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Free Flapping Vibration of Rotating Inclined Euler Beams

Authors: Chih-Ling Huang, Wen-Yi Lin, Kuo-Mo Hsiao


A method based on the power series solution is proposed to solve the natural frequency of flapping vibration for the rotating inclined Euler beam with constant angular velocity. The vibration of the rotating beam is measured from the position of the corresponding steady state axial deformation. In this paper the governing equations for linear vibration of a rotating Euler beam are derived by the d'Alembert principle, the virtual work principle and the consistent linearization of the fully geometrically nonlinear beam theory in a rotating coordinate system. The governing equation for flapping vibration of the rotating inclined Euler beam is linear ordinary differential equation with variable coefficients and is solved by a power series with four independent coefficients. Substituting the power series solution into the corresponding boundary conditions at two end nodes of the rotating beam, a set of homogeneous equations can be obtained. The natural frequencies may be determined by solving the homogeneous equations using the bisection method. Numerical examples are studied to investigate the effect of inclination angle on the natural frequency of flapping vibration for rotating inclined Euler beams with different angular velocity and slenderness ratio.

Keywords: natural frequency, inclination angle, Flapping vibration, Rotating beam

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[1] M. J. Schilhansl Bending frequency of a rotating cantilever beamASME Journal of Applied Mechanics vol. 25 pp. 28-30, 1958.
[2] J. T. S. Wang, O. Mahrenholtz and J. Bohm, Extended Galerkin's method for rotating beam vibrations using Legendre polynomials SolidMechanics Archives, vol. 1, pp. 341-365, 1976.
[3] A. Leissa Vibrational aspects of rotating turbomachinery bladesASME Applied Mechanics Reviews vol. 34 pp. 629-635, 1981.
[4] D. H. Hodges and M. J. Rutkowski, Free-vibration analysis of rotating beams by a variable-order finite-element method, AIAA Journal, vol. 19, pp. 1459-1466, 1981.
[5] T. Yokoyama Free vibration characteristics of rotating Timoshenko beam International Journal of Mechanical Science vol. 30 pp. 743-755, 1988.
[6] H. H. Yoo and S. H. Shin, Vibration analysis of rotating cantilever beams Journal of Sound and vibration, vol. 212, pp. 807-828, 1988.
[7] S. Y. Lee and Y. H. Kuo Bending frequency of a rotating beam with an elastically restrained root ASME Journal of Applied Mechanics vol. 58pp. 209-214, 1991.
[8] H. P. Lee, Vibration on an inclined rotating cantilever beam with tip mass, ASME Journal of Vibration and Acoustics, vol. 115, pp. 241245,1993.
[9] S. C. Lin and K. M. Hsiao Vibration analysis of rotating Timoshenko beam Journal of Sound and Vibration vol. 240, pp. 303-322, 2001.
[10] A. A. Al-Qaisia, Non-linear dynamics of a rotating beam clamped with an attachment angle and carrying an inertia element, The Arabian Journal for Science and Engineering, vol. 29, pp. 81-98, 2004.
[11] S. Y. Lee and J. J. Sheu, Free vibrations of a rotating inclined beam, ASME Journal of Applied Mechanics vol. 74 pp. 406-414, 2007.
[12] S. Y. Lee and J. J. Sheu, Free vibration of an extensible rotating inclined Timoshenko beam, Journal of Sound and Vibration, vol. 304, pp. 606-624, 2007.
[13] J. C. Simo and K. Vu-Quac The role of non-linear theories in transient dynamic analysis of flexible structures Journal of Sound and Vibrationvol. 119, pp. 487-508, 1987.
[14] K. M. Hsiao, Corotational total Lagrangian formulation for three-dimensional beam element, AIAA Journal, vol. 30, pp. 797-804, 1992.
[15] K. M. Hsiao, R. T. Yang, and A. C. Lee, A cosistent finite element formulation for nonlinear dynamic analysis of planar beam, International Journal for Numerical Methods in Engineering, vol. 37, pp. 75-89, 1994
[16] P. W. Likins, Mathematical modeling of spinning elastic bodies for model analysis AIAA Journal, vol. 11, pp. 1251-1258, 1973.