnum.2004.12.002.  Google Scholar [9] R. W. Dickey, Free vibrations and dynamic buckling of the extensible beam, J. Math. Anal. Appl., 29 (1970), 443-454.  Google Scholar [10] G. Duvaut and J.-L. Lions, "Inequalities in Mechanics and Physics," Grundlehren der Mathematischen Wissenschaften, 219, Springer-Verlag, Berlin-New York, 1976.  Google Scholar [11] C. Eck, J. Jarušek and M. Krbec, "Unilateral Contact Problems. Variational Methods and Existence Theorems," Pure and Applied Mathematics (Boca Raton), 270, Chapman & Hall/CRC, Boca Raton, FL, 2005. doi: 10.1201/9781420027365.  Google Scholar [12] M. Frémond, "Non-Smooth Thermomechanics," Springer-Verlag, Berlin, 2002.  Google Scholar [13] H. Gao and J. E. Muñoz Rivera, Global existence and decay for the semilinear thermoelastic contact problem, J. Differential Equations, 186 (2002), 52-68. doi: 10.1016/S0022-0396(02)00016-5.  Google Scholar [14] W. Han and M. Sofonea, "Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity," AMS/IP Studies in Advanced Mathematics, 30, American Mathematical Society, Providence, RI; International Press, Somerville, MA, 2002.  Google Scholar [15] N. J. Hoff, The dynamics of the buckling of elastic columns, J. Appl. Mech., 18 (1951), 68-74.  Google Scholar [16] N. Kikuchi and J. T. Oden, "Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods," SIAM Studies in Applied Mathematics, 8, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1988.  Google Scholar [17] J. U. Kim, A one-dimensional dynamic contact problem in linear viscoelasticity, Math. Methods Appl. Sci., 13 (1990), 55-79. doi: 10.1002/mma.1670130106.  Google Scholar [18] K. L. Kuttler, A. Park, M. Shillor and W. Zhang, Unilateral dynamic contact of two beams, Math. Comput. Modelling, 34 (2001), 365-384. doi: 10.1016/S0895-7177(01)00068-1.  Google Scholar [19] K. L. Kuttler and M. Shillor, Vibrations of a beam between two stops, Dyn. Contin. Discrete Impuls. Syst. Ser. B Appl. Algorithms, 8 (2001), 93-110.  Google Scholar [20] J. A. C. Martins and J. T. Oden, A numerical analysis of a class of problems in elastodynamics with friction, Comput. Methods Appl. Mech. Engrg., 40 (1983), 327-360. doi: 10.1016/0045-7825(83)90105-6.  Google Scholar [21] J. E. Muñoz Rivera and D. Andrade, Existence and exponential decay for contact problems in thermoelasticity, Appl. Anal., 72 (1999), 253-273. doi: 10.1080/00036819908840741.  Google Scholar [22] J. E. Muñoz Rivera and M. de Lacerda Oliveira, Exponential stability for a contact problem in thermoelasticity, IMA J. Appl. Math., 58 (1997), 71-82. doi: 10.1093/imamat/58.1.71.  Google Scholar [23] J. E. Muñoz Rivera and S. Jiang, The thermoelastic and viscoelastic contact of two rods, J. Math. Anal. Appl., 217 (1998), 423-458. doi: 10.1006/jmaa.1997.5717.  Google Scholar [24] J. E. Muñoz Rivera and H. Portillo Oquendo, Existence and decay to contact problems for thermoviscoelastic plates, J. Math. Anal. Appl., 233 (1999), 56-76. doi: 10.1006/jmaa.1998.6236.  Google Scholar [25] J. E. Muñoz Rivera and H. Portillo Oquendo, Exponential decay for a contact problem with local damping, Funkcial. Ekvac., 42 (1999), 371-387.  Google Scholar [26] J. E. Muñoz Rivera and H. Portillo Oquendo, Exponential stability to a contact problem of partially viscoelastic materials, J. Elasticity, 63 (2001), 87-111. doi: 10.1023/A:1014091825772.  Google Scholar [27] J. E. Muñoz Rivera and J. B. Sobrinho, Existence and uniform rates of decay for contact problems in viscoelasticity, Appl. Anal., 67 (1997), 175-199. doi: 10.1080/00036819708840604.  Google Scholar [28] M. Nakao and J. E. Muñoz Rivera, The contact problem in thermoviscoelastic materials, J. Math. Anal. Appl., 264 (2001), 522-545. doi: 10.1006/jmaa.2001.7686.  Google Scholar [29] F. G. Pfeiffer, Applications of unilateral multibody dynamics. Non-smooth mechanics, R. Soc. Lond. Phil. Trans. Ser. A Math. Phys. Eng. Sci., 359 (2001), 2609-2628. doi: 10.1098/rsta.2001.0912.  