OC3-Hywind eigenanalysis

Hi,

I am analyzing floating wind turbine. The floating wind turbine is similar to OC3 Hywind report 41958 and 47535. I have got the same stiffness matrix resulting from the mooring system and hydrostatic restoring stiffness. The mass matrix includes the structural matrix of the spar and the added mass matrix.
The results I have got in the surge, sway, heave, yaw are the same in reference: Jonkman J., Musial W.: Offshore code comparison collaboration (OC3) for IEA Wind Task 23 offshore wind technology and deployment. Technical Report No. NREL/TP-5000-48191,National Renewable Energy Lab. (NREL), Golden, CO, USA(2010) in paragraph2, section 5.4.4 page 65/74. However, the pitch and roll natural frequency (0.0374 and 0.0375 Hz) differs from the reference (0.034 Hz).
If I use the diagonal mass matrix (no cross term), I will get an exact similar natural frequency to the above reference.
Did anyone do eigenanalysis of the OC3-Hywind system before, and did you get the same results? Also, can you please explain to me which mass matrix I d better use? Thank you.

Best regards,
Hoa Nguyen

Dear Hao Nguyen,

I would think you’d need to include the off-diagonal coupling terms to predict the correct natural frequencies of the NREL 5-MW turbine atop the OC3-Hywind spar buoy.

The full-system mass, center of mass, and inertia of the OC3-Hywind system, from which the 6x6 mass matrix can be formed, can be found in the following forum topic: http://forums.nrel.gov/t/the-frequency-component-of-flapwise/1730/1.

Best regards,

Dear Jason.Jonkman,

Thank you so much for your prompt response. I rechecked my calculation, and the difference might result from the added mass matrix, which is computed by strip theory and the method of computing mass matrix. My structural mass is spar mass with the lumped mass matrix from the superstructure. I don’t use the same technique like you in http://forums.nrel.gov/t/oc3-hywind-raos/1085/23. Below are my mass matrices and stiffness matrices.

M_structural =
[8065975.57351049 0 0 0 96047931.0553377 0
0 8065975.57351049 0 -96047931.0553377 0 -96101
0 0 8065975.57351049 0 96101 0
0 -96047931.0553377 0 19922216225.0856 0 0
96047931.0553377 0 96101 0 19925853675.2147 0
0 -96101 0 0 0 169608940.129100]

M_addedmass =
[7981568.18556329 0 0 0 222274267.649286 0
0 7981568.18556329 0 -222274267.649286 0 0
0 0 0 0 0 0
0 -222274267.649286 0 15164062100.0025 0 0
222274267.649286 0 0 0 15164062100.0025 0
0 0 0 0 0 0]

K_mooring =
[41181.1776176891 0 0 0 -2906608.23246224 0
0 41181.1776176891 0 2906608.23246224 0 0
0 0 11941.5064526553 0 0 0
0 2906608.23246224 0 310785043.914555 0 0
-2906608.23246224 0 0 0 310785043.914555 0
0 0 0 0 0 11566677.9908399]

K_hydrostatic =
[0 0 0 0 0 0
0 0 0 0 0 0
0 0 333550.146410852 0 0 0
0 0 0 1575800183.54207 0 0
0 0 0 0 1575800183.54207 0
0 0 0 0 0 98340000]

f=
[0.00801929726094901
0.00801929746514019
0.0323647474682099
0.0374444408562521
0.0374468019191301
0.121188604319519]

My mooring stiffness matrix is similar to report 47535, page 23/31. Hydrostatic restoring stiffness considers the restoring effects of body weight. There is a slight difference in the natural frequency, f(Hz). Do you have any idea that helps me improve my code? Thank you so much for your time.

Best regards,
Hoa Nguyen

Dear Hao,

Well, your structural mass looks good, but your center of mass and inertias are different. From your mass matrix, I see that your center of mass is located at +11.9 m, but it should be located at -78 m. Your inertias about (0,0,0) are also quite different, e.g, Ixx = 1.99E10 in yours and 6.8E10 in mine.

