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Vibration analysis of rotating Timoshenko beams by means of the differential quadrature method

  • Bambill, D.V. (Department of Engineering, Universidad Nacional del Sur) ;
  • Felix, D.H. (Department of Engineering, Universidad Nacional del Sur) ;
  • Rossi, R.E. (Department of Engineering, Universidad Nacional del Sur)
  • Received : 2009.04.22
  • Accepted : 2009.10.23
  • Published : 2010.01.30

Abstract

Vibration analysis of rotating beams is a topic of constant interest in mechanical engineering. The differential quadrature method (DQM) is used to obtain the natural frequencies of free transverse vibration of rotating beams. As it is known the DQM offers an accurate and useful method for solution of differential equations. And it is an effective technique for solving this kind of problems as it is shown comparing the obtained results with those available in the open literature and with those obtained by an independent solution using the finite element method. The beam model is based on the Timoshenko beam theory.

Keywords

References

  1. Al-Ansary, M.D. (1998), "Flexural vibrations of rotating beams considering rotary inertia", Comput. Struct., 69, 321-328. https://doi.org/10.1016/S0045-7949(98)00134-5
  2. ALGOR V. 20.3. (2007), Linear Mode Shapes and Natural Frequencies with Load Stiffening Module.
  3. Banerjee, J.R. (2000), "Free vibration of centrifugally stiffened uniform and tapered beams using the dynamic stiffness method", J. Sound Vib., 233, 857-875. https://doi.org/10.1006/jsvi.1999.2855
  4. Banerjee, J.R. (2001), "Dynamic stiffness formulation and free vibration analysis of centrifugally stiffened Timoshenko beam", J. Sound Vib., 247, 97-115. https://doi.org/10.1006/jsvi.2001.3716
  5. Bellman, R. and Casti, J. (1971), "Differential quadrature and long-term integration", J. Math. Anal. Appl., 34, 235-238. https://doi.org/10.1016/0022-247X(71)90110-7
  6. Bellman, R.E. and Roth, R.S. (1986), Methods in Approximation: Techniques for Mathematical Modeling, Editorial D. Reidel Publishing Company, Dordrecht, Holland.
  7. Bert, C.W. and Malik, M. (1996), "Differential quadrature method in computational mechanics: A review", Appl. Mech. Rev., 49, 1-28. https://doi.org/10.1115/1.3101882
  8. Choi, S.T., Wu, J.D. and Chou, Y.T. (2000), "Dynamic analysis of a spinning Timoshenko beam by the differential quadrature method", AIAA J., 38, 851-856. https://doi.org/10.2514/2.1039
  9. Chung, J. and Yoo, H.H. (2002), "Dynamic analysis of a rotating cantilever beam by using the finite element method", J. Sound Vib., 249, 147-164. https://doi.org/10.1006/jsvi.2001.3856
  10. Du, H., Lim, M.K. and Liew, K.M. (1994), "A power solution for vibration of a rotating Timoshenko beam", J. Sound Vib., 175, 505-523. https://doi.org/10.1006/jsvi.1994.1342
  11. Gunda, J.B. and Ganguli, R. (2008), "Stiff-string basis functions for vibration analysis of high speed rotating beams", J. Appl. Mech. - T. ASME, 75(2), 0245021-0245025.
  12. Gunda, J.B. and Ganguli, R. (2008), "New rational interpolation functions for finite element analysis of rotating beams", Int. J. Mech. Sci., 50, 578-588. https://doi.org/10.1016/j.ijmecsci.2007.07.014
  13. Gunda, J.B., Gupta, R.K. and Ganguli, R. (2009), "Hybrid stiff-string-polynomial basis functions for vibration analysis of high speed rotating beams", Comput. Struct., 87(3-4), 254-265. https://doi.org/10.1016/j.compstruc.2008.09.008
  14. Gunda, J.B., Singh, A.P., Chhabra, P.S. and Ganguli, R. (2007), "Free vibration analysis of rotating tapered blades using Fourier-p superelement", Struct. Eng. Mech., 27(2), 243-257. https://doi.org/10.12989/sem.2007.27.2.243