Google Scholar [30] A. Rodríguez-Arós, J. M. Viaño and M. Sofonea, Numerical analysis of a frictional contact problem for viscoelastic materials with long-term memory, Numer. Math., 108 (2007), 327-358. doi: 10.1007/s00211-007-0117-7.  Google Scholar [31] M. Shillor, M. Sofonea and J. J. Telega, Quasistatic viscoelastic contact with friction and wear diffusion, Quart. Appl. Math., 62 (2004), 379-399.  Google Scholar [32] J. Simon, Compact sets in the space $L^p(0,T;B)$, Ann. Mat. Pura Appl. (4), 146 (1987), 65-96. doi: 10.1007/BF01762360.  Google Scholar [33] M. Sofonea, W. Han and M. Shillor, "Analysis and Approximation of Contact Problems with Adhesion or Damage," Pure and Applied Mathematics (Boca Raton), 276, Chapman & Hall/CRC, Boca Raton, FL, 2006.  Google Scholar [34] M. E. Stavroulaki and G. E. Stavroulakis, Unilateral contact applications using FEM software. Mathematical modelling and numerical analysis in solid mechanics, Int. J. Appl. Math. Comput. Sci., 12 (2002), 115-125.  Google Scholar [35] S. Woinowsky-Krieger, The effect of an axial force on the vibration of hinged bars, J. Appl. Mech., 17 (1950), 35-36.  Google Scholar

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##### References:
 [1] K. T. Andrews, M. Shillor and S. Wright, On the dynamic vibrations of an elastic beam in frictional contact with a rigid obstacle, J. Elasticity, 42 (1996), 1-30. doi: 10.1007/BF00041221.  Google Scholar [2] J. M. Ball, Initial-boundary value problems for an extensible beam, J. Math. Anal. Appl., 42 (1973), 61-90.  Google Scholar [3] A. Berti and M. G. Naso, Unilateral dynamic contact of two viscoelastic beams, Quart. Appl. Math., 69 (2011), 477-507.  Google Scholar [4] E. Bonetti, G. Bonfanti and R. Rossi, Global existence for a contact problem with adhesion, Math. Methods Appl. Sci., 31 (2008), 1029-1064. doi: 10.1002/mma.957.  Google Scholar [5] G. Bonfanti, M. Fabrizio, J. E. Muñoz Rivera and M. G. Naso, On the energy decay for a thermoelastic contact problem involving heat transfer, J. Thermal Stresses, 33 (2010), 1049-1065. doi: 10.1080/01495739.2010.511903.  Google Scholar [6] G. Bonfanti, J. E. Muñoz Rivera and M. G. Naso, Global existence and exponential stability for a contact problem between two thermoelastic beams, J. Math. Anal. Appl., 345 (2008), 186-202. doi: 10.1016/j.jmaa.2008.04.003.  Google Scholar [7] G. Bonfanti and M. G. Naso, A dynamic contact problem between two thermoelastic beams, in "Applied and Industrial Mathematics in Italy III," Ser. Adv. Math. Appl. Sci., 82, World Sci. Publ., Hackensack, NJ, (2010), 123-133. doi: 10.1142/9789814280303_0011.  Google Scholar [8] M. I. M. Copetti and D. A. French, Numerical approximation and error control for a thermoelastic contact problem, Appl. Numer. Math., 55 (2005), 439-457. doi: 10.1016/j.apnum.2004.12.002.  Google Scholar [9] R. W. Dickey, Free vibrations and dynamic buckling of the extensible beam, J. Math. Anal. Appl., 29 (1970), 443-454.  Google Scholar [10] G. Duvaut and J.-L. Lions, "Inequalities in Mechanics and Physics," Grundlehren der Mathematischen Wissenschaften, 219, Springer-Verlag, Berlin-New York, 1976.  Google Scholar [11] C. Eck, J. Jarušek and M. Krbec, "Unilateral Contact Problems. Variational Methods and Existence Theorems," Pure and Applied Mathematics (Boca Raton), 270, Chapman & Hall/CRC, Boca Raton, FL, 2005. doi: 10.1201/9781420027365.  Google Scholar [12] M. Frémond, "Non-Smooth Thermomechanics," Springer-Verlag, Berlin, 2002.  Google Scholar [13] H. Gao and J. E. Muñoz Rivera, Global existence and decay for the semilinear thermoelastic contact problem, J. Differential Equations, 186 (2002), 52-68. doi: 10.1016/S0022-0396(02)00016-5.  Google Scholar [14] W. Han and M. Sofonea, "Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity," AMS/IP Studies in Advanced Mathematics, 30, American Mathematical Society, Providence, RI; International Press, Somerville, MA, 2002.  Google Scholar [15] N. J. Hoff, The dynamics of the buckling of elastic columns, J. Appl. Mech., 18 (1951), 68-74.  