The three-dimensional (3D) nonlinear dynamics of an axially accelerating beam is examined numerically taking into account all of the longitudinal, transverse, and lateral displacements and inertia. Hamilton’s principle is employed in order to derive the nonlinear partial differential equations governing the longitudinal, transverse, and lateral motions. These equations are transformed into a set of nonlinear ordinary differential equations by means of the Galerkin discretization technique. The nonlinear global dynamics of the system is then examined by time-integrating the discretized equations of motion. The results are presented in the form of bifurcation diagrams of Poincaré maps, time histories, phase-plane portraits, Poincaré sections, and fast Fourier transforms (FFTs).

References

References
1.
Chen
,
L.-Q.
, and
Wang
,
B.
,
2009
, “
Stability of Axially Accelerating Viscoelastic Beams: Asymptotic Perturbation Analysis and Differential Quadrature Validation
,”
Eur. J. Mech.-A/Solids
,
28
(
4
), pp.
786
791
.10.1016/j.euromechsol.2008.12.002
2.
Ding
,
H.
, and
Chen
,
L.-Q.
,
2010
, “
Galerkin Methods for Natural Frequencies of High-Speed Axially Moving Beams
,”
J. Sound Vib.
,
329
(
17
), pp.
3484
3494
.10.1016/j.jsv.2010.03.005
3.
Kong
,
L.
, and
Parker
,
R. G.
,
2004
, “
Approximate Eigensolutions of Axially Moving Beams With Small Flexural Stiffness
,”
J. Sound Vib.
,
276
(
1–2
), pp.
459
469
.10.1016/j.jsv.2003.11.027
4.
Stylianou
,
M.
, and
Tabarrok
,
B.
,
1994
, “
Finite Element Analysis of an Axially Moving Beam, Part I: Time Integration
,”
J. Sound Vib.
,
178
(
4
), pp.
433
453
.10.1006/jsvi.1994.1497
5.
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.1996.0935
6.
Pakdemirli
,
M.
,
Ulsoy
,
A. G.
, and
Ceranoglu
,
A.
,
1994
, “
Transverse Vibration of an Axially Accelerating String
,”
J. Sound Vib.
,
169
(
2
), pp.
179
196
.10.1006/jsvi.1994.1012
7.
Ding
,
H.
,
Yan
,
Q.-Y.
, and
Zu
,
J. W.
,
2014
, “
Chaotic Dynamics of an Axially Accelerating Viscoelastic Beam in the Supercritical Regime
,”
Int. J. Bifurcation Chaos Appl. Sci. Eng.
,
24
(
5
), p.
1450062
.10.1142/S021812741450062X
8.
Yan
,
Q.-Y.
,
Ding
,
H.
, and
Chen
,
L.-Q.
,
2014
, “
Periodic Responses and Chaotic Behaviors of an Axially Accelerating Viscoelastic Timoshenko Beam
,”
Nonlinear Dyn.
,
78
(
2
), pp.
1577
1591
.10.1007/s11071-014-1535-6
9.
Ghayesh
,
M. H.
,
Kafiabad
,
H. A.
, and
Reid
,
T.
,
2012
, “
Sub- and Super-Critical Nonlinear Dynamics of a Harmonically Excited Axially Moving Beam
,”
Int. J. Solids Struct.
,
49
(
1
), pp.
227
243
.10.1016/j.ijsolstr.2011.10.007
10.
Ghayesh
,
M. H.
,
2011
, “
Nonlinear Forced Dynamics of an Axially Moving Viscoelastic Beam With an Internal Resonance
,”
Int. J. Mech. Sci.
,
53
(
11
), pp.
1022
1037
.10.1016/j.ijmecsci.2011.08.010
11.
Ghayesh
,
M. H.
,
Yourdkhani
,
M.
,
Balar
,
S.
, and
Reid
,
T.
,
2010
, “
Vibrations and Stability of Axially Traveling Laminated Beams
,”
Appl. Math. Comput.
,
217
(
2
), pp.
545
556
.10.1016/j.amc.2010.05.088
12.
