Some examples of friction-induced vibrations and instabilities

  • Franck Moirot
  • Quoc-Son Nguyen
Part of the International Centre for Mechanical Sciences book series (CISM, volume 457)


Some examples of friction-induced vibrations and instabilities of elastic structures in unilateral contact are discussed in this paper. The case of an elastic solid in contact with a moving (rigid or elastic) obstacle is considered. The onset of instability and the transition to a dynamic regime is the underlying centre of interest of the discussion.

Part 1 gives an overview of some classical and basic results concerning the stability analysis of an equilibrium. The linearization method and Liapunov’s theorem are first recalled. The possibility of Hopf’s bifurcation is considered when there is flutter instability. For time-independent standard dissipative systems, the description of the rate problem and the criteria of static stability and rate uniqueness in the sense of Hill are discussed.

Part 2 deals with the instability of the steady sliding of an elastic structure in contact with friction with a rigid or elastic obstacle. Closed-form solutions are discussed for some simple systems and for the problem of frictional contact of elastic layers.

The possibility of stick-slip vibrations is discussed in Part 3 in an analytical example of two encased cylinders. The existence of a family of stick-slip waves propagating at constant velocity and with positive slip is discussed.

Part 4 addresses the problem of brake squeal as a direct application of the theoretical analysis. This phenomenon is interpreted here as a consequence of the flutter instability of the steady sliding solution. A numerical analysis by the finite element method is performed to compute the steady sliding solution and to discuss its stability for an automotive disk brake.


