Multi-blade Coordinate Transformation

For linearisation calculations or Campbell diagrams it is recommended to select the multi-blade coordinate transformation, which generates coupled modes referring to the non-rotating coordinate system including the backward and forward whirling modes of the rotor. This is based on theory developed in (Bir, 2008) and (Hansen, 2003). The linearised model is significantly azimuth-dependent, but when transformed to non-rotating coordinates the resulting model of the structural dynamics should be only weakly azimuth-dependent. However, for 2-bladed turbines there is still a strong azimuth dependency.

Single mode transformation

The transformation matrix of displacements of a 3-blade system with azimuths \(\psi_1\) to \(\psi_3\) from non-rotating to rotating coordinates is given by

\[ \begin{align} \begin{bmatrix} q_{1} \\ q_{2} \\ q_{3} \\ \end{bmatrix} &= \widetilde{\bmatrix{t}}_{NR \rightarrow R} \begin{bmatrix} q_{0} \\ q_{c} \\ q_{s} \\ \end{bmatrix}, \label{eq:NRtoR} \end{align} \]

with

\[ \begin{align} \widetilde{\bmatrix{t}}_{NR \rightarrow R} &= \begin{bmatrix} 1 & \cos\psi_{1} & \sin\psi_{1} \\ 1 & \cos\psi_{2} & \sin\psi_{2} \\ 1 & \cos\psi_{3} & \sin\psi_{3} \\ \end{bmatrix}. \end{align} \]

Multi-blade coordinate transformations are often quoted in the above form, but the primary aim is to go the other way and transform from rotating to non-rotating coordinates. The transformation matrix of displacements of a 3-blade system from rotating to non-rotating coordinates is the inverse of the above matrix given by

\[ \begin{align} \bmatrix{t}_{R \rightarrow NR} = \frac{1}{3}\begin{bmatrix} 1 & 1 & 1 \\ 2\cos\psi_{1} & 2\cos\psi_{2} & 2\cos\psi_{3} \\ 2\sin\psi_{1} & 2\sin\psi_{2} & 2\sin\psi_{3} \\ \end{bmatrix}. \end{align} \]

Note, that the inverse relation does not hold for the derivatives of this matrix.

The general transformation matrix for a turbine with an arbitrary number of blades (\(n\)) is

\[ \begin{align} \bmatrix{t}_{R \rightarrow NR} = \frac{1}{n} \begin{bmatrix} 1 & 1 & 1 & \cdots & 1 \\ 2\cos\psi_{1} & 2\cos\psi_{2} & 2\cos\psi_{3} & \cdots & 2\cos\psi_{n} \\ 2\sin\psi_{1} & 2\sin\psi_{2} & 2\sin\psi_{3} & \cdots & 2\sin\psi_{n} \\ 2\cos{j\psi}_{1} & 2\cos{j\psi}_{2} & 2\cos{j\psi}_{3} & \cdots & 2\cos{j\psi}_{n} \\ 2\sin{j\psi}_{1} & 2\sin{j\psi_{2}} & 2\sin{j\psi}_{3} & \cdots & 2\sin{j\psi}_{n} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 1 & - 1 & 1 & \cdots & ( - 1)^{n} \\ \end{bmatrix}, \end{align} \]

where the last row is the transformation to the differential mode and exists only if there is an even number of blades. For odd bladed turbines, the last row will be a sine cyclic row. The counter \(j\) goes from \(1\) to \((n-1)/2\) if \(n\) is odd, and from \(2\) to \((n-2)/2\) if \(n\) is even.

