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The vibrations of pre-twisted rotating Timoshenko beams by the Rayleigh–Ritz method

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Abstract

A modeling method for flapwise and chordwise bending vibration analysis of rotating pre-twisted Timoshenko beams is introduced. In the present modeling method, the shear and the rotary inertia effects on the modal characteristics are correctly included based on the Timoshenko beam theory. The kinetic and potential energy expressions of this model are derived from the Rayleigh–Ritz method, using a set of hybrid deformation variables. The equations of motion of the rotating beam are derived from the kinetic and potential energy expressions introduced in the present study. The equations thus derived are transmitted into dimensionless forms in which main dimensionless parameters are identified. The effects of dimensionless parameters such as the hub radius ratio, slenderness ration, etc. on the natural frequencies and modal characteristics of rotating pre-twisted beams are successfully examined through numerical studies. Finally the resonance frequency of the rotating beam is evaluated.

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References

  1. Southwell R, Gough F (1921) The free transverse vibration of airscrew blades. Br ARC Rep Memoranda o.766

  2. Liebers F (1930) Contribution to the the ory of propeller vibration. NACA TM No. 568

  3. Theodorsen T (1950) Propeller vibrations and the effect of centrigual force. NACA TN No. 516

  4. Schilhansl M (1958) Bending frequency of rotating cantilever beam. J Appl Mech Trans Am Soc Mech Eng 25: 28–30

    MATH  MathSciNet  Google Scholar 

  5. Putter S, Manor H (1978) Natural frequencies of radial rotating beams. J Sound Vib 56: 175–185

    Article  MATH  Google Scholar 

  6. Hodges DH, Dowell EH (1974) Nonlinear equations of motion for the elastic bending and torsion of twisted nonuniform rotor blades. NASA TN D-7818

  7. Hodges D (1979) Vibration and response of nonuniform rotating beams with discontinuities. J Am Helicopter Soc 24: 43–50

    Article  Google Scholar 

  8. Hoa J (1979) Vibration of a rotating beam with tip mass. J Sound Vib 67: 369–381

    Article  MATH  Google Scholar 

  9. Bhat R (1986) Transverse vibrations of a rotating uniform cantilever beam with tip mass as predicted by using beam characteristic orthogonal polynomials in the Rayleigh–Ritz method. J Sound Vib 105: 199–210

    Article  Google Scholar 

  10. Fox C, Burdress J (1979) The natural frequencies of a thin rotating cantilever with offset root. J Sound Vib 65:151–158

    Article  MATH  Google Scholar 

  11. Yokoyama T (1988) Free vibration characteristics of rotating Timoshenko beams. Int J Mech Sci 30: 743–755

    Article  MATH  Google Scholar 

  12. Lee S, Lin S (1994) Bending vibrations of rotating nonuniform Timoshenko beams with an elastically restrained root. J Appl Mech 61: 949–955

    Article  MATH  Google Scholar 

  13. Yoo HH, Shin SH (1998) Vibration analysis of rotating cantilever beams. J Sound Vib 212: 807–828

    Article  Google Scholar 

  14. Ozgumus OO, Kaya MO (2008) Flapwise bending vibration analysis of a rotating double-tapered Timoshenko beam. Arch Appl Mech 78: 379–392

    Article  MATH  Google Scholar 

  15. Dawson B (1968) Coupled bending vibrations of pretwisted cantilever blading treated by Rayleigh–Ritz method. J Mech Eng Sci 10(5): 381–386

    Article  Google Scholar 

  16. Dawson B, Carnegie W (1969) Modal curves of pretwisted beams of rectangular cross-section. J Mech Eng Sci 11(1): 1–13

    Article  Google Scholar 

  17. Carnegie W, Thomas J (1972) The coupled bending-bending vibration of pre-twisted tapered blading. J Eng Indus 94: 255–266

    Article  Google Scholar 

  18. Dokumaci E, Thomas J, Carnegie W (1967) Matrix displacement analysis of coupled bending-bending vibrations of pre-twsited blading. J Mech Eng Sci 9: 247–251

