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Higher Level Physics Internal Assessment
The Relationship Between Hole Radius and Fatigue Life in Aluminium Eyeglass Hinge Under Continuous Bending Stress
Zirui Guo SPH4U9-1: HL Physics
Mr. James Weekes Nov 23, 2018
Introduction
I wear glasses but I have always been troubled with my eyeglass frame breaking as shown in Figure 1. It
is always the hinge between the frame front and temple that is broken as shown in Figure 2.
Figure 1. (Left) My broken eyeglass frame. Figure 2. (Right) A close-up view of the broken joint.
Sometimes it broke because a basketball hit it, but more often it just suddenly broke without any
enormous external force applied. I realized it is the long-term process that the aluminium hinge (parts
around the screw) became fatigued and eventually broke. I became curious about what caused the fatigue.
It turns out that the rotational motions (closing and opening in an acceptable range) performed
horizontally as illustrated in Figure 3 do not affect the fatigue. It is the motions applied in the vertical
direction as illustrated in Figure 4 that cause the fatigue, which can happen when taking off the glasses
with one hand or holding the end of the temples. Repetitions of such action cause bending stress applied
on the hinge and over time the aluminium around the screws in the hinge will fatigue and fail. I also
noticed that the eyeglass frames I own have different radii of screws and I wonder if the different radii can
affect the fatigue process of the hinge. If there is a radius that can endure most cycles of fatigues, then it
can be the optimal radius to make the most durable eyeglasses.
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Figure 3. (Left) Horizontal motions when opening and closing the temples.
Figure 4. (Right) Vertical force applied when taking off eyeglasses.
This investigation will examine the effect of hole radius in aluminium eyeglass hinges on the fatigue life
(number of stress cycles withstood until failure). A hypothesis is proposed, and an experiment is designed
and performed to be evaluated with detailed quantitative and qualitative analysis supported by theories
and calculations.
Background
Metal Fatigue
Metals have elastic properties, varied by the arrangement and strength of the bonds between the atoms in
the material; Deformation caused by elasticity is reversible in a certain limit (elastic limit), and the
material will return to its initial shape. (Burnley, 2018) However, repeatedly applied loads will cause the
formation of microscopic cracks, or cracks to be initiated from notch under localized stress, and overtime
propagate suddenly, and the structure will fracture. (COSMOL)
Stress Concentration
The presence of a notch created by the environment or during the machinery process quickly draws the
stress to concentrate instead of being uniformly distributed. As shown in Figure 5, when the metal is
deformed, the rows of atoms are forced to move, which leads to an alteration in structure, breaking bonds
between atoms. This results in small cracks forming along the metal’s surface, cracks which eventually
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migrate deeper inside the metal with each subsequent bend. With time the metal will become so
compromised by the cracks that breakage occurs. (O'Keefe, 2012)
Figure 5. (Left) Atoms being forced to move and create local stress concentration. (O'Keefe, 2012)
Figure 6. (Right) Stress concentration distribution around a circular hole in a plate. (McGinty)
The geometry of structure significantly affects the fatigue life. As shown in Figure 6, in a circular hole
structure, at the hole’s edge there is 3-time concentration compared to one diameter away. The stress
dissipates away as it becomes away from the hole. This illustrates that there is more stress near the edge
of the circle and there is a higher chance that local stress will initiate a crack. (McGinty)
The Relationship between Hole Radius and Stress Ratio
The Stress Concentration Factor, Kt, is the ratio of maximum stress at a hole, fillet, or notch, (but not a
crack) to the remote stress. . (McGinty) The stress concentration around a circular hole is K t = σnomσmax
directly affected by , the ratio between the hole diameter and plate or strip width, as shown in Figure 7.dW
As increases, the stress concentration decreases, as shown in Figure 8. This can be explained by thedW
larger circumference of the hole which relieves the stress to more area. The larger the radius, the less
stress concentration there will be.
