Nonlinear parametric vibration of an axially accelerating viscoelastic string is investigated. The string is constituted by the fractional Kelvin model. The principal parametric resonance is analyzed by using an asymptotic approach. The modulation equation is derived from the solvability condition. Closed-form expressions of the amplitudes and the existence conditions of steady-state responses are obtained from the modulation equation. Numerical examples are presented to highlight the effects of fractional order and other system parameters on the responses.

References

References
1.
Mote
,
C.
,
1965
, “
A Study of Band Saw Vibrations
,”
J. Franklin Inst.
,
279
, pp.
430
444
.10.1016/0016-0032(65)90273-5
2.
Wickert
,
J.
, and
Mote
,
C.
,
1990
, “
Classical Vibration Analysis of Axially Moving Continua
,”
ASME J. Appl. Mech.
,
57
, pp.
738
744
.10.1115/1.2897085
3.
Wickert
,
J.
, and
Mote
,
C.
,
1991
, “
Travelling Load Response of an Axially Moving String
,”
J. Sound Vib.
,
149
, pp.
267
284
.10.1016/0022-460X(91)90636-X
4.
Chen
,
L.-Q.
,
Zhang
,
W.
, and
Zu
,
J. W.
,
2009
, “
Nonlinear Dynamics for Transverse Motion of Axially Moving Strings
,”
Chaos Solitons Fractals.
,
40
, pp.
78
90
.10.1016/j.chaos.2007.07.023
5.
Parker
,
R. G.
,
1999
, “
Supercritical Speed Stability of the Trivial Equilibrium of an Axially-Moving String on an Elastic Foundation
,”
J. Sound Vib.
,
221
, pp.
205
219
.10.1006/jsvi.1998.1936
6.
Ghayesh
,
M. H.
, “
Parametric Vibrations and Stability of an Axially Accelerating String Guided by a Non-Linear Elastic Foundation
,”
Int. J. Non-Linear Mech.
,
45
, pp.
382
394
.10.1016/j.ijnonlinmec.2009.12.011
7.
Miranker
,
W. L.
,
1960
, “
The Wave Equation in a Medium in Motion
,”
J. Res. Dev.
,
4
(
1
), pp.
36
42
.10.1147/rd.41.0036
8.
Chen
,
L.
,
2005
, “
Analysis and Control of Transverse Vibrations of Axially Moving Strings
,”
ASME Appl. Mech. Rev.
,
58
, pp.
91
116
.10.1115/1.1849169
9.
Marynowski
,
K.
,
2004
, “
Non-Linear Vibrations of an Axially Moving Viscoelastic Web With Time-Dependent Tension
,”
Chaos Solitons Fractals.
,
21
, pp.
481
490
.10.1016/j.chaos.2003.12.020
10.
Marynowski
,
K.
, and
Kapitaniak
,
T.
,
2007
, “
Zener Internal Damping in Modelling of Axially Moving Viscoelastic Beam With Time-Dependent Tension
,”
Int. J. Non-Linear Mech.
,
42
, pp.
118
131
.10.1016/j.ijnonlinmec.2006.09.006
11.
Chen
,
L. Q.
,
Zu
,
J.
, and
Wu
,
J.
,
2003
, “
Steady-State Response of the Parametrically Excited Axially Moving String Constituted by the Boltzmann Superposition Principle
,”
Acta Mech.
,
162
, pp.
143
155
.10.1007/s00707-002-1000-3
12.
Chen
,
L. Q.
,
Zhao
,
W. J.
, and
W Zu
,
J.
,
2004
, “
Transient Responses of an Axially Accelerating Viscoelastic String Constituted by a Fractional Differentiation Law
,”
J. Sound Vib.
,
278
(
4–5
), pp.
861
871
.10.1016/j.jsv.2003.10.012
13.
Mote
,
C. D.
,
1968
, “
Parametric Excitation of an Axially Moving String
,”
ASME J. Appl. Mech.
,
35
, pp.
171
172
.10.1115/1.3601138
14.
Mockensturm
,
E. M.
,
Perkins
,
N. C.
, and
Ulsoy
,
A. G.
,
1996
, “
Stability and Limit Cycles of Parametrically Excited, Axially Moving Strings
,”
ASME J. Vib. Acoust.
