Civil Engineering Reference
In-Depth Information
where
i
is an arbitrary mode number and
N
is the total number of modes, it then
mod
follows from Eq. 7.62 that
(
)
(
)
(
)
( )
x
,
t
x
x
t
F
=
T
⋅
⎡
βη
(7.68)
⋅
⎤
⎣
⎦
r
r
r
Taking the Fourier transform on either side
T
(
)
⎡
⎤
(
)
(
)
(
)
x
,
a
a
a
a
a
x
⎡
x
⎤
a
ω
=
=
T
⋅
β
⋅
a
ω
(7.69)
⎣
⎦
Fr
V
V
M
M
M
r
r
⎣
⎦
η
y
z
x
y
z
T
()
⎡
⎤
a
a
a
where
a
ω
=
⎢
""
, and defining the matrix
η
η
η
η
⎥
⎣
1
i
N
⎦
mod
S
S
S
S
S
⎡
⎤
VV
VV
VM
VM
VM
yy
yz
y x
y y
y z
⎢
⎥
S
S
S
S
⎢
⎥
VV
VM
VM
VM
zz
z x
z y
z z
⎢
⎥
(
)
S
S
S
x
,
S
ω
=
⎢
⎥
(7.70)
MM
MM
MM
Fr
x
x
x
y
x
z
⎢
⎥
Sym
.
S
S
⎢
⎥
MM
MM
y
y
y
z
⎢
⎥
S
⎢
⎥
⎣
MM
z
⎦
z
containing auto spectral densities and cross spectral densities of all force components,
then the following is obtained:
1
1
(
)
T
(
)
(
)
*
T
⎡
*
⎤
⎡
⎤
S
x
,
ω
=
lim
a
⋅
a
=
lim
T β a Tβ a
⋅
⋅
⋅
⋅
⋅
Fr
F F
⎣
⎦
⎣
η
⎦
η
π
T
π
T
T
→∞
T
→∞
1
⎡
(
)
⎤
(
)
*
T
T
T
T
T
x
,
lim
⇒
S
ω
=⋅
T β
⋅
aa β TTβ S β T
(7.71)
⋅
⋅
⋅
=⋅
⋅
⋅
⋅
⎢
⎥
Fr
ηη
η
T
π
⎣
T
→∞
⎦
where
S
contains the entire dynamic
response, i.e. background as well as resonant, it requires reduction to include only the
resonant part. The extraction of the resonant part is equivalent to a white noise type of
load assumption, and thus
S
is given in Eq. 4.74. However, because
η
η
ˆ
ˆ
T
()
*
()
()
S
ω
=
H
ω
⋅
S
⋅
H
ω
(7.72)
ˆ
η
η
η
Q
R
R
where
ˆ
H
is given in Eqs. 4.69 and where (see Eq. 4.75)
η
