Civil Engineering Reference
In-Depth Information
⎡
⎣⎦
2
EX
2
σ
k
X
k
()
S
ω
=
=
(2.51)
xk
Δ
ω
Δ
ω
which, when
T
becomes large, is given by
T
11
2
()
(
)
S
lim
∫
c
cos
t
dt
ω
=
⋅
⎡
ω
+
ϕ
⎤
(2.52)
⎣
⎦
xk
k
k
k
T
Δ
ω
T
→∞
0
Introducing the period of the harmonic component,
, and replacing
T
with
T
=
2/
πω
k
k
,
n
→∞
, then the following is obtained
nT
⋅
k
2
T
k
2
⎡
⎤
11
⎛
2
⎞
c
T
π
()
k
∫
S
ω
=
lim
⋅
⋅
n
⋅
c
cos
⋅
t
+
ϕ
dt
=
(2.53)
⎢
⎜
⎟
⎥
xk
k
k
Δ⋅
ω
nT
2
Δ
ω
n
→∞
⎢
⎥
⎝
⎠
⎣
⎦
k
k
0
()
In Fig. 2.11, the arrival at
is shown via the amplitude spectrum (or the Fourier
amplitude diagram) to ease the understanding of the concept of spectral density
representations. It is seen from this illustration that it is not possible to retrieve the parent
time domain variable from the spectral density function alone, because it does not
contain the necessary phase information (unless a corresponding phase spectrum is also
established). From its very definition the spectrum contains information about the
variance distribution in frequency domain, and from Eqs. 2.51 and 2.53 it is seen that
S
ω
x
k
2
N
N
N
c
2
∑
2
∑
∑
()
k
σ
=
lim
σ
=
lim
=
lim
S
ω
⋅ Δ
ω
(2.54)
x
X
x
k
k
2
N
→∞
N
→∞
N
→∞
k
1
k
1
k
1
=
=
=
In a continuous format, i.e. in the limit of both
N
and
T
approaching infinity, the single-
sided auto-spectral density is defined by
2
(
)
⎡
⎤
E Xt
,
ω
⎣
⎦
()
S
ω
=
lim lim
(2.55)
x
Δ
ω
TN
→∞
→∞
(
)
()
where
is an arbitrary Fourier component of
x t
. In the limit
ωω
, and
X
ω
,
t
Δ→
thus, the variance of the process may be calculated from
∞
2
()
∫
S
d
σ
=
ω
ω
(2.56)
x
x
0