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Figure 7. (Left) A finite width plate with a circular hole in the centre. (McGinty)
Figure 8. (Right) Stress concentration versus the ratio between diameter and width. (McGinty)
Bending Stress on a cantilever beam
The cantilever beam is a projecting beam fixed only at one end (MATHTAB), which is the type of beam
in an eyeglass temple. As shown in Figure 9, when a cantilever is being bent by a force applied from the
top, tension and compression occur on top and bottom of the neutral axis. Also, note that bending stress
increases linearly away from the neutral axis. The maximum bending stress is , where M isσb.max = IcMc
the bending moment, c is the centroidal distance of the cross section, and Ic is the centroidal moment of
inertia (MechaniClac). Figure 10 shows equations to calculate property related to a cantilever beam.
Figure 9. (Left) A cantilever beam under bending stress. (MechaniCalc)
Figure 10. (Right) Equations to calculate cantilever beam property. (MechaniCalc)
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Hypothesis
The hypothesis is that the radius with the longest fatigue life is not the largest or the smallest one. For the
smallest radius, although there is more metal left, the stress is more concentrated and can cause an initial
crack to propagate faster. For the largest radius, although stress is less concentrated, there is not much
metal to withstand the cycle. This investigation hypothesizes that there is a radius in between that can
withstand most stress cycles. The number of stress cycles that the structure can withstand increases and
then decreases as the radius increases.
Experiment Design
System Design
To examine the effect of different hole radius in aluminium strips on fatigue life, and to simulate the
situations of eyeglass temple, an experiment is designed after several early prototypes (Appendix A). A
full list of materials and tools can be found in Appendix B. Full engineering process can be found in
Appendix C. Five distinct diameters of holes are examined: ″, ″, 1″, ″, ″ (1.27cm, 2.22cm,21
87 1 8
1 1 41
2.54cm, 2.86cm, 3.18cm). The holes are drilled on 2.00 ± 0.05 ″ (5.08 ± 0.13cm) ×8″ (20.32cm) ×0.025″
(0.064 cm) aluminium metal strips that are cut. Each diameter is given a group number from 1-5, 1 being
the smallest radius and 5 the largest. Three metal strips are drilled with each diameter, numbered from 1-3
and placed after the group number and a decimal, as shown in Figure 11. The holes are drilled at 6.00 ±
0.05cm from the top and centred in width to allow room to be clamped on a table.
Figure 11. Numbered metal strips drilled with the corresponding diameter prepared for the experiment.
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The metal strip simulates the aluminium temple of eyeglasses and hole simulates the hole for the screws
in the hinge. A mechanism is designed to simulate the repeated bending stress caused by taking off
glasses.
Figure 12. (Left) Schematic illustration of the complete set up of the experiment.
Figure 13. (Right) A complete set up of the experiment.
As shown in Figure 12, the aluminium strip (1) is clamped by a trigger clamp (3) on a wooden desk (2)
where exactly half of the hole is on the desk. There is an iron G-clamp (4) clamped on the bottom of the
trigger clamp, and its usage will be explained later. Next to the metal, there is a woodwork vice (5)
clamped on the desk and holds a drill (6) equipped with an eccentric gear (7) that can repetitively hit the
metal strip. The drill is connected to a variac transformer (8) for speed control. Also clamped on the desk,
there is an iron vice (9) that is used to flatten metal stripes before experiments. Figure 13 demonstrates the
complete set up before an experiment.
Procedure
The eccentric gear during the experiment repeatedly hit the metal at the location of 5.50 ± 0.05 cm from
the bottom of the strip. The point of contact is horizontally centred. There are usually three stages until
the strip failure happens (a crack completely forms from one side to the circle) and some equipment needs
to be adjusted to accommodate the three situations.
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In the first stage, the iron G-clamp on the trigger clamp faces down and has no contact with the metal
strip. As shown in Figure 14, the strip will be hit by the eccentric wheel. Because of elasticity of the sheet
metal, it will bounce back. This process of being hit and bounce back on one side is the starting condition
and when it no longer bounces, as the elasticity on one side vanishes, time is recorded by a timer.
