Civil Engineering Reference
In-Depth Information
Thus, all of Eq. 9.143 has been defined. A time domain solution of this equation is dealt
with in chapter 9.9 below.
The procedure for a frequency domain solution in modal coordinates is identical to
that which has been shown in original coordinates in chapter 9.6. Taking the Fourier
transform throughout Eq. 9.143, i.e. setting
()
∑
( )
it
()
∑
( )
it
e
ω
e
ω
t
t
η
=
a
ω
⋅
and
R
=
a
ω
⋅
(9.149)
η
R
ω
ω
where
()
()
N
by 1 vectors
a
ω
and
a
ω
contain the Fourier coefficients of the
mod
η
R
modal coordinates and the modal and load, and where
∑
indicates summation over all
ω
ω -settings. Then Eq. 9.141 is satisfied for each ω -setting if
(
)
(
)
2
⎡
⎤
MCC
i
KKa
a
−
ω
+
−
ω
+
−
⋅
=
(9.150)
⎣
ae
ae
⎦
η
R
Thus,
()
R
⇒
aH
=
ω
⋅
a
(9.151)
η
η
where
−
1
(
)
(
)
()
⎡
2
⎤
i
H
ω
=−
M
ω
+−
C
C
ω
+−
K
K
(9.152)
η
⎣
ae
ae
⎦
The cross spectral density matrix of the modal coordinates is defined by
S
S
S
"
"
⎡
⎤
ηη
ηη
ηη
11
1
j
1
N
mod
⎢
⎥
⎢
#
%#
$ #
⎥
⎢
⎥
1
(
)
S
"
S
"
S
()
*
T
lim
S
ω
=
a
⋅
a
=
⎢
⎥
ηη
ηη
ηη
i
1
i
j
i N
ηη
η
η
mod
T
π
T
→∞
⎢
⎥
#
$#
% #
⎢
⎥
⎢
⎥
S
S
S
"
"
η
η
η
η
η
η
⎢
⎥
N
1
N
N
N
⎣
⎦
mod
mod
j
mod
mod
(9.153)
Introducing Eq. 9.151, then
1
1
(
)
(
)
*
(
)
T
⎡
⎤
()
*
T
lim
lim
S
ω
=
a
⋅
a
=
H a
⋅
H a
⎢
⎥
ηη
η
η
η
R
η
R
T
T
π
π
⎣
⎦
T
T
→∞
→∞
(9.154)
1
(
)
*
*
TT
*
T
lim
=⋅
H
a
⋅
a
⋅
H
=⋅
H
S
⋅
H
η
RR
η
η
R
η
T
π
T
→∞
