HydroDyn Linearization: Does the Stiffness Matrix Include More Than Hydrostatic Effects

[menglin912]

Hi everyone,

I performed the following steps in OpenFAST:

Step 1: I enabled only the ElastoDyn module and obtained the mass, stiffness, and damping matrices from the output .lin file.

Step 2: Based on Step 1, I additionally enabled the HydroDyn module and obtained a new set of mass, stiffness, and damping matrices.

I initially assumed that the difference between the two stiffness matrices (i.e., the stiffness matrix with HydroDyn minus the one without HydroDyn) would represent the hydrostatic stiffness matrix. However, the result differs significantly from the hydrostatic stiffness matrix provided in the official documentation at the equilibrium position.

On the other hand, if I directly extract a submatrix from the C matrix in the .lin file—corresponding to the hydrostatic output channels and the six-degree-of-freedom displacement states—the result matches very well with the official hydrostatic stiffness matrix.

So my question is:

Does the stiffness contribution from the HydroDyn module include more than just the hydrostatic stiffness matrix?

Thank you in advance for your help!

[jjonkman]

Dear @menglin91,

In HydroDyn, structural displacements are used in various calulations depending on your model set-up, but I would expect the effect of buoyancy contribution to hydrostatics to be the dominant term.

When you obtain the hydrostatic stiffness matrix your extract from the C-matrix of the .HD.lin file, I gather you are using HydroDyn outputs B1HdSFxi to B1HdSMzi- is that correct? Do you get the same matrix when using HydroDyn outputs HydroFxi to HydroMzi? This will tell you if there are any additional terms included, based on your model set-up.

I understand your methodology to isolate the buoyancy term from the others by subtracting the two sets of stiffness matrices, but the issue may be in the stiffness matrix from ElastoDyn. When only ElastoDyn is enabled in OpenFAST and the platform DOFs and gravity are enabled in the ElastoDyn module, the ElastoDyn model cannot be in static equilibrium, and so, the stiffness matrix associated with gravitational restoring would not be computed properly through a linearization process.

Best regards,

[menglin912]

Dear Jonkman,

Thank you very much for your previous response and guidance. I have a few follow-up questions regarding the hydrostatic stiffness matrix and hydrodynamic added mass in OpenFAST, and I would greatly appreciate your clarification.

First,regarding the hydrostatic stiffness matrix: yes, I extracted it from the C-matrix using the HydroDyn outputs B1HdSFxi to B1HdSMzi. However,when I use HydroFxi to HydroMzi, the resulting matrix is not the same.The two matrix I obtained are shown below:

In my understanding, I assumed that the total hydrodynamic force could be written as:

hydrodynamic force = wave radiation force + diffraction force + hydrostatic force,

with the radiation force contributing the added mass and radiation damping. Based on this understanding, I expected the two stiffness matrices above to be identical. Besides, I found that HydroFxi does not equal B1RdtFxi + B1WvsF1xi + B1HdSFxi . Could you please clarify why this is the case, and whether there are any additional terms that I should exclude when extracting the hydrodynamic stiffness contribution?

Second, regarding your earlier statement that you understood my methodology as isolating the buoyancy term by subtracting two sets of stiffness matrices: I am not sure I fully understand why the buoyancy term is being emphasized. In my understanding, within the linearized equations of motion, buoyancy and structural gravity should already balance each other at equilibrium. My objective is to obtain the Jacobian of the hydrodynamic force with respect to displacement. I would be very grateful if you could explain whether my understanding here is incomplete.

Third, regarding your point that enabling only the ElastoDyn module does not produce the correct structural contributions to the mass, damping, and stiffness matrices: I tried an alternative approach. I referred to your previous forum posts, in which you provided the structural mass and stiffness matrices for the NREL 5-MW OC4-DeepCwind semisubmersible wind turbine.

Using the mass and stiffness matrices in Step 2 from the previous question(ElastoDyn + HydroDyn), I subtracted those structural contributions to obtain updated hydrodynamic mass and stiffness matrices. The resulting hydrodynamic mass agrees very well with the zero-frequency added mass matrix reported in the official documentation, but the hydrodynamic stiffness matrix still remains inconsistent.

As I understand it, the hydrodynamic added mass and damping matrices are generated from the WAMIT .1 and .3 files. If I want to construct a simplified equation of motion for a floating wind turbine, would it be reasonable to treat the hydrodynamic added mass and damping matrices as constants? Also, to what extent are these matrices influenced by wave conditions, such as wave height, wave period, or the choice of wave spectrum (for example, JONSWAP or Pierson–Moskowitz)?

Thank you again for your continued guidance. I sincerely appreciate your help in clarifying these hydrodynamic contributions.

Best regards,

[jjonkman]

Dear @menglin912,

Here are my responses to your questions:

1. Regarding your force summation, there can be other forces such as from 2nd-order potential-flow or from the strip-theory solution, depending on how you set-up your HydroDyn model. Can you share your HydroDyn input file, which may help me provide better guidance?
2. The term "hydrostatics" often implies the combination of weight terms and buoyancy terms and I assumed based on your description that you were trying to isolate the buoyancy term. Regardless, I would expect buoyancy to be the largest contributor from HydroDyn to the change in hydrodynamic loads with displacement.
3. Because of the the frequency-dependence of added mass and damping in the potential-flow solution, it is generally not possible to express the equations of motion of a floating body in the time domain as you describe, without approximations (unless, the frequency-dependence is negligible). Rather, the frequency-dependence in the time-domain is captured by using a fixed added mass (typically the infinite frequency limit) and expressing the damping in terms of a convolution that captures the radiation "memory effect". In first-order hydrodynamics theory, the waves directly affect the force on the right-hand side, but not the added mass, damping, or hydrostatic terms, although in the frequency domain formulation the wave frequency and body motion frequency would be assumed identical, so, the wave frequency (period) does implicitly define the frequency of the added mass and damping matrices.

