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Boundary control of an axially moving string: actuator dynamics included

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Abstract

In this paper, an active vibration control of a translating tensioned string with the use of an electro-hydraulic servo mechanism at the right boundary is investigated. The equations of motion of the string are derived by using Hamilton’s principle for the systems with changing mass. The control objective is to suppress the transverse vibrations of the string via a rightboundary control. An energy-based right-boundary control law, generating a specific current input to the servo-valve, is derived. It is revealed that a time-varying boundary force, as a function of the slope of the string at the right end and a suitably chosen damping coefficient of the actuator, can successfully suppress the transverse vibrations. The exponential stability of the closed loop system is proved. The effectiveness of the proposed control law is demonstrated via simulations.

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References

  • Alleyne, A. and Liu, R., 2000, “A Simplified Approach to Force Control for Electro-Hydraulic Systems,”Control Engineering Practice, Vol. 8, No. 12, pp. 1347–1356.

    Article  Google Scholar 

  • Araki, M. and Taguchi, H., 2003, “Two-De-gree-of-Freedom PID Controllers,”International Journal of Control, Automation, and Systems, Vol. 1, No. 4, pp. 401–411.

    Google Scholar 

  • Bapat, V. A. and Srinivasan, P., 1967, “Nonlinear Transverse Oscillations in Traveling Strings by the Method of Harmonic Balance,”Journal of Applied Mechanics, Vol. 34, pp. 775–777.

    Google Scholar 

  • Benaroya, H., 1998,Mechanical Vibration: Analysis, Uncertainties, and Control, Prentice Hall, New Jersey.

  • Carrier, G. F., 1945, “On the Nonlinear Vibration Problem of the Elastic String,”Quarterly of Applied Mathematics, Vol. 3, pp. 157–165.

    MATH  MathSciNet  Google Scholar 

  • 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,”Journal of Vibration and Control, Vol. 10, No. 5, pp. 661–682.

    Article  MATH  MathSciNet  Google Scholar 

  • Fard, M. P. and Sagatun, S. I., 2001, “Exponential Stabilization of a Transversely Vibrating Beam via Boundary Control,”Journal of Sound and Vibration, Vol. 240, No. 4, pp. 613–622.

    Article  MathSciNet  Google Scholar 

  • Fung, R. F., Wu, J. W. and Wu, S. L., 1999a, “Exponential Stability of an Axially Moving String by Linear Boundary Feedback,”Automatica, Vol. 35, No. 1, pp. 177–181.

    Article  MATH  MathSciNet  Google Scholar 

  • Fung, R. F., Wu, J. W. and Wu, S. L., 1999b, “Stabilization of an Axially Moving String by Non-Linear Boundary Feedback,”ASME Journal of Dynamic Systems, Measurement, and Control, Vol. 121, No. 1, pp. 117–120.

    Article  MathSciNet  Google Scholar 

  • Fung, R. F., Chou, J. H. and Kuo, Y. L., 2002a, “Optimal Boundary Control of an Axially Moving Material System,”ASME Journal of Dynamic Systems, Measurement, and Control, Vol. 124, No. 1, pp. 55–61.

    Article  Google Scholar 

  • Fung, R. F., Wu, J. W. and Lu, P. Y., 2002b, “Adaptive Boundary Control of an Axially Moving String System,”Journal of Vibration and Acoustics, Vol. 124, No. 3, pp. 435–440.

    Article  Google Scholar 

  • Goodwin, G. C. and Quevedo, D. E., 2003, “Finite Alphabet Control and Estimation,”International Journal of Control, Automation, and Systems, Vol. 1, No. 4, pp. 412–430.

    Google Scholar 

  • 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,”International Journal of Control, Automation, and Systems, Vol. 2, No. 1, pp. 55–67.

    MathSciNet  Google Scholar 

  • Hong, K. T., Huh, C. D. and Hong, K. S., 2003, “Command Shaping Control for Limiting the Transient Sway Angle of Crane Systems,”International Journal of Control, Automation, and Systems, Vol. 1, No. 1, pp. 43–53.

    Google Scholar 

  • Kim, J. Y., Hyun, S. H. and Yoo, H. H., 2002, “Nonlinear Modeling and Dynamic Analysis of Flexible Structures Undergoing Overall Motions Employing Mode Approximation Method,”KSME International Journal, Vol. 16, No. 7, pp. 896–901.

    Google Scholar 

  • Kim, S. K. and Yoo, H. H., 2002, “Vibration Analysis of Rotating Composite Cantilever Plates,”KSME International Journal, Vol. 16, No. 3, pp. 320–326.

    Google Scholar 

  • Laousy, H., Xu, C. Z. and Sallet, G., 1996, “Boundary Feedback Stabilization of a Rotating Body-Beam System,”IEEE Transactions on Automatic Control, Vol. 41, No. 2, pp. 241–244.

    Article  MATH  MathSciNet  Google Scholar 

  • 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, No. 1, pp. 66–74.

    Article  MATH  Google Scholar 

  • Lee, S. Y. and Mote, C. D., 1999, “Wave Characteristics and Vibration Control of Translating Beams by Optimal Boundary Damping,”Journal of Vibration and Acoustics, Vol. 121, No. 1, pp. 18–25.

    Article  Google Scholar 

  • Lee, W. H., 2002, “Mathematical Model for Cold Rolling and Temper Rolling Process of Thin Steel Strip,”KSME International Journal, Vol. 16, No. 10, pp. 1296–1302.

