Civil Engineering Reference
In-Depth Information
q
⎡⎤
⎢⎥
y
(
)
xt
,
q
q
θ
q
=
=
B v C r K
⋅
+
⋅
+
⋅
r
(5.16)
⎢⎥
⎢
⎣⎦
z
q
ae
ae
Bv
is the dynamic loading associated with turbulence (
u
and
w
) in the
oncoming flow, while
where
⋅
q
Cr
and
Kr
are motion induced loads associated with
structural velocity and displacement. It is seen that linearity has been obtained, and thus,
the theory is applicable in time domain as well as in frequency domain. The frequency
domain amplitudes of the dynamic load are obtained by taking the Fourier transform
throughout Eq. 5.16. Thus,
⋅
⋅
ae
ae
(
)
i
aBa CKa
(5.17)
=⋅
+
ω
+
⋅
q
q
v
ae
ae
r
where:
T
⎫
(
)
⎡
⎤
a
x
,
ω
=
a
a
a
⎪
q
q
q
q
⎣
⎦
⎪
⎪
y
z
θ
T
⎡
⎤
(
)
a
x
,
ω
=
a
a
a
(5.18)
⎬
r
r
r
r
⎣
⎦
y
z
T
θ
⎪
⎪
(
)
[
]
x
,
a
a
a
ω
=
v
u
w
⎪
⎭
and where
i
is the imaginary unit. Taking it for granted that the theory will primarily be
applied in a modal frequency domain approach it is favourable to introduce two major
improvements. First, for the purpose of frequency domain calculations it has been
suggested to include frequency dependent flow induced dynamic loads, i.e. to replace
()
in Eq. 5.12 with
B
q
x
(
)
(
)
(
)
⎡
⎤
2/
DBCA
DBC
/
′ −
C A
D
yu
D
L
yw
⎢
⎥
VB
ρ
(
)
(
)
⎢
(
)
⎥
B
x
,
2
C A
C
′
D B C
/
A
ω
=
+
(5.19)
q
L
zu
L
D
zw
⎢
⎥
2
⎢
⎥
2
BC
A
BC
′
A
⎢
Mu
θ
Mw
θ
⎥
⎣
⎦
where:
myz
,,
,
nuw
θ
=
⎧
⎨
()
A
ω
(5.20)
mn
=
⎩
