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Modal Dynamics of Wind Turbines with Anisotropic Rotors Peter F. Skjoldan 7 January 2009

Modal Dynamics of Wind Turbines with Anisotropic Rotors Peter F. Skjoldan 7 January 2009. Presentation. Ph.D. project ”Aeroservoelastic stability analysis and design of wind turbines” Collaboration between Siemens Wind Power A/S Risø DTU - National Laboratory for Sustainable Energy. Outline.

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Modal Dynamics of Wind Turbines with Anisotropic Rotors Peter F. Skjoldan 7 January 2009

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  1. Modal Dynamics of Wind Turbines with Anisotropic RotorsPeter F. Skjoldan7 January 2009

  2. Presentation • Ph.D. project ”Aeroservoelastic stability analysis and design of wind turbines” • Collaboration betweenSiemens Wind Power A/SRisø DTU - National Laboratory for Sustainable Energy

  3. Outline • Motivations • Wind turbine model • Modal analysis • Results for isotropic rotor • Analysis methods for anisotropic rotor • Results for anisotropic rotor • Conclusions and future work

  4. Motivations • Far goal: build stability tool compatible with aeroelastic model used in industry • Conventional wind turbine stability tools consider isotropic conditions • Load calculations are performed in anisotropic conditions • Method of Coleman transformation works only in isotropic conditions • Alternative 1: Floquet analysis • Alternative 2: Hill’s method • Effect of anisotropy on the modal dynamics

  5. Model of wind turbine • 3 DOF on rotor (blade flap), 2 DOF on support (tilt and yaw) • Structrual model (no aerodynamics), no gravity • Blade stiffnesses can be varied to give rotor anisotropy

  6. Modal analysis • Modal analysis of wind turbine in operation • Operating point defined by a constant mean rotor speed • Time-invariant system needed for eigenvalue analysis • Coordinate transformation to yield time-invariance • Modal frequencies, damping, eigenvectors / periodic mode shapes • Describes motion for small perturbations around operating point

  7. Floquet theory • Solution to a linear system with periodic coefficients: periodic mode shape oscillating term • Describes solution form for all methods in this paper

  8. Coleman transformation • Introduces multiblade coordinates on rotor • Describes rotor as a whole in the inertial frame instead of individual blades in the rotating frame • Yields time-invariant system if rotor is isotropic • Modal analysis performed by traditional eigenvalue analysis of system matrix

  9. Results for isotropic rotor • 1st forward whirling modal solution Time domain Frequency domain

  10. Floquet analysis • Numerical integration of system equations gives fundamental solution and monodromy matrix • Lyapunov-Floquet transformation yields time-invariant system • Modal frequencies and damping found from eigenvalues of Rwith non-unique frequency • Periodic mode shapes

  11. Hill’s method • Solution form from Floquet theory • Fourier expansion of system matrix and periodic mode shape(in multiblade coordinates) • Inserted into equations of motion • Equate coefficients of equal harmonic terms

  12. Hill’s method • Hypermatrix eigenvalue problem

  13. Hill’s method • Eigenvalues of hypermatrix • Multiple eigenvalues for each physical mode 2 additional harmonic terms(n = 2)

  14. Identification of modal frequency • Non-unique frequencies and periodic mode shapes • Modal frequency is chosen such that the periodic mode shape isas constant as possible in multiblade coordinates Floquet analysis Hill’s method n = 2 Amplitude Amplitude j j

  15. Comparison of methods • Convergence of eigenvalues Floquet analysis Hill’s method

  16. Comparison of methods • Floquet analysis: Mode shapes in time domain + Nonlinear model can be used directly to provide fundamental solutions – Slow (numerical integration) • Hill’s method: Mode shapes in frequency domain + Fast (pure eigenvalue problem) + Accuracy increased by using Coleman transformation – Eigenvalue problem can be very large • Frequency non-uniqueness can be resolved using a common approach

  17. Results for anisotropic rotor • Blade 1 is 16% stiffer than blades 2 and 3 • Small change in frequencies compared to isotropic rotor • Larger effect on damping of some modes

  18. Results for anisotropic rotor • 1st backward whirling mode, Fourier coefficients Blade 116% stiffer than blades 2 and 3

  19. Results for anisotropic rotor • Symmetric mode, Fourier coefficients Blade 116% stiffer than blades 2 and 3

  20. Results for anisotropic rotor • 2nd yaw mode, Fourier coefficients Blade 116% stiffer than blades 2 and 3

  21. Conclusions • Isotropic rotor: Coleman transformation yields time-invariant systemMotion with at most three harmonic components • Anisotropic rotor: Floquet analysis or Hill’s methodMotion with many harmonic components • These methods give similar resultsFrequency non-uniqueness resolved using a common approach • Anisotropy affects some modes more:whirling / low damping / low frequency ? • Additional harmonic components on anisotropic rotor are smallbut might have significant effect when coupled to aerodynamics

  22. Further work • Set up full finite element model and obtain linearized system • Apply Floquet analysis or Hill’s method to full model • Compare anisotropy in the rotating frame (rotor imbalance) and in the inertial frame (wind shear, yaw/tilt misalignment, gravity, tower shadow)

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