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Research Papers

Approximate Floquet Analysis of Parametrically Excited Multi-Degree-of-Freedom Systems With Application to Wind Turbines

[+] Author and Article Information
Gizem D. Acar

Dynamics and Control Laboratory,
Department of Mechanical Engineering,
University of Maryland,
College Park, MD 20742
e-mail: gizem@umd.edu

Brian F. Feeny

Professor
Department of Mechanical Engineering,
Michigan State University,
East Lansing, MI 48824
e-mail: feeny@egr.msu.edu

1Corresponding author.

Contributed by the Technical Committee on Vibration and Sound of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received March 14, 2018; final manuscript received June 4, 2018; published online July 24, 2018. Assoc. Editor: Stefano Lenci.

J. Vib. Acoust 141(1), 011004 (Jul 24, 2018) (10 pages) Paper No: VIB-18-1107; doi: 10.1115/1.4040522 History: Received March 14, 2018; Revised June 04, 2018

General responses of multi-degrees-of-freedom (MDOF) systems with parametric stiffness are studied. A Floquet-type solution, which is a product between an exponential part and a periodic part, is assumed, and applying harmonic balance, an eigenvalue problem is found. Solving the eigenvalue problem, frequency content of the solution and response to arbitrary initial conditions are determined. Using the eigenvalues and the eigenvectors, the system response is written in terms of “Floquet modes,” which are nonsynchronous, contrary to linear modes. Studying the eigenvalues (i.e., characteristic exponents), stability of the solution is investigated. The approach is applied to MDOF systems, including an example of a three-blade wind turbine, where the equations of motion have parametric stiffness terms due to gravity. The analytical solutions are also compared to numerical simulations for verification.

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References

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Figures

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Fig. 1

A 2DOF spring-mass chain

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Fig. 2

Stability regions for the 2DOF mass spring chain for n = 2

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Fig. 3

Response frequencies as a function of excitation frequency for the 2DOF system with δ = 0.4

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Fig. 4

Fast Fourier transforms (FFT) of displacements of m1 for the two “modal responses” excited by the symmetric and the anti-symmetric initial conditions, for ω = 2.3 and δ = 0.4

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Fig. 5

Response and FFT plots for n = 2, ω = 2.3, δ = 0.4, x(0) = [1 5]T, and x˙(0)=[0 0]T

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Fig. 6

A 3DOF mass-spring system

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Fig. 7

Stability regions for the 3DOF mass spring system for β = 1, γ3 = 1, and n = 2

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Fig. 8

Response and FFT plots for n = 2, ω = 0.7, δ = 0.5, γ3 = 0.4, x(0) = [0 0 0]T, and x˙(0)=[1 1 1]T

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Fig. 9

Response and FFT plots for n = 2, ω = 3.5, δ = 0.3, γ3 = 0.4, x(0) = [1 −1 0.5]T, and x˙(0)=[0 0 0]T

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Fig. 10

In-plane vibrations of a three-blade wind turbine

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Fig. 11

Response orders as a function of rotor speed for a system with mb = 1, k0 = 1.5, k1 = 0.2, k2 = 0.4, Jr = 10, and e = 1, using n = 2 harmonics in the Floquet solution

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Fig. 12

Time response and FFT plots of the blades with parameters mb = 1, k0 = 1, k1 = 0.1, k2 = 0.2, Jr = 10, e = 0.1, and Ω = 0.4. The periodic part of the solution was truncated at n = 2.

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Fig. 13

Time response and FFT plots of the blades with parameters mb = 1, k0 = 1.5, k1 = 0.2, k2 = 0.4, Jr = 10, e = 1, and Ω = 1.45. The periodic part of the solution was truncated at n = 2.

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Fig. 14

Stability of the solutions for the system with mb = 1, k0 = 1.5, k1 = 0.2, and Jr = 10. The periodic part of the solution was truncated at n = 2. (a) e = 0, (b) e = 0.5, and (c) e = 1.

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