We consider the dynamics of a closed loop of inextensible string which is undergoing an axial motion. At each instant, one material point of the string is in contact with a singular supply of linear momentum (also known as an external constraint). Several peculiar features of this problem which have not been previously discussed are presented. These include the possible presence of an arbitrary number of kinks, the vanishing nature of the singular supply of momentum, and the critical nature of the tension in the string. When the linear momentum is supplied by a mass-spring-dashpot system, we are also able to establish an exact expression for the frequency of the resulting vibrations, prove that dissipation cannot be present, show that these vibrations only occur for discrete speeds of axial motion, and establish that Coulomb friction is absent.

Issue Section:
Technical Papers
Keywords:
linear momentum, classical mechanics
Topics:
Momentum, Shock absorbers, Springs, String, Linear momentum, Vibration
1.
Schajer
,
G. S.
,
1984
, “
The Vibration of a Rotating Circular String Subject to a Fixed Elastic Restraint
,”
J. Sound Vib.
,
92
(
1
), pp.
11
–
19
.
2.
Perkins
,
N. C.
, and
Mote
, Jr.,
C. D.
,
1986
, “
Comments on Curve Veering in Eigenvalue Problems
,”
J. Sound Vib.
,
106
(
3
), pp.
451
–
463
.
3.
Xiong
,
Y.
, and
Hutton
,
S. G.
,
1994
, “
Vibration and Stability Analysis of a Multi-Guided Rotating String
,”
J. Sound Vib.
,
169
(
5
), pp.
669
–
683
.
4.
Yang
,
L.
, and
Hutton
,
S. G.
,
1995
, “
Interaction Between an Idealized Rotating String and Stationary Constraints
,”
J. Sound Vib.
,
185
(
1
), pp.
139
–
154
.
5.
Tian
,
J.
, and
Hutton
,
S. G.
,
1998
, “
Lateral Vibration Instability Mechanisms in a Constrained Rotating String
,”
ASME J. Appl. Mech.
,
65
(
3
), pp.
774
–
776
.
6.
Tian
,
J.
, and
Hutton
,
S. G.
,
1999
, “
On the Mechanisms of Vibrational Instability in a Constrained Rotating String
,”
J. Sound Vib.
,
225
(
1
), pp.
111
–
126
.
7.
Chen
,
J.-S.
,
1997
, “
Natural Frequencies and Stability of an Axially-Travelling String in Contact With a Stationary Load System
,”
ASME J. Vibr. Acoust.
,
119
(
2
), pp.
152
–
157
.
8.
Cheng
,
S.-P.
, and
Perkins
,
N. C.
,
1991
, “
The Vibration and Stability of a Friction-Guided, Translating String
,”
J. Sound Vib.
,
144
(
2
), pp.
281
–
292
.
9.
Lamb, H., 1929, Dynamics, 2nd Reprinted Ed., Cambridge University Press, Cambridge, UK.
10.
Miranker
,
W. L.
,
1960
, “
The Wave Equation in a Medium in Motion
,”
IBM J. Res. Dev.
,
4
(
1
), pp.
36
–
42
.
11.
Wickert
,
J. A.
, and
Mote
, Jr.,
C. D.
,
1990
, “
Classical Vibration Analysis of Axially Moving Continua
,”
ASME J. Appl. Mech.
,
57
(
3
), pp.
738
–
744
.
12.
O’Reilly
,
O. M.
,
1996
, “
On Steady Motions of a Drawn Cable
,”
ASME J. Appl. Mech.
,
63
(
1
), pp.
180
–
189
.
13.
Healey
,
T. J.
,
1990
, “
Stability and Bifurcation of Rotating Nonlinearly Elastic Loops
,”
Q. Appl. Math.
,
48
(
4
), pp.
679
–
698
.
14.
O’Reilly
,
O. M.
, and
Varadi
,
P. C.
,
1999
, “
A Treatment of Shocks in One-Dimensional Thermomechanical Media
,”
Continuum Mech. Thermodyn.
,
11
(
6
), pp.
339
–
352
.
15.
O’Reilly
,
O. M.
, and
Varadi
,
P. C.
,
2003
, “
On Energetics and Conservations for Strings in the Presence of Singular Sources of Momentum and Energy
,”
Acta Mech.
,
126
, pp.
27
–
45
.
16.
Healey
,
T. J.
, and
Papadopoulos
,
J. N.
,
1990
, “
Steady Axial Motions of Strings
,”
ASME J. Appl. Mech.
,
57
(
3
), pp.
785
–
787
.
17.
Reeken
,
M.
,
1977
, “
On the Equation of Motion of a Chain
,”
Math. Z.
,
155
(
3
), pp.
219
–
237
.
18.
Purohit
,
P. K.
, and
Bhattacharya
,
K.
,
2003
, “
Dynamics of Strings Made of Phase-Transforming Materials
,”
J. Mech. Phys. Solids
,
51
, pp.
393
–
424
.
19.
Abeyaratne, R., and Knowles, J. K., 1993, “Nucleation, Kinetics and Admissibility Criteria for Propagating Phase Boundaries,” Shock Induced Transitions and Phase Structures in Generalized Media, J. E. Dunn, R. Fosdick, and M. Slemrod, eds., Springer-Verlag, Berlin, pp. 1–33.
20.
Truskinovsky, L. M., 1993, “Shocks and Kinks,” Shock Induced Transitions and Phase Structures in Generalized Media, J. E. Dunn, R. Fosdick, and M. Slemrod, eds., Springer-Verlag, Berlin, pp. 185–229.
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