# 5MW FOWT linearization issue

Dear all,

After reading many of the post related with this topic, I have another issue for linearizing the 5MW FOWT system.

I would like to linearize said system in a specific operation point (for example: 13m/s steady wind, 1173.7rpm GenSpeed, 43093.55Nm GenTorque and 6.6º collective pitch angle) but I cant. If I try having ‘CalcStdy = False’ my GenTorque overcomes the nominal value (43093.55Nm), and if I have ‘CalcStdy = True’ then my collective blade pitch gets overcome to the rated angle (6.6º) for this wind speed.

How can I do the linearization process in mentioned op?

I attached both .lin files in .txt format to see both cases.

JOE
NREL5MW_FOWT_ITI_13ms_CalcStdyTrue.txt (38.9 KB)
NREL5MW_FOWT_ITI_13ms_CalcStdyFalse.txt (38.9 KB)

Dear Joannes,

Your linearization results with CalcStdy = False are resulting an in operating point generator torque of 53.5 Nm. Your linearization results with CalcStdy = True are resulting an in operating point generator torque of -43093 Nm. What setting of VSContrl and GenModel are you using to get this? Are the corresponding input file parameters for the selected torque-speed curve set appropriately for the result you desire?

Best regards,

Dear Jason,

Yes was my mistake. After many changes I was taking another .fst file with VSContrl parameters wrong.

Best regards,
JOE

Dear all,

According to FAST User’s Guide are two ways for linearize systems, with CalcStdy true or false right? I did it both ways, and the results it is the same. But when I was checking I realized that the result is not very accurate.

For example, if I am doing the linearization in the operational point of: Generator torque 43093.55Nm and Blade collective pitch 6.6º, 13m/s constant wind speed and 0.2 vertical shear. The resulting open loop bode diagram from rotor collective blade pitch to generator speed is…

Where at very low frecuencies the generator speed it would have to be the rated speed, right?
But the achieved generator speed is higher than the rated: 10^(62/20) = 1258.9rpm (rated = 1173.7rpm)

It can be said that the linearization is correct? I check it both ways (CalcStdy true and false), and the results is the same but with different blade pitch angles. I attached both .lin files. Also, is it relevant to have the same results with two different blade angles (CalcStdy(true) = 0.109216rad and CalcStdy(false) = 0.115192rad)?

Best regards,
JOE
NRELOffshrBsline5MW_Floating_ITIBarge4_CalcStdyFalse.txt (148 KB)
NRELOffshrBsline5MW_Floating_ITIBarge4_CalcStdyTrue.txt (148 KB)

Dear Joannes,

I can’t really comment on your Bode diagram because (1) the figure is too blurry to decipher, (2) I’m not sure how you generated it, and (3) I don’t have much experience looking at them.

I’m not sure where you are getting 1258.9 rpm. According to the linearization output files you attached, the generator speed is 1173.7 rpm in both.

Regardless, CalcStdy in FAST v7 is used to set how the linearization operating point is determined i.e. CalcStdy = False results in the linearization about initial conditions and CalcStdy = True tells find to calculate the steady-state solution, perhaps with trim of the control inputs. The resulting linearized model is based on the resulting operating states and inputs, not on how they are derived.

Best regards,

Dear Jason,

Ok, let’s forget about the blurry bode. But according to the linearization output files I attached, is it logical to achieve the same rotor speed and same generator torque with different blade pitch angle at 13m/s constant wind speed?

With TrimCase = 3 → Blade pitch = 6.25º
With initial cond. → Blade pitch = 6.6º

Best regards,
JOE

Dear Joannes,

You are getting the same rotor speed in both cases because you’ve set the same RotSpeed in both:

• With CalcStdy = False, you’re linearizing about your user-specified initial values of RotSpeed and BlPitch; the system is not necessarily in equilibrium.
• With CalcStdy = True and TrimCase = 3, FAST is trimming blade pitch to obtain the target rotor speed defined by your user-specified RotSpeed; the system should be in equilibrium.

Best regards,

Dear Jason,

I agree with you. But if I do with CalcStdy = True and TrimCase = 3 the generator torque also it varies and it should not, right? I mean, above rated wind speed as it is the case (13m/s) torque should be at rated value (43093.55Nm) for achieving nominal output power and only blade pitch would have to trim to obtain the target rotor speed. Thus I have been adjusting DispTol and VelTol to achieve accurate torque result because if these numbers are very small the solution does not appear to converge.

Thank you again.
Kind regards,
JOE

Dear JOE,

The FAST v7 User’s Guide explains how to set the generator torque to be fixed during TrimCase = 3. If you’ve followed this approach, you should be able to set the fixed torque to 43093.55 Nm.

Best regards,

Dear Jason

I linearized test 13 at t = 150 s , torque = 8188 Nm , pitch = 7.786 deg . after that I use mbc3 and get system matrices. The problem is when I check eigen values I found one of them have positive real part = 0.013 which mean the system unstable. Do you have any explanation for this? Non linear system is stable how come the system become unstable after linearization

Regards

Dear Rana,

I’m sorry, but I’m not sure I can comment without knowing more about your simulation set-up and results. I would expect the linear and nonlinear response to be consistent around the operating point.

Best regards,

Dear Jason
what you mean with simulation set up ? i use open loop .mdl provide with matlab/Fast v8 without change any initials. simulation time equals Tmax = 250 sec, system linearized at 150 sec. I use step function for torque 8300 Nm to 8188 Nm, step finction for pitch 7.786

this is output .lin file

Linearized model: Predictions were generated on 26-Apr-2017 at 02:17:15 using FAST (v8.16.00a-bjj, 27-Jul-2016), compiled as a DLL S-Function for Simulink as a 64-bit application using single precision
linked with NWTC Subroutine Library (v2.09.00, 23-Jul-2016); ElastoDyn (v1.04.00a-bjj, 26-Jul-2016); InflowWind (v3.03.00, 26-Jul-2016); AeroDyn (v15.03.00, 27-Jul-2016); ServoDyn (v1.06.00a-bjj, 26-Jul-2016)

Description from the FAST input file: FAST Certification Test #13: WindPACT 1.5 MW Baseline with many DOFs with VS and VP and FF turbulence.

