Civil Engineering Reference
In-Depth Information
In a frequency domain formulation Eq. D.2 may be replaced by (see chapter 5.2)
*
*
*
½
PPP
B
HHH
BA
ª
º
1
5
2
2
ρ
«
»
°
*
*
*
C
=
ω
°
«
»
ae
i
5
1
2
2
°
«
*
*
2
*
»
BA
B A
¬
¼
°
°
¾
°
5
1
2
(D.3)
PPP
B
HHH
BA
*
*
*
ª
º
°
4
6
3
2
ρ
«
»
°
2
*
*
*
K
=
ω
«
»
°
ae
i
6
4
3
2
«
»
°
*
BA
*
B A
2
*
¬
¼
¿
6
4
3
P
,
*
H
,
*
A
(
*
k
=
1,2,...,6
where
is the
frequency of motion. Theoretically, Eq. D.2 is applicable in time domain as well as in
frequency domain. However, the basic hypothesis behind the quasi-steady theory was
that fluctuations in the oncoming flow or in the motion of the structure will
instantaneously give rise to corresponding fluctuations in the cross sectional loads. Such
a hypothesis will not render reliable results in a time domain solution, and therefore, Eq.
D.1 needs to be formulated at an incremental level
) are the aerodynamic derivatives and
ω
i
()
()
()
()
d
r
d
r
d
q
τ
τ
C
s
K
s
ae
=
+
(D.4)
ae
ae
d
d
d
τ
τ
τ
is a dummy time history variable,
st
τ
where (see Fig. 9.8)
τ
=−
and
()
(
)
½
ª
()
º
2
DC
s
DC
′
BC
s
0
−
Φ−
−
Φ
DDy
D L z
°
«
»
°
V
ρ
()
(
)
«
»
()
()
C
s
=
−
2
BC
Φ−
s
BC
′
+
DC
Φ
s
0
ae
L
Ly
L
D
Lz
°
2
«
»
°
«
()
()
»
2
2
−
2
BC
Φ
s
−
BC
Φ
′
s
0
«
MMy
MMz
»
°
¬
¼
¾
°
°
(D.5)
()
()
()
00
00
DC
′
s
ª
Φ
º
DD
°
θ
V
2
ρ
«
»
()
s
BC
′
s
K
=
Φ
°
«
»
ae
L
L
2
θ
°
«
»
00
BC
2
′
Φ
s
¬
¼
¿
MM
θ
mDLM
nyz
, or
, or
-
®
=
()
mn
s
where
Φ
(D.6)
=
θ
¯
are the indicial memory functions associated with interaction between drag, lift or
moment forces and the velocity of motion in
y
,
z
or θ directions. A displacement
increment and its corresponding load increment in the remote part of the response
history must comply to the quasi-steady solution, and thus
