Technical Briefs

Resonances of a Forced Mathieu Equation With Reference to Wind Turbine Blades

[+] Author and Article Information
Venkatanarayanan Ramakrishnan

e-mail: venkat@msu.edu

Brian F. Feeny

e-mail: feeny@egr.msu.edu
Dynamics Systems Laboratory,
Vibration Research Department of Mechanical Engineering,
Michigan State University,
East Lansing, MI 48824

Contributed by the Desing Rngineering Division of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received August 8, 2011; final manuscript received January 24, 2012; published online October 29, 2012. Assoc. Editor: Walter Lacarbonara.

J. Vib. Acoust 134(6), 064501 (Oct 29, 2012) (5 pages) doi:10.1115/1.4006183 History: Received August 08, 2011; Revised January 24, 2012

A horizontal axis wind turbine blade in steady rotation endures cyclic transverse loading due to wind shear, tower shadowing and gravity, and a cyclic gravitational axial loading at the same fundamental frequency. These direct and parametric excitations motivate the consideration of a forced Mathieu equation. This equation with cubic nonlinearity is analyzed for resonances by using the method of multiple scales. Superharmonic and subharmonic resonances occur. The effect of various parameters on the response of the system is demonstrated using the amplitude-frequency curve. The order-two superharmonic resonance persists for the linear system. While the order-two subharmonic response level is dependent on the ratio of parametric excitation and damping, nonlinearity is essential for the order-two subharmonic resonance. Order-three resonances are present in the system as well and, to first order, they are similar to those of the Duffing equation.