Simulation information:
Simulation time: 150.0000 s
Number of continuous states: 30
Number of discrete states: 0
Number of constraint states: 0
Number of inputs: 9
Number of outputs: 27
Jacobians included in this file? No

Order of continuous states:
Row/Column Operating Point Rotating Frame? Description

``````      1    2.367E-01                                 F        ED 1st tower fore-aft bending mode DOF (internal DOF index = DOF_TFA1), m
2    2.816E-02                                 F        ED 1st tower side-to-side bending mode DOF (internal DOF index = DOF_TSS1), m
3   -5.682E-05                                 F        ED 2nd tower fore-aft bending mode DOF (internal DOF index = DOF_TFA2), m
4    6.658E-04                                 F        ED 2nd tower side-to-side bending mode DOF (internal DOF index = DOF_TSS2), m
5    1.880E+00                                 F        ED Variable speed generator DOF (internal DOF index = DOF_GeAz), rad
6    1.286E-04                                 F        ED Drivetrain rotational-flexibility DOF (internal DOF index = DOF_DrTr), rad
7    1.288E+00                                 T        ED 1st flapwise bending-mode DOF of blade 1 (internal DOF index = DOF_BF(1,1)), m
8    1.431E+00                                 T        ED 1st flapwise bending-mode DOF of blade 2 (internal DOF index = DOF_BF(2,1)), m
9    1.604E+00                                 T        ED 1st flapwise bending-mode DOF of blade 3 (internal DOF index = DOF_BF(3,1)), m
10    1.032E-01                                 T        ED 1st edgewise bending-mode DOF of blade 1 (internal DOF index = DOF_BE(1,1)), m
11    1.657E-01                                 T        ED 1st edgewise bending-mode DOF of blade 2 (internal DOF index = DOF_BE(2,1)), m
12   -9.531E-02                                 T        ED 1st edgewise bending-mode DOF of blade 3 (internal DOF index = DOF_BE(3,1)), m
13    2.560E-02                                 T        ED 2nd flapwise bending-mode DOF of blade 1 (internal DOF index = DOF_BF(1,2)), m
14    4.059E-02                                 T        ED 2nd flapwise bending-mode DOF of blade 2 (internal DOF index = DOF_BF(2,2)), m
15    5.924E-03                                 T        ED 2nd flapwise bending-mode DOF of blade 3 (internal DOF index = DOF_BF(3,2)), m
16    2.839E-04                                 F        ED First time derivative of 1st tower fore-aft bending mode DOF (internal DOF index = DOF_TFA1), m/s
17    3.414E-04                                 F        ED First time derivative of 1st tower side-to-side bending mode DOF (internal DOF index = DOF_TSS1), m/s
18   -3.882E-05                                 F        ED First time derivative of 2nd tower fore-aft bending mode DOF (internal DOF index = DOF_TFA2), m/s
19    3.074E-06                                 F        ED First time derivative of 2nd tower side-to-side bending mode DOF (internal DOF index = DOF_TSS2), m/s
20    2.119E+00                                 F        ED First time derivative of Variable speed generator DOF (internal DOF index = DOF_GeAz), rad/s
21   -1.368E-08                                 F        ED First time derivative of Drivetrain rotational-flexibility DOF (internal DOF index = DOF_DrTr), rad/s
22   -2.010E-01                                 T        ED First time derivative of 1st flapwise bending-mode DOF of blade 1 (internal DOF index = DOF_BF(1,1)), m/s
23    3.188E-01                                 T        ED First time derivative of 1st flapwise bending-mode DOF of blade 2 (internal DOF index = DOF_BF(2,1)), m/s
24   -1.192E-01                                 T        ED First time derivative of 1st flapwise bending-mode DOF of blade 3 (internal DOF index = DOF_BF(3,1)), m/s
25    3.183E-01                                 T        ED First time derivative of 1st edgewise bending-mode DOF of blade 1 (internal DOF index = DOF_BE(1,1)), m/s
26   -2.428E-01                                 T        ED First time derivative of 1st edgewise bending-mode DOF of blade 2 (internal DOF index = DOF_BE(2,1)), m/s
27   -7.548E-02                                 T        ED First time derivative of 1st edgewise bending-mode DOF of blade 3 (internal DOF index = DOF_BE(3,1)), m/s
28    4.228E-02                                 T        ED First time derivative of 2nd flapwise bending-mode DOF of blade 1 (internal DOF index = DOF_BF(1,2)), m/s
29   -2.799E-02                                 T        ED First time derivative of 2nd flapwise bending-mode DOF of blade 2 (internal DOF index = DOF_BF(2,2)), m/s
30   -1.448E-02                                 T        ED First time derivative of 2nd flapwise bending-mode DOF of blade 3 (internal DOF index = DOF_BF(3,2)), m/s
``````

