Civil Engineering Reference
In-Depth Information
This equation may be used to calculate dynamic response in time domain as well as in
frequency domain. In a time domain approach it will be necessary to perform a time
domain simulation of the fluctuating flow components
(
)
(
)
uX Z t
,
, ,
kk
vX Z t
and
, ,
kk
(
)
wX Z t
in every node
k
. In a frequency domain approach it will be necessary to
introduce the stochastic properties of the flow components and to perform spanwise
averaging to obtain the corresponding stochastic properties of the relevant response
quantities.
If a time domain solution strategy with indicial functions is chosen, then everything
above is still applicable, except that Eq. 9.65 must be introduced for the determination of
motion induced loads. Thus, the aerodynamic load given in Eq. 9.78 must be replaced
by:
, ,
kk
t
t
N
⎧
⎫
⎪
⎪
()
∑
∫
T
()
( )
T
( )
( )
∫
R
t
=
A c
s
⋅
A r
ττ
d
+
A k
s
⋅
A r
ττ
d
(9.81)
⎨
⎬
ae
n
ae
n
n
ae
n
n
n
⎪
⎪
n
1
⎩
⎭
=
0
0
Recalling that
ds d
τ =−
1
(
)
(
)
and assuming initial conditions
00
and
r
00
r
τ ==
τ ==
then Eq. 9.81 can be further developed by integration by parts into
()
⎧
⎪
⎡
t
ds
N
c
ds
τ
=
t
ae
()
∑
T
()
( )
T
()
⎤
n
R
t
=
A c
s
⋅
A r
τ
−
∫
A
⋅
⋅
⋅
A r
τ
d
τ
⎨
⎣
ae
n
ae
n
⎦
n
n
n
ds
d
τ
0
τ
=
⎪
⎩
n
1
=
0
()
t
ds
k
⎫
⎪
ds
t
τ
τ
=
=
ae
T
()
()
T
()
⎡
⎤
n
s
∫
d
+
Ak
⋅
Ar
τ
−
A
⋅
⋅
⋅
Ar
τ
τ
⎬
⎣
nae
n
⎦
n
n
n
ds
d
0
τ
⎪
⎭
0
(9.82)
t
⎧
⎪
N
∑
T
(
)
( )
T
( )
( )
s
0
t
∫
′
s
d
=
Ac
=
⋅
Ar
+
A c
⋅
⋅
Ar
ττ
⎨
⎪
⎩
nae
n
n
ae
n
n
n
n
=
1
0
t
⎫
⎪
T
(
)
( )
T
()
()
s
0
t
∫
′
s
ττ
+
Ak
=
⋅
Ar
+
A k
⋅
⋅
Ar
⎬
⎪
⎭
nae
n
n
ae
n
n
n
0
where the prime on
c
′
k
′
and
mean derivation with respect to
s
. Introducing
ae
n
ae
n
(
)
T
(
)
⎡
⎤
s
s
0
0
∑
Ac
N
⋅
s
=
0
⋅
A
⎡
C
K
=
⎤
ae
ae
=
⎢
⎥
(9.83)
⎢
⎥
(
)
T
(
)
=
Ak
s
0
A
⎢
⋅
=
⋅
⎥
⎣
⎦
ae
n
1
⎣
⎦
=
ae
n
and
t
N
()
∑
∫
T
()
()
T
()
()
⎡
⎤
t
′
s
′
s
d
(9.84)
Δ
R
=
A
⋅
c
⋅
A
⋅
r
τ
+
A
⋅
k
⋅
A
⋅
r
τ
τ
ae
⎣
n
ae
n
n
ae
n
⎦
n
n
n
=
1
0
