Author: YehLiang Hsu, MingSho Hsu, ShuGen Wang, TzuChi Liu (20040127);
recommended: YehLiang Hsu (20041209).
Note: This paper is published in Journal of the Chinese Institute of
Industrial Engineers, Vol. 21, No. 6, pp. 551558.
Prediction of Fatigue Failures of
Aluminum Disc Wheels Using the Failure Probability Contour Based on Historical
Test Data
Abstract
Disc wheels
intended for normal use on passenger cars have to pass three tests before going
into production: the dynamic cornering fatigue test, the dynamic radial fatigue
test, and the impact test. This paper describes a probability model for
prediction of fatigue failures of aluminum disc wheels, which intends to better
link the prediction using simulation results with historical test data. Finite
element models of 54 aluminum wheels, which are already physically tested, are
constructed to simulate the dynamic cornering fatigue test. Their mean stresses
and stress amplitudes during the fatigue loading cycle are calculated and
plotted on a twodimensional plane. Matching with historical test data, the
failure probability contour can be drawn. For a new wheel, the failure
probability of dynamic cornering fatigue test can be read directly from this
probability contour. The test result of the new wheel can be added into the set
of historical test data and the failure probability contour is updated. Same
procedure is directly applied to the fatigue prediction of dynamical radial
fatigue test. At this point we only have 20 historical test data to construct
the failure contour. The prediction will become more and more reliable as the
number of historical test data increases.
Keywords: Fatigue test, aluminum
disc wheel, failure probability
1.
Introduction
Disc wheels
intended for normal use on passenger cars have to pass three tests before going
into production: the dynamic cornering fatigue test, the dynamic radial fatigue
test, and the impact test. Fatigue prediction has been an important issue to
the design of aluminum disc wheels. Karandikar and Fuchs [1990] developed a
computerbased system, including a CAD package, a finite element analysis
program, and a fatigue life computation program, for predicting the fatigue
life of wheels. Kaumle and Schnell [1998] also developed a technique for
fatigue testing using a rapidprototyping system. On the rapidprototyping
wheel the stresses will be measured, areas with high stress can be detected
early, and the fatigue behavior can be tested with the rapidprototyping wheel.
The randomness
of fatigue prediction due to the inherent uncertainties in loading,
manufacturing variability, and material properties has been commonly
recognized. Probabilistic approaches are proposed to account for the
uncertainties in fatigue prediction models for various industrial applications.
For example, Shen and Nicholas [2001] described the probabilistic analysis for
high cycle fatigue design of gas turbine; De Lorenzo and Hull [1999] used a
fully instrumented test bicycle to quantified the loads input to an offroad
bicycle as a result of surfaceinduced loads to provide data for fatigue
prediction; Sheikh et al. [1995] proposed a method for calculating the
reliability of an assembly of rotating parts subjected to fatigue failure;
Adrov [1994] presented a probabilistic approach to predict airframe fatigue
damage using a load spectrum.
In such models,
probability distributions of the factors with uncertainties are often obtained
from measurements in actual operating conditions. Given the probability
distributions of these factors, the distribution of fatigue life is derived
analytically or fitting with experimental data. Tallian [1996] presented the
results of a project aimed at improving the fatigue prediction of rolling
bearings. A large experimental bearing life database is assembled, a
mathematical fatigue prediction model is formulated, and the model is fitted to
the historical experimental database. We will use similar approach in this
research.
This paper
describes a probability model for prediction of fatigue failures of aluminum disc
wheels, which intends to better link the prediction using simulation results
with historical test data. In various fatigue criterions, the mean stress and
the stress amplitude are the two critical variables for fatigue prediction. In
this paper, finite element models of aluminum wheels are constructed to
simulate the dynamic cornering fatigue test. Fiftyfour wheels, which are
already physically tested, are analyzed and their mean stresses and stress
amplitudes during the fatigue loading cycle are calculated and plotted on a
twodimensional plane. Instead of using the fatigue failure criterions commonly
seen in literature, such as Goodman and Gerber’s criterions [Bannantine et.
al., 1990], we match the analysis results with historical test data, and a mathematical
model is used to fit the historical data to construct the “failure probability
contour.” For a new wheel, the failure probability of dynamic cornering fatigue
test can be read directly from this probability contour. The test result of the
new wheel can be added into the set of historical test data and the failure
