In this paper we consider the most basic multi-impact system, the so-called “Newton’s Cradle.” The task of developing an analytical method to predict the post impact velocities of the balls in the cradle has baffled investigators in the field of impact research for many years. The impulse-based rigid-body body as well as the alternative compliance-based time-base approaches have failed to produce valid solutions to this problem. Here, we present a new method that produces energetically consistent solutions to the problem. Our method is based on the traditional impulse-momentum-based rigid-body approach. We do, however, resolve the nonuniqueness difficulty in the rigid-body approach by introducing a new constant called the Impulse Transmission Ratio. Finally, we verify our method by conducting a set of experiments and comparing the theoretical predictions with the experimental outcomes.

1.
Marghitu
,
D. B.
, and
Hurmuzlu
,
Y.
,
1995
, “
Three-Dimensional Rigid-Body Collisions With Multiple Contact Points
,”
ASME J. Appl. Mech.
,
62
, pp.
725
732
.
2.
Johnson
,
W.
,
1976
, “
Simple Linear Impact
,”
ImechE. IJMEE
,
4
, No.
2
, pp.
167
181
.
3.
Han
,
I.
, and
Gilmore
,
B. J.
,
1993
, “
Multi-Body Impact Motion With Friction—Analysis, Simulation, and Experimental Validation
,”
ASME J. Mech. Des.
,
115
, pp.
412
422
.
4.
Brogliato, B., 1996, Nonsmooth Impact Mechanics: Models, Dynamics and Control, Springer-Verlag, New York, LNCIS 220.
5.
Smith
,
E. A. L.
,
1955
, “
Impact and Longitudinal Wave Transmission
,”
Trans. ASME
,
77
, pp.
963
973
.
6.
Walkiewicz
,
T. A.
, and
Newby
,
N. D.
, Jr.
,
1972
, “
Linear Collisions
,”
Am. J. Phys.
,
40
, pp.
133
137
.
7.
Newby
,
N. D.
, Jr.
,
1979
, “
Linear Collisions With Harmonic Oscillator Forces: The Inverse Scattering Problem
,”
Am. J. Phys.
,
47
, No.
2
, pp.
161
165
.
8.
Hinch
,
E. J.
, and
Saint-Jean
,
S.
,
1999
, “
The Fragmentation of a Line of Balls by an Impact
,”
Proc. R. Soc. London, Ser. A
,
455
, pp.
3201
3220
.
9.
Cholet, C., 1998, “Chocs de Solids Rigid,” Ph.D. thesis, University of Paris 6 and LCPC-CNRS.
10.
Moreau
,
J. J.
,
1994
, “
Some Numerical Methods in Multibody Dynamics: Application to Granular Materials
,”
Eur. J. Mech. A/Solids
,
13
, No.
4
, pp.
93
114
.
11.
Fremond
,
M.
,
1995
, “
Rigid Bodies Collisions
,”
Phys. Lett. A
,
204
, pp.
33
41
.
12.
Stronge
,
W. J.
,
1990
, “
Rigid Body Collisions with Friction
,”
Proc. R. Soc. London, Ser. A
,
431
, pp.
169
181
.
You do not currently have access to this content.