# State space model from linearization in Fast v7

Dear Jason,

I have done a Linearization analysis in FAst v7 and from that I obtained the M, K and C matrix. In this analysis I have used only the tower FA and SS dof ( 4 dof).

Having these matrices is it possible, in your opinion, to simulate in matlab a step_force input at the tower top and obtain the tower_top displacement ?
I was thinking about converting the “Physiscal” step force of 800N into a modal one( beacuse the state space is in terms of modal coordinates) and then apply this force at the system in matlab.

Dear Alessandro,

Yes, this would be possible. What you want is an input (u) or disturbance (ud) that represents a point force at the tower top that could be defined through the control-input matrix (F) or wind-input disturbance matrix (Fd). A point force applied at the tower top is not a standard input or disturbance, but you could derive the equivalent term yourself or use one of the built-in terms to mimic the effect e.g. the rotor-collective blade pitch input or the horizontal hub-height wind speed disturbance, the perturbations of which effectively introduce a change in thrust etc.

Best regards,

Dear Jason,

I would like to take the M, K and C matrices and apply in my SS model an equivalent modal force derived from the physical one applied at the tower top, possibly without using the wind disturbance. I write the formulas because it is not easy to explain:

I made a Linearization Analysis with the following DOFs actived: q1=Tower FA mode 1 , q2= Tower SS mode 1, q3= Tower FA mode 2, q4= Tower SS mode 2 and from it I obtained the matrices M, K, C.

I can write the linearized equations of motions as:

`````` [M] q" + [C] q' +[K] q =F(t)modal        ,               with "F(t)modal" the force vector ( referred to modal coordinates q1,q2,q3,q4)   (**)
``````

I can transform this equations into a SS model written as:

z"=[A]z+[B] F(t)modal, where z is the vector of dof “q” and dof velocities "q’ " : z=(q; q’);

and the state matrix [A]= [, [I] ; -[M]^-1*[k], -[M]^-1*[C]]; and [B]=[  ; [M]^-1];

the output SS equation is on the form Y=[C]z+[D]F(t)modal

I want as output the modal coordinates so [C]=[ [ I ], [ 0 ] ] and [D]=;

``````  (**) the problem is that I want to apply a tower-top IMPULSIVE force of 800N   but  the F(t)modal  vector defined in the above equation is expressed  in a "modal base" , because the equations are in terms of modal coordinates qi(t). So i decided to "Transform" a real Force vector( the 800N IMPULSE I want to apply: F(t)real)  into the Modal force vector this way:
``````

F(t)modal= [fiTOP,FA,1; fiTOP,SS,1; fiTOP,FA,2; fiTOP,SS,2]* F(t)real

where, for example, fiTOP,FA,1 is the mode shape value at the tower top in FA direction for the first mode that is “1”( because of normalization).

This notation is similar to the one used in the FEA method when I create a SS model in matlab and I want to apply a force at a node (the tower-top one if I imagine to discretize the tower).

Where in this case (being the mode shapes normalized so that they have value one at the tower top ) will be:

F(t)modal= [1 ; 1; 1; 1]* F(t)real

The tower-top displacement I obtained is the one shown in the picture below for a tower 1 meter high and with a 0.08 mof external diameter.

Do you think this method is reasonable?

( I also tried to decompose the coupled equations of motions: [M] q" + [C] q’ +[K] q =F(t)modal and then went on with the SS and I obtained the same result as the one in the picture below).

Dear Alessandro,

Yes, that sounds reasonable. However, you have the same force (F(t)real) applied in both directions. I would think you’d want separate fore-aft (FA) and side-to-side (SS) forces e.g.

F(t)modal = [ 1, 0; 0, 1; 1, 0; 0, 1 ]*[F_FA(t)real; F_SS(t)real ]

Best regards,

Dear Jason,