Abstract
Nonlinear vibrations of axially moving plates partially immersed in fluid are investigated in this paper. The system has time dependency in velocity and tension in axial direction. The Galerkin method is used to solve the nonlinear vibration differential equation. The method of multiple scales and Runge–Kutta method are applied to solve the nonlinear vibration response of the system. Additionally, the stability conditions of trivial and nontrivial solutions are analyzed using the Routh–Hurwitz criterion. The effects of mean velocity, amplitude of pulsating velocity, mean tension, amplitude of pulsating tension, and pulsating frequency on the complex dynamics of the system are obtained. The study results reveal rich dynamic behaviors of fluid–structure coupling system.
References
1.
Wang
,
L.
, and
Ni
,
Q.
, “
Vibration and Stability of an Axially Moving Beam Immersed in Fluid
,” Int. J. Solids Struct.
,
45
, pp. 1445
–1457
.10.1016/j.ijsolstr.2007.10.0152.
Banichuk
,
N.
,
Jeronen
,
J.
,
Neittaanmäki
,
P.
, and
Tuovinen
,
T.
, 2010
, “
On the Instability of an Axially Moving Elastic Plate
,” Int. J. Solids Struct.
,
47
(1
), pp. 91
–99
.10.1016/j.ijsolstr.2009.09.0203.
Ding
,
H.
, and
Chen
,
L. Q.
, 2011
, “
Natural Frequencies of Nonlinear Vibration of Axially Moving Beams
,” Nonlinear Dyn.
,
63
(1–2
), pp. 125
–134
.10.1007/s11071-010-9790-74.
Nguyen
,
Q. C.
, and
Hong
,
K. S.
, 2012
, “
Transverse Vibration Control of Axially Moving Membranes by Regulation of Axial Velocity
,” IEEE Trans. Control Syst. Technol.
,
20
(4
), pp. 1124
–1131
.10.1109/TCST.2011.21593845.
Marynowski
,
K.
, and
Kapitaniak
,
T.
, 2014
, “
Dynamics of Axially Moving Continua
,” Int. J. Mech. Sci.
,
81
, pp. 26
–41
.10.1016/j.ijmecsci.2014.01.0176.
Zhou
,
Y. F.
, and
Wang
,
Z. M.
, 2019
, “
Dynamic Instability of Axially Moving Viscoelastic Plate
,” Eur. J. Mech. A
,
73
, pp. 1
–10
.10.1016/j.euromechsol.2018.06.0097.
Pellicano
,
F.
, and
Vestroni
,
F.
, 2002
, “
Complex Dynamics of High-Speed Axially Moving Systems
,” J. Sound Vib.
,
258
(1
), pp. 31
–44
.10.1006/jsvi.2002.50708.
Chen
,
L. Q.
, and
Yang
,
X. D.
, 2007
, “
Nonlinear Free Transverse Vibration of an Axially Moving Beam: Comparison of Two Models
,” J. Sound Vib.
,
299
(1–2
), pp. 348
–354
.10.1016/j.jsv.2006.06.0459.
Yang
,
X. D.
, and
Zhang
,
W.
, 2014
, “
Nonlinear Dynamics of Axially Moving Beam With Coupled Longitudinal–Transversal Vibrations
,” Nonlinear Dyn.
,
78
(4
), pp. 2547
–2556
.10.1007/s11071-014-1609-510.
Ghayesh
,
M. H.
,
Amabili
,
M.
, and
Païdoussis
,
M. P.
, 2012
, “
Nonlinear Vibrations and Stability of an Axially Moving Beam With an Intermediate Spring Support: Two-Dimensional Analysis
,” Nonlinear Dyn.
,
70
(1
), pp. 335
–354
.10.1007/s11071-012-0458-311.
Ghayesh
,
M. H.
, and
Amabili
,
M.
, 2013
, “
Non-Linear Global Dynamics of an Axially Moving Plate
,” Int. J. Non-Linear Mech.
,
57
, pp. 16
–30
.10.1016/j.ijnonlinmec.2013.06.00512.
