# 6x6 Stiffness Matrix of NREL5 MW Turbine

Hello everyone,

I am trying to simulate the NREL 5MW Turbine in an aerolastic simulation which abstracts the turbine(-blades) using beam elements.
The program handling the structural simulation requires a 6x6 stiffness matrix (M) for the beam elements.
The matrix represents a “standard” beam section constitutive law; it relates deformations to forces / moments in the following way:
F_x = f(E A)
F_y = f(G A_y)
F_z = f(G A_z)
M_x = f(G J)
M_y = f(E J_y)
M_z = f(E J_z)

E is E-Modul, G is shear modulus, A is section area, J (as well as J_y abd J_z) are area moments of inertia

In my setup only the diagonal matrix entries will be needed. To my understanding most of these values can be found in “Definition of a 5-MW Reference

E A → EAStff
G A_y → —
G A_z → —
G J → GJStff
E J_y → EdgStff
E J_z → FlpStff

Is this correct ?
How do I obtain the values for shear stiffness ?

Best Regards,

Tobias

Dear Tobias,

I agree with your interpretation of the diagonal entries, assuming the local coordinate system is oriented with the local principle axes of bending (based on the structural twist). If the local coordinate system is not oriented with the local principle axes of bending, then you would see off-diagonal bending-bending coupling terms (K56 and K65). The shear stiffness terms (for Timeshenko beams) where not included in the NREL 5-MW specifications report, but should not be important for the high aspect-ratio NREL 5-MW blade. In our FAST model of the NREL 5-MW turbine that makes use of BeamDyn (Test26 in the FAST v8.12 CertTest), which also uses full 6x6 cross-sectional stiffness and mass matrices, we assumed GA_y and GA_z were equal to EAStff, but again, the solution was not found to be sensitive to these values.

If you plan to capture the edgewise sectional CG offset from the pitch axis in your model, your mass matrix will have off-diagonal terms (based on EdgcgOf). Our FAST model of the NREL 5-MW turbine that makes use of BeamDyn (Test26 in the FAST v8.12 CertTest) has these terms.

Best regards,

Hi Jason,

many thanks for confirming my train of thought and giving additional input !
It is also very nice to know that there already is a “Test Model” for the NREL 5 MW turbine in FAST since I plan to use FAST for validation

Best Regards,

Tobias

Hello again,

concerning the inertia values specified “Definition of a 5-MW Reference Wind Turbine for Offshore System Development”:
Is it right to assume that said inertia values are given in inertia per sectionlength?
Probably quite the trivial question but I want to make sure!

Best Regards

Tobias,

Yes, that’s correct.

Best regards,

Dear Jason,
I’m trying to simulate beam model with build-in sweep. I obtained the cross sectional properties for the beam from VABS.
Since VABS diagonal elements are EA, K_shrflap, K_shredge, GJ, EI_flap, EI_edge (1,2,3,4,5,6) are in this way, in order to use it in
Beamdyn i swapped the position of those values in this way {K_shrflap, K_shredge, EA, EI_edge, EI_flap, GJ (2,3,1,6,5,4) }. can i do it this way or I have to use GlbPos as 2,3,1.

If its correct, I used this swapped stiffness and mass matrices, i simulated it only with g load (Gz), but BeamDyn exits with a warning “Load may be too large”.
What could be the issue?.

Regards,
Subramani.

Dear Subramani,

My guess is you that the order of the stiffness terms is incorrect. If 1=EA, 2=K_ShrFlap, and 3=K_ShrEdge, than I would guess 4=GJ, 5=EI_Edge, and 6=EI_Flap. That is, you’ve either swapped Flap/Edge for either the shear or bending… This is because K_ShrFlap is the shear stiffness along the flapwise direction and EI_Flap is the bending stiffness about the edgewise direction; likewise, K_ShrEdge is the shear stiffness along the edgewise direction and EI_Edge is the bending stiffness about the flapwise direction.

Best regards,

Hi NREL team,

I am trying to simulate the DTU 10MW reference turbine. I have written a small script to take the values out of the available excel file for the sectional properties, but I have a few small questions about which terms to use:
I have been following the defintions outlined on page 15 of the Beamdyn user guide, however I am being confused a little by the definitions used.

1. How are Xcm and Ycm defined (+ve X_cm is towards TE?, +ve Y_cm towards Suction side etc)
2. How exactly is i_cp defined? The definition (sectional cross product of inertia) is not really so helpful as I have had difficulty finding this term in other sources. How can I take I calculate this from the other sectional properties?

I have already had an attempt at defining these terms but it appears that my blade appears to have a very strong flap/edge coupling, whence my assumption that my implementation is erroneous.

Thanks in advance for any help,
Regards,
Joe

Dear Joe,

We are working on an update to the draft BeamDyn manual that will hopefully clarify its proper usage. But to answer your direct questions:

1. The cross-sectional properties are defined in a local blade coordinate (x_l/y_l/z_l) fixed in a cross section, with x_l directed nominally towards the suction surface, y_l directed nominally towards the trailing edge, and z_l directed nominally down the blade. X_cm and Y_cm are the x_l and y_l coordinates of the sectional center of mass.
2. I_cp is the sectional cross-product inertia, calculated as the integral of the density times x_l*y_l over the cross-sectional area.

Best regards,

Hi NREL team & others,

I am modelling the 5MW reference W.T. turbine using another aeroelastic software. I have to include offset.

It is mentioned in previous entrances that Test 26 includes the edgewise sectional C.G. offset. It is not clear to me how this is included.
I assume it should be defined in the BeamDyn structural file ( mass matrix) but I find the value as 0.

It is also not clear to me how the x_l and y_l coordinates are related to the airfoil geometry in each section.
It is defined in relation to the Pitch axis? Aerodynamic center?

