A spatial string model incorporating a nonlinear (and nonconservative) material law is proposed using finite deformation continuum mechanics. The resulting model is shown to reduce to the classical nonlinear string when a linear material law is used. The influence of material nonlinearities on the string’s dynamic response to excitation near a transverse natural frequency is shown to be small due to their appearance at high orders only. Material nonlinearities appear at low order in the equations for excitation near a longitudinal natural frequency, and a solution for this case is developed by applying a second order multiple scales method directly to the partial differential equations. The material nonlinearities are found to influence both the degree of nonlinearity in the response and its softening or hardening nature.

Issue Section:
Technical Papers
Keywords:
dynamics, partial differential equations, vibrations, nonlinear dynamical systems
Topics:
Excitation, String, Partial differential equations
1.
Narasimha
,
R.
,
1968
, “
Non-linear Vibration of an Elastic String
,”
J. Sound Vib.
,
8
, pp.
134
–
146
.
2.
Miles
,
J. W.
,
1984
, “
Resonant Nonplanar Motion of a Stretched String
,”
J. Acoust. Soc. Am.
,
75
, pp.
1505
–
1510
.
3.
Johnson
,
J. M.
, and
Bajaj
,
A. K.
,
1989
, “
Amplitude Modulated and Chaotic Dynamics In Resonant Motion of Strings
,”
J. Sound Vib.
,
128
, pp.
87
–
107
.
4.
Molteno
,
T. C. A.
, and
Tufillaro
,
N. B.
,
1990
, “
Torus Doubling and Chaotic String Vibrations: Experimental Results
,”
J. Sound Vib.
,
137
, pp.
327
–
330
.
5.
O’Reilly
,
O.
, and
Holmes
,
P. J.
,
1992
, “
Non-linear, Non-planar, Non-periodic Vibrations of a String
,”
J. Sound Vib.
,
153
, pp.
413
–
435
.
6.
Bajaj
,
A. K.
, and
Johnson
,
J. M.
,
1992
, “
On the Amplitude Dynamics and Crisis in Resonant Motions of Stretched Strings
,”
Philos. Trans. R. Soc. London, Ser. A
,
338
, pp.
1
–
41
.
7.
O’Reilly
,
O.
,
1993
, “
Global Bifurcations in the Forced Vibration of a Damped String
,”
Int. J. Non-Linear Mech.
,
28
, pp.
337
–
351
.
8.
Rubin
,
M. B.
, and
Gottlieb
,
O.
,
1996
, “
Numerical Solutions of Forced Vibration and Whirling of a Non-Linear String Using the Theory of a Cosserat Point
,”
J. Sound Vib.
,
197
, No.
1
, pp.
85
–
101
.
9.
Lee
,
C.
, and
Perkins
,
N. C.
,
1992
, “
Nonlinear Oscillations of Suspended Cables Containing a Two-to-One Internal Resonance
,”
Nonlinear Dyn.
,
3
, pp.
465
–
490
.
10.
Nayfeh, Samir A., Nayfeh, Ali H., and Mook, Dean T., 1995, “Nonlinear Response of a Taut String to Longitudinal and Transverse End Excitation,” Journal of Vibration and Control, pp. 307–334.
11.
Green, A. E., and Zerna, W., 1968, Theoretical Elasticity, Second Edition, Dover Publications, New York.
12.
Oden, John T., 1972, Finite Elements of Nonlinear Continua, McGraw-Hill, New York.
13.
Meirovitch, Leonard, 1997, Principles and Techniques of Vibrations, Prentice-Hall, London, Section 7.18.
14.
Nayfeh
,
A. H.
,
Nayfeh
,
J. F.
, and
Mook
,
D. T.
,
1992
, “
On Methods for Continuous Systems with Quadratic and Cubic Nonlinearities
,”
Nonlinear Dyn.
,
1
, No.
3
, pp.
145
–
162
.
15.
Pakdemirli
,
M.
,
Nayfeh
,
S. A.
, and
Nayfeh
,
A. H.
,
1995
, “
Analysis of One-to-One Autoparametric Resonances in Cables—Discretization vs. Direct Treatment
,”
Nonlinear Dyn.
,
8
, pp.
65
–
83
.
16.
Boyaci
,
H.
, and
Pakdemirli
,
M.
,
1997
, “
A Comparison of Different Versions of The Method of Multiple Scales for Partial Differential Equations
,”
J. Sound Vib.
,
204
, pp.
595
–
607
.
17.
Luongo, A., and Paolone, A., 1999, “On the Reconstitution Problem in the Multiple Time Scale Method,” Nonlinear Dyn., to appear.
18.
Doedel, E. J., and Wang, X. J., 1995, “AUTO94: Software for Continuation and Bifurcation Problems in Ordinary Differential Equations,” Technical Report, Center for Research on Parallel Computing, California Institute of Technology, Pasadena, CA 91125. CRPC-95-2.
19.
Nayfeh, A. H., and Mook, D. T., 1979, Nonlinear Oscillations, Wiley, New York.
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