Hi all,
I am studying the dynamics of the OC3 phase IV Spar. Since the MATLAB code that I developed for frequency domain analysis gives me an incorrect pitch natural frequency (0.11 rad/s instead of around 0.21 rad/s expected; by ‘natural frequency’ I mean its first approximation by the location of the pitch RAO peak), I am doing a simple check by calculating the natural frequency with the simple formula (square root of the ratio of the total system stiffness and the sum of the system mass and added mass), using the same input as that used within the code. The error persists.
Hence my question: would anyone be able to share the correct stiffness and mass matrices of the OC3 Spar, please?
I have been able to verify the pitch inertia (IYY = 6.8E+10 kg-m2) based on another post in this forum, so the error would most probably be caused by the wrong stiffness. My value for total stiffness including the effects of the waterplane, volume/buoyancy, gravity and mooring is 1.48E+09 Nm/rad, and added mass computed with Nemoh and verified against WAMIT is 3.90E+10 kg-m2.
I would be grateful for any hints.
Best wishes
Dear Kasia,
The BModes input file we have provided for the OC3-Hyind spar includes the 6x6 added mass matrix, the 6x6 hydrostatic stiffness matrix (augmented with the pitch and roll restoring from body weight / gravity), and the 6x6 mooring stiffness matrix–see the following forum post: BModes : Input parameters about tower support subsystem - #2 by Jason.Jonkman. (Please note that the link to BModes_JJ.exe and the OC3Hywind.bmi input file is now: drive.google.com/drive/folders/ … sp=sharing.)
The full-system mass, center of mass, and inertia of the rigid NREL 5-MW baseline wind turbine atop the OC3-Hywind spar is given in the following forum topic: Inertial Moments of OC3-Hywind Components - #2 by Jason.Jonkman.
Best regards,
Dear Jason,
Thank you very much for your quick response, I appreciate it. Taking the data as provided:
- Full-system mass (M): 8066048.1545 kg
- Gravitational acceleration (g): 9.81 m/s**2
- Full-system vertical centre of gravity (z): -78.001 m
I calculate the gravitational stiffness as -Mgz = 6.17E+09. The mooring stiffness in pitch (5,5) is given as 3.111E+08 Nm/rad. Therefore, the mooring+gravity pitch stiffness seems to be 6.48E+09 Nm/rad. In the OC3Hywind.bmi file this value is given to be 6.89E+09 instead. I can see that this number is obtained when only the platform is considered; why only the platform and not the full-system is considered when calculating the gravitational restoring?
Also, in the Definition of the Floating System for Phase IV of OC3 document, it is stated that
However, in the OC3Hywind.bmi file the reference point for platform inertia is given as 0 (i.e. at MWL). Which reference point does the value of 4,229,230,000 kg•m2 refer to?
Lastly, if I omit the RNA inertia, the natural frequency calculates to 0.21 which is about the same as provided in many references. Why do we not consider the inertia of the tower and the RNA?
Kind regards,
Kasia
Dear Kasia,
Good point. I don’t recall why the (4,4) and (5,5) elements of mooring_K in this BModes_JJ model of the OC3-Hywind spar were computed based on the platform mass and center of mass rather than the full-system mass and center of mass, especially since the forum post mentions that the full-system values were used. This looks to be an error. (This BModes file was only provided as an example, and not used by NREL to derive the mode shapes of the ElastoDyn model.)
Actually, i_matrix_pform in BModes is specified about the platform CM. The roll and pitch inertia of the OC3-Hywind spar only (including ballast, but not including the tower or RNA) is 4229E6 kgm^2 as stated.
I’m not understanding your last question. I would expect the tower and RNA mass and inertia to be used when calculating the full-system natural frequencies.
Best regards,
Dear Jason,
Thank you a lot for looking into this. My last question was not very clear, sorry, let me explain it better.
To verify my calculations, I am looking at the RAOs and the natural frequencies reported in ‘Investigation of Response Amplitude Operators for Floating Offshore Wind Turbines’ by NREL published in 2013. The table titled ‘OC3-Hywind Platform and Tower Natural Frequencies’ claims the pitch natural frequency to be 0.034 Hz, i.e., 0.2136 rad/s. This is also the approximate location of the pitch RAO (Figure 3).
When I included all subsystems (platform, tower, RNA) in the calculation of inertia moments and stiffness, I got the full-system natural frequency of 0.1183 rad/s. Hence, I tried to find the possible set up that could be used to obtain the value of 0.2136 rad/s instead (reverse-engineering). It happens that if you only include the platform, and you keep the platform pitch inertia as 4,229,230,000 kg•m2 without translating the moment to MWL, the result is correct. This is why I asked about which subsystems were included.
Could you please confirm which turbine elements (platform/tower/RNA) were included when deriving the plot and the table in the above-mentioned report, please? As far as I understand, the plot was derived based on the entire system but with no consideration of aerodynamic/gyroscopic loads- is that correct? Are the natural frequencies included in the table related to the full-system? If not, where can I find the information about the full-system natural frequencies?
FYI, I am attaching a spreadsheet with my calculations.
