Civil Engineering Reference
In-Depth Information
adopted then
c
and
k
from Eqs. 9.52 and 9.53 are applicable in time domain as
ae
ae
0
0
well as in frequency domain, while an approach based on
a
k
containing
aerodynamic derivatives given in Eqs. 9.54 and 9.55 is only applicable in a modal
frequency domain approach. It is a physical requirement that loads obtained from
aerodynamic derivatives should converge towards quasi-static loads as the frequency of
motion is approaching zero, i.e. when the motion becomes very slow.
It should also be noted that a basic hypothesis behind the development of the
buffeting theory in Chapter 5 was that fluctuations in the oncoming flow are
instantaneously giving rise to corresponding fluctuations in the cross sectional loads. If a
time domain solution in original coordinates (i.e. in the element degrees of freedom) is
chosen then such a hypothesis will no longer be justifiable, in which case indicial
memory functions will have to be introduced. Thus, it is necessary to introduce a dummy
time history variable
c
and
ae
0
0
, a relative time
st
τ
τ
=−
(see Fig. 9.8) and a set of indicial
functions
iDL M
jyz
, or
, or
=
⎧
⎨
()
ij
s
Φ
where
(9.56)
⎩
=
θ
associated with interaction between drag, lift or moment forces and velocity of motion in
y
,
z
or
directions. These functions describe how an incremental structural motion is
giving rise to a corresponding change of motion induced load, i.e. the basic motion
induced load hypothesis is given by
θ
0
0
⎡
⎤
⎡
⎤
⎢
⎥
⎢
⎥
(
)
(
)
rx
,
rx
,
τ
τ
⎢
y
⎥
⎢
y
⎥
el
el
⎢
⎥
⎢
⎥
(
)
(
)
(
)
rx
,
rx
,
dxt
q
,
τ
τ
d
d
z
z
ae
()
⎢
⎥
()
⎢
⎥
s
el
s
el
=
c
⋅
+
k
⋅
(9.57)
ae
ae
d
0
d
⎢
(
)
⎥
0
d
⎢
(
)
⎥
τ
τ
rx
,
τ
rx
,
τ
τ
θ
θ
el
el
⎢
⎥
⎢
⎥
⎢
0
⎥
⎢
0
⎥
⎢
⎥
⎢
⎥
0
0
⎣
⎦
⎣
⎦
As time goes towards infinity it is a physical requirement that loads obtained from
indicial functions will asymptotically be approaching quasi-static loads. Adopting
functions
()
s
1
Φ
⎯ →
, i.e. similar to that which has been indicated in Fig. 9.8,
ij
s
→∞
()
()
then
c
s
and
k
s
are simply obtained by expanding the expressions in Eqs.
ae
ae
0
0
9.52 and 9.53 into
