# Side-to-side tower top deflection

Dear @Jason.Jonkman,

I derived the equations of motion for the wind turbine using the Lagrange equations. And programmed it, in comparison with FAST I have the following questions.

I would like to know how FAST takes into account the effect of rotor torque on the side to side displacement of the tower. Is there a specific formula and theory? The unit corresponding to the generalized coordinate q_tss should be N. Assuming that the torque of the rotor is known how should it be converted to the force acting on the tower in the equation?

Best regards,

Dear @Huajian.Xiao,

Presumably you are referring to the ElastoDyn module of FAST/OpenFAST.

The tower top moment is not converted to an equivalent shear force in ElastoDyn; rather, both the shear forces and moments are used to deflect the tower. Related to this, the displacement of the tower-top degree of freedom results in both translation and rotation of the tower top (where rotation is based on the slope of the deflection).

The theory basis of ElastoDyn is documented here: 4.2.7. ElastoDyn Users Guide and Theory Manual — OpenFAST v3.5.3 documentation.

Best regards,

Dear @Jason.Jonkman,

I see what you are saying about the FAST directly applying force and bending moment. May I ask if the torque M1 of the wind turbine rotor(Fig. 1) is the bending moment M2 (Fig. 2) acting on the top of the tower? In other words, what is the relationship (equation) between the wind turbine torque and the bending moment at the top of the tower?

Best regards,

Dear @Huajian.Xiao,

Indeed those moments will be the same unless there is also a force applied at M1, which may add to the moment at M2 (based on the moment arm distance).

Best regards,

Dear @Jason.Jonkman,

I have some questions about the stiffness matrix of a floating wind turbine, mainly in the PITCH and ROLL directions.

In the figure below, I calculated the contribution of gravity to the restoring stiffness, but the result is not as expected. Can you give me some suggestions?

Best regards,

Dear @Huajian.Xiao,

I’m not sure what values you are using for all terms, but your equation for K_COG is missing the contribution of Gravity. That is, the right-hand side should be multiplied by 9.80665 m/s^2.

Best regards,

Dear @Jason.Jonkman

I apologize for not writing the acceleration of gravity due to a clerical error on my part. Here is my explanation of the terms involved.

3*m_b + m_hub + m_nac is the weight of the wind turbine which is calculated as 3*16844 + 56780 + 240000 = 347314kg.

SG is the distance from the center of gravity of the platform to sea level and its value is 89.9155m.

Hh is the distance from the hub to the sea level and its value is 90m.

*m_tower (Gt + SG) represents the tower’s contribution to K_COG, which is calculated as 249718kg * (43.4 + 89.9155) = 33291280.

K_COG = 347314 * (89.9155 + 90) * 9.8 + 33291280 * 9.8 = 938628829 = 9.4e8.

Did I miss something in my calculations?

Best regards,

Dear @Huajian.Xiao,

I still don’t agree with your calculation of K_COG. The rotor, nacelle, and tower actually result in a negative K_COG because their COGs are above SWL (you shouldn’t add Gt and Hh with SG). You are also missing the contribution of the spar mass, which is centered at SG and will result in positive K_COG (and which is not already included in K_HS).

Best regards,

Dear @Jason.Jonkman

I got the correct K_COG based on your suggestion. you are referring to the intersection of the SWL with the spar platform and not the center of gravity of the spar platform as the center. I have several questions as follows.

Is the origin of the global coordinate system selected by OpenFast the intersection of SWL and spar platform and is the K_hs related to the selection of the global coordinate system? Is the add mass matrix due to hydrodynamic inertial forces related to the selection of the global coordinate system?

Assuming that K_hs has nothing to do with the selection of the global coordinate system, then I add the stiffness matrix obtained by map++ adjusting the reference point to the position of the center of gravity of the spar to what I mentioned in my previous reply. This in turn gives me the stiffness matrix of the system, do you think this is a reasonable approach? The origin of the global coordinates I set up to build the dynamics equations is the center of gravity of the spar platform.

Best regards,

Dear @Huajian.Xiao,

The stiffness matrices reported in the OC3-Hywind specifications report, are reported relative to the OpenFAST origin, which is the intersection of the tower centerline and SWL. And my comments about K_COG were also made in reference to this origin.

You can always convert the matrices from one coordinate system to another (such as the full-system COG), e.g., as discussed in the following forum topic: OC3-Hywind RAOs.

Best regards,

Dear @Jason.Jonkman,

Is hydrodynamic-added-mass(Aij) related to the selection of the global coordinate system? This will determine if I need to convert it.

TransMat = [ [ 1 0 0 0 78.001301464 1.1132995542E-002 ];
[ 0 1 0 -78.001301464 0 1.3918125391E-002 ];
[ 0 0 1 -1.1132995542E-002 -1.3918125391E-002 0 ];
[ 0 0 0 1 0 0 ];
[ 0 0 0 0 1 0 ];
[ 0 0 0 0 0 1 ] ]

The x_cg and y_cg are sized(although smaller, but also have some effect on the matrix transformation if the non-diagonal entries of the matrix to be transformed have larger values). Couldn’t the center of gravity of the platform and the intersection of the SWL with the platform be on the same plumb line?

Best regards,

Dear @Huajian.Xiao,

Yes, all matrices should be transformed if you shift the coordinate system, including added mass.

I’m not sure I fully understand your question about the “plumb line”, but if you are asking whether you can neglect the x_cg and y_cg offsets, in most cases I would expect that to be a reasonable thing to do. The CG of the RNA, though, is typically offset from the tower centerline, which shifts the CG of the entire FOWT a bit.

Best regards,

Dear @Jason.Jonkman,

I’m sorry to bother you again.

I got the linearization file (.lin) of OC3 Spar through OpenFast and ran MBC3 successfully and got the following result.

My question is this.

1. A seems to be a matrix of first order (i.e. consists of 0 matrix, unit matrix, inverse matrix of mass matrix, stiffness matrix, damping matrix), what is the best way for me to get the mass matrix, damping matrix, stiffness matrix (i.e. matrices of second order system).

2. How does the result of eigSol.NaturalFreqs_Hz correspond to the modal forms (e.g. surge, sway, 1st flap, etc.).

3. The results (Freq) of my run differ from those calculated by others, is there an official document describing the intrinsic frequency of OC3 Spar.

Best regards,

Dear @Huajian.Xiao,

Here are my responses:

1. I agree with your interpretation of the state matrix A. Regarding the distinct mass, stiffness, and damping matrices, these can be found for 6x6 rigid-body mass, stiffness and damping matrices as outlined in the following forum post: OpenFast 2nd order Linearization. But this process would not work for structurally flexible models. I’m not aware of a simple way to separate out the mass, stiffness, and damping matrices for structurally flexible models derived through an OpenFAST linearization analysis.

2. Interpretation of the eigensolution to identify the specific modes has been discussed in many other forum posts. Our new ACDC tool under development is targeted at automating and simplifying the process: GitHub - OpenFAST/acdc: ACDC: Automated Campbell Diagram Code.

3. The natural frequencies of the NREL 5-MW baseline wind turbine atop the OC3-Hywind spar are documented in the OC3 Phase IV results paper: https://www.nrel.gov/docs/fy10osti/47534.pdf.

Best regards,