Copyright © 2012 by ASME
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Ramakrishnan, V., and Feeny, B. F., 2011, “In-Plane Nonlinear Dynamics of Wind Turbine Blades,” ASME International Design Engineering Technical Conferences, 23rd Biennial Conference on Vibration and Noise, Paper No. DETC2011-48219 (CDROM).
Rhoads, J. F., and Shaw, S. W., 2010, “The Impact of Nonlinearity on Degenerate Parametric Amplifiers,” Appl. Phys. Lett., 96(23), p. 234101. [CrossRef]
Rhoads, J. F., Miller, N. J., Shaw, S. W., and Feeny, B. F., 2008, “Mechanical Domain Parametric Amplification,” ASME J. Vibr. Acoust., 130(6), p. 061006. [CrossRef]
Pandey, M., Rand, R. H., and Zehnder, A. T., 2007, “Frequency Locking in a Forced Mathieu-van der Pol-Duffing System,” Nonlinear Dyn., 54(1–2), pp. 3–12. [CrossRef]
Month, L. A., and Rand, R. H., 1982, “Bifurcation of 4-1 Subharmonics in the Non-Linear Mathieu Equation,” Mech. Res. Commun., 9(4), pp. 233–240. [CrossRef]
Newman, W. I., Rand, R. H., and Newman, A. L., 1999, “Dynamics of a Nonlinear Parametrically Excited Partial Differential Equation,” Chaos, 9(1), pp. 242–253. [CrossRef] [PubMed]
Ng, L., and Rand, R., 2002, “Bifurcations in A Mathieu Equation With Cubic Nonlinearities,” Chaos, Solitons Fractals, 14(2), pp. 173–181. [CrossRef]
Belhaq, M., and Houssni, M., 1999, “Quasi-Periodic Oscillations, Chaos and Suppression of Chaos in a Nonlinear Oscillator Driven by Parametric and External Excitations,” Nonlinear Dynamics, 18(1), pp. 1–24. [CrossRef]
Veerman, F., and Verhulst, F., 2009, “Quasiperiodic Phenomena in the van der Pol-Mathieu Equation,” J. Sound Vib., 326(1–2), pp. 314–320. [CrossRef]
Arrowsmith, D. K., and Mondragón, R. J., 1999, “Stability Region Control for a Parametrically Forced Mathieu Equation,” Meccanica, 34, pp. 401–410. [CrossRef]
Marathe, A., and Chatterjee, A., 2006, “Asymmetric Mathieu Equations,” Proc. R. Soc. London, 462, pp. 1643–1659. [CrossRef]
McLachlan, N., 1964, Theory and Application of Mathieu Functions, Dover, New York.
Hodges, D., and Dowell, E., 1974, “Nonlinear Equations of Motion for Elastic Bending and Torsion of Twisted Nonuniform Rotor Blades,” NASA Technical NoteD-7818.
Bhat, S. R., and Ganguli, R., 2004, “Validation of Comprehensive Helicopter Aeroelastic Analysis With Experimental Data,” Def. Sci. J., 54(4), pp. 419–427. [CrossRef]
Wang, Y. R., and Peters, D. A., 1996, “The Lifting Rotor Inflow Mode Shapes and Blade Flapping Vibration System Eigen-Analysis,” Comput. Methods Appl. Mech. Eng., 134(1–2), pp. 91–105. [CrossRef]
Rand, O., and Barkai, S. M., 1997, “A Refined Nonlinear Analysis of Pre-Twisted Composite Blades,” Compos. Struct., 39(1–2), pp. 39–54. [CrossRef]
Wendell, J., 1982, “Simplified Aeroelastic Modeling of Horizontal-Axis Wind Turbines,” Technical Report No. DOE/NASA/3303-3 NASA-CR-168109, Massachusetts Institute of Technology, Cambridge, MA.
Kallesøe, B. S., 2007, “Equations of Motion for a Rotor Blade, Including Gravity and Pitch Action,” Wind Energy, 10(3), pp. 209–230. [CrossRef]
Caruntu, D. I., 2008, “On Nonlinear Forced Response of Nonuniform Blades,” Proceedings of the 2008 ASME Dynamic Systems and Control Conference, Paper No. DSCC2008-2157, ASME.
Chopra, I. and Dugundji, J., 1979, “Nonlinear Dynamic Response of a Wind Turbine Blade,” J. Sound Vibr., 63(2), pp. 265–286. [CrossRef]
Jonkman, J., 2010, NWTC Design Codes (FAST), November.
Jonkman, J., 2003, “Modeling of the UAE Wind Turbine for Refinement of Fast Ad,” Technical Report TP-500-34755, NREL, Golden, CO.
Ishida, Y., Inoue, T., and Nakamura, K., 2009, “Vibration of a Wind Turbine Blade (Theoretical Analysis and Experiment Using a Single Rigid Blade Model),” J. Environ. Eng., 4(2), pp. 443–454. [CrossRef]
Inoue, T., Ishida, Y., and Kiyohara, T., 2012, “Nonlinear Vibration Analysis of the Wind Turbine Blade (Occurrence of the Superharmonic Resonance in the Out-Of-Plane Vibration of the Elastic Blade),” ASME J. Vibr. Acousti., 134, p. 031009. [CrossRef]
Nayfeh, A. H. and Mook, D. T., 1979, Nonlinear Oscillations, Wiley Interscience, New York.
Rand, R., 2005, Lecture Notes on Nonlinear Vibration, version 52, Cornell University, Ithaca, NY, available online at http://audiophile.tam.cornell.edu/randdocs/nlvibe52.pdf


Grahic Jump Location
Fig. 1

Amplitudes of simulated responses of Eq. (1) showing superharmonic resonances at orders 1/2 and 1/3, with ɛ=0.1,μ=0.5, α=3, F=0.05, γ=3. The frequency ratio sweeps up.

Grahic Jump Location
Fig. 2

Amplitudes of simulated responses of the linear case of Eq. (1) showing superharmonic resonances at orders 1/2 and 1/3, with ɛ=0.1, μ=0.5, α=0, F=0.5, γ=3

Grahic Jump Location
Fig. 3

Amplitudes of simulated responses of Eq. (1) showing the effect of the parametric forcing amplitude, with ɛ=0.1, μ=0.5,α=0, F=0.5. Different curves depict γ = 0.5, 1, and 3.

Grahic Jump Location
Fig. 4

Amplitudes of simulated responses of Eq. (1) showing the effect of the direct forcing amplitude, with ɛ=0.1, μ=0.5,α=0, γ=3. Different curves depict F = 0.5, 1, and 2.

Grahic Jump Location
Fig. 5

Simulated sweep-up response of Eq. (1) showing the effect of the nonlinear term, with ɛ=0.1, μ=0.5, F=0.5,γ=3. Different curves depict α = 0.1, 0.5, and 1.



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