Pakdemirli
,
M.
, and
Ulsoy
,
A. G.
, 1997
, “
Stability Analysis of an Axially Accelerating String
,” J. Sound Vib.
,
203
(5
), pp. 815
–832
.10.1006/jsvi.1996.093513.
Özkaya
,
E.
, and
Pakdemirli
,
M.
, 2000
, “
Vibrations of an Axially Accelerating Beam With Small Flexural Stiffness
,” J. Sound Vib.
,
234
(3
), pp. 521
–535
.10.1006/jsvi.2000.289014.
Yang
,
K. J.
,
Hong
,
K. S.
, and
Matsuno
,
F.
, 2005
, “
Energy-Based Control of Axially Translating Beams: Varying Tension, Varying Speed, and Disturbance Adaptation
,” IEEE Trans. Control Syst. Technol.
,
13
, pp. 1045
–1054
.10.1109/TCST.2005.85436815.
Ponomareva
,
S. V.
, and
van Horssen
,
W. T.
, 2007
, “
On Transversal Vibrations of an Axially Moving String With a Time-Varying Velocity
,” Nonlinear Dyn.
,
50
(1–2
), pp. 315
–323
.10.1007/s11071-006-9160-716.
Ghayesh
,
M. H.
, and
Amabili
,
M.
, 2013
, “
Steady-State Transverse Response of an Axially Moving Beam With Time-Dependent Axial Speed
,” Int. J. Non-Linear Mech.
,
49
, pp. 40
–49
.10.1016/j.ijnonlinmec.2012.08.00317.
Mao
,
X. Y.
,
Ding
,
H.
, and
Chen
,
L. Q.
, 2017
, “
Dynamics of a Super-Critically Axially Moving Beam With Parametric and Forced Resonance
,” Nonlinear Dyn.
,
89
(2
), pp. 1475
–1487
.10.1007/s11071-017-3529-718.
Öz
,
H. R.
, and
Pakdemirli
,
M.
, 1999
, “
Vibrations of an Axially Moving Beam With Time-Dependent Velocity
,” J. Sound Vib.
,
227
(2
), pp. 239
–257
.10.1006/jsvi.1999.224719.
Ghayesh
,
M. H.
, 2012
, “
Coupled Longitudinal–Transverse Dynamics of an Axially Accelerating Beam
,” J. Sound Vib.
,
331
(23
), pp. 5107
–5124
.10.1016/j.jsv.2012.06.01820.
Bağdatli
,
S. M.
,
Özkaya
,
E.
, and
Öz
,
H. R.
, 2013
, “
Dynamics of Axially Accelerating Beams With Multiple Supports
,” Nonlinear Dyn.
,
74
(1–2
), pp. 237
–255
.10.1007/s11071-013-0961-121.
Sahoo
,
B.
,
Panda
,
L. N.
, and
Pohit
,
G.
, 2014
, “
Stability and Bifurcation Analysis of an Axially Accelerating Beam
,” Vib. Eng. Technol. Mach.
,
23
, pp. 915
–928
.10.1007/978-3-319-09918-7_8122.
Sahoo
,
B.
,
Panda
,
L. N.
, and
Pohit
,
G.
, 2016
, “
Combination, Principal Parametric and Internal Resonances of an Accelerating Beam Under Two Frequency Parametric Excitation
,” Int. J. Non-Linear Mech.
,
78
, pp. 35
–44
.10.1016/j.ijnonlinmec.2015.09.01723.
Zhang
,
D. B.
,
Tang
,
Y. Q.
, and
Chen
,
L. Q.
, 2017
, “
Irregular Instability Boundaries of Axially Accelerating Viscoelastic Beams With 1:3 Internal Resonance
,” Int. J. Mech. Sci.
,
133
, pp. 535
–543
.10.1016/j.ijmecsci.2017.08.05224.
Lv
,
H. W.
,
Li
,
L.
, and
Li
,
Y. H.
, 2018
, “
Non-Linearly Parametric Resonances of an Axially Moving Viscoelastic Sandwich Beam With Time-Dependent Velocity
,” Appl. Math. Modell.
