The flapwise bending vibrational equations of tapered Rayleigh beam are derived based on Hamilton’s principle. The corresponding vibrational characteristics of rotating tapered Rayleigh beams are investigated via variational iteration method (VIM). Natural frequencies and corresponding mode shapes are examined under various rotation speed, taper ratio and slenderness ratio focusing on two types of tapered beam. The convergence of VIM is examined as part of the paper. Validation of VIM solution is made by referring to results available in other literature and corresponding results show that VIM is capable of yielding precise results in a very efficient way.
This is a preview of subscription content, access via your institution.
Buy single article
Instant access to the full article PDF.
Tax calculation will be finalised during checkout.
Banerjee, J.R., 2000. Free vibration of centrifugally stiffened uniform and tapered beams using the dynamic stiffness method, Journal of Sound and Vibration, 233(5), 857–875.
Banerjee, J.R., 2001. Dynamic stiffness formulation and free vibration analysis of centrifugally stiffened Timoshenko beams, Journal of Sound and Vibration, 247(1), 97–115.
Banerjee, J.R. and Jackson, D.R., 2013. Free vibration of a rotating tapered Rayleigh beam: a dynamic stiffness method of solution, Computers & Structures, 124, 11–20.
Banerjee, J.R. and Sobey, A.J., 2002. Energy expressions for rotating Tapered Timoshenko beams, Journal of Sound and Vibration, 254(4), 818–822.
Bazoune, A. and Khulief, Y.A., 1992. A finite beam element for vibration analysis of rotating tapered Timoshenko beams, Journal of Sound and Vibration, 156(1), 141–164.
Chen, Y.F., Zhang, J. and Zhang, H., 2016. Flapwise bending vibration of rotating tapered beams using variational iteration method, Journal of Vibration and Control, 22(15), 3384–3395.
Chen, Y.F., Zhang, J. and Zhang, H., 2017. Free vibration analysis of rotating tapered Timoshenko beams via variational iteration method, Journal of Vibration and Control, 23(2), 220–234.
Du, H., Lim, M.K. and Liew, K.M., 1994. A power series solution for vibration of a rotating Timoshenko beam, Journal of Sound and Vibration, 175(4), 505–523.
El-Sayed, T.A. and El-Mongy, H.H., 2018. Application of variational iteration method to free vibration analysis of a tapered beam mounted on two-degree of freedom subsystems, Applied Mathematical Modelling, 58, 349–364.
Eroglu, U., 2016. Large deflection analysis of planar curved beams made of functionally graded materials using variational iterational method, Composite Structures, 136, 204–216.
Gere, J.M. and Timoshenko, S.P., 1997. Mechanics of Materials, Fifth ed., PWS Publishing Company, Boston.
He, J.H., 2007. Variational iteration method-Some recent results and new interpretations, Journal of Computational and Applied Mathematics, 207(1), 3–17.
He, J.H., 2012. Notes on the optimal variational iteration method, Applied Mathematics Letters, 25(10), 1579–1581.
Hoa, S.V., 1979. Vibration of a rotating beam with tip mass, Journal of Sound and Vibration, 67(3), 369–381.
Hodges, D.H. and Rutkowski, M.J., 1981. Free-vibration analysis of rotating beams by a variable-order finite-element method, AIAA Journal, 19(11), 1459–1466.
Huang, C.H., Lin, W.Y. and Hsiao, K.M., 2010. Free vibration analysis of rotating Euler beams at high angular velocity, Computers & Structures, 88(17–18), 991–1001.
Huang, Y., Yang, L.E. and Luo, Q.Z., 2013. Free vibration of axially functionally graded Timoshenko beams with non-uniform cross-section, Composites Part B: Engineering, 45(1), 1493–1498.
Lee, J.W., 2020. Free vibration analysis of tapered Rayleigh beams using the transfer matrix method, Journal of the Brazilian Society of Mechanical Sciences and Engineering, 42(11), 612.
Lee, S.Y. and Kuo, Y.H., 1993. Bending frequency of a rotating Timoshenko beam with general elastically restrained root, Journal of Sound and Vibration, 162(2), 243–250.
Lee, J.W. and Lee, J.Y., 2018. An exact transfer matrix expression for bending vibration analysis of a rotating tapered beam, Applied Mathematical Modelling, 53, 167–188.
