pres rainflow

8/11/2019 Pres Rainflow

1/23

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RAINFLOW COUNTS

and composite signals


2/23

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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 !


3/23

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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).


4/23

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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 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).


5/23

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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.


6/23

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RAINFLOW COUNTS

The same ASTM algorithm is sometimes also called rainfill.


7/23

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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.


8/23

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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 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.


9/23

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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.


10/23

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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.


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12/23

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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.


13/23

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RAINFLOW COUNTS

Some ideas about how to deal with composite signals:


14/23

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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


15/23

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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


16/23

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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


17/23

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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)


18/23

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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 ?


19/23

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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.


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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).


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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 ?


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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.

pres rainflow

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