In the second stage, the strip is flipped, so the eccentric wheel hits this other side (the side without
number). Due to existing elasticity in this other direction, the strip will still bounce back, as shown in
Figure 15. However, after some time, it no longer bounces back thus all elasticity in the strip is consumed,
and a timer records the time.
Figure 14. (Left) First stage. Figure 15. (Right) Second stage.
In the third stage, the G-clamp is adjusted to a location that the top of the clamp is 4.00 ± 0.05 cm from
the table surface (where the top end of the strip is), as shown in Figure 16. This is necessary because,
without any elasticity, the strip will not come back after being hit. As shown in Figure 17, with the clamp
on the bottom, the strip hit the clamp after being hit by the eccentric gear, and the reaction force from the
clamp forces the strip to bounce back, thus completing a fatigue cycle. Fatigue failure usually happens in
this stage, and a timer records the time of failure. However, fatigue sometimes happens in the previous
two stages without entering the next stage and the time of failure will be recorded as well.
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Figure 16. (Left) G-clamp is lifted to be 4.00 ± 0.05 cm from the table surface.
Figure 17. (Right) Metal strip hits the clamp and is forced to bounce back.
List of Variables
Table 1 shows the independent and dependent variables investigated in this investigation. Table 2 shows a
list of controlled variables, the importance for them to be controlled, and the means of control.
Independent variables
The radius of the hole is the independent variable. In practice, it would be the diameter of the drill bit used. Five different diameters are assessed. Various radii mean that there are different cross-sectional areas and aluminium left in the test strips. The change in hole size changes the stress concentration.
Dependent variables
The cycles that the aluminium strip can withstand until fatigue failure is the dependent variable. This number includes the number of cycles in all three stages (deformation on one side, deformation on another side, fatigue cycles). The cycles are recorded by a timer and calculated by multiplying revolutions per second observed from a video.
Table 1. The independent and dependent variables.
Variable controlled Importance to be controlled Means of control
Force exerted by the eccentric wheel which is driven by a drill; influenced by the power supplied, the position of the drill, and the position of the test strip.
Different drill speed among trials can affect the speed of cycles applied in a given time. Variations in the drill location relative to the metal strip affect the amount of force applied, and thus the bending stress exerted.
Use a variac transformer to power the drill thus maintaining the same potential difference supplied and speed. To secure location, clamp the drill in woodwork vice and clamp the metal strip where the edge of the desk lies halfway in the hole.
Iron G-clamp's location in stage 3.
Variation in G-clamp locations could affect the bending force exerted on the test strip.
G-clamp is always clamped on the trigger clamp, 4.00 ± 0.05 cm to the desk surface.
Table 2. The controlled variables and the importance and means of control.
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Experiment Results
Fatigue Time
Table 3 shows the raw data collected using the designed experiment procedure. An uncertainty of ± 2
seconds is chosen rather than using the centisecond uncertainty of the timer. This is because that human
experimenter takes about 2 seconds to observe the end of stage and press stop on a timer.
Strip No. Stage 1 End (± 2") 1 Stage 2 End (± 2") Fatigue Failure (± 2")
1.1 0:07′38"13 N/A 0:08′51"54
1.2 N/A N/A 0:04′53"46
1.3 0:04′53"84 0:05′24"18 0:11′43"93
2.1 0:06′08"07 0:06′34"96 0:15′20"56
2.2 0:05′31"98 0:05′51"71 0:09′53"69
2.3 0:11′28"71 0:12′00"75 0:19′57"70
3.1 0:05′44"12 0:05′51"03 0:07′07"76
3.2 0:03′08"07 0:03′53"69 0:32′57"97
3.3 0:06′59"61 N/A 0:09′20"51
4.1 0:06′02"81 0:06′24"50 0:40′28"93
4.2 0:02′27"12 0:04′94"72 0:42′33"70
4.3 0:06′34"31 0:13′19"18 1:15′50"62
5.1 0:07′45"68 0:12′12"11 0:35′05"88
5.2 N/A N/A 0:04′23"53
5.3 0:05′18"22 N/A 0:13′24"93
Table 3. Raw data recorded on the timer.
