counting and histogramming

8/12/2019 Counting and Histogramming

1/10

Theory

LMS proprietary information: reproduction or distribution

of this document requires permission in writing from LMS

Counting and histogramming.doc

Category:Time data processing

Topic: Counting and histogramming

Introduction

In fatigue analysis, real life measurements of mechanical or thermal loads are used to assess and

predict the damage inflicted by such loads over the life time of a product. Figure 7-1 shows such

measurements made on a vehicle part over a period of around 5 minutes (330 seconds).

acceleration

time(s)

(g)

0.4

-0.4

Figure 7-1 Typical load/time data

In terms of fatigue analysis it is the occurrence of specific events that are of more significance than the

frequency content of the loads. The approach used is to scan such time histories looking for typical

fatigue-generating events and then to register how often they occur. These typical events can be

demonstrated with a zoomed-in section of a load time history, shown in Figure 7-2.


2/10

Figure 7-2 Typical events in a data trace

The interesting events are:-

The occurrence of peaks at specific levels

These are represented by the circles and are determined using Peak

counting methods described in section 13.2.1.

The exceedence or crossing of specific levels.

These are represented by the squares and are determined using

Level cross counting methods described in section 13.2.2.

The occurrence of signal changes of a certain size.

These are represented by the arrows and are determined using Range

count methods described in section 13.2.3The determination of the signal characteristics based on the events mentioned above is a two stage

process

Stage 1, counting

The data is scanned for the occurrence of one of the events listed above.

This in effect reduces the full time history to a set of mechanical or thermal

load events.

Stage 2 histogramming

This involves dividing the counted occurrences into classes where for each

event, its number of occurrences is specified.

One dimensional counting methods

The procedures described above deal with the counting of single events or occurrences which are

further explored in this section.

Section 13.3 describes a number of methods used to examine the occurrence of additional event

circumstances. These methods are termed Two dimensional counting methods.

Peak count methods

The turning points in a data trace are termed peaks(maximums ) and valleys (minimums ). Thenumber of times that peaks and valleys occur at specific levels is counted as shown below. You can

choose to count both the peaks and the valleys (extrema) or just the peaks (maxima), or just the

valleys (minima).


3/10

0

-1

-2

1

2

Figure 7-3 Counting of peaks and valleys

A histogram is then created by calculating the distribution of the number of occurrences as a function

of the level at which the occurrence appeared. The Figure 7-4 shows the results of processing the

above peak-valley reduction according to the three types of counting methods.

0

-2

1

2

3

4

-1 0 1 2level

Nrofoccurrences

0

-2

1

2

3

4

-1 0 1 2level

Nrofoccurrences

0

-2

1

2

3

4

-1 0 1 2level

Nrofoccurrences

ExtremaMinima Maxima

Figure 7-4 Histograms of peaks (maxima), valleys (minima) and both (extrema)

Level cross counting methods

This procedure counts the number of times that the signal crosses various levels. Distinctions can be

made between an upward (positive ) and a downward (negative ) crossing as illustrated below.

You can choose to count both the positive (up) crossings, the negative (down) crossings or both types.

Figure 7-5 Counting of level crossings


4/10

Peak counts and level cross counts are closely related. The number of positive crossings of a certain

level is equal of the number of peaks above that level minus the number of valleys above it. This

implies that a level cross count can be derived from a peak-valley count.

A level crossing count is typically initiated by specifying a grid on top of the signal to determine the

levels. The grid can be specified in ordinate units or as a percentage of the ordinate range. The

resulting histograms for the above signal when up, down and both types of crossings are counted are

shown below.

0-2

2

4

6

8

-1 0 1 2level

Nr

ofoccurrences

u + crossin s

10

-2 -1 0 1 2level

Nr

ofoccurrences

down - crossin s

0

2

4

6

8

10

0-2

2

4

6

8

-1 0 1 2

level

Nr

ofoccurrences

up (+) & down (-) crossings

10

Figure 7-6 Histograms of level crossing counts

Range counting methods

A range count method will determine the number of times that a specific range change is observed

between successive peak-valley sequences.

Counting of single ranges

The range between successive peak-valley pairs is counted. Ranges are considered positive when

the slope is rising and negative when the slope is falling.

-1

+1

+4

-4

+1

-1

-1

+1

+1

-1

Figure 7-7 Counting of single peak-valley ranges

A histogram of the number of occurrences, as a function of the range, is generated.


5/10

0

-2

1

2

3

4

-1 0 1 2

Nrofoccurrence

s

Range

3 4-4 -3

Figure 7-8 Histogram of single peak-valley ranges

Counting of range-pairs

The counting of single ranges (usually indicated as a range-count), is both simple and straightforward

but sensitive to small variations of the signal. Thus in the analysis of the left hand signal illustrated in

Figure 7-9, single range counting would result in a large number of relatively small ranges.

Figure 7-9 Sensitivity of single range counting to signal variation

If this signal were passed through a filter, suppressing the small load variations, the resulting signal

would reveal a count of only one very large range. As a consequence the two analysis results are

completely different and the method is very sensitive to small signal variations.

The range-pair counting method overcomes this sensitivity. Rather then splitting up the signal into

consecutive ranges, it is interpreted in terms of a main signal variation (or range) with a smaller cycle

(range pair) superimposed on it.

Figure 7-10 Range pair counting

If a pair of extremities are separated by a range that is less than the defined range of interest (R), then

they are filtered out of the range count.


6/10

Two-dimensional counting methods

The counting methods described so far, consider the occurrence of single events in isolation from anyother circumstances which may affect these events. However, it is also meaningful to count events

differently, depending on other circumstances using two-dimensional methods. Such methods are

discussed in this section.

From-to-counting

Such a combined event can be the occurrence of a peak at level j followed by a valley at level i. As

an example, consider the combination of a valley at level A followed by a peak at level C as illustrated

in Figure 7-11.

A

B

C

D2

3

4

11

12

Figure 7-11 From-to counting

In this example, the From!to sequence (1!2) is counted separately from the sequences (3!4) and

(11!12), although the ranges involved are identical (C-A=D-B).

The result of such from!to counting can be presented in a so called Markov-Matrix A[i,j]. The

element aijgives the number of peaks at level j followed by a valley at level i. The matrix of results of

counting the events in Figure 7-11 are shown below.

A B C D

A

B

C

D

X

X

X

X

0 1

2

0

1

2

0

1 1

1 12

From j

T

o

i

X

0

4

2

1

3

0

2

The lower left triangle of the Markov matrix contains the positive from!to events, the upper right triangle

summarizes the negative transitions. The additional separate columns contain the counting results for

peaks and valleys at a particular level. These results are easily obtained for the triangles of the

Markov matrix.


7/10

Range-mean counting

Another example of a two-dimensional counting method results in the so-called Range-mean matrix.The variation or range (i-j) is associated with its corresponding mean value (i+j)/2.

A

B

C

D

2

3

4

11

12

D-B D-B

B

C C

Figure 7-12 Range mean counting

Instead of considering the actual values of A and C, the Range-mean method will consider the values

C!A (the range) and B (= A+C / 2 the mean). Ranges, means and the number of occurrences can be

displayed in a 3D format.

Number of events

RangeMean

Figure 7-13 Display of range-mean counting

Range pair-range or Rainflow method

A two-dimensional counting method of special interest, especially for fatigue damage calculations, is

the range pair-range method. Such a method was also developed, simultaneously and

independently in Japan, known as the Rainflow method. Both methods yield exactly the same

results, i.e. they extract the same range-pairs and ranges from the signal, by combining the range-pair

counting principle and the single range counting principle into one method. For further details see the

references listed on page 91.

Essentially the signal is split into separate cycles, having a specific amplitude (or range) and a mean.

The result can be put directly into cumulative fatigue damage calculations according to Miners rule

and into simple crack growth calculations. Three steps are involved in the complete procedure.

1 Conversion of the load history into a peak-valley sequence.


8/10

As the counting procedure considers only the values of successive peaks and valleys, the complete

signal may first be reduced to a peak-valley sequence. In doing this it is usual to apply a specific

range-filter or gate. For a range filter of size R, a peak (or valley) at a certain level is only recognizedas such if the signal has dropped (or risen) to a level which is R lower (or higher) then the previous

peak (or valley) level.

Figure 7-14 Conversion of a load history to a peak valley sequence

In the above example e1is counted as a peak because the signal drops by more then the range filter

size R after it.

After counting the first peak, the next valid valley is looked for, which in this case is e2. This point is

validated as a valley as the signal rises by more then R to go to e3. The algorithm then searches for

the next valid peak. The first peak encountered is e3, but this is not counted as a valid peak as the

signal does not drop sufficiently before reaching the next extremum in the signal (e 4). So the algorithm

checks whether the following peak is a valid one. Peak e5is regarded as valid since the drop in signal

level following it, is greater than R.

In this example the range filter eliminated the small signal variation (e 3,e4) from the peak-valley

sequence.

Note that increasing the range filter eliminates only those transitions from the histogram for which the

range is smaller than the new value of R. This is important for fatigue purposes since it proves that the

filtering is not that sensitive to the range filter size.

2 Scanning of the entire signal for range-pairs.

This phase of the counting procedure consists of taking a set of four consecutive points, and check

whether a range-pair is contained in it. If not, the search through the peak-valley sequence continues

by shifting one data point ahead. Once a range-pair is detected, the pair is counted and removed from

the sequence. After this, the next new set of four points is formed by adding the closest two previously

scanned points, to the two remaining after removal of the range pair. The fact that earlier scanned

points are re-considered, clearly distinguishes Range-pair range counting from single range counting.

3 Counting the Residue


9/10

At the end of the second phase, a residue of peaks and valleys is left which is analyzed according to

the single range principle. It can be shown that this residue has a specific shape, namely a diverging

part followed by a converging part.

Example

The following example shows how the range-pair range method operates.

The second phase (scanning of the range-pair occurrences) starts by looking at the 4 first extremes.

In this group (S1,S2,S3,S4), a pair is counted if the two inner extremes (S2,S3,) fall within the range

covered by the two outer extremes (S1, and S4),. If this is not (as in this example), then the algorithm

moves one step forward and considers the extremes S2,S3,S4, and S5. These do not satisfy thecondition either, so the extremes S3,S4,S5, and S6 are considered and this time a range pair is counted.

Counting a range-pair implies deleting the counted extremes from the signal.

Stepping backwards, the extremes S1,S2,S3, and S6 are now considered and another pair (S2,S3) is

found.


10/10

From the remaining four extremes, no pairs can be subtracted. This forms the residue which is

further counted as single from-to-ranges.

Further considerations

The result of the range pair-range counting depends on the length of the data record being analyzed at

one time because the largest range counted will be between the lowest valley and the highest peak.

This largest variation is often referred to as the half load cycle. If the lowest valley occurs near the

beginning of a very long load cycle, and the highest peak near the end, you should consider whether it

makes physical sense to combine such occurrences, so remote in time into one cycle.

The counting method is insensitive to the size of the range filter applied. The only effect of increasing

the range filter size from R to 3R, for example, is that all elements in a From-to counting for which

|from-to|

counting and histogramming

Documents