Your rotational add-mass terms are also quite different than mine, e.g., A(4,4) = 1.5E10 in yours and 3.9E10 in mine.

I also calculate different hydrostatic stiffness (including body weight restoring) in the (4,4) and (5,5) elements.

Is your reference origin for these 6x6 matrices the same as mine, i.e., (0,0,0), which is the intersection of the undisplaced tower centerline and the still water level?

Best regards,

Dear Jason.Jonkman,

Thank you for your help.

  • ‘‘e.g, Ixx = 1.99E10 in yours and 6.8E10 in mine.’’ This is taken concerning the spar centre of mass at -78m.

  • ‘‘A(4,4) = 1.5E10 in yours and 3.9E10 in mine.’’ I used strip theory to compute added mass matrix. addedmasspaper.png[/attachment]

-’‘I also calculate different hydrostatic stiffness (including body weight restoring) in the (4,4) and (5,5) elements’’. If I just use platform’s water-plane shape, displaced volume, and COB I will get C(4,4) = C(5,5) = -5.01E9 while yours are -4.999E9 (page 12/31 47535). when the mass is considered, the added term mgz_G = 74663309.8178 = -5.7132e+09.

-’’ Is your reference origin for these 6x6 matrices the same as mine, i.e., (0,0,0)’’, mine is at the spar’s centre of mass. Mass matrices below are taken concerning the centre of mass

M_spar =
[7466330 0 0 0 0 0
0 7466330 0 0 0 0
0 0 7466330 0 0 0
0 0 0 4229230000.00000 0 0
0 0 0 0 4229230000.00000 0
0 0 0 0 0 164230000]

M_lumpedmass =
[599645.573510489 0 0 0 88903634.6413001 0
0 599645.573510489 0 -88903634.6413001 0 -96101
0 0 599645.573510489 0 96101 0
0 -88903634.6413001 0 13489436551.8318 0 0
88903634.6413001 0 96101 0 13493074001.9609 0
0 -96101 0 0 0 5378940.12910000]

M_addedmass (z_G = -78) =
[7981568.18556329 0 0 0 127180279.657864 0
0 7981568.18556329 0 -127180279.657864 0 0
0 0 0 0 0 0
0 -127180279.657864 0 11000591244.0771 0 0
127180279.657864 0 0 0 11000591244.0771 0
0 0 0 0 0 0]

M_total =
[16047543.7590738 0 0 0 216083914.299164 0
0 16047543.7590738 0 -216083914.299164 0 -96101
0 0 8065975.57351049 0 96101 0
0 -216083914.299164 0 28719257795.9089 0 0
216083914.299164 0 96101 0 28722895246.0380 0
0 -96101 0 0 0 169608940.129100]

f =
[0.00801562463344613
0.00801562524548552
0.0264210386431131
0.0264228990394938
0.0323647515190774
0.121188578543521]

Could you please have a look? Thank you so much, I really appreciate your help.

Best regards,
Hoa Nguyen

Dear Hao,

I haven’t checked all of your numbers, but just a few comments:

  • All of the 6x6 matrices, for mass, added mass, and stiffness, must be expressed relative to the same origin. I would suggest using (0,0,0) or the full-system center of mass (0,0,-78 m). The transformation from one system to another has been discussed in the following forum topic: http://forums.nrel.gov/t/time-domain-analysis-by-aqwa/1946/2.
  • The spar alone center of mass is (0,0,-89.9155 m).
  • The body weight restoring term (-mgz_CG) should be based on the full-system mass and center of mass, not just the mass and center of mass of the spar. It looks like you are mixing the mass of the spar alone (7.466E6 kg) and the full-system center of mass (-78 m).

Best regards,

Dear Jason.Jonkman,

Thank you for your support. I have figured it out. Have a great day.

Best regards,
Hoa Nguyen