This paper presents the active vibration control of an axially moving string system through a mass-damper-spring (MDS) controller at its right-hand side (RHS) boundary. A nonlinear partial differential equation (PDE) describes a distributed parameter system (DPS) and directly selected as the object to be controlled. A new boundary control law is designed by sliding mode associated with Lyapunov method. It is shown that the boundary feedback states only include the displacement, velocity, and slope of the string at RHS boundary. Asymptotical stability of the control system is proved by the semigroup theory. Finally, finite difference scheme is used to validate the theoretical results.

Issue Section:
Technical Papers
Topics:
Lyapunov methods, String, Control equipment, Control systems, Dampers, Displacement, Distributed parameter systems, Feedback, Partial differential equations, Springs, Stability, Vibration control
1.
Abhyankar
N. S.
,
Hall
E. K.
, and
Hanagud
S. V.
,
1993
, “
Chaotic Vibrations of Beams: Numerical Solution of Partial Differential Equations
,”
ASME Journal of Applied Mechanics
, Vol.
60
, No.
1
, pp.
167
–
174
.
2.
Baily, T., and Hubbard, J. E., Jr., 1985, “Distributed Piezoelectric-Polymer Active Vibration Control of a Cantilever Beam,” Journal of Guid. Control. Vol. 8, Sept./Oct.
3.
Balas
M. J.
,
1978
, “
Feedback Control of Flexible System
,”
IEEE Transactions on Automatic Control
,
AC-23
, No.
4
, pp.
673
–
679
.
4.
Carrier
G. F.
,
1945
, “
On the Nonlinear Vibration Problem of the Elastic String
,”
Quarterly of Applied Mathematics
, Vol.
3
, pp.
157
–
165
.
5.
Chen
G.
,
1979
, “
Energy Decay Estimates and Exact Boundary Value Controllability for the Wave Equation in a Bounded Domain
,”
Journal de Mathematiques Pures Appliquees.
Vol.
58
, pp.
249
–
273
.
6.
Chen, G., and Zhou, J., 1993, Vibration and Damping in Distributed Systems, CRC Press, Florida.
7.
Chen
J. S.
,
1997
, “
Natural Frequencies and Stability of an Axially-Traveling String in Contact with a Stationary Local System
,”
ASME Journal of Vibration and Acoustics
, Vol.
119
, pp.
152
–
157
.
8.
Chung
C. H.
, and
Tan
C. A.
,
1995
, “
Active Vibration Control of the Axially Moving String by Wave Cancellation
,”
ASME Journal of Vibration and Acoustics
, Vol.
117
, pp.
49
–
55
.
9.
Habib
M. S.
, and
Radcliffe
C. J.
,
1991
, “
Active Parametric Damping of Distributed Parameter Beam Transverse Vibration
,”
ASME JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL
, Vol.
113
, pp.
295
–
299
.
10.
Jai
A. E.
, and
Pritchard
A. J.
,
1987
, “
Sensors and Actuators in Distributed Systems
,”
International Journal of Control
, Vol.
46
, pp.
1139
–
1153
.
11.
Ko¯mura
Y.
,
1967
, “
Nonlinear Semi-Groups in Hilbert Space
,”
Journal of the Mathematical Society of Japan
, Vol.
19
, pp.
493
–
507
.
12.
Lee
S. Y.
, and
Mote
C. D.
,
1996
, “
Vibration Control of an Axially Moving String by Boundary Control
,”
ASME JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL
, Vol.
118
, pp.
66
–
74
.
13.
Lee
S. Y.
, and
Mote
C. D.
,
1997
, “
A Generalized Treatment of the Energetics of Translating Continua, Part I: Strings and Second Order Tensioned Pipes
,”
Journal of Sound and Vibration
, Vol.
204
, No.
5
, pp.
717
–
734
.
14.
Meirovitch
L.
, and
Baruh
H.
,
1982
, “
Control of Self-Adjoint Distributed-Parameter Systems
,”
Journal of Guidance, Control, and Dynamics
, Vol.
5
, pp.
60
–
66
.
15.
Meirovitch
L.
, and
Baruh
H.
,
1983
, “
On the Problem of Observation Spillover in Self-Adjoint Distributed-Parameter Systems
,”
Journal of Optimization Theory and Applications
, Vol.
39
, No.
2
, pp.
269
–
291
.
16.
Mote
C. D.
,
1972
, “
Dynamic Stability of Axially Moving Materials
,”
Shock and Vibration Digest
, Vol.
4
, pp.
2
–
11
.
17.
Orlov
Yu. V.
,
1983
, “
Application of Lyapunov Method in Distributed System
,”
Automation and Remote Control
, Vol.
44
, pp.
426
–
431
.
18.
Orlov
Yu. V.
, and
Utkin
V. I.
,
1982
, “
Use of Sliding Modes in Distributed System Control Problems
,”
Automation and Remote Control
, Vol.
43
, Iss.
9
, pp.
1127
–
1135
.
19.
Orlov
Yu. V.
, and
Utkin
V. I.
,
1987
, “
Sliding Mode Control in Indefinite-Dimensional Systems
,”
Automatica
, Vol.
23
, No.
6
, pp.
753
–
757
.
20.
Pazy, A., 1983, Semigroups of Linear Operators and Applications to Partial Differential Equations, New York.
21.
Wickert
J. A.
, and
Mote
C. D.
,
1989
, “
On the Energetics of Axially Moving Continua
,”
Journal of Acoustical Society of America
, Vol.
85
, pp.
1365
–
1368
.
22.
Wickert
J. A.
, and
Mote
C. D.
,
1990
, “
Classical Vibration Analysis of Axially Moving Continua
,”
ASME Journal of Applied Mechanics
, Vol.
57
, pp.
738
–
744
.
23.
Wu, S. L., 1997, “Boundary Feedback Control of the Axially Moving String System,” Master’s thesis, University of Chung Yuan, Chung-Li, Taiwan.
24.
Yang
B.
, and
Mote
C. D.
,
1991
, “
Active Vibration Control of the Axially Moving String in the S Domain
,”
ASME Journal of Applied Mechanics
, Vol.
58
, pp.
189
–
196
.
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