Table of Contents Author Guidelines Submit a Manuscript
Journal of Applied Mathematics
Volume 2014, Article ID 982574, 7 pages
http://dx.doi.org/10.1155/2014/982574
Research Article

A Numerical Method of the Euler-Bernoulli Beam with Optimal Local Kelvin-Voigt Damping

1Laboratory of Information & Control Technology, Ningbo Institute of Technology, Zhejiang University, Ningbo 315100, China
2The State Key Laboratory of Industrial Control Technology and Institute of Cyber-Systems & Control, Zhejiang University, Hangzhou 310027, China

Received 12 February 2014; Accepted 28 May 2014; Published 23 June 2014

Academic Editor: Magdy A. Ezzat

Copyright © 2014 Xin Yu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. H. T. Banks, R. C. Smith, and Y. Wang, “The modeling of piezoceramic patch interactions with shells, plates, and beams,” Quarterly of Applied Mathematics, vol. 53, no. 2, pp. 353–381, 1995. View at Google Scholar · View at MathSciNet · View at Scopus
  2. H. T. Banks and Y. Zhang, “Computational methods for a curved beam with piezoceramic patches,” Journal of Intelligent Material Systems and Structures, vol. 8, no. 3, pp. 260–278, 1997. View at Publisher · View at Google Scholar · View at Scopus
  3. K. H. Ip and P. C. Tse, “Optimal configuration of a piezoelectric patch for vibration control of isotropic rectangular plates,” Smart Materials and Structures, vol. 10, no. 2, pp. 395–403, 2001. View at Publisher · View at Google Scholar · View at Scopus
  4. S. J. Dyke, B. F. Spencer Jr., M. K. Sain, and J. D. Carlson, “Modeling and control of magnetorheological dampers for seismic response reduction,” Smart Materials and Structures, vol. 5, no. 5, pp. 565–575, 1996. View at Publisher · View at Google Scholar · View at Scopus
  5. Y. Kim, R. Langari, and S. Hurlebaus, “Semiactive nonlinear control of a building with a magnetorheological damper system,” Mechanical Systems and Signal Processing, vol. 23, no. 2, pp. 300–315, 2009. View at Publisher · View at Google Scholar · View at Scopus
  6. C. Y. Lai and W. H. Liao, “Vibration control of a suspension system via a magnetorheological fluid damper,” Journal of Vibration and Control, vol. 8, no. 4, pp. 527–547, 2002. View at Publisher · View at Google Scholar · View at Scopus
  7. G. Yang, B. F. Spencer Jr., H. J. Jung, and J. D. Carlson, “Dynamic modeling of large-scale magnetorheological damper systems for civil engineering applications,” Journal of Engineering Mechanics, vol. 130, no. 9, pp. 1107–1114, 2004. View at Publisher · View at Google Scholar · View at Scopus
  8. Z. Y. Liu and S. M. Zheng, Semigroups Associated with Dissipative Systems, vol. 398, CRC Press, New York, NY, USA, 1999. View at MathSciNet
  9. C. Zhang, “Boundary feedback stabilization of the undamped Timoshenko beam with both ends free,” Journal of Mathematical Analysis and Applications, vol. 326, no. 1, pp. 488–499, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  10. H. L. Zhao, K. S. Liu, and C. G. Zhang, “Stability for the Timoshenko beam system with local Kelvin-Voigt damping,” Acta Mathematica Sinica (English Series), vol. 21, no. 3, pp. 655–666, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  11. K. S. Liu and Z. Y. Liu, “Exponential decay of energy of the Euler-Bernoulli beam with locally distributed Kelvin-Voigt damping,” SIAM Journal on Control and Optimization, vol. 36, no. 3, pp. 1086–1098, 1998. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  12. B. Z. Guo, “Riesz basis property and exponential stability of controlled Euler-Bernoulli beam equations with variable coefficients,” SIAM Journal on Control and Optimization, vol. 40, no. 6, pp. 1905–1923, 2002. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  13. K. Ammari and M. Tucsnak, “Stabilization of Bernoulli-Euler beams by means of a pointwise feedback force,” SIAM Journal on Control and Optimization, vol. 39, no. 4, pp. 1160–1181, 2000. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  14. G. Chen, S. G. Krantz, D. W. Ma, and C. E. Wayne, “The Euler-Bernoulli beam equation with boundary energy dissipation,” in Operator Methods for Optimal control Problems, S. J. Lee, Ed., vol. 108 of Lecture Notes in Pure and Appl. Math., pp. 67–96, Louisiana, Baton Rouge, La, USA, 1987. View at Google Scholar · View at MathSciNet
  15. B. Guo and K. Yang, “Dynamic stabilization of an Euler-Bernoulli beam equation with time delay in boundary observation,” Automatica, vol. 45, no. 6, pp. 1468–1475, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  16. I. Y. Shen, “Stability and controllability of Euler-Bernoulli beams with intelligent constrained layer treatments,” Journal of Vibration and Acoustics, vol. 118, no. 1, pp. 70–77, 1996. View at Publisher · View at Google Scholar · View at Scopus
  17. N. Tanaka and Y. Kikushima, “Optimal vibration feedback control of an Euler-Bernoulli beam: toward realization of the active sink method,” Journal of Vibration and Acoustics, Transactions of the ASME, vol. 121, no. 2, pp. 174–182, 1999. View at Publisher · View at Google Scholar · View at Scopus
  18. J. M. Ball, J. E. Marsden, and M. Slemrod, “Controllability for distributed bilinear systems,” SIAM Journal on Control and Optimization, vol. 20, no. 4, pp. 575–597, 1982. View at Publisher · View at Google Scholar · View at MathSciNet
  19. K. Beauchard, “Local controllability and non-controllability for a 1D wave equation with bilinear control,” Journal of Differential Equations, vol. 250, no. 4, pp. 2064–2098, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  20. M. Slemrod, “Stabilization of bilinear control systems with applications to nonconservative problems in elasticity,” SIAM Journal on Control and Optimization, vol. 16, no. 1, pp. 131–141, 1978. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  21. Q. Lin, R. Loxton, and K. L. Teo, “The control parameterization method for nonlinear optimal control: a survey,” Journal of Industrial and Management Optimization, vol. 10, no. 1, pp. 275–309, 2014. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  22. Q. Lin, R. Loxton, K. L. Teo, and Y. H. Wu, “Optimal control computation for nonlinear systems with state-dependent stopping criteria,” Automatica, vol. 48, no. 9, pp. 2116–2129, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  23. R. C. Loxton, K. L. Teo, V. Rehbock, and K. F. Yiu, “Optimal control problems with a continuous inequality constraint on the state and the control,” Automatica, vol. 45, no. 10, pp. 2250–2257, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  24. K. L. Teo, C. J. Goh, and K. H. Wong, A Unified Computational Approach to Optimal Control Problems, Longman Scientific and Technical, London, UK, 1991.
  25. M. S. Bazaraa, H. D. Sherali, and C. M. Shetty, Nonlinear Programming: Theory and Algorithms, John Wiley & Sons, New York, NY, USA, 2013.
  26. D. G. Luenberger and Y. Y. Ye, Linear and Nonlinear Programming, vol. 116, Springer, New York, NY, USA, 2008.
  27. R. C. Loxton, K. L. Teo, and V. Rehbock, “Optimal control problems with multiple characteristic time points in the objective and constraints,” Automatica, vol. 44, no. 11, pp. 2923–2929, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  28. R. Loxton, K. L. Teo, and V. Rehbock, “Robust suboptimal control of nonlinear systems,” Applied Mathematics and Computation, vol. 217, no. 14, pp. 6566–6576, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus