Author: YehLiang Hsu, ChiaChieh Yu, ShangChieh Wu, Ming Hsiu Hsu (20061027);
recommended: YehLiang Hsu (20070802)
Note: This paper is published in Proceedings of the I MECH E Part B
Journal of Engineering Manufacture, Vol. 221, No. 3, March 2007, pp. 447456.
Developing
an automated design modification system for aluminum disk wheels
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. Fatigue prediction has been an important issue in
the design of aluminum disc wheels. This paper presents an automated design
modification system that integrates engineering analysis results and common
redesign strategies of engineers with a fuzzy logic algorithm in order to
provide effective strategies to reduce the failure probability in the fatigue
test.
A probability
model which provides quantitative information of “how likely the wheel is going
to fail in the fatigue test” using simulation results is described first,
followed by providing engineers’ common strategies for redesigning the wheels.
Sensitivities of the strategies with respect to the probability of fatigue
failures have been calculated to evaluate these strategies. A set of fuzzy
rules on how to reduce the failure probability using these strategies is established,
and an automated design modification system is constructed. Finally this system
is validated using 3 real wheel design examples. Using this system, designers’
previous experience can be properly transformed and utilized by all designers.
Keywords: aluminum disk wheels; fatigue; probability model; fuzzy logic.
1.
Introduction
Wheels are one
of the most critical components in automotive engineering. Their function is of
vital importance to human safety. 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 in the design of aluminum
disc wheels. Kocabicak and Firat developed numerical methods for the prediction
of cornering fatigue tests for passenger car wheels [1, 2]. The correlation
between the numerical simulation results and the cornering test results was
good. Kaumle and Schnell also developed a technique for fatigue testing using a
rapidprototyping system, on which the fatigue behavior of the wheel can be
tested [3]. Karandikar and Fuchs 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 [4].
The randomness
of fatigue prediction due to the inherent uncertainties in loading, manufacturing
variability and material properties has been commonly recognized [5].
Probabilistic approaches are proposed to account for the uncertainties in
fatigue prediction models for various industrial applications, such as the
fatigue prediction of rolling bearings [6], the airframe fatigue test [7], and
fatigue failure probability prediction of the vehicle axle of Shinkansen
highspeed train [8], to name just a few.
The authors also
presented a probability model for the prediction of fatigue failures of aluminum
disc wheels, which intended to better link the prediction using simulation
results with historical experimental data [9, 10]. In the previous work, finite
element models of aluminum wheels were constructed to simulate the dynamic
cornering fatigue test and dynamic radial fatigue test. The analysis results were
compared with the historical experimental data, and a mathematical model was
used to fit this data to construct the “failure probability contour.” This
failure probability contour was then used to predict the failure probability of
a new wheel. The failure probability contour can be updated when the number of
historical experimental data increases. This probability
model provides an excellent tool for the engineers to decide whether the wheel
needs to be redesigned. The next question is, based on the prediction of
the probability model, how to redesign the aluminum disk wheel.
When a wheel
fails in the fatigue test, engineers in wheel manufactures often use their
previous experiences to redesign the wheel according to the type of the wheel
and the various failure phenomena in the fatigue test. Many researchers used
fuzzy logic to help with the design decisions based on engineers’ experience in
the product design stage. Deciu et al. proposed an integrated approach of
configurable product design based on multiple fuzzy models. The transition from
customer specifications into physical solutions is performed by the help of
multiple fuzzy models [11]. Huang presented a fuzzy approach to model imprecise
information and requirements of customers in modular product development; and
to evaluate the design alternatives for designers [12]. Hsiao proposed a
decisionmaking method by quantifying the correlations between human sensations
and the physical characteristics of products using fuzzy logic analysis
methodology to create products that satisfy the needs of customers [13]. From
the researches discussed above, fuzzy logic seems to be a suitable approach to
assist engineering decisions in the design stage when there is not enough
quantitative, precise information.
This paper
presents an automated design modification system that integrates engineering
analysis results and engineers’ common redesign strategies with a fuzzy logic algorithm
in order to provide effective strategies to reduce the failure probability.
Section 2 of
this paper describes the probability model in detail. In Section 3, engineers’
common strategies for redesigning the wheels were collected. Sensitivities of
the strategies with respect to the probability of fatigue failures were
calculated to evaluate these strategies. In Section 4, a set of fuzzy rules on
how to reduce the failure probability using these strategies was established,
and an automated design modification system was constructed. This system was
validated using three real wheel design examples in section 5. Finally the
paper is concluded in section 6.
2.
The probability model for
fatigue failure prediction of aluminum wheels
