Hydrostatic restoring matrix

Hi Jason,
I am a bit confused about the hydrostatic restoring matrix. In the manual, it is highlighted that wamit only calculate the hydrostatic restoring provided by buoyancy, because gravatitional restoring would be intrinsicly considered in FAST in elastodyn. At the same time, the .hst file is a 6*6 matrix. This matrix seems exactly equal to the Cij in the picture?
image

I feel that this part is the change of buoyancy, rather than the net buoyancy. So the rougVo is omitted here because it would be balanced by the gravity? And the Cij here is also the final hydrostatic restoring matrix in the equation of motion? I tried to solve the equation using wamit output file, and got confused about the matrix. I check the theory about hydrostatic in Newman’s book and find the expression. For the specific application in terms of FAST and WAMIT, i get lost.

Thanks
Lulu

Dear @Lulu.Liu,

I agree that the WAMIT .hst file to be used in HydroDyn should equal [C_ij]^Hydrostatic from your first equation. And I agree that this restoring matrix should not include the effects of body weight because body weight is intrinsically included in the structural modules of OpenFAST (ElastoDyn, SubDyn, BeamDyn), which would otherwise be double counted if included in [C_ij]^Hydrostatic. This restoring matrix will be different than the restoring matrix in your last equation (from Newman) because the restoring in your last equation must include the contribution from both buoyancy and body weight–i.e., the total restoring.

The undisplaced buoyancy term (rho * g * V_0) does not appear in your last equation (from Newman) because it should cancel with the weight of the floating system and vertical undisplaced mooring pretension.

Best regards,

Hi Jason
Thank you for your time and quick reply. I checked wamit manuual with the matrix. I was wondering the relationship between the C(i,j) in wamit mannual and the Cij^hydrostatic (that is the output file) in the defination of the semi-sub. Why there is an additional term (rho * g * V_0) ? This term is the most confusing one.
If I want to change the Cij from .hst file to the equation of motion (considering the contribution from weight), I just need to add the term -mg*Zg back to the element. Is that right? Because we assume Xg,Yg,Zg=0 in wamit. In this case, the structure is symmetry(Xg=Yg=0) and regardless of the different mode sequence.

image

Best regards

Dear @Lulu.Liu,

The term rho * g * V_0 is not part of the hydrostatic restoring matrix. This is the undisplaced buoyancy of the body. That is, the total contribution from buoyancy is rho * g *V_0 - [C_ij]^Hydrostatis * q_j, as in your first equation. Often the rho * g * V_0 term is neglected because it is assumed to cancel with the body weight and vertical mooring pretension. In OpenFAST, however, the rho * g * V_0 term is included in HydroDyn because the body weight and vertical mooring pretension are present in other modules.

Regarding the z_g term, I agree that you can add that term back into the restoring matrix if you need the total restoring matrix.

Best regards,

Dear Jason
Many thanks. It is very helpful.

Regards