Civil Engineering Reference
In-Depth Information
the variance
T
1
2
2
(
)
lim
∫
⎡
c
cos
t
⎤
dt
σ
=
⋅
ω
⎣
⎦
x
x
x
T
T
→∞
0
(2.47)
2
T
x
2
⎡
⎛ ⎞
⎤
c
1
2
π
x
lim
nc
∫
cos
t
t
=
⋅
⋅
⋅
=
⎢
⎜ ⎟
⎥
x
nT
T
2
⋅
n
→∞
⎢
⎝ ⎠
⎥
⎣
⎦
x
x
0
x
c
2
and thus, for such a process
== ⋅
σ
. Therefore, Eq. 2.45 is only applicable
max
x
x
for fairly broad banded processes.
2.5 Auto spectral density
The auto spectral density contains the frequency domain properties of the process, i.e. it
is the frequency domain counterpart to the concept of variance. The various steps in the
development of an auto spectral density function are illustrated in Fig. 2.11.
Given a zero mean time variable
()
x t
with length
T
and performing a Fourier
()
x t
implies that it may be approximated by a sum of harmonic
transformation of
(
)
components
X
,
kk
t
, i.e.
ω
N
k
⎧
⎨
ω
ω
ωπ
=⋅ Δ
()
∑
(
)
k
x t
lim
X
,
t
=
ω
where
(2.48)
kk
2/
T
Δ=
N
→∞
⎩
k
=
1
The harmonic components in Eq. 2.48 are given by
(
)
(
)
Xt
,
c
cos
t
ω
=⋅
ω
+
ϕ
(2.49)
kk
k
k
k
22
(
)
where the amplitudes
c
ab
and phase angles
arc tan
ba
/
, and where
=
+
ϕ
=
k
k
k
k
kk
a
and
b
are given by
the constants
T
a
cos
t
⎡⎤
2
⎡ ⎤
ω
ω
k
()
k
∫
x t
dt
=
(2.50)
⎢⎥
⎢ ⎥
b
T
sin
t
⎣⎦
⎣ ⎦
k
k
0
()
As shown in Fig. 2.11 the auto-spectral density of
x t
is intended to represent its
variance density distribution in the frequency domain. Hence, the definition of the
single-sided auto-spectral density
S
associated with the frequency
ω
is
x
k
