Jason,
I am currently linearizing a turbine model and expect that the state and its time derivative will have poles in the form of a complex conjugate (i.e. the pole for the deflection state and the pole for the time derivative state will have identical real values but imaginary values reflected across the real axis). However, for the linearization’s I have produced (this is a floating platform) I am not seeing this trend, but I am seeing complex conjugate polls for surge and sway displacement states. Am I confused on my underlying theory, expecting that the displacement and its corresponding time derivative state should be complex conjugate eigenvalues, or is it possible that the method OpenFAST uses to perform linearizations (numeric perturbation) could result in less than analytically accurate LTI systems?
Dear Martin,
I generally agree that the eigenvalues of a linear second-order system form complex conjugate pairs. This is summarized a bit in my post dated Aug 20, 2020 in the following forum topic: BeamDyn: add point masses and show eigenfrequencies - #7 by Jason.Jonkman. Of course, you won’t have the complex eigenvalue pair if you have rigid-body modes (whereby omega_n = 0) or an overly damped system (ABS(zeta) >= 1), where the imaginary part is zero.
And I also agree that an OpenFAST linearization analysis will produce a linear model that should generate such eigenvalues. I gather you are seeing that for the surge and sway modes, but not other modes? Can you clarify what you are seeing?
Note that if HydroDyn states are included in the OpenFAST linearization, then these states don’t include second time derivatives. This is explained a bit in our DeepWind 2019 paper: iopscience.iop.org/article/10.1 … 012022/pdf.
Best regards,
Jason,
For the linearizations I am working with, I am seeing that the platform states corresponding to the displacements have a natural frequency of 0, which as you said above can be attributed to the rigid body modes used in this version of OpenFAST. The tower 1st fore-aft bending and 1st tower side to side bending displacement states form a complex conjugate eigen value pair. Similar trends are observed for 2nd fore-aft and side to side tower displacement states. However, looking at their corresponding derivative states the eigenvalues are completely different for the platform and similar to that of tower derivative states. They do not form complex conjugate pairs among themselves or relative to their displacement states.
So in my process to compute natural frequencies and damping ratios, the result I get varies from the displacement state to the derivative state. And this is perplexing that there could possibly be a different natural frequency and damping ratio for the 1st tower fore-aft bending as computed from either of the two states, and similarly across other components as well.
Dear Martin,
Just a couple comments:
- I would not expect the floating platform modes to be rigid-body modes with zero frequency unless you don’t have proper hydrostatic or mooring restoring set.
- I would expect the complex conjugate pair to be between a DOF and its first derivative, not between different DOFs, although the first fore-aft and side-side modes are likely to have similar natural frequencies. If you are not seeing this, I think I’d need to see the eigensolution to understand what you are referring to.
As you learn to interpret an OpenFAST linearization analysis and eigensolution, I would suggest starting with a simple linear model first–e.g., a model with only one tower DOF (two states) enabled. Do you then get the result you expect?
Best regards,