
-
Previous Article
Existence for a quasistatic variational-hemivariational inequality
- EECT Home
- This Issue
-
Next Article
Topological optimization and minimal compliance in linear elasticity
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 |
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 [
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. Google Scholar |
[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. Google Scholar |
[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. Pozzolini, Y. 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. Google Scholar |
[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. Google Scholar |
[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. Pozzolini, Y. 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. |

[1] |
Manuel de León, Víctor M. Jiménez, Manuel Lainz. Contact Hamiltonian and Lagrangian systems with nonholonomic constraints. Journal of Geometric Mechanics, 2020 doi: 10.3934/jgm.2021001 |
[2] |
Onur Şimşek, O. Erhun Kundakcioglu. Cost of fairness in agent scheduling for contact centers. Journal of Industrial & Management Optimization, 2020 doi: 10.3934/jimo.2021001 |
[3] |
Michiel Bertsch, Danielle Hilhorst, Hirofumi Izuhara, Masayasu Mimura, Tohru Wakasa. A nonlinear parabolic-hyperbolic system for contact inhibition and a degenerate parabolic fisher kpp equation. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3117-3142. doi: 10.3934/dcds.2019226 |
[4] |
Franck Davhys Reval Langa, Morgan Pierre. A doubly splitting scheme for the Caginalp system with singular potentials and dynamic boundary conditions. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 653-676. doi: 10.3934/dcdss.2020353 |
[5] |
Fang Li, Bo You. On the dimension of global attractor for the Cahn-Hilliard-Brinkman system with dynamic boundary conditions. Discrete & Continuous Dynamical Systems - B, 2021 doi: 10.3934/dcdsb.2021024 |
[6] |
Patrick W. Dondl, Martin Jesenko. Threshold phenomenon for homogenized fronts in random elastic media. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 353-372. doi: 10.3934/dcdss.2020329 |
[7] |
Niklas Kolbe, Nikolaos Sfakianakis, Christian Stinner, Christina Surulescu, Jonas Lenz. Modeling multiple taxis: Tumor invasion with phenotypic heterogeneity, haptotaxis, and unilateral interspecies repellence. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 443-481. doi: 10.3934/dcdsb.2020284 |
[8] |
Peter Frolkovič, Viera Kleinová. A new numerical method for level set motion in normal direction used in optical flow estimation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 851-863. doi: 10.3934/dcdss.2020347 |
[9] |
Fioralba Cakoni, Pu-Zhao Kow, Jenn-Nan Wang. The interior transmission eigenvalue problem for elastic waves in media with obstacles. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020075 |
[10] |
Jianli Xiang, Guozheng Yan. The uniqueness of the inverse elastic wave scattering problem based on the mixed reciprocity relation. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2021004 |
[11] |
Xiaoxiao Li, Yingjing Shi, Rui Li, Shida Cao. Energy management method for an unpowered landing. Journal of Industrial & Management Optimization, 2020 doi: 10.3934/jimo.2020180 |
[12] |
Elvio Accinelli, Humberto Muñiz. A dynamic for production economies with multiple equilibria. Journal of Dynamics & Games, 2021 doi: 10.3934/jdg.2021002 |
[13] |
Nicolas Rougerie. On two properties of the Fisher information. Kinetic & Related Models, 2021, 14 (1) : 77-88. doi: 10.3934/krm.2020049 |
[14] |
Xinfu Chen, Huiqiang Jiang, Guoqing Liu. Boundary spike of the singular limit of an energy minimizing problem. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3253-3290. doi: 10.3934/dcds.2020124 |
[15] |
Marcello D'Abbicco, Giovanni Girardi, Giséle Ruiz Goldstein, Jerome A. Goldstein, Silvia Romanelli. Equipartition of energy for nonautonomous damped wave equations. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 597-613. doi: 10.3934/dcdss.2020364 |
[16] |
Jia Cai, Guanglong Xu, Zhensheng Hu. Sketch-based image retrieval via CAT loss with elastic net regularization. Mathematical Foundations of Computing, 2020, 3 (4) : 219-227. doi: 10.3934/mfc.2020013 |
[17] |
Chao Xing, Zhigang Pan, Quan Wang. Stabilities and dynamic transitions of the Fitzhugh-Nagumo system. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 775-794. doi: 10.3934/dcdsb.2020134 |
[18] |
P. K. Jha, R. Lipton. Finite element approximation of nonlocal dynamic fracture models. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1675-1710. doi: 10.3934/dcdsb.2020178 |
[19] |
Marcos C. Mota, Regilene D. S. Oliveira. Dynamic aspects of Sprott BC chaotic system. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1653-1673. doi: 10.3934/dcdsb.2020177 |
[20] |
Yongjie Wang, Nan Gao. Some properties for almost cellular algebras. Electronic Research Archive, 2021, 29 (1) : 1681-1689. doi: 10.3934/era.2020086 |
2019 Impact Factor: 0.953
Tools
Metrics
Other articles
by authors
[Back to Top]