In this paper, an initial-boundary value problem for a linear-homogeneous axially moving tensioned beam equation is considered. One end of the beam is assumed to be simply-supported and to the other end of the beam a spring and a dashpot are attached, where the damping generated by the dashpot is assumed to be small. In this paper only boundary damping is considered. The problem can be used as a simple model to describe the vertical vibrations of a conveyor belt, for which the velocity is assumed to be constant and relatively small compared to the wave speed. A multiple time-scales perturbation method is used to construct formal asymptotic approximations of the solutions, and it is shown how different oscillation modes are damped.

References

References
1.
Suweken
,
G.
, and
van Horssen
,
W. T.
, 2003, “
On the Transversal Vibrations of a Conveyor Belt With a Low and Time Varying Velocity. Part I: The String-Like Case
,”
J. Sound Vib.
,
264
(
1
), pp.
117
133
.
2.
Ponomareva
,
S. V.
, and
van Horssen
,
W. T.
, 2007, “
On the Transversal Vibrations of an Axially Moving String With a Time-Varying Velocity
,”
Nonlinear Dyn.
,
50
(
1–2
), pp.
315
323
.
3.
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
.
4.
Ponomareva
,
S. V.
, and
van Horssen
,
W. T.
, 2009, “
On the Transversal Vibrations of an Axially Moving Continuum With a Time-Varying Velocity: Transient from String to Beam Behavior
,”
J. Sound Vib.
,
325
(
4–5
), pp.
959
973
.
5.
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
.
6.
Zhu
,
W. D.
,
Ni
,
J.
, and
Huang
,
J.
, 2001, “
Active Control of Translating Media With Arbitrarily Varying Length
,”
ASME J. Vibr. Acoust.
,
123
(
3
), pp.
347
358
.
7.
Zhu
,
W. D.
, and
Chen
,
Y.
, 2005, “
Forced Response of Translating Media With Variable Length and Tension: Application to High-Speed Elevators
,”
Proc. Inst. Mech. Eng. Part K: J. Multi-Body Dyn.
,
219
(
1
), pp.
35
53
.
8.
Zhu
,
W. D.
, and
Chen
,
Y.
, 2006, “
Theoretical and Experimental Investigation of Elevator Cable Dynamics and Control
,”
ASME J. Vibr. Acoust.
,
128
(
1
), pp.
66
78
.
9.
Darmawijoyo
, and
van Horssen
,
W. T.
, 2002, “
On Boundary Damping for a Weakly Nonlinear Wave Equation
,”
Nonlinear Dyn.
,
30
(
2
), pp.
179
191
.
10.
Mahalingam
,
S.
, 1957, “
Transverse Vibrations of Power Transmission Chains
,”
Br. J. Appl. Phys.
,
8
(
4
), pp.
145
148
.
11.
Kuiper
,
G. L.
, and
Metrikine
,
A. V.
, 2004, “
On Stability of a Clamped-Pinned Pipe Conveying Fluid
,”
Heron
,
49
(
3
), pp.
211
232
.
12.
Öz
,
H. R.
, and
Boyaci
,
H.
, 2000, “
Transverse Vibrations of Tensioned Pipes Conveying Fluid With Time-Dependent Velocity
,”
J. Sound Vib.
,
236
(
2
), pp.
259
276
.
13.
Chen
,
L. Q.
, 2005, “
Analysis and Control of Transverse Vibrations of Axially Moving Strings
,”
Appl. Mech. Rev.
,
58
(
2
), pp.
91
116
.
14.
Ulsoy
,
A. G.
,
Mote
,
C. D.
, Jr.
, and
Szymni
,
R.
, 1978, “
Principal Developments in Band Saw Vibration and Stability Research
,”
Holz Roh-Werkst.
,
36
(
7
), pp.
273
280
.
15.
Xu
,
M.
, 2006, “
Free Transverse Vibrations of Nano-to-Micron Scale Beams
,”
Proc. R. Soc. London
,
462
(
2074
), pp.
2977
2995
.
