Dear Jason,

Thank for your answer.

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]= [[0], [I] ; -[M]^-1*[k], -[M]^-1*[C]]; and [B]=[ [0] ; [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]=[0];

```
(**) 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).