This section
describes the four major steps for establishing the probability model for the fatigue
failure prediction of aluminum wheels, using cornering fatigue test as an example.
(1)
Establishing the finite element
model
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 test, according to SAE J32 [14]. In the figure, the downside
outboard flange of the rim is clamped securely to the test device, and a rigid
load arm shaft is attached to its mounting surface. A test load is applied 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 to sustain load
Figure 1. Typical setup of 90 degree loading
method of cornering fatigue [14]
Figure 2 shows
the finite element model that simulates the cornering fatigue test. All degrees
of freedom of the nodes on the downside outboard flange of the rim are fixed.
The dynamic load is represented by 20 discrete loads that are 18 degrees apart.
Figure
2. The finite element model of a wheel
(2)
Interpreting the analysis
results
Several
different criteria are commonly used in predicting fatigue failure. For
example, Goodman’s criterion can be expressed as follows [15]:
Goodman’s
criterion: _{} (1)
where s_{m} is the mean stress, s_{a} is the stress amplitude, S_{e} is the endurance
limit, and S_{u} is the ultimate strength of the material.
In the finite
element model in Figure 2, the maximum and minimum Von Mises stresses of each
node during the load cycle are extracted to obtain s_{m} and 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 top 1% nodes is
calculated to obtain (s_{m}, s_{a})_{1%} of the wheels. This (s_{m}, s_{a})_{1%} is used as the
index to represent the wheel in fatigue prediction.
Figure 3. The top 1% nodes those are closest to
the Goodman’s line
(3)
Fitting with historical
experimental data
Finite element
models of the 159 aluminum wheels, which were already physically tested by a
local wheel manufacturer, were constructed to simulate the cornering fatigue
test. 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 or failed the
cornering fatigue test.
Figure 4. The 159 historical experimental data of
the cornering fatigue test
(4)
Constructing the failure
probability contour
An algorithm was
developed to construct the failure probability contour based on Figure 4 in the authors’ previous work [9].
Note that in Figure 4, historical data points towards the upper right corner
are more likely to fail. Given a new data point (s_{m}, s_{a})_{1%}, a circle of
radius r, which is centered on this new data point, is defined. The
historical data points that fall in the circle are used 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.
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, a “probability contour” is drawn on the
twodimensional domain s_{m}s_{a} first. To do this, the x(s_{m}) and y(s_{a}) axes of the 159 historical data
points are normalized between 0 and 1, then this domain is divided into m´m rectangular grids. On each grid point,
a circle of radius r=1/m, which is the length of the grid, is drawn.
Then the failure probability of this grid point can be calculated from the
historical data points that fall in this circle. Figure 5 shows the failure
probabilities of the grid points for m=10.
Figure 5. Failure probability on the grid points
for m=10
The historical
data points scatter along the diagonal of the s_{m}s_{a} domain, and only the 24 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
24 original probability figures in Figure 5 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
_{}, if _{} and _{} (2)
All probability
figures generated during the process have to satisfy Equation (2). Moreover, since
the probability value lies between 0 and 1, Equation (2) also implies
_{}_{} and _{}, _{} if _{} (3)
_{}_{} and _{}, _{} if _{} (4)
These two
equations are also necessary when extrapolating the probability figures to the
boundary of the domain.
Figure 6 shows
the failure probability contour based on the 159 historical experimental data
points. A new wheel can be simulated in the same way described in steps (1) and
(2) to obtain its (s_{m}, s_{a})_{1%}, then the failure probability of this new wheel can
be read directly from the failure probability contour. After obtaining the experimental
test result, this wheel became a “historical experimental data.” The failure
probability contour can then be updated to include this extra historical
experimental data.
Figure 6. The failure probability contour
This probability
contour provides quantitative information on “how likely a new wheel is going
to fail” based on the historical experimental data. There is also a mechanism
to update the failure probability contour to improve the prediction as the
number of historical experimental data points increases.
3.
Redesign strategies for
different types of wheels
From a series of
interviews of several senior engineers in the local wheel manufacturer, it was
concluded that engineers commonly use 5 redesign strategies when a new wheel
fails in the fatigue test:
A.
Lathe spokes
B.
Modify the “xfactor”, which is the space of caliper clearance (or
brake clearance) that can be reduced
C.
Raise the front surface of the center of the wheel
D.
Change the taper angle of the spoke
E.
Adjust the depth of the weight reduction hole
Figure 7 marks
the positions of modifications on the wheel in these 5 common strategies. Basically,
strategies A and D change the shape of the spoke, strategies B and C change the
stiffness of the center of the wheel, and strategy E adjusts the depth of weight reduction holes. Table 1 lists the various
discrete values for each strategy that are commonly used by the wheel
manufacturer. More than one strategy can be used simultaneously when
redesigning a wheel.
Table 1. The discrete values for each strategy that
are commonly used by the wheel manufacturer
A: Lathe
spokes

