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

2009-01-07 2/22Modal Dynamics of Wind Turbines with Anisotropic Rotors, Peter F. Skjoldan

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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

2009-01-07 3/22Modal Dynamics of Wind Turbines with Anisotropic Rotors, Peter F. Skjoldan

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Outline

Motivations

Wind turbine model

Modal analysis

Results for isotropic rotor

Analysis methods for anisotropic rotor

Results for anisotropic rotor

Conclusions and future work

2009-01-07 4/22Modal Dynamics of Wind Turbines with Anisotropic Rotors, Peter F. Skjoldan

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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

2009-01-07 5/22Modal Dynamics of Wind Turbines with Anisotropic Rotors, Peter F. Skjoldan

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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

2009-01-07 6/22Modal Dynamics of Wind Turbines with Anisotropic Rotors, Peter F. Skjoldan

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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

2009-01-07 7/22Modal Dynamics of Wind Turbines with Anisotropic Rotors, Peter F. Skjoldan

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Floquet theory

Solution to a linear system with periodic coefficients:

periodic mode shape oscillating term

Describes solution form for all methods in this paper

2009-01-07 8/22Modal Dynamics of Wind Turbines with Anisotropic Rotors, Peter F. Skjoldan

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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

2009-01-07 9/22Modal Dynamics of Wind Turbines with Anisotropic Rotors, Peter F. Skjoldan

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Results for isotropic rotor

1st forward whirling modal solution

Time domain Frequency domain

2009-01-07 10/22Modal Dynamics of Wind Turbines with Anisotropic Rotors, Peter F. Skjoldan

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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

2009-01-07 11/22Modal Dynamics of Wind Turbines with Anisotropic Rotors, Peter F. Skjoldan

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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

2009-01-07 12/22Modal Dynamics of Wind Turbines with Anisotropic Rotors, Peter F. Skjoldan

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Hill’s method

Hypermatrix eigenvalue problem

2009-01-07 13/22Modal Dynamics of Wind Turbines with Anisotropic Rotors, Peter F. Skjoldan

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Hill’s method

Eigenvalues of hypermatrix

Multiple eigenvalues for each physical mode

2 additional harmonic terms(n = 2)

2009-01-07 14/22Modal Dynamics of Wind Turbines with Anisotropic Rotors, Peter F. Skjoldan

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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

j j

Amplitude

Amplitude

Floquet analysis Hill’s method

n = 2

2009-01-07 15/22Modal Dynamics of Wind Turbines with Anisotropic Rotors, Peter F. Skjoldan

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Comparison of methods

Convergence of eigenvalues

Floquet analysis Hill’s method

2009-01-07 16/22Modal Dynamics of Wind Turbines with Anisotropic Rotors, Peter F. Skjoldan

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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

2009-01-07 17/22Modal Dynamics of Wind Turbines with Anisotropic Rotors, Peter F. Skjoldan

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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

Mode 1st BW 1st FW Symmetric 2nd yaw 2nd tilt

Frequency, Hz 0.447 0.749 0.860 1.471 1.590

Deviation from isotropic case, % 0.20 0.41 0.45 0.03 0.006

Damping, s-1 0.0101 0.0125 0.0127 0.0733 0.0681

Deviation from isotropic case, % 4.1 0.36 2.7 0.08 0.03

2009-01-07 18/22Modal Dynamics of Wind Turbines with Anisotropic Rotors, Peter F. Skjoldan

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Results for anisotropic rotor

1st backward whirling mode, Fourier coefficients

Blade 116% stiffer thanblades 2 and 3

2009-01-07 19/22Modal Dynamics of Wind Turbines with Anisotropic Rotors, Peter F. Skjoldan

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Results for anisotropic rotor

Symmetric mode, Fourier coefficients

Blade 116% stiffer thanblades 2 and 3

2009-01-07 20/22Modal Dynamics of Wind Turbines with Anisotropic Rotors, Peter F. Skjoldan

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Results for anisotropic rotor

2nd yaw mode, Fourier coefficients

Blade 116% stiffer thanblades 2 and 3

2009-01-07 21/22Modal Dynamics of Wind Turbines with Anisotropic Rotors, Peter F. Skjoldan

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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

2009-01-07 22/22Modal Dynamics of Wind Turbines with Anisotropic Rotors, Peter F. Skjoldan

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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)