Lee, J.W. and Lee, J.Y., 2020. Free vibration analysis of a rotating double-tapered beam using the transfer matrix method, Journal of Mechanical Science and Technology, 34(7), 2731–2744.
Lee, S.Y. and Lin, S.M., 1994. Bending vibrations of rotating nonuniform Timoshenko beams with an elastically restrained root, Journal of Applied Mechanics, 61(4), 949–955.
Li, X.F., Tang, A.Y. and Xi, L.Y., 2013. Vibration of a Rayleigh cantilever beam with axial force and tip mass, Journal of Constructional Steel Research, 80, 15–22.
Li, Y.D. and Yang, Y.R., 2017. Vibration analysis of conveying fluid pipe via He’s variational iteration method, Applied Mathematical Modelling, 43, 409–420.
Lin, S.C. and Hsiao, K.M., 2001. Vibration analysis of a rotating Timoshenko beam, Journal of Sound and Vibration, 240(2), 303–322.
Liu, W.H. and Liu, D.S., 1989. Natural frequencies of a restrained Timoshenko beam with a tip body at its free end, Journal of Sound and Vibration, 128(1), 167–173.
Martin, O., 2016. A modified variational iteration method for the analysis of viscoelastic beams, Applied Mathematical Modelling, 40(17–18), 7988–7995.
Mei, C., 2006. Differential transformation approach for free vibration analysis of a centrifugally stiffened Timoshenko beam, Journal of Vibration and Acoustics, 128(2), 170–175.
Mei, C., 2008. Application of differential transformation technique to free vibration analysis of a centrifugally stiffened beam, Computers & Structures, 86(11–12), 1280–1284.
Ozgumus, O.O. and Kaya, M.O., 2006. Flapwise bending vibration analysis of double tapered rotating Euler-Bernoulli beam by using the differential transform method, Meccanica, 41(6), 661–670.
Ozgumus, O.O. and Kaya, M.O., 2008. Flapwise bending vibration analysis of a rotating double-tapered Timoshenko beam, Archive of Applied Mechanics, 78(5), 379–392.
Rajasekaran, S., 2013. Free vibration of centrifugally stiffened axially functionally graded tapered Timoshenko beams using differential transformation and quadrature methods, Applied Mathematical Modelling, 37(6), 4440–4463.
Tang, A.Y., Li, X.F. and Wu, J.X., 2015. Flapwise bending vibration of rotating tapered Rayleigh cantilever beams, Journal of Constructional Steel Research, 112, 1–9.
Tian, J.J., Su, J.P., Zhou, K. and Jia, H.X., 2018. A modified variational method for nonlinear vibration analysis of rotating beams including Coriolis effects, Journal of Sound and Vibration, 426, 258–277.
Wright, A.D., Smith, C.E., Thresher, R.W. and Wang, J.L.C., 1982. Vibration modes of centrifugally stiffened beam, Journal of Applied Mechanics, 49(1), 197–202.
Xi, L.Y., Li, X.F. and Tang, G.J., 2013. Free vibration of standing and hanging gravity-loaded Rayleigh cantilevers, International Journal of Mechanical Sciences, 66, 233–238.
The work was financially supported by the National Natural Science Foundation of China (Grant Nos. 51779265 and 52171285), Open Project Program of State Key Laboratory of Structural Analysis for Industrial Equipment (Grant No. GZ19119), Science Foundation of China University of Petroleum, Beijing (Grant No. 2462020YXZZ045), Open Project Program of Beijing Key Laboratory of Pipeline Critical Technology and Equipment for Deepwater Oil & Gas Development (Grant No. BIPT2018002), Special Funding for Promoting Economic Development in Guangdong Province (Grant No. GDOEA39), and Opening fund of State Key Laboratory of Hydraulic Engineering Simulation and Safety (Grant No. HESS-1411).
About this article
Cite this article
Chen, Yf., Zang, Zp., Dong, Sh. et al. Flapwise Bending Vibration Analysis of Rotating Tapered Rayleigh Beams for the Application of Offshore Wind Turbine Blades. China Ocean Eng 35, 544–553 (2021). https://doi.org/10.1007/s13344-021-0049-5
- tapered Rayleigh beam
- flapwise bending vibration
- variational iteration
- natural frequencies