E: Adjust the weight reduction hole

A1:
Decrease 2mm

E1:
Decrease 2mm in depth

A2:
Decrease 4mm

E2:
Decrease 4mm in depth

B: Modify the “xfactor”

E3:
Decrease 2mm in width

B1:
Decrease 2mm

E4:
Increase 2mm in
depth

C: Raise the front
surface of the center

E5: Increase 4mm in depth

C1: Increase 2mm


D: Change the taper angle of the spoke

D1: Increase _{}

Figure 7. Five redesign strategies commonly used
by wheel manufacturers
The geometry of
the wheel also restricts the selection of different strategies. Table 2 shows
three types of wheels with different number of spokes and therefore with different
widths of spokes. Type 1 wheels have less than 5 spokes. The spokes are wide
enough to contain a large weight reduction hole. Thus, strategy E is often used for Type 1 wheels. Type
2 wheels have 610 spokes, which are narrower than those of Type 1 wheels. Their
weight reduction holes are smaller and cannot be redesigned freely. Type 3
wheels have many thin spokes and do not have any weight reduction holes. Thus,
strategy E cannot be used at all.
Table 2. Three types of the wheel
Type

Wheel

1

1
2
3

2

1
2

3

1

Six sample
wheels (three of Type 1, two of Type 2, and one of Type 3) were chosen to test
the effect of the 5 strategies using the failure probability model described in
the previous section. The first column in Table 3 shows the initial failure
probabilities of each wheel. The differences in failure probability (DP%) and in weight (DW%) after applying the
strategies are listed. Note that in Table 3, some of the strategies are not
effective in reducing the failure probability (DP%>0), and some others will increase
the weight of the wheel (DW%>0).
Table 3. The change in failure probability and
weight using redesign strategies A1~E5


A1

A2

B1

C1

D1

E1

E2

E3

E4

E5

Type 11
82.7%

DP%

1.1

5.3

20.8

13.1

1.4

11.9

49.4

12.8

10.4

0.9

DW%

1.0

1.7

0.9

2.1

0.2

0.5

1.2

0.4

0.7

1.2

Type 12
89.6%

DP%

5.1

10.2

10.1

31.9

17.1

30.4

36.4

10.9

11.9

12.1

DW%

0.5

1.1

0.4

5.0

1.1

1.0

2.2

0.4

1.1

2.2

Type 13
58.2%

DP%

2.3

7.9

18.4

30.4

7.3

10.0

191.