The raw data is recorded in hours, minutes, seconds, and centiseconds format and needed to be transferred
to only seconds, as shown in Table 4. A Python program (Appendix D) is written to translate.
Strip No. Stage 1 End (± 2.00s) Stage 2 End (± 2.00s) Fatigue Failure (± 2.00s)
1.1 458.13 N/A 531.54
1.2 N/A N/A 293.46
1.3 293.84 324.18 703.93
2.1 368.07 394.96 920.56
1 Although the instrumental uncertainty is 1 centisecond, the actual uncertainty can reach about 2 seconds due to human reaction time to notice the fatigue, process information, and press the button.
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2.2 331.98 351.71 593.69
2.3 688.71 720.75 1197.70
3.1 344.12 351.03 427.76
3.2 188.07 233.69 1977.97
3.3 419.61 N/A 560.51
4.1 362.81 384.50 2428.93
4.2 147.12 334.72 2553.70
4.3 394.31 799.18 4550.62
5.1 465.68 732.11 2105.88
5.2 N/A N/A 263.53
5.3 318.22 N/A 804.93
Table 4. Raw data converted into seconds using a Python script (Appendix D).
The timestamp endpoints in seconds are then split into the length of three stages, as shown in Table 5. The
centiseconds are no longer relevant due to the considerable uncertainty; however, they are still carried for
calculations in further steps.
Strip No. Stage 1 (± 2s) Fatigue Failure Stage 2 (± 4s) Fatigue Failure Stage 3 (± 4s) Fatigue Failure
1.1 458 73 ✕ 0
1.2 293 ✕ 0 0
1.3 294 30 380 ✕
2.1 368 27 526 ✕
2.2 332 20 242 ✕
2.3 689 32 477 ✕
3.1 344 7 77 ✕
3.2 188 46 1744 ✕
3.3 420 141 ✕
4.1 363 22 2044 ✕
4.2 147 198 2219 ✕
4.3 394 405 4081 ✕
5.1 466 267 1374 ✕
5.2 264 ✕ 0 0
5.3 318 487 ✕ 0
Table 5. Time of three stages and indication of fatigue failure for all test strips.
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Notice that most of the fatigue happen in stage 3, where the elasticity of the metal strip has vanished.
Also, note that most fatigue failure that doesn’t happen in stage 3 belongs to smallest and largest radii.
Calculate RPS and Fatigue Life
To count the fatigue life, the rotation per minute must be found. Due to the different settings in different
stages, the revolutions per seconds of eccentric gear is different.
A Google Pixel XL is used to record a sample of video footage in different stages under 240 frames per
second. The video is then analyzed by slowing down to a quarter speed in video player so that the
revolutions can be counted. The length of video footage is acquired through an audio editor. The original
video files can be found in Appendix E. The revolutions is then calculated using:
evolutions per second / rev r · s−1 = video footage lengthnumber of revolutions observed
The data and revolutions per second for all three stages can be found in Table 6.
Stage Video footage length / s Number of cycles observed / rev Revolutions per second / rev•s-1
1 1.79 ± 0.01 21.0 ± 0.5 12.00 ± 0.34
2 2.14 ± 0.01 24.0 ± 0.5 11.00 ± 0.29
3 2.88 ± 0.01 34.0 ± 0.5 12.00 ± 0.21
Table 6. Video length, number of cycles observed, revolutions per second observed for three stages.
With the calculated revolutions per second and time of each stage, the revolutions or fatigue cycles
completed during different stages can be calculated with the equation:
umber of cycles time evolutions per secondN = × r
Table 7 demonstrates the number of cycles completed in each stage using the equation above processed
by a Python script. (Appendix F). The number of cycles is kept to the nearest integer.
