Civil Engineering Reference
In-Depth Information
where
C
and
a
K
are defined in Eqs. 9.75 (see also 9.63). Once frequency domain
has been entered, then the content of
ae
C
and
a
K
may be taken from experimentally
determined aerodynamic derivatives (see Eqs. 9. 54 and 9.55) as well as from quasi
static properties (see Eqs. 9. 52 and 9.53).
Taking the Fourier transform throughout Eq. 9.103, i.e. setting
ae
()
∑
( )
it
()
∑
( )
it
e
ω
e
ω
r
t
=
a
ω
⋅
and
R
t
=
a
ω
⋅
(9.104)
r
R
ω
ω
where
()
()
a
ω
and
a
ω
are the Fourier coefficients of the displacement and load
r
R
processes and where
∑
indicates summation over all ω -settings, then Eq. 9.103 is
ω
satisfied for each
ω
-setting if
2
(
)
(
)
⎡
i
⎤
−
MCC
ω
+
−
ω
+
KKa
−
⋅
=
a
(9.105)
⎣
ae
ae
⎦
r
R
Thus,
()
⇒
aH
=
ω
⋅
a
(9.106)
r
r
R
where
1
−
()
2
(
)
(
)
⎡
i
⎤
H
ω
=−
M
ω
+−
C
C
ω
+−
K
K
(9.107)
r
⎣
ae
ae
⎦
The cross spectral density matrix of the response quantities corresponding to the chosen
degrees of freedom is defined by
S
S
S
S
⎡
"
"
"
⎤
rr
rr
rr
rN
1 1
1
i
1
j
1
r
⎢
⎥
# %#
#
#
⎢
⎥
⎢
⎥
S
S
S
"
"
#
⎢
⎥
rr
rr
rr
1
i
i i
i j
1
(
)
⎢
⎥
()
*
T
lim
S
ω
=
a
⋅
a
=
⎢
#
# % #
#
(9.108)
⎥
rr
r
r
T
π
T
→∞
⎢
S
S
S
⎥
"
"
#
rr
rr
rr
1
j
j i
j j
⎢
⎥
⎢
⎥
#
% #
"""""
⎢
⎥
S
S
⎢
⎥
Nr
r r
⎣
1
r
N
rr
⎦
