SYNTHESIS OF TIME HISTORIES FROM SPECTRA
161
Then from équations (6.16) and (6.32)
=
=
<6-36)
Equation (6.35) then gives
(6.37)
4BN
(6.38)
Now determinedare thepositionsof thepartitionfrequenciesW2, • •. ,unFrom équation (6.36) where w = F, it follows that
^=eB/F‘$(F)
(6.39)
Because of the equal area partition
S(un) = -J 5(F) =
e~B^ = S(F) eB'F*e~B^
(6.40)
ZV
42>
It follows that
^eS/F4=eB/Wl
(641)
When équation (6.41) is solved for un, the resuit is
/
B
\
Wn = Vln(2V/n) + B/F4 J
’ n = ’'2........A
which détermines the partition frequencies.
The first-order simulation can be obtained by arbitrarily choosing the coordinate x equal to zéro. Thus équation (6.34) becomes
N
7j(t) = a 52
+ £n)
ns=l
(6.43)
The random phase angles en can be generated using one of the many available
codes, but once selected, they are identified with a spécifie set of frequency
components on a one-to-one basis. On the basis of some unpublished work by
this writer, it has been found that an N value of at least 15 to 20 is necessary to
produce time historiés with statistical features similar to thoee predicted from
low-order spectral moments. Time historiés of other parameters such as water
particle velocity, accélération, and pressure can also be derived following the
above approach.