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. 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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Presentation Transcript
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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