×

An investigation on the nonlinear free vibration analysis of beams with simply supported boundary conditions using four engineering theories. (English) Zbl 1235.74167

Summary: The objective of this study is to present a brief survey on the geometrically nonlinear free vibrations of the Bernoulli-Euler, the Rayleigh, shear, and the Timoshenko beams with simple end conditions using the Homotopy Analysis Method (HAM). Expressions for the natural frequencies, the transverse deflection, postbuckling load-deflection relation to, and critical buckling load are presented. The results of nonlinear analysis are validated with the published results, and excellent agreement is observed. The effects of some parameters, such as slender ratio, the rotary inertia, and the shear deformation, are examined as other parameters are fixed.

MSC:

74K10 Rods (beams, columns, shafts, arches, rings, etc.)
74H45 Vibrations in dynamical problems in solid mechanics
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] L. Azrar, R. Benamar, and R. G. White, “A semi-analytical approach to the non-linear dynamic response problem of S-S and C-C beams at large vibration amplitudes part I: general theory and application to the single mode approach to free and forced vibration analysis,” Journal of Sound and Vibration, vol. 224, no. 2, pp. 183-207, 1999.
[2] M. I. Qaisi, “Application of the harmonic balance principle to the nonlinear free vibration of beams,” Applied Acoustics, vol. 40, no. 2, pp. 141-151, 1993. · doi:10.1016/0003-682X(93)90087-M
[3] W. C. Xie, H. P. Lee, and S. P. Lim, “Normal modes of a non-linear clamped-clamped beam,” Journal of Sound and Vibration, vol. 250, no. 2, pp. 339-349, 2002. · doi:10.1006/jsvi.2001.3918
[4] G. Singh, A. K. Sharma, and G. Venkateswara Rao, “Large-amplitude free vibrations of beams-a discussion on various formulations and assumptions,” Journal of Sound and Vibration, vol. 142, no. 1, pp. 77-85, 1990. · doi:10.1016/0022-460X(90)90583-L
[5] Q. Guo and H. Zhong, “Non-linear vibration analysis of beams by a spline-based differential quadrature method,” Journal of Sound and Vibration, vol. 269, no. 1-2, pp. 413-420, 2004. · Zbl 1236.74294 · doi:10.1016/S0022-460X(03)00328-6
[6] G. V. Rao, I. S. Raju, and K. K. Raju, “Nonlinear vibrations of beams considering shear deformation and rotary inertia,” AIAA Journal, vol. 14, no. 5, pp. 685-687, 1976. · Zbl 0343.73034 · doi:10.2514/3.7138
[7] G. V. Rao, K. M. Saheb, and G. R. Janardhan, “Fundamental frequency for large amplitude vibrations of uniform Timoshenko beams with central point concentrated mass using coupled displacement field method,” Journal of Sound and Vibration, vol. 298, no. 1-2, pp. 221-232, 2006. · doi:10.1016/j.jsv.2006.05.014
[8] S. Liao, Beyond Perturbation-Introduction to the Homotopy Analysis Method, Chapman & Hall/CRC, New York, NY, USA, 2004. · Zbl 1051.76001 · doi:10.1201/9780203491164
[9] H. Abramovich and I. Elishakoff, “Application of the Krein’s method for determination of natural frequencies of periodically supported beam based on simplified Bresse-Timoshenko equations,” Acta Mechanica, vol. 66, no. 1-4, pp. 39-59, 1987. · Zbl 0606.73070 · doi:10.1007/BF01184284
[10] S. Gopalakrishnan, M. Martin, and J. F. Doyle, “A matrix methodology for spectral analysis of wave propagation in multiple connected Timoshenko beams,” Journal of Sound and Vibration, vol. 158, no. 1, pp. 11-24, 1992. · Zbl 0925.73322 · doi:10.1016/0022-460X(92)90660-P
[11] S. M. Han, H. Benaroya, and T. Wei, “Dynamics of transversely vibrating beams using four engineering theories,” Journal of Sound and Vibration, vol. 225, no. 5, pp. 935-988, 1999. · Zbl 1235.74075 · doi:10.1006/jsvi.1999.2257
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.