Dropping the matrix representation the non-rotating coordinates can be calculated as

\[ \begin{align} q_{0} &= \frac{1}{n}\sum_{i=1}^{n} q_{i} \\ q_{cj} &= \frac{2}{n}\sum_{i=1}^{n} q_{i}\cos\left( j\psi_{i} \right) \\ q_{sj} &= \frac{2}{n}\sum_{i=1}^{n} q_{i}\sin\left( j\psi_{i} \right) \\ q_{d} &= \frac{1}{n}\sum_{i=1}^{n}q_{i}( - 1)^{n} \\ \end{align} \]

Returning to the specific case of 3-bladed turbines as an example, the derivative transformation matrices are now calculated. Each azimuth angle \(\psi_i\) can be expressed in terms of the (assumed constant) rotor speed \(\Omega\) and initial azimuth angle \(\Psi_i\) as linear relationship

\[ \begin{align} \psi_{i} = \Omega t + \Psi_{i}. \end{align} \]

Taking the time-derivatives of the transformation matrix gives

\[ \begin{align*} \dot{\bmatrix{t}}_{R \rightarrow NR} = \frac{\Omega}{3} \begin{bmatrix} 0 & 0 & 0 \\ - 2\sin\psi_{1} & - 2\sin\psi_{2} & - 2\sin\psi_{3} \\ 2\cos\psi_{1} & 2\cos\psi_{2} & 2\cos\psi_{3} \\ \end{bmatrix} \end{align*} \]

and

\[ \begin{align*} \ddot{\bmatrix{t}}_{R \rightarrow NR} = - \frac{\Omega^{2}}{3} \begin{bmatrix} 0 & 0 & 0 \\ 2\cos\psi_{1} & 2\cos\psi_{2} & 2\cos\psi_{3} \\ 2\sin\psi_{1} & 2\sin\psi_{2} & 2\sin\psi_{3} \\ \end{bmatrix} \end{align*} \]

for the first and second derivatives, respectively.

System transformation matrix

A transformation matrix for the whole state list, including both displacement and velocity states, is required. For the displacement states we have already established in Equation \(\eqref{eq:NRtoR}\) that

\[ \begin{align*} \bvector{q}_{NR} = \bmatrix{t}_{R \rightarrow NR}\bvector{q}_{R} \end{align*} \]

holds. Taking the time-derivative of Equation \(\eqref{eq:NRtoR}\) gives

\[ \begin{align} \dot{\bvector{q}}_{NR} = \bmatrix{t}_{R \rightarrow NR} \dot{\bvector{q}}_{R} + \dot{\bmatrix{t}}_{R \rightarrow NR} \bvector{q}_{R} \end{align} \]

for the velocity states.

Combining \(\bvector{q}_{NR}\) and \(\dot{\bvector{q}}_{NR}\) to a vector of all states (both displacements and velocities) allows us to define a common transformation matrix \(\bmatrix{T}\) that is of the same dimensions as \(\bmatrix{A}\). We define

\[ \begin{align} \bmatrix{T} &:= \begin{bmatrix} \bmatrix{t}_{R \rightarrow NR} & 0 \\ \dot{\bmatrix{t}}_{R \rightarrow NR} & \bmatrix{t}_{R \rightarrow NR} \\ \end{bmatrix} \end{align} \]

allowing us to express the combined vector as

\[ \begin{align} \begin{bmatrix} \bvector{q}_{NR} \\ \dot{\bvector{q}}_{NR} \\ \end{bmatrix} &= \begin{bmatrix} \bmatrix{t}_{R \rightarrow NR} & 0 \\ \dot{\bmatrix{t}}_{R \rightarrow NR} & \bmatrix{t}_{R \rightarrow NR} \\ \end{bmatrix} \begin{bmatrix} \bvector{q}_{R} \\ \dot{\bvector{q}}_{R} \end{bmatrix}. \end{align} \]

Note that in general the displacement and velocity states are not ordered in this way and a permutation of the system transformation matrix \(\bmatrix{T}\) will occur. The system transformation matrix is not singular and the inverse can be calculated.
The derivative of the system transformation matrix is trivially inferred as

\[ \begin{align} \dot{\bmatrix{T}}= \begin{bmatrix} \dot{\bmatrix{t}}_{R \rightarrow NR} & 0 \\ \ddot{\bmatrix{t}}_{R \rightarrow NR} & \dot{\bmatrix{t}}_{R \rightarrow NR} \\ \end{bmatrix}. \end{align} \]

