Skip to main content
SpringerLink
Log in

Constraints in structural and rigid body mechanics: a frictional contact problem

  • Original Article
  • Published:
Annals of Solid and Structural Mechanics

Abstract

In this work, we propose some basic approaches towards a unification of the theories for deformable and rigid bodies. This unification process is based on two fundamental mechanical concepts, which are the principle of virtual work and the principle of d’Alembert–Lagrange. The basic idea is to initially look upon structural elements as general continua, and to endow them later with specific properties like rigidity, imposed by perfect bilateral constraints. It is shown by the example of a simple flexible multibody system how this unifying and systematic approach has to be carried out. The system under consideration consists of a rigid disk and a nonlinear elastic string, which may come into contact with each other. The contact is modeled as a hard unilateral geometric constraint combined with a one-dimensional Coulomb friction element. The contact interactions are formulated as set-valued force laws and impact laws, and the system is consequently treated within the framework of nonsmooth dynamics. The model of the string allows for large deformations in time and for a nonlinear elastic material response. By constraining the kinematics of the string to finite dimensions, a nonlinear finite element formulation is achieved in a very natural way.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

Similar content being viewed by others

References

  1. Acary V, Brogliato B (2008) Numerical methods for nonsmooth dynamical systems: applications in mechanics and electronics. Lecture notes in applied and computational mechanics, vol 35. Springer, Berlin

  2. Alart P, Curnier A (1991) A mixed formulation for frictional contact problems prone to Newton like solution methods. Comput Methods Appl Mech Eng 92:353–375

    Article  MathSciNet  MATH  Google Scholar 

  3. Antman SS (1981) Material constraints in continuum mechanics. Atti Della Accademia Nazionale dei Lincei, Rendiconti, Classe di Scienze Fisiche, Matematiche e Naturali 70(8):256–264

    MATH  Google Scholar 

  4. Antman SS (2005) Nonlinear problems of elasticity, applied mathematical sciences, vol 107, 2nd edn. Springer, New York

    Google Scholar 

  5. Antman SS, Marlow RS (1991) Material constraints, Lagrange multipliers, and compatibility. Applications to rod and shell theories. Arch Ration Mech Anal 116:257–299

    Article  MathSciNet  MATH  Google Scholar 

  6. Ballard P, Millard A (2009) Poutres et arcs élastiques. Les Éditions de l’École Polytechnique

  7. Bertails F, Audoly B, Cani MP, Querleux B, Leroy F, Lévêque JL (2006) Super-helices for predicting the dynamics of natural hair. In: ACM transactions on graphics (proceedings of the ACM SIGGRAPH ’06 conference), ACM, pp 1180–1187

  8. Bertails-Descoubes F, Cadoux F, Daviet G, Acary V (2011) A nonsmooth newton solver for capturing exact coulomb friction in fiber assemblies. ACM Trans Graph 30(1):6–1614

    Article  Google Scholar 

  9. Biot M (1974) Non-linear thermoelasticity, irreversible thermodynamics and elastic instability. Indiana Univ Math J 23:309–335

    Article  Google Scholar 

  10. Brogliato B, Ten Dam A, Paoli L, Génot F, Abadie M (2002) Numerical simulation of finite dimensional multibody nonsmooth mechanical systems. Appl Mech Rev 55:107

    Article  Google Scholar 

  11. Chamekh M, Mani-Aouadi S, Moakher M (2009) Modeling and numerical treatment of elastic rods with frictionless self-contact. Comput Methods Appl Mech Eng 198(47–48):3751–3764

    Article  MathSciNet  MATH  Google Scholar 

  12. Cho YH (2008) Numerical simulation of the dynamic responses of railway overhead contact lines to a moving pantograph, considering a nonlinear dropper. J Sound Vib 315(3):433–454

    Article  Google Scholar 

  13. Collina A, Bruni S (2002) Numerical simulation of pantograph-overhead equipment interaction. Veh Syst Dyn 38(4):261–291

    Article  Google Scholar 

  14. Dufva K, Kerkkänen K, Maqueda L, Shabana A (2007) Nonlinear dynamics of three-dimensional belt drives using the finite-element method. Nonlinear Dyn 48:449–466

    Article  MATH  Google Scholar 

  15. Eugster SR (2009) Dynamics of an elevator: 2-dimensional modeling and simulation. Master’s thesis, ETH Zurich

  16. Frémond M (1989) Internal constraints and constitutive laws. In: Rodrigues JF (ed) Mathematical models for phase change problems. pp 3–18

  17. Frémond M (2001) Internal constraints in mechanics. Philos Trans R Soc Lond A: Math Phys Eng Sci 359(1789):2309–2326

    Article  MATH  Google Scholar 

  18. Frémond M, Gormaz R, San Martín JA (2003) Collision of a solid with an incompressible fluid. Theoret Comput Fluid Dyn 16:405–420

    Article  MATH  Google Scholar 

  19. Glocker Ch (2001) Set-valued force laws, dynamics of non-smooth systems. Lecture notes in applied mechanics, vol 1. Springer, Berlin

  20. Glocker Ch (2004) Concepts for modeling impacts without friction. Acta Mech 168:1–19

    Article  MATH  Google Scholar 

  21. Glocker Ch (2006a) An introduction to impacts. In: Haslinger J, Stavroulakis G (eds) Nonsmooth mechanics of solids. CISM courses and lectures, vol 485. Springer, Wien, pp 45–101

