Civil Engineering Reference
In-Depth Information
A.2 Simulation of single point time series
The mathematical development from a single time series to its auto-spectral density is
presented in chapter 2.5. In principle, the process is illustrated on Fig. 2.11. A time
domain simulation is obtained by the reverse process.
Let
()
x
S
ω
be the single-sided single point auto spectral density of an arbitrary
stochastic variable
x
, for simplicity with zero mean value. A time domain
representative,
()
x t
, can then be obtained by subdividing
S
into
N
blocks along the
frequency axis, each centred at
ω
(
k
1, 2,
…
) and covering a frequency segment
,
N
=
Δω
, as shown in Fig. A.1.
k
Fig. A.1
Spectral decomposition
()
S
ω
is the variance of each harmonic component per
frequency segment, as defined in Eq. 2.53 (see also Fig. 2.11), i.e.
On a discrete form
x
k
()
2
/(2
S
c
)
ω
=
Δω
(A.1)
x
k
k
k
A time series representative of
x
is then obtained by
N
()
¦
(
)
xt
c
cos
t
=
ω ψ
+
(A.2)
k
k
k
k
=
1
1/2
()
c
2
S
where
=⋅
ª
ω ω
⋅
º
¼
(A.3)
¬
k
x
k
k
