This paper discusses the boundary stabilization of a beam in free transverse vibration. The dynamics of the beam is presented by a nonlinear partial differential equation (PDE). Based on this model a nonlinear control law is constructed to stabilize the system. The control law is a nonlinear function of the slopes and velocity at the boundary of the beam. The novelty of this article is that it has been possible to exponentially stabilize a free transversely vibrating beam via boundary control without restoring to truncation of the model. This result is achieved while the coupling between longitudinal and transversal displacements has been taken into account.
Issue Section:
Technical Papers
Keywords:
asymptotic stability,
nonlinear control systems,
distributed parameter systems,
control system synthesis,
Lyapunov methods,
dynamics
Topics:
Computer simulation,
Design,
Displacement,
Distributed parameter systems,
Dynamic systems,
Dynamics (Mechanics),
Equations of motion,
Partial differential equations,
Stability,
Theorems (Mathematics),
Vibration,
Boundary-value problems,
Control systems,
Lyapunov methods,
Nonlinear control systems,
Tension,
Marine drilling risers
1.
Balas, M. J., 1977, “Active Control of Flexible Systems,” Proceedings of the AIAA Symposium on Dynamic and Control of Large Flexible Spacecraft, Blacksburg, Va., pp. 217–236.
2.
Meirovitch
, L.
, and Baruh
, H.
, 1983
, “On the Problem of Observation Spillover in Self-adjoint Distributed-parameter Systems
,” Journal of Optimization and Applications
, 30
, No. 2
, pp. 269
–291
.3.
Fung
, Rong-Fong
, and Tseng
, Chun-Chang
, 1999
, “Boundary Control of an Axially Moving String via Lyapunov Method
,” ASME J. Dyn. Syst., Meas., Control
, 121
, pp. 105
–110
.4.
Shahruz
, S. M.
, and Krishna
, L. G.
, 1996
, “Boundary Control of a Nonlinear String
,” J. Sound Vib.
195
, No. 1
, pp. 169
–174
.5.
Shahruz
, S. M.
, and Narasimha
, C. A.
, 1997
, “Suppression of Vibration in Stretched Strings by the Boundary Control
,” J. Sound Vib.
204
, No. 5
, pp. 835
–840
.6.
Shahruz, S. M., 1997, “Suppression of Vibration in Stretched String by the Boundary Control,” Proceedings of the 36th Conference on Decision and Control, San Diego, CA pp. 535–536.
7.
Fung
, Rong-Fong
, Wu
, Jinn-Wen
, and Wu
, Sheng-Luong
, 1999
, “Stabilization of an Axially Moving String by Nonlinear Boundary Feedback
,” ASME J. Dyn. Syst., Meas., Control
, 121
, pp. 117
–121
.8.
Luo
, Z. H.
, 1993
, “Direct Strain Feedback Control of Flexible Robot Arms: New Theoretical and Experimental Results
,” IEEE Trans. Autom. Control
, 38
, No. 11
, pp. 1610
–1622
.9.
Luo
, Zheng-Hua
, Kitamura
, Nobuyuki
, and Bao-Zhu
, Guo
, 1995
, “Shear Feedback Control of Flexible Robot Arms
,” IEEE Trans. Rob. Autom.
, 11
, No. 5
, pp. 760
–765
.10.
Habib
, M. S.
, and Radcliffe
, C. J.
, 1991
, “Active Parametric Damping of Distributed Parameter Beam Transverse Vibration
,” ASME J. Dyn. Syst., Meas., Control
, 113
, pp. 295
–299
.11.
Wang
, P. K.
, 1967
, “Stability Analysis of Simplified Flexible Vehicle via Liapuniv’s Direct Method
,” AIAA J.
, 3
, No. 9
, pp. 1764
–1766
.12.
Leipholtz, H., 1966, “Application of Liapunov’s Direct Method to the Stability Problem of Rods Subject to Follower Forces,” Instability of Continuous Systems, Symposium Herrenlab, Germany, Sept. 8–12.
13.
Zubov, V. L., 1964, Methods of A. M. Liapunov and Their Application, Leningrad 1957 (English Translation), P. Noordhoff Ltd., Gorning, The Netherlands.
14.
Wang
, P. K.
, 1966
, “Stability Analysis of Elastic and Aeroelastic System via Liapuniv’s Direct Method
,” J. Franklin Inst.
, 281
, No. 1
, pp. 51
–72
.1.
Khalil
, H. K.
, 1996
, Nonlinear Systems, Second Edition, Prentic Hall, NJ; 2.
Lee
, E. W.
, 1957
, “Non-Linear Forced Vibration of a Stretched String
,” Br. J. Appl. Phys.
, 8
, pp. 411
–413
.1.
Fard, Mehrdad, 2000, “Modelling and Control of Mechanical Flexible Systems,” Dr. Eng. Thesis, Department of Engineering Cybernetics, The Norwegian University of Science and Technology, Trondheim, Norway.
Copyright © 2001
by ASME
You do not currently have access to this content.