Strip No. Stage 1 Cycles Stage 2 Cycles Stage 3 Cycles Fatigue Failure Stage Total Fatigue Life Cycles
1.1 5497 ± 179 807 ± 65 0 2 6305 ± 245
1.2 3521 ± 123 0 0 1 3521 ± 123
1.3 3526 ± 123 333 ± 52 4559 ± 127 3 8419 ± 304
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2.1 4416 ± 149 295 ± 51 6307 ± 158 3 11019 ± 359
2.2 3983 ± 136 217 ± 49 2903 ± 98 3 7104 ± 285
2.3 8264 ± 258 352 ± 53 5723 ± 148 3 14340 ± 459
3.1 4129 ± 141 76 ± 46 920 ± 64 3 5126 ± 251
3.2 2256 ± 87 501 ± 57 20931 ± 414 3 23690 ± 559
3.3 5035 ± 166 1549 ± 84 0 2 6585 ± 251
4.1 4353 ± 147 238 ± 50 24533 ± 477 3 29125 ± 674
4.2 1765 ± 74 2173 ± 101 26627 ± 513 3 30566 ± 689
4.3 4731 ± 158 4453 ± 161 48977 ± 905 3 58162 ± 1224
5.1 5588 ± 182 2931 ± 121 16485 ± 336 3 25005 ± 640
5.2 3162 ± 113 0 0 1 3162 ± 113
5.3 3818 ± 132 5353 ± 185 0 2 9172 ± 317
Table 7. Individual stage and total fatigue life cycles calculated through a Python script (Appendix F).
Calculate RPM and the Force
Given the revolutions per second, the revolutions per minute can be calculated, and so can the angular
velocity of the eccentric wheel (when it is not hitting metal). When the side of eccentric gear that has the
largest radius hits the metal, the gear is completely stopped by the metal, but the drill keeps spinning to
accelerate the gear to its angular velocity by accelerating. During the acceleration, force is generated to hit
the metal downwards, and that tangential force can be calculated. With the angular velocity, the tangential
acceleration can be found using information of the eccentric gear and thus the applied force can be
calculated. All data calculated can be found in Table 8.
Revolutions per Minute / rev•min 60 Revolutions per Second / rev•s −1 = × −1
hange in Angular V elocity Δω / rad •s Revolutions per Minute / rev•s / rad •s • s •rev C −1 = −1 × 602π −1 −1
The expression of tangential acceleration is: at = ΔtΔv = Δt
Δ(rω) = Δtr×Δω
The longest distance from the screw to the gear edge in the eccentric gear
2.60 cm 0.05 cm 0.0260m 0.0005m r = ± = ±
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Time of the gear hitting the strip, analyzed in slow motion using video in Appendix E
t T (time to complete one revolution) Δ = 51 = 1
5f = 15 × revolutions per second
As shown in Figure 18 and 19, the mass of gear contains the gear itself and shaft and lock plates parts.
The force applied can then be calculated using Newton’s Second Law.
a (mass of gear mass of shaf ts and lock plates) (23.16g ± 0.01g 3.24 ± 0.01g) F app = m t = + × at = + 1 × at
36.4g .02g) 0.0364kg .00002kg)F app = ( ± 0 × at = ( ± 0 × at
Figure 18. (Left) Mass of the plastic gear. Figure 19. (Right) Mass of the shafts, collars and lock plate
Stage Revolutions per second / rev•s-1
Revolutions per minute / rev•min-1
Change in angular velocity Δω / rad •s-1
Change in time Δt /s
Tangential acceleration at / m•s-2
Force / N
1 12.00 ± 0.34 720 ± 20 75.4 ± 2.1 0.01600 ± 0.00043
122.52 ± 9.37 4.45 ± 0.34
2 11.00 ± 0.29 660 ± 17 69.1 ± 1.8 0.01800 ± 0.00047
99.81 ± 7.12 3.63 ± 0.26
3 12.00 ± 0.21 720 ± 12 75.4 ± 1.3 0.01600 ± 0.00028
122.52 ± 6.61 4.45 ± 0.24
Table 8. RPS, RPM, change angular velocity change, change in time, tangential acceleration and force
applied by the eccentric gear.
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Analysis
Process of Fatigue
During the experiment, the activities around the hole are monitored and notable ones are taken pictures.