probability contour is updated.
Same procedure
can be directly applied to the fatigue prediction of dynamical radial fatigue
test. At this point we only have 20 historical test data to construct the
failure contour. The prediction will become more and more reliable as the
number of historical test data increases.
2.
Predicting the dynamic
cornering fatigue test
The dynamic
cornering fatigue test simulates the loading condition of the wheels in normal
driving. Figure 1 shows a typical setup of the 90degree loading method of
cornering fatigue, according to SAE J32 [SAE, 1992]. In the Figure, the
downside outboard flange of rim of the wheel is clamped securely to the test
device, and a rigid load arm shaft is attached to the mounting surface of the
wheel. A test load applies on the arm shaft to provide a constant cyclical
rotation bending moment. After being subjected to the required number of test
cycles, there shall be no evidence of failure of the wheel, as indicated by
propagation of a crack existing prior to test, new visible cracks penetrating
through a section, or the inability of the wheel sustain load.
Figure 1. Typical
setup of 90 degree loading method of cornering fatigue [SAE, 1992]
The finite
element model used to simulate the dynamic cornering fatigue test has been
described in great details in the authors’ previous work [Hsu et al, 2001].
Figure 2 shows the finite element model of an aluminum wheel. All degrees of
freedom of the nodes on the downside outboard flange of the rim are fixed. The
dynamic load is represented by 24 discrete loads 15 degrees apart. For each
node, the maximum and minimum Von Mises stresses during the load cycle are
extracted to obtain the mean stress _{} and the stress amplitude _{} of the node. The wellknown Goodman and
Gerber’s criterions were first used for fatigue prediction in this study.
Goodman’s
criterion: _{} (1)
Gerber’s
criterion: _{} (2)
where _{} is the endurance
limit, _{} is the ultimate
strength of the material, and n is
the safety factor [Bannantine et. al., 1990].
Figure 3 shows
the two curves with safety factor n =
1. Each node of the finite element model is represented by its (s_{m}, s_{a}). The top 1% nodes that are
closest to the Goodman’s line, which have the greatest possibility to fail in
the fatigue test, are plotted in Figure 3. The average (s_{m}, s_{a}) of these 1% top stress nodes
are calculated, as shown by the solid circle (s_{m}, s_{a})_{1%} in Figure 3. This
point is used to represent the wheel when checking with Goodman and Gerber’s
lines to predict whether the wheel will pass the cornering fatigue test.
Finite element
models of the first 28 aluminum wheels, which were already physically tested by
a local manufacturer, were constructed to simulate the cornering fatigue tests.
The (s_{m}, s_{a})_{1%} for each wheel was calculated and plotted in Figure 4,
where “o” and “´” represent whether the wheel actually passed of failed the
cornering fatigue test.
Figure 2. Simulating
the dynamic cornering fatigue test
Figure 3. The
Goodman and Gerber’s lines for safety factor n = 1
Figure 4. The
Goodman and Gerber’s lines for safety factor n = 2.6
These
“historical data points” provide good references for choosing the safety factor
n for this specific case. As shown in
Figure 4, we found that for n = 2.6,
the Goodman and Gerber lines best fit the 28 historical data points. Using this
safety factor, we can then define the “safe zone,” “dangerous zone,” and “failure
zone” for predicting whether a new aluminum wheel can pass the dynamic
cornering fatigue test.
Following this
study, another batch of 26 new aluminum wheels were predicted using this procedure.
The prediction results were then matched with the physical test results. As
shown in Figure 5, all 10 wheels that are in the safe zone did pass the
cornering fatigue test. Three wheels fall in the dangerous zone and 1 of them
did not pass the cornering fatigue test. Thirteen wheels fall in the failure zone,
and 12 of them did not pass the cornering fatigue test. In this sample of 26
wheels, this procedure has a 96% success rate in correctly predicting whether
the wheels will pass the dynamic cornering fatigue test.
Figure 5. Prediction
results of 26 new aluminum wheels
3.
Predicting the dynamic cornering
fatigue test using the probability model
Though the procedure
described above works well for practical purposes, there are still several
fundamental problems. First, this procedure only provides three qualitative
predictions: safe, dangerous, and fail. Referring to Figure 5, in the
neighborhood of the lines between safe, dangerous, and failure zones, a small
deviation may result in completely different predictions. Quantitative
information of “how likely the wheel is going to fail” is desired, especially
for the data points in the dangerous zone. Secondly, in this procedure, the
prediction of a new aluminum wheel is made based on historical test data, and
the safety factor n is the only “mathematical relation” used to
fit the prediction with historical test data. A better mathematical model
should be construct to fit the historical data. As the number of historical data