Best regards.

[menglin912]

Dear Dr. Jonkman,

Thank you very much for your prompt response.

1. I have attached my hydrodynamic input files below for your reference:
   hydro_input.zip

2. I would like to confirm the following: if I solve the motion using the time-domain equation shown below, are the expressions of the
   structural mass matrix M_struc and the gravity-related restoring matrix K_struc (as pasted in my previous question) correct?

   In particular, the term z_g is defined as the full-system center of mass location—is it defined in the global coordinate system,right?（Also,I have seen another statement saying that z_g represents the vertical distance between the center of buoyancy and the center of
   mass),could you please help clarify this?
   Anyway, regardless of what z_g represents,if it varies continuously with the platform motion during the simulation, does that imply that
   M_struc and K_struc are also time-varying?

   My current understanding is that this may depend on whether the above equation is interpreted as a linearized equation of motion.

3. In addition, I found that by dividing the (1,5) element of the M_struc matrix by the (1,1) element, I obtain z_g = -9.89m. However, the value given in ElastoDyn.dat for PtfmCMzt is −8.6588 m. Similarly, PtfmCMxt does not match either. Could you please explain the reason for this discrepancy?

Thank you very much for your time and assistance.
Best regards,

[jjonkman]

Dear @menglin912,

Thanks for sharing your HydroDyn input file.

Regarding (1), I see a few entries that will cause displacements to affect the hydrodynamic loads separate from the hydrostatic restoring output from WAMIT used in the potential flow solution:

• You have set ExctnDisp = 1, so, the wave-excitation from the potential-flow solution will depend on displacement
• You have set WaveDisp = 1, so, the wave-induced loads from the strip-theory solution (in your case viscous terms because PropPot = TRUE) will depend on displacement
• You have filled members set in the strip-theory solution, so, the ballast water in the strip-theory solution will influence the hydrostatic restoring

Regarding (2), when the equations of motion are linearized, the mass, damping, and stiffness are assumed to be constant (and independent of displacement). This assumption is only valid for small motion. And in K_struct, z_g should be defined globally at the undisplaced position, and the center of buoyancy in C_hydrostatic would also be defined globally at the undisplaced position, so, the difference between the two is captured by the sum of C_struct and C_hydrostatic. Note that your equation for K_struct misses the m * g * x term in the (4,6) position, where x is the x-offset of the structural center of mass (a similar m * g * y term would appear in the (5,6) position if y-offset of the structural center of mass was nonzero).

Regarding (3), you mention obtaining the numerical values of M_struct and K_struct from a forum post. Can you clarify exactly where you obtained these values?

Best regards,

[menglin912]

Dear Dr. Jonkman,

I have two follow-up questions based on your previous guidance.

First, according to your explanation, the resulting formulation suggests that the total hydrodynamic load consists only of the hydrostatic force, radiation force, and diffraction force calculated using potential-flow theory:

HydroFxi = B1HdSFxi + B1RdtFxi + B1WvsF1xi

This leads to another question for me. For a semi-submersible platform, my understanding is that its hydrodynamic loads can be divided into two categories. The first category includes the radiation, diffraction, and hydrostatic loads provided by WAMIT based on potential-flow theory. The second category consists of the loads on Morison members supplemented through strip theory, such as viscous forces and additional hydrostatic effects associated with ballast water. Is this understanding correct?

From an engineering practice perspective, how large are the viscous forces and similar contributions in the second category compared with the forces provided by the first category? To put it more directly: when constructing a rough equation of motion, would neglecting viscous forces seriously impair the accuracy of the predicted motion response?

Second, I am referring to the following link:

https://forums.nlr.gov/t/the-frequency-component-of-flapwise/1730/7 I assumed that all elements in the structural stiffness matrix are zero except for K44 and K55.

Following your suggestion, I re-derived the structural stiffness matrix (see the attached figure). However, in my derivation, the position vector of the center of mass is defined in the local coordinate system. This seems somewhat inconsistent with your earlier explanation.

In addition, is this center-of-mass position vector explicitly defined in the ElastoDyn input file? Parameters such as PtfmCMxt seem to be defined in the global coordinate system.

I would greatly appreciate your clarification on these points.

Best regards,

[jjonkman]

Dear @menglin912,

Regarding (1), your equation for HydroFxi only includes terms from the first-order potential flow solution and neglects terms from second-order potential-flow and strip-theory solutions. In the model you are using, you have disabled second-order potential flow, but you have enabled viscous-drag and water ballast in the strip-theory solution, so, these terms must be included for your equation to be valid. Viscous effects can be quite important, depending on the platform.

Regarding (2), I had previously mentioned "globally at the undisplaced position", which means the same thing as a body-fixed local coordinate system (global and local are the same for an undisplaced body). The center of mass of the platform is defined in the ElastoDyn file (and is defined locally in the tower-base coordinate system), but this is not the same as the full-system center of mass, which is not specified directly. In ElastoDyn, different components (blades, hub, drivetrain, nacelle, tower, platform) are specified separately and the contribution from all of these terms represents the "full system".

Best regards,

[menglin912]

Dear Dr. Jonkman,

Thank you very much for your patient and prompt response to my questions. I truly appreciate the time and effort you took to clarify my doubts.

Best regards,
Meng Lin