Civil Engineering Reference
In-Depth Information
(
)
( )
( )
rxt
,
x
t
=
φ
⋅
η
(4.16)
i
i
i
As mentioned above, it is assumed that the corresponding instantaneous cross sectional
load contains the sum of flow induced and motion induced contributions. Thus, the total
load per unit length (horizontal, vertical or torsion) is given by
(
)
(
)
qqx t
,
qx t r r r
, , , ,
(4.17)
=
+
tot
ae
(
)
(
)
a
q xtrrr
is the additional load associ-
ated with interaction between flow and structural motion. The modal time domain equi-
librium equation for mode number
i
is then given by
qxt
is the flow induced part and
,
,, , ,
where
()
()
()
()
(
)
Mt
Ct
Kt
Qt
Qt
,,,
(4.18)
⋅
η
+
⋅
η
+
⋅
η
=
+
η η η
i
i
i
i
i
i
i
e
i
i
i
i
where
2
⎫
⎡
∫
⎤
φ
mdx
i
⎪
⎢
⎥
⎡⎤
M
C
L
i
⎪
⎢⎥
⎢
⎥
2
M
⎪
=
⎢
ωζ
⎥
⎢⎥
i
i
i
i
⎪
⎢⎥
⎢
⎥
⎪
2
K
M
ω
⎢
⎥
(4.19)
⎢⎥
⎣⎦
⎢
i
⎬
i
i
⎥
⎣
⎦
⎪
⎪
()
⎡
⎤
Qt
q
⎡⎤
⎪
i
∫
dx
⎢
⎥
=
φ
⋅
⎢⎥ ⎪
i
(
)
q
Qt
,,,
ηηη
⎢
⎥
⎣⎦
ae
⎣
ae
⎦
⎪
L
i
⎭
exp
(
)
L
is the flow exposed part of the structure and
Qt
,,,
η ηη
is the modal motion
exp
e
i
i
i
()
mx
in the equation above will
either be translational or rotational (with respect to the shear centre), depending on
whether the mode shape involves displacements in the y or z directions or if it involves
pure torsion. Transition into the frequency domain is obtained by taking the Fourier
transform on either side of Eq. 4.18. Thus,
induced load. It should be noted that structural mass
(
)
2
()
()
(
)
−
M
ωω
+
i
+
Ka
⋅
ω
=
a
ω
+
a
,,,
η η η
(4.20)
i
i
i
i
i
i
η
i
Q
Qae
i
i
()
()
()
where
a
η
,
a
and
a
are the Fourier amplitudes of
η
i
t
,
i
Qt
and
ae
Qt
,
Q
i
Qae
i
i
respectively. (Index
i
is the mode shape number and the symbol
i
is the imaginary unit
1
i
=−
.)
It
is
now
assumed
that
the
Fourier
amplitude
of
the
