December  2020, 9(4): 1133-1151. doi: 10.3934/eect.2020057

Vibrations of a beam between stops: Collision events and energy balance properties

University of Lyon, F-42023 Saint-Etienne, Institut Camille Jordan, UMR CNRS 5208, 23 rue Paul Michelon, 42023 Saint-Etienne Cedex 2, France

* Corresponding author: Laetitia Paoli

Received  October 2019 Revised  February 2020 Published  December 2020 Early access  May 2020

We consider the model problem of an elastic beam vibrating between two stops. More precisely the beam is clamped at its left end while its right end may undergo contact and collision events with two stops. We model the interaction between the beam and the stops either with Signorini complementarity conditions when the stops are perfectly rigid or with a normal compliance contact law allowing some penetration within the stops and given by a linear relationship between the shear stress and the penetration at some positive power $ \beta $ when contact occurs.

Motivated by computational issues we study the evolution of the energy functional defined as the sum of the kinetic energy and the potential energy of elastic deformation of the beam. When contact is modelled with a normal compliance law we prove an energy conservation property. Then we interpret the relationship between the shear stress and the penetration in case of contact as a penalization of the non-penetration condition. We show that the solutions of the penalized problems converge to a strong solution of the problem with Signorini conditions as defined in [26] and we prove that the limit satisfies an energy conservation property through instantaneaous collision events.

Citation: Laetitia Paoli. Vibrations of a beam between stops: Collision events and energy balance properties. Evolution Equations and Control Theory, 2020, 9 (4) : 1133-1151. doi: 10.3934/eect.2020057
References:
[1]

J. Ahn and D. E. Stewart, An Euler-Bernoulli beam with dynamic contact: Discretization, convergence, and numerical results, SIAM J. Numer. Anal., 43 (2005), 1455-1480.  doi: 10.1137/S0036142903432619.

[2]

L. Amerio and G. Prouse, Study of the motion of a string vibrating against an obstacle, Rend. Mat., 8 (1975), 563-585. 

[3]

L. Amerio, Su un problema di vincoli unilaterali per l'equazione non omogenea della corda vibrante, Pubbl. I. A. C., Ser Ⅲ, 109 (1976), 1-11. 

[4]

L. Amerio, On the motion of a string vibrating through a moving ring with a continuously variable diameter, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat., 62 (1977), 134-142. 

[5]

L. Amerio, A unilateral problem for a nonlinear vibrating string equation, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat., 64 (1978), 8-21. 

[6]

A. Bamberger and M. Schatzman, New results on the vibrating string with a continuous obstacle, SIAM J. Math. Anal., 14 (1983), 560-595.  doi: 10.1137/0514046.

[7]

V. Barbu and T. Precupanu, Convexity and optimization in Banach spaces, in Mathematics and its Applications (East European Series), 10, D. Reidel Publishing Co., Dordrecht; Editura Academiei Republicii Socialiste România, Bucharest, 1986.

[8]

I. Bock and J. Jarusěk, Solvability of dynamic contact problems for elastic von Kármán plates, SIAM J. Math. Anal., 41 (2009), 37-45.  doi: 10.1137/080712179.

[9]

H. Brézis, Opérateurs Maximaux Monotones et Semi-groupes de Contractions dans les Espaces de Hilbert, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1973.

[10]

C. Citrini, Sull'urto parzialmente elastico o anelastico di una corda vibrante contro un obstacolo, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat., 59 (1975), 368-376. 

[11]

C. Citrini, Sull'urto parzialmente elastico o anelastico di una corda vibrante contro un obstacolo. Ⅱ, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat., 59 (1975), 667-676. 

[12]

C. Citrini, The energy theorem in the impact of a string vibrating against a pointshaped obstacle, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat., 62 (1977), 143-149. 

[13]

C. Citrini, Discontinuous solutions of a nonlinear hyperbolic equation with unilateral constraints, Manuscripta Math., 29 (1979), 323-352.  doi: 10.1007/BF01303634.

[14]

C. Citrini, The motion of a vibrating string in the presence of a point-shaped obstacle, Rend. Sem. Mat. Fis. Milano, 52 (1982), 353-362.  doi: 10.1007/BF02925018.