10.7

16.8

54.2

DW%

1.0

1.7

0.8

1.8

0.5

0.3

0.9

0.3

0.5

1.2

Type 21
90.4%

DP%

1.1

10.8

17.7

29.9

12.7

47.1

57.3

20.4

4.9

2.1

DW%

0.5

0.9

0.6

2.8

1.0

0.9

1.4

0.3

0.4

0.4

Type 22
63.1%

DP%

33.4

22.3

15.9

23.7

6.3

NA

NA

NA

15.0

15.1

DW%

1.0

1.8

0.8

1.5

0.7

NA

NA

NA

0.3

0.5

Type 31
55.4%

DP%

4.9

6.4

79.9

38.9

36.8

NA

NA

NA

NA

NA

DW%

1.4

1.9

1.6

2.8

1.0

NA

NA

NA

NA

NA

In the
simulations, Strategies B, D, and E1~E3 are effective in
reducing the failure probability for all types of wheels (Strategies B and C have negative effects on Type 31 wheel), with an increase in
their weights. On the other hand, Strategies A, E4 and E5 reduce the weight of the wheels, but
have negative effects on failure probability. These strategies can be used when
a wheel is overdesigned.
Empirically, choosing
an appropriate strategy to use often depends on the location of cracks occurred
in the fatigue test. For example, if cracks occur near the spoke, the engineer
will change the taper angle to increase the stiffness (Strategy D). Engineers may reduce the size of the
weight reduction holes simultaneously. Similar idea can be used in choosing
redesign strategies using the simulation results. For example, Figure 8 shows
the simulated stress distribution of Type 11 wheel. High stress occurs at the
end of the weight reduction hole near the xfactor. To reduce the
failure probability, Strategy B
(modify the xfactor) and Strategy E
(adjust the weight reduction hole) can be effective. The simulation results
also confirm that Strategies B1 and E2 can reduce the failure probabilities
of Type 11 wheel significantly.
Figure 8. The stress contribution of the Type 11
wheel from simulations
From the
discussion above, engineers should consider the probability of failure, type of
the wheel (the width of the spokes, type of weight reduction hole), and location
of high stress in the fatigue test simulation using the finite element model, and
then manipulate these 10 redesign strategies (A1~E5) trying to reduce
the probability of failure or to reduce the weight of the wheel. This decision
process is integrated into a fuzzylogic based automated system for redesigning
the aluminum disk wheel.
Figure 9 shows
the flowchart of the automated design system for redesigning the aluminum disk
wheel developed in this research. An initial wheel design is simulated and both
the high stress location and failure probability of the wheel are found by the probability
model described in the previous section. The failure probability, the location
of high stress, and the two statements (width of the spoke and size of the
weight reduction hole) defined by the user are input to the fuzzylogic engine.
Figure 9. The flowchart of an automated design
modification system
There are two decision
stages in the fuzzylogic engines. First, the engine takes four inputs (the
failure probability, the location of high stress, width of the spoke, and size
of the weight reduction hole) and generates the locations of the wheel (such as
spoke, reduction hole and center of the wheel) that need to be redesigned. In
the second decision stage, the user defines the design restrictions of the
wheel where the geometry cannot be redesigned. The fuzzylogic engine takes the
output from the first stage and design restrictions, and then maps to the 10
redesign strategies A1~E5. The output redesign strategies are
then applied to the wheel and the procedure is repeated until the system
indicates that no redesign is necessary.
Table 4 lists
the linguistic terms of inputs. Figure 10 shows the membership function and linguistic
terms of the failure probability. Typical triangleshaped and trapezoidshape
membership functions are used. The linguistic terms of the failure probability
is “very good” (0%20%), “good” (10%45%), “acceptable” (35%60%), “bad”
(45%80%), and “very bad” (60%100%).These values are determined by the
experience of the senior engineers of the local manufacturer. From the
experience of the local manufacturer, 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 fatigue test. These trials are
also the major cause of the development cost and time.
Figure 10. The membership function of failure
probability
Table 4. The linguistic terms of inputs
Four inputs

