Civil Engineering Reference
In-Depth Information
Again, if
()
()
x t
and
y t
are realisations of the same stationary and ergodic process,
()
()
S
S
then
ω
=
ω
and the real part of the cross-spectrum is given by
x
y
ˆ
()
()
()
Re
⎡
S
⎤
S
Co
ω
=
ω
⋅
ω
(2.89)
⎣
⎦
xy
x
xy
2.9 The spectral density of derivatives of processes
It may in some cases be of interest to calculate the spectral density of the time
derivatives [e.g.
()
()
x t
and
x t
] of processes. In structural engineering this is
()
particularly relevant if
x t
is a response displacement of such a character that it is
necessary to evaluate as to whether or not it is acceptable with respect to human
perception, in which case the design criteria most often will contain acceleration
requirements. Since (see Eq. 2.59)
∞
∞
()
(
)
(
)
i
t
∑
∑
e
ω
⋅
xt
X
,
t
i
d
k
=
ω
=
ω
⋅
ω
⋅
(2.90)
kk
kkk
−∞
−∞
∞
∞
2
()
(
)
(
)
(
)
i
t
∑
∑
e
ω
⋅
xt
X
,
t
i
d
k
=
ω
=
ω
⋅
ω
⋅
(2.91)
kk
k
kk
−∞
−∞
and the double sided spectral density in general is given by the complex Fourier
amplitude multiplied by its conjugated counterpart (see Eq. 2.62), then
*
⎫
()
()
id
id
*
⎡
ωω
⎤ ⎡
⋅
ωω
⎤
dd
⎣
⎦ ⎣
⎦
kk
k
kk
k
(
)
2
kk
2
(
)
⎪
S
S
±=
ω
=
ω
=
ω
±
ω
x
k
k
k
x
k
⎪
Δ
ω
Δ
ω
⎪
⎬
(2.92)
*
2
2
⎡
(
)
(
)
⎤ ⎡
(
)
(
)
⎤
i
ω
d
ω
⋅
i
ω
d
ω
⎪
*
dd
⎢
k
k
k
⎥ ⎢
k
k
k
⎥
⎣
⎦ ⎣
⎦
(
)
4
kk
4
(
)
⎪
S
S
±=
ω
=
ω
=
ω
±
⎪
ω
x
k
k
k
x
k
Δ
ω
Δ
ω
⎭
Similarly, cross spectral densities between a fluctuating displacement and its
corresponding velocity and acceleration are given by
*
⎫
()
()
d
i
d
*
⎡
ω
⎤ ⎡
⋅
ω
ω
⎤
dd
⎣
⎦ ⎣
⎦
kk
kkk
(
)
kk
(
)
⎪
S
i
i
S
±=
ω
=
ω
=
ω
±
ω
xx
k
k
k
x
k
⎪
Δ
ω
Δ
ω
⎪
*
⎡
2
⎤
() ( ) ()
d
i
d
⎡
ω
⎤
⋅
ω
ω
⎪
*
⎣
⎦
⎢
dd
kk
k kk
⎥
⎪
⎣
⎦
(
)
2
kk
2
(
)
(2.93)
S
S
±
ω
=
=−
ω
=−
ω
±
ω
⎬
xx
k
k
k
x
k
Δ
ω
Δ
ω
⎪
⎪
*
⎡
2
⎤
() ( ) ()
⎡
id
ωω
⎤
⋅
i
ω
d
ω
*
⎣
⎦
⎢
dd
kk
k
k
k
k
⎥
⎪
⎣
⎦
(
)
3
kk
3
(
)
S
i
i
S
±=
ω
=
ω
=
ω
±
ω
⎪
xx
k
k
k
x
k
Δ
ω
Δ
ω
⎪
⎭
