Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-11-30T21:18:29.030Z Has data issue: false hasContentIssue false
  • English
  • Français

A model of a spring-mass-damper system with temperature-dependent friction†

Published online by Cambridge University Press:  10 September 2013

S. MIGORSKI ,
A. OCHAL ,
M. SHILLOR  and
M. SOFONEA
S. MIGORSKI
Affiliation:
Faculty of Mathematics and Computer Science, Institute of Computer Science, Jagiellonian University, 30348 Krakow, Poland emails: [email protected]; [email protected]
A. OCHAL
Affiliation:
Faculty of Mathematics and Computer Science, Institute of Computer Science, Jagiellonian University, 30348 Krakow, Poland emails: [email protected]; [email protected]
M. SHILLOR
Affiliation:
Department of Mathématics and Statistics, Oakland University, Rochester, MI 48309, USA email: [email protected]
M. SOFONEA
Affiliation:
Laboratoire de Mathematicques et Physique, University of Perpignan Via Domitia, 66860 Perpignan, France email: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

This work models and analyses the dynamics of a general spring-mass-damper system that is in frictional contact with its support, taking into account frictional heat generation and a reactive obstacle. Friction, heat generation and contact are modelled with subdifferentials of, possibly non-convex, potential functions. The model consists of a non-linear system of first-order differential inclusions for the position, velocity and temperature of the mass. The existence of a global solution is established and additional assumptions yield its uniqueness. Nine examples of conditions arising in applications, for which the analysis results are valid, are presented.

Keywords

Spring-mass-damper systemContactTemperature-dependent frictionNormal complianceHeat generationDifferential inclusionClarke subdifferentialExistence and uniqueness
Type
Papers
Information
European Journal of Applied Mathematics , Volume 25 , Issue 1 , February 2014 , pp. 45 - 64
Copyright
Copyright © Cambridge University Press 2013 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

†

Research supported by the Marie Curie IRSES Fellowship within the 7th European Community Framework Programme under Grant Agreement No. 295118 and the National Science Center of Poland under Maestro Project no. DEC-2012/06/A/ST1/00262. The first two authors are also partially supported by the National Science Center of Poland under grant no. N N201 604640 and by the Ministry of Science and Higher Education of Republic of Poland under the International Cofinanced Project no. W111/7.PR/2012.

References

[1]Adly, S., Hantoute, A. & Thera, M. (2012) Nonsmooth Lyapunov pairs for infinite-dimensional first-order differential inclusions. Nonlinear Anal. 75, 985–1008.CrossRefGoogle Scholar
[2]Bajkowski, J., Fernandez, J. R., Kuttler, K. L. & Shillor, M. (2004) A thermoviscoelastic beam model for brakes. Euro. J. Appl. Math. 15, 181–202.CrossRefGoogle Scholar
[3]Cameron, D. S. (1999) Platform stabilization with a novel active suspension. Doctoral Dissertation, Oakland University, Rochester, MI.Google Scholar
[4]Chang, K. C. (1981) Variational methods for nondifferentiable functionals and applications to partial differential equations. J. Math. Anal. Appl. 80, 102–129.CrossRefGoogle Scholar
[5]Chipman, J. C., Roux, A., Shillor, M. & Sofonea, M. (2011) A damageable spring. Mach. Dyn. Res. 35, 82–96.Google Scholar
[6]Clarke, F. H. (1983) Optimization and Nonsmooth Analysis, Wiley, New York, NY.Google Scholar
[7]Denkowski, Z., Migórski, S. & Papageorgiou, N. S. (2003) An Introduction to Nonlinear Analysis: Theory, Kluwer, Boston, MA.CrossRefGoogle Scholar
[8]Drozdov, A. D. (1996) Finite Elasticity and Viscoelasticity. A Course in the Nonlinear Mechanics of Solids, World Scientific, Singapore.CrossRefGoogle Scholar
[9]Dumont, Y., Goeleven, D., Rochdi, M. & Shillor, M. (2000) Frictional contact of a nonlinear spring. Math. Computer Modelling 31, 83–97.CrossRefGoogle Scholar
[10]Eck, C., Jarusek, J. & Krbec, M. (2005) Unilateral Contact Problems, Pure and Applied Mathematics, vol. 270, Chapman/CRC Press, New York, NY.CrossRefGoogle Scholar
[11]Han, W. & Sofonea, M. (2002) Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity, Studies in Advanced Mathematics, vol. 30, American Mathematical Society, Providence, RI.CrossRefGoogle Scholar
[12]Klarbring, A., Mikelic, A. & Shillor, M. (1988) Frictional contact problems with normal compliance. Int. J. Eng. Sci. 26, 811–832.CrossRefGoogle Scholar
[13]Kuttler, K. L., Shillor, M. & Fernandez, J. R. (2004) Existence for a thermoviscoelastic beam model for brakes. Nonlinear Analysis 5, 857–880.Google Scholar
[14]Migórski, S., Ochal, A. & Sofonea, M. (2011) Analysis of lumped models with contact and friction. Z. Angew. Math. Phys. 62, 99–113.CrossRefGoogle Scholar
[15]Migórski, S., Ochal, A. & Sofonea, M. (2013) Nonlinear Inclusions and Hemivariational Inequalities. Models and Analysis of Contact Problems, Advances in Mechanics and Mathematics, vol. 26, Springer, New York, NY.CrossRefGoogle Scholar
[16]Oden, J. T. & Martins, J. A. C. (1985) Models and computational methods for dynamic friction phenomena, Comput. Methods Appl. Mech. Eng. 52, 527–634.CrossRefGoogle Scholar
[17]Panagiotopoulos, P. D. (1993) Hemivariational Inequalities, Applications in Mechanics and Engineering, Springer-Verlag, Berlin, Germany.CrossRefGoogle Scholar
[18]Papageorgiou, N. S. & Kyritsi-Yiallourou, S. Th. (2009) Handbook of Applied Analysis, Advances in Mechanics and Mathematics, vol. 19, Springer, New York, NY.Google Scholar
[19]Rabinowicz, E. (1995) Friction and Wear of Materials, 2nd ed., Wiley, New York, NY.Google Scholar
[20]Shillor, M., Sofonea, M. & Telega, J. J. (2004) Models and Analysis of Quasistatic Contact, Lecture Notes in Physics, 655, Springer, Berlin, Germany.CrossRefGoogle Scholar
[21]Sofonea, M. and Matei, A. (2009) Variational Inequalities with Applications. A~Study of Antiplane Frictional Contact Problems, Advances in Mechanics and Mathematics 18, Springer, New York, NY.Google Scholar
[22]Strömberg, N., Johansson, L. and Klarbring, A. (1996) Derivation and analysis of a generalized standard model for contact friction and wear. Int. J. Solids Struct. 13, 1817–1836.CrossRefGoogle Scholar