Research Papers

Energetically Consistent Calculations for Oblique Impact in Unbalanced Systems With Friction

[+] Author and Article Information
W. J. Stronge

Department of Engineering,
University of Cambridge,
Cambridge CB2 1PZ, UK

When required, a static coefficient of friction μ¯s, where |μ¯s|>|μ¯| can be introduced for “periods” of stick [6,9].

Equations (27a) and (29) apply also to jam where ψ0<0.

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received February 24, 2015; final manuscript received April 25, 2015; published online June 9, 2015. Editor: Yonggang Huang.

J. Appl. Mech 82(8), 081003 (Aug 01, 2015) (12 pages) Paper No: JAM-15-1109; doi: 10.1115/1.4030459 History: Received February 24, 2015; Revised April 25, 2015; Online June 09, 2015

Analytical mechanics is used to derive original 3D equations of motion that represent impact at a point in a system of rigid bodies. For oblique impact between rough bodies in an eccentric (unbalanced) configuration, these equations are used to compare the calculations of energy dissipation obtained using either the kinematic, the kinetic, or the energetic coefficient of restitution (COR); eN,eP, or e*. Examples demonstrate that for equal energy dissipation by nonfrictional sources, either eNe*eP or ePe*eN depending on whether the unbalance of the impact configuration is positive or negative relative to the initial direction of slip. Consequently, when friction brings initial slip to rest during the contact period, calculations that show energy gains from impact can result from either the kinematic or the kinetic COR. On the other hand, the energetic COR always correctly accounts for energy dissipation due to both hysteresis of the normal contact force and friction, i.e., it is energetically consistent.

Copyright © 2015 by ASME
Your Session has timed out. Please sign back in to continue.


Keller, J. B., 1986, “Impact With Friction,” ASME J. Appl. Mech., 53(1), pp. 1–4. [CrossRef]
Routh, E. J., 1905, Dynamics of a System of Rigid Bodies, Dover Publications, New York.
Nordmark, A., Dankowicz, H., and Champneys, A., 2009, “Discontinuity-Induced Bifurcations in Systems With Impact and Friction: Discontinuities in the Impact Law,” Int. J. Nonlinear Mech., 44(10), pp. 1022–1023. [CrossRef]
Stronge, W. J., 1990, “Rigid-Body Collisions With Friction,” Proc. R. Soc. A, 431(1881), pp. 169–181. [CrossRef]
Kane, T. R., 1984, “A Dynamics Puzzle,” Stanford Mechanics Alumni Club Newsletter, p. 6.
Kane, T. R., and Levinson, D. A., 1985, Dynamics: Theory and Applications, McGraw-Hill, New York.
Wang, Y., and Mason, M. T., 1992, “Two Dimensional Rigid-Body Collisions With Friction,” ASME J. Appl. Mech., 59(3), pp. 635–642. [CrossRef]
Pereira, M. S., and Nikravesh, P., 1996, “Impact Dynamics of Multibody Systems With Frictional Contact Using Joint Coordinates and Canonical Equations of Motion,” Nonlinear Dyn., 9(1–2), pp. 53–71. [CrossRef]
Shen, H., and Townsend, M. A., 1998, “Collisions of Constrained Rigid Body Systems With Friction,” Shock Vib., 5(3), pp. 141–151. [CrossRef]
Ahmed, M. S., Lankarani, H. M., and Pereira, M. S., 1999, “Frictional Impact Analysis in Open-Loop Multi-Body Mechanical Systems,” ASME J. Mech. Des., 121(1), pp. 119–127. [CrossRef]
Djerassi, S., 2009, “Collision With Friction: Part A: Newton's Hypotheses,” Multibody Syst. Dyn., 21(1), pp. 37–54. [CrossRef]
Glocker, Ch., 2014, “Energetic Consistency Conditions for Standard Impacts, Part II: Poisson-Type Inequality Impact Laws,” Multibody Syst. Dyn., 32(4), pp. 445–509. [CrossRef]
Lankarani, H. M., and Pereira, M. S., 2001, “Treatment of Impact With Friction in Planar Multibody Mechanical Systems,” Multibody Syst. Dyn., 6(3), pp. 203–227. [CrossRef]
Djerassi, S., 2009, “Collision With Friction: Part B: Poisson's and Stronge's Hypotheses,” Multibody Syst. Dyn., 21(1), pp. 55–70. [CrossRef]
Stronge, W. J., 2000, Impact Mechanics, Cambridge University Press, Cambridge, UK.
Batlle, J. A., 1996, “The Sliding Velocity Flow of Rough Collisions in Multibody Systems,” ASME J. Appl. Mech., 63(3), pp. 804–809. [CrossRef]
Batlle, J. A., 1996, “Rough Balanced Collisions,” ASME J. Appl. Mech., 63(1), pp. 168–172. [CrossRef]
Batlle, J. A., 1993, “On Newton's and Poisson's Rules of Percussive Dynamics,” ASME J. Appl. Mech., 60(2), pp. 376–381. [CrossRef]
Shen, Y., and Stronge, W. J., 2011, “Painlevé Paradox During Oblique Impact With Friction,” Eur. J. Mech. A/Solids, 30(4), pp. 457–467. [CrossRef]
Stronge, W. J., 2013, “Smooth Dynamics of Oblique Impact With Friction,” Int. J. Impact Eng., 51, pp. 36–49. [CrossRef]
Zhen, Z., Liu, C., and Chen, B., 2007, “The Painlevé Paradox Studied at a 3D Slender Rod,” Multibody Syst. Dyn., 19(4), pp. 323–343. [CrossRef]
Han, I., and Gilmore, B. J., 1993, “Multibody Impact Motion With Friction—Analysis, Simulation and Experimental Validation,” ASME J. Appl. Mech., 115(3), pp. 412–422. [CrossRef]
Souchet, R., 1994, “Restitution and Friction Laws in Rigid Body Collisions,” Int. J. Eng. Sci., 32(5), pp. 863–876. [CrossRef]
Hurmuzlu, Y., 1998, “An Energy Based Coefficient of Restitution for Planar Impacts of Slender Bar With Massive External Surfaces,” ASME J. Appl. Mech., 65(4), pp. 952–962. [CrossRef]
Paoli, L., and Schatzman, M., 2007, “Numerical Simulation of the Dynamics of an Impacting Bar,” Comput. Methods Appl. Mech. Eng., 196(29–30), pp. 2839–2851. [CrossRef]
Payr, M., and Glocker, Ch., 2005, “Oblique Frictional Impact of a Bar: Analysis and Comparison of Different Impact Laws,” Nonlinear Dyn., 41(4), pp. 361–383. [CrossRef]
Lubarda, V. A., 2010, “The Bounds on the Coefficients of Restitution for Frictional Impact of a Rigid Pendulum Against a Fixed Surface,” ASME J Appl. Mech., 77(1), p. 011006. [CrossRef]


