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8/11/2019 Pres Rainflow
1/23
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ww
.ifr
emer.
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lfremer
Michel Olagno n
RAINFLOW COUNTS
and composite signals
8/11/2019 Pres Rainflow
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.ifr
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lfremer
Michel Olagno n
RAINFLOW COUNTSThe purpose of rainflow
counting: consider a range with a
wiggle on the way, it can be split
into two half-cycles in several
ways.
Not that interesting
+
MUCH BETTER !
8/11/2019 Pres Rainflow
3/23
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Michel Olagno n
RAINFLOW COUNTS
Rainflow counting:Let not small oscillations (small
cycles) stop the flow of large
amplitude ones.
Rainflow was originally definedby T. Endo as an algorithm (1968,
1974), another equivalent
algorithm (de Jonge, 1980), that
is easier to implement, is
generally recommended(ASTM,1985), and a pure
mathematical definition was first
found by I. Rychlik (1987).
8/11/2019 Pres Rainflow
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Michel Olagno n
RAINFLOW COUNTS
The Endo algorithm:
turn the signal around 90
make water flow from its upper
corner on each of the pagoda
roofs so defined, until either a
roof on the other side extends
opposite beyond the starting
point location, or the flow
reaches a point that is alreadywet.
Each time, a half-cycle is so
defined, and most of them can be
paired into full cycles (all could
be if the signal was infinite).
8/11/2019 Pres Rainflow
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Michel Olagno n
RAINFLOW COUNTS
The ASTM algorithm: for any set of 4 consecutive turning
points,
Compute the 3 corresponding ranges (absolute values)If the middle range is smaller than the two other ones,
extract a cycle of that range from the signal and proceed
with the new signal
The remaining ranges when no more cycle can be extracted
give a few additional half cycles.
8/11/2019 Pres Rainflow
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Michel Olagno n
RAINFLOW COUNTS
The same ASTM algorithm is sometimes also called rainfill.
8/11/2019 Pres Rainflow
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Michel Olagno n
RAINFLOW COUNTS
The mathematical definition: Cons ider a local maximum M.
The corresponding min imum in the extracted ra inf low count
of c ycles is max(L, R) where L and R are the overal l min ima
on the intervals to the left and to the r ight u nt i l the signal
reaches on ce again the level of M.
8/11/2019 Pres Rainflow
8/23
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Michel Olagno n
RAINFLOW COUNTS
Rainflow is a nice way to separate small, uninteresting
oscillations from the large ones, without affecting turning
points by the smoothing effect of a filter nor interruptinga large range before it is actually completed.
Especial ly, in fat igue damage calculat ions , smal l
ampl i tud e ranges can often be neglected because they do
not c ause the cracks to grow.
8/11/2019 Pres Rainflow
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Michel Olagno n
RAINFLOW COUNTS
The mathematical definition allows, for discrete levels, to
calculate the rainflow transition matrix from the min-max
transition matrix of a Markov process. Thus, for instance, if
the min-max only were recorded on a past experiment, a
rainflow count can still be computed.
8/11/2019 Pres Rainflow
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Michel Olagno n
RAINFLOW COUNTS
When dealing with stationary Gaussian processes, a
number of theoretical results are available that enable inmost cases to compute the rainflow count (and the
corresponding fatigue damage) exactly though not
always quickly.
The turning points normalized by have distribution
sqrt(1-
2)R+
N where N is a normalized normaldistribution, R a normalized Rayleigh distribution, is
the standard deviation of the signal andis the spectral
bandwidth parameter.
When the narrow-band approximation can be used (=0),
and damage is of the Miner form (D=
iim, where
i is
the number of cycles of range i), damage can becomputed in closed-form since the moments of theRayleigh distribution are related to the function andranges can be taken as twice the amplitude of maxima.
8/11/2019 Pres Rainflow
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8/11/2019 Pres Rainflow
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.ifr
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Michel Olagno n
RAINFLOW COUNTS
If a power spectral density is given for the signal, therainflow transition probabilities and range densities can be
computed exactly,.. Why that ?
The interesting joint probability is p(x
, x
, x
| x0, x0, x0).
It allows to find the probability distribution for the troughassociated to a given peak level in the signal.
For a gaussian process, all those distributions are
multidimensional gaussian ones.
The spectrum is FT(E(x(t-
)x(t))), it gives all the
correlation functions implying the signal and its
derivatives.
