Civil Engineering Reference
In-Depth Information
As explained in chapter 4.3 (see Eq. 4.41), only diagonal
ˆ
a
K
and
ˆ
a
C
will maintain the
presupposition that no modal coupling will occur. Flow induced coupling will occur if
ˆ
a
K
and
ˆ
a
C
are not diagonal.
Multi mode buffeting response calculations
The general solution to a multi mode approach is given by the three by three response
matrix shown in Eqs. 4.80 - 4.82. The corresponding three by three response covariance
matrix
⎡
2
⎤
σ
Cov
Cov
rr
rr
rr
⎢
yy
yz
y
⎥
θ
⎢
⎥
()
2
Cov
x
=
⎢
ov
σ
ov
(6.47)
rr
r
r r
r r
r r
⎥
zy
zz
z
θ
⎢
⎥
2
Cov
Cov
σ
⎢
⎥
rr
rr
rr
⎣
⎦
y
z
θ
θ
θ θ
which contains the variance of each response displacement component
r
,
r
and
r
θ
at
x
= on its diagonal and cross covariance on its off-diagonal terms, is obtained by
frequency domain integration. Thus,
r
∞
⎡
∞
⎤
ˆ
ˆ
( )
(
)
( )
() () ()
( )
*
T
T
x
∫
x
,
d
x
∫
d
x
Cov
=
S
ωω
=
Φ HSH Φ
ω
ω
ωω
(6.48)
⎢
⎥
ˆ
rr
rr
rr
η
η
r r
Q
⎢
⎥
⎣
⎦
0
0
ˆ
()
()
where
and
are
N
by
N
matrices given in Eqs. 4.69 and 4.75, and
H
ω
S
ω
Q
mod
mod
η
()
x
is a three by
N
matrix defined in Eq. 4.79. What remains is to bring the
Φ
rr
mod
ˆ
()
()
results from chapter 5.1 into
and
. Disregarding any aerodynamic mass
H
ω
S
ω
Q
η
ˆ
()
effects, the frequency response matrix
in Eq. 4.69 is reduced to
H
ω
η
1
−
⎧
2
⎫
⎛
⎞
⎡⎤
1
⎡⎤
1
⎪
⎪
ˆ
()
(
)
diag
2
i
diag
H
ω
=− − ⋅
I κ
ω
+
ω
⋅
⋅
ζ ζ
−
(6.49)
⎜
⎟
⎨
⎢⎥
⎢⎥
⎬
η
ae
⎜
⎟
⎣⎦
ae
ω
ω
⎪
⎝
⎠
⎣⎦
⎪
i
i
⎩
⎭
where
I
is the
identity matrix (
N
by
N
), and where
ζ
,
a
ζ
and
a
κ
are defined
mod
mod
(
)
K
−
1
⎡
2
⎤
in Eq. 4.68. By introducing the modal stiffness matrix
diag
1/
M
, the
=
ω
0
i
i
⎣
⎦
m
in Eq. 6.41 and the notation in Eqs. 5.24 and 5.25, then the content of
definition of