Grahic Jump Location
Fig. 1

Impact configurations with positive unbalance β2 > 0 or negative unbalance β2 < 0

Grahic Jump Location
Fig. 2

Normal and tangential components of relative velocity at C as functions of normal impulse during collision. The shaded areas represent work done by the normal impulse during compression and restitution.

Grahic Jump Location
Fig. 3

Relative velocity components as function of normal impulse at C for all slip processes: (a) slip-stick for μ≥|μ¯|; (b) slip-reverse slip for β2 > 0 and μ<|μ¯|; (c) gross-slip for β2 < 0 and μ<|μ¯|; and (d) jam-stick for β2 < 0 and a large coefficient of friction μ≥-β3/β2

Grahic Jump Location
Fig. 4

Slip processes for impact of a slender rod against a half-space as functions of the angle of inclination θ and the coefficient of friction μ

Grahic Jump Location
Fig. 5

Kinematic (solid line) and kinetic (dashed line) CORs for elastic impact, e*=1, for normalized angle of incidence Ψ∧0 if unbalance is either positive, β2 > 0, or negative, β2 < 0

Grahic Jump Location
Fig. 6

Slender bar with angle of inclination θ when it impacts at angle of incidence ψ0≡tan -1(-v1(0)/v3(0))=π/4 against a rough half-space. Configurations are shown with both (a) positive unbalance β2 > 0 and (b) negative unbalance β2 < 0.

Grahic Jump Location
Fig. 7

(a) Kinematic and (b) kinetic COR for elastic impact (e*=1) of rigid rod inclined at angle θ that impacts at angle of incidence ψ0=π/4, against a rough half-space

Grahic Jump Location
Fig. 8

Ratio of final to initial kinetic energy Tf/T0 as function of coefficient of friction μ for CORs eN = eP = e* = 1 used in analysis of oblique impact by rigid rod against rough elastic half-space. Rod inclination is (a) θ = π/6 rad (β2 > 0), or (b) θ = -π/6 rad (β2 < 0) and the angle of incidence is ψ0 = π/4.

Grahic Jump Location
Fig. 9

Double pendulum that impacts against rough half space. Configurations are shown with either (a) positive unbalance, β2 > 0 or (b) negative unbalance, β2 < 0. In both cases the incident relative velocity at C is v(0) = 0.5343 n1-0.2684 n3.

Grahic Jump Location
Fig. 10

(a) Kinematic and (b) kinetic COR for elastic impact (e*=1) of double pendulum that impacts at angle of incidence ψ0=1.105 rad, against a rough half-space



Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In