×

On coupled bending-torsional vibrations of beams with initial loads. (English) Zbl 1258.74112

Summary: The objective of this paper is to analyse the free vibration and mode shapes of straight beams where the coupling between the bending and torsion is induced by steady state lateral loads. The governing differential equations and boundary conditions for coupled vibrations of Euler-Bernoulli-Vlasov beams are performed by using the virtual work principle which includes the second order terms of finite beam rotations. Closed form solution is derived for the coupled frequencies and mode shapes of a symmetric beam with simply supported ends under uniform bending. A finite element model with seven degrees of freedoms per node is also presented. To illustrate the accuracy of this formulation, numerical solutions are presented and compared with available solutions.

MSC:

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

References:

[1] Arpaci, A.; Bozdag, E.: On the free vibration analysis of thin-walled beams with nonsymmetric open cross sections, Computers and structures 80, 691-695 (2002)
[2] Attard, M.: Lateral buckling analysis of beams by the FEM, Computers and structures 23, No. 2, 217-231 (1986) · Zbl 0582.73044 · doi:10.1016/0045-7949(86)90214-2
[3] Banerjee, J. R.; Williams, F. W.: Coupled bending – torsional dynamic stiffness matrix of an axially loaded Timoshenko beam element, International journal of solids and structures 31, 749-762 (1994) · Zbl 0810.73022 · doi:10.1016/0020-7683(94)90075-2
[4] Bishop, R. E. D.; Johnson, D. C.: The mechanics of vibration, (1979) · Zbl 1220.74001
[5] Bishop, R. E. D.; Cannon, S. M.; Miao, S.: On coupled bending and torsional vibration of uniform beams, Journal of sound and vibration 131, 457-464 (1989) · Zbl 1235.74158
[6] Chen, H. H.; Hsiao, K. M.: Coupled axial – torsional vibration of thin walled Z-section beam induced by boundary conditions, Thin-walled structures 45, 573-583 (2007)
[7] Dokumaci, E.: An exact solution for coupled bending and torsion vibrations of uniform beams having single cross-sectional symmetry, Journal of sound and vibration 119, 443-449 (1987)
[8] Hijmissen, J. W.; Horssen, W. T.: On transverse vibration of a vertical Timoshenko beam, Journal of sound and vibration 314, 161-179 (2008)
[9] Kim, M. Y.; Chang, S. P.; Park, H. G.: Spatial postbuckling analysis of nonsymmetric thin-walled frames. I: theoretical considerations based on semitangential property, Journal of engineering mechanics (ASCE) 127, No. 8, 769-778 (2001)
[10] Kim, M. Y.; Chang, S. P.; Kim, S. B.: Spatial postbuckling analysis of nonsymmetric thin-walled frames. II: geometrically nonlinear FE procedures, Journal of engineering mechanics (ASCE) 127, No. 8, 779-790 (2001)
[11] Kim, M. Y.; Yun, H. T.; Kim, N. I.: Exact dynamic and static element stiffness matrices of nonsymmetric thin-walled beam-columns, Computers and structures 81, No. 7 – 8, 1425-1448 (2003)
[12] Kollár, P. L.: Flexural – torsional vibration of open section composite beams with shear deformation, International journal of solids and structures 38, 7543-7558 (2001) · Zbl 1022.74014 · doi:10.1016/S0020-7683(01)00025-7
[13] Li, J.; Li, W.; Shen, R.; Hua, H.: Coupled bending and torsional vibration of nonsymmetrical axially loaded thin-walled Bernoulli – Euler beams, Mechanics research communications 31, 697-711 (2004) · Zbl 1098.74595 · doi:10.1016/j.mechrescom.2004.04.005
[14] Orloske, K.; Leamy, M. J.; Parker, R. G.: Flexural-torsional buckling of misaligned axially moving beams, part I: Three-dimensional modeling, equilibria, and bifurcations, International journal of solids and structures 43, 4297-4322 (2006) · Zbl 1120.74583 · doi:10.1016/j.ijsolstr.2005.08.014
[15] Orloske, K.; Parker, R. G.: Flexural-torsional buckling of misaligned axially moving beams, part ii: Vibration and stability analysis, International journal of solids and structures 43, 4323-4341 (2006) · Zbl 1120.74584 · doi:10.1016/j.ijsolstr.2005.08.015
[16] Prokić, A.: On fivefold coupled vibrations of Timoshenko thin-walled beams, Engineering structures 28, 54-62 (2006)
[17] Sapountzakis, E. J.; Tsiatas, G. C.: Flexural-torsional vibrations of beams by BEM, Computational mechanics 39, 409-417 (2007) · Zbl 1188.74079 · doi:10.1007/s00466-006-0039-8
[18] Tanaka, M.; Bercin, A. N.: Free vibration solution for uniform beams of nonsymmetric cross section using Mathematica, Computers and structures 71, No. 7 – 8, 1-8 (1999)
[19] Vörös, G. M.: An improved formulation of space stiffeners, Computers and structures 85, No. 7 – 8, 350-359 (2007)
[20] Vörös, G. M.: Buckling and free vibration analysis of stiffened panels, Thin-walled structures (2008)
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.