  15. Hodges, D.H. and Rutkowski, M.J. (1981), "Free vibration analysis of rotating beams by a variable order finite method", AIAA J., 19(11), 1459-1466. https://doi.org/10.2514/3.60082
  16. Karami, G., Malekzadeh, P. and Shahpari, S.A. (2003), "DQEM for vibration of shear deformable nonuniform beams with general boundary conditions", Eng. Struct., 25, 1169-1178. https://doi.org/10.1016/S0141-0296(03)00065-8
  17. Kumar, A. and Ganguli, R. (2009), "Rotating Beams and Nonrotating Beams with Shared Eigenpair", J. Appl. Mech., 76(5) Article Number: 051006 (14 pages).
  18. Laura, P.A.A. and Gutiérrez, R.H. (1993), "Analysis of vibrating Timoshenko beams using the method of differential quadrature", Shock Vib. Digest, 1, 9-93.
  19. Lee, S.Y. and Sheu, J.J. (2007), "Free vibration of an extensible rotating inclined Timoshenko beam", J. Sound Vib., 304(3-5), 606-624. https://doi.org/10.1016/j.jsv.2007.03.005
  20. Lin, S.C. and Hsiao, K.M. (2001), "Vibration analysis of a rotating Timoshenko beam", J. Sound Vib., 240, 303- 322. https://doi.org/10.1006/jsvi.2000.3234
  21. Liu, G.R. and Wu, T.Y. (2001), "Vibration analysis of beams using the generalized differential quadrature rule and domain decomposition", J. Sound Vib., 246(3), 461-481. https://doi.org/10.1006/jsvi.2001.3667
  22. Mei, C. (2008), "Application of differential transformation technique to free vibration analysis of a centrifugally stiffened beam", Comput. Struct., 86, 1280-1284. https://doi.org/10.1016/j.compstruc.2007.10.003
  23. Naguleswaran, S. (1994), "Lateral vibration of a centrifugally tensioned uniform Euler-Bernoulli beam", J. Sound Vib., 176(5), 613-624. https://doi.org/10.1006/jsvi.1994.1402
  24. Ouyang, H. and Wang, M. (2007), "A dynamic model for a rotating beam subjected to axially moving forces", J. Sound Vib., 308(3-5), 674-682. https://doi.org/10.1016/j.jsv.2007.03.082
  25. Rao, S.S. and Gupta, R.S. (2001), "Finite element vibration analysis of rotating Timoshenko beams", J. Sound Vib., 242(1), 103-124. https://doi.org/10.1006/jsvi.2000.3362
  26. Senatore, A. (2006), "Measuring the natural frequencies of centrifugally tensioned beam with laser doppler vibrometer", Measurement, 39, 628-633. https://doi.org/10.1016/j.measurement.2006.01.006
  27. Seon Han, M., Benaroya, H. and Wei, T. (1999), "Dynamics of transversely vibrating beams using four engineering theories", J. Sound Vib., 225, 935-988. https://doi.org/10.1006/jsvi.1999.2257
  28. Shu, C. and Chen, W. (1999), "On optimal selection of interior points for applying discretized boundary conditions in DQ vibration analysis of beams and plates", J. Sound Vib., 222(2), 239-257. https://doi.org/10.1006/jsvi.1998.2041
  29. Shu, C. (2000), Differential Quadrature and Its Application in Engineering. Editorial Springer-Verlag London Limited, Great Britain.
  30. Singh, A.P., Mani, V. and Ganguli, R. (2007) "Genetic programming metamodel for rotating beams", CMES - Comput. Model. Eng. Sci., 21(2), 133-148.
  31. Vinod, K.G., Gopalakrishnan, S. and Gangul, R. (2007), "Free vibration and wave propagation analysis of uniform and tapered rotating beams using spectrally formulated finite elements", Int. J. Solids Struct., 44, 5875-5893. https://doi.org/10.1016/j.ijsolstr.2007.02.002
  32. Yoo, H.H. and Shin, S.H. (1998), "Vibration analysis of rotating cantilever beams", J. Sound Vib., 212(5), 807- 828. https://doi.org/10.1006/jsvi.1997.1469
  33. Wolfram, S. (1996), Mathematica: A System for Doing Mathematics by Computer. Third Ed. Addison-Wesley.

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