Modal Dynamics of Wind Turbines with Anisotropic Rotors Peter F. Skjoldan 7 January 2009 - PowerPoint PPT Presentation

Modal dynamics of wind turbines with anisotropic rotors peter f skjoldan 7 january 2009
Download
1 / 22

  • 101 Views
  • Uploaded on
  • Presentation posted in: General

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.

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.

Download Presentation

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

An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript


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)


  • Login