Environmental Engineering Reference
In-Depth Information
Thus, the solutions correspond to small motions about a prestressed state.
Aeroelastic
effects (
i.e.
,
aerodynamic loading caused by structural motions) have not been included in
this analysis. While strongly influencing the aerodynamic stability of a VAWT [
e.g.,
Popelka 1982], aeroelasticity seems to have little effect on the natural frequencies and
modes of the turbine.
Eigenvalue Solution
Obtaining mode shapes and frequencies from Equation (11-44) involves solving what
is known as a
characteristic-value problem
,
which has the standard form
a
11
x
1
+
a
12
x
2
+
…
+
a
1
n
x
n
= w
x
1
a
21
x
1
+
a
22
x
2
+
…
+
a
2
n
x
n
= w
x
2
. ... .... ... ... ...
a
n
1
x
1
+
a
n
2
x
2
+
¼
+
a
nn
x
n
= w
x
n
é
ê
ë
(11-45)
where, for our modal analysis problem
n
= number of degrees of freedom; number of finite elements times three
displacement components per element
X
j
...x
n
= eigenvector
or
characteristic vector
;
defines one mode shape
w =
eigenvalue
or
characteristic value
;
defines one modal frequency (rad/s)
Each non-trivial solution of this characteristic-value problem has an eigenvector and a
corresponding eigenvalue, and the set of eigenvector/eigenvalue pairs comprises the results
of a theoretical modal analysis of our structural system. Modes are usually presented in
ascending order of frequency, from the lowest or
fundamental
mode to the highest fre-
quency of interest. The latter seldom needs to exceed ten times the maximum rotational
speed of the turbine (10
P
).
To find the eigenvalues of the system of equations represented by Equation (11-44),
it is more convenient to use
state vectors
instead of the displacements
u.
Hence, we will
introduce the state vector
u
- - -
u
é
ê
ë
(11-46a)
v
=
and the matrices
M
| 0
- - | - -
0 |
K
é
ê
é
ê
ë
A
=
(11-46b)
C
|
K
- - | - -
-
K
|
0
B
=
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