[15]

C. Citrini and C. Marchionna, On the problem of the point shaped obstacle for the vibrating string equation $cmy = f(x, \, t, \, y, \, y_{x}, \, y_{t})$, Rend. Accad. Naz Sci. XL Mem. Mat., 5 (1981/82), 53-72. 

[16]

Y. Dumont and L. Paoli, Vibrations of a beam between obstacles. Convergence of a fully discretized approximation, M2AN Math. Model. Numer. Anal., 40 (2006), 705-734.  doi: 10.1051/m2an:2006031.

[17]

Y. Dumont and L. Paoli, Numerical simulation of a model of vibrations with joint clearance, International Journal of Computer Applications in Technology, 33 (2008), 41-53.  doi: 10.1504/IJCAT.2008.021884.

[18]

N. Dunford and J. Schwartz, Linear Operators, Interscience, New-York, 1958.

[19]

C. Eck, J. Jarušek and M. Krbec, Unilateral contact problems in mechanics. Variational methods and existence theorems, in Pure and Applied Mathematics (Boca Raton), 270, Chapman & Hall/CRC, Boca Raton, FL, 2005. doi: 10.1201/9781420027365.

[20]

H. Hertz, Ueber die Berührung fester elastischer Körper, J. Reine Angew. Math., 92 (1882), 156-171.  doi: 10.1515/crll.1882.92.156.

[21]

N. Kikuchi and J. T. Oden, Contact problems in elasticity: A study of variational inequalities and finite element methods, in SIAM Studies in Applied Mathematics, 8, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1988. doi: 10.1137/1.9781611970845.

[22]

J. U. Kim, A boundary thin obstacle problem for a wave equation, Comm. Partial Differential Equations, 14 (1989), 1011-1026.  doi: 10.1080/03605308908820640.

[23]

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. 

[24]

G. Lebeau and M. Schatzman, A wave problem in a half-space with a unilateral constraint at the boundary, J. Differential Equations, 53 (1984), 309-361.  doi: 10.1016/0022-0396(84)90030-5.

[25]

J.-L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications. Vol. 2, in Travaux et Recherches Mathématiques, 18, Dunod, Paris, 1968.

[26]

L. Paoli and M. Shillor, Vibrations of a beam between two rigid stops: Strong solutions in the framework of vector-valued measures, Appl. Anal., 97 (2018), 1299-1314.  doi: 10.1080/00036811.2017.1344226.

[27]

C. Pozzolini and M. Salaun, Some energy-conservative schemes for vibro-impacts of a beam on rigid obstacles, ESAIM Math. Model. Numer. Anal., 45 (2011), 1163-1192.  doi: 10.1051/m2an/2011008.

[28]

C. PozzoliniY. Renard and M. Salaun, Vibro-impact of a plate on rigid obstacles: Existence theorem, convergence of a scheme and numerical simulations, IMA J. Numer. Anal., 33 (2013), 261-294.  doi: 10.1093/imanum/drr057.

[29]

R. T. Rockafellar, Integrals which are convex functionals, Pacific J. Math., 24 (1968), 525-539.  doi: 10.2140/pjm.1968.24.525.

[30] R. T. Rockafellar, Convex analysis, Princeton Mathematical Series, 28, Princeton University Press, Princeton, NJ, 1970. 
[31]

R. T. Rockafellar, Integrals which are convex functionals. Ⅱ, Pacific J. Math., 39 (1971), 439-469.  doi: 10.2140/pjm.1971.39.439.

[32]

M. Schatzman, Un problème hyperbolique du 2ème ordre avec contrainte unilatérale: La corde vibrante avec obstacle ponctuel, J. Differential Equations, 36 (1980), 295-334.  doi: 10.1016/0022-0396(80)90068-6.

[33]

M. Schatzman, A hyperbolic problem of second order with unilateral constraints: The vibrating string with a concave obstacle, J. Math. Anal. Appl., 73 (1980), 138-191.  doi: 10.1016/0022-247X(80)90026-8.

[34]

M. Schatzman, The penalty method for the vibrating string with an obstacle, in Analytical and Numerical Approaches to Asymptotic Problems in Analysis (Proc. Conf., Univ. Nijmegen, Nijmegen, 1980), North-Holland Math. Stud., 47, North-Holland, Amsterdam-New York, 1981,345–357.

