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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.

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Figures

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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