# Numerical characterization and computation of dynamic instabilities for frictional contact problems

• Michel Raous
• Serge Barbarin
• Didier Vola
Chapter
Part of the International Centre for Mechanical Sciences book series (CISM, volume 457)

## Abstract

This chapter focuses on the numerical aspects of the characterization of friction-induced instabilities and their dynamic computation for linear and nonlinear problems. We begin by presenting briefly basic formulations and several computational methods for solving unilateral frictional contact problems, in quasi-statics and dynamics, and in elasticity and hyper-elasticity. The above specific dynamic formulations will be used to compute the flutter solutions presented in the last sections. Numerical schemes are then given for computing the various sufficient or necessary conditions for instability established together with Professor J.A.C. Martins Finally, the stability analysis and the computation of flutter solutions are carried out for two examples: the sliding of a Polyurethane block on a plane and the squeal of a rubber waist seal sliding on a car window.

In Section 1, a variational inequality formulation and numerical methods for solving quasistatic problems in elasticity are briefly recalled. Details can be found in a previous CISM course volume (see Raous (1999)).

This approach is extended to dynamic problems in Section 2. The formulation is written in terms of differential measures in order to deal with the non-smooth character of the solutions. It is an extension of those developed by J.J. Moreau and M. Jean.

In Section 3, the above formulations are extended to hyper-elastic problems and a method for computing directly the steady sliding solution is given.

Numerical analysis of the stability of quasistatic solutions in the context of linear elasticity is carried out in Section 4. The example of the sliding of a Polyurethane block is studied.

In Section 5, the stability analysis is carried out for a steady sliding solution in the context of hyper-elasticity and used to characterize the squeal of a waist seal sliding on a car window.

## Keywords

Friction Coefficient Complementarity Problem Dynamic Instability Generalize Eigenvalue Problem Unilateral Contact
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