Google Scholar [16] N. Kikuchi and J. T. Oden, "Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods," SIAM Studies in Applied Mathematics, 8, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1988.  Google Scholar [17] J. U. Kim, A one-dimensional dynamic contact problem in linear viscoelasticity, Math. Methods Appl. Sci., 13 (1990), 55-79. doi: 10.1002/mma.1670130106.  Google Scholar [18] K. L. Kuttler, A. Park, M. Shillor and W. Zhang, Unilateral dynamic contact of two beams, Math. Comput. Modelling, 34 (2001), 365-384. doi: 10.1016/S0895-7177(01)00068-1.  Google Scholar [19] K. L. Kuttler and M. Shillor, Vibrations of a beam between two stops, Dyn. Contin. Discrete Impuls. Syst. Ser. B Appl. Algorithms, 8 (2001), 93-110.  Google Scholar [20] J. A. C. Martins and J. T. Oden, A numerical analysis of a class of problems in elastodynamics with friction, Comput. Methods Appl. Mech. Engrg., 40 (1983), 327-360. doi: 10.1016/0045-7825(83)90105-6.  Google Scholar [21] J. E. Muñoz Rivera and D. Andrade, Existence and exponential decay for contact problems in thermoelasticity, Appl. Anal., 72 (1999), 253-273. doi: 10.1080/00036819908840741.  Google Scholar [22] J. E. Muñoz Rivera and M. de Lacerda Oliveira, Exponential stability for a contact problem in thermoelasticity, IMA J. Appl. Math., 58 (1997), 71-82. doi: 10.1093/imamat/58.1.71.  Google Scholar [23] J. E. Muñoz Rivera and S. Jiang, The thermoelastic and viscoelastic contact of two rods, J. Math. Anal. Appl., 217 (1998), 423-458. doi: 10.1006/jmaa.1997.5717.  Google Scholar [24] J. E. Muñoz Rivera and H. Portillo Oquendo, Existence and decay to contact problems for thermoviscoelastic plates, J. Math. Anal. Appl., 233 (1999), 56-76. doi: 10.1006/jmaa.1998.6236.  Google Scholar [25] J. E. Muñoz Rivera and H. Portillo Oquendo, Exponential decay for a contact problem with local damping, Funkcial. Ekvac., 42 (1999), 371-387.  Google Scholar [26] J. E. Muñoz Rivera and H. Portillo Oquendo, Exponential stability to a contact problem of partially viscoelastic materials, J. Elasticity, 63 (2001), 87-111. doi: 10.1023/A:1014091825772.  Google Scholar [27] J. E. Muñoz Rivera and J. B. Sobrinho, Existence and uniform rates of decay for contact problems in viscoelasticity, Appl. Anal., 67 (1997), 175-199. doi: 10.1080/00036819708840604.  Google Scholar [28] M. Nakao and J. E. Muñoz Rivera, The contact problem in thermoviscoelastic materials, J. Math. Anal. Appl., 264 (2001), 522-545. doi: 10.1006/jmaa.2001.7686.  Google Scholar [29] F. G. Pfeiffer, Applications of unilateral multibody dynamics. Non-smooth mechanics, R. Soc. Lond. Phil. Trans. Ser. A Math. Phys. Eng. Sci., 359 (2001), 2609-2628. doi: 10.1098/rsta.2001.0912.  Google Scholar [30] A. Rodríguez-Arós, J. M. Viaño and M. Sofonea, Numerical analysis of a frictional contact problem for viscoelastic materials with long-term memory, Numer. Math., 108 (2007), 327-358. doi: 10.1007/s00211-007-0117-7.  Google Scholar [31] M. Shillor, M. Sofonea and J. J. Telega, Quasistatic viscoelastic contact with friction and wear diffusion, Quart. Appl. Math., 62 (2004), 379-399.  Google Scholar [32] J. Simon, Compact sets in the space $L^p(0,T;B)$, Ann. Mat. Pura Appl. (4), 146 (1987), 65-96. doi: 10.1007/BF01762360.  Google Scholar [33] M. Sofonea, W. Han and M. Shillor, "Analysis and Approximation of Contact Problems with Adhesion or Damage," Pure and Applied Mathematics (Boca Raton), 276, Chapman & Hall/CRC, Boca Raton, FL, 2006.  Google Scholar [34] M. E. Stavroulaki and G. E. Stavroulakis, Unilateral contact applications using FEM software. Mathematical modelling and numerical analysis in solid mechanics, Int. J. Appl. Math. Comput. Sci., 12 (2002), 115-125.  Google Scholar [35] S. Woinowsky-Krieger, The effect of an axial force on the vibration of hinged bars, J. Appl. Mech., 17 (1950), 35-36.  Google Scholar
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