Civil Engineering Reference
In-Depth Information
are now given by
(
)
ˆ
T
i
∫
dx
φ K φ
⋅
⋅
ae
j
2
K
()
2
⎡
V
⎤
B
m
ω
ρ
ae
L
ij
i
exp
κ
=
=
⋅
⋅
(8.3)
⎢
⎥
(
)
ae
ij
2
2
M
ω
T
ω
∫
dx
φφ
⋅
⎣
⎦
i
i
i
i
i
i
L
(
)
ˆ
T
i
∫
dx
φ C φ
⋅
⋅
ae
j
C
()
2
V
B
m
ω
ω
ρ
ae
L
ij
exp
i
i
ζ
=
=
⋅
⋅
(8.4)
ae
ij
(
)
2
2
4
ω
T
ω
M
∫
φφ
⋅
dx
i
i
i
i
i
i
L
()
where
ω
i
V
is the mean wind velocity dependent resonance frequency associated with
(
)
mode
i
and
= =
(or as calculated in vacuum). The solution to Eq. 8.1 is an
eigen-value problem with
mo
N
roots. Each of these eigen-values represents a limiting
behaviour in which the structural response is nominally infinitely large (or irrelevant).
I.e., the condition
i
V
0
ωω
i
(
)
ˆ
(
)
det
,
V
0
E
ω
=
(8.5)
η
will formally reveal
N
stability limits associated with all the relevant mode shapes
mod
()
contained in
ˆ
ˆ
E
, static or dynamic. In general
det
E
will contain complex quantities,
η
η
()
ˆ
and therefore
det
0
implies the simultaneous conditions that
E
=
η
(
)
(
)
()
()
ˆ
ˆ
Re det
0
Im det
0
E
=
and
E
=
(8.6)
η
η
As shown above,
ˆ
E
is a function of the frequency and of the mean wind velocity, and
η
and
V
values which may be used to identify the
relevant stability problem. For a static stability limit
thus, each root will contain a pair of
ω
ω = , and thus, such a limit may
simply be identified by a critical wind velocity
c
V
. For a dynamic stability limit the
response is narrow-banded and centred on an in-wind preference or resonance frequency
associated with a certain mode or combination of modes. Thus, the outcome of the
eigen-value solution to Eq. 8.5 will identify a dynamic stability limit by a critical
velocity
0
V
and the corresponding in-wind preference or resonance frequency
ω
.
cr
r
