Dear Jason,

Thank you for your answer and the URL.

I found the report “Floquet Modal Analysis of a Teetered-Rotor Wind Turbine” and I’m trying to understand the method. I linearized the turbine model with FAST and used the state matrices A from the .lin-output to recreate the Floquet approach with Matlab, i.e. computing the transition matrix Φ after one period. But unfortunately the results weren’t correct yet, so now I’m trying to understand the linearization process of FAST first, to get the correct A matrices.

I linearized a 3-blade rotor without control and wind disturbance inputs, so the state space model should be: x’ =A(t) * x, where x= [Δq, Δq’] and x’ = [Δq’, Δq’’].

The model has 9 DOFs, which are the 1st & 2nd flapwise and 1st edgewise bending modes of each of the 3 blades.

The states are:

```
Row/column 1 = 1st flapwise bending-mode DOF of blade 1 (internal DOF index = DOF_BF(1,1))
Row/column 2 = 1st flapwise bending-mode DOF of blade 2 (internal DOF index = DOF_BF(2,1))
Row/column 3 = 1st flapwise bending-mode DOF of blade 3 (internal DOF index = DOF_BF(3,1))
Row/column 4 = 1st edgewise bending-mode DOF of blade 1 (internal DOF index = DOF_BE(1,1))
Row/column 5 = 1st edgewise bending-mode DOF of blade 2 (internal DOF index = DOF_BE(2,1))
Row/column 6 = 1st edgewise bending-mode DOF of blade 3 (internal DOF index = DOF_BE(3,1))
Row/column 7 = 2nd flapwise bending-mode DOF of blade 1 (internal DOF index = DOF_BF(1,2))
Row/column 8 = 2nd flapwise bending-mode DOF of blade 2 (internal DOF index = DOF_BF(2,2))
Row/column 9 = 2nd flapwise bending-mode DOF of blade 3 (internal DOF index = DOF_BF(3,2))
Row/column 10 to 18 = First derivatives of row/column 1 to 9.
```

From the.lin-file at Azimuth = 0.00°:

x’ and x have the following values:

[code]op State | op

Derivativs | States

2.547E-01 | -4.121E-04

-1.286E-01 | 1.693E-01

-1.258E-01 | -1.689E-01

-5.541E-01 | 8.301E-04

2.745E-01 | -3.749E-01

2.795E-01 | 3.741E-01

-6.048E-02 | 5.423E-05

3.007E-02 | -4.097E-02

3.041E-02 | 4.091E-02

-1.928E-03 | 2.547E-01

-2.253E-01 | -1.286E-01

2.272E-01 | -1.258E-01

-3.705E-03 | -5.541E-01

5.848E-01 | 2.745E-01

-5.811E-01 | 2.795E-01

-2.667E-04 | -6.048E-02

6.377E-02 | 3.007E-02

-6.350E-02 | 3.041E-02

[/code]

But now my question is: If I am multiplying the state matrix A with the current state vector x (‘op States’) manually, I am getting different second derivatives as opposed to the ones in the x’-vector from the .lin-output (‘op State Derivatives’ from line 10/after the blank line). It seems that I misinterpreted something here. Are there perhaps conversions involved, which I missed?

I posted the A-matrix and attached the .lin, .fst and linear.dat files at the end of this post. I hope someone can explain this discrepancy, thank you in advance!

x’ = A(t) * x → manually multiplied

[code] 0.2547

-0.1286

-0.1258

-0.5541

0.2745

0.2795

-0.0604

0.0300

0.0304

-0.0019

-4.2694

4.2714

-0.0036

18.2634

-18.2609

-0.0002

2.9202

-2.9185

[/code]

==================================================

Linearization.zip (465 KB)

The A-Matrix has the following form:

`Removed (see .lin file)`