Frictional Contact Disk Brake Generalize Eigenvalue Problem Unilateral Contact Dissipation Potential 
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  1. Adams, G. (1995). Self-excited oscillations of two elastic half-spaces sliding with a constant coefficient of friction. J. Appl. Mech. 62: 867–872.ADSCrossRefMATHGoogle Scholar
  2. Bazant, Z., and Cedolin, L. (1991). Stability of structures. Elastic, plastic, fracture and damage theories. Oxford: Oxford University Press.Google Scholar
  3. Biot, M. (1965). Mechanics of incremental deformation. New York: Wiley.Google Scholar
  4. Budiansky, B. (1974). Theory of buckling and post-buckling behaviour of elastic structures. In Advances in Applied Mechanics, volume 14. New York: Academic Press. 1–65.Google Scholar
  5. Chambrette, P., and Jezequel, L. (1992). Stability of a beam rubbed against a rotating disk. Eur. J. Mech., A/Solids 11: 107–138.MATHGoogle Scholar
  6. Chateau, X., and Nguyen, Q. (1991). Buckling of elastic structures in unilateral contact with or without friction. Eur. J. Mech., A/Solids 10: 71–89.MATHMathSciNetGoogle Scholar
  7. Cho, H., and Barber, J. (1999). Stability of the three-dimensional Coulomb friction law. Phil. Trans. R. Soc. London 455: 839–861.MATHMathSciNetGoogle Scholar
  8. Cochard, A., and Madariaga, R. (1995). Dynamic faulting under rate-independent friction. Pure & Appl. Geophys. 142: 419–445.ADSCrossRefGoogle Scholar
  9. Cocu, M., Pratt, E., and Raous, M. (1996). Analysis of an incremental formulation for frictional contact problems. In Contact mechanics, Marseille, 1995. New York: Plenum Press.Google Scholar
  10. Coddington, E., and Levinson, N. (1955). Theory of ordinary differential equations. New York: McGraw-Hill.MATHGoogle Scholar
  11. Cottle, R., Pang, J., and Stone, R. (1992). The linear complementarity problem. New York: Academic Press.MATHGoogle Scholar
  12. Durand, S. (1996). Dynamique des systèmes à liaisons unilatérales avec frottement sec. Thèse, Ecole Nationale des Ponts et Chaussées, Paris.Google Scholar
  13. Girardot, D. (1997). Stabilité et bifurcation dynamiques des systèmes discrets. Thèse, Ecole Polytechnique, Paris.Google Scholar
  14. Hale, J., and Kocak, H. (1991). Dynamics and bifurcation. New York: Springer-Verlag.CrossRefGoogle Scholar
  15. Hill, R. (1958). A general theory of uniqueness and stability in elastic/plastic solids. J. Mech. Phys. Solids 6: 236–249.ADSCrossRefMATHGoogle Scholar
  16. Hlavacek, I., Haslinger, J., Necas, J., and Lovisek, J. (1988). Solution of variational inequalities in Mechanics. Berlin: Springer-Verlag.CrossRefMATHGoogle Scholar
  17. Hutchinson, J. (1974). Plastic buckling. In Advances in Applied Mechanics, volume 14. New York: Academic Press. 67–114.Google Scholar
  18. Iooss, G., and Joseph, D. (1981). Elementary stability and bifurcation theory. New York: Springer-Verlag.Google Scholar
  19. Isac, G. (1992). Complementary problems. New York: Springer-Verlag, Lecture Note in Mathematics.Google Scholar
  20. Jean, M., and Moreau, J. (1987). Dynamics in the presence of unilateral contact and dry friction: a numerical approach. In Unilateral problems in structural analysis. Wien: CISM Course 304, Springer-Verlag. 1–50.Google Scholar
  21. Klarbring, A. (1986). A mathematical programming approach to three-dimensional contact problems with friction. Comp. Meth. Appl. Mech. Engng. 58: 175–200.ADSCrossRefMATHMathSciNetGoogle Scholar
  22. Klarbring, A. (1990). Derivation and analysis of rate boundary value problems of frictional contact. Eur. J. Mech. A/Solids 9: 53–85.MATHMathSciNetGoogle Scholar
  23. Klarbring, A. (1997). Contact, friction, discrete mechanical structures and mathematical programming. In Contact problems: theory, methods, applications. Wien: CISM Course, Springer-Verlag. 1–50.Google Scholar
  24. Koiter, W. (1945). Over de stabiliteit van het elastisch evenwicht. Thesis, University of Delft. English translation AFFDL TR 70–25 (1970).Google Scholar
  25. Marsden, J., and McCracken, M. (1976). The Hopf bifurcation and its applications. New York: Springer-Verlag.CrossRefMATHGoogle Scholar
  26. Martins, J., Guimaraes, J., and Faria, L. (1995). Dynamic surface solutions in linear elasticity and viscoelasticity with frictional boundary conditions. J. Vibration and Acoustics 117: 445–451.CrossRefGoogle Scholar
  27. Martins, J., Barbarin, S., Raous, M., and Pinto da Costa, A. (1999). Dynamic stability of finite dimensional linear elastic system with unilateral contact and Coulomb’s friction. Comp. Meth. Appl. Mech. Engng. 177: 298–328.MathSciNetGoogle Scholar
  28. Moirot, F., and Nguyen, Q. (2000a). Brake squeal: a problem of flutter instability of the steady sliding solution ? Arch. Mech. 52: 645–662.MATHGoogle Scholar
  29. Moirot, F., and Nguyen, Q. (2000b). An example of stick-slip waves. C. R. Acad. Sc. IIb, 328: 663–669.ADSMATHGoogle Scholar
  30. Moirot, F. (1998). Etude de la stabilité d’un équilibre en présence du frottement de Coulomb. Application au crissement des freins à disque. Thèse, Ecole Polytechnique, Paris.Google Scholar
  31. Nguyen, Q. (1994). Bifurcation and stability in dissipative media (plasticity, friction, fracture). Appl. Mech. Rev. 47: 1–31.ADSCrossRefGoogle Scholar
  32. Nguyen, Q. (2000). Stability and Nonlinear Solid Mechanics. Chichester: Wiley.Google Scholar
  33. Oancea, V., and Laursen, T. (1997). Stability analysis of state-dependent dynamic frictional sliding. Int. J. Nonlinear Mech. 32: 837–853.ADSCrossRefMATHGoogle Scholar
  34. Oden, J., and Martins, J. (1985). Models and computational methods for dynamic friction phenomena. Comp. Meth. Appl. Mech. Engng. 52: 527–634.ADSCrossRefMATHMathSciNetGoogle Scholar
  35. Oestreich, M., Hinrichs, N., and Popp, K. (1996). Bifurcation and stability analysis for a non-smooth friction oscillator. Arch. Appl. Mech. 66: 301–314.ADSCrossRefMATHGoogle Scholar
  36. Popp, K., and Stelter, P. (1990). Stick-slip vibrations and chaos. Phil. Trans. R. Soc. Lond A, 332: 89–105.ADSCrossRefMATHGoogle Scholar
  37. Renard, Y. (1998). Modélisation des instabilités liées au frottement sec des solides, aspects théoriques et numériques. Thèse de doctorat, Université de Grenoble.Google Scholar
  38. Troger, H., and Steindl, A. (1991). Nonlinear stability and bifurcation theory. Wien: Springer-Verlag.CrossRefMATHGoogle Scholar
  39. Vola, D., Raous, M., and Martins, J. (1999). Friction and instability of steady sliding squeal of a glass/rubber contact. Int. J. Num. Meth. Engng. 45: 301–314.Google Scholar
  40. Zharii, O. (1996). Frictional contact between the surface wave and a rigid strip. J. Appl. Mech. 63: 15–20.ADSCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Wien 2002

Authors and Affiliations

  • Franck Moirot
    • 1
  • Quoc-Son Nguyen
    • 2
  1. 1.Centre TechniquePSAParisFrance
  2. 2.Laboratoire de Mécanique des Solides, Ecole PolytechniqueCNRSParisFrance

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