    Google Scholar 

  • Li, Y. and Rahn, C. D., 2000, “Adaptive Vibration Isolation for Axially Moving Beams,”IEEE/ASME Transactions on Mechatronics, Vol. 5, No. 4, pp. 419–428.

    Article  Google Scholar 

  • Li, Y., Aron, D. and Rahn, C. D., 2002, “Adaptive Vibration Isolation for Axially Moving Strings: Theory and Experiment,”Automatica, Vol. 38, No. 3, pp. 379–389.

    Article  MATH  MathSciNet  Google Scholar 

  • Matsuno, F., Ohno, T. and Orlov, Y., 2002, “Proportional Derivative and Strain (PDS) Boundary Feedback Control of a Flexible Space Structure with a Closed-Loop Chain Mechanism,”Automatica, Vol. 38, No. 7, pp. 1201–1211.

    Article  MATH  MathSciNet  Google Scholar 

  • Mclver, D. B., 1973, “Hamiltion’s Principle for Systems of Changing Mass,”Journal of Engineering Mathematics, Vol. 7, No. 3, pp. 249–261.

    Article  Google Scholar 

  • Morgul, O., 1992, “Dynamics Boundary Control of an Euler-Bernoulli Beam,”IEEE Transactions on Automatic Control, Vol. 37, No. 5, pp. 639–642.

    Article  MathSciNet  Google Scholar 

  • Mote, C. D., 1965, “A Study of Band Saw Vibration,”Journal of the Franklin Institute, Vol. 279, No. 6, pp. 430–444.

    Article  Google Scholar 

  • Oostveen, J. C. and Curtain, R. F., 2000, “Robustly Stabilization Controllers for Dissipative Infinite-Dimensional Systems with Collocated Actuators and Sensor,”Automatica, Vol. 36, No. 3, pp. 337–348.

    Article  MATH  MathSciNet  Google Scholar 

  • Oshima, K., Takigami, T. and Hayakawa, Y., 1997, “Robust Vibration Control of a Cantilever Beam Using Self-Sensing Actuator,”JSME International Journal Series C, Vol. 40, No. 4, pp. 681–687.

    Google Scholar 

  • Pellicano, F. and Zirilli, F., 1998, “Boundary Layers and Non-Linear Vibrations in an Axially Moving Beam,”International Journal of Non-Linear Mechanics, Vol. 33, No. 4, pp. 691–694.

    Article  MATH  MathSciNet  Google Scholar 

  • Qu, Z., 2002, “An Iterative Learning Algorithm for Boundary Control of a Stretched Moving String,”Automatica, Vol. 38, No. 5, pp. 821–827.

    Article  MATH  MathSciNet  Google Scholar 

  • Rao, S. S., 1990,Mechanical Vibrations, Addison Wesley.

  • Shahruz, S. M., 1998, “Boundary Control of the Axially Moving Kirchhoff String,”Automatica, Vol. 34, No. 10, pp. 1273–1277.

    Article  MATH  Google Scholar 

  • Shahruz, S. M., 2000, “Boundary Control of a Non-Linear Axially Moving String,”International Journal of Robust Nonlinear Control, Vol. 10, No. 1, pp. 17–25.

    Article  MATH  MathSciNet  Google Scholar 

  • Sohn, J. H., Yoo, W. S., Hong, K. S. and Kim, K. S., 2002, “Massless Links with External Forces and Bushing Effect for Multibody Dynamic Analysis,”KSME International Journal, Vol. 16, No. 6, pp. 810–818.

    Google Scholar 

  • Wickert, J. A. and Mote, C. D., 1988, “On the Energetics of Axially Moving Continua,”Journal of Acoustic Society of America, Vol. 85, No. 3, pp. 1365–1368.

    Article  Google Scholar 

  • Wickert, J. A. and Mote, C. D., 1990, “Classical Vibration Analysis of Axially Moving Continual,”Journal of Applied Mechanics, Vol. 57, No. 3, pp. 738–744.

    Article  MATH  Google Scholar 

  • Wickert, J. A., 1992, “Non-Linear Vibration of a Traveling Tensioned Beam,”International Journal of Non-Linear Mechanics, Vol. 27, No. 3, pp. 503–517.

    Article  MATH  Google Scholar 

  • Yang, K. J., Hong, K. S. and Matsuno, F., 2004a, “Robust Adaptive Boundary Control of an Axially Moving String under a Spatiotemporally Varying Tension,”Journal of Sound and Vibration, Vol. 273, No. 4-5, pp. 1007–1029.

    Article  MathSciNet  Google Scholar 

  • Yang, K. J., Hong, K. S. and Matsuno, F., 2004b, “Boundary Control of an Axially Moving Steel Strip under a Spatiotemporally Varying Tension,”JSME International Journal, Series C, Vol. 47, No. 2, pp. 665–674.

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Mechanical and Intelligent Systems Engineering, Pusan National University, Busan 609-735, San 30, Jangjeon-dong Gumjeong-gu, Korea

    Chang Won Kim & Hahn Park

  2. School of Mechanical Engineering, Pusan National University, Busan 609-735, San 30, Jangjeon-dong Gumjeong-gu, Korea

    Keum Shik Hong

Authors
  1. Chang Won Kim
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  2. Keum Shik Hong
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  3. Hahn Park
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Correspondence to Keum Shik Hong.

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Kim, C.W., Hong, K.S. & Park, H. Boundary control of an axially moving string: actuator dynamics included. J Mech Sci Technol 19, 40–50 (2005). https://doi.org/10.1007/BF02916103

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  • DOI: https://doi.org/10.1007/BF02916103

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