points increases, there should be a mechanism for updating this “relation” to
improve the prediction.
Based on the
experience of predicting the fatigue failure of aluminum disc wheel, this paper
presents a probability model that intends to better link the prediction using
simulation results with historical test data. Instead of the three qualitative
predictions, this probability model generates the probability of failure based
on historical test data.
In our case, we
now have 28+26=54 historical data points, as shown in Figure 6. Note that
historical data points towards the upper right corner are more likely to fail.
Given a new data point (s_{m}, s_{a})_{1%}, we can define a circle of radius r, which is
centered on this new data point. We can then use the historical data points in
the circle to predict the failure probability of the new data point. The
failure probability can be easily calculate as _{}, where N_{fail} is the number of historical
data points in the circle that actually failed, and N_{total} is
the total number of historical data points in the circle.
Figure 6. 54
historical data points of the cornering fatigue test
While this is a straightforward
way to calculate the failure probability of a new data point using historical
test data, there is a practical problem of how to define the radius r.
If r is large, the circle may be too big to adequately represent the new
data point. If the radius r is small (or the historical data points are
sparse in the neighborhood of the new data point), the number of historical
data points in the circle will be too few to generate a meaningful probability
value.
Therefore,
instead of calculating failure probability directly from the historical data
points that fall in the circle, we try to draw a “probability contour” on the
twodimensional domain s_{m}s_{a} first. To do this, the x(s_{m}) and y(s_{a}) axes of the 54 historical data
points are normalized between 0 and 1, then we divided this domain into m´m rectangular grids. On each grid point,
we can draw a circle of radius r=1/m, which is the length of the grid,
then the failure probability of this grid point can be calculated from the
historical data points that fall in this circle. Figure 7 shows the failure
probabilities of the grid points for m=10.
Figure 7. Failure
probability on the grid points for m=10
The historical
data points scatter along the diagonal of the s_{m}s_{a} domain in this case, and only
the 38 grid points along the diagonal have failure probability figures. These
probability figures are extrapolated to the whole domain in order to draw the
failure probability contour. The extrapolation is done in an iterative manner,
and the 38 original probability figures in Figure 7 remain fixed during the
extrapolation.
Several assumptions
are considered during the extrapolation. Grid points toward the upper right
corner should have a higher failure probability. Therefore we have the
following assumption:
_{}, if _{} and _{} (3)
All probability
figures generated during the process have to satisfy Eq. (3). Moreover, since
the probability value lies between 0 and 1, Eq. (3) also implies
_{}_{} and _{}, _{} if _{} (4)
_{}_{} and _{}, _{} if _{} (5)
These two
equations are also necessary when extrapolating the probability figures to the
boundary of the domain.
Figure 8 shows
the results of the extrapolation, and finally Figure 9 shows the failure
probability contour based on the 54 historical data points. Note that the
Goodman and Gerber criterion are also plotted on the Figure for comparison. The
failure probability of a new data point can be read directly from this Figure,
even for data points in the “dangerous zone” between the Goodman and Gerber’s
lines.
Figure 8. The
results of the extrapolation
Figure 9. The
failure probability contour based on 54 historical test data
4.
Predicting the dynamic
cornering fatigue test using the failure probability contour
Table 1 shows
the predictions of 8 new aluminum wheels using the failure probability contour.
For example, a finite element model is constructed to calculate the (s_{m}, s_{a})_{1%} for wheel No. 1,
and its failure probability predicted by the failure probability contour (Figure
9) is 67.1%. This wheel actually failed in the cornering fatigue test. After
the physical test result was obtained, this wheel became a “historical test
data.” The failure probability contour is then updated using 54+1=55 historical
test data. Table 1 also shows that, after updating the failure probability
contour, the failure probability predicted for the same (s_{m}, s_{a})_{1%} increases from
67.1% to 73.3%. Similarly, the predicted failure probability for wheel No.2
using the probability contour obtained from 55 historical data is 23.9%. Wheel
No.2 actually passed the cornering fatigue test, and after updating the failure
probability contour using 55+1=56 historical test data, the predicted failure
probability for the same (s_{m}, s_{a})_{1%} dropped to 20.2%. Finally, Figure 10 shows the
updated failure probability contour based on 54+8=62 historical test data.
Table 1. Failure
probability prediction of 8 new wheels
Wheel No.

