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
1.
Mote
, C.
, 1965
, “A Study of Band Saw Vibrations
,” J. Franklin Inst.
, 279
, pp. 430
–444
.10.1016/0016-0032(65)90273-52.
Wickert
, J.
, and Mote
, C.
, 1990
, “Classical Vibration Analysis of Axially Moving Continua
,” ASME J. Appl. Mech.
, 57
, pp. 738
–744
.10.1115/1.28970853.
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-X4.
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.2007.07.0235.
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.1998.19366.
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.2009.12.0117.
Miranker
, W. L.
, 1960
, “The Wave Equation in a Medium in Motion
,” J. Res. Dev.
, 4
(1
), pp. 36
–42
.10.1147/rd.41.00368.
Chen
, L.
, 2005
, “Analysis and Control of Transverse Vibrations of Axially Moving Strings
,” ASME Appl. Mech. Rev.
, 58
, pp. 91
–116
.10.1115/1.18491699.
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.2003.12.02010.
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.2006.09.00611.
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/s00707-002-1000-312.
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.2003.10.01213.
Mote
, C. D.
, 1968
, “Parametric Excitation of an Axially Moving String
,” ASME J. Appl. Mech.
, 35
, pp. 171
–172
.10.1115/1.360113814.
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
.10.1115/1.288818915.
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.2002.503116.
Mote
, C. D.
, 1975
, “Stability of Systems Transporting Accelerating Axially Moving Materials
,” J. Dyn. Syst. Meas. Control.
, 97
, pp. 96
–98
.10.1115/1.342688017.
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.1994.101218.
Wickert
, J. A.
, 1996
, “Transient Vibration of Gyroscopic Systems With Unsteady Superposed Motion
,” J. Sound Vib.
, 195
, pp. 797
–807
.10.1006/jsvi.1996.046219.
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.1999.265120.
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)01281-621.
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.ca22.
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.2003.10.03723.
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.2009.05.00424.
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.1058725.
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.2007.06.00327.
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/s00419-010-0446-328.
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.2008.11.013Copyright © 2013 by ASME
You do not currently have access to this content.