Abstract
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.
Issue Section:
Research Papers
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.
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
.Copyright © 2012
by American Society of Mechanical Engineers
You do not currently have access to this content.