,
53
, pp. 83
–105
.10.1016/j.apm.2017.05.04825.
Parker
,
P. G.
, and
Lin
,
Y.
, 2001
, “
Parametric Instability of Axially Moving Media Subjected to Multifrequency Tension and Speed Fluctuations
,” ASME J. Appl. Mech.
,
68
(1
), pp. 49
–57
.10.1115/1.134391426.
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
(1
), pp. 118
–131
.10.1016/j.ijnonlinmec.2006.09.00627.
Lv
,
H. W.
,
Li
,
Y. H.
,
Li
,
L.
, and
Liu
,
Q. K.
, 2014
, “
Transverse Vibration of Viscoelastic Sandwich Beam With Time-Dependent Axial Tension and Axially Varying Moving Velocity
,” Appl. Math. Modell.
,
38
(9–10
), pp. 2558
–2585
.10.1016/j.apm.2013.10.05528.
Chen
,
L. Q.
,
Tang
,
Y. Q.
, and
Zu
,
J. W.
, 2014
, “
Nonlinear Transverse Vibration of Axially Accelerating Strings With Exact Internal Resonances and Longitudinally Varying Tensions
,” Nonlinear Dyn.
,
76
(2
), pp. 1443
–1468
.10.1007/s11071-013-1220-129.
Tang
,
Y. Q.
,
Zhang
,
D. B.
, and
Gao
,
J. M.
, 2016
, “
Parametric and Internal Resonance of Axially Accelerating Viscoelastic Beams With the Recognition of Longitudinally Varying Tensions
,” Nonlinear Dyn.
,
83
(1–2
), pp. 401
–418
.10.1007/s11071-015-2336-230.
Tang
,
Y. Q.
,
Zhang
,
Y. X.
, and
Yang
,
X. D.
, 2018
, “
On Parametric Instability Boundaries of Axially Moving Beams With Internal Resonance
,” Acta Mech. Solida Sin.
,
31
(4
), pp. 470
–483
.10.1007/s10338-018-0032-831.
Sahoo
,
B.
, 2019
, “
Nonlinear Dynamics of a Viscoelastic Traveling Beam With Time-Dependent Axial Velocity and Variable Axial Tension
,” Nonlinear Dyn.
,
97
(1
), pp. 269
–296
.10.1007/s11071-019-04969-932.
Kim
,
C. W.
,
Park
,
H.
, and
Hong
,
K. S.
, 2005
, “
Boundary Control of Axially Moving Continua: Application to a Zinc Galvanizing Line
,” Int. J. Control, Autom., Syst.
,
3
, pp. 601
–611
.https://www.researchgate.net/publication/24137660333.
Li
,
P.
, and
Chen
,
H.
, 2013
, “
Vibration Analysis of Steel Strip in Continuous Hot-Dip Galvanizing Process
,” J. Appl. Math. Phys.
,
1
(6
), pp. 31
–36
.10.4236/jamp.2013.1600734.
Li
,
H. Y.
,
Li
,
J.
, and
Liu
,
Y. J.
, 2015
, “
Internal Resonance of an Axially Moving Unidirectional Plate Partially Immersed in Fluid Under Foundation Displacement Excitation
,” J. Sound Vib.
,
358
, pp. 124
–141
.10.1016/j.jsv.2015.07.03035.
Li, J., Guo, X. H., Luo, J., Li, H. Y., and Wang, Y. Q., 2013, “Analytical Study on Inherent Properties of a Unidirectional Vibrating Steel Strip Partially Immersed in Fluid,”
Shock Vib
., 20(4), pp. 793–807.10.3233/SAV-13078536.
Kerboua, Y., Lakis, A. A., Thomas, M., and Marcouiller, L., 2008, “Vibration Analysis of Rectangular Plates Coupled With Fluid,”
Appl. Math. Modell.
s, 32(12), pp. 2570–2586.10.1016/j.apm.2007.09.004Copyright © 2021 by ASME
You do not currently have access to this content.
