Hydrostatic Stiffness Matrix with Different Reference Frame

[Parrotyeon]

Dear all,
I have a question regarding the transformation of the stiffness matrix between different reference points in ANSYS AQWA.

I am currently analyzing the OC4 5MW Semi-submersible platform. According to the NREL definition document, there are two distinct stiffness matrices provided with respect to the Center of Gravity (C.O.G.) and the Still Water Level (S.W.L.):

 w.r.t C.O.G
C_33 = 3794443 N/m, C_44 = - C_55 = 1.49282e+9 Nm/rad

w.r.t S.W.L
C_33 = 3.836e6 N/m, C_44 = - C_55 = -3.776e+8 Nm/rad

I could get the same result if i change the reference point in AQWA but i want to know the transformation matrix.

I attempted to derive the C.O.G. matrix from the S.W.L. matrix using the standard transformation $[C_{COG}] = [T]^T [C_{SWL}] [T]$. Given $z_{cg} = -13.46$ m, I used the following transformation matrix $[T]$ (as suggested in a previous NREL forum thread, link: https://forums.nrel.gov/t/time-domain-analysis-by-aqwa/1946/2):

[TransMat] = [
[1 0 0 0 -z_cg y_cg];
[0 1 0 z_cg 0 -x_cg];
[0 0 1 -y_cg x_cg 0];
[0 0 0 1 0 0];
[0 0 0 0 1 0];
[0 0 0 0 0 1];
]

z_cg = -13.46 m so,

[TransMat] = [
[1 0 0 0 13.46 0];
[0 1 0 -13.46 0 0];
[0 0 1 0 0 0];
[0 0 0 1 0 0];
[0 0 0 0 1 0];
[0 0 0 0 0 1];
]

However, this geometric transformation alone does not account for the significant change in $C_{44}$ and $C_{55}$. After consulting the WAMIT Theory Manual, I found that the hydrostatic stiffness terms include a gravity component (e.g., $-mgz_g$).

When I manually adjusted the value by calculating $C_{44_COG} - (m \cdot g \cdot 13.46)$, the result was approximately $-3.901 \times 10^8$, which is much closer to the NREL S.W.L. value.Could you please confirm if this additional adjustment for the gravity moment is the correct theoretical approach for the reference point change?
Thank you for your help.
Best regards,
Yeon

[jjonkman]

Dear @Parrotyeon,

Could you clarify where your COG value of C_44 = 1.49282e+9 Nm/rad comes from? Is that what are you getting from AQWA? I don't believe this value is stated in the OC4-DeepCwind semisubmersible specifications document. I also don't understand why you say C_44 = - C_55; I would think you'd say C_44 = C_55 for this specific floater.

Best regards,

[Parrotyeon]

Dear @jjonkman ,

Thank you for quick reply.

As shown in the figure below, the results appear to be identical.

Could you kindly let me know if there is any mistake in my calculation or interpretation?

I would greatly appreciate your feedback.

Best regards,

[jjonkman]

Dear @Parrotyeon,

I see; that makes sense. Your error is that you redefined the center of buoyancy in your calculation of C₄₄,COG.

Best regards,

[Parrotyeon]

Dear @jjonkman,

I have been thinking about the hydrostatic stiffness with respect to different reference points.

My understanding is that the hydrostatic rotational stiffness should depend on the reference point, because the unit rotational displacement used to evaluate the restoring moment changes depending on the chosen origin. Based on this reasoning, I expect the rotational stiffness values to differ when calculated about different reference points.

In AQWA, when I compute the hydrostatic stiffness with respect to the still water level (SWL), the results agree well with the values reported in the NREL documentation.

Could you kindly advise whether my understanding is correct and what would be the correct equation to calculate hydro static stiffness w.r.t C.O.G ?
I would greatly appreciate any clarification you can provide.

Best regards,

[jjonkman]

Dear @Parrotyeon,

I believe the issue you are facing is that the equations you reference from the forum are correct for the full equations of motion, but do not properly express the transform of each individual term. At the end of the day, for hydrostatic restoring, it is the difference between the center of buoyancy and center of gravity that matters, whether you are expressing the equations of motion relative to the SWL or COG.

Best regards,

[Parrotyeon]

Dear @jjonkman ,

Thank you very much for your explanation.

If I understand correctly, the reference-point transformation should be applied to the complete linearized equations of motion, rather than attempting to transform individual restoring terms separately.

That is, once the full system matrices ([M], [A], [B], [C]) are consistently defined at a given reference point (including hydrostatic restoring formulated in a consistent manner), then applying the transformation to the entire system (e.g.,
( M' = T^T M T ),
( C' = T^T C T ), etc.)
should yield the correct equations expressed about the new reference point.

Thank you again for your guidance.

Best regards,