Failure probability

Width of spoke

Size of weight reduction hole

Location of high stress

Linguistic terms

Very good

Thin

None

Two sides of weight reduction

Good

Normal

Narrow and shallow

Near xfactor

Acceptable

Large

Wide and deep

Between two spokes

Bad

NA

Very bad

Given a
predicted failure probability P_{fail}
of a new wheel, the “expected value” of the number of trials of making and
revising the die can be estimated. For example, if the failure probability of a
new wheel P_{fail}=0.2, the
chance for passing the fatigue test with just one trial is (1 P_{fail}). If the wheel does not
pass in the first trial and is redesigned using the procedure described in this
paper, the failure probability P’_{fail}
after redesign should be less than P_{fail}.
The chance for passing the fatigue test with 2 trials is P_{fail}×(1 P’_{fail})_{}P_{fail}, if
the higher order terms are neglected for small P_{fail}. Therefore the expected value of the number of trials
of making and revising the die is approximately 1×(1 P_{fail})+2×P_{fail}=1.2,
which is a very good number for the local wheel manufacturer. Therefore failure
probability 0~20% is defined as “very good”.
Values of
failure probability are mapped to the linguistic terms through membership
functions in the fuzzification step. The other inputs are also mapped in the
same way. After the crisp inputs are mapped to the linguistic terms, inference
rules are applied to determine the output in the step 1 fuzzy engine. The rules
are constructed from experienced engineers’ expertise, with help from computer
simulation results listed in Table 3. These rules are written in following “IFTHEN”
form, and Table 5 lists the linguistic terms of the outputs of the step 1 fuzzy
engine.
Rule 1: IF
the failure probability is “Good”, THEN the geometry of spoke is “No change”,
AND the geometry of weight reduction is “No change”, AND the geometry of the
center is “No change”.
Rule 2: IF
the failure probability is “Very good”, AND the width of spoke is “Thin”, AND
the size of weight reduction hole is “None”, AND the location of high stress is
“Between two spokes”, THEN geometry of spoke is to “Reduce stiffness slightly”.
Rule 3: IF
the failure probability is “Very good”, AND the width of spoke is “Normal”, AND the size of
weight reduction hole is “Narrow and shallow”, AND location of high stress is “Near
xfactor”, THEN the geometry of
weight reduction hole is to be “Enlarged slightly”.
………
Table 5. The linguistic terms of first step outputs
1^{st}step outputs

Geometry of spoke

Geometry of weight reduction hole

Geometry of the center

Linguistic terms

Reduce stiffness
greatly

Enlarged greatly

No change

Reduce stiffness
slightly

Enlarged slightly

Increase stiffness
slightly

No change

No change

Increase stiffness
greatly

Increase stiffness
slightly

Reduced slightly


Increase stiffness
greatly

Reduced greatly

In the system
constructed in this research, there are 19 such rules in the first step fuzzy
engine. These redesign rules are not generally true. They are specific to the
production line of the local wheel manufacture. These rules are only useful
(and confidential) to the local wheel manufacturer and thus are not listed
here.
In the second
decision step, the fuzzylogic engine takes the output from the first step and
adds the design restrictions, and then maps to the 10 redesign strategies
A1~E5. The linguistic terms of design restrictions are “No restriction”, “xfactor cannot be changed”, and “Geometry
of the center cannot be changed”. The rules in the step 2 fuzzy engine are as
follows:
Rule 1: IF the geometry of spoke is to “Reduce stiffness
slightly”, AND the geometry of weight reduction hole is “No change”, AND the
geometry of the center is “No change”, AND design restrictions are “No
restriction”, THEN “Lathe spoke 2mm”.
Rule 2: IF the geometry of spoke is to “Increase
stiffness slightly”, AND the geometry of weight reduction hole is “No change”,
AND the geometry of the center is to “Increase stiffness slightly”, AND design
restrictions are “Geometry of the center cannot be changed”, THEN “Increase the
taper angle is 3^{o}”.
Rule 3: IF the geometry of spoke is to “Increase stiffness
greatly”, AND the geometry of weight reduction hole is to be “Reduced greatly”,
AND the geometry of the center is to “Increase stiffness slightly”, AND design
restrictions are “xfactor cannot be
changed”, THEN “Increase the taper angle is 3^{o}”, AND “Raise the
front surface of the center is 2mm”,
AND “Decrease the weight reduction hole 4mm in depth”.
………
There are 14 such
rules in the second step fuzzy engine. Again these redesign rules are specific
to the production line of the local wheel manufacture, and thus are not listed
here.
According to
inputs and collected fuzzy rules, an automated design modification system
generates a crisp output of redesigned strategies that is shown in Table 1.
Engineers use this system to obtain the redesigned strategies A1~E5, and try to
reduce the probability of failure and to prevent over design.
5.
Application examples
In this section,
the automated design modification system is applied on three aluminum disk
wheel design examples to validate the effectiveness of the system.
Case I
Table 6 shows the iteration results of Case I wheel. The initial failure probability of cornering fatigue
simulation is 43.5%, which needs to be reduced further. The width of the spoke
and size of the weight reduction hole of the wheel, and the failure probability
and high stress location from the simulation are input to the automated design
modification system. The system suggests strategy D1 (increase the taper angle
of the spoke by 3^{o}) to modify the wheel.
Table 6. The iterations results of Case I wheel