Order of continuous state derivatives:
Row/Column Operating Point Rotating Frame? Description

``````      1    2.839E-04                                 F        First time derivative of ED 1st tower fore-aft bending mode DOF (internal DOF index = DOF_TFA1), m/s
2    3.414E-04                                 F        First time derivative of ED 1st tower side-to-side bending mode DOF (internal DOF index = DOF_TSS1), m/s
3   -3.882E-05                                 F        First time derivative of ED 2nd tower fore-aft bending mode DOF (internal DOF index = DOF_TFA2), m/s
4    3.074E-06                                 F        First time derivative of ED 2nd tower side-to-side bending mode DOF (internal DOF index = DOF_TSS2), m/s
5    2.119E+00                                 F        First time derivative of ED Variable speed generator DOF (internal DOF index = DOF_GeAz), rad/s
6   -1.368E-08                                 F        First time derivative of ED Drivetrain rotational-flexibility DOF (internal DOF index = DOF_DrTr), rad/s
7   -2.010E-01                                 T        First time derivative of ED 1st flapwise bending-mode DOF of blade 1 (internal DOF index = DOF_BF(1,1)), m/s
8    3.188E-01                                 T        First time derivative of ED 1st flapwise bending-mode DOF of blade 2 (internal DOF index = DOF_BF(2,1)), m/s
9   -1.192E-01                                 T        First time derivative of ED 1st flapwise bending-mode DOF of blade 3 (internal DOF index = DOF_BF(3,1)), m/s
10    3.183E-01                                 T        First time derivative of ED 1st edgewise bending-mode DOF of blade 1 (internal DOF index = DOF_BE(1,1)), m/s
11   -2.428E-01                                 T        First time derivative of ED 1st edgewise bending-mode DOF of blade 2 (internal DOF index = DOF_BE(2,1)), m/s
12   -7.548E-02                                 T        First time derivative of ED 1st edgewise bending-mode DOF of blade 3 (internal DOF index = DOF_BE(3,1)), m/s
13    4.228E-02                                 T        First time derivative of ED 2nd flapwise bending-mode DOF of blade 1 (internal DOF index = DOF_BF(1,2)), m/s
14   -2.799E-02                                 T        First time derivative of ED 2nd flapwise bending-mode DOF of blade 2 (internal DOF index = DOF_BF(2,2)), m/s
15   -1.448E-02                                 T        First time derivative of ED 2nd flapwise bending-mode DOF of blade 3 (internal DOF index = DOF_BF(3,2)), m/s
16    1.433E-03                                 F        First time derivative of ED First time derivative of 1st tower fore-aft bending mode DOF (internal DOF index = DOF_TFA1), m/s/s
17    1.862E-03                                 F        First time derivative of ED First time derivative of 1st tower side-to-side bending mode DOF (internal DOF index = DOF_TSS1), m/s/s
18   -4.108E-04                                 F        First time derivative of ED First time derivative of 2nd tower fore-aft bending mode DOF (internal DOF index = DOF_TFA2), m/s/s
19   -3.519E-05                                 F        First time derivative of ED First time derivative of 2nd tower side-to-side bending mode DOF (internal DOF index = DOF_TSS2), m/s/s
20   -1.958E-04                                 F        First time derivative of ED First time derivative of Variable speed generator DOF (internal DOF index = DOF_GeAz), rad/s/s
21    1.084E-06                                 F        First time derivative of ED First time derivative of Drivetrain rotational-flexibility DOF (internal DOF index = DOF_DrTr), rad/s/s
22    8.996E-01                                 T        First time derivative of ED First time derivative of 1st flapwise bending-mode DOF of blade 1 (internal DOF index = DOF_BF(1,1)), m/s/s
23    5.181E-02                                 T        First time derivative of ED First time derivative of 1st flapwise bending-mode DOF of blade 2 (internal DOF index = DOF_BF(2,1)), m/s/s
24   -7.864E-01                                 T        First time derivative of ED First time derivative of 1st flapwise bending-mode DOF of blade 3 (internal DOF index = DOF_BF(3,1)), m/s/s
25   -1.936E-01                                 T        First time derivative of ED First time derivative of 1st edgewise bending-mode DOF of blade 1 (internal DOF index = DOF_BE(1,1)), m/s/s
26   -4.867E-01                                 T        First time derivative of ED First time derivative of 1st edgewise bending-mode DOF of blade 2 (internal DOF index = DOF_BE(2,1)), m/s/s
27    6.944E-01                                 T        First time derivative of ED First time derivative of 1st edgewise bending-mode DOF of blade 3 (internal DOF index = DOF_BE(3,1)), m/s/s
28    4.124E-03                                 T        First time derivative of ED First time derivative of 2nd flapwise bending-mode DOF of blade 1 (internal DOF index = DOF_BF(1,2)), m/s/s
29   -7.557E-02                                 T        First time derivative of ED First time derivative of 2nd flapwise bending-mode DOF of blade 2 (internal DOF index = DOF_BF(2,2)), m/s/s
30    8.049E-02                                 T        First time derivative of ED First time derivative of 2nd flapwise bending-mode DOF of blade 3 (internal DOF index = DOF_BF(3,2)), m/s/s
``````

Order of inputs:
Column Operating Point Rotating Frame? Description

``````      1    1.200E+01                                 F        IfW Extended input: horizontal wind speed (steady/uniform wind), m/s
2    2.000E-01                                 F        IfW Extended input: vertical power-law shear exponent, -
3    0.000E+00                                 F        IfW Extended input: propagation direction, rad
7    0.000E+00                                 F        ED Yaw moment, Nm
8    8.188E+03                                 F        ED Generator torque, Nm
``````