,
118
, pp.
346
351
.10.1115/1.2888189
15.
Liu
,
Z. S.
, and
Huang
,
C.
,
2002
, “
Evaluation of the Parametric Instability of an Axially Translating Media Using a Variational Principle
,”
ASME J. Sound Vib.
,
257
, pp.
985
995
.10.1006/jsvi.2002.5031
16.
Mote
,
C. D.
,
1975
, “
Stability of Systems Transporting Accelerating Axially Moving Materials
,”
J. Dyn. Syst. Meas. Control.
,
97
, pp.
96
98
.10.1115/1.3426880
17.
Pakdemirli
,
M.
,
Ulsoy
,
A. G.
, and
Ceranoglu
,
A.
,
1994
, “
Transverse Vibration of an Axially Accelerating String
,”
J. Sound Vib.
,
169
, pp.
179
196
.10.1006/jsvi.1994.1012
18.
Wickert
,
J. A.
,
1996
, “
Transient Vibration of Gyroscopic Systems With Unsteady Superposed Motion
,”
J. Sound Vib.
,
195
, pp.
797
807
.10.1006/jsvi.1996.0462
19.
Ozkaya
,
E.
, and
Pakdemirli
,
M.
,
2000
, “
Lie Group Theory and Analytical Solutions for the Axially Accelerating String Problem
,”
J. Sound Vib.
,
230
, pp.
729
742
.10.1006/jsvi.1999.2651
20.
Chen
,
L.-Q.
,
Zhang
,
N.-H.
, and
Zu
,
J. W.
,
2003
, “
The Regular and Chaotic Vibrations of an Axially Moving Viscoelastic String Based on Fourth Order Galerkin Truncation
,”
J. Sound Vib.
,
261
, pp.
764
773
.10.1016/S0022-460X(02)01281-6
21.
Chen
,
L. Q.
,
Wu
,
J.
, and
Zu
,
J. W.
,
2004
, “
Asymptotic Nonlinear Behaviors in Transverse Vibration of an Axially Accelerating Viscoelastic string
,”
Nonlinear. Dyn.
,
35
, pp.
347
360
.10.1023/B:NODY.0000027744.15436.ca
22.
Chen
,
L. Q.
,
Wu
,
J.
and
Zu
,
J. W.
,
2004
, “
The Chaotic Response of the Viscoelastic Traveling String: An Integral Constitutive Law
,”
Chaos Solitons Fractals.
,
21
, pp.
349
357
.10.1016/j.chaos.2003.10.037
23.
Ghayesh
,
M. H.
,
2009
, “
Stability Characteristics of an Axially Accelerating String Supported by an Elastic Foundation
,”
Mech. Mach. Theory.
,
44
, pp.
1964
1979
.10.1016/j.mechmachtheory.2009.05.004
24.
Drozdov
,
A.
, and
Kalamkarov
,
A.
,
1996
, “
A Constitutive Model for Nonlinear Viscoelastic Behavior of Polymers
,”
Polym. Eng. Sci.
,
36
, pp.
1907
1919
.10.1002/pen.10587
25.
Oldham
,
K.
, and
Spanier
,
J.
,
1974
,
The Fractional Calculus. 1974
,
Academic Press
,
New York
.
26.
Chen
,
L. Q.
, and
Zu
,
J. W.
,
2008
, “
Solvability Condition in Multi-Scale Analysis of Gyroscopic Continua
,”
J. Sound Vib.
,
309
, pp.
338
342
.10.1016/j.jsv.2007.06.003
27.
Ghayesh
,
M. H.
, and
Moradian
,
N.
,
2011
, “
Nonlinear Dynamic Response of Axially Moving, Stretched Viscoelastic Strings
,”
Arch. Appl. Mech.
,
81
. pp.
781
799
.10.1007/s00419-010-0446-3
28.
Yang
,
T.
,
Fang
,
B.
,
Chen
,
Y.
, and
Zhen
,
Y.
,
2009
, “
Approximate Solutions of Axially Moving Viscoelastic Beams Subject to Multi-Frequency Excitations
,”
Int. J. Non-Linear Mech.
,
44
, pp.
230
238
.10.1016/j.ijnonlinmec.2008.11.013
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