These pictures help explain the process of fatigue in this structure. As shown in Figure 20, the hole drilled
before the experiment has rough edges due to the drilling process. These uneven edges give the possibility
for concentrated local stress to take place and eventually initiate a crack. Because the strip is bent only up
and down, the bending stress acts mostly on the edge of the table or along the horizontal line that passes
the origin of the circular hole. In Figure 21, a crack was initiated about that location due to repetitive
stress. Then over time, the crack propagated rapidly from edges of the circle to the edge of the strips as
shown in Figure 22, 23, 24. It was also noticed that the crack propagates less rapidly when moving away
from the circle, because of the decreasing stress concentration away from the hole described in the
background section. Eventually, the crack propagated entirely to the edge of the strip and a fatigue failure
happened instantaneously, and the metal strip broke on one side, as shown in Figure 25.
Figure 20. (Left) The hole drilled before the experiment on strip No 2.2. Notice the rough edges.
Figure 21. (Medium) A crack initiated at the edge of the circle on strip No. T1.
Figure 22. (Right) A crack propagated from the circular hole on strip No 1.1.
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Figure 23. (Left) A crack further propagated from the circular hole to edge of the strip on strip No.1.1.
Figure 24. (Medium) The crack propagated through most of the strip width on strip No. 1.3.
Figure 25. (Right) At the instance that the fatigue failure happened on strip No. T2.
Appendix G shows pictures of all metals after fatigue failure and demonstrates their patterns of cracks.
Stage Time Comparison
The time of all stages are shown in Figure 26. There is relatively no extensive difference in stage 1 time
because elasticity is the property of the metal, and the different surface area in different strips have some
but not large difference on the time to make the elasticity property disappear. Stage 2, generally, is short
because that most of the elasticity property has vanished after stage 1, and it takes only a little time to
remove the elasticity property in the other direction. Note that stage 3 is significantly longer than stage 1
and 2 in cases that stage 3 exists. As the diameter of the hole increases, more cases with stage 3 appear,
but then for the largest diameter, two cases do not have stage 3. In the cases with stage 3, as diameter
increases, length of stage 3 increases and then decreases after peaking in the fourth group. The cases
without stage 3 are usually the smallest or the largest diameter. There are two possible causes for that: the
drill bit used for the smallest diameter is very rough, and the drilling process of the largest diameter was
very challenging, thus creating more scratches and notches around the hole and make cracks easier to
initiate; or it matches up with the hypothesis: the case with small radius has more metal but larger stress
concentration to allow notch and crack to propagate in fewer cycles, and the case with large radius has
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smaller stress concentration, but not much metal and a crack can quickly propagate through the shorter
distance and make it break.
Figure 26. Numbers of cycles in 3 stages for all aluminium strips with a moving average trendline for
each stage. Due to limitations of graphing software, uncertainty is not shown. Used the data in Table 7.
Fatigue Life
The cross-sectional area of the hole at the edge of the table is:
(all in inches).025 2 ), where d is the diameter of the holeA = 0 × ( − d
onvert f rom square inches to cm .025 2 ) .4516, where d is the radius of the hole in inches C 2 : A = 0 × ( − d × 6
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Figure 27. Fatigue life versus cross-sectional area. Used the data in Table 7.
In Figure 27, the relationship between the hole radius, cross-sectional area, and the total fatigue life is
shown. As a hole diameter increases, the cross-sectional area decreases (note that the x-axis in the graph
is purposely arranged in decreasing order). The fatigue life increases and then decreases as a general
trend. The fourth largest diameter ( ″ or 2.86cm) has larger fatigue life than any other diameter. Thus1 81
radius about 28.13% of the strip length withstands most cycles of fatigue and could be considered an
optimal radius for the hinge hole. The smallest and largest diameter have the shortest fatigue life, because
of the two reasons previously stated. It might even be a combination of the two reasons: since the smallest
and largest hole are hard to drill, more scratches and notches are created in the process, making cracks
easier to form and propagates.
