Abstract
The mathematical modelling of a rotating tapered Timoshenko beam with preset and pre-twist angles is constructed. The partial differential equations governing the six degrees, i.e., three displacements in the axial, flapwise, and edgewise directions and three cross-sectional angles of torsion, flapwise bending, and edgewise bending, are obtained by the Euler angle descriptions. The power series method is then used to investigate the natural frequencies and the corresponding complex mode functions. It is found that all the natural frequencies are increased by the centrifugal stiffening except the twist frequency, which is slightly decreased. The tapering ratio increases the first transverse, torsional, and axial frequencies, while decreases the second transverse frequency. Because of the pre-twist, all the directions are gyroscopically coupled with the phase differences among the six degrees.
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References
Wright, A. D., Smith, C. E., Thresher, R.W., and Wang, J. L. C. Vibration modes of centrifugally stiffened beams. Journal of Applied Mechanics, 49, 197–202 (1982)
Kane, T. R., Ryan, R. R., and Banerjee, A. K. Dynamics of a cantilever beam attached to a moving base. Journal of Guidance Control and Dynamics, 10, 139–151 (1987)
Naguleswaran, S. Lateral vibration of a centrifugally tensioned uniform Euler-Bernoulli beam. Journal of Sound and Vibration, 176, 613–624 (1994)
Huang, Y., Deng, Z., and Yao, L. Dynamic analysis of a rotating rigid-flexible coupled smart structurewith large deformations. Applied Mathematics and Mechanics (English Edition), 28, 1349–1360 (2007) DOI 10.1007/s10483-007-1008-z
Surace, G., Anghel, V., and Mares, C. Coupled bending-bending-torsion vibration analysis of rotating pretwisted blades: an integral formulation and numerical examples. Journal of Sound and Vibration, 206, 473–486 (1997)
Liu, K. C., Friend, J., and Yeo, L. The axial-torsional vibration of pretwisted beams. Journal of Sound and Vibration, 321, 115–136 (2009)
Ghafarian, M. and Ariaei, A. Free vibration analysis of a system of elastically interconnected rotating tapered Timoshenko beams using differential transform method. International Journal of Mechanical Sciences, 107, 93–109 (2016)
Huo, Y. and Wang, Z. Dynamic analysis of a rotating double-tapered cantilever Timoshenko beam. Archive of Applied Mechanics, 86, 1147–1161 (2015)
Huang, J. L. and Zhu, W. D. Nonlinear dynamics of a high-dimensional model of a rotating Euler-Bernoulli beam under the gravity load. Journal of Applied Mechanics, 81, 101007 (2014)
Kim, H., Yoo, H., and Chung, J. Dynamic model for free vibration and response analysis of rotating beams. Journal of Sound and Vibration, 332, 5917–5928 (2013)
Sinha, S. K. and Turner, K. E. Natural frequencies of a pre-twisted blade in a centrifugal force field. Journal of Sound and Vibration, 330, 2655–2681 (2011)
Banerjee, J. R., Papkov, S. O., Liu, X., and Kennedy, D. Dynamic stiffness matrix of a rectangular plate for the general case. Journal of Sound and Vibration, 342, 177–199 (2015)
Banerjee, J. R. and Kennedy, D. Dynamic stiffness method for inplane free vibration of rotating beams including Coriolis effects. Journal of Sound and Vibration, 333, 7299–7312 (2014)
Banerjee, J. R. Dynamic stiffness formulation and free vibration analysis of centrifugally stiffened Timoshenko beams. Journal of Sound and Vibration, 247, 97–115 (2001)
Banerjee, J. R. Free vibration of centrifugally stiffened uniform and tapered beams using the dynamic stiffness method. Journal of Sound and Vibration, 233, 857–875 (2000)
Chung, J. and Yoo, H. H. Dynamic analysis of a rotating cantilever beam by using the finite element method. Journal of Sound and Vibration, 249, 147–164 (2002)
Hashemi, S. M. and Richard, M. J. Natural frequencies of botating uniform beams with coriolis effects. Journal of Vibration and Acoustics, 123, 444 (2001)
Du, H., Lim, M. K., and Liew, K.M. A power-series solution for vibration of a rotating Timoshenko beam. Journal of Sound and Vibration, 175, 505–523 (1994)
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Project supported by the National Natural Science Foundation of China (Nos. 11672007, 11402028, 11322214, and 11290152), the Bei**g Natural Science Foundation (No. 3172003), and the Key Laboratory of Vibration and Control of Aero-Propulsion System Ministry of Education, Northeastern University (No.VCAME201601)
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Yang, X., Wang, S., Zhang, W. et al. Dynamic analysis of a rotating tapered cantilever Timoshenko beam based on the power series method. Appl. Math. Mech.-Engl. Ed. 38, 1425–1438 (2017). https://doi.org/10.1007/s10483-017-2249-6
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DOI: https://doi.org/10.1007/s10483-017-2249-6