I do not really understand the relationship. I.e. how is “EdgcgOf” obtained in the 5MW reference wind turbine report?

Dear Albert,

Looking back at Test26 from the FAST v8.12 archive, indeed the edgewise sectional CG offsets are missing from the BeamDyn blade input file containing the sectional 6x6 stiffness and mass matrices. I don’t remember the exact reasoning for this, but we must have simplified the blade (by removing the edgewise sectional CG offsets) to better match Test18, which is the similar to Test26, but using ElastoDyn in place of BeamDyn. Sorry for the confusion, However, a BeamDyn model of the same NREL 5-MW baseline blade with the edgewise sectional CG offsets included is provided in the CertTest directory of the standalone BeamDyn archive: nwtc.nrel.gov/BeamDyn.

For an explanation of the local blade coordinate system (x_l, y_l, z_l), see the following forum topic: http://forums.nrel.gov/t/section-coordinate-system-in-beamdyn/1285/1.

The NREL 5-MW specifications report identifies the source of the EdgecgOf data.

Best regards,

Hi Jason,

Based on the link you attached , It is more clear the direction of the blade local coordinate system.
However I am still a little bit confused how the center coordinate is defined in BeamDyn.

Let’s imagine we have a completly straight blade where kp_x and kp_y is 0.
Regardless the direction of the axis, is the position of x_i and y_i matching half chord of the airfoil?

Thank you

Dear Albert,

The (potentially curved) reference axis of BeamDyn defined by the key points is not tied to a specific location within the chord of the airfoil. Indeed, the beam model of BeamDyn does not care where the airfoil leading edge, trailing edge, or other aerodynamic surfaces are. Instead the BeamDyn reference axis identifies the origin and orientation of the cross-sectional 6x6 stiffness and mass matrices. It should not really matter where in the cross section the 6x6 stiffness and mass matrices are defined relative to, as long as the reference axis is defined consistently and closely follows the natural geometry of the blade.

When coupled to FAST, the aerodynamic geometry and discretization is defined in AeroDyn, independent from the geometry and discretization of BeamDyn.

Best regards,

Dear Jason,

I have read an old post where it is stated that ( in older versions) the blade assumes zero offset of elastic and shear center.

Does the new BeamDyn module consider a possible offset of shear and elastic center?
I do not see in the 5MW reference model any information about that.

Thank you again,

Dear Albert,

Yes, BeamDyn can include sectional offsets of the shear center, tension center, etc. not possible with the structural model of ElastoDyn and older versions of FAST. Basically, these sectional offsets, plus other material couplings, may be included in the 6x6 cross-sectional stiffness matrices.

The blade for the NREL 5-MW baseline turbine, however, does not include these complexities.

Best regards,

Dear Jason,

In an older post you mentioned about the local principal axes:

In the ‘Definition of the NREL 5-MW reference wind turbine’ the values FlpStff, EdgStff are provided about the principal axes oriented by the twist angle. I have seen some transformation formulas that converts these stiffness terms from the principal axes to the out-of-plane/in-plane coordinate system as follows (e.g. Eqs. 5 and 6 from ro.uow.edu.au/cgi/viewcontent.cg … =eispapers):

```OoP-Stff = FlpStff * cos^2(StrcTwst) + EdgStff * sin^2(StrcTwst) IP-Stff = FlpStff * sin^2(StrcTwst) + EdgStff * cos^2(StrcTwst)```

How do you convert in FAST the stiffness matrix from the principal axes to this coordinate system (OoP/IP)? do you use similar formulas that I can find them somewhere?

Furthermore, regarding the FlpIner and EdgIner values that are in the NREL 5-MW documentation, I am not sure in which direction should I use each of these two values? is it FlpIner (as FlpStff) for the edgewise direction? and secondly do I need to apply some transformation as for the stiffness to orientate them with the OoP/IP coordinate system?

Thanks a lot for the great help.

Best regards,
Koen

Dear Koen,

Actually, FAST never converts the flapwise and edgewise stiffness from the principle axes to the OoP/IP directions. If you wish to express this stiffness in the OoP/IP directions, you’ll need to transform the stiffness and introduce cross-coupling terms.

In the ElastoDyn module of FAST v7, FlpIner is the flapwise inertia nominally about the chordline and EdgIner is the edgewise inertia about an axis nominally normal to the chordline (but oriented by the structural twist, StrcTwst, instead of the aerodynamic twist, AeroTwst). Here, the principle axes of bending are assumed to be coincident with the principle axes of inertia. If you wish to express this inertia in the OoP/IP directions, you’ll need to transform the inertia and introduce cross-coupling terms.

Best regards,

Dear Jason,

Thank you for the prompt response. Does this mean that when FAST solves the equations of motion that all the terms are oriented accordingly with the principal axes? In other words so that I clearly understand it, are the aerodynamic loads applied in the direction of the structural principal axes and then the generated deflections and motion are also pointing to these coordinates? and only for post-processing you convert them to other coordinate systems?

Best regards,
Koen

Dear Koen,

Different parts of FAST are solved in different coordinate systems, but the generalized elastic stiffness of the blade is formulated using the principle axes of bending. More information on the theory basis of the structural model of FAST v7 and the ElastoDyn module of FAST v8 is provided in the following forum topic: http://forums.nrel.gov/t/coupled-blade-modes-in-fast/314/1.

I hope that helps.

Best regards,

Dear Jason,

in this topic in one of questions you mentioned to :

my question is that what will be the sign of each terms? if we assume that in VABS: x direction is along blade from root to tip, y pointing nominally towards to suction side and z towards to leading Edg direction. in swapping data from VABS to Beamdyn ,what should be the sign of both mass and stiffness matrix terms while have coupling torsion-flap bending?

Best regards,
Tohid.