# Frictional Impact in Mechanisms

## Abstract

This paper obtains smooth (continuous) dynamics for impact between two rough (frictional) mechanisms. The mechanisms are composed of rigid bodies joined together by ideal non-dissipative pinned joints. The system has a configuation described in terms of generalized coordinates *q* _{ i } and time *t*, and it has kinetic energy *T*(*q̇* _{ i } *, q* _{ i } *, t*). Generalized momentum of the system is defined as a vector, *∂T / ∂q̇* _{ i }. During collision the system is subject to a set of constraint and friction forces that give another vector — the differential of generalized impulse d*Π* _{ i }. If the applied forces act impulsively, then the differentials of generalized momentum and generalized impulse are equal, d(∂*T / ∂q̇* _{ i } *)* = d*Π* _{ i }.

When applied to impact between systems of hard bodies where there is slip that changes direction during contact, this differential relation is required. If the direction of slip is constant, however, it is more convenient to use an integrated form of this generalized impulse-momentum relation. In either case, terminal reaction impulse is obtained from the energetic coefficient of restitution.

## Key words

Rigid-body dynamics Impact Energetic coefficient of restitution Friction## Preview

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## References

- Ahmed, S., Lankarani, H.M., & Pereira, M.F.O.S. (1999) Frictional impact analysis in open-loop multibody mechanical systems,
*ASME J. Mech. Design***121**, 119–127.CrossRefGoogle Scholar - Batlle, J.A. (1996) Rough balanced collisions,
*ASME J. Appl. Mech.***63**, 168–172.zbMATHCrossRefGoogle Scholar - Bahar, L. (1994) On use of quasi-velocities in impulsive motion,
*Int. J. Engr. Sci.***32**, 1669–1686.MathSciNetzbMATHCrossRefGoogle Scholar - Calladine, C.R. (1990) Teaching of some aspects of the theory of inelastic collisions,
*Int. J. Mech. Engng. Ed.***18**(4), 301–310.Google Scholar - Drazetic, P., Level, P., Canaple, B. & Mongenie, P. (1996) Impact on planar kinematic chain of rigid bodies: application to movement of anthropomorphic dummy in crash,
*Int. J. Impact Engng.***18**(5), 505–516.CrossRefGoogle Scholar - Hurmuzlu, Y. & Marghitu, D. (1994) Rigid body collisions of planar kinematic chains with multiple contact points,
*Int. J. Robotics Research***13**(1), 82–92.CrossRefGoogle Scholar - Johnson, W. (1991) Simple model of jack-knifing of a train of coaches and Samuel Vince (1749–1821)
*Int. J. Mech Engng. Ed*.**19**(3), 159–170.Google Scholar - Kane, T. & Levinson, D.A. (1985)
*Dynamics: Theory and Application*, McGraw-Hill, New York.Google Scholar - Pereira, M.S. & Nikravesh, P. (1996) Impact dynamics of systems with frictional contact using joint coordinates and canonical equations of motion,
*Nonlinear Dynamics***9**, 53–71.MathSciNetCrossRefGoogle Scholar - Souchet, R. (1993) Analytical dynamics of rigid body impulsive motions.
*Int. J. Engng. Sci.***31**, 85–92.MathSciNetzbMATHCrossRefGoogle Scholar - Stronge, W.J. (1990) Rigid body collisions with friction,
*Proc. Roy. Soc. Lond*. A**431**, 169–181.MathSciNetzbMATHCrossRefGoogle Scholar - Synge, J.L. & Griffith, B.A. (1959)
*Principles of Mechanics*. McGraw-Hill, New York.Google Scholar - Wittenburg, J. (1977)
*Dynamics of System of Rigid Bodies*, Teubner, Stuttgart.CrossRefGoogle Scholar - Zhao, W. (1999) Kinetostatics and analysis methods for the impact problem,
*Eur. J. Mech. A/Solids***18**, 319–329.zbMATHCrossRefGoogle Scholar