Sources of Errors and Uncertainties
1. The drill bits used can introduce uncertainty due to its difficulty to be controlled and hence lack
of high precision. The drilling process also introduces different initial crack patterns for each
strip, and in some case because of the imprecise operations, the high-speed spinning drill bit can
introduce a large amount of force on some part of the hole and introduce high local stress to
initiate a crack.
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2. The eccentric gear has lost some material on some of its teeth over many cycles of hitting
aluminium strips. This can affect the force applied on the strip due to a slight shift of its moment
of inertia.
3. Although the metal strips are acquired from the same sheet metal and stored together in the same
condition, sand and dust in the environment rubbed against the sheet metal can cause cracks on a
micro-level, thus affecting the fatigue process.
4. After the drilling process, because of the large force applied by the drill bit, some strips are not
flat anymore. Although they are flattened again with an iron vice before experiment, this
deformation affects the elasticity and internal structure of the strips, thus affecting all stages.
Limitations of the experiment
1. The experiment only records cracks from the hole to one edge of the strip. In a real eyeglass hinge
that has a disconnect around the screws, there will be other stresses and movements rather than
bending stress and bending up and down to let the other side break.
2. The aluminium strip simulating the situation is very thin (0.025″ or 0.064cm) to save time in
fatigue testings. In real eyeglass temples, the temple is a lot thicker, and the fatigue life will be
different.
3. Another type of common actions that damages eyeglass temples is to take off by rotating it
outwards using one of the temples as the centre. This action can be split into action in the
horizontal, vertical, and rotational axis, and is not simulated in this experiment.
4. The experiment only examined five different diameters of holes. More diameters can be
examined in the future, and thus help determine the optimal hinge hole radius
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Conclusion & Reflection
The investigation researched the nature of metal fatigue process in eyeglass hinges and performed sets of
experiments to simulate the situation to investigate the relationship between hole radius in aluminium
eyeglass hinge and fatigue life under continuous bending stress. The experiments matched the initial
hypothesis that as radius increases, the fatigue life increases and then decreases. The fatigue life, in this
case, is affected by both stress concentration, which decreases as the radius increases, and the amount of
metal left, which also decreases as the radius increases. The fatigue process includes three stages
involving breaking elasticity and then the fatigue process. Because of the complexity of this subject, only
part of the theoretical calculation is performed, and the evaluations are mostly based on experimental
results and high-level explanations. Though the results match the hypothesis, more samples should have
been examined because of the many error and uncertainty sources of the experiment.
Based on the data collected and analyzed, the optimal radius for the screws used in eyeglass hinge will be
a radius not smallest nor largest but in between to help it to withstand the maximum cycles of loads or
have the longest lifespan for the users. The best way to take off a pair of glasses is to hold the hinges on
both sides with both hands gently, thus minimizing the amplitude of bending stress. The circular hole
should be drilled or laser cut with the finest precision possible thus initial cracks can be reduced.
Preferably the hinge can be enclosed by a protective structure to prevent sands and dust from rubbing
against the joint, causing initial cracks.
This exploration also helped me realize the importance of physics, material science, and mechanical
engineering in everyday object design. Fatigue is not only crucial in aircraft and bridge designs, but it also
exists in every object around us and should be noticed. I enjoy the process of using physics to find
explanations to problems I have and use it to design solutions. This is my first experiment investigating
metal fatigue and I plan to investigate other types of fatigues in everyday objects.
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Reference
Burnley, P. (2018, April 24). Tensors, Stress, Strain, Elasticity. Retrieved from
https://serc.carleton.edu/NAGTWorkshops/mineralogy/mineral_physics/tensors.html
COMSOL. (n.d.). Material Fatigue. Retrieved from
https://www.comsol.com/multiphysics/material-fatigue
MATHTAB. (n.d.). Deflection Of Beams. Retrieved from
https://mathtab.com/page.php?page_id=29
McGinty, B. (n.d.). Stress Concentrations at Holes. Retrieved from
http://www.fracturemechanics.org/hole.html
MechaniCalc. (n.d.). Stresses & Deflections in Beams. Retrieved from
https://mechanicalc.com/reference/beam-analysis
O'Keefe, P. (2012, May 13). Mechanical Power Transmission – The Centrifugal Clutch and
Metal Fatigue. Retrieved from
http://www.engineeringexpert.net/Engineering-Expert-Witness-Blog/tag/metal-atoms
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Appendices
Appendix A - Early Designs
Figure 28. (Left) A motor-driven eccentric gear that repetitively hits the metal strip that can be located
above or underneath it. The motor was not powerful enough.
