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Research Papers

Parametric Instability and Localization of Vibrations in Three-Blade Wind Turbines

[+] Author and Article Information
Takashi Ikeda

Department of Mechanical Systems Engineering,
Hiroshima University,
1-4-1, Kagamiyama,
Higashi-Hiroshima, Hiroshima 739-8527, Japan
e-mail: tikeda@hiroshima-u.ac.jp

Yuji Harata

Department of Mechanical Engineering,
Aichi Institute of Technology,
1247 Yachigusa, Yakusa-cho,
Toyota, Aichi 470-0392, Japan
e-mail: y-harata@aitech.ac.jp

Yukio Ishida

Institute of International Education and Exchange,
Nagoya University,
Fro-cho, Chikusa-ku,
Nagoya Aichi, 464-8601 Japan
e-mail: ishida@nuem.nagoya-u.ac.jp

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received July 6, 2017; final manuscript received March 22, 2018; published online May 17, 2018. Assoc. Editor: Brian Feeny.

J. Comput. Nonlinear Dynam 13(7), 071001 (May 17, 2018) (11 pages) Paper No: CND-17-1297; doi: 10.1115/1.4039899 History: Received July 06, 2017; Revised March 22, 2018

Nonlinear vibration characteristics of three-blade wind turbines are theoretically investigated. The wind turbine is modeled as a coupled system, consisting of a flexible tower with two degrees-of-freedom (2DOF), and three blades, each with a single degree of freedom (SDOF). The blades are subjected to steady winds. The wind velocity increases proportionally with height due to vertical wind shear. The natural frequency diagram is calculated with respect to the rotational speed of the wind turbine. The corresponding linear system with parametric excitation terms is analyzed to determine the rotational speeds where unstable vibrations appear and to predict at what rotational speeds the blades may vibrate at high amplitudes in a real wind turbine. The frequency response curves are then obtained by applying the swept-sine test to the equations of motion for the nonlinear system. They exhibit softening behavior due to the nonlinear restoring moments acting on the blades. Stationary time histories and their fast Fourier transform (FFT) results are also calculated. In the numerical simulations, localization phenomena are observed, where the three blades vibrate at different amplitudes. Basins of attraction (BOAs) are also calculated to examine the influence of a disturbance on the appearance of localization phenomena.

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References

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Figures

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Fig. 2

Analytical model: (a) top view and (b) side view

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Fig. 3

Coordinate system for blade “i

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Fig. 4

Definition of wind velocity

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Fig. 5

Natural frequency diagram for the three-blade wind turbine when μT = 0.94, μi = 0.02 (i = 1,2,3), kx = kz = 1.0, ki = 0.5, li = 40.0, bi = 2.0, α1 = 0 deg, α2 = 120 deg, and α3 = 240 deg

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Fig. 6

Unstable regions showing the influence of blade length li

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Fig. 7

Unstable regions showing the influence of spring stiffness ki

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Fig. 8

Frequency response curves calculated by the swept-sine test when μΑ = 3.2 × 10−8, μT = 0.94, μi = 0.02 (i = 1,2,3), kx = kz = 1.0, ki = 0.5, li = 40.0, cx = cz = 0.03, ci = 0.04, bi = 2.0, α1 = 0 deg, α2 = 120 deg, α3 = 240 deg, V0 = 6.0, Vi = 0.6, and λ = ±3.0 × 10−8. (a) Translation x0 of the 2DOF system; (b) translation z0 of the 2DOF system; and (c)–(e) blade inclinations θi (i = 1, 2, 3).

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Fig. 9

Stationary time histories at ω = 0.180 in Fig. 8: (a) pattern I; (b) pattern II-1; (c) pattern III-1; and (d) pattern IV

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Fig. 10

Fast Fourier transform results of pattern II-1 at ω = 0.180 in Fig. 9(b)

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Fig. 11

Frequency response curves demonstrating pattern II-1

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Fig. 12

Frequency response curves demonstrating pattern III-1

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Fig. 13

Basins of attraction when blade 1 is disturbed at (a) ω = 0.170 and (b) ω = 0.180 on branches A1A2, and at (c) ω = 0.170 and (d) ω = 0.180 on branches A2A3 in Fig. 8. BOA for pattern III in (a) and (b), and those for pattern II in (c) and (d).

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Fig. 14

Stationary time histories and FFT results of pattern III-1 at ω = 0.116 in Fig. 13: (a) time histories and (b) FFT results

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Fig. 15

Basins of attraction when blade 1 is disturbed at ω = 0.116: (a) on branches B1B2, (b) on branches B2B3 in Fig. 8. BOA for patterns III and II in (a) and (b), respectively.

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