Environmental Engineering Reference
In-Depth Information
in which
K
=
K
+
K
G
-
S
Equation (11-43) can now be written as
(11-47)
A v
+
Bv
= 0
Here
A
is
real symmetric
,
and
B
is
real skew-symmetric.
Seeking a solution for Equation
(11-46) in the form
v
(
t
) = f
e
i
w
t
(11-48a)
where
i
is the square root of -1, the resulting eigenvalue problem is
(11-48b)
[
B
+
i
w
A
] f = 0
We now define an eigenvector
x
as
x
=
A
1
/
2
φ
(11-48c)
assuming that
A
is now positive definite. We can now transform Equation (11-47b) into
the standard eigenvalue form of Equation (11-44), with all terms on the left side, as follows:
(11-49a)
[
G
- w
I
]
x
= 0
G
=
i
A
-
1
/
2
BA
-
1
/
2
(11-49b)
Because of the skew-symmetry of the matrix
B
,
the matrix
G
is
Hermetian.
Consequently,
it can be shown that the eigenvectors are, in general, complex; but the eigenvalues are real
[Franklin 1968].
2
If structural damping is included, the system loses its Hermetian character
and the eigenvalues as well as the eigenvectors are complex. Again, in the present analysis,
structural damping has been ignored because VAWTs are lightly damped. The Hermetian
character of the eigenvalue system has important ramifications to the rotating modal test.
Even if the modes of the non-rotating structure are perfectly real, the modes of the rotating
structure will be complex.
2
If
A
is not positive-definite, then
G
is no longer Hermetian and the eigenvalues are not
necessarily real. This condition can lead to dynamic instability.
Search WWH ::
Custom Search