[35]

M. Shillor, M. Sofonea and J. J. Telega, Models and analysis of quasistatic contact, in Lecture Notes in Physics, 655, Springer, Berlin, Heidelberg, 2004. doi: 10.1007/b99799.

[36]

A. Signorini, Sopra alcune questioni di statica dei sistemi continui, Ann. Scuola Norm. Super. Pisa Cl. Sci., 2 (1933), 231-251. 

[37]

J. Simon, Compact sets in the space $L^p(0, T;B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96.  doi: 10.1007/BF01762360.

[38]

R. Temam, Navier-Stokes equations, in Studies in Mathematics and its Applications, 2, North-Holland Publishing Co., Amsterdam-New York, Oxford, 1979.

show all references

References:
[1]

J. Ahn and D. E. Stewart, An Euler-Bernoulli beam with dynamic contact: Discretization, convergence, and numerical results, SIAM J. Numer. Anal., 43 (2005), 1455-1480.  doi: 10.1137/S0036142903432619.

[2]

L. Amerio and G. Prouse, Study of the motion of a string vibrating against an obstacle, Rend. Mat., 8 (1975), 563-585. 

[3]

L. Amerio, Su un problema di vincoli unilaterali per l'equazione non omogenea della corda vibrante, Pubbl. I. A. C., Ser Ⅲ, 109 (1976), 1-11. 

[4]

L. Amerio, On the motion of a string vibrating through a moving ring with a continuously variable diameter, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat., 62 (1977), 134-142. 

[5]

L. Amerio, A unilateral problem for a nonlinear vibrating string equation, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat., 64 (1978), 8-21. 

[6]

A. Bamberger and M. Schatzman, New results on the vibrating string with a continuous obstacle, SIAM J. Math. Anal., 14 (1983), 560-595.  doi: 10.1137/0514046.

[7]

V. Barbu and T. Precupanu, Convexity and optimization in Banach spaces, in Mathematics and its Applications (East European Series), 10, D. Reidel Publishing Co., Dordrecht; Editura Academiei Republicii Socialiste România, Bucharest, 1986.

[8]

I. Bock and J. Jarusěk, Solvability of dynamic contact problems for elastic von Kármán plates, SIAM J. Math. Anal., 41 (2009), 37-45.  doi: 10.1137/080712179.

[9]

H. Brézis, Opérateurs Maximaux Monotones et Semi-groupes de Contractions dans les Espaces de Hilbert, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1973.

[10]

C. Citrini, Sull'urto parzialmente elastico o anelastico di una corda vibrante contro un obstacolo, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat., 59 (1975), 368-376. 

[11]

C. Citrini, Sull'urto parzialmente elastico o anelastico di una corda vibrante contro un obstacolo. Ⅱ, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat., 59 (1975), 667-676. 

[12]

C. Citrini, The energy theorem in the impact of a string vibrating against a pointshaped obstacle, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat., 62 (1977), 143-149. 

[13]

C. Citrini, Discontinuous solutions of a nonlinear hyperbolic equation with unilateral constraints, Manuscripta Math., 29 (1979), 323-352.  doi: 10.1007/BF01303634.

[14]

C. Citrini, The motion of a vibrating string in the presence of a point-shaped obstacle, Rend. Sem. Mat. Fis. Milano, 52 (1982), 353-362.  doi: 10.1007/BF02925018.

[15]

C. Citrini and C. Marchionna, On the problem of the point shaped obstacle for the vibrating string equation $cmy = f(x, \, t, \, y, \, y_{x}, \, y_{t})$, Rend. Accad. Naz Sci. XL Mem. Mat., 5 (1981/82), 53-72. 

[16]

Y. Dumont and L. Paoli, Vibrations of a beam between obstacles. Convergence of a fully discretized approximation, M2AN Math. Model. Numer. Anal., 40 (2006), 705-734.  doi: 10.1051/m2an:2006031.

[17]

Y. Dumont and L. Paoli, Numerical simulation of a model of vibrations with joint clearance, International Journal of Computer Applications in Technology, 33 (2008), 41-53.  doi: 10.1504/IJCAT.2008.021884.