1

2

3

4

5

6

7

8

(s_{m}, s_{a})_{1%}

(40.0,
32.2)

(32.3,
26.3)

(43.6,
31.5)

(41.8,
30.8)

(33.3,
28.5)

(70.1,
52.7)

(40.2,
28.2)

(45.9,
38.3)

P_{fail}

67.1%

23.9%

71.2%

66.4%

27.2%

100%

39.5%

73.5%

Updated P_{fail}

73.3%

20.2%

73.9%

59.1%

23.0%

100%

37.3%

75.7%

Test result

Fail

Pass

Fail

Pass

Pass

Fail

Pass

Fail

Figure 10. The failure probability contour based
on 62 historical test data
Using this
probability contour, we can now provide quantitative information of “how likely
the wheel is going to fail,” and we also have a mechanism to update the model
to improve the prediction as the number of historical data points increases.
In the batch of
8 wheels shown in Table 1, the 3 wheels that were predicted to have failure
probabilities higher than 70% actually failed the cornering fatigue test. The 3
wheels that were predicted to have failure probabilities lower than 40%
actually passed the cornering fatigue test. The predicted failure probabilities
for wheel No. 1 and wheel No. 4 were close, but wheel No. 1 passed the
cornering fatigue test, while wheel No. 4 failed. When the predicted failure
probability of a new aluminum wheel is high, we should certainly redesign the
wheel to decrease its failure probability before actually making the die to
manufacture a wheel for physical testing. In the mean time, it may not be
practical if we accept the design only when it has zero failure probability.
From the
experience of local manufacturers, without careful computer simulations and
predictions, 3~4 trials of making or revising the die are common for a new
aluminum wheel until it can finally pass the physical test. These trials are
also the major cause of the development cost and time. Given a predicted
failure probability of a new wheel, we can use the expected value of the number
of trials of making and revising the die to determine whether the designer
should accept this new design or redesigns the wheel. If the failure
probability of a new wheel is P_{fail}, the expected value of trials
is approximately (1P_{fail})+2P_{fail}, neglecting
the higher order terms. Therefore, for example, if a manufacturer has a policy
that average number of trials of making and revising the die for all wheels
should be lower than, say, 1.3, then P_{fail}=0.3. That is,
designers should only accept the wheels whose predicted failure probabilities
are lower than 0.3.
5.
Predicting the Dynamic Radial
Fatigue Test Using the Failure Probability Contour
Figure 11 is a
typical setup of the dynamic radial fatigue test, according to SAE J328 [SAE,
1992]. In the Figure, the test wheel is constrained by bolts through the PCD.
The driven rotatable drum, whose axis is parallel to the axis of the test
wheel, presents a smooth surface wider than the section width of the loaded
test tire section width. The test wheel and tire provide loading normal to the
surface of drum and in line radially with the center of test wheel and the
drum. After being subjected to the required number of test cycles, there shall
be no evidence of failure of the wheel, as indicated by propagation of a crack
existing prior to test, new visible cracks penetrating through a section, or
the inability of the wheel to sustain load.
Finite element analysis
is used to simulate the dynamic radial fatigue test. As shown in Figure 12, the
wheel is glued to a base through the PCD, and all degrees of freedom of the
bottom of the base are constrained. The dynamic load is represented by 20
discrete loads 18 degrees apart. Similar to the simulation of dynamic cornering
fatigue test, (s_{m}, s_{a}) during the load cycle are calculated for each node of the finite
element model. The average of the top 1% nodes that are closest to the
Goodman’s line (s_{m}, s_{a})_{1%} is used to represent the wheel when predicting
whether the wheel will pass the radial fatigue test.
Figure 11. Typical setup of dynamic radial fatigue
test
Figure 12. Simulating the dynamic radial fatigue
test
Finite element models
of 20 aluminum wheels, which were already physically tested by a local
