Calculation of negative damping in pitch mode of OC4 semi-5MW

[yangyangyang98]

Dear @jjonkman :
I want to calculate the negative damping of the OC4 semi-5MW， I adopt the simplified single degree of freedom floating body pitch mode model like this paper: #Influence of Control on the Pitch Damping of a Floating Wind Turbine
Influence of Control on the Pitch.pdf
And I found most of the parameters in the report of “Definition of the
Semisubmersible Floating System for Phase II of OC4”, except for the damping associated with hydrodynamic radiation in pitch. But I found the B55=0 at the platform-pitch natural frequency, Wxn=0.037Hz in page 18 of this report. So, I would like to verify if the damping associated with hydrodynamic radiation in pitch is zero.
Additionally, I found that the units given for damping coefficient and stiffness coefficient in this report are Nms^2/rad^2 and Nm/rad, instead of kgm^2/s and kgm^2/s^2. So I cannot derive that the natural frequency of the pitch mode of the semi OC4-5MW wind turbine is 0.037Hz, May I know how to convert units, or could you please tell me the converted values?

[jjonkman]

Dear @yangyangyang98,

I agree that the hydrodynamic radiation damping is quite small at 0.037 Hz based on Figure 4-3 of the OC4-DeepCwind semisubmersible specifications report.

I'm not sure I understand your question about units, but Nm/rad is equivalent to kgm^2/s^2 and Nm/(rad/s) is equivalent to kgm^2/s because 1 N = 1 kg*m/s^2 and rad is unitless.

Best regards,

[yangyangyang98]

Dear @jjonkman :

I am a control engineer of wind turbine who is not familiar with the theory of hydrodynamics, which is far beyond my professional ability. I have been searching for the above parameters for a long time and have struggled to achieve the expected results. But no results.

May I ask if it is convenient for you to directly provide the specific values of the above three parameters I_Mass、C_Hydrostatic and B_Quad of OC4-DeepCwind semisubmersible 5MW wind turbine? I would greatly appreciate it. Or if there are any specific parameters where I can reference them.

Thank you again!

[jjonkman]

Dear @yangyangyang98,

Regarding I_Mass for the NREL 5-MW baseline wind turbine atop the OC4-DeepCwind semisubmersible, this is available in my post dated Feb 13, 2020 in the following topic on our forum: https://forums.nrel.gov/t/oc4-deepcwind-semisubmersible-wamit-files/2203.

Regarding C_Hydrostatic, this can be computed as -m * g * z_cg, where m is the full-system mass, g is gravity, and z_cg is vertical center of mass. m * z_cg is also available in the aforementioned forum topic.

I have not personally computed B_Quad for the OC4-DeepCwind semisubmersible.

Best regards,

[yangyangyang98]

Dear @jjonkman :
Thank you for your reply.

I have carefully reviewed the content of the link you mentioned above. And it can seen that I_Mass=1.269 * 10 ^ 10kgm ². My understanding of the hydrodynamic stiffness coefficient KT is that the influence of the full system body weight / center of mass is should add to pitch mode stiffness, KT = C_Hydrostatic + C_Line - mgz = -3.77610^8 + 8.7310^7 -mgz，where m is the full-system mass, g is gravity, and z is the full-system center of mass location (negative valued)，The m = 1.37*10^7 kg, g=9.8m/s^2, but I cannot find the full-system center of mass location. But based on the pitch mode natural frequency of 0.037Hz, I can deduce that the center of mass position of the full system is -2.365m. May I ask if this value is correct？

Looking forward to your reply.

[jjonkman]

Dear @yangyangyang98,

I agree with your determination of I_Mass.

The 6x6 mass matrix contains the mass (upper-left quadrant), inertias (lower-right quadrant) and mass times center of mass (off-diagonal quadrants). The mass (m) reported in that forum post is 1.407E+07 kg and the vertical center of mass (z_cg) is -1.392E+08 / 1.407E+07 = -9.893 m.

Best regards,

[yangyangyang98]

Dear @jjonkman :

Thank you for your early reply! It is a good time to communicate with you!

Based on the data you provided above, Based on the data you provided, I can calculate the natural frequency of the pitch mode of the OC4 semi submersible FOWT as W_xn = sqrt( K_x /M_x ) = sqrt( (C_Hydrostatic+C_Line-mgz)/ (I_Mass+A_Radiation) ) = sqrt( (-3.77610^8+8.7310^7-1.40779.8*(-9.893))/ (1.26910^10+7.2110^9) ) = 0.23229 rad/s, this is consistent with the 0.037Hz in the report of “Definition of the Semisubmersible Floating System for Phase II of OC4”. That's really good news！

Thank you again！

Besides, I still have doubts about the calculation of FOWT pitch mode damping, mainly because the value of B_Viscous cannot be obtained. I found the quadratic viscous drag coefficient of pitch = 3.7*10^10 Nms^2/rad^2 in Page24 of “Definition of the Semisubmersible Floating System for Phase II of OC4”, but I couldn't find the linearized damping associated with hydrodynamic viscous drag in pitch B_Viscous. Do you have any suggestions for this？

[jjonkman]

Dear @yangyangyang98,

As I mentioned before, you could use e.g., Borgman linearization to compute B_Viscous, but I have not done this myself to provide detailed guidance.

Best regards,

[yangyangyang98]

Dear @jjonkman :

Thank you for your reply. I will continue to study according to your suggestions.