Civil Engineering Reference
In-Depth Information
from which the following is obtained:
%
$
⎡
⎤
⎢
⎥
(
)
(
)
(
)
(
)
T
(
)
S
x
,
ω
=
Sx
,
ω
=
φ
xS
⋅
ω
⋅
φ
x
(4.44)
r
r
i
r
⎢
nmr
⎥
r
r
η
i
i
i
⎢
⎥
$
%
⎣
⎦
i
n
1
⎫
(
)
()
*
where:
=
rrr
,,
and
S
ω
=
lim
T
a
⋅
a
is given in Eqs. 4.26-4.27, i.e.
⎬
⎭
yz
η
ηη
θ
m
i
T
i
i
π
→∞
2
ˆ
()
H
ω
i
(
)
(
)
()
T
( )
S
x
,
ω
=
φ
x
⋅
⋅
S
ω
⋅
φ
x
(4.45)
r
r
i
r
r
r
i
2
Q
i
K
i
i
The response covariance matrix is obtained by the frequency domain integration of
(
)
x
,
, and thus
S
ω
i
r
⎡
2
()
()
()
⎤
σ
x
ov
x
ov
x
rr
r
rr
r
rr
r
⎢
yy
yz
y
⎥
θ
∞
⎢
⎥
(
)
()
2
()
()
∫
S
xd
,
ωω
=
⎢
vx
σ
x
vx
(4.46)
i
r
rr
r
rr
r
rr
r
⎥
zy
zz
z
θ
0
⎢
⎥
()
()
2
()
Cov
x
Cov
x
x
σ
⎢
⎥
rr
r
rr
r
rr
r
⎣
⎦
y
z
θ
θ
θ θ
i
However, the three components of each mode shape are fully correlated and therefore all
cross-covariance coefficients that may be extracted from Eq. 4.46 are equal to unity.
Thus, it is only the terms on the diagonal of Eq, 4.46 that are of any interest, and then the
calculations simplify into
⎡
2
()
()
()
⎤
S
x
⎡ ⎤
φ
2
r
yy
yr
ˆ
()
⎢
⎥
H
⎢ ⎥
ω
i
(
)
⎢
2
⎥
()
x
,
S
x
S
S
ω
=
⎢ ⎥
=
φ
⋅
⋅
ω
(4.47)
i
r
rr
z r
Q
zz
2
⎢
⎥
K
i
⎢ ⎥
i
2
x
S
⎢
φ
⎥
⎢
⎣ ⎦
⎣
rr
θ
r
⎦
θθ
i
i
and
2
⎡ ⎤
⎢ ⎥
⎢ ⎥
σ
r
yy
∞
()
2
(
)
∫
Var
x
=
σ
=
S
xd
,
ω
ω
(4.48)
i
r
rr
i
r
⎢ ⎥
⎢
⎢
⎣ ⎦
zz
0
2
σ
rr
θθ
i
The total response may be obtained by adding up variance contributions from all modes,
i.e.
2
()
()
()
⎡
⎤
x
σ
rr
r
yy
⎢
⎥
N
mod
⎢
⎥
()
2
∑
Var
x
=
σ
x
=
Var
(4.49)
r
r r
r
i
⎢
⎥
zz
i
1
=
⎢
⎥
2
x
σ
⎢
rr
r
⎥
⎣
⎦
θθ
