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

Vibrations of a damped extensible beam between two stops

Alessia Berti 1, and  Maria Grazia Naso 2,

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.
Keywords: extensible elastic beam, Dynamic contact, asymptotic behavior of solutions, normal compliance., Signorini condition.
Mathematics Subject Classification: Primary: 74H40, 74M15; Secondary: 35B4.
Citation: Alessia Berti, Maria Grazia Naso. Vibrations of a damped extensible beam between two stops. Evolution Equations and 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.

[2]

J. M. Ball, Initial-boundary value problems for an extensible beam, J. Math. Anal. Appl., 42 (1973), 61-90.

[3]

A. Berti and M. G. Naso, Unilateral dynamic contact of two viscoelastic beams, Quart. Appl. Math., 69 (2011), 477-507.

[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.

[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.

[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.

[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.

[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.

[9]

R. W. Dickey, Free vibrations and dynamic buckling of the extensible beam, J. Math. Anal. Appl., 29 (1970), 443-454.

[10]

G. Duvaut and J.-L. Lions, "Inequalities in Mechanics and Physics," Grundlehren der Mathematischen Wissenschaften, 219, Springer-Verlag, Berlin-New York, 1976.

[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.

[12]

M. Frémond, "Non-Smooth Thermomechanics," Springer-Verlag, Berlin, 2002.

[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.

[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.

[15]

N. J. Hoff, The dynamics of the buckling of elastic columns, J. Appl. Mech., 18 (1951), 68-74.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[25]

J. E. Muñoz Rivera and H. Portillo Oquendo, Exponential decay for a contact problem with local damping, Funkcial. Ekvac., 42 (1999), 371-387.

[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.

[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.

[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.

[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.

[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.

[31]

M. Shillor, M. Sofonea and J. J. Telega, Quasistatic viscoelastic contact with friction and wear diffusion, Quart. Appl. Math., 62 (2004), 379-399.

[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.

[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.

[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.

[35]

S. Woinowsky-Krieger, The effect of an axial force on the vibration of hinged bars, J. Appl. Mech., 17 (1950), 35-36.

show all references

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.

[2]

J. M. Ball, Initial-boundary value problems for an extensible beam, J. Math. Anal. Appl., 42 (1973), 61-90.

[3]

A. Berti and M. G. Naso, Unilateral dynamic contact of two viscoelastic beams, Quart. Appl. Math., 69 (2011), 477-507.

[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.

[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.

[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.

[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.

[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.

[9]

R. W. Dickey, Free vibrations and dynamic buckling of the extensible beam, J. Math. Anal. Appl., 29 (1970), 443-454.

[10]

G. Duvaut and J.-L. Lions, "Inequalities in Mechanics and Physics," Grundlehren der Mathematischen Wissenschaften, 219, Springer-Verlag, Berlin-New York, 1976.

[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.

[12]

M. Frémond, "Non-Smooth Thermomechanics," Springer-Verlag, Berlin, 2002.

[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.

[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.

[15]

N. J. Hoff, The dynamics of the buckling of elastic columns, J. Appl. Mech., 18 (1951), 68-74.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[25]

J. E. Muñoz Rivera and H. Portillo Oquendo, Exponential decay for a contact problem with local damping, Funkcial. Ekvac., 42 (1999), 371-387.

[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.

[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.

[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.

[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.

[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.

[31]

M. Shillor, M. Sofonea and J. J. Telega, Quasistatic viscoelastic contact with friction and wear diffusion, Quart. Appl. Math., 62 (2004), 379-399.

[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.

[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.

[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.

[35]

S. Woinowsky-Krieger, The effect of an axial force on the vibration of hinged bars, J. Appl. Mech., 17 (1950), 35-36.

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