×

zbMATH — the first resource for mathematics

A set-valued force law for spatial Coulomb–Contensou friction. (English) Zbl 1038.74513
Summary: The aim of this paper is to develop a set-valued contact law for combined spatial Coulomb–Contensou friction, taking into account a normal friction torque (drilling friction) and spin. The set-valued Coulomb–Contensou friction law is derived from a non-smooth velocity pseudo potential. A higher-order Runge–Kutta time-stepping method is presented for the numerical simulation of rigid bodies with Coulomb–Contensou friction. The algebraic inclusion describing the contact problem is solved with an Augmented Lagrangian approach. The theory and numerical methods are applied to the Tippe-Top. The analysis and numerical results on the Tippe-Top illustrate the importance of Coulomb–Contensou friction for the dynamics of systems with friction.

MSC:
74A55 Theories of friction (tribology)
74M10 Friction in solid mechanics
Software:
RODAS; Meschach
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Alart, P.; Curnier, A., A mixed formulation for frictional contact problems prone to Newton like solution methods, Comput. methods appl. mech. engrg., 92, 353-375, (1991) · Zbl 0825.76353
[2] Anitescu, M.; Potra, F.A., Formulating dynamic multi-rigid-body contact problems with friction as solvable linear complementarity problems, Nonlinear dynamics, 14, 3, 231-247, (1997) · Zbl 0899.70005
[3] Brogliato, B., Nonsmooth mechanics, (1999), Springer London · Zbl 0917.73002
[4] Bronstein, I.N.; Semendjajew, K.A., Taschenbuch der Mathematik, (1984), Harri Deutsch Thun · Zbl 0085.00103
[5] Contensou, P., Couplage entre frottement de glissement et frottement de pivotement dans la théorie de la toupie, (), 201-216 · Zbl 0118.41001
[6] Friedl, C., 1997. Der Stehaufkreisel. Master’s thesis. Universität Augsburg
[7] Glocker, Ch., Dynamik von starrkörpersystemen mit reibung und stößen, Fortschr.-ber. VDI., 18, 182, (1995)
[8] Glocker, Ch., Formulation of spatial contact situations in rigid multibody systems, Comput. methods appl. mech. engrg., 177, 199-214, (1998) · Zbl 0952.70007
[9] Glocker, Ch., Set-valued force laws, dynamics of non-smooth systems, Lecture notes in appl. mech., 1, (2001), Springer-Verlag Berlin
[10] Hairer, E.; Wanner, G., Solving ordinary differential equations II; stiff and differential-algebraic problems, Springer ser. comput. math., 14, (2002), Springer Berlin
[11] Johnson, K.L., Contact mechanics, (1985), Cambridge University Press Cambridge · Zbl 0599.73108
[12] Kuypers, F., Klassische mechanik, (1990), VCH-Verlagsgesellschaft Weinheim · Zbl 0717.70002
[13] Laursen, T.A.; Simo, J.C., Algorithmic symmetrization of Coulomb frictional problems using augmented Lagrangians, Comput. methods appl. mech. engrg., 108, 133-146, (1993) · Zbl 0782.73076
[14] Leine, R.I.; Glocker, Ch.; Van Campen, D.H., Nonlinear dynamics and modeling of various wooden toys with impact and friction, J. vib. control, 9, 1, 25-78, (2003) · Zbl 1045.70008
[15] Magnus, K., Kreisel; theorie und anwendungen, (1971), Springer-Verlag Berlin
[16] Moreau, J.J., 1988. Unilateral contact and dry friction in finite freedom dynamics. In: Moreau and Panagiotopoulos (1988), pp. 1-82 · Zbl 0703.73070
[17] ()
[18] Murty, K.G., Linear complementarity, linear and nonlinear programming, Sigma series in appl. math., 3, (1988), Heldermann Berlin
[19] Pfeiffer, F.; Glocker, Ch., Multibody dynamics with unilateral contacts, (1996), Wiley New York · Zbl 0922.70001
[20] Rockafellar, R.T., Augmented Lagrangians and applications of the proximal point algorithm in convex programming, Math. oper. res., 1, 2, 97-116, (1976) · Zbl 0402.90076
[21] Simo, J.C.; Laursen, T.A., An augmented Lagrangian treatment of contact problems involving friction, Comput. & structures, 42, 1, 97-116, (1992) · Zbl 0755.73085
[22] Stewart, D.E.; Trinkle, J.C., An implicit time-stepping scheme for rigid body dynamics with inelastic collisions and Coulomb friction, Int. J. numer. methods engrg., 39, 15, 2673-2691, (1996) · Zbl 0882.70003
[23] Stiegelmeyr, A., Zur numerischen berechnung strukturvarianter mehrkörpersysteme, Fortschr.-ber. VDI., 18, 271, (2001) · Zbl 0984.70001
[24] Zhuravlev, V.G., The model of dry friction in the problem of the rolling of rigid bodies, J. appl. math. mech., 62, 5, 705-710, (1998)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.