8/11/2019 Pres Rainflow
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Michel Olagno n
RAINFLOW COUNTS
Some ideas about how to deal with composite signals:
8/11/2019 Pres Rainflow
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Michel Olagno n
RAINFLOW COUNTS
Separate the turning points into two sets:* The maximum (and resp. minimum) over each interval
where the low frequency signal is positive (resp. negative)
* The other turning points
8/11/2019 Pres Rainflow
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RAINFLOW COUNTS
The other turning points:Their number is clearly (NH-NL)
The average rainflow damage per cycle of the high frequency signal
is conservative with respect to that of those remaining turning points.
Proof: The average range in the high frequency signal is E(HP)-E(HT)
where HPand HTare the values of the peaks and troughs in that
signal. When the (uncorrelated) low frequency signal is added, theexpectancies are not modified, since the low frequency signal has 0
mean. When removing the red turning points, E(HP) is decreased,
E(HT) is increased and the remaining ranges imply the same
turning points as their original counterparts. The ranges are thus
reduced, and the new damage per cycle is lower than originally.
CONSERVATIVE DAMAGE ESTIMATE: (NH-NL)/NH DH
8/11/2019 Pres Rainflow
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RAINFLOW COUNTS
The maximum (and resp. minimum) over each interval where the
low frequency signal is positive (resp. negative)Estimates of the corresponding damage can use the narrow-band
approximation. The DNV formula by Lotsberg (2005) uses a sum of
sine waves approach, and thus D=NL((DH/NH)1/m + (DL/NL)
1/m)m
Yet, narrow-band does not mean sine wave, nor is a sum of only 2
sine waves a gaussian process. Following Rychlik, the red processextracted from a gaussian process is still a gaussian process, with
turning points probability density given by the Rayleigh formula of
the full process, and thus a (less) conservative approximation is
D=NL/K(2sqrt(2M0))m(1+m/2)or D=NL((DH/NH)
2/m + (DL/NL)2/m)m/2
8/11/2019 Pres Rainflow
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RAINFLOW COUNTS
The SS formula D = NH/NH DH+ NL( (DL/NL)1/m)m
The CS formula D = NH+L((DH/NH)2/m + (DL/NL)
2/m)m/2
The DNV formula D = (NH-NL)/NH DH+ NL((DH/NH)1/m + (DL/NL)
1/m)m
The new formula D = (NH-NL)/NH DH+ NL((DH/NH)2/m + (DL/NL)2/m)m/2
The exact formula D = WAFO (Combined Spectrum)
8/11/2019 Pres Rainflow
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RAINFLOW COUNTS
A few topics:
A. How conservative is the narrow-band approximation
for a unimodal spectrum ?
B. How good are MC simulations wrt exact WAFO
calculations ?
C. How good are the various formulas ?D. What is a conservative climate ?
8/11/2019 Pres Rainflow
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RAINFLOW COUNTS
A. How conservative is the narrow-band approximation
for a unimodal spectrum ?
It may be noted that M0and M2are normalizing factors,
and thus normalized damage for a Pierson-Moskowitz
depends only on m, for a Jonswap on m and
, etc.
Sensitivity
to spectral bandwidth.Sensitivity to spectral shape.
Sensitivity to cut-off frequency.
8/11/2019 Pres Rainflow
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RAINFLOW COUNTS
B. How good are MC simulations wrt exact WAFO
calculations ?
Sensitivity to the number of simulations.
Sensitivity to alea modeling (random phases/complex
spectrum).
Sensitivity to stationarity (high- or low-frequency
consisting of impulse-like responses from time totime).
8/11/2019 Pres Rainflow
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C. How good are the various formulas ?
vs. the ratio of M0
vs. the ratio of TP
vs. the bandwidth of the high-frequency signal.
8/11/2019 Pres Rainflow
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RAINFLOW COUNTS
D. What is a conservative climate ?
It is clear that combining spectra gives more severe
damage than having them occur separately. What
about combining 3 swells + 1 wind sea until the least
frequent component is exhausted, then 2 swells + 1
wind sea, and so on ?
8/11/2019 Pres Rainflow
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RAINFLOW COUNTS
Example:
S1 = 16% of the time
S2 = 4% of the time
S3 = 33% of the time
W = 50% of the time
A conservative climate would be:
S1+S2+S3+W for 4% of the time
S1+S3+W for 12% of the time
S3+W for 17% of the time
W for 17% of the time
Nothing for 50% of the time.
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