Analysis of linearized plant


I am linearizing with FAST on a two-bladed turbine, and observe following things:

  1. If I unlock only the fist flap DOF, I get following eigenfrequencies (in Hz) from the linearized AvgAMat:
    0.98458 - blade 1 position
    0.98458 - blade 1 velocity
    0.98638 - blade 2 position
    0.98638 - blade 2 velocity.

  2. If I unlock the first and the second flap DOF, I get following eigenfrequencies:
    0.95432 - blade 1 position
    0.95432 - blade 1 velocity
    0.95631 - blade 2 position
    0.95631 - blade 2 velocity.
    2.5812 - blade 1 position (second mode)
    2.5812 - blade 1 velocity (second mode)
    2.5833 - blade 2 position (second mode)
    2.5833 - blade 2 velocity (second mode)

  3. If I unlock the first two flap DOFs, and the first edge DOF, I get following:
    0.94701 - blade 1 position
    0.94701 - blade 1 velocity
    0.94905 - blade 2 position
    0.94905 - blade 2 velocity
    1.2851 - blade 1 position (edge)
    1.2851 - blade 1 velocity (edge)
    1.2864 - blade 2 position (edge)
    1.2864 - blade 2 velocity (edge)
    2.7014 - blade 1 position (second mode)
    2.7014 - blade 1 velocity (second mode)
    2.7033 - blade 2 position (second mode)
    2.7033 - blade 2 velocity (second mode)

Now, things get strange when I unlock the drivetrain mode, because it breakes the symmetry of the two blades, but only in the edge mode:

  1. Drivetrain DOF unlocked ONLY:
    1.12 - DT position
    1.12 - DT velocity

  2. Drivetrain DOF and edge DOF unlocked
    0.8617 - DT position
    0.8617 - DT velocity
    1.2817 - blade 1 position (edge)
    1.2817 - blade 1 velocity (edge)
    4.1722 - blade 2 position (edge) ?!
    4.1722 - blade 2 velocity (edge) ?!

Can somebody explain me why this happens?

Thanks in advance,

Dear Mario,

I’m not sure I can explain the results you are getting, but you should be aware that eigenanalysis of the linearized state matrix “A” is of little use to a 2-bladed spinning rotor. Floquet theory is required for proper analysis of a 2-bladed spinning rotor.

Best regards,

Dear Jason,

thank you for your quick reply. I am actually planning to analyze the plant by using the Floquet theory, but I came to the idea of using the simple analysis by reading Wright and Fingersh - Advanced Control Design for Wind Turbines, where they control a linearized model of the two bladed turbine by using just the averaged dynamic matrix A. And somehow it makes sense to me. Do you know if any work has been done on “edge mode control”? It looks to me that it can not be neglected, because it has direct impact on the loads experienced by the drivetrain. Am I wrong in my assumptions?

Best regards,

Hi, although I can’t say for certain, my initial response to your question of whether edge mode control has been attempted, is yes and no:

A drivetrain damper controls the coupled drivetrain torsion/edgewise collective mode, and this is quite typically included in wind turbine control. For the edgewise asymmetric modes, I can’t personally think of any publications which document controlling these modes. Hope this helps!