Michel Olagnon

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RAINFLOW COUNTS

and composite signals

Michel Olagnon

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RAINFLOW COUNTS

The 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 !

Michel Olagnon

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RAINFLOW COUNTS

Rainflow counting:Let not small oscillations (small cycles) stop the “flow” of large amplitude ones.

Rainflow was originally defined by 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).

Michel Olagnon

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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 already wet.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).

Michel Olagnon

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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.

Michel Olagnon

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RAINFLOW COUNTS

The same ASTM algorithm is sometimes also called “rainfill”.

Michel Olagnon

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RAINFLOW COUNTS

The mathematical definition: Consider a local maximum M. The corresponding minimum in the extracted rainflow count of cycles is max(L, R) where L and R are the overall minima on the intervals to the left and to the right until the signal reaches once again the level of M.

Michel Olagnon

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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 interrupting a large range before it is actually completed.Especially, in fatigue damage calculations, small amplitude ranges can often be neglected because they do not cause the cracks to grow.

Michel Olagnon

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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.

Michel Olagnon

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RAINFLOW COUNTS

When dealing with stationary Gaussian processes, a number of theoretical results are available that enable in most 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 normal

distribution, R a normalized Rayleigh distribution, is the standard deviation of the signal and is 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 be computed in closed-form since the moments of the Rayleigh distribution are related to the function and ranges can be taken as twice the amplitude of maxima.

Michel Olagnon

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RAINFLOW COUNTS

When the narrow-band approximation cannot be used, the following results apply:• The narrow-band approximation provides conservative estimates of the actual rainflow damage.• If a power spectral density is given for the signal, the rainflow transition probabilities and range densities can be computed exactly, and thus also the fatigue damage.• If the original process is not gaussian, the use of an appropriate transformation to bring its probability density to the normal distribution is generally sufficient to have the gaussian process results apply to the transformed process (though “gaussian process” implies more than just normal probability density for the signal).

Michel Olagnon

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RAINFLOW COUNTS

If a power spectral density is given for the signal, the rainflow transition probabilities and range densities can be computed exactly,.. Why that ?

The interesting joint probability is p(x, x’, x” | x0, x’0, x”0).It allows to find the probability distribution for the trough associated 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.

Michel Olagnon

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RAINFLOW COUNTS

Some ideas about how to deal with composite signals:

Michel Olagnon

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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

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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 HP and HT are the values of the peaks and troughs in that signal. When the (uncorrelated) low frequency signal is added, the expectancies 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

Michel Olagnon

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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 process extracted 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

Michel Olagnon

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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)

Michel Olagnon

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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” ?

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RAINFLOW COUNTS

A. How conservative is the narrow-band approximation for a unimodal spectrum ?

It may be noted that M0 and M2 are 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.

Michel Olagnon

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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 to time).

Michel Olagnon

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RAINFLOW COUNTS

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.

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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 ?

Michel Olagnon

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RAINFLOW COUNTS

Example:S1 = 16% of the timeS2 = 4% of the timeS3 = 33% of the timeW = 50% of the time

A conservative climate would be:S1+S2+S3+W for 4% of the timeS1+S3+W for 12% of the timeS3+W for 17% of the timeW for 17% of the timeNothing for 50% of the time.