Civil Engineering Reference
In-Depth Information
⎡
⎤
⎢
() ()
()
()
⎥
2
D S
∫
x
x dx
S
∫
x
x dx
λ
φ
φ
+
φ
φ
q
i
j
q
i
j
z
z
z
⎢
θ
θ
θ
⎥
L
L
⎣
⎦
exp
exp
()
S
ω
≈
(6.75)
ˆˆ
(
) (
)
Q
ij
2
2
ω
MM
⋅
ω
i
i
j
j
Again, due to the orthogonal properties of the mode shapes this implies that
S
becomes diagonal, i.e.
diag S
⎡ ⎤
S
=
(6.76)
ˆ
⎢
⎣ ⎦
ˆ
Q
Q
i
where
⎡
⎤
⎢
()
2
()
2
⎥
2
D S
∫
dx
S
∫
dx
λ
ω
φ
+
ω
φ
q
i
q
i
z
z
⎢
θ
θ
⎥
L
L
⎣
⎦
exp
exp
()
S
ω
=
(6.77)
ˆ
Q
i
2
(
)
2
M
ω
i
i
The calculation of the spectral response matrix is given in Eqs. 4.80 - 4.82, though, it
should be noted that if the simplifications above hold then both
ˆ
H
and
S
are diagonal,
η
in which case
N
mod
(
)
()
⎡
()
⎤
T
()
∑
()
T
()
()
x
,
x
diag S
x
x
x
S
S
ω
=
Φ
⋅
ω
⋅
Φ
=
φ
⋅
φ
⋅
ω
rr
r
r
r
⎣
η
⎦
r
r
i
r
i
r
η
i
i
i
=
1
2
()
() ()
() ()
⎡
⎤
x
x
x
x
x
φ
φ
⋅
φ
φ
⋅
φ
y r
yr zr
yr
r
θ
⎢
⎥
N
mod
∑
2
()
() ()
()
()
⎢
⎥
=
φ
x
φ
x
⋅
φ
x
⋅
S
ω
z
r
z
r
r
η
θ
i
⎢
⎥
i
=
1
2
Sym
.
x
⎢
φ
⎥
θ
r
⎣
⎦
i
(6.78)
where
2
ˆ
()
()
()
S
H
S
ω
=
ω
⋅
ω
(6.79)
η
η
ˆ
i
i
Q
i
ˆ
H
η
is given by (see Eq. 6.65)
i
−
1
2
⎡
⎤
⎛
⎞
ω
ω
(
)
ˆ
()
⎢
⎥
H
1
2
i
ω
=−
+⋅
ζ
−
ζ
⋅
(6.80)
⎜⎟
η
i
ae
i
i
ω
ω
⎢
⎥
⎝⎠
i
i
⎣
⎦
and
is given in Eq. 6.68 (see also 5.37).
ζ
ae
i