Transforming the A,B,C,D matrices

We consider the linear model equations in the rotating frame of reference and define

\[ \begin{align} \bvector{x}_{R} := \begin{bmatrix} \bvector{q}_{R} \\ \dot{\bvector{q}}_{R} \end{bmatrix} \end{align} \]

to express the principal system as

\[ \begin{align} {\dot{\bvector{x}}}_{R} &= \bmatrix{A}_{R}\bvector{x}_{R} + \bmatrix{B}_{R} \bvector{u} \label{eq:StateSpaceRotating}\\ \bvector{y} &= \bmatrix{C}_{R} \bvector{x}_{R} + \bmatrix{D}_{R}\bvector{u} \label{eq:OutputRotating} \end{align} \]

with respect to rotating blade coordinates.
The transformation of the state vector from rotating to non-rotating coordinates is given as

\[ \begin{align*} \bvector{x}_{NR}= \bmatrix{T} \bvector{x}_{R} \end{align*} \]

and its derivative follows as

\[ \begin{align} \dot{\bvector{x}}_{NR} = \bmatrix{T}\dot{\bvector{x}}_{R} + \dot{\bmatrix{T}}\bvector{x}_{R}. \label{eq:DerivativeNonRotating} \end{align} \]

By combining Equation \(\eqref{eq:DerivativeNonRotating}\) with Equation \(\eqref{eq:StateSpaceRotating}\) we infer

\[ \begin{align} \dot{\bvector{x}}_{NR} &= \bmatrix{T}\left( \bmatrix{A}_{R}\bvector{x}_{R} + \bmatrix{B}_{R}\bvector{u} \right) +\dot{\bmatrix{T}}\bvector{x}_{R} \\ &= \left( \bmatrix{T}\bmatrix{A}_{R} + \dot{\bmatrix{T}} \right)\bvector{x}_{R} + \bmatrix{T}\bmatrix{B}_{R}\bvector{u} \\ &= \left( \bmatrix{T} \bmatrix{A}_{R} + \dot{\bmatrix{T}} \right) \bmatrix{T^{-1}} \bvector{x}_{NR} + \bmatrix{T}\bmatrix{B}_{R} \bvector{u} \end{align} \]

and conclude

\[ \begin{align} \bmatrix{A}_{NR} &= \left( \bmatrix{T}\bmatrix{A}_{R} + \dot{\bmatrix{T}} \right)\bmatrix{T^{-1}} \\ \bmatrix{B}_{NR} &= \bmatrix{T}\bmatrix{B}_{R} \end{align} \]

from there.

Similar transformation in Equation \(\eqref{eq:OutputRotating}\) gives

\[ \begin{align} \bvector{y} &= \bmatrix{C}_{R}\bvector{x}_{R} + \bmatrix{D}_{R}\bvector{u} \\ &= \bmatrix{C}_{R}\bmatrix{T^{-1}}\bvector{x}_{NR} + \bmatrix{D}_{R}\bvector{u} \end{align} \]

for the output \(\bvector{y}\) of the linear system. We now define

\[ \begin{align} \bmatrix{C}_{NR} &:= \bmatrix{C}_{R}\bmatrix{T^{-1}}, \quad \text{ and } \bmatrix{D}_{NR} := \bmatrix{D}_{R} \end{align} \]

for the matrices concerned with the output of the linear model.
This completes the derivation of a linear model with respect to a non-rotating frame.

Rotational transformations are exclusively applied to states, which represent the degress of freedom in a mathematical model defined for all blades. These states include blade mode states as well as dynamic stall states, whereas any other individual-blade states such as pitch positions, rates, actuator internal states etc. are not transformed. The matrix \(\bmatrix{T}\) just has unit diagonal elements for rows and columns corresponding to the states and state derivatives which are not transformed. For other rows and columns, the elements of \(\bmatrix{T}\) represent the basic transformation defined above for each group of modes. Note that the elements connecting states and state derivatives also need to be defined by differentiating the equations of the basic transformation, bearing in mind that the derivative of the azimuth angle is equal to the rotor speed (which is assumed constant for this purpose). Model inputs and outputs are not transformed.

Last updated 26-11-2024