6x6 Stiffness Matrix of NREL5 MW Turbine

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.

Dear Tohid,

From your description, it doesn’t sound like you have a right-handed coordinate system in VABS, or your rotor is spinning backwards from most. Do you mean that y points towards the leading edge and z points towards the suction side?

Best regards,

Dear Jason,
Thanks for your reply. you are right that the y points toward leading edg and Z points towards to suction side.
I would like to know if we want to swap data position from VABS coordinates to BeamDyn coordinates, what will be the sign of each terms in both Mass and Stiffness matrix? because of that I assumed the coordinate system in Beamdyn is : z points along the blade from root to tip, y point towards to trailing edg side and x points towards to suction side. we will have in VABS 1=EA,2=K_shredg,3=K_shrflap,4=GJ,5=EI_flap,6=EI_edg, then swapping position to BeamDyn coordinates system will be K_shrflap-3,K_shredg=-2,EA=1, EI_edg=6,EI_flap=-5, GJ=4. I then should change the Mass matrix according this way. please correct me if I am wrong.

Best Regards,
Tohid.

Dear Tohid,

Here is how I would solve your problem.

Write out the matrix that transforms the coordinate system in VABS to the coordinate system in BeamDyn i.e.:

T = [ [ 0 0 1 0 0 0 ];
[ 0 -1 0 0 0 0 ];
[ 1 0 0 0 0 0 ];
[ 0 0 0 0 0 1 ];
[ 0 0 0 0 -1 0 ];
[ 0 0 0 1 0 0 ] ]

Then, the mass and stiffness matrices necessary for BeamDyn can be derived from the mass and stiffness matrices from VABS as follows:

Mass_BeamDyn = TMass_VABST’
Stiff_BeamDyn = TStiff_VABST’

where T’ is the transpose (inverse) of T. This transformation will swap rows and columns and change signs of the off-diagonal terms as necessary.

Best regards,

Dear jason,

Thanks very much for your understanding. Now it’s work well.

Best Regards,
Tohid

Dear Dr. Jason,

fallowing my question related stiffness and mass matrices, I am using BECAS to compute blade distributed properties. what I understand from BECAS is that this calculate mass and stiffness matrix respect to reference point which is defined in geometry input (0,0) point in each section. I am going to calculate the structure properties respect to pitch axis coordinate. I would like to know how can I translate my computation from reference point to pitch axis?

Best Regards,
Tohid.

Dear Tohid,

Section 1.4.1 of the SectionBuilder documentation from Bauchau provides a good description for how to shift the reference axis for 6x6 stiffness and mass matrices that can be decomposed into uncoupled axial-bending and shear-torsion problems e.g. for isotropic materials: soliton.ae.gatech.edu/people/oba … Manual.pdf. I don’t think that there is a direct way for changing the reference axis for fully populated matrices – in this case you would have to change the reference axis and rerun your sectional analysis tool.

Best regards,