General Foundations

It has
proven advantageous to assume the cumulative distribution of the failures as
the basis for calculations. The distribution used in the Weibull calculation is
especially suited to this field of application. In general the Weibull
distribution is obtained through an exponential distribution. Calculations are
done in this way because :

·
Many
forms of distribution can be created through the Weibull distribution.

·
The
Weibull functions are user-friendly.

·
Time-dependant
fault mechanisms are depicted on a line graph.

·
The
method has proven itself to be reliable.

When presented on a linear diagram, the failure probability of the
entire vehicle life span will be shown in a complicated "S"-form.
Distortion of the vertical scale (double logarithms) and the horizontal scale
(logarithm) will enable this "S" to be shown as a straight line
(fitting line). The Weibull distribution function in this simplified 2-variable
form is known as Equation 1.

_{}_{
}H = failure
probability or failure frequency (standardised at 1 (%) in 100), t = lifetime
variables (distance driven, length of time in operation, load cycles, etc.), T
= typical service life, the point at which 63.2% of the units were faulty (for
t = T, H =100% (1-1/e) = 63.2%) and b = the parameter form, i.e. the rise
(slope) of the fault line within the Weibull chart. In the 3-parametrical form,
the so-called fault-free time period to is taken into consideration (Equation
2).

_{}_{
}

In most cases to = 0 can be assumed. This corresponds with the
2-parametrical form. Despite stress, some component faults behave in such a way
that they are first detected after the operation-time to. In this situation the
co-ordinates on the Weibull graph will lie above the service life line most
often curving to the right. When analysing the steep and falling left-hand
curve, it is possible to pre-position the point of intersection of the curve
with the infinite "zero-line" using to .

Generally the reliability of components, assemblies and vehicles can be
determined only once failures exist, i.e. once the service life of the unit in
question is available. In order to be able to comment on reliability, one must
first establish the service life e.g. during testing, in the laboratory or in
the field. The service life line (variable t) is in most cases :

·
Distance
driven

·
Length
of time in operation

·
Load
cycles

·
Number
of load changes

In order for reliability analysis to be possible, data for at least one
of the above must be available for the "damaged part".

This information is charted on the X-axis of the Weibull graph. For a
spot check it is usually necessary to carry out classification of the failures
from n>50, resulting in certain service life groups being placed together.
Such classification results in a roughly even curve on the Weibull graph. The
range of classification can be estimated using Sturges Equation 3.

_{}

In practice it is often appropriate to use a range of ‘000’s, rounding
either up or down. This is especially useful in cases where field data in
conjunction with kilometres is used for analysis, e.g. 1000 km, 2000 km, 5000
km etc.

Information is also lost through classification which means that there
are slight deviations in the results between the different classes. The same
parameters and or methods of classification should therefore be selected when
attempting a comparison between various analyses.

If the damaged parts are sorted by increasing service life then it is very
simple to establish the corresponding failure probability H using Equation 4
for values n < 50.

_{
}

This means I = the ordinal number of the damaged parts, n = the spot
check or number of defective parts. For n ³ 50 the
mid-range value is used for calculations in accordance with Gumbel (Equation
5).

_{
}

The simple relation H=100*i/n can frequently be found in the literature,
but Equation 5 is more suitable to the lower area. This data can be entered
directly into the Weibull chart .

In this classical interpretation, Weibull parameters are provided through
defining the adjustment line on the Weibull failure probability line graph. The
points for the adjustment line are obtained by altering the 2-parametrical
Weibull-function. The relationship for the X-axis and the axis of ordinates is
provided in accordance with Equation 6.

_{
}

The scale is calculated using the acknowledged method of taking the
lowest failure quadrant and the values for X and Y. This scale corresponds to
Equation 7.

_{
}

b is the rise and b.In(T) is the vertical position of the adjustment
line.

In practice b and T are often calculated using the Gumbel method, in
which the points on the Weibull chart are weighted differently (Equations 8 and
9).

_{
}_{}_{
}

Through the Gumbel method, the values provided for b are greater than
those given when using the standard method. This should be noted when
interpreting the results.

A further possible way of determining points b and T is the Maximum-Likelihood
Estimate (Equation 10):

_{
}

In order to ascertain b Equation 10 must be repeatedly solved. Once b is
obtained T can be calculated (Equation 11)

_{}

Due the regular dispersion in statistics for the service life, it is
clear to see that there is little point in inputting the average figure for the
"running time". The Weibull evaluation reaches a sufficient
conclusion regarding the cause of failures of the relevant part. The figure at
which the average lies is usually taken as the typical-service-life T, at which
point 63.2% of the parts show faults. This is most often illustrated with the
perpendicular in the graph.