1. S. Barbarin (1997) Instabilité et frottement en élasticité linéaire. Application à un problème d’ondes de contrainte, Thesis, Université de Provence, Marseille.Google Scholar
2. G. Björkman, A. Klarbring, B. Sjödin, T. Larsson and M. Rönnqvist (1995). Sequential quadratic programming for non-linear elastic contact problems, Int. J. Numer. Meth. Engng., 38, 137–165.
3. P. Chabrand, F. Dubois and M. Raous (1998). Various numerical methods for solving unilateral contact problems with friction, Mathl. Comput. Modelling., 28, 97–108.
4. X. Chateau and Q.S. Nguyen (1991). Buckling of elastic structures in unilateral contact with or without friction, Eur. J. Mech., A/Solids, 10, 71–89.
5. R.W. Clough and J. Penzien (1975). Dynamics of structures, Int. Stud Eds, Mc Graw Hill, Kogakuska.
6. M. Cocu, E. Pratt and M. Raous (1996). Formulation and approximation of quasistatic frictional contact, Int. J. Engng. Sci., 34, 7, 783–798.
7. R. W. Cottle, J. S. Pang and R. Stone (1992). The linear complementarily problem, Computer Science and Scientific Computing, Academic press, New-York.Google Scholar
8. A. Curnier and P. Alart (1988). A generalized Newton method for contact problems with friction, J. Méca. Th. Appl., 7, 67–82.
9. B. Feeny, A. Guran, N. Hinrichs and K. Popp (1998). A historical review on dry friction and stick-slip phenomena, Appl. Mech. Rev., 51, 321–341.
10. M. S. Gadala (1986). Numerical solutions of nonlinear problems of continua–II. A survey of incompressibility constraints and software aspects, Comput. Struct., 22, 841–855.
11. Q. C. He, J. J. Telega and A. Curnier (1996). Unilateral contact of two solids subject to large deformations: formulation and existence results, Proc. R. Soc. Lond. A, 452, 2691–2717.
12. R.A. Ibrahim (1994). Friction-induced vibration, chatter, squeal and chaos: Part I–Mechanics of friction, Part II–Dynamics and modeling, Appl. Mech. Rev, 47, 209–253.
13. G. Isac (1992). Complementarity problems, Lecture notes in Mathematics, 1528, Springer Verlag.
14. M. Jean and J.J. Moreau (1987). Dynamics in the presence of unilateral contact and dry friction: a numerical approach, in Del Piero and Maceri (Eds.), Unilateral problems in structural analysis - 2, CISM lectures vol. 304, Springer Verlag, 151–196.Google Scholar
15. M. Jean (1999). The non smooth contact dynamics method, Comput. Meth. Appl. Mech. Engng., 177, 235–257.
16. N. Josephy (1979). Newton’s method for generalized equations, Report TSR 1965, Mathematics research center, University of Wisconsin.Google Scholar
17. A. Klarbring (1988). On discrete and discretized non-linear elastic structures in unilateral contact (stability, uniqueness and variational principles), Int. J. Solids Structures, 24, 459–479.
18. A. Klarbring (1990). Derivation and analysis of rate boundary-value problems of frictional contact, Eur. J. Mech., A/Solids, 9, 53–85.
19. A. Klarbring and G. Björkmann (1992). Solution of large displacement contact problems with friction using Newton’s method for generalized equations, Int. J. Numer. Meth. Engng., 34, 249–269.
20. A. Klarbring (1997). Steady sliding and linear complementarity, in M. Ferris and J. C. Pang (Eds.), Complementarity and variational problems: state of the art, SIAM publication, Philadelphia, 132–147.Google Scholar
21. A. Klarbring (1999). Contact, friction, discrete mechanical structures and mathematical programming, in P. Wriggers–P. Panagiotopoulos (Eds.), New developments in contact problems, CISM Courses and Lectures, 384, Springer Verlag, 55–100.Google Scholar
22. A.M. Lang and D.A. Crolla (1991). Brake noise and vibration, the state of art, Vehicle Tribology, Tribologies series, 18, 165–173.
23. C.E. Lemke (1980). A survey of complementarity theory, in Cottle-Gianessi-Lions (Eds), Variational Inequalities and Complementarily Problems, John Wiley, New York, 213–235.Google Scholar
24. C. Licht, E. Pratt and M. Raous (1991). Remarks on a numerical method for unilateral contact including friction, Int. Series Num. Math., 101, 129–144.
25. C. H. Liu, G. Hofstetter and H. A. Mang (1994). 3D finite element analysis of rubber-like materials at finite strains, Engng. comput., 11, 111–128.Google Scholar
26. J. A. C. Martins, J. T. Oden and F. M. F. Simbes (1990). A study of static and kinetic friction, Int. J. Engng. Sci., 28, 29–92.
27. J. A. C. Martins, S. Barbarin, M. Raous and A. Pinto da Costa (1999). Dynamic stability of finite dimensional linearly elastic systems with unilateral contact and Coulomb friction, Comput. Meth. Appl. Mech. Engng., 177, n 3–4, 289–328.
28. J. A. C. Martins and A. Pinto da Costa (2000). Stability of finite dimensional systems with unilateral contact and friction: non-linear elastic behaviour and obstacle curvature, Int. Journal of Solids and Structures, 37, 2519–2564.
29. F. Moirot (1998). Etude de la stabilité d’un équilibre en présence de frottement de Coulomb, application au crissement des freins à disques, Ph.D. Thesis, Ecole Polytechnique, Paris, France.Google Scholar
30. J.-J. Moreau (1988a). Unilateral contact and dry friction in finite freedom dynamics, in Non Smooth Mechanics and Applications, CISM Courses and Lectures, 302, J.-J. Moreau, P.D. Panagiotopoulos (Eds), Springer-Verlag, Wien, 1–82.
