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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 $ψ∧0<0$.

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

## Abstract

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 $eN≤e*≤eP$ or $eP≤e*≤eN$ 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.

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## Figures

Fig. 1

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

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.

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

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

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

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.

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

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.

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.

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

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