Environmental Engineering Reference
In-Depth Information
Fig. 13.13 Lumped-mass
structural model of a wind
turbine
δ
0
δ
r
f
dm
m
r
δ
r
-1
k
r
, c
r
m
r
-1
f
d
3
h
f
d
2
m
2
m
1
k
1
, c
1
α
k
s
f
d
1
l
s
l
d
When the control forces are not present, in the typical hypothesis of neglecting
damping, the eigenvector analysis of the system described by Eq.
13.4
, namely the
search for solutions of the equation
x
2
I
M
1
K
¼
0
ð
dim I
¼
n
n
Þ
ð
13
:
5
Þ
where I is the identity matrix, yields real eigenvalues x
i
(angular frequencies),
each associated to real eigenvectors u
i
, that can be ordered in the columns of a
matrix U:
ð
13
:
6
Þ
The linear system of n time-invariant differential equations of the second order
(Eq.
13.4
) can be also written as a set of 2n linear time-invariant first order
differential equations in the state-space representation:
z
¼
Az
þ
Bf
d
ð
13
:
7
Þ
where
0
n
n
I
n
n
M
1
K
M
1
C
A
¼
ð
dim A
¼
2n
2n
Þ
ð
13
:
8
Þ
0
n
1
M
1
P
B
¼
ð
dim B
¼
2n
m
Þ
