Practical Ship Hydrodynamics

134 Practical Ship Hydrodynamics
The average frequency (occurrences/time) of upward zero crossings and also
as the above definition of amplitudes of r is derived from the r spectrum to:
�
� 1 � 2
ω2 eSr⊲ω, ⊳ d dω
f0 D 1
2 r
0
0
Together with the formula for 1 F⊲rA⊳ this yields the average occurrence of
r amplitudes which exceed a limit rA during a period T:
�
z⊲rA⊳ D Tf0 exp
2
rA �
2 2 r
Often we are interested in questions such as, ‘How is the probability that during
aperiodT a certain stress is exceeded in a structure or an opening is flooded?’
Generally, the issue is then the probability P0⊲rA⊳ that during a period T the
limit rA is never exceeded. In other words, P0⊲rA⊳ is the probability that the
maximum amplitude during the period T is less than rA. Thisisgivenbythe
sum function of the distribution of the maximum or r during T. Wemaketwo
assumptions:
ž z⊲rA⊳ − Tf0; this is sufficiently well fulfilled for rA ½ 2 r.
ž An amplitude rA is statistically nearly independent of its predecessors. This
is true for most seakeeping responses, but not for the weakly damped amplitudes
of elastic ship vibration excited by seaway, for example.
Under these assumption we have:
P0⊲rA⊳ D e z⊲rA⊳
If we insert here the above expression for z⊲rA⊳ we obtain the ‘double’ exponential
distribution typical for the distribution of extreme values:
P0⊲rA⊳ D e Tf0 exp⊲ r 2 A /⊲2 2 r ⊳⊳
The probability of exceedence is then 1 P0⊲rA⊳. Under the (far more limiting)
assumption that z⊲rA⊳ − 1 we obtain the approximation:
1 P0⊲rA⊳ ³ z⊲rA⊳
The equations for P0⊲rA⊳ assume neither a linear correlation of the response r
from the wave amplitude nor a stationary seaway. They can therefore also be
applied to results of non-linear simulations or long-term distributions.
4.4.7 Simulation methods
The appropriate tool to investigate strongly non-linear ship reactions are
simulations in the time domain. The seaway itself is usually linearized,
i.e. computed as superposition of elementary waves. The frequencies of the
individual elementary waves ωj may not be integer multiples of a minimum
frequency ωmin. In this case, the seaway would repeat itself after 2 /ωmin
unlike a real natural seaway. Appropriate methods to chose the ωj are: