Assessing the complete solution set of the planar frictional wedging problem (original) (raw)

Abstract

This paper examines the computation of all the (infinitely many) solutions of the frictional wedging problem in the non-coercive context by an algorithm applied to its complementarity formulation. Our analysis offers insights regarding the mechanical and the geometrical meaning of the solution set. We find that the solution structure depends much on the value of the coefficient of friction. The evidence indicates that non-coercivity implies much longer computation times. Coefficients of friction at the onset of wedging are computed and related.

Loading...

Loading Preview

Sorry, preview is currently unavailable. You can download the paper by clicking the button above.

References (40)

  1. P. Alart, A. Curnier, Contact discret avec frottement: unicité de la solution - convergence de lalgorithme, Laboratoire de Mécanique Appliquée, Département de Mécanique, École Polytechnique Fédérale de Lausanne, 1987.
  2. L.-E. Andersson, Quasistatic frictional contact problems; relations between the wedge problem and the rate problem, in: A. Ferreira, I.N. Figueiredo, J. Videman (Eds.), CIM, Proceedings of the International Conference on Mathematics and Continuum Mechanics, Porto, Portugal, vol. 30, 2008, pp. 61-66.
  3. J.R. Barber, P. Hild, On wedged configurations with Coulomb friction, in: P. Wriggers, U. Nackenhorst (Eds.), Analysis and Simulation of Contact Problems, Lecture Notes in Applied and Computational Mechanics, vol. 27, Springer, 2006, pp. 205-213.
  4. H. Béghin, Sur certains problèmes de frottement, Nouv. Ann. Math. 5e série 2 (1923) 305-312.
  5. H. Béghin, Sur l'indétermination de certains problèmes de frottement, Nouv. Ann. Math. 5e série 3 (1924) 343-347.
  6. S.L. Campbell, C.D. Meyer Jr., Generalized Inverses of Linear Transformations, Dover, 1979.
  7. R.W. Cottle, J.-S. Pang, R.E. Stone, The Linear Complementarity Problem, Academic Press, Inc., 1992.
  8. Ét. Delassus, Considérations sur le frottement de glissement, Nouv. Ann. Math. 4e série 20 (1920) 485-496.
  9. Ét. Delassus, Sur les lois du frottement de glissement, Bull. Soc. Math. France 51 (1923) 22-33.
  10. B. De Moor, Mathematical Concepts and Techniques for Modelling of Static and Dynamic Systems, PhD Thesis, Department of Electrical Engineering, Katholieke Universiteit Leuven, Belgium, 1988.
  11. B. De Moor, L. Vandenberghe, J. Vandewalle, The generalised linear complementarity problem and an algorithm to find all its solutions, Math. Program. Ser. A 57 (3) (1992) 415-426.
  12. U. Fingberg, A model of wheel-rail squealing noise, J. Sound Vib. 143 (3) (1990) 365-377.
  13. R. Fujita, Y. Kanno, Enumeration of all wedged equilibrium configurations in contact problem with Coulomb friction, Comput. Methods Appl. Mech. Engrg. (2010), doi:10.1016/j.cma.2009.12.008.
  14. F. Génot, B. Brogliato, New results on Painlevé paradoxes, Internal Report, INRIA -RR n 3366, 1998.
  15. Ch. Glocker, E. Cataldi-Spinola, R.I. Leine, Curve squealing of trains: measurement, modelling and simulation, J. Sound Vib. 324 (2009) 365-386.
  16. R. Hassani, P. Hild, I.R. Ionescu, N.-D. Sakki, A mixed finite element method and solution multiplicity for Coulomb frictional contact, Comput. Methods Appl. Mech. Engrg. 192 (2003) 4517-4531.
  17. R. Hassani, P. Hild, I.R. Ionescu, Sufficient conditions of non-uniqueness for the Coulomb friction problem, Math. Methods Appl. Sci. 27 (2004) 47-67.
  18. R. Hassani, I.R. Ionescu, E. Oudet, Critical friction for wedged configurations, Int. J. Solids Struct. 44 (2007) 6187-6200.
  19. P. Hild, On finite element uniqueness studies for Coulomb's frictional contact model, Int. J. Appl. Math. Comput. Sci. 12 (1) (2002) 41-50.
  20. P. Hild, Non-unique slipping in the Coulomb friction model in two- dimensional linear elasticity, Q. J. Mech. Appl. Math. 57 (2) (2004) 225-235.
  21. P. Hild, Solution multiplicity and stick configurations in continuous and finite element friction problems, Comput. Methods Appl. Mech. Engrg. 196 (2006) 57-65.
  22. R.P. Jarvis, B. Mills, Vibrations induced by dry friction, Proc. Inst. Mech. Engrs. 178-Pt1 32 (1963-1964) 847-866.
  23. J.H. Jellet, A Treatise on the Theory of Friction, Hodges, Foster and Co., 1872.
  24. A. Klarbring, Contact Problems in Linear Elasticity. Friction Laws and Mathematical Programming Applications, PhD Thesis 133, Linkping Studies in Science and Technology, 1985.
  25. A. Klarbring, Examples of non-uniqueness and non-existence of solutions to quasistatic contact problems with friction, Ing. Arch. 56 (1990) 529-541.
  26. P. Lancaster, M. Tismenetsky, The Theory of Matrices, Academic Press, 1985.
  27. L. Lecornu, Sur la loi de Coulomb, C.R. Acad. Sci. Paris 140 (1905) 847-848.
  28. H. Lütkepohl, Handbook of Matrices, John Wiley & Sons, 1996.
  29. J.A.C. Martins, S. Barbarin, M. Raous, A. Pinto da Costa, Dynamic stability of finite dimensional linearly elastic systems with unilateral contact and Coulomb friction, Comput. Methods Appl. Mech. Engrg. 177 (1999) 289-328.
  30. J.J. Moreau, Unilateral contact and dry friction in finite freedom dynamics, in: J.J. Moreau, P.D. Panagiotopoulos (Eds.), Nonsmooth Mechanics and Applications, CISM Courses and Lectures, vol. 302, Springer-Verlag, Wien, New York, 1988, pp. 1-81.
  31. P. Painlevé, Sur les lois du frottement de glissement, C.R. Acad. Sci. Paris 141 (1905) 401-405.
  32. P. Painlevé, Sur les lois du frottement de glissement, C.R. Acad. Sci. Paris 141 (1905) 546-552.
  33. J.-S. Pang, J.C. Trinkle, Complementarity formulations and existence of solutions of dynamic multi-rigid-body contact problems with Coulomb friction, Math. Program. 73 (1996) 199-226.
  34. A. Pinto da Costa, J.A.C. Martins, A numerical study on multiple rate solutions and onset of directional instability in quasi-static frictional contact problems, Comput. Struct. 82 (2004) 1485-1494.
  35. W.H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flannery, Numerical Recipes in Fortran. The Art of Scientific Computing, Cambridge University Press, 1992.
  36. B. de Schutter, Max-Algebraic System Theory for Discrete Event Systems, PhD Thesis, Department of Electrical Engineering, Katholieke Universiteit Leuven, 1996.
  37. M. de Sparre, Sur le frottement de glissement, C.R. Acad. Sci. Paris 141 (1905) 310-312.
  38. D.E. Stewart, J.C. Trinkle, An implicit time-stepping scheme for rigid body dynamics with inelastic collisions and Coulomb friction, Int. J. Numer. Methods Engrg. 39 (1996) 2673-2691.
  39. J.C. Trinkle, J.-S. Pang, S. Sudarsky, G. Lo, On dynamic multi-rigid-body contact problems with Coulomb friction, ZAMM 77 (4) (1997) 267-279.
  40. L. Vandenberghe, B. De Moor, J. Vandewalle, The generalised linear complementarity problem applied to the complete analysis of resistive piecewise-linear circuits, IEEE Trans. Circ. Syst. 36 (11) (1989) 1382-1391.