31. J.-J. Moreau (1988b). Bounded variation in time, In Moreau-Paniagiatopoulos-Strang (Eds), Topics in non-smooth mechanics, Birkhauser Verlag, 1–74.Google Scholar
32. J.-J. Moreau (1994). Some numerical methods in multibody dynamics: application to granular materials, Eur. J. Mech. A/Solids, 13 (4), 93–114.
33. Q.S. NGuyen (1994). Bifurcation and stability in dissipative media (plasticity, friction, fracture), Appl. Mech. Rev., 47, 1–31.
34. J.T. Oden and N. Kikuchi (1982). Finite element methods for constrained problems in elasticity, Int. J. Num. Meth. Eng., 19, 701–725.
35. J. T. Oden and J. A. C. Martins (1985). Models and computational methods for dynamic friction phenomena, Comput. Meth. Appl. Mech. Engng., 52, 527–634.
36. R.W. Ogden (1976). Volume change associated with the deformation of rubber-like solids, J. Mech. Phys. Solids, 24, 323–338.
37. S. Pandit and S. Deo (1982). Differential systems involving impulses, Lectures notes in mathematics, Springer Verlag.
38. A.M.F. Pinto da Costa (2001). Instabilidade e bifurcaçaöes em sistemas de comportamento nâo-suave, Thesis, Istituto Supérior Tecnico, Lisbon, Portugal.Google Scholar
39. E. B. Pires and L. Trabucho (1990). The steady sliding problem with nonlocal friction, Int. J. Engng. Sci., 28, 631–641.
40. P. Rabier, J. A. C. Martins, J. T. Oden and L. Campos (1986). Existence and local uniqueness of solutions for contact problems with non-linear friction laws, Int. J. Engng. Sci., 24, 1755–1768.
41. C. Rajakumar and C. Rogers (1991). The Lanczos algorithm applied to unsymmetric generalized eigenvalue problem, Int. J. Numer. Meth. Engng., 32, 1009–1026.
42. A. Ralston and P. Rabinowitz (1978). A first course in numerical analysis, McGRAW-HILL, New-York.
43. M. Raous (Ed) (1988). Numerical methods in mechanics of contact involving friction,J. de Mécanique Théorique et Appliquée, Special Issue, Supp. 1 to vol. 7, Gauthier-Villars.Google Scholar
44. M. Raous, P. Chabrand and F. Lebon (1988). Numerical methods for solving unilateral contact problems with friction, in [Raous (Ed.), 1988], 111–128.Google Scholar
45. M. Raous and S. Barbarin (1992). Conjugate Gradient for Frictional Contact, In Curnier A. (Ed), Proceedings of Contact Mechanics International Symposium, Presses Polytech. et Univ. Romandes, Lausanne, 423–432.Google Scholar
46. M. Raous, J.-J. Moreau and M. Jean (Eds) (1995). Contact Mechanics, Plenum Publisher, New York.Google Scholar
47. M. Raous, S. Barbarin, D. Vola and J.A.C. Martins (1995). Friction induced instabilities and sound generation, Proceed. ASME Design Engineering Technical Conference, 18–21 september 1995, Boston, USA, 799–802.Google Scholar
48. M. Raous and S. Barbarin (1996). Stress waves in a sliding contact. Part 2: modelling, in D. Dowson et al. (Eds.), Proceedings of the 22 nd Leeds Lyon Symposium on Tribology, Elsevier Science, 39–44.Google Scholar
49. M. Raous (1999). Quasistatic Signorini problem with Coulomb friction and coupling to adhesion, in P. Wriggers, P. Panagiotopoulos (Eds.), New developments in contact problems, CISM Courses and Lectures, 384, Springer Verlag, 101–178.Google Scholar
50. M. Raous (2001), Constitutive models and numerical methods for frictional contact, In J.Lemaitre (Ed.), Handbook of material behavior–Non linear models and properties, Section 8.5, Academic Press, 777–786.Google Scholar
51. R. T. Rockafellar (1970). Convex analysis, Princeton University Press.Google Scholar
52. J. C. Simo and R. L. Taylor (1991). Quasi-incompressible finite elasticity in principal stretches. Continuum basis and numerical algorithms, Comput. Meth. Appl. Mech. Engng., 85, 272–310.
53. T. Sussman and K.J. Bathe (1987). A finite element formulation for nonlinear incompressible elastic and inelastic analysis, Comp. Stryct., 26, 1–2.
54. D. Vola (1998). Frottement et instabilités en dynamique: bruit de crissement, thesis, Université de la Méditerranée, Marseille, France.Google Scholar
55. D. Vola, E. Pratt, M. Jean and M. Raous (1998). Consistent time discretization for a dynamical frictional contact problem and complementarity techniques, Revue Européenne des Eléments Finis, 7, 149–162.
56. D. Vola, A. Pinto da Costa, S. Barbarin, J.A.C. Martins and M. Raous (1999). Bifurcations and instabilities in some finite dimensional frictional contact problems, in F. Pfeiffer, Ph. Glocker (Eds), Proceedings of 1998 IUTAM symposium: unilateral multibody dynamics, Kluwer, 179–190.Google Scholar
57. D. Vola, A. M. Raous and J.A.C. Martins (1999). Friction and instability of steady sliding: squeal of a rubber/glass contact, International Journal for Numerical Methods in Engineering, 46, 1699–1720.
58. P. Wriggers, T. Vu Van and E. Stein (1990). Finite element formulation for large deformation impact-contact problems with friction, Comp. Struct., 37 (3), 319–331.
59. T. Zeghloul and B. Villechaise (1996). Stress waves in a sliding contact. Part 1: experimental study, in D. Dowson et al. (Eds.), Proceedings of the 22 nd Leeds Lyon Symposium on Tribology, Elsevier Science, 33–37.Google Scholar