Civil Engineering Reference
In-Depth Information
then the dynamic equilibrium condition is defined by
()
(
) ()
(
) ()
()
()
t
s
0
t
s
0
t
t
t
Mr
+−
⎡
C C
=
⎤
r
+−
⎡
K K
=
⎤
r
=
R
+Δ
R
(9.85)
⎣
⎦
⎣
⎦
ae
ae
ae
Since the motion induced load contributions are wind velocity dependant the system
contains static as well as dynamic singularities, i.e. the static or the dynamic response
will become infinitely large at some critical mean wind velocity.
9.5 The time invariant static solution
The time invariant static solution is given by Eq. 9.79, i.e.:
1
(
)
−
r
=−
KK R
(9.86)
ae
The corresponding cross sectional element forces may be obtained from Eq. 9.25, i.e.:
Fkd kAr
=
=
(9.87)
n
n
n
n
n
9.6 The quasi-static solution
If the lowest eigen frequency of the structure is high, say beyond 4 Hz, then the structure
i
s
quasi-static, and the solution may be obtained as a sum of the time invariant solution
r
(given in Eq. 9.83) and a slowly varying part
1
−
() (
)
()
t
t
r
=−
KKR
(9.88)
ae
where
()
t
R
is the fluctuating load due to turbulence in the oncoming flow, i.e.:
N
()
∑
T
nn
()
t
t
R
=
A R
(9.89)
n
=
1
where
()
n
R
is given in Eq. 9.49. There are two alternative solutions strategies. A time
domain solution may be pursued, or alternatively a stochastic solution based on the
covariance properties of the turbulence components may be chosen.
