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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 391584, 13 pages
http://dx.doi.org/10.1155/2012/391584
Research Article

Analysis of the Nonlinear Structural-Acoustic Resonant Frequencies of a Rectangular Tube with a Flexible End Using Harmonic Balance and Homotopy Perturbation Methods

Department of Civil and Architectural Engineering, City University of Hong Kong, Kowloon Tong, Kowloon, Hong Kong

Received 13 August 2012; Accepted 9 November 2012

Academic Editor: Lan Xu

Copyright © 2012 Y. Y. Lee. 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. Y. Y. Lee and C. F. Ng, “Sound insertion loss of stiffened enclosure plates using the finite element method and the classical approach,” Journal of Sound and Vibration, vol. 217, no. 2, pp. 239–260, 1998.
  2. Y. Y. Lee, “Insertion loss of a cavity-backed semi-cylindrical enclosure panel,” Journal of Sound and Vibration, vol. 259, no. 3, pp. 625–636, 2003.
  3. W. Frommhold, H. V. Fuchs, and S. Sheng, “Acoustic performance of membrane absorbers,” Journal of Sound and Vibration, vol. 170, no. 5, pp. 621–636, 1994.
  4. S. Nakanishi, K. Sakagami, M. Daido, and M. Morimoto, “Effect of an air-back cavity on the sound field reflected by a vibrating plate,” Applied Acoustics, vol. 56, pp. 241–256, 1999.
  5. R. H. Lyon, “Noise reduction of rectangular enclosures with one flexible wall,” Journal of the Acoustical Society of America, vol. 35, pp. 1791–1797, 1963.
  6. A. J. Pretlove, “Free vibrations of a rectangular panel backed by a closed rectangular cavity,” Journal of Sound and Vibration, vol. 2, no. 3, pp. 197–209, 1965.
  7. J. Pan, S. J. Elliott, and K. H. Baek, “Analysis of low frequency acoustic response in a damped rectangular enclosure,” Journal of Sound and Vibration, vol. 223, no. 4, pp. 543–566, 1999.
  8. J. A. Esquivel-Avila, “Dynamic analysis of a nonlinear Timoshenko equation,” Abstract and Applied Analysis, Article ID 724815, 36 pages, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  9. M. L. Santos, J. Ferreira, and C. A. Raposo, “Existence and uniform decay for a nonlinear beam equation with nonlinearity of Kirchhoff type in domains with moving boundary,” Abstract and Applied Analysis, no. 8, pp. 901–919, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  10. R. Ma, J. Li, and C. Gao, “Existence of positive solutions of a discrete elastic beam equation,” Discrete Dynamics in Nature and Society, Article ID 582919, 15 pages, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  11. Y. Y. Lee, W. Y. Poon, and C. F. Ng, “Anti-symmetric mode vibration of a curved beam subject to autoparametric excitation,” Journal of Sound and Vibration, vol. 290, no. 1-2, pp. 48–64, 2006.
  12. C. K. Hui, Y. Y. Lee, and C. F. Ng, “Use of internally resonant energy transfer from the symmetrical to anti-symmetrical modes of a curved beam isolator for enhancing the isolation performance and reducing the source mass translation vibration: theory and experiment,” Mechanical Systems and Signal Processing, vol. 25, no. 4, pp. 1248–1259, 2011.
  13. W. Y. Poon, C. F. Ng, and Y. Y. Lee, “Dynamic stability of curved beam under sinusoidal loading,” Journal of Aerospace Engineering, Proceeding of the Institution of Mechanical Engineers G, vol. 216, pp. 209–217, 2002.
  14. C. S. Chen, C. P. Fung, and R. D. Chien, “Nonlinear vibration of an initially stressed laminated plate according to a higher-order theory,” Composite Structures, vol. 77, no. 4, pp. 521–532, 2007.
  15. H.-N. Chu and G. Herrmann, “Influence of large amplitudes on free flexural vibrations of rectangular elastic plates,” vol. 23, pp. 532–540, 1956. View at Zentralblatt MATH
  16. Y. Wei and R. Vaicaitis, “Nonlinear models for double-wall systems for vibrations and noise control,” Journal of Aircraft, vol. 34, no. 6, pp. 802–810, 1997.
  17. C. K. Hui, Y. Y. Lee, and J. N. Reddy, “Approximate elliptical integral solution for the large amplitude free vibration of a rectangular single mode plate backed by a multi-acoustic mode cavity,” Thin-Walled Structures, vol. 49, no. 9, pp. 1191–1194, 2011.
  18. Y. Y. Lee, “Structural-acoustic coupling effect on the nonlinear natural frequency of a rectangular box with one flexible plate,” Applied Acoustics, vol. 63, no. 11, pp. 1157–1175, 2002.
  19. Y. Y. Lee, X. Guo, and E. W. M. Lee, “Effect of the large amplitude vibration of a finite flexible micro-perforated panel absorber on sound absorption,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 8, no. 1, pp. 41–44, 2007.
  20. Y. Y. Lee, Q. S. Li, A. Y. T. Leung, and R. K. L. Su, “The jump phenomenon effect on the sound absorption of a nonlinear panel absorber and sound transmission loss of a nonlinear panel backed by a cavity,” Nonlinear Dynamics, vol. 69, pp. 99–116, 2012.
  21. M. K. Yazdi, A. Khan, Y. Madani, M. Askari, H. Saadatnia, and Z . Yildirim, “Analytical solutions for autonomous conservative nonlinear oscillator,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 11, no. 11, pp. 975–980, 2010.
  22. J.-H. He, “New interpretation of homotopy perturbation method,” International Journal of Modern Physics B, vol. 20, no. 18, pp. 2561–2568, 2006. View at Publisher · View at Google Scholar
  23. J.-H. He, “Some asymptotic methods for strongly nonlinear equations,” International Journal of Modern Physics B, vol. 20, no. 10, pp. 1141–1199, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  24. J.-H. He, “Hamiltonian approach to nonlinear oscillators,” Physics Letters A, vol. 374, no. 23, pp. 2312–2314, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  25. J. H. He, “Preliminary report on the energy balance for nonlinear oscillations,” Mechanics Research Communications, vol. 29, no. 2-3, pp. 107–111, 2002.
  26. A. Beléndez, E. Gimeno, M. L. Alvarez, and D. I. Méndez, “Nonlinear oscillator with discontinuity by generalized harmonic balance method,” Computers & Mathematics with Applications, vol. 58, no. 11-12, pp. 2117–2123, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  27. J. F. Chu and T. Xia, “The Lyapunov stability for the linear and nonlinear damped oscillator with time-periodic parameters,” Abstract and Applied Analysis, vol. 2010, Article ID 286040, 12 pages, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  28. Y. Q. Liu, “Variational homotopy perturbation method for solving fractional initial boundary value problems,” Abstract and Applied Analysis, vol. 2012, Article ID 727031, 10 pages, 2012. View at Zentralblatt MATH
  29. H. A. Zedan and E. El Adrous, “The application of the homotopy perturbation method and the homotopy analysis method to the generalized Zakharov equations,” Abstract and Applied Analysis, vol. 2012, Article ID 561252, 19 pages, 2012. View at Zentralblatt MATH
  30. A. Beléndez, E. Gimeno, M. L. Álvarez, D. I. Méndez, and A. Hernández, “Application of a modified rational harmonic balance method for a class of strongly nonlinear oscillators,” Physics Letters A, vol. 372, no. 39, pp. 6047–6052, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  31. H. R. Srirangarajan, “Nonlinear free vibrations of uniform beams,” Journal of Sound Vibration, vol. 175, no. 3, pp. 425–427, 1994.
  32. W. Han and M. Petyt, “Geometrically nonlinear vibration analysis of thin, rectangular plates using the hierarchical finite element method .2. 1st mode of laminated plates and higher modes of isotropic and laminated plates,” Computers & Structures, vol. 63, no. 2, pp. 309–318, 1997.
  33. M. I. McEwan, J. R. Wright, J. E. Cooper, and A. Y. T. Leung, “A combined modal/finite element analysis technique for the dynamic response of a non-linear beam to harmonic excitation,” Journal of Sound and Vibration, vol. 243, no. 4, pp. 601–624, 2001.
  34. J. E. Locke, “Finite element large deflection random response of thermally buckled plates,” Journal of Sound and Vibration, vol. 160, no. 2, pp. 301–312, 1993.
  35. K. M. Liew, Y. Y. Lee, T. Y. Ng, and X. Zhao, “Dynamic stability analysis of composite laminated cylindrical panels via the mesh-free kp-Ritz method,” International Journal of Mechanical Sciences, vol. 49, pp. 1156–1165, 2007.
  36. J.-H. He, “Homotopy perturbation method with an auxiliary term,” Abstract and Applied Analysis, vol. 2012, Article ID 857612, 7 pages, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  37. M. Akbarzade and J. Langari, “Determination of natural frequencies by coupled method of homotopy perturbation and variational method for strongly nonlinear oscillators,” Journal of Mathematical Physics, vol. 52, no. 2, Article ID 023518, 2011. View at Publisher · View at Google Scholar