Analysis of the nonlinear structural-acoustic resonant frequencies of a rectangular tube with a flexible end using harmonic balance and homotopy perturbation methods. (English) Zbl 1257.35176

Summary: The structural acoustic problem considered in this study is the nonlinear resonant frequencies of a rectangular tube with one open end, one flexible end, and four rigid side walls. A multiacoustic single structural modal formulation is derived from two coupled partial differential equations which represent the large amplitude structural vibration of the flexible end and acoustic pressure induced within the tube. The results obtained from the harmonic balance and homotopy perturbation approaches verified each other. The effects of vibration amplitude, aspect ratio, the numbers of acoustic modes and harmonic terms, and so forth, on the first two resonant natural frequencies, are examined.


35Q74 PDEs in connection with mechanics of deformable solids
74H45 Vibrations in dynamical problems in solid mechanics
Full Text: DOI


[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. · Zbl 1217.35184
[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. · Zbl 1092.35068
[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. · Zbl 1188.39008
[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. · Zbl 0074.20001
[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. · Zbl 06942240
[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. · Zbl 06942705
[22] J.-H. He, “New interpretation of homotopy perturbation method,” International Journal of Modern Physics B, vol. 20, no. 18, pp. 2561-2568, 2006.
[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. · Zbl 1102.34039
[24] J.-H. He, “Hamiltonian approach to nonlinear oscillators,” Physics Letters A, vol. 374, no. 23, pp. 2312-2314, 2010. · Zbl 1237.70036
[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. · Zbl 1048.70011
[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. · Zbl 1189.65159
[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. · Zbl 1204.34073
[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. · Zbl 1246.65191
[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. · Zbl 1246.65193
[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. · Zbl 1223.34055
[31] H. R. Srirangarajan, “Nonlinear free vibrations of uniform beams,” Journal of Sound Vibration, vol. 175, no. 3, pp. 425-427, 1994. · Zbl 1147.74347
[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. · Zbl 0933.74535
[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. · Zbl 0925.73787
[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. · Zbl 1235.65096
[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. · Zbl 1314.34076
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.