    Chapter  Google Scholar 

  22. Glocker Ch (2006b) Simulation von harten Kontakten mit Reibung: Eine iterative Projektionsmethode. In: VDI-Berichte No. 1968: Schwingungen in Antrieben 2006 Tagung, Fulda, VDI Verlag, Düsseldorf, vol 1968

  23. Goyal S, Perkins N, Lee CL (2008) Non-linear dynamic intertwining of rods with self-contact. Int J Non-Linear Mech 43(1):65–73

    Article  MATH  Google Scholar 

  24. Jean M (1999) The non-smooth contact dynamics method. Comput Methods Appl Mech Eng 177(3-4):235–257

    Article  MathSciNet  MATH  Google Scholar 

  25. Kerkkänen K, García-Vallejo D, Mikkola A (2006) Modeling of belt-drives using a large deformation finite element formulation. Nonlinear Dyn 43:239–256

    Article  MATH  Google Scholar 

  26. Leine RI, Nijmeijer H (2004) Dynamics and bifurcations of non-smooth mechanical systems. Lecture notes in applied and computational mechanics, vol 18. Springer, Berlin

  27. Möller M (2012) Consistent integrators for non-smooth dynamical systems. PhD thesis, ETH Zurich

  28. Moreau JJ (1985) Standard inelastic shocks and the dynamics of unilateral constraints. In: Del Piero G, Maceri F (eds) Unilateral problems in structural analysis. Proceedings of the second meeting on unilateral problems in structural analysis ravello, September 22–24, 1983, pp 173–221

  29. Moreau JJ (1988a) Bounded variation in time. In: Moreau JJ, Panagiotopoulos PD, Strang G (eds) Topics in nonsmooth mechanics. Birkhäuser Verlag, Basel, pp 1–74

    Google Scholar 

  30. Moreau JJ (1988b) Unilateral contact and dry friction in finite freedom dynamics. In: Moreau JJ, Panagiotopoulos PD (eds) Non-smooth mechanics and applications CISM courses and lectures. Springer, Wien, pp 1–82

    Google Scholar 

  31. Moreau JJ (2003) Fonctionnelles convexes, Séminaire sur les équations aux dérivées partielles, Collège de France, 1966, et Edizioni del Dipartimento di Ingeneria Civile dell’Università di Roma Tor Vergata, Roma

  32. Naghdi PM, Rubin MB (1984) Constrained theories of rods. J Elast 14:343–361

    Article  MATH  Google Scholar 

  33. Papastavridis JG (2002) Analytical mechanics: a comprehensive treatise on the dynamics of constrained systems: for engineers, physicists, and mathematicians. Oxford University Press, Oxford

    Google Scholar 

  34. Rajagopal KR, Srinivasa AR (2005) On the nature of constraints for continua undergoing dissipative processes. Proc R Soc A: Math Phys Eng Sci 461(2061):2785–2795

    Article  MathSciNet  MATH  Google Scholar 

  35. Rauter FG, Pombo J, Ambrósio J, Pereira M (2007) Multibody modeling of pantographs for pantograph-catenary interaction. In: Eberhard P (ed) IUTAM symposium on multiscale problems in multibody system contacts, IUTAM bookseries, vol 1. Springer, Dordrecht, pp 205–226

  36. Rockafellar R (1970) Convex analysis. Princeton mathematical series. Princeton University Press, Princeton, NJ

    Google Scholar 

  37. Romano G, Sacco E (1987) Convex problems in structural mechanics. In: Del Piero G, Maceri F (eds) Unilateral problems in structural analysis-2: proceedings of the second meeting on unilateral problems in structural analysis prescudin, June 17–20, 1985, pp 279–297

  38. Rubin MB (2000) Cosserat theories: shells, rods, and points. Solid mechanics and its applications. Kluwer Academic Publishers, Dordrecht

    Book  Google Scholar 

  39. Salençon J (2001) Handbook of continuum mechanics: general concepts, thermoelasticity. Physics and astronomy online library. Springer, Berlin

    Book  Google Scholar 

  40. Studer C (2009) Numerics of unilateral contacts and friction: modeling and numerical time integration in non-smooth dynamics. Lecture notes in applied and computational mechanics, vol 47. Springer, Berlin

  41. Truesdell C (1968) Essays in the history of mechanics. Springer, New York

    Book  MATH  Google Scholar 

  42. Truesdell C, Noll W (2003) The non-linear field theories of mechanics, 3rd edn. Springer, Berlin

    Google Scholar 

  43. Yi L, Ji-Shun L, Guo-Ding C, Yu-Jun X, Ming-De D (2010) Multibody dynamics modeling of friction winder systems using absolute nodal coordination formulation. In: Huang G, Mak K, Maropoulos P (eds) Proceedings of the 6th CIRP-sponsored international conference on digital enterprise technology, advances in intelligent and soft computing, vol 66. Springer, Berlin, Heidelberg, pp 533–546

Download references

Author information

Authors and Affiliations

  1. Department of Mechanical and Process Engineering, Center of Mechanics, Institute for Mechanical Systems, ETH Zurich, 8092, Zurich, Switzerland

    S. R. Eugster & Ch. Glocker

Authors
  1. S. R. Eugster
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ch. Glocker
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to S. R. Eugster.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Eugster, S.R., Glocker, C. Constraints in structural and rigid body mechanics: a frictional contact problem. Ann. Solid Struct. Mech. 5, 1–13 (2013). https://doi.org/10.1007/s12356-013-0032-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12356-013-0032-9

keywords

Search

Navigation