manufacturer, were constructed. The (s_{m}, s_{a})_{1%} for each wheel was
calculated and plotted in Figure 13, where “o” and “´” represent whether the wheel actually
passed or failed the radial fatigue test. Using the same procedure presented in
the previous sections, Figure 14 shows the failure probability contour for m=10,
based on the 20 historical data points. The failure probability of dynamic
radial fatigue test of a new wheel can be read directly from this figure. In
this case, we used much fewer historical data points to construct the failure probability
contour. The prediction will become more and more reliable as the number of
historical data points increases.
Figure 13. 20 history data points of the radial
fatigue test
Figure 14. The failure probability contour based
on 20 historical test data
6.
Discussions and Conclusions
This paper
presents a procedure that predicts the fatigue failure probability based on
historical test data. Using this probability model, we can now provide
quantitative information of “how likely the wheel is going to fail” using
simulation results, and we also have a mechanism to update the model to improve
the prediction as the number of historical data points increases.
Computer
simulations are often used to predict the performance of a new product, but the
results from computer simulation still have to be confirmed by physical
testing. Therefore historical test results are very valuable to manufacturers
and should be systematically preserved and utilized. Based on the experience of
predicting the fatigue failure of aluminum disc wheel, we can summarize a
probability model that intends to better link the simulation results with
historical test data. The whole idea is rather straightforward. There are three
basic assumptions in this model:
(1)
The desired prediction using simulation
results is a passfail type of prediction.
(2)
The essential indices that
affect performance being predicted, evaluated from computer simulation, have
been identified.
(3)
There are enough historical
data points as references of the prediction.
Under these
three assumptions, a design is expressed by a data point (I, p)
in this model, where I is the vector of performance indices obtained
from computer simulation, and p is the probability of whether this data
point will fail in the physical test. Assume that there is a set of q
historical data points (I_{i}, p_{i}), i =
1, 2, …, q, that are already physically tested. Note that for a
historical data point, p_{i} is either zero (the corresponding
design passes) or one (the corresponding design fails).
Now a new candidate
design j is evaluated using computer simulation to obtain its vector of
performance indices I_{j}. From I_{j}, we wish to
predict p_{j}, the probability of whether this candidate design j
will fail in the physical test, from the set of n historical data
points. To do this, we define a hypersphere whose center is I_{j}
and radius is r. The n historical data points are checked to see
if I_{i}  I_{j} < r, that is,
whether the historical data point i falls in the hypersphere. Assume
that there are m data points in the hypersphere, (I_{k}, p_{k}),
k = 1, 2, …, m, we can obtain _{}. After this candidate design j is physically tested,
it becomes a “historical data point,” and the set of historical data points is
updated.
A practical
problem of using this model is how to define a proper radius r for the
hypersphere. For the problems having only 2 or less components in I, as
in our aluminum wheel case, a probability contour can be drawn on the twodimensional
domain formed by the 2 components of I, using the set of n
historical data points (I_{i}, p_{i}), i =
1, 2, …, q. Then p_{j} can be read directly from the
probability contour.
7.
Acknowledgements
This research is
sponsored by Industrial Development Bureau, Ministry of Economic Affairs,
Taiwan, ROC, grant number 9001010109, and by Ensure Co., Ltd. This support is
gratefully acknowledged.
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