Civil Engineering Reference
In-Depth Information
and where
are the velocity dependent damping coefficients equivalent to
those defined by Vickery & Basu [18, 19]. (However, if appropriate experimental
evidence is available, there is no reason why
a
C
and
a
K
should not be full three by
three matrices, also in the region of distinct vortex shedding excitation.) Assuming that
()
K
and
K
a
z
a
θ
(
)
0
, then the aerodynamic damping term in Eq. 5.35 may be taken
from Eq. 4.73, and thus,
V
V
ω
≈
ω
=
i
i
T
i
⎫
∫
dx
φ C φ
⋅
⋅
ae
i
⎪
⎪
C
L
ae
i
exp
ζ
=
=
ae
i
T
⎪
2
M
∫
ω
2
ω
m
φφ
⋅
dx
i
i
ii
i
i
⎪
⎪
⎪
⎬
⎪
L
(5.38)
(
)
*2
2 *2
∫
H
BA
dx
φ
+
φ
⎪
⎪
1
z
2
θ
2
B
m
ρ
L
exp
⇒
ζ
=
⋅
⎪
ae
i
(
)
4
2
2
2
∫
φφφ
++
dx
⎪
⎪
⎭
i
y
z
θ
L
where
M
M
i
i
m
=
=
(5.39)
i
(
)
T
i
∫
2
2
2
φφ
⋅
dx
∫
dx
φφφ
++
i
y
z
θ
L
L
are the evenly distributed and modally equivalent masses associated with mode
i
.
K
a
m
(
mz
=
) are the coefficients that account for the accelerating part of the motion
induced load when
V
is close to
R
V
. Apart from being cross sectional characteristics,
they are functions of
V
and the resonance frequency of the mode in question (see right
hand side diagram in Fig. 5.5).
or
are quantities associated with the self-
limiting nature of vortex shedding, i.e. they represent upper displacement or rotation
limits at which the aerodynamic damping becomes insignificant.
It should be noted that in Eq. 5.37 the damping coefficients are defined such that
consistency is obtained with the general definition of aerodynamic derivatives in Eqs.
5.24 and 5.25 rather than the definition adopted by Vickery & Basu [18, 19]. [Thus, the
aD
and
a
θ
K
values given by Vickery & Basu in references [18, 19] are applicable in the
a
z
2
(
)
expressions given above if they are multiplied by
4
D B
. Vickery & Basu have not
given any recommendations regarding the
K
coefficient.]
a
θ
