Environmental Engineering Reference
In-Depth Information
Fig. 13.14 Structural model
of the tested mock-up
δ
rig
δ
el
k
T
c
T
h
f
d
α
k
s
l
s
l
d
d
el
is the elastic part of the top displacement;
M, C, and K
are the mass, damping, and stiffness matrices;
p
is the vector describing the position of the dampers;
d = [d
el
d
rig
]
T
is the vector collecting the DOFs of the system.
Parameters c
T
and k
T
have been defined according to a preliminary identifi-
cation campaign of the fixed base tower, which turned out to show a natural period
of vibration of 0.92 s and a damping ratio equal to 0.8 % [
12
,
13
]. Once the tower
is mounted on the rotating support, the resulting 2 DOFs free (f
d
= 0) system
described by Eq.
13.28
has modal periods equal to:
1st mode :
T
1
¼
2
:
09 s;
2nd mode :
T
2
¼
0
:
92 s
:
The first mode is dominated by a rigid rotation around the base hinge, whereas the
second replies the elastic motion of the tower alone on a fixed base, as shown in
Eq.
13.30
where the undamped modal shapes are ordered as columns of the matrix U
and normalized, for clarity, so that one component of each eigenvector is set to 1:
:
1
:
000
0
:
000
d
rig
d
el
U
¼
ð
13
:
30
Þ
0
:
239
1
:
000
By considering the dampers in their ''off'' state, they are equivalent to linear
viscous damper with a constant c
d
= 6,900 Ns/m [
17
]. In this case, the force in
each damper is equal to (the sign is already considered in the model of Fig.
13.14
).
f
d
¼
c
d
l
d
d
rig
=
h
ð
13
:
31
Þ
