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

2. 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
Goodmorning, my name is Peter Skjoldan. I am a ph.d. student at the Technical University of Denmark and employed with Siemens Wind Power. My project is about stability analysis of wind turbines.

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
My ph.d. project is about developing a stability tool for an inhouse aeroelastic model we use at Siemens. So I have been looking at different methods for approaching stability analysis. Most stability tools that have been produced for wind turbines consider only the case of isotropic rotor and inflow. However, load calculations that are the basis for design are done with anisotropic conditions, such as wind shear, gravity, tower shadow, and possibly an imbalance on the rotor. The method of Coleman transformation for modal analysis works only in isotropic conditions. I would like to not be bound by the limitations of only isotropic conditions, so I have been examining other more general methods. An alternative that works in anisotropic conditions is Floquet analysis. It has been extensively used on helicopters, and also for wind turbines. Another alternative is Hill’s method; I call it like that because it is simlar to the approach of setting up Hill’s infinite determinant. In this presentation I will review these methods and examine the effect of the 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
To get a better picture of what this is about, I will first present the model that I have used. It is a very simple model of wind turbine with 5 degrees of freedom. Each of the three blades can flap and the nacelle can tilt (figure a) and yaw (figure b). The model is purely structural, gravity and aerodynamic forces are neglected. The inertia is the same for all blades but the stiffnesses can be varied to give an anisotropic rotor. The equations of motion are linearized and written in state space form like this with the state vector containing positions and velocities. The system matrix is determined by the mass, damping and stiffness matrices. It is time-variant because the rotor rotates.

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
We are interested in doing modal analysis of the wind turbine in operation, that is when the rotor is rotating. Each analysis is carried out for an operating point defined by a constant mean rotor speed. In order to do eigenvalue analysis a time-invariant system of equations is needed. This can be obtained by a coordinate transformation. Then modal frequencies, damping from the eigenvalues. The eigenvectors are given in the transformed coordinates and periodic mode shapes are obtained when transforming back to physical coordinates. It is important to remember that the linearized model is valid only for small perturbations around the 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
Floquet theory is the theoretical foundation for all the solution methods used here. It states that the solution to a linear system with periodic coefficients is a sum of modal solutions, each a product of a oscillating term and a periodic mode shape with the same period as the system matrix. Here, the q denotes the content of mode k in the solution. So the oscillation contained in the solution derives from both the eigenfrequency and the mode shape. And from Floquet theory there is no constraint on the distribution of the frequency content between the oscillating and the periodic mode shape. This gives rise to an ambiguity in the results. It can however be resolved as we shall see later.

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
The Coleman transformation can be used for modal analysis of the wind turbine with an isotropic rotor. It introduces multiblade coordinates for the rotor states, where the rotor is described as a whole in the intertial, that is non-rotating, frame instead of as individual blades in the rotating frame. The transformation includes cosines and sines to the azimuth angles of the individual blades. So if you remember the model I showed before, it means that instead of having three DOFs describing the flap of the individual blades we get a symmetric flap, a tilt of the rotor, and a yaw of the rotor. The Coleman transformation yields a time-invariant system matrix, and eigenvalue analysis of this matrix gives damping and frequencies from the eigenvalues and the periodic mode shapes which are the transformation matrix times the eigenvectors.

9. Results for isotropic rotor
1st forward whirling modal solution
Time domain
Frequency domain
Now we’ll have a look at what results can be obtained for an isotropic rotor. This is a time series of a modal solution, that is a pure excitation of the first forward whirling mode. The blade DOFs are shown in blue, green and red, and the support DOFs are in cyan and magenta. The states on the support oscillate harmonically but the blades oscillate with a modulation. This can be seen more easily in the frequency domain. The periodic mode shape contains three harmonics which are zero and plus/minus the rotor speed. The frequency spectrum of the solution is found simply by adding the modal frequency. It is seen that the support states have only one component at the modal frequency whereas the blade motion is a sum of two harmonics. This is because only the rotor states are transformed with the Coleman transformation.

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 R with non-unique frequency
Periodic mode shapes
Because of lack of symmetry on an anisotropic rotor, one that has an imbalance, the Coleman transformation does not remove all the time-variance in the system matrix. Subsequently Floquet analysis can be used. It uses numerical integration over one period of the system for N different initial conditions where N is the number of states of the system, 10 in this case. These integrated solutions are collected in the fundamental solution matrix. The monodromy matrix is the fundamental solution after one period of rotation. It appears that a time-invariant transformed system matrix R can be obtained from the monodromy matrix like this and the coordinate transformation that gives this system is called the Lyapunov-Floquet transformation. The imaginary part of the eigenvalues of R, that is the frequency, is non-unique, because the complex logarithm has an infinite number of soluitons. I will come back to this later. The periodic mode shapes are determined from the transformation matrices and the eigenvectors and are calculated like this.