    Article  Google Scholar 

  19. Rao JS (1992) Advanced theory of vibration. Wiley, New York

    Google Scholar 

  20. Gupta RS, Rao JS (1978) Finite element eigenvalue analysis of tapared and twisted timoshenko beams. J Sound Vib 56(2): 187–200

    Article  MATH  Google Scholar 

  21. Celep Z, Turhan D (1986) On the influence of pretwisting on the vibration of beams including the shear and rotatory inertia effects. J Sound Vib 110(3): 523–528

    Article  Google Scholar 

  22. Lin SM (1997) Vibrations of elastically restrained nonuniform beams with arbitrary pretwist. AIAA J 35(11): 1681–1687

    Article  MATH  Google Scholar 

  23. Ramamurti V, Kielb R (1984) Natural frequencies of twisted rotating plate. J Sound Vib 97: 429–449

    Article  Google Scholar 

  24. Subrahmanyam KB, Kaza KRV (1986) Vibration and buckling of rotating, pretwisted, preconed beams including coriolis effects. J Vib Acoustics Stress Reliab Des 108(2): 140–149

    Google Scholar 

  25. Subrahmanyam KB, Kulkarni SV, Rao JS (1982) Analysis of lateral vibrations of rotating cantilever blades allowing for shear deflection and rotary inertia by reissner and potential energy methods. Mech Mach Theory 17(4): 235–241

    Article  Google Scholar 

  26. Lin SM (2001) The instability and vibration of rotating beams with arbitrary pretwist and an elastically restrained root. ASME J Appl Mech 68: 844–853

    Article  MATH  Google Scholar 

  27. Lin SM, Wu CT, Lee SY (2006) Analysis of rotating nonuniform pretwisted beams with an elastically restrained root and a tip mass. Int J Mech Sci 45: 741–755

    Article  Google Scholar 

  28. Frisch H (1975) A vector-dynamic development of the equations of motion for N-coupled flexible bodies and point masses. NASA TN D-8047

  29. Ho J (1977) Direct path method for flexible multibody spacecraft dynamics. J Spacecrafts Rockets 14: 102–110

    Article  Google Scholar 

  30. Kane T, Ryan R, Banerjee A (1987) Dynamics of a cantilever beam attached to a moving base. J Guidance Control Dyn 10: 139–151

    Article  Google Scholar 

  31. Yoo H, Ryan R, Scott R (1995) Dynamics of flexible beam undergoing overall motions. J Sound Vib 181: 261–278

    Article  Google Scholar 

  32. Yoo HH, Park JH, Park J (2001) Vibration analysis of rotating pre-twisted blades. Comput Struct 79: 1811–1819

    Article  Google Scholar 

  33. Yoo HH, Lee SH, Shin SH (2005) Flapwise bending vibration analysis of rotating multi-layered composite beams. J Sound Vib 286: 745–761

    Article  Google Scholar 

  34. Eisenhart L (1947) An Introduction to differential geometry. Princeton University Press, Princeton

    MATH  Google Scholar 

  35. Timoshenko S (1990) Vibration problems in engineering, 5th edn, Wiley. New York

    Google Scholar 

  36. Craig RR (1981) Structural dynamics: an introduction to computer methods. Wiley, New York

    Google Scholar 

  37. Cowper GR (1966) The shear coefficient in Timoshenko’s beam theory. J Appl Mech 33(2): 335–340

    MATH  Google Scholar 

  38. Reissner E (1945) The effect of transverse shear deformation on the bending of elastic plates. J Appl Mech 12: A68–A77

    MathSciNet  Google Scholar 

  39. Vlachoutsis S (1992) Shear correction factors for plates and shells. Int J Numer Methods Eng 33(7): 1537–1552

    Article  MATH  Google Scholar 

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  1. Department of Mechanical Engineering, Widener University, One University Place, Chester, PA, 19013, USA

    T.-L. Zhu

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  1. T.-L. Zhu
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Correspondence to T.-L. Zhu.

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Zhu, TL. The vibrations of pre-twisted rotating Timoshenko beams by the Rayleigh–Ritz method. Comput Mech 47, 395–408 (2011). https://doi.org/10.1007/s00466-010-0550-9

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  • DOI: https://doi.org/10.1007/s00466-010-0550-9

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