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.
Issue Section:
Technical Papers
1.
Abarbanel
H. D. I.
Holm
D. D.
Marsden
J. E.
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.
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.
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.
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.
Papadopoulos
J. N.
1990
, “Steady Axial Motions of Strings
,” ASME JOURNAL OF APPLIED MECHANICS
, Vol. 57
, pp. 785
–787
.11.
Knops
R. J.
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.
Varadi
P.
1995
, “Elastic Equilibria of Translating Cables
,” Acta Mechanica
, Vol. 118
, pp. 189
–206
.18.
Perkins
N. C.
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.
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.
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