Vibration control can improve the performance of many axially moving material systems (e.g., web handling machines and tape drives). Researchers have used Lyapunov analysis to develop vibration stabilizing controllers for distributed parameter models of axially moving material systems. Both the material and regular time derivatives have been used in these analyses despite the fact that they give different results. This paper proves that for a pinned axially moving string model: (i) Lyapunov stability analysis using the material derivative incorrectly predicts that a time-varying functional is constant and (ii) neglect of the coupled domain in boundary control analysis is ill posed and incorrectly predicts bounded forced response and exponentially decaying transients.
1.
Wang
, J.
, and Li
, Q.
, 2004, “Active Vibration Control Methods of Axially Moving Materials—A Review
,” J. Vib. Control
1077-5463, 10
(4
), pp. 475
–491
.2.
Kim
, C. W.
, Hong
, K. S.
, and Park
, H.
, 2005, “Boundary Control of an Axially Moving String: Actuator Dynamics Included
,” J. Mech. Sci. Technol.
, 19
(1
), pp. 40
–50
.3.
Hong
, K. S.
, Kim
, C. W.
, and Hong
, K. T.
, 2004, “Boundary Control of an Axially Moving Belt System in a Thin-Metal Production Line
,” Int. J. Control Autom.
, 2
(1
), pp. 55
–67
.4.
Choi
, J. Y.
, Hong
, K. S.
, and Yang
, K. J.
, 2004, “Exponential Stabilization of an Axially Moving Tensioned Strip by Passive Damping and Boundary Control
,” J. Vib. Control
1077-5463, 10
(5
), pp. 661
–682
.5.
Yang
, K. J.
, Hong
, K. S.
, and Matsuno
, F.
, 2004, “Robust Adaptive Boundary Control of an Axially Moving String Under a Spatio Temporally Varying Tension
,” J. Sound Vib.
0022-460X, 273
(4–5
), pp. 1007
–1029
.6.
Fung
, R. F.
, Wu
, J. W.
, and Lu
, P. Y.
, 2002,“Adaptive Boundary Control of an Axially Moving String System
,” ASME J. Vibr. Acoust.
0739-3717, 124
(3
), pp. 435
–440
.7.
Fung
, R. F.
, Chou
, J. H.
, and Kuo
, Y. L.
, 2002, “Optimal Boundary Control of an Axially Moving Material System
,” ASME J. Dyn. Syst., Meas., Control
0022-0434, 124
(1
), pp. 55
–61
.8.
Fung
, R. F.
, and Tseng
, C. C.
, 1999, “Boundary Control of an Axially Moving String via Lyapunov Method
,” ASME J. Dyn. Syst., Meas., Control
0022-0434, 121
(1
), pp. 105
–110
.9.
Fung
, R. F.
, Wu
, J. W.
, and Wu
, S. L.
, 1999, “Stabilization of an Axially Moving String by Nonlinear Boundary Feedback
,” ASME J. Dyn. Syst., Meas., Control
0022-0434, 121
(1
), pp. 117
–121
.10.
Lee
, S. Y.
, and Mote
, C. D.
, 1996, “Vibration Control of an Axially Moving String by Boundary Control
,” ASME J. Dyn. Syst., Meas., Control
0022-0434, 118
(1
), pp. 66
–74
.11.
Lee
, S. Y.
, and Mote
, C. D.
, 1999, “Wave Characteristics and Vibration Control of Translating Beams by Optimal Boundary Damping
,” ASME J. Vibr. Acoust.
0739-3717, 121
(1
), pp. 18
–25
.12.
Shahruz
, S. M.
, 1998, “Boundary Control of the Axially Moving Kirchhoff String
,” Automatica
0005-1098, 34
(10
), pp. 1273
–1277
.13.
Shahruz
, S. M.
, and Kurmaji
, D. A.
, 1997, “Vibration Suppression of a Nonlinear Axially Moving String by Boundary Control
,” J. Sound Vib.
0022-460X, 201
(1
), pp. 145
–152
.14.
Chao
, P. C. P.
, and Lai
, C. L.
, 2003, “Boundary Control of an Axially Moving String via Fuzzy Sliding-Mode Control and Fuzzy Neural Network Methods
,” J. Sound Vib.
0022-460X, 262
(4
), pp. 795
–813
.15.
Renshaw
, A.
, Rahn
, C.
, Wickert
, J.
, and Mote
, C. D.
, 1998, “Energy and Conserved Functionals for Axially Moving Materials
,” ASME J. Vibr. Acoust.
0739-3717, 120
(2
), pp. 634
–636
.16.
de Queiroz
, M.
, Dawson
, D.
, Rahn
, C.
, and Zhang
, F.
, 1999, “Adaptive Vibration Control of an Axially Moving String
,” ASME J. Vibr. Acoust.
0739-3717, 121
, pp. 41
–49
.17.
Li
, Y.
, and Rahn
, C.
, 2000, “Adaptive Vibration Isolation for Axially Moving Beams
,” IEEE/ASME Trans. Mechatron.
1083-4435, 5
(4
), pp. 491
–428
.18.
Li
, Y.
, Aron
, D.
, and Rahn
, C.
, 2002, “Adaptive Vibration Isolation for Axially Moving Strings: Theory and Experiment
,” Automatica
0005-1098, 38
(3
), pp. 379
–390
.19.
Huang
, J. S.
, Chao
, P. C. P.
, Fung
, R. F.
, Lai
, Cheng-Liang
, 2003, “Parametric Control of an Axially Moving String via Fuzzy Sliding-Mode and Fuzzy Neural Network Methods
,” J. Sound Vib.
0022-460X, 264
(1
), pp. 177
–201
.20.
Lee
, S. Y.
, and Mote
, C. D.
, 1997, “A Generalized Treatment of the Energetics of Translating Continua. 1. Strings and Second Order Tensioned Pipes
,” J. Sound Vib.
0022-460X, 204
(5
), pp. 717
–734
.21.
Nagarkatti
, S.
, Zhang
, F.
, Costic
, B.
, Dawson
, D.
, and Rahn
, C.
, 2002, “Speed Tracking and Transverse Vibration Control of an Axially Accelerating Web
,” Mech. Syst. Signal Process.
0888-3270, 16
(2/3
), pp. 337
–356
.22.
Rahn
, C.
, Mechatronic Control of Distributed Noise and Vibration: A Lyapunov Approach
, Springer-Verlag
, Berlin
, 2001.23.
Wickert
, J. A.
, and Mote
, C. D.
, Jr., 1989, “On the Energetics of Axially Moving Continua
,” J. Acoust. Soc. Am.
0001-4966, 85
(3
), pp. 1365
–1368
.24.
Zhu
, W. D.
, 2002, “Control Volume and System Formulations for Translating Media and Stationary Media With Moving Boundaries
,” J. Sound Vib.
0022-460X, 254
(1
), pp. 189
–201
.25.
Wickert
, J. A.
, and Mote
, C. D.
, Jr., 1990, “Classical Vibration Analysis of Axially-Moving Continua
,” ASME J. Appl. Mech.
0021-8936, 57
(3
), pp. 738
–744
.Copyright © 2006
by American Society of Mechanical Engineers
You do not currently have access to this content.