16.
van Horssen
,
W. T.
, and
Ponomareva
,
S. V.
, 2005, “
On the Construction of the Solution of an Equation Describing an Axially Moving String
,”
J. Sound Vib.
,
287
(
1–2
), pp.
359
366
.
17.
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
.
18.
Wickert
,
J. A.
, and
Mote
,
C. D.
, Jr.
, 1990, “
Classical Vibration Analysis of Axially Moving Continua
,”
ASME J. Appl. Mech.
,
57
(
3
), pp.
738
744
.
19.
Chakraborty
,
G.
,
Mallik
,
A. K.
, and
Hatwal
,
H.
, 1999, “
Non-Linear Vibration of a Travelling Beam
,”
Int. J. Non-Linear Mech.
,
34
(
4
), pp.
655
670
.
20.
Thurman
,
A. L.
, and
Mote
,
C. D.
, Jr.
, 1969, “
Free, Periodic, Nonlinear Oscillation of an Axially Moving Strip
,”
ASME J. Appl. Mech.
,
36
, pp.
83
91
.
21.
Wickert
,
J. A.
, 1992, “
Non-Linear Vibration of a Travelling Tensioned Beam
,”
Int. J. Non-Linear Mech.
,
27
(
3
), pp.
503
517
.
22.
Pellicano
,
F.
, and
Vestroni
,
F.
, 2000, “
Nonlinear Dynamics and Bifurcations of an Axially Moving Beam
,”
ASME J. Vibr. Acoust.
,
122
(
1
), pp.
21
30
.
23.
Miranker
,
W. L.
, 1960, “
The Wave Equation in a Medium in Motion
,”
IBM J. Res. Dev.
,
4
(
1
), pp.
36
42
.
24.
Spelsberg-Korspeter
,
G.
,
Kirrilov
,
O. N.
, and
Hagedorn
,
P.
, 2008, “
Modeling and Stability Analysis of an Axially Moving Beam With Frictional Contact
,”
ASME J. Appl. Mech.
,
75
(
3
), p.
0310011
.
25.
Chen
,
L. Q.
, and
Ding
,
H.
, 2010, “
Steady-State Transverse Response in Coupled Planar Vibration of Axially Moving Viscoelastic Beams
,”
ASME J. Vibr. Acoust.
,
132
(
1
), p.
0110091
.
26.
Bağdatli
,
S. M.
,
Özkaya
,
E.
, and
Öz
,
H. R.
, 2011, “
Dynamics of Axially Accelerating Beams With an Intermediate Support
,”
ASME J. Vibr. Acoust.
,
133
(
3
), p.
0310131
.
27.
Nayfeh
,
A. H.
, 2000,
Perturbation Methods
,
John Wiley and Sons,
New York.
28.
Kevorkian
,
J.
, and
Cole
,
J. D.
, 1996,
Multiple Scale and Singular Perturbation Methods
,
Springer-Verlag
,
New York.
29.
Wickert
,
J. A.
, and
Mote
,
C. D.
, Jr.
, 1989, “
On the Energetics of Axially Moving Continua
,”
J. Acoust. Soc. Am.
,
85
(
3
), pp.
1365
1368
.
30.
Zhu
,
W. D.
, and
Ni
,
J.
, 2000, “
Energetics and Stability of Translating Media With an Arbitrary Varying Length
,”
ASME J. Vibr. Acoust.
,
122
(
3
), pp.
295
304
.
31.
Chen
,
L. Q.
, 2006, “
The Energetics and the Stability of Axially Moving Strings Undergoing Planar Motion
,”
Int. J. Eng. Sci.
,
44
(
18–19
), pp.
1346
1352
.
32.
Chen
,
L. Q.
, and
Zu
,
J. W.
, 2004, “
Energetics and Conserved Functional of Axially Moving Materials Undergoing Transverse Nonlinear Vibration
,”
ASME J. Vibr. Acoust.
,
126
(
3
), pp.
452
455
.
33.
Haberman
,
R.
, 2004,
Applied Partial Differential Equations
,
Pearson Prentice-Hall
,
New Jersey
.
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