Order of outputs:
Row Operating Point Rotating Frame? Description

``````      1    1.200E+01                                 F        IfW Wind1VelX, (m/s)
2    0.000E+00                                 F        IfW Wind1VelY, (m/s)
3    0.000E+00                                 F        IfW Wind1VelZ, (m/s)
4    1.450E+03                                 F        SrvD GenPwr, (kW)
5    7.786E+00                                 T        ED BldPitch2, (deg)
6   -1.984E-01                                 T        ED IPDefl1, (m)
7   -1.718E-01                                 T        ED IPDefl2, (m)
8    0.000E+00                                 T        ED TwstDefl1, (deg)
9    0.000E+00                                 T        ED TwstDefl2, (deg)
10    0.000E+00                                 T        ED TwstDefl3, (deg)
11   -2.330E+02                                 T        ED RootMxb2, (kN·m)
12    1.124E+03                                 T        ED RootMyb2, (kN·m)
13    8.812E+00                                 T        ED RootMzb2, (kN·m)
14   -1.460E+00                                 F        ED LSShftFys, (kN)
15   -2.633E+02                                 F        ED LSShftFzs, (kN)
16    1.320E+02                                 F        ED LSSTipMys, (kN·m)
17    4.071E+01                                 F        ED LSSTipMzs, (kN·m)
18    2.366E-01                                 F        ED YawBrTDxp, (m)
19   -2.882E-02                                 F        ED YawBrTDyp, (m)
20    7.242E+02                                 F        ED YawBrMxn, (kN·m)
21   -5.372E+02                                 F        ED YawBrMyn, (kN·m)
22   -1.737E+01                                 F        ED YawBrMzn, (kN·m)
23    2.023E+01                                 F        ED RotSpeed, (rpm)
24    7.786E+00                                 T        ED BldPitch1, (deg)
25    7.202E+02                                 F        ED RotTorq, (kN·m)
26    1.526E+03                                 F        ED RotPwr, (kW)
27    5.327E+00                                 F        AD RtSkew, (deg)
``````

Linearized state matrices:

A: 30 x 30
0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 1.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00
0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 1.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00
0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 1.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00
0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 1.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00
0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 1.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00
0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 1.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00
0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 1.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00
0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 1.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00
0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 1.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00
0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 1.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00
0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 1.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00
0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 1.000E+00 0.000E+00 0.000E+00 0.000E+00
0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 1.000E+00 0.000E+00 0.000E+00
0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 1.000E+00 0.000E+00
0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 1.000E+00
-7.159E+00 -1.581E-03 1.332E+02 -8.643E-02 -8.175E-03 -4.201E+02 3.173E-01 4.970E-01 4.169E-01 1.283E-01 2.496E-01 4.163E-01 -1.556E+00 -2.092E+00 -1.835E+00 -1.043E-01 4.770E-03 4.040E-01 5.150E-02 1.565E-01 -5.937E-01 -5.721E-04 1.783E-03 9.272E-04 3.870E-03 -2.262E-03 1.629E-03 -9.674E-03 -1.442E-02 -1.273E-02
-2.040E-03 -6.869E+00 -2.295E+00 1.437E+02 -2.599E-01 6.667E+02 -1.031E-01 1.058E-01 -6.895E-03 9.614E-01 -7.365E-01 -2.007E-01 4.357E-01 -2.620E-01 -1.757E-01 2.359E-05 -6.964E-02 -1.219E-02 4.942E-01 3.296E-03 1.193E+00 -2.811E-03 -3.952E-03 6.740E-03 1.861E-02 1.286E-02 -3.130E-02 3.899E-03 3.140E-03 -7.005E-03
2.189E-01 -4.028E-03 -3.334E+02 -2.582E-01 -1.839E-02 -3.724E+01 -1.802E-01 2.183E-01 7.696E-02 -2.116E-01 1.241E-01 1.602E-01 4.340E-01 -6.802E-01 -2.663E-01 -2.121E-03 5.680E-04 -1.158E+00 -1.648E-04 1.557E-02 -5.091E-02 -1.967E-03 1.607E-03 4.419E-04 9.612E-04 -1.393E-03 6.817E-04 3.066E-03 -4.584E-03 -1.957E-03
-3.249E-04 2.376E-01 -2.604E-01 -3.446E+02 -2.448E-02 1.693E+03 -1.782E-02 -3.084E-03 2.045E-02 9.721E-02 -9.229E-02 -2.126E-03 6.508E-02 1.133E-02 -7.679E-02 3.668E-04 1.877E-03 4.124E-03 -1.178E+00 2.635E-03 3.023E+00 -3.407E-04 -5.155E-04 8.550E-04 1.944E-03 1.296E-03 -3.221E-03 5.231E-04 5.796E-04 -1.107E-03
1.868E-06 9.280E-04 1.704E-03 1.274E+00 1.626E-04 1.364E+04 9.542E-05 -1.656E-05 -7.609E-05 -6.268E-04 5.481E-04 6.159E-05 -3.643E-04 2.703E-05 3.395E-04 -1.401E-06 1.143E-05 -1.212E-05 4.357E-03 -1.657E+00 2.436E+01 2.049E-06 3.018E-06 -5.056E-06 -1.237E-05 -8.368E-06 2.062E-05 -3.034E-06 -3.044E-06 6.087E-06
-5.132E-02 8.179E-02 -2.481E+00 1.109E+02 1.647E-02 -2.652E+04 5.263E-01 5.204E-01 5.117E-01 -4.700E+00 -4.584E+00 -4.626E+00 -1.328E+00 -1.304E+00 -1.277E+00 -4.164E-02 2.462E-03 -6.731E-02 3.944E-01 1.698E+00 -4.731E+01 4.610E-03 5.167E-03 4.492E-03 -4.758E-02 -4.717E-02 -4.498E-02 -5.694E-03 -5.167E-03 -4.957E-03
1.350E+01 -4.723E+00 -4.702E+03 -4.539E+02 -4.909E+00 2.049E+05 -7.983E+01 -6.053E+00 -7.913E+00 7.767E+01 7.656E+01 7.739E+01 3.847E+01 1.450E+01 1.976E+01 -5.969E+00 -4.642E-01 4.363E+01 9.964E+00 1.103E+01 3.769E+02 -7.203E+00 -7.768E-02 -2.775E-02 -1.461E+00 8.803E-01 6.306E-01 -2.365E+00 5.411E-02 6.846E-02
2.131E+01 4.058E+00 5.532E+03 -1.355E+02 1.047E+01 2.038E+05 -6.304E+00 -8.268E+01 -1.127E+01 7.890E+01 7.695E+01 7.164E+01 1.866E+01 4.844E+01 3.262E+01 -2.009E+01 -4.698E-01 -1.287E+02 3.718E+00 1.265E+01 3.766E+02 -1.057E-02 -7.229E+00 -1.296E-01 5.933E-01 -1.521E+00 8.894E-01 5.636E-02 -2.377E+00 1.842E-01
1.837E+01 5.685E-01 1.922E+03 5.844E+02 -5.309E+00 1.998E+05 -8.016E+00 -1.128E+01 -7.902E+01 7.673E+01 7.459E+01 7.790E+01 2.220E+01 3.134E+01 3.541E+01 -7.692E+00 1.961E+00 4.379E+01 -2.194E+00 1.023E+01 3.669E+02 -8.664E-02 -6.274E-02 -7.408E+00 8.704E-01 6.403E-01 -1.542E+00 1.052E-01 1.388E-01 -2.489E+00
1.718E+00 1.442E+01 -1.796E+03 8.008E+02 2.021E+01 -6.122E+05 2.486E+01 2.633E+01 2.567E+01 -3.767E+02 -2.257E+02 -2.270E+02 -5.064E+00 -6.624E+01 -6.421E+01 -2.537E+00 2.274E-01 -2.111E+01 7.357E+00 9.181E+00 -1.084E+03 6.415E-01 2.592E-01 2.257E-01 -3.781E+00 -2.329E+00 -2.219E+00 3.980E-01 -2.667E-01 -2.556E-01
3.782E+00 -1.122E+01 1.117E+03 -7.752E+02 -1.585E+01 -5.987E+05 2.571E+01 2.454E+01 2.484E+01 -2.261E+02 -3.738E+02 -2.295E+02 -6.316E+01 -2.755E+00 -6.192E+01 -2.034E+00 3.948E-01 -8.463E+00 -7.719E-01 1.001E+01 -1.059E+03 2.196E-01 7.880E-01 2.562E-01 -2.252E+00 -3.683E+00 -2.378E+00 -2.617E-01 4.936E-01 -2.653E-01
6.290E+00 -2.920E+00 1.275E+03 -6.815E+00 -3.828E+00 -6.039E+05 2.588E+01 2.404E+01 2.465E+01 -2.274E+02 -2.293E+02 -3.732E+02 -6.361E+01 -5.845E+01 -3.759E+00 3.001E-01 1.740E-01 2.734E+01 3.459E+00 9.970E+00 -1.068E+03 2.285E-01 2.280E-01 7.393E-01 -2.297E+00 -2.264E+00 -3.763E+00 -2.679E-01 -1.986E-01 4.451E-01
-1.627E+01 4.493E+00 2.539E+03 3.722E+02 6.926E+00 -1.177E+05 6.941E+00 4.246E+00 5.077E+00 -1.820E+00 -4.291E+01 -4.332E+01 -5.928E+02 -1.133E+01 -1.364E+01 2.639E+00 -4.862E-01 5.571E+01 -1.635E+00 2.926E+00 -2.073E+02 -4.542E+00 3.849E-02 4.787E-02 -1.287E+00 -4.113E-01 -4.362E-01 -8.218E+00 -4.125E-02 -5.663E-02