Cornering

Initial failure probability

43.5%

Initial weight

8.63 Kg


1^{st} iteration: Using strategy D1 to
redesign the wheel


Failure Probability

37.3%

Weight

8.77 Kg


2^{nd} iteration: Using strategy D1 to
redesign the wheel


Failure Probability

33.8%

Weight

8.82 Kg


As shown in
Table 6, the failure probability reduces to 37.3% after the redesign strategy
D1 is applied, and the weight of the wheel increases from 8.63kg to 8.77kg. This result is input
to the system again, and the system suggests strategy D1 again. In the second
iteration, the failure probability further reduces to 33.8%, but the weight of
the wheel increases to 8.82 Kg.
In the 3^{rd}
iteration, the system suggests no design change. According to the suggestions
by the automated design modification system, the taper angle of the initial
wheel was increased by 6^{o}. Later in a physical test, the modified
design actually passed the cornering fatigue test.
Case II
Table 7 shows the iteration results of Case II wheel. The initial failure probability of cornering fatigue
simulation is 80.3%, which needs to be reduced further. The width of the spoke
and size of the weight reduction hole of the wheel, and the failure probability
and high stress location from the simulation are input to the automated design
modification system. The system suggests strategy C1 (raise in the front surface of the center by 2mm), D1 (increase the taper angle of the spoke
by 3^{o}), and E4 (increase in the depth of the weight reduction hole
by 2mm) simultaneously to modify
the wheel.
Table 7. The iterations results of Case II wheel

Cornering

Initial failure probability

80.3%

Initial weight

7.43 Kg

1^{st} iteration: Using strategy C1, D1,
and E4 to redesign the wheel

Failure Probability

51.3%

Weight

7.80 Kg

2^{nd} iteration: Using strategy C1 to
redesign the wheel

Failure Probability

26.4%

Weight

8.17 Kg

As shown in
Table 7, the failure probability reduces to 51.3% after applying redesign
strategies C1, D1, and E4. This result is input to the system again, and the
system suggests strategy C1. In this second iteration, the failure probability
further reduces to 26.4%.
In the 3^{rd}
iteration, the system suggests no design change. According to the suggestions
by the automated design modification system, the front surface of the center of
the initial wheel was increased by 4mm,
the taper angle of the initial wheel was increased by 3^{o}, and the
depth of the weight reduction hole of the initial wheel was increased by 2mm. The modified design actually passed
the physical test.
Case III
Table 8 shows the iteration results of Case III wheel. The initial failure probability of cornering fatigue
simulation is 24.7%, and this wheel could be overly designed. The width of the
spoke and size of the weight reduction hole of the wheel, the failure
probability, and high stress location from the simulation are input to the
automated design modification system. The system suggests strategy E4 (increase
depth of the weight reduction hole by 2mm) to modify the wheel.
Table 8. The iterations results of Case III wheel

Cornering

Initial failure probability

24.7%

Initial weight

7.17 Kg


1^{st} iteration: Using strategy E4 to
redesign the wheel


Failure Probability

24.3%

Weight

7.10 Kg


2^{nd} iteration: Using strategy E4 to
redesign the wheel


Failure Probability

26.4%

Weight

7.03 Kg


As shown in
Table 8, the failure probability reduces to 24.3% after redesign strategy E4 is
applied, and the weight of the wheel reduces from 7.17kg to 7.10kg.
This result is input to the system again, suggesting strategy E4. In the second
iteration, the failure probability increases to 26.4%, but the weight of the
wheel reduces to 7.03 Kg.
In the 3^{rd}
iteration, the system suggests no design change. According to the suggestions
by the automated design modification system, and the depth of the weight
reduction hole of initial wheel was increased by 4mm. The weight of the modified design is reduced by 0.14kg, and it also passed the physical
test.
6.
Conclusions
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 and engineers’ experience on how to
modify the design are very valuable to manufacturers and should be systematically
preserved and utilized.
This paper first
presents a procedure that predicts the fatigue failure probability of aluminum
disk wheels based on historical test data. Then an automated design
modification system that integrates the engineering analysis results and
engineers’ common redesign strategies with a fuzzy logic algorithm in order to
provide effective strategies to reduce the failure probability of aluminum disk
wheels is presented. This automated design modification system is constructed
and validated using 3 real wheel design examples.
This automated
design modification system is now being used in a local wheel manufacturer. The
system provides effective design modification suggestions, which help the
engineers to reduce the failure probability or reduce weight of the aluminum
disc wheels. Using this system, historical test results and designers’ previous
experience in how to modify the aluminum wheel to pass the fatigue test can be
properly transformed and utilized by all designers.
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