Structural Engineering and Mechanics

  • 제55권3호
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  • Pages.655-674
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  • 2015
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  • 1225-4568(pISSN)
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  • 1598-6217(eISSN)
테크노프레스 (Techno-Press)
권호 표지
DOI QR코드

DOI QR Code

A new analytical approach for determination of flexural, axial and torsional natural frequencies of beams

  • Mohammadnejad, Mehrdad (Department of Civil Engineering, Faculty of Engineering, Birjand University of Technology)
  • 투고 : 2015.03.26
  • 심사 : 2015.07.07
  • 발행 : 2015.08.10
⟨ 이전 논문 다음 논문 ⟩

초록

In this paper, a new and simplified method is presented in which the natural frequencies of the uniform and non-uniform beams are calculated through simple mathematical relationships. The various vibration problems such as: Rayleigh beam under variable axial force, axial vibration of a bar with and without end discrete spring, torsional vibration of a bar with an attached mass moment of inertia, flexural vibration of the beam with laterally distributed elastic springs and also flexural vibration of the beam with effects of viscose damping are investigated. The governing differential equations are first obtained and then; according to a harmonic vibration, are converted into single variable equations in terms of location. Through repetitive integrations, the governing equations are converted into weak form integral equations. The mode shape functions of the vibration are approximated using a power series. Substitution of the power series into the integral equations results in a system of linear algebraic equations. The natural frequencies are determined by calculation of a non-trivial solution for system of equations. The efficiency and convergence rate of the current approach are investigated through comparison of the numerical results obtained with those obtained from other published references and results of available finite element software.

키워드

참고문헌

  1. Bambill, D.V., Felix, D.H. and Rossi, R.E. (2010), "Vibration analysis of rotating Timoshenko beams by means of the differential quadrature method", Struct. Eng. Mech., 34(2), 231-245. https://doi.org/10.12989/sem.2010.34.2.231
  2. Banerjee, J.R. and Jackson, D.R. (2013), "Free vibration of a rotating tapered Rayleigh beam: a dynamic stiffness method of solution", J. Comput. Struct., 124, 11-20. https://doi.org/10.1016/j.compstruc.2012.11.010
  3. Chalah, F., Djellab, S.E., Chalah-Rezgui, L., Falek, K. and Bali, A. (2013), "Tapered beam axial vibration frequency: linear cross-area variation case", 2nd International Conference on Civil Engineering - ICCEN 2013, Stockholm, Sweden, December.
  4. Clough, R.W. and Penzien, J. (1975), Dynamics of Structures, McGraw-Hill Book Company, New York.
  5. Hijmissen, J.W. and Horssen, W.T.V. (2008), "On transverse vibrations of a vertical Timoshenko beam", J. Sound Vib., 314(1-2), 161-179. https://doi.org/10.1016/j.jsv.2007.12.039
  6. Huang, Y., Yang, L.E. and Luo, Q.Z. (2013), "Free vibration of axially functionally graded Timoshenko beams with non-uniform cross-section", Compos. Part B, 45(1), 1493-1498. https://doi.org/10.1016/j.compositesb.2012.09.015
  7. Huang, Y. and Li, X.F. (2010), "A new approach for free vibration of axially functionally graded beams with non-uniform cross-section", J. Sound Vib., 329, 2291-2303. https://doi.org/10.1016/j.jsv.2009.12.029
  8. Kelly, S.G. (2007), Advanced Vibration Analysis, CRC Press, Taylor & Francis Group, Boca Raton, FL, USA.
  9. Li, X.F., Tang, A.Y. and Xi, L.Y. (2013), "Vibration of a Rayleigh cantilever beam with axial force and tip mass", J. Construct. Steel Res., 80, 15-22. https://doi.org/10.1016/j.jcsr.2012.09.015
  10. Li, Q.S. (2003), "Torsional vibration of multi-step non-uniform rods with various concentrated elements", J. Sound Vib., 260(4), 637-651. https://doi.org/10.1016/S0022-460X(02)01010-6
  11. Maiz, S., Bambill, D.V., Rossit, C.A. and Laura, P.A.A. (2007), "Transverse vibration of Bernoulli-Euler beams carrying point masses and taking into account their rotatory inertia: exact solution", J Sound Vib, 303(3-5), 895-908. https://doi.org/10.1016/j.jsv.2006.12.028
  12. Pouyet, J.M. and Lataillade, J.L. (1981), "Torsional vibration of a shaft with non-uniform cross-section", J. Sound Vib., 76(1), 13-22. https://doi.org/10.1016/0022-460X(81)90287-X
  13. Pradhan, K.K. and Chakraverty, S. (2013), "Free vibration of Euler and Timoshenko functionally graded beams by Rayleigh-Ritz method", Compos. Part B, 51, 175-184. https://doi.org/10.1016/j.compositesb.2013.02.027
  14. Rayleigh, L. (1877), Theory of Sound, Macmillan, Second Edition, New York.
  15. Saffari, H., Mohammadnejad, M. and Bagheripour, M.H. (2012), "Free vibration analysis of non-prismatic beams under variable axial forces", Struct. Eng. Mech., 43(5), 561-582. https://doi.org/10.12989/sem.2012.43.5.561
  16. Sapountzakis, E.J. and Dourakopoulos, J.A. (2010), "Shear deformation effect in flexural-torsional buckling analysis of beams of arbitrary cross section by BEM", Struct. Eng. Mech., 35(2), 141-173. https://doi.org/10.12989/sem.2010.35.2.141
  17. Saapountzakis, E.J. (2000), "Solutions of non-uniform torsion of bars by an integral equation method", J. Comput. Struct., 77(6), 659-667. https://doi.org/10.1016/S0045-7949(00)00020-1
  18. Stojanovic, V. and Kozic, P. (2012), "Forced transverse vibration of Rayleigh and Timoshenko double-beam system with effect of compressive axial load", Int. J. Mech. Sci., 60(1), 59-71. https://doi.org/10.1016/j.ijmecsci.2012.04.009
  19. Timoshenko, S.P. (1922), "On the transverse vibrations of bars of uniform cross-section", Phil. Mag., Series 6, 43(253), 125-131. https://doi.org/10.1080/14786442208633855
  20. Timoshenko, S.P. (1921), "On the correction for shear of the differential equation for transverse vibration of prismatic bars", Phil. Mag., Series 6, 41(245), 744-746. https://doi.org/10.1080/14786442108636264
  21. Wright, A.D., Smith, C.E., Thresher, R.W. and Wang, J.L.C. (1982), "Vibration modes of centrifugally stiffened beams", ASME J. Appl. Mech., 49(2), 197-202. https://doi.org/10.1115/1.3161966
  22. Yan, S.X., Zhang, Z.P., Wei, D.J. and Li, X.F. (2011), "Bending vibration of rotating tapered cantilevers by the integral equation method", AIAA J., 49(4), 872-876. https://doi.org/10.2514/1.J050572
  23. Yesilce, Y. (2015), "Differential transform method and numerical assembly technique for free vibration analysis of the axial-loaded Timoshenko multiple-step beam carrying a number of intermediate lumped masses and rotary inertias", Struct. Eng. Mech., 53(3), 537-573. https://doi.org/10.12989/sem.2015.53.3.537
  24. Yesilce, Y. and Demirdag, O. (2008), "Effect of axial force on free vibration of Timoshenko multi-span beam carrying multiple spring-mass systems", Int. J. Mech. Sci., 50(6), 995-1003. https://doi.org/10.1016/j.ijmecsci.2008.03.001

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