-2.261E+01 -2.896E+00 -3.986E+03 5.828E+01 -2.885E+00 -1.160E+05 3.475E+00 9.752E+00 7.248E+00 -4.523E+01 -2.478E-01 -3.969E+01 -1.140E+01 -6.006E+02 -2.190E+01 -3.543E-01 8.015E-02 -7.623E+00 6.285E+00 3.310E+00 -2.039E+02 1.810E-02 -4.669E+00 4.887E-02 -4.320E-01 -1.325E+00 -4.020E-01 -4.347E-02 -8.420E+00 -1.196E-01
-1.909E+01 -1.628E+00 -1.562E+03 -4.334E+02 -3.890E+00 -1.135E+05 4.635E+00 7.504E+00 6.420E+00 -4.381E+01 -4.214E+01 -3.865E-01 -1.368E+01 -2.187E+01 -5.922E+02 -4.829E+00 6.738E-01 -6.191E+01 -1.942E+00 2.132E+00 -2.005E+02 2.012E-02 6.559E-02 -4.768E+00 -3.779E-01 -4.736E-01 -1.308E+00 -5.480E-02 -1.117E-01 -8.446E+00
B: 30 x 9
0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00
0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00
0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00
0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00
0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00
0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00
0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00
0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00
0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00
0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00
0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00
0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00
0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00
0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00
0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00
2.986E-02 2.973E-04 -1.573E-02 -1.629E-01 -9.777E-02 -6.545E-01 0.000E+00 1.483E-10 -9.152E-01
-1.556E-04 1.339E-02 -1.877E-03 -4.513E-01 3.591E-01 1.512E-01 0.000E+00 1.189E-07 5.900E-02
2.837E-03 7.169E-03 -7.228E-03 7.397E-02 -1.774E-02 -1.527E-01 0.000E+00 4.061E-10 -9.645E-02
5.803E-06 1.523E-03 -4.804E-04 -4.577E-02 4.186E-02 -7.995E-03 0.000E+00 3.056E-07 -1.190E-02
1.790E-08 -9.377E-06 2.334E-06 2.948E-04 -2.553E-04 -9.837E-06 0.000E+00 -2.143E-04 2.963E-05
3.973E-02 -2.950E-03 -5.031E-03 2.232E+00 2.523E+00 2.481E+00 0.000E+00 2.142E-04 7.236E+00
1.021E+01 -4.836E+01 1.492E+01 -6.137E+02 -4.139E+01 -4.173E+01 0.000E+00 5.830E-07 -6.968E+02
1.122E+01 3.095E+01 1.031E+01 -3.655E+01 -6.187E+02 -3.557E+01 0.000E+00 -1.211E-08 -6.908E+02
1.143E+01 9.052E+00 -2.912E+01 -3.587E+01 -4.124E+01 -6.095E+02 0.000E+00 -5.635E-07 -6.866E+02
1.660E+00 2.531E+00 2.904E+00 1.407E+02 1.244E+02 1.215E+02 0.000E+00 -1.231E-06 3.865E+02
1.266E+00 -2.408E+00 -5.027E+00 1.076E+02 1.655E+02 1.237E+02 0.000E+00 1.096E-06 3.968E+02
1.469E+00 -6.580E-01 5.183E-01 1.083E+02 1.257E+02 1.725E+02 0.000E+00 1.079E-07 4.064E+02
2.270E-01 -1.728E+01 3.725E+00 -1.875E+02 2.337E+01 2.285E+01 0.000E+00 -4.941E-07 -1.413E+02
7.376E-01 9.656E+00 4.135E+00 2.081E+01 -1.769E+02 1.899E+01 0.000E+00 4.921E-08 -1.371E+02
7.679E-01 2.918E+00 -9.424E+00 2.040E+01 2.357E+01 -2.244E+02 0.000E+00 4.492E-07 -1.805E+02
C: 27 x 30
2.181E-10 9.247E-10 3.163E-09 1.320E-08 3.906E-08 3.906E-08 5.518E-11 2.083E-10 0.000E+00 -4.475E-10 -6.430E-10 0.000E+00 2.259E-11 1.263E-10 0.000E+00 8.431E-12 -3.251E-10 1.038E-10 -4.088E-09 -1.191E-08 -1.191E-08 0.000E+00 -1.491E-11 0.000E+00 0.000E+00 3.449E-10 0.000E+00 0.000E+00 5.491E-11 0.000E+00
0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00
0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00
0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 2.053E+03 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00
0.000E+00 -1.761E-06 1.001E-09 -2.513E-05 -7.499E-05 -7.499E-05 0.000E+00 -3.428E-07 0.000E+00 0.000E+00 2.108E-06 0.000E+00 0.000E+00 9.727E-08 0.000E+00 0.000E+00 -2.495E-06 0.000E+00 -1.557E-05 -3.926E-05 -3.926E-05 0.000E+00 -3.280E-07 0.000E+00 0.000E+00 1.073E-06 0.000E+00 0.000E+00 -1.814E-07 0.000E+00
0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 -2.302E-01 0.000E+00 0.000E+00 9.669E-01 0.000E+00 0.000E+00 -6.524E-02 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00
0.000E+00 4.531E-08 -2.577E-11 6.467E-07 1.930E-06 1.930E-06 0.000E+00 -2.302E-01 0.000E+00 0.000E+00 9.669E-01 0.000E+00 0.000E+00 -6.524E-02 0.000E+00 0.000E+00 6.421E-08 0.000E+00 4.007E-07 1.010E-06 1.010E-06 0.000E+00 8.441E-09 0.000E+00 0.000E+00 -2.761E-08 0.000E+00 0.000E+00 4.668E-09 0.000E+00
0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00
0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00
0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00
-1.974E+00 6.438E+01 1.211E+03 3.983E+03 9.246E+01 1.350E+06 -5.627E+01 5.534E+01 -5.831E+01 5.123E+02 -1.488E+03 5.229E+02 1.381E+02 -1.066E+02 1.486E+02 1.083E+00 1.682E-01 1.602E+01 1.305E+01 -6.750E+00 2.404E+03 -4.586E-01 9.196E-01 -6.378E-01 5.003E+00 -1.499E+01 5.631E+00 5.454E-01 1.436E-01 6.968E-01
6.278E+01 1.848E+00 8.233E+03 -3.082E+02 6.394E+00 4.364E+04 5.591E-01 7.415E+02 -7.663E+00 1.873E+01 4.085E+02 5.566E+00 6.270E+00 -1.886E+03 2.884E+01 -1.149E+01 8.099E-01 -6.702E+01 4.450E+00 2.481E+01 1.027E+02 5.612E-02 4.693E+00 -4.785E-02 2.961E-02 3.123E+00 1.553E-01 2.081E-02 -1.090E+01 1.959E-01
-1.342E-01 4.398E-01 -5.900E+00 2.785E+01 8.016E-01 1.092E+04 -4.607E-01 1.579E+00 -4.595E-01 4.131E+00 3.354E+01 4.222E+00 1.123E+00 -3.758E-02 1.160E+00 -8.814E-01 4.368E-02 -3.595E+00 6.385E-01 8.352E-01 2.033E+01 -4.179E-03 -1.421E-01 -6.231E-03 3.458E-02 5.033E-01 5.035E-02 6.864E-03 1.026E-01 -7.774E-04
1.310E-01 -1.713E+02 1.542E+02 -1.025E+04 1.728E+01 8.414E+04 6.219E+00 -8.063E+00 2.111E+00 -6.341E+01 4.711E+01 1.472E+01 -2.707E+01 2.026E+01 6.975E+00 2.763E-02 -1.746E+00 1.246E+00 -3.516E+01 -3.750E-02 1.501E+02 1.806E-01 2.518E-01 -4.307E-01 -1.221E+00 -8.498E-01 2.060E+00 -2.467E-01 -1.866E-01 4.306E-01