A further important measurement is the parameter form b. This is nothing
else but the rise in the line on the Weibull graph. Conclusions can be drawn on
the behaviour of the faults by using rise b. Usually the Weibull chart will
show a scale on the right-hand side which includes rise (slope) b. Taking the
guiding line (bottom right) and the increase in the Weibull line, it is
possible to make a simple estimate as to the real increase in the faults.

In practice it is often the case that the fault data has a variety of
causes. This means that, after a certain "running time", there is a
pronounced change in the fault increase due to other influences on the part
also being present. The various sections should therefore be looked at
separately and it is advantageous to connect the individual points on the graph
(faults) to one another when an entire fitting line is available.

Before a part goes into production it is necessary to have a prediction
indicating its reliability. In order to obtain this prediction, long-term tests
and simulation tests are done under laboratory conditions, in which the part is
put under a certain load stress in order to obtain an estimate of its life
span. This is most often required due to safety reasons, e.g. factor 2..3.
Entering the service life characteristics into the Weibull-Chart will result in
the fitting line lying to the left of a test line in which a normal load had
been used. If the progression of the fitting line is not parallel, but
increases at a different rate, then this shows a variety of fault mechanisms in
relation to the test and real applications. The test is therefore not suitable.

In cases where a calculation of faulty parts is required for parts
already in production cars (so-called in-field faults), then the failure probability
is calculated in the same way as previously shown. The relevant production
figure is taken along with the production period for that figure. The fault
figure is then used in relation with this information.

It is clearly necessary to presume that all faults that relate to this
production figure have been recorded and that none have been exchanged due to
incorrect findings. In the case of incorrect findings, parts are wrongly
exchanged by customer services although they are not the actual reason for the
fault. These parts have no failures are cannot be counted within the failure
probability, therefore they are not included in the analysis. Apart from this
it is important to note the service life. It is pointless to include parts in
the analysis which have failed due to other influences (for example due to an
accident). This means that damage analysis should always be carried out before
the individual data analysis begins.

The Weibull calculation is based on what may be viewed as a random
sample. This in turn means that the line on the Weibull chart reflects a
sample. The points on the Weibull line will become increasingly scattered as
the number of parts included in the calculation rises. It is however possible
to make a statistical estimation as to the area on the chart which will make up
the basic entirety of the failure probability. A so-called "confidence
range" is taken in this case. This is generally given with a statement
probability, mostly at PA=90%. This means that the upper confidence
range is at 95%, and the lower at 5%. The confidence range is calculated using
a Beta-distribution with the relevant rank numeration as parameter. Tabled
values are generally available for the F-distribution. The confidence range may
be confirmed through a transformation using this distribution (Equations 12 and
13).

_{
}

The example shown in the Diagram shows the two limit lines, within which
the 90% basic failure entirety may be found. There will, to a varying extent,
be a split between the path of the failure probability line and the upper and
lower lines which define the limits of the confidence range. This shows that
the failure probability prediction is far less accurate within these areas than
in the upper middle section of the chart. It is not possible to lengthen the
confidence range, or the fitting line, so that either lies very far outside of
the co-ordinate points plotted on the chart.

** **

**Confirmation of fault-free period t _{o
}**

There are various methods used to confirm to. One
mathematical formula which is clearly best suited in this case does not exist.
One can always say that to must lie between time point 0 and the time at which
the first part became faulty. to usually falls very close to the first
occurrence of a faulty part. The following method is possible : apply to in
small, regular steps in the interval from 0 up to the first fault (Tmin) and
calculate the coefficient correlation to the fitting line. The better the value
of the coefficient correlation, the more precise the points on the Weibull
chart line. to is the greatest value and therefore permits a good approximation with
the fitting line. The confirmation of to is described later on in this text.

In
cases where 2 different designs or systems are to be compared, perhaps with
different finishing processes, then the typical service life can be examined.
The relevant service life is often given at the point where a 10% failure
probability has been reached. A comparison between the service life values
alone is not enough because a statistical statement on the likelihood of
further faults is not included. An example: In a test, design A is compared
with design B which has a longer service life. For example: A design “A” around
5 units at 50% service life, whilst “B” has a considerably longer service life,
showing ca. 6.5 at this point. A response to the failure probability in this
case would therefore be that B is better than A. Additionally to this
information, a relationship is shown as (Equation 14…17):

_{
}

_{
}

_{
}

_{
}

In this case z, y, y’ = help-functions for
obtaining the statement of probability, q = sum-%-failure area, t_{q,a}
= service life of design A at q-%-failures, t_{q,b} = service life of
design B at q-%-failures, b_{A} = increase parameter of the Weibull
distribution for design A, b_{B} = increase parameter of the Weibull
distribution for design B, n_{A} = sample size for design A, n_{B}
= sample size for design B. AW = Probability of different distributions. Is AW
= 50% there is no difference between the two curves.

** **

See also Consideration of faults not yet occurred

Software for Weibull
-> Visual-XSel www.crgraph.com