FINITE ELEMENT ANALYSIS OF SHEAR-DEFORMATION AND ROTATORY INERTIA FOR BEAM VIBRATION

Autores

  • Ana Carolina Azevedo Vasconcelos Universidade Federal do Piauí
  • Anderson Soares da Costa Azevêdo
  • Simone dos Santos Hoefel

DOI:

https://doi.org/10.26512/ripe.v2i34.21810

Palavras-chave:

Finite element method. Timoshenko. Critical frequency.

Resumo

Vibration analysis of a beam is an important subject of study in engineering. All real physical structures, when subjected to loads or displacements, behave dynamically. In case of structure with large aspect ratio of height and length the Timoshenko beam theory (TBT) is used, instead of the Euler-Bernoulli theory (EBT), since it takes both shear and rotary inertia into account. Shear effect is extremely large in higher vibration modes due to reduced mode half wave length. In this paper, the full development and analysis of TBT for the transversely vibrating uniform beam are presented for classical boundary condition. Finally, a finite element is developed in terms of dimensionless parameters of rotatory and shear. The stiffness and mass matrices for a two-node beam element with two degree of freedom per node is obtained based upon Hamilton’s principle. Cubic and quadratic Lagrangian polynomials are made interdependent by requiring them to satisfy both of the homogeneous differential equations associated with TBT. Numerical examples are given for some boundary conditions. The results showed that for frequencies above critical frequency, Timoshenko beams presents distinct mode shapes behavior including the presence of double eigenvalues, shear mode or remarkably modes.

Downloads

Não há dados estatísticos.

Referências

Anderson, R. A., 1953. Flexural vibration in uniform beams according to the Timoshenko theory. Journal of Applied Mechanics, vol. 20, pp. 504-510.

Archer, J. S., 1965. Consistent matrix formulations for structural analysis using finite element techniques. AIAA Journal, vol, 3, pp. 1910-1918.

Azevedo, A. C., Soares, A., & Hoefel, S. S., 2016. The second spectrum of Timoshenko beam. In Proceedings of the IX Congresso Nacional de Engenharia Mecˆanica - CONEM 2016, Fortaleza, Brazil.

Carnegie, W., Thomas, J., & Dokumci, E., 1969. An improved method of matrix displacement analysis in vibration problems. Aeronautical Quarterly Journal, vol. 20, pp. 321-332.

Cowper, G. R., 1966. The shear coefficient in Timoshenko’s beam theory. Journal of Applied Mechanics, vol. 33, pp. 335-340.

Davis, R., Henshelland, R. D., & Warburton, G. B., 1972. A Timoshenko beam element. Journal of Sound and Vibration, vol. 22, pp. 475-487.

Dolph, C., 1954. On the Timoshenko theory of transverse beam vibrations. Quarterly of Applied Mathematics, vol. 12, pp. 175-187.

Dong, S. B., & Secor, G. A., 1973. Effect of transverse shear deformation on vibrations of planar structures composed of beam-type elements. The Journal of the Acoustical Society of

America, vol. 53, pp. 12-127.

Downs, B., 1976. Vibration of a uniform, simply supported Timoshenko beam without transverse deflection. Journal of Applied Mechanics, vol. 43, pp. 671-674.

Egle, D. M., 1969. An approximate theory for transverse shear deformation and rotary inertia effects in vibrating beams. NASA-1317.

Euler, L., & Bousquet, M. M. C., 1744. Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes, sive Solutio problematis isoperimetrici latissimo sensu accepti. Apud

Marcum-Michaelem Bousquet & Socios

Friedman, Z., & Kosmatka, J. B., 1993. An improved two-node Timoshenko beam finite element. Computers & Structures, vol. 47, pp. 473-481.

Han, S. M., Benaroya, H., & Wei, T., 1999. Dynamics of transversely vibrating beams using four engineering theories. Journal of Sound and Vibration, vol. 225, pp. 935-988.

Huang, T. C., 1961. The effect of rotatory inertia and of shear deformation on the frequency and normal mode equations of uniform beams with simple end conditions. Journal of Applied

Mechanics, vol. 28, pp. 579-584.

Hughes, T. J. R., Taylor, R. L., & Kanoknukulchoii, W., 1977. A simple and efficient plate element for bending. International Journal for Numerical Methods in Engineering, vol. 11, pp. 1529-1943.

Levinson, M., & Cooke, D. W., 1982. On the two frequency spectra of Timoshenko beams. Journal of Sound and Vibration, vol. 84, pp. 319-326.

McCalley, R. B., 1963. Rotary inertia correction for mass matrices. General Electric Knolls Atomic Power Laboratory, vol. Report DIG/SA, pp. 63-73.

Nickell, R. E., & Secor, G. A., 1972. Convergence of consistently derived Timoshenko beam finite elements. International Journal for Numerical Methods in Engineering, vol. 5, pp. 243-253.

Przemieniecki, J. S., 1968. Theory of matrix structural analysis.

Rayleigh, 1877. On progressive waves. In Proceedings of the London Mathematical Society, London, vol. IX, pp. 21-26.

Severn, R. T., 1970. Inclusion of shear deformation in the stiffness matrix for a beam element. Strain Analysis, vol. 5, pp. 239-241.

Smith, R. W. M., 2008. Graphical representation of Timoshenko beam modes for clampedclamped boundary conditions at high frequency: Beyond transverse deflection. Wave Motion, vol. 45, pp. 785-794.

Soares, A., & Hoefel, S. S., 2015. Modal analysis for free vibration of four beam theories. In Proceedings of the 23rd International Congress of Mechanical Engineering - COBEM 2015, Rio de Janeiro, Brazil.

Tessler, A., & Dong, S. B., 1981. On a hierarchy of conforming Timoshenko beam elements. Computers and Structures, vol. 14, pp. 335-344.

Thomas, D. L., Wilson, J. M. W., & Wilson, R. R., 1973. Timoshenko beam finite elements. Journal of Sound and Vibration, vol. 31, pp. 315-330.

Thomas, J., & Abbas,B. A. H., 1975. Finite element model for dynamic analysis of Timoshenko beam. Journal of Sound and Vibration, vol. 41, pp. 291-299.

Timoshenko, S. P., 1921. On the correction for shear of the differential equation for transverse vibration of prismatic bars. Philosophical Magazine, vol. 41, pp. 744-746.

Traill-Nash, R. W., & Collar, A. R., 1953. The effects of shear flexibility and rotatory inertia on bending vibrations beams. Quaterly Journal of Mechanics and Applied Mathematics, vol. 6, pp. 186-213.

Downloads

Publicado

2017-08-07

Como Citar

Vasconcelos, A. C. A., Costa Azevêdo, A. S. da, & Hoefel, S. dos S. (2017). FINITE ELEMENT ANALYSIS OF SHEAR-DEFORMATION AND ROTATORY INERTIA FOR BEAM VIBRATION. Revista Interdisciplinar De Pesquisa Em Engenharia, 2(34), 86–103. https://doi.org/10.26512/ripe.v2i34.21810

Artigos mais lidos pelo mesmo(s) autor(es)