The steady motions of nonlinearly elastic and inextensible strings which are being drawn between two fixed points and are subject to a gravitational load are examined. It is shown that, dependent on the boundary conditions, constitutive equations and a reference drawing speed, multiple co-existant steady motions are possible in certain situations. Using a variational method, stability criteria are also established for some of these motions.

1.
Abarbanel
H. D. I.
,
Holm
D. D.
,
Marsden
J. E.
, and
Ratiu
T. S.
,
1986
, “
Nonlinear Stability Analysis of Stratified Fluid Equilibria
,”
Philosophical Transactions of the Royal Society of London
, Vol.
A318
, pp.
349
409
.
2.
Abarbanel
H. D. I.
, and
Holm
D. D.
,
1987
, “
Nonlinear Stability Analysis of Inviscid Flows in Three Dimensions: Incompressible Fluids and Barotropic Fluids
,”
Physics of Fluids
, Vol.
30
, pp.
3369
3382
.
3.
Antman
S. S.
,
1979
, “
Multiple Equilibrium States of Nonlinearly Elastic Strings
,”
SIAM Journal of Applied Mathematics
, Vol.
37
, pp.
588
604
.
4.
Antman
S. S.
, and
Reeken
M.
,
1987
, “
The Drawing and Whirling of Strings: Singular Global Multiparameter Bifurcation Problems
,”
SIAM Journal on Mathematical Analysis
, Vol.
18
, pp.
337
365
.
5.
Casey
J.
, and
Naghdi
P. M.
,
1991
, “
A Lagrangian Description of Vorticity
,”
Archive for Rational Analysis and Mechanics
, Vol.
115
, pp.
1
15
.
6.
Clebsch
A.
,
1860
, “
Ueber die Gleichgewichtsfigur eines biegsamen Fadens
,”
Crelle Journal fu¨r die reine und angewandte Mathematik
, Vol.
57
, pp.
93
110
.
7.
Dickey
R. W.
,
1969
, “
The Nonlinear String under a Vertical Load
,”
SIAM Journal of Applied Mathematics
, Vol.
17
, pp.
172
178
.
8.
Dickey, R. W., 1976, Bifurcation Problems in Nonlinear Elasticity, Pitman, London.
9.
Healey
T. J.
,
1990
, “
Stability and Bifurcation of Rotating Nonlinearly Elastic Loops
,”
Quarterly of Applied Mathematics
, Vol.
48
, pp.
679
698
.
10.
Healey
T. J.
, and
Papadopoulos
J. N.
,
1990
, “
Steady Axial Motions of Strings
,”
ASME JOURNAL OF APPLIED MECHANICS
, Vol.
57
, pp.
785
787
.
11.
Knops
R. J.
, and
Wilkes
E. W.
,
1966
, “
On Movchan’s Theorems for Stability of Continuous Systems
,”
International Journal of Engineering Science
, Vol.
4
, pp.
303
329
.
12.
Lamb, H., 1929, Dynamics, 2nd reprinted ed., Cambridge University Press, Cambridge, U.K.
13.
Lamb, H., 1945, Hydrodynamics, 6th ed., Dover, New York.
14.
Love, A. E. H., 1897, Theoretical Mechanics. An Introductory Treatise on the Principles of Dynamics, Cambridge University Press, Cambridge, U.K.
15.
Naghdi, P. M., 1982, “Finite Deformation of Rods and Shells,” Proceedings of the IUTAM Symposium on Finite Elasticity, D. E. Carlson, and R. T. Shield, eds., Martinus Nijhoff, The Hague, pp. 47–103.
16.
Nashed
M. Z.
,
1966
, “
Some Remarks on Variations and Differentials
,”
American Mathematical Monthly
, Vol.
73
, No.
4
, pp.
63
76
.
17.
O’Reilly
O. M.
, and
Varadi
P.
,
1995
, “
Elastic Equilibria of Translating Cables
,”
Acta Mechanica
, Vol.
118
, pp.
189
206
.
18.
Perkins
N. C.
, and
Mote
C. D.
,
1987
, “
Three Dimensional Vibration of Travelling Elastic Cables
,”
Journal of Sound and Vibration
, Vol.
114
, pp.
325
340
.
19.
Perkins
N. C.
, and
Mote
C. D.
,
1989
, “
Theoretical and Experimental Stability of Two Translating Cable Equilibria
,”
Journal of Sound and Vibration
, Vol.
128
, pp.
397
410
.
20.
Rowth, E. J., 1882, The Advanced Part of a Treatise on the Dynamics of Rigid Bodies, 4th ed., Macmillan, London.
21.
Simpson
A.
,
1972
, “
On the Oscillatory Motions of Translating Elastic Cables
,”
Journal of Sound and Vibration
, Vol.
20
, pp.
177
189
.
22.
Troutman, J. L., 1983, Variational Calculus with Elementary Convexity, Springer-Verlag, New York.
This content is only available via PDF.
You do not currently have access to this content.