[18]

N. Dunford and J. Schwartz, Linear Operators, Interscience, New-York, 1958.

[19]

C. Eck, J. Jarušek and M. Krbec, Unilateral contact problems in mechanics. Variational methods and existence theorems, in Pure and Applied Mathematics (Boca Raton), 270, Chapman & Hall/CRC, Boca Raton, FL, 2005. doi: 10.1201/9781420027365.

[20]

H. Hertz, Ueber die Berührung fester elastischer Körper, J. Reine Angew. Math., 92 (1882), 156-171.  doi: 10.1515/crll.1882.92.156.

[21]

N. Kikuchi and J. T. Oden, Contact problems in elasticity: A study of variational inequalities and finite element methods, in SIAM Studies in Applied Mathematics, 8, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1988. doi: 10.1137/1.9781611970845.

[22]

J. U. Kim, A boundary thin obstacle problem for a wave equation, Comm. Partial Differential Equations, 14 (1989), 1011-1026.  doi: 10.1080/03605308908820640.

[23]

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. 

[24]

G. Lebeau and M. Schatzman, A wave problem in a half-space with a unilateral constraint at the boundary, J. Differential Equations, 53 (1984), 309-361.  doi: 10.1016/0022-0396(84)90030-5.

[25]

J.-L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications. Vol. 2, in Travaux et Recherches Mathématiques, 18, Dunod, Paris, 1968.

[26]

L. Paoli and M. Shillor, Vibrations of a beam between two rigid stops: Strong solutions in the framework of vector-valued measures, Appl. Anal., 97 (2018), 1299-1314.  doi: 10.1080/00036811.2017.1344226.

[27]

C. Pozzolini and M. Salaun, Some energy-conservative schemes for vibro-impacts of a beam on rigid obstacles, ESAIM Math. Model. Numer. Anal., 45 (2011), 1163-1192.  doi: 10.1051/m2an/2011008.

[28]

C. PozzoliniY. Renard and M. Salaun, Vibro-impact of a plate on rigid obstacles: Existence theorem, convergence of a scheme and numerical simulations, IMA J. Numer. Anal., 33 (2013), 261-294.  doi: 10.1093/imanum/drr057.

[29]

R. T. Rockafellar, Integrals which are convex functionals, Pacific J. Math., 24 (1968), 525-539.  doi: 10.2140/pjm.1968.24.525.

[30] R. T. Rockafellar, Convex analysis, Princeton Mathematical Series, 28, Princeton University Press, Princeton, NJ, 1970. 
[31]

R. T. Rockafellar, Integrals which are convex functionals. Ⅱ, Pacific J. Math., 39 (1971), 439-469.  doi: 10.2140/pjm.1971.39.439.

[32]

M. Schatzman, Un problème hyperbolique du 2ème ordre avec contrainte unilatérale: La corde vibrante avec obstacle ponctuel, J. Differential Equations, 36 (1980), 295-334.  doi: 10.1016/0022-0396(80)90068-6.

[33]

M. Schatzman, A hyperbolic problem of second order with unilateral constraints: The vibrating string with a concave obstacle, J. Math. Anal. Appl., 73 (1980), 138-191.  doi: 10.1016/0022-247X(80)90026-8.

[34]

M. Schatzman, The penalty method for the vibrating string with an obstacle, in Analytical and Numerical Approaches to Asymptotic Problems in Analysis (Proc. Conf., Univ. Nijmegen, Nijmegen, 1980), North-Holland Math. Stud., 47, North-Holland, Amsterdam-New York, 1981,345–357.

[35]

M. Shillor, M. Sofonea and J. J. Telega, Models and analysis of quasistatic contact, in Lecture Notes in Physics, 655, Springer, Berlin, Heidelberg, 2004. doi: 10.1007/b99799.

[36]

A. Signorini, Sopra alcune questioni di statica dei sistemi continui, Ann. Scuola Norm. Super. Pisa Cl. Sci., 2 (1933), 231-251. 

[37]

J. Simon, Compact sets in the space $L^p(0, T;B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96.  doi: 10.1007/BF01762360.

[38]

R. Temam, Navier-Stokes equations, in Studies in Mathematics and its Applications, 2, North-Holland Publishing Co., Amsterdam-New York, Oxford, 1979.

Figure 1.  The mechanical setting
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