Sze
,
K. Y.
,
Chen
,
S. H.
, and
Huang
,
J. L.
,
2005
, “
The Incremental Harmonic Balance Method for Nonlinear Vibration of Axially Moving Beams
,”
J. Sound Vib.
,
281
(
3–5
), pp.
611
626
.10.1016/j.jsv.2004.01.012
13.
Huang
,
J. L.
,
Su
,
R. K. L.
,
Li
,
W. H.
, and
Chen
,
S. H.
,
2011
, “
Stability and Bifurcation of an Axially Moving Beam Tuned to Three-to-One Internal Resonances
,”
J. Sound Vib.
,
330
(
3
), pp.
471
485
.10.1016/j.jsv.2010.04.037
14.
Wickert
,
J. A.
,
1992
, “
Non-Linear Vibration of a Traveling Tensioned Beam
,”
Int. J. Non-Linear Mech.
,
27
(
3
), pp.
503
517
.10.1016/0020-7462(92)90016-Z
15.
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.2006.09.006
16.
Chen
,
L.-Q.
, and
Yang
,
X.-D.
,
2006
, “
Vibration and Stability of an Axially Moving Viscoelastic Beam With Hybrid Supports
,”
Eur. J. Mech.-A/Solids
,
25
(
6
), pp.
996
1008
.10.1016/j.euromechsol.2005.11.010
17.
Riedel
,
C. H.
, and
Tan
,
C. A.
,
2002
, “
Coupled, Forced Response of an Axially Moving Strip With Internal Resonance
,”
Int. J. Non-Linear Mech.
,
37
(
1
), pp.
101
116
.10.1016/S0020-7462(00)00100-1
18.
Ghayesh
,
M.
,
2012
, “
Stability and Bifurcations of an Axially Moving Beam With an Intermediate Spring Support
,”
Nonlinear Dyn.
,
69
(
1–2
), pp.
193
210
.10.1007/s11071-011-0257-2
19.
Suweken
,
G.
, and
Van Horssen
,
W. T.
,
2003
, “
On the Transversal Vibrations of a Conveyor Belt With a Low and Time-Varying Velocity. Part II: The Beam-Like Case
,”
J. Sound Vib.
,
267
(
5
), pp.
1007
1027
.10.1016/S0022-460X(03)00219-0
20.
Öz
,
H. R.
,
Pakdemirli
,
M.
, and
Özkaya
,
E.
,
1998
, “
Transition Behaviour From String to Beam for an Axially Accelerating Material
,”
J. Sound Vib.
,
215
(
3
), pp.
571
576
.10.1006/jsvi.1998.1572
21.
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.2000.2890
22.
Pakdemirli
,
M.
, and
Öz
,
H. R.
,
2008
, “
Infinite Mode Analysis and Truncation to Resonant Modes of Axially Accelerated Beam Vibrations
,”
J. Sound Vib.
,
311
(
3–5
), pp.
1052
1074
.10.1016/j.jsv.2007.10.003
23.
Xu
,
G. Y.
, and
Zhu
,
W. D.
,
2010
, “
Nonlinear and Time-Varying Dynamics of High-Dimensional Models of a Translating Beam With a Stationary Load Subsystem
,”
ASME J. Vib. Acoust.
,
132
(
6
), p.
061012
.10.1115/1.4000464
24.
Ghayesh
,
M. H.
,
2012
, “
Subharmonic Dynamics of an Axially Accelerating Beam
,”
Arch. Appl. Mech.
,
82
(
9
), pp.
1169
1181
.10.1007/s00419-012-0609-5
25.
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.2012.06.018
26.
Öz
,
H. R.
,
Pakdemirli
,
M.
, and
Boyacı
,
H.
,
2001
, “
Non-Linear Vibrations and Stability of an Axially Moving Beam With Time-Dependent Velocity
,”
Int. J. Non-Linear Mech.
,
36
(
1
), pp.
107
115
.10.1016/S0020-7462(99)00090-6
27.
Pakdemirli
,
M.
,
2001
, “
A General Solution Procedure for Coupled Systems With Arbitrary Internal Resonances
,”
Mech. Res. Commun.
,
28
(
6
), pp.
617
622
.10.1016/S0093-6413(02)00213-6
28.
