Civil Engineering Reference
In-Depth Information
()
()
()
()
x
x
....
x
....
x
Φ
=
⎣
⎡
φ
φ
φ
⎤
⎦
rr
1
r
i
r
Nr
⎡
()
()
()
()
()
()
()
()
()
⎤
x
x
x
⎡
φ
⎤
⎡
φ
⎤
⎡
φ
⎤
yr
yr
yr
⎢
⎥
(4.79)
⎢
⎥
⎢
⎥
⎢
⎥
x
....
x
....
x
=
⎢
φ
φ
φ
⎥
⎢
⎥
⎢
⎥
⎢
⎥
zr
zr
zr
⎢
⎥
⎢
⎥
⎢
⎥
⎢
⎥
x
x
x
φ
φ
φ
⎢
⎥
⎣
θ
r
⎦
⎣
θ
r
⎦
⎣
θ
r
⎦
⎣
⎦
1
i
N
then the three by three cross spectral density matrix of the unknown modal displace-
ments
r
,
r
and
r
θ
at
x
=
x
r
⎡
⎤
SSS
rr
rr
rr
⎢
yy
yz
y
⎥
θ
(
)
⎢
⎥
S
x
,
ω
=
⎢
S S S
SSS
(4.80)
rr
r
r r
r r
r r
zy
zz
z
⎥
θ
⎢
⎥
rr
rr
rr
⎣
⎦
y
z
θ
θ
θ θ
is given by
(
)
() ()
T
()
x
,
x
x
S
ω
=
Φ
⋅
S
ω
⋅
Φ
(4.81)
rr
r
r
r
η
r
r
()
where
S
ω
is given in Eq. 4.74, i.e.:
η
ˆ
ˆ
(
)
( )
*
()
()
()
( )
⎡
T
⎤
T
x
,
x
x
S
ω
=
Φ HSHΦ
⋅
ω
⋅
ω
⋅
ω
⋅
(4.82)
ˆ
rr
r
r
r
⎣
η
η
⎦
r
r
Q
This equation is applicable to any linear load on a line-like structure. If all mechanical
properties of the structure are known, then an eigen-value analysis will provide the basic
input to
ˆ
H
and
Φ
. What then remains is the set-up of
S
and the motion induced
η
contributions to
ˆ
. This is shown in chapters 5 and 6.
η
