Hybrid Analytical Technique for Nonlinear Vibration Analysis of Thin‐Walled Beams
Publication: Journal of Engineering Mechanics
Volume 119, Issue 4
Abstract
A two‐step hybrid analytical technique is presented for the nonlinear vibration analysis of thin‐walled beams. The first step involves the generation of various‐order perturbation functions using the Linstedt‐Poincaré perturbation technique. The second step consists of using the perturbation functions as coordinate (or approximation) functions and then computing both the amplitudes of these functions and the nonlinear frequency of vibration via a direct variational procedure. The analytical formulation is based on a form of the geometrically nonlinear beam theory with the effects of in‐plane inertia, rotatory inertia, and transverse shear deformation included. The effectiveness of the proposed technique is demonstrated by means of a numerical example of thin‐walled beam with a doubly symmetric I‐section. The solutions obtained using a single‐spatial mode were compared with those obtained using multiple‐spatial modes. The standard of comparison was taken to be the frequencies obtained by the direct integration/fast Fourier transform (FFT) technique. The nonlinear frequencies obtained by the hybrid technique were shown to converge to the corresponding ones obtained by the direct integration/fast Fourier transform (FFT) technique well beyond the range of applicability of the perturbation technique. The frequencies and total strain energy of the beam were overestimated by using a single‐spatial mode.
Get full access to this article
View all available purchase options and get full access to this article.
References
1.
Bennett, J. A., and Eisley, J. G. (1969). “A multiple degree‐of‐freedom approach to nonlinear beam vibrations.” AIAA J., 8(4), 734–739.
2.
Bhashyam, G. R., and Prathap, G. (1980). “Galerkin finite element method for nonlinear beam vibrations.” J. Sound and Vibration, 72(2), 191–203.
3.
Dowell, E. H., Traybar, J., and Hodges, D. H. (1977). “An experimental‐theoretical correlation study of nonlinear bending and torsion deformations of a cantilever beam.” J. Sound and Vibration, 50(4), 533–544.
4.
Evensen, D. A. (1967). “Nonlinear vibrations of beams with various boundary conditions.” AIAA J., 6(2), 370–372.
5.
Gjelsvik, A. (1981). The theory of thin‐walled beams. John Wiley & Sons, Inc., New York, N.Y.
6.
Hasan, S. A., and Barr, A. D. S. (1974). “Nonlinear and parametric vibration of thin‐walled beams of equal angle‐section.” J. Sound and Vibration, 32(1), 25–47.
7.
Iu, V. P., Cheung, Y. K., and Lau, S. L. (1985). “Nonlinear vibration analysis of multilayer beams by incremental finite elements, part I: Theory and numerical formulation.” J. Sound and Vibration, 100(3), 359–372.
8.
Kapania, R. K., and Raciti, S. (1989). “Recent advances in analysis of laminated beams and plates, part II: Vibrations and wave propagation.” AIAA J., 27(7), 935–946.
9.
Lewandowski, R. (1987). “Application of the Ritz method to the analysis of nonlinear free vibrations of beams.” J. Sound and Vibration, 114(1), 91–101.
10.
Luongo, A., Rega, G., and Vestroni, F. (1984). “Accurate nonlinear equations and a perturbation solution for the free nonplanar vibrations of inextensional beams.” Proc., Int. Conf. on Recent Advances in Struct. Dynamics, vol. 1, University of Southampton, Southampton, England, 341–350.
11.
Nayfeh, A. H., Mook, D. T., and Lobitz, D. W. (1974). “Numerical‐perturbation method for the nonlinear analysis of structural vibrations.” AIAA J., 12(9), 1222–1228.
12.
Nayfeh, A. H., and Mook, D. T. (1979). Nonlinear oscillations. John Wiley & Sons, Inc., New York, N.Y.
13.
Nishino, F., Kasemset, C., and Lee, S. L. (1973). “Variational formulation of stability problems for thin‐walled members.” Ingenieur‐Archiv, Springer‐Verlag, Berlin, Germany, 43(1), 58–68.
14.