2.369E+01 1.584E-01 9.508E+03 -8.851E-01 -6.240E-01 2.974E+03 1.064E+01 -4.101E+00 -1.546E+01 -2.279E+01 -6.949E+01 8.680E+01 -1.411E+01 -2.388E+00 5.426E+01 7.884E-01 -9.825E-02 3.724E+01 -4.663E-01 -1.703E+00 3.606E+00 -5.229E-01 4.883E-01 2.757E-02 2.376E+00 -2.701E+00 3.097E-01 4.743E-01 -3.760E-01 1.638E-01
1.193E+01 -1.353E+01 1.394E+04 -8.093E+02 -5.718E+01 9.016E+03 -7.529E+02 5.784E+02 1.667E+02 -6.851E+02 5.575E+02 1.164E+02 2.050E+03 -1.566E+03 -4.431E+02 -3.497E+00 6.182E-01 -1.015E+01 -1.640E+01 4.041E+00 2.023E+01 -4.752E+00 3.225E+00 1.087E+00 -5.791E+00 4.830E+00 7.350E-01 1.216E+01 -9.289E+00 -2.759E+00
2.372E+00 4.521E+00 7.544E+02 1.534E+03 1.327E+02 -6.825E+03 -2.438E+02 -5.334E+02 7.754E+02 -2.504E+02 -4.733E+02 7.286E+02 6.594E+02 1.456E+03 -2.112E+03 1.321E+01 5.239E-01 2.006E+02 8.032E-01 2.353E-01 -1.218E+01 -1.418E+00 -3.176E+00 4.636E+00 -2.212E+00 -3.792E+00 6.045E+00 3.824E+00 8.806E+00 -1.278E+01
1.000E+00 -1.554E-06 9.998E-01 -7.772E-06 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00
1.425E-06 -1.000E+00 1.943E-05 -9.999E-01 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00
1.378E-01 8.212E+02 -3.195E+01 4.522E+04 4.037E-01 5.129E+06 -2.362E+01 -3.863E+01 6.195E+01 1.779E+01 -6.697E+01 5.053E+01 7.007E+01 1.065E+02 -1.764E+02 1.081E+00 8.277E+00 1.593E+01 1.535E+02 7.270E+00 9.160E+03 -2.245E-01 -4.100E-01 6.368E-01 5.539E-01 2.037E-01 -7.486E-01 4.542E-01 8.332E-01 -1.302E+00
8.529E+02 -1.212E+01 8.441E+04 -7.640E+02 -5.536E+01 8.321E+03 -6.616E+02 5.370E+02 1.169E+02 -7.054E+02 3.112E+02 3.843E+02 1.832E+03 -1.511E+03 -2.863E+02 6.180E+00 3.583E-01 2.473E+02 -1.588E+01 1.941E+00 1.689E+01 -6.049E+00 4.544E+00 1.119E+00 1.992E+00 -3.850E+00 1.672E+00 1.258E+01 -1.011E+01 -2.335E+00
1.947E+00 6.192E+02 2.602E+02 3.793E+04 7.763E+01 -7.896E+05 -2.623E+02 -5.057E+02 7.654E+02 -4.941E+01 -6.198E+02 6.790E+02 7.419E+02 1.386E+03 -2.125E+03 1.329E+01 6.775E+00 1.991E+02 1.255E+02 -3.696E-01 -1.409E+03 -1.981E+00 -3.956E+00 5.973E+00 1.645E+00 -1.097E+00 -4.738E-01 4.586E+00 9.357E+00 -1.408E+01
0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 9.549E+00 9.549E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00
0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00
-1.444E-03 -1.178E-03 -4.183E-02 -2.288E-02 -7.535E-03 5.600E+06 1.149E-03 -2.692E-03 -7.520E-04 -2.619E-04 -1.309E-04 4.446E-04 3.576E-03 4.821E-03 1.506E-03 -3.621E-03 1.739E-03 -2.002E-02 1.184E-03 1.394E-03 1.000E+04 3.516E-04 -1.641E-03 -1.702E-04 5.479E-04 -4.255E-04 -1.180E-04 -6.181E-03 -6.302E-03 6.702E-03
-2.887E-03 -2.355E-03 -8.365E-02 -4.577E-02 -1.507E-02 1.186E+07 2.297E-03 -5.384E-03 -1.504E-03 -5.238E-04 -2.617E-04 8.891E-04 7.152E-03 9.642E-03 3.013E-03 -7.241E-03 3.477E-03 -4.005E-02 2.368E-03 7.202E+02 2.191E+04 7.031E-04 -3.282E-03 -3.405E-04 1.096E-03 -8.509E-04 -2.359E-04 -1.236E-02 -1.260E-02 1.340E-02
1.350E+00 -4.895E-05 1.927E+01 -7.742E-04 -8.940E-03 -8.940E-03 -2.315E-01 2.362E-01 -5.647E-03 1.080E+00 -8.411E-01 -2.388E-01 1.030E-01 -2.899E-02 -7.497E-02 -3.091E-01 3.051E-03 -4.475E+00 2.549E-02 1.390E-02 1.391E-02 -1.665E-02 -1.940E-02 1.711E-01 1.758E-01 3.777E-01 -5.113E-01 3.033E-03 5.944E-02 -2.565E-02
D: 27 x 9
1.000E+00 0.000E+00 0.000E+00 0.000E+00 7.158E-10 0.000E+00 0.000E+00 0.000E+00 3.343E-10
0.000E+00 0.000E+00 -1.200E+01 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00
0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00
0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00
0.000E+00 0.000E+00 0.000E+00 0.000E+00 5.730E+01 0.000E+00 0.000E+00 0.000E+00 5.730E+01
0.000E+00 0.000E+00 0.000E+00 -1.305E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 -1.305E+00
0.000E+00 0.000E+00 0.000E+00 0.000E+00 -1.475E+00 0.000E+00 0.000E+00 0.000E+00 -1.475E+00
0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00
0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00
0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00
-2.328E-01 3.354E+00 5.716E+00 -2.434E+02 -3.318E+02 -2.807E+02 0.000E+00 -5.930E-06 -8.559E+02
6.056E+00 1.077E+01 -5.315E+00 -7.126E+00 -2.512E+02 3.032E+00 0.000E+00 8.785E-07 -2.553E+02
6.954E-01 1.155E+00 -2.663E-02 -2.092E+00 -2.077E+01 -2.278E+00 0.000E+00 1.579E-07 -2.514E+01
1.172E-02 -8.703E-01 1.015E-01 2.976E+01 -2.322E+01 -1.183E+01 0.000E+00 1.530E-05 -5.291E+00
-2.300E-01 -7.204E-01 8.567E-01 7.784E+00 3.937E+01 -3.894E+01 0.000E+00 2.196E-07 8.213E+00
1.060E-02 2.646E+01 -2.177E+01 2.810E+02 -1.536E+02 -1.641E+02 0.000E+00 1.318E-06 -3.675E+01
8.007E-01 2.651E+00 -9.990E+00 1.119E+02 1.158E+02 -8.972E+02 0.000E+00 -1.620E-06 -6.695E+02
0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00
0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00
5.952E-02 7.344E-01 -8.825E-01 -8.821E+00 2.348E+01 -6.608E+01 0.000E+00 9.269E-04 -5.143E+01
-4.174E-02 2.243E+01 -1.800E+01 2.820E+02 -2.414E+01 -2.869E+02 0.000E+00 8.785E-07 -2.906E+01
7.603E-01 5.382E+00 -1.027E+01 1.756E+01 1.885E+02 -8.559E+02 0.000E+00 -1.432E-04 -6.499E+02
0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00
0.000E+00 0.000E+00 0.000E+00 5.730E+01 0.000E+00 0.000E+00 0.000E+00 0.000E+00 5.730E+01
6.590E-04 1.288E-02 -7.580E-03 8.235E-02 -6.757E-02 -1.001E-01 0.000E+00 -4.393E-07 -8.536E-02
1.318E-03 2.577E-02 -1.516E-02 1.647E-01 -1.351E-01 -2.003E-01 0.000E+00 -8.785E-07 -1.707E-01
8.043E-03 -1.048E-03 0.000E+00 -1.256E+00 1.035E+00 2.677E-01 0.000E+00 0.000E+00 4.712E-02

