Environmental Engineering Reference
In-Depth Information
values of damping ratios and natural periods can be achieved [
10
]. Therefore, the
matrix G can be designed so as to achieve the first objective of maintaining the
modal periods while increasing the modal damping ratios.
As said before, within the experimental setup herein focused, only one inde-
pendent control force was available, provided by the 2 SA MR dampers. Should
more independent control forces be available, the gain matrix G could be designed
to arbitrarily modify the modal shapes of the controlled system, too. Actually, this
is the second objective of the control strategy adopted, i.e., to modify the modal
behavior of the tower so as to obtain a dominant, highly damped mode corre-
sponding to a rigid rotation of the structure around the base hinge and a secondary
mode combining a rigid rotation to an elastic deformation of the tower bounded by
a damping ratio significantly higher than that of the tower alone (0.8 %). Due to
the presence of only one independent controller, the procedure described before
cannot be applied. However, also the pole placement technique modifies, through
G, the CL system matrix and, in turn, its complex eigenvectors (i.e., the modal
shapes of the controlled tower). Therefore, when the desired control force u is
designed through Eq.
13.14
so as to obtain the given values of periods of vibration
and modal damping ratios for the CL system, also the modal shapes of the con-
trolled system change compared to the original, uncontrolled one. Based on a trial
and error iterative procedure, authors ended up with the following feedback
control law:
u
¼
2
½
g
1
g
2
g
3
g
4
z
ð
13
:
38
Þ
where:
g
1
= 597 N/m; g
2
= 0; g
3
= 408 Ns/m; g
4
= 1,154 Ns/m.
In this case, the CL controlled tower shows the following periods and damping
ratios:
1st mode :
T
1
¼
2
:
09 s
n
1
¼
20 %;
2nd mode :
T
2
¼
0
:
92 s
n
2
¼
5 %
:
Correspondingly, the complex eigenvectors of the controlled towers, ordered as
columns of the matrix W* and normalized as before, are:
ð
13
:
39
Þ
j
\ 0
j
\93
j
1
:
000
j
0
:
084
d
rig
d
el
W
¼
j
\
27
j
\ 0
j
0
:
234
j
1
:
000
Equation
13.39
shows only a portion (W*) of the eigenvector matrix W of the
controlled tower. Indeed, in the present case the eigenvectors come in complex
conjugate pairs carrying, each pair, exactly the same piece of information. Fur-
thermore, each eigenvector has four components, two related to displacements
(shown in Eq.
13.39
) and two relates to velocities (out of interest in the present
case). Complex eigenvectors in Eq.
13.39
are described through their module and