Civil Engineering Reference
In-Depth Information
2
⎡
⎤
⎛
⎞
⎛
⎞
2
Δ
x
Δ
x
ˆ
(
)
⎢
⎥
Co
x
cos
exp
Δ
=
⋅
−
(5.34)
⎜
⎟
⎜
⎟
q
m
3
D
3
D
λ
⎢
λ
⎥
⎝
⎠
⎝
⎠
m
m
⎣
⎦
where
mz
or
,
2
f
,
ˆ
q
m
is the non-dimensional root mean square lift or
=
θ
ωπ
=
σ
s
torsion moment coefficient,
b
is a non-dimensional load spectrum band width
parameter,
is a non-dimensional coherence length scale and
x
is span-wise
λ
Δ
m
separation. [By substituting
Δ
x
D
,
a
=
13
and
b
=
23
, and using the known integral
=
α
λ
∞
π
⎡
2
⎤
⎡
2
⎤
()
()
(
)
∫
cos
b
exp
a
d
exp
b
2
a
it may readily be shown that
α
⋅
−
α
α
=
⋅
−
⎢
⎥
⎢
⎥
⎣
⎦
2
a
⎣
⎦
0
∞
3
π
ˆ
(
)
(
)
1
∫
Co
x d
x
e
−
D
0.9778
D
D
.]
ΔΔ
=
⋅
⋅
λ
≈
⋅
λ
≈
λ
q
m
2
0
In general,
ˆ
q
z
increases with increasing bluffness of the cross section,
b
attains
σ
values between 0.1 and 0.3, while
λ
is typically in the order of 2 to 5. Similar
z
properties may be expected of
q
θ
.
For the description of the characteristic motion induced load effects at “lock-in”
Vickery & Basu [18, 19] have suggested that this may be accounted for by a negative
motion dependent aerodynamic modal damping ratio,
, such that the total modal
ζ
ae
i
damping ratio associated with mode
i
is given by
=−
(5.35)
This is equivalent to the introduction of motion dependent aerodynamic derivatives as
described in chapter 5.2 above. Adopting the notation given in Eqs. 5.24 and 5.25, it is
the aerodynamic derivatives
ζ
ζ
ζ
tot
i
ae
i
i
*
*
A
that are responsible for aerodynamic damping
exclusively effective in the across wind vertical (
z
) direction or in torsion (
H
and
). Assuming
that in the vicinity of a distinct vortex shedding type of response all other motion
induced effects may be ignored, then
θ
00
0
⎡
⎤
2
B
⎢
⎥
ρ
()
*
1
VH
B A
0
0
0
C
≈
ω
and
K
≈
(5.36)
⎢
⎥
ae
i
ae
2
⎢
⎥
2*
2
00
⎣
⎦
where
2
2
⎡
⎤
⎡
⎤
⎛
⎞
⎛
⎞
σ
σ
*
1
*
2
⎢
z
⎥
⎢
a
θ
⎥
HK
1
AK
1
=
−
and
=
−
(5.37)
⎜ ⎟
⎜ ⎟
a
z
a
aD
θ
⎢
⎥
⎢
⎥
⎝ ⎠
⎝ ⎠
z
θ
⎣
⎦
⎣
⎦
