# Natural frequency and damping ratio calculation

Dear @Jason.Jonkman,

I have replaced the tapered tower in the WP1.5MW with a uniform tower, with EI= 1.3710e+11 Nm^2 at all stations along the length ( 82.4 m). Accordingly, I have calculated the mode shape coefficients using “Modes” code and incorporated in the tower structural properties used for OpenFAST simulations.

Regarding M_eqTwr = 33/140 * M_twr:

Eq 3 and Eq 10 from the paper below has answers for choosing 33/140 factor;

https://www.researchgate.net/publication/238951371_On_the_representation_of_a_cantilevered_beam_carrying_a_tip_mass_by_an_equivalent_spring-mass_system

The paper doesn’t say anything about the damping though.

In OpenFAST model, I have set the “TwrAero” to False, meaning aerodynamic forces on tower are not calculated, just like the case in my in-house model.

Talking of the ‘tower motion’, I have considered it in the aerodynamic force calculations, which could provide aerodynamic damping. The relative wind velocity at each instant at each airfoil is calculated as,

• V_rel^2 = [ (V_wind – V_twr)(1-a) ]^2 + [ r(omega)*(1+a’) ]^2
• Thrust = 0.5rhoAV_rel^2Cx (at each section)

Infact calculated induction factors, are calculated using BEMT and the algorithm in the the paper ( Ning, S. A., “A simple solution method for the blade element momentum equations with guaranteed convergence,” Wind
Energy, Vol. 17, No. 9, 2014, pp. 1327–1345 ).

Am I missing to capture anything more that could add aerodynamic damping?

I have used a PI controller, with rotor speed error ( = omega_t – omega_rated ) as the feedback for the controller. ( NO tower motion measurements are considered for feedback in controller ) Again, in OpenFAST controller also, tower acceleration feedback is NOT considered.

One difference between controllers (FAST vs In-house) is that FAST controller has implemented gain scheduling but I have used a single set of gains (Kp, Ki) values for the controller across the operating region in my model. But, I have verified through step disturbance analysis that the chosen gains are capable of maintaining the system at the operating point for the entire region3 (V_rated to V_cutoff) with a reasonable step response. (Fig1 - in previous post).
I agree that the pitch action (rate of it ) could play a vital role in tower oscillations, but, the picture (Fig3 in the previous post) suggest the pitch action is almost same between OpenFAST and in-house code.
I will have to look at in more detail.

Regards,
Kumara

Dear @KumaraRaja.Eedara,

OK, it sounds like you’ve incorporated some amount of aerodynamic damping; I didn’t understand that from your original post.

I would suggest simplifying the model further to debug the differences in damping. For example, how does the response between OpenFAST and your in-house model compare if you disable the controller and fix the generator speed, with and without tower structural damping?

Best regards,

Dear @Jason.Jonkman ,

Sorry for being demanding, but i think i did not clarify my point of view.

For me, when we talk about modal damping, we are in the basis of eigenvectors and the formula
c =2damping_ratiocircular_natural_frequency*modal_mass is valid.
Note that in the basis of eigenvectors, the mass matrix of a system is called modal mass and so on for the stiffness matrix.
In order to get the damping matrix in the basis (i,j,k), we should transform the damping matrix from the basis of eigenvectors to the (i,j,k).

Did you do that in OpenFast ? or what i’ve said is incorrect ?
Because when i opened for example “ElastoDyn tower”, i found damping ratio in fore aft and another one in side side so i said to myself you did in OpenFast the transformation that i was talking about right ?

What’s the difference between " structural damping ratio " and " modal damping ratio " ?

Best Regards,