Thank you for your time, I really appreciate your help!
Best,
Kasia
natural_frequency-1.xlsx (45.3 KB)
Dear Kasia,
The results given in that table (with a pitch natural frequency of 0.034 Hz) are for the full system (RNA + tower + spar + moorings).
I haven’t reviewed your spreadsheet in great detail, but I would guess the big problem is that you are considering the solution as a one-DOF problem with respect to (0,0,0), but the pitch and surge modes are coupled at this location. The spar mostly pitches (uncoupled from surge) about the location where the moorings attach to the spar (0,0,-70). While I haven’t tried this myself, I would assume your one-DOF approach to calculating the pitch natural could work OK if you referenced the the inertia and stiffness about this point. Alternatively, you can perform an eigenanalysis on the linearized full-system matrices (including multiple DOFs), which is what NREL did to generate the published result.
Best regards,
Dear Jason,
All clear now, thank you very much for your help. Using the input you provided my frequency-domain coupled model outputs the frequencies and RAOs matching those reported in the document mentioned, so all works fine now. Again, thank you for your support.
Best wishes,
Kasia
Dear Jason,
I’m having the same problem with constructing the full system equations of motion, mainly with a large error in the natural frequency of the pitch. I am new to water motion and after reading the threads I still don’t understand how to get the correct results.
My query is to calculate the result with reference point (0,0,-70):
K_hydrost[5,5] = -4.89892E9 . Reference point (0,0,0) and K_hydrost[5,5] = 650383000. reference point (0,0,-70)
K_moor[5,5] = 3.11E+08. Reference point (0,0,-70)
M_add_mass[5 ,5] = 3.9E10. The value of add_mass should be independent of the reference point, right?
M_sys[5 ,5] = m_rotor * (z_Hub - sg) ** 2 + m_tower * (z_tower - sg) ** 2 + m_platform * (z_platform - sg) ** 2 + I_platform
K_grav[5,5] = -(m_rotor * (z_Hub - sg) * 9.81) - (m_tower * (z_tower - sg) * 9.81) - (m_platform * (z_platform - sg) * 9.81)
where sg is -70.
M = M_sys[5,5] + M_add_mass[5 ,5]
K = K_moor[5,5] + K_hydrost[5,5] + K_grav[5,5]
f = (np.sqrt(K/M))2/pi
The final result is f = 0.02614hz, which does not agree with the reported 0.034hz. Did I make some mistake in the calculation?
Kind regards,
Dear Jason,
Thank you very much for your reply.
I was solving first using the full system equations and I got incorrect results, it should be a problem with the calculation of the element values of the main diagonal of the matrix, so I verified that the separate calculations were in the pitch direction. And the previous calculation used also calculated at the unified reference point (0,0,0), but the result is not correct, the mooring system is connected to the floating platform at (0,0,-70), so I thought at that time that the stiffness matrix calculated by map++ is based on (0,0,-70). The following shows the results of my previous calculations at the reference point (0,0,0).
K_moor = 3.11E+08
K_hydrost = -4999184000
K_grav = -(m_rotor * z_Hub * 9.81) - (m_tower * z_tower * 9.81) + (m_platform * z_platform * 9.81)
K = K_moor + K_hydrost + K_grav
add_mass = 3.79E+10
m_sys = m_rotor * (z_Hub) ** 2 + m_tower * (z_tower) ** 2 + m_platform * (z_platform) ** 2 + I_platform + I_tower + I_rotor
M = m_sys + add_mass
f = (np.sqrt(K/M))2/pi
In the above equation, I_rotor = 2607890 and I_tower = 530350000. about whether these two values are correct.？But based on the calculations, the order of magnitude of I_rotor and I_tower should have less impact on the calculation of the result, right?
In the end, my calculation results in f = 0.0213hz. what am I missing or what is the problem in my calculation?
Best regards,
Dear @Huajian.Xiao,
Just a few comments / questions:
- Your equation for K_grav and m_sys seems to be missing the nacelle mass/inertia.
- What values are you using for m_rotor, z_Hub (and likewise for the tower and platform)?
- Where did you obtain I_rotor and I_tower? I looked back at one my spreadsheets where I noted that I_tower (the transverse inertia of the tower about its CM) is 119,000,000 kg*m^2, so, your value is off by a factor of 4.45.
- Your equation for the natural frequency (f) only considers [5,5], neglecting the coupling between pitch and surge and pitch and heave.
Best regards,
Dear Jason,
Thank you very much for your reply. I figured out the problem, thanks again!
One small question for you, is this inertia like IXY calculated or is it an inherent property of the platform? Do you have any suggestions on how to get it?
IXY : 9.9135213851E+006 kg-meter** 2
Best regards,
Dear @Huajian.Xiao,
I’m glad you solved your issue.
Presumably you are referring to the IXY value of 9.9135213851E+006 kg-meter** 2 from the following forum topic: Inertial Moments of OC3-Hywind Components. I’m not sure I understand your question about it, but this represents the cross-inertia of the full OC3-Hywind system (platform + ballast + tower + nacelle + rotor) with respect to the (0,0,0) reference point.
Best regards,