Dear Rana,

A few more clarifying questions:

• What do you mean when you say you use a step function for torque and pitch?
• Is the time-series solution in equilibrium (periodic steady state) when you linearize?
• How did you post-process the results to extract the damping ratio–did you apply MBC followed by azimuth averaging and eigenanalysis?
• Which mode is negative damped?

Best regards,

Dear Jason,

Getting back to the previous topic, yes I did as FAST v7 User’s Guide explains for linearization, TrimCase to 3, VSContrl to 1, VS_RtTq to the desired constant generator torque (43093.55Nm in this case), and VS_RtGnSp, VS_Rgn2K, and VS_SlPc to 9999.9E-9 (very small don’t cares > 0.0). But doing this, the system does not converge.
To solve this convergence issue I increase the TMax value as it is explained in the Guide up to 9999s. Also I try changing DispTol and VelTol, but the pitch angle varies between 8 and 4 degrees aprox., which is a large variation I think.
Last option would be to increase the system platform pitch damping, but if I do this, I would be changing the system behaviour and I want to obtain as much accurate linear model as possible. There is no other way?

JOE

Dear JOE,

I haven’t played around much with TrimCase = 3 for a floating system. Perhaps the gain on the proportional feedback control law for the trim solution is set too high, exciting the platform-pitch motion? You could change this gain, but this will require a recompile of FAST v7.

Best regards,

Dear Jason,

That was my thought. Ok, I will try it. I will tell you if I get it.

Best regards,
JOE

Dear Jason

Really i am thankful for you, the answers are as following

1- desired torque entered system as in step input function and the same thing for blade pitch angle
2- yes the time-series solution was in equilibrium
3- i execute GetMats-f8 and after that i execute mbc3, one of the results is MBC_ eigen vals. i don’t execute cce. m
4- no one is negative damped but number 13 equals 0 .

i attached the MBC-eigien- vals
eigen-vals.txt (633 Bytes)

Dear Rana,

Ah, I see now. The negative damping is showing up for a zero-frequency i.e. rigid-body mode. The damping of the rigid-body mode does not necessarily indicate that the system is unstable. As described in this forum topic: http://forums.nrel.gov/t/learizing-baseline-5mw-wind-turbine-with-fast/494/1, rigid-body modes show up in MBC3 as a pair of zero-valued (or near-zero-valued) frequencies with +/- inf damping (i.e., eigenvalues with real values only). That is, each rigid-body mode will introduce an additional mode beyond the number of enabled DOFs and the damping is unphysical.

Best regards,

Dear Jason,

I linearized the OC3-Hywind Model in FAST 7.02 for different Windspeeds. All Eigenfrequencies match the one I expected but the PtfmYaw Sensor (0.08 Hz instead of 0.12 Hz).
I have not used the trim option because of the not converging simulation and instead used the average for Windspeed, RotSpeed and RtTq that I found in time series calculations to define a point of operation.
I don’t think I misunderstood the values from CampbellDiagramm.xls (with GetMats.m and MBC.m) because there is no mode around 0.12 Hz found at all.
In time series calculation the peaks in the ROA´s (with Wave excitation) and powerspectra (with turbulent wnd excitation) match the 0.12Hz acceptable well. So I don’t think the model itself has errors.
I attached the Main file as well as the linearization setting file.

Do you have any idea where this error could evolve from?

Best regards,

Simon
NRELOffshrBsline5MW_Linear.txt (2.05 KB)
FAST.7.02_v11.00 _Main.txt (26.3 KB)

Dear Simon,

I’m not sure I understand your problem. Are you saying the time series shows excitation of the platform-yaw mode at 0.12 Hz, but linearization shows the natural frequency of the platform-yaw mode to be 0.08 Hz for the same conditions?

It’s a bit troubling that the trim option is not finding a converged solution. I did see that you have set VSContrl to 2 instead of 1 in your FAST primary input file; could this be causing a problem?

I would also suggest simplifying the model e.g. by eliminating degrees of freedom (DOFs) to better isolate the problem.

Best regards,