Noor, A. K., and Balch, C. D. (1984). “Hybrid perturbation/Bubnov‐Galerkin technique for nonlinear thermal analysis.” AIAA J., 22(2), 287–294.
15.
Noor, A. K. (1985). “Hybrid analytical technique for nonlinear analysis of structures.” AIAA J., 23(6), 938–946.
16.
Novozhilov, V. V. (1961). Theory of elasticity. Pergamon Press, New York, N.Y.
17.
Rao, G. V. (1975). “Nonlinear torsional vibrations of thin‐walled beams of open section.” J. Appl. Mech., 42(1), 240–241.
18.
Rao, G. V., Raju, I. S., and Raju, K. K. (1976). “Nonlinear vibrations of beams considering shear deformation and rotary inertia.” AIAA J., 14(5), 685–687.
19.
Ray, J. D., and Bert, C. W. (1969). “Nonlinear vibrations of a beam with pinned ends.” J. Engrg. for Industry, 91(Nov.), 997–1004.
20.
Reddy, J. N., and Singh, I. R. (1981a). “Large deflections and large amplitude free vibrations of straight and curved beams.” Int. J. Numerical Methods in Engrg., 17(6), 829–852.
21.
Reddy, J. N., Huang, C. L., and Singh, I. R. (1981b). “Large deflection and large amplitude vibrations of axisymmetric circular plates.” Int. J. Numerical Methods in Engrg., 17(4), 527–541.
22.
Reddy, J. N., and Huang, C. L. (1981c). “Large amplitude free vibrations of annular plates of varying thickness.” J. Sound and Vibration, 79(3), 387–396.
23.
Rozmarynowski, B., and Szymczak, C. (1984). “Nonlinear free torsional vibrations of thin‐walled beams with bisymmetric cross‐section.” J. Sound and Vibration, 97(1), 145–152.
24.
Sarma, B. S., and Varadan, T. K. (1984). “Ritz finite element approach to nonlinear vibrations of beams.” Int. J. Numerical Methods in Engrg., 20(2), 353–367.
25.
Sarma, B. S., and Varadan, T. K. (1985), “Ritz finite element approach to nonlinear vibrations of a Timoshenko beam.” Communications in Appl. Numerical Methods, 1(1), 23–32.
26.
Sarma, B. S., Varadan, T. K., and Prathap, G. (1988). “On various formulations of large amplitude free vibrations of beams.” Comp. and Struct., 29(6), 959–966.
27.
Sathyamoorthy, M. (1982a). “Nonlinear analysis of beams, part I: A survey of recent advances.” The Shock and Vibration Dig., 14(8), 19–35.
28.
Sathyamoorthy, M. (1982b). “Nonlinear analysis of beams, part II: Finite element methods.” The Shock and Vibration Dig., 14(9), 7–18.
29.
Singh, G., Sharma, A. K., and Rao, G. V. (1990). “Large‐amplitude free vibrations of beams—a discussion on various formulations and assumptions.” J. Sound and Vibration, 142(1), 77–85.
30.
Srinivasan, A. V. (1966). “Nonlinear vibrations of beams and plates.” Int. J. Nonlinear Mech., 1(3), 179–191.
31.
Szemplinska‐Stupnicka, W. (1983). “Nonlinear normal modes and the generalized Ritz method in the problems of vibrations of nonlinear elastic continuous systems.” Int. J. Nonlinear Mech., 18(2), 149–165.
32.
Vlasov, V. Z. (1961). “Thin‐walled elastic beams.” Israel Program for Scientific Translations. Office of Technical Services, U.S. Dept. of Commerce, Washington, DC.
33.
Wagner, H. (1965). “Large‐amplitude free vibrations of a beam.” J. Appl. Mech., 32(4), 887–892.
Information & Authors
Information
Published In
Journal of Engineering Mechanics
Volume 119 • Issue 4 • April 1993
Pages: 786 - 800
Copyright
Copyright © 1993 American Society of Civil Engineers.
History
Received: Nov 24, 1992
Published online: Apr 1, 1993
Published in print: Apr 1993
Authors
Metrics & Citations
Metrics
Citations
Download citation
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.