Civil Engineering Reference
In-Depth Information
6.4 Vortex shedding
As shown in chapter 5.3, the vortex shedding induced load effects at or in the vicinity of
lock-in are dependent on the dynamic response of the structure, i.e. the total damping in
each mode is unknown prior to any knowledge about the actual structural displacements.
Thus, the calculation of vortex shedding induced dynamic response will inevitably
involve iterations.
It should be acknowledged that the peak factor for vortex shedding response does not
comply with the theory behind what may be obtained from Eq. 2.45. For an ultra-
narrow-banded vortex shedding response the peak factor is close to 1.5 (theoretically
2
, see Eq. 2.47). For broad-banded response Eq. 2.45 will most often render
conservative results. Some time domain simulations of response spectra (see Appendix
A) will give a good indication on what peak factor should be chosen.
Multi mode response calculations
The general solution of a multi mode approach to the problem of calculating vortex
shedding induced dynamic response is identical to that which has been presented above
for buffeting response calculations. I.e., the general solution to the calculation of the
three by three cross spectra response matrix
(
)
x
S
is given in Eq. 4.80-4.82, while
the corresponding covariance matrix is given in Eqs. 6.47 and 6.48. The
,
rr
r
N
by
N
mod
mod
ˆ
()
()
frequency response matrix
Q
S
are given in Eqs.
4.69 and 4.75, except that for vortex shedding the motion induced load is assumed
exclusively related to structural velocity, and its effect applies to the actual modal
response and not to the individual Fourier components. As shown in Eq. 5.36, this
implies that
and the modal load matrix
H
ω
η
(
)
2
()
*
2
*
and
⎡
⎤
, and thus
K
=
0
C
=
ρ
B
/2
⋅
ω
Vdiag
⋅
0
H BA
ae
ae
i
⎣
1
2
⎦
−
1
2
⎧
⎫
⎛
⎞
⎡⎤
1
⎡⎤
1
⎪
⎪
ˆ
()
(
)
H
ω
=− ⋅
I
ω
diag
+
2
i
ω
⋅
diag
⋅
ζζ
−
(6.65)
⎜
⎟
⎨
⎬
⎢⎥
⎢⎥
ae
η
⎜
⎟
⎣⎦
ω
ω
⎣⎦
⎪
⎝
⎠
⎪
i
i
⎩
⎭
[]
where
ζ
=
diag
ζ
and the content of
a
ζ
is given by
i
