Civil Engineering Reference
In-Depth Information
Example A.1
The top diagram in Fig. A.2 shows the single point single sided spectrum of a process
x
of which we wish to portray two representatives in time domain. As shown, the
frequency span of the spectrum is first divided into five equal frequency segments, and
the corresponding values
()
and
,
, are read off. Thus the process is
ω
S
ω
k
=
1,2,...,5
k
x
k
()
represented by five harmonic components whose amplitudes
c
2
S
⋅Δ
are
=⋅
ωω
k
x
k
given in the far right hand side column in the table of Fig. A.2. Thus
5
()
¦
( )
(
)
xt
=
2
S
ωΔω ω ψ
⋅
cos
t
+
x
k
k
k
k
1
=
What then remains is to choose five arbitrary value of
ψ . In Fig. A.2 the five cosine
components are first shown by fully drawn lines, representing a certain choice of
k
ψ
values. The sum of these components shown in the lower diagram in Fig. A.2 is an
arbitrary representation of the process
k
()
x t
. If the second and the fourth of these
components are moved an arbitrary time shift, then together with the remaining
unchanged components they sum up to become another arbitrary representation of the
process shown by the broken line in Fig. A.2. As can be seen, the two simulated
representatives look quite different in time domain, although they come from the same
spectral density. What is important is that they both have zero mean and the same
variance, i.e. they have identical statistical properties up to and including the variance.
A.3 Simulation of spatially non-coherent time series
While the procedure presented above may be used to simulate single point time series
representatives of
x
, it is not applicable if we wish to simulate multiple point time series
whose properties are expected to be distributed according to certain coherence
properties. E.g., let us assume that we wish to simulate the turbulence components
u
-
°
=
®
°
¯
(
)
x yzt
,,
x
v
w
(A.9)
f
f
of a stationary and homogeneous wind field at a chosen number of points
M
in a plane
perpendicular to the main flow direction. It is then important to capture the fact that
these time series are representatives of simultaneous events, and therefore, they must
contain the appropriate spatial coherence properties that are characteristic to the process.