Pakdemirli
,
M.
, and
Boyacı
,
H.
,
2003
, “
Non-Linear Vibrations of a Simple–Simple Beam With a Non-Ideal Support in Between
,”
J. Sound Vib.
,
268
(
2
), pp.
331
341
.10.1016/S0022-460X(03)00363-8
29.
Pakdemirli
,
M.
, and
Özkaya
,
E.
,
2003
, “
Three-to-One Internal Resonances in a General Cubic Non-Linear Continuous System
,”
J. Sound Vib.
,
268
(
3
), pp.
543
553
.10.1016/S0022-460X(03)00364-X
30.
Ghayesh
,
M.
,
2014
, “
Nonlinear Size-Dependent Behaviour of Single-Walled Carbon Nanotubes
,”
Appl. Phys. A
,
117
(
3
), pp.
1393
1399
.10.1007/s00339-014-8561-6
31.
Ghayesh
,
M. H.
,
2012
, “
Nonlinear Dynamic Response of a Simply-Supported Kelvin–Voigt Viscoelastic Beam, Additionally Supported by a Nonlinear Spring
,”
Nonlinear Anal.: Real World Appl.
,
13
(
3
), pp.
1319
1333
.10.1016/j.nonrwa.2011.10.009
32.
Chen
,
L.-Q.
,
Ding
,
H.
, and
Lim
,
C. W.
,
2012
, “
Principal Parametric Resonance of Axially Accelerating Viscoelastic Beams: Multi-Scale Analysis and Differential Quadrature Verification
,”
Shock Vib.
,
19
(
4
), pp.
527
543
.10.1155/2012/948459
33.
Chen
,
L.-Q.
,
Tang
,
Y.-Q.
, and
Lim
,
C. W.
,
2010
, “
Dynamic Stability in Parametric Resonance of Axially Accelerating Viscoelastic Timoshenko Beams
,”
J. Sound Vib.
,
329
(
5
), pp.
547
565
.10.1016/j.jsv.2009.09.031
34.
Chen
,
L.-Q.
, and
Tang
,
Y.-Q.
,
2011
, “
Combination and Principal Parametric Resonances of Axially Accelerating Viscoelastic Beams: Recognition of Longitudinally Varying Tensions
,”
J. Sound Vib.
,
330
(
23
), pp.
5598
5614
.10.1016/j.jsv.2011.07.012
35.
Ghayesh
,
M. H.
,
2011
, “
On the Natural Frequencies, Complex Mode Functions, and Critical Speeds of Axially Traveling Laminated Beams: Parametric Study
,”
Acta Mech. Solida Sin.
,
24
(
4
), pp.
373
382
.10.1016/S0894-9166(11)60038-4
36.
Ghayesh
,
M. H.
,
Kazemirad
,
S.
, and
Darabi
,
M. A.
,
2011
, “
A General Solution Procedure for Vibrations of Systems With Cubic Nonlinearities and Nonlinear/Time-Dependent Internal Boundary Conditions
,”
J. Sound Vib.
,
330
(
22
), pp.
5382
5400
.10.1016/j.jsv.2011.06.001
37.
Gholipour
,
A.
,
Farokhi
,
H.
, and
Ghayesh
,
M.
,
2014
, “
In-Plane and Out-of-Plane Nonlinear Size-Dependent Dynamics of Microplates
,”
Nonlinear Dyn.
,
79
(
3
), pp.
1771
1785
.1007/s11071-014-1773-7
38.
Ghayesh
,
M. H.
, and
Farokhi
,
H.
,
2015
, “
Nonlinear Dynamics of Microplates
,”
Int. J. Eng. Sci.
,
86
, pp.
60
73
.10.1016/j.ijengsci.2014.10.004
39.
Farokhi
,
H.
, and
Ghayesh
,
M. H.
,
2015
, “
Nonlinear Dynamical Behaviour of Geometrically Imperfect Microplates Based on Modified Couple Stress Theory
,”
Int. J. Mech. Sci.
,
90
, pp.
133
144
.10.1016/j.ijmecsci.2014.11.002
You do not currently have access to this content.