Modal dynamics of wind turbines with anisotropic rotors peter f skjoldan 7 january 2009
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Modal Dynamics of Wind Turbines with Anisotropic Rotors Peter F. Skjoldan 7 January 2009 PowerPoint PPT Presentation


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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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Modal dynamics of wind turbines with anisotropic rotors peter f skjoldan 7 january 2009

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


Presentation

Presentation

  • Ph.D. project ”Aeroservoelastic stability analysis and design of wind turbines”

  • Collaboration betweenSiemens Wind Power A/SRisø DTU - National Laboratory for Sustainable Energy


Outline

Outline

  • Motivations

  • Wind turbine model

  • Modal analysis

  • Results for isotropic rotor

  • Analysis methods for anisotropic rotor

  • Results for anisotropic rotor

  • Conclusions and future work


Motivations

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


Model of wind turbine

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


Modal analysis

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


Floquet theory

Floquet theory

  • Solution to a linear system with periodic coefficients:

    periodic mode shape oscillating term

  • Describes solution form for all methods in this paper


Coleman transformation

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


Results for isotropic rotor

Results for isotropic rotor

  • 1st forward whirling modal solution

Time domain

Frequency domain


Floquet analysis

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


Hill s method

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


Hill s method1

Hill’s method

  • Hypermatrix eigenvalue problem


Hill s method2

Hill’s method

  • Eigenvalues of hypermatrix

  • Multiple eigenvalues for each physical mode

2 additional harmonic terms(n = 2)


Identification of modal frequency

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


Comparison of methods

Comparison of methods

  • Convergence of eigenvalues

Floquet analysis

Hill’s method


Comparison of methods1

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


Results for anisotropic rotor

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


Results for anisotropic rotor1

Results for anisotropic rotor

  • 1st backward whirling mode, Fourier coefficients

Blade 116% stiffer than

blades 2 and 3


Results for anisotropic rotor2

Results for anisotropic rotor

  • Symmetric mode, Fourier coefficients

Blade 116% stiffer than

blades 2 and 3


Results for anisotropic rotor3

Results for anisotropic rotor

  • 2nd yaw mode, Fourier coefficients

Blade 116% stiffer than

blades 2 and 3


Conclusions

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


Further work

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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