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
Another method for solving periodic systems is Hill’s method. It uses the fact from Floquet theory that the mode shapes are periodic and expresses it and the system matrix as a Fourier series. These are put into the equations of motion and are

12. Hill’s method Hypermatrix eigenvalue problem
These resulting equations are assembled into a hypermatrix system where the system matrix contains the Fourier coefficients and therefore is time-invariant. The eigenvector contains Fourier coefficients of the mode shape. The system has infinite size, but in practice it is truncated to include n harmonic terms. So the number of eigenvalues is 2n+1 times the number of states.

13. Hill’s method Eigenvalues of hypermatrix
Multiple eigenvalues for each physical mode
2 additional harmonic terms (n = 2)
Here is an example of the eigenvalues obtained for n=2, so there are five eigenvalues for each physical mode. Here is the real part of the eigenvalue, the damping, and here is the imaginary part, the frequency. It is seen that the eigenvalues that belong to the same mode have the same damping and the frequency differs by the rotor speed. There are some deviations to this because the system is truncated. Now is the question how to find the eigenvalue with the modal frequency.

14. Identification of modal frequency
Non-unique frequencies and periodic mode shapes
Modal frequency is chosen such that the periodic mode shape is as constant as possible in multiblade coordinates
Floquet analysis
Hill’s method
n = 2
Amplitude
The non-uniqueness of the frequencies is a problem both for Floquet analysis and Hill’s method. In Floquet analysis you can add any integer multiple of the rotor speed to the principal frequency which is the one closest to zero. The periodic mode shape is adjusted accordingly. In Hill’s method there are several sets of eigenvalues with different frequencies and correspondent eigenvectors with the Fourier coefficients of the periodic mode shapes. The modal frequency is chosen such that the periodic mode shape is as constant as possible. In Floquet anlysis the mode shape is expressed as a Fourier series. It contains discrete frequencies of j times the rotor speed, the lines are only for illustration. The mode shape calculated with the principal frequency is the dashed one and the dominant component is at two times the rotor speed. Thus the modal frequency is found by adding two times the rotor speed to the principal frequency and this oscillation is taken out of the mode shape so its dominant harmonic is constant. The same mode shape is seen in five different versions for Hill’s method. The mode shape is chosen as the one where the dominating component is at zero frequency (the blue one) and the modal frequency from the corresponding eigenvalue. In this example the mode shape contains only five harmonics and and some of them are quite bad representations, which explains why some eigenvalues have different damping. So, it is better to use a few more terms than shown here.
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
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
4.1
0.36
2.7
0.08

18. Results for anisotropic rotor
1st backward whirling mode, Fourier coefficients
Blade 1 16% stiffer than
blades 2 and 3
Now I will show some results for an anisotropic rotor which has a difference in stiffness of the blades. This is the first forward whirling mode. The isotropic rotor had harmonics only at -1, 0 and 1 times the rotor speed. There is not much change to these dominating harmonics compared to the isotropic rotor. But now we see that there are more harmonic components. That means that the motion is more modulated. The magnitude of the additional blade harmonics is of a few percent of the dominating harmonic, notice the logarithmic scale.

19. Results for anisotropic rotor
Symmetric mode, Fourier coefficients
Blade 1 16% stiffer than
blades 2 and 3
This is the symmetric mode which for the isotropic rotor had no coupling to the support. We see that the dominating motion is still the symmetric flap of the blades but now the support moves as well. And the blade motion is modulated. A thing to notice is that the rotor and the support never moves with the same frequency.

20. Results for anisotropic rotor
2nd yaw mode, Fourier coefficients
Blade 1 16% stiffer than
blades 2 and 3
Now for the second yaw mode we see similar results, but the additional blade harmonics are somewhat smaller, around 1%.

21. Conclusions
Isotropic rotor: Coleman transformation yields time-invariant system Motion with at most three harmonic components
Anisotropic rotor: Floquet analysis or Hill’s method Motion with many harmonic components
These methods give similar results Frequency non-uniqueness resolved using a common approach
Anisotropy affects some modes more: whirling / low damping / low frequency ?
Additional harmonic components on anisotropic rotor are small but might have significant effect when coupled to aerodynamics
The conclusions of this study are that modal analysis of a rotating wind turbine with an isotropic rotor can be done using the Coleman transformation, and the results are motion with at most three harmonic components. For an anisotropic rotor Floquet analysis or Hill’s method can be used. This gives motion with many harmonic components. These methods give similar results and the frequency non-uniqueness can be resolved in a similar manner. The rotor anisotropy has a larger effect on some modes than others, but it is difficult to say from only this simple model whether they are the modes that are whirling or the ones with low damping or low frequency. The additional harmonic components on the anisotropic rotor are quite small but it will be interesting to see if the effect is larger when coupled to the aerodynamics in an aeroelastic model.
Thank you!

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)