The general control laws for pointwise controllers to dissipate vibratory energies of translating beams and strings with arbitrarily varying length are presented. Special domain and boundary control laws that can be easily implemented result as a special case. Sufficient conditions for uniform stability and uniform exponential stability of controlled systems are established via Lyapunov stability criteria. Numerical simulations demonstrate the effectiveness of the active controllers in stabilizing translating media during both extension and retraction. Optimal gains leading to the fastest rates of decay of vibratory energies of controlled systems are identified. It is shown that under the optimal control gains, translating media can be completely stabilized during extension and retraction.
Issue Section:
Technical Papers
1.
Mote
, Jr., C. D.
, 1971
, “Dynamic Stability of Axially Moving Materials
,” Shock Vib. Dig.
, 4
, No. 4
, pp. 2
–11
.2.
Yuh
, J.
, and Young
, T.
, 1991
, “Dynamic Modeling of an Axially Moving Beam in Rotation: Simulation and Experiment
,” ASME J. Dyn. Syst., Meas., Control
, 113
, pp. 34
–40
.3.
Tsuchiya
, K.
, 1983
, “Dynamics of a Spacecraft During Extension of Flexible Appendages
,” AIAA J.
, 6
, No. 2
, pp. 100
–103
.4.
Terumichi
, Y.
, Ohtsuka
, M.
, Yoshizawa
, M.
, Fukawa
, Y.
, and Tsujioka
, Y.
, 1997
, “Nonstationary Vibrations of a String with Time-Varying Length and a Mass-Spring System Attached at the Lower End
,” Nonlinear Dyn.
, 12
, No. 1
, pp. 39
–55
.5.
Misra
, A. K
, and Modi
, V. J.
, 1985
, “Depolyment and Retrieval of Shuttle Supported Tethered Satellite
,” AIAA Journal of Guidance, Control, and Dynamics
, 5
, No. 3
, pp. 278
–285
.6.
Mansfield
, L.
, and Simmonds
, J. G.
, 1987
, “The Reverse Spaghetti Problem: Drooping Motion of an Elastica Issuing from a Horizontal Guide
,” ASME J. Appl. Mech.
, 54
, pp. 147
–150
.7.
Stolte
, J.
, and Benson
, R. C.
, 1992
, “Dynamic Deflection of Paper Emerging from a Channel
,” ASME J. Vibr. Acoust.
, 114
, No. 2
, pp. 187
–193
.8.
Stolte
, J.
, and Benson
, R. C.
, 1993
, “An Extending Dynamic Elastica—Impact with a Surface
,” ASME J. Vibr. Acoust.
, 115
, No. 3
, pp. 308
–313
.9.
Sato
, K.
, and Sakawa
, Y.
, 1988
, “Modelling and Control of a Flexible Rotary Crane
,” Int. J. Control
, 48
, No. 5
, pp. 2085
–2105
.10.
Kumaniecka
, K.
, and Niziok
, J.
, 1994
, “Dynamic Stability of a Rope with Slow Variability of the Parameters
,” J. Sound Vib.
178
, No. 2
, pp. 211
–226
.11.
Carrier
, G. F.
, 1949
, “The Spaghetti Problem
,” Am. Math. Monthly
, 56
, pp. 669
–672
.12.
Tabarrok
, B.
, Leech
, C. M.
, and Kim
, Y. I.
, 1974
, “On the Dynamics of an Axially Moving Beam
,” J. Franklin Inst.
, 297
, No. 3
, pp. 201
–220
.13.
Zajaczkowski
, J.
, and Lipinski
, J.
, 1979
, “Instability of the Motion of a Beam of Periodically Varying Length
,” J. Sound Vib.
63
, pp. 9
–18
.14.
Zajaczkowski
, J.
, and Yamada
, G.
, 1980
, “Further Results on Instability of the Motion of a Beam of Periodically Varying Length
,” J. Sound Vib.
, 68
, No. 2
, pp. 173
–180
.15.
Zhu
, W. D.
, and Ni
, J.
, 2000
, “Energetics and Stability of Translating Media with an Arbitrarily Varying Length
,” ASME J. Vibr. Acoust.
, 122
, pp. 295
–304
.16.
Vu-Quoc
, L.
, and Li
, S.
, 1995
, “Dynamics of Sliding Geometrically-Exact Beams: Large Angle Maneuver and Parametric Resonance
,” Comput. Methods Appl. Mech. Eng.
, 120
, pp. 65
–118
.17.
Yang
, B.
, and Mote
, Jr., C. D.
, 1991
, “Active Vibration Control of the Axially Moving String in the S Domain
,” ASME J. Appl. Mech.
, 58
, pp. 189
–196
.18.
Zhu
, W. D.
, Mote
, Jr., C. D.
, and Guo
, B. Z.
, 1997
, “Asymptotic Distribution of Eigenvalues of a Constrained Translating String
,” ASME J. Appl. Mech.
, 64
, pp. 613
–619
.19.
Zhu
, W. D.
, Guo
, B. Z.
, and Mote
, Jr., C. D.
, 2000
, “Stabilization of a Translating Tensioned Beam Through a Pointwise Control Force
,” ASME J. Dyn. Syst., Meas., Control
, 122
, pp. 322
–331
.20.
Yang
, B.
, and Mote
, Jr., C. D.
, 1990
, “Vibration Control of Band Saws: Theory and Experiment
,” Wood Sci.
, 24
, pp. 355
–373
.21.
Huang
, D.
, Fan
, Q.
, and Tan
, C. A.
, 1998
, “Experimental Investigations on the Active Vibration Control of Chain Drives
,” Noise Control Eng. J.
, 46
, No. 4
, pp. 139
–145
.22.
Rahn, C. D., “Parametric Control of Vibration in Flexible and Axially-Moving Materials Systems,” 1992, Ph.D. Dissertation, University of California, Berkeley.
23.
Rahn
, C. D.
, and Mote
, Jr., C. D.
, 1994
, “Parametric Control of Flexible Systems
,” ASME J. Vibr. Acoust.
, 116
, pp. 379
–385
.24.
Chung
, C. A.
, and Tan
, C. A.
,1995
, “Active Vibration Control of the Axially Moving String by Wave Cancellation
,” ASME J. Vibr. Acoust.
, 117
, No. 1
, pp. 49
–55
.25.
Lee
, S. Y.
, and Mote
, Jr., C. D.
, 1996
, “Vibration Control of an Axially Moving String by Boundary Control
,” ASME J. Dyn. Syst., Meas., Control
, 118
, pp. 66
–74
.26.
Renshaw
, A. A.
, Rahn
, C. D.
, Wickert
, J. A.
, and Mote
, Jr., C. D.
, 1998
, “Energy and Conserved Functionals for Axially Moving Materials
,” ASME J. Vibr. Acoust.
, 120
, pp. 634
–636
.27.
Rugh, W. J., 1996, Linear System Theory, Prentice Hall, pp. 116–117.
28.
Wang
, P. K. C.
, and Wei
, J. D.
, 1987
, “Vibrations in a Moving Flexible Robot Arm
,” J. Sound Vib.
, 116
, pp. 149
–160
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