Civil Engineering Reference
In-Depth Information
ˆ
ˆ
2
11
2
21
2
22
⎫
⎪
⎪
J
J
J
(
)
1exp
ˆ
ˆ
+
−
ω
ω
()
()
2
4
2
=⋅
ψ
ω
where
ψω
=
+
π
⋅
⎬
⎪
⎪
⎭
2
2
2
ˆ
(
)
ωπ
+
2
2
ˆ
ωπ
+
ˆ
ˆ
ux
CLV
and where
ω
=
ω
. The normalised modal load matrix
S
is then given by
exp
2
⎡
⎤
⎛⎞
m
ω
ω
2
2
2
⎢
C
′
BC C
′
′
⎥
⎜⎟
S
S
L
L
M
⎡
⎤
2
(
)
() ()
m
VB
S
⎢
⎥
ρ
⋅
ω
⋅
ψ
ω
ˆˆ
ˆˆ
⎝⎠
QQ
QQ
1
1
()
11
12
⎢
⎥
w
S
ω
=
=
⎢
⎥
(
) (
)
ˆ
Q
S
S
2
2
⎢
⎥
2
ω
mm
⋅
ω
⎢
⎥
⎛⎞
m
ˆˆ
ˆˆ
ω
ω
⎣
QQ
QQ
⎦
11
22
21
2 2
2
(
)
BC C
′′
BC
′
1
1
⎢
⎥
⎜⎟
LM
M
m
⎢
⎥
⎣
⎝⎠
⎦
2
2
x
L
2
And thus, the spectral density response matrix at
=
is given by (see Eqs. 4.81 and 4.82)
r
SS
⎡
⎤
rr
rr
(
)
zz
z
(
)
( )
(
)
SS
θ
T
L
2,
L
2
L
2
S
ω
=
⎢
⎥
=
Φ
⋅
S
ω
⋅
Φ
rr
r
r
η
⎢
⎥
⎣
rr
rr
⎦
θ
z
θ θ
10
⎡ ⎤
ˆ
ˆ
T
()
()
()
()
(
)
*
r
L
2
where:
S
ω
=
H
ω
⋅
S
ω
⋅
H
ω
and
Φ
=
⎢
⎣ ⎦
ˆ
η
η
η
Q
01
EE
EE
⎡
⎤
()
11
12
E
Introducing the impedance matrix
ω
=
where
⎢
⎥
⎣
⎦
21
22
2
⎛⎞
=−
ω
ω
(
)
E
1
2
i
E
+
ζζ
−
,
=−
κ
,
⎜
⎝⎠
11
1
ae
12
ae
11
12
ω
ω
1
1
2
⎛⎞
i
ω
ω
ω
(
)
E
2
E
1
2
i
=−
ζ
and
=−
κ
−
+
ζ
−
ζ
⎜
⎝⎠
21
ae
22
ae
2
ae
21
22
22
ω
ω
ω
2
2
2
ˆ
ˆ
⎡
⎤
HH
E E
EE
1
⎡
−
⎤
ˆ
()
11
12
22
12
−
1
H
E
Then
ω
=
⎢
⎥
=
=
⎢
⎥
η
ˆ
ˆ
det
−
E
⎢
HH
⎥
⎣
⎦
⎣
⎦
21
11
21
22
x
L
2
rendering the following expression for the spectral density response matrix at
=
r
ˆ
ˆ
2
2
⎡
⎤
SS
()
2
4
S
BB V
⎛ ⎞ ⎛ ⎞
V
ω
ρρ
η
η
(
)
w
()
11
12
2
⎢
⎥
S
L
2,
I
ω
=
⋅
⋅
⋅
⋅
⋅
⋅
ψ
ω
⋅
⎜ ⎟ ⎜ ⎟
rr
w
2
ˆ
ˆ
mm B
B
⎢
⎥
ω
ω
σ
SS
⎝ ⎠ ⎝ ⎠
1
2
1
2
w
⎣
⎦
η
η
21
22
where:
(
)
(
)
(
)
ˆ
ˆˆ
ˆˆ
ˆˆ
ˆˆ
()
*
*
*
*
S
H
H
H
H
H
H
H
H
ωγ
=⋅
+ ⋅
γ
+
+ ⋅
γ
LL
11
11
LM
12
11
11
12
MM
12
12
η
11
(
)
(
)
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
()
*
*
*
*
S
HH
HH HH
HH
ωγ
=⋅
⋅
+ ⋅
γ
⋅
+⋅
+ ⋅
γ
⋅
LL
11
21
LM
12
21
11
22
MM
12
22
η
12
(
)
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
()
*
*
*
*
S
HH
HH HH
HH
ωγ
=⋅
⋅
+ ⋅
γ
⋅
+ ⋅
+ ⋅
γ
⋅
LL
11
21
LM
21
12
22
11
MM
22
12
η
21