Figure 29. (Right) A drill secured in a wooden vice with a wooden eccentric wheel that repetitively hits
the metal clamped on a wooden desk. The metal has a hole to hang a mass to accelerate the fatigue
process, but the mass was too easy to fall off and the wooden wheel quickly lost contents.
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Figure 30. (Left) A motor-driven rubberband mechanism that repetitively hits the test strip. The force was
too large that the metal strips seriously deformed, which makes it not a good simulator of the problem.
Figure 31. (Right) A drill with a wooden eccentric wheel that repetitively hits the metal clamped on a
plastic desk. The plastic desk was too weak to support the woodwork vice and the drill, and
overtime the desk deformed.
Appendix B - Full List of Materials and Tools Used
2 - 24″×8″×0.025″ (60.96cm×20.32cm×0.064cm) Aluminium Sheet Metal
1 - Drill Press
5 - Drill Bits: ″, ″, 1″, ″, ″ (1.27cm, 2.22cm, 2.54cm, 2.86cm, 3.18cm)21
87 1 8
1 1 41
1 - Safety Goggle
1 - Oven Glove
1 - Drill
1 - Variac Transformer
1 - VEX Robotics 60T High Strength Gear
1 - VEX Robotics Pillow Block Lock Plate
1 - ″ (0.64cm) × 1.750″ (4.445cm) VEX Robotics Button Head Screw832
1 - ″ (0.64cm) VEX Robotics Keps Nut832
1 - ″ (0.24cm) VEX Robotics Hex Screwdriver332
1 - ″ (0.20cm) VEX Robotics Hex Key564
2 - 4mm VEX Robotics Shaft Collars
1 - 2" (5.08cm) VEX Robotics Square Bar Drive Shaft
1 - Wooden Table
1 - 18" (45.72cm) Trigger Clamp
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1 - 4" (10.16cm) Iron G-clamp
1 - Woodwork Vice
1 - Iron Vice
1 - MyChron Timer
1 - Calliper
1 - 20 cm Plastic Clear Ruler
1 - Yardstick
1 - Digital Balance
1 - Google Pixel XL Mobile Phone
1 - Paper Trimmer
1 - Scissor
1 - Permanent Marker
Electric Tape
Tissue Paper
Newspaper
Appendix C - Engineering Procedure
A series of procedures were designed and performed to create the experiments, including preparing the
metal strips, drilling holes, eccentric wheel design, and experiment set-up.
First, as shown in Figure 32, two 24″×8″×0.025″ (60.96cm×20.32cm×0.064cm) aluminium sheet metals
are prepared and marked with a permanent marker in 2 ± 0.05 inches (5.08 ± 0.13cm) interval measured
in a yardstick. Then, the sheet metals are cut along the labelled line by a paper trimmer into 24 aluminium
metal strips, as shown in Figure 33. A paper trimmer is used to ensure precision and minimize
deformation and notch-creation in this process.
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Figure 32. (Left) Marked aluminium sheet metals in 2-inch intervals.
Figure 33. (Right) Sheet metals being cut to metal strips using a paper trimmer.
15 metal strips are labelled with 1.1, 1.2, 1.3, 2.1, 2.2, 2.3, 3.1, 3.2, 3.3, 4.1, 4.2, 4.3, 5.1, 5.2, 5.3 with a
permanent marker. Three metals are labelled with T1, T2, T3 for testing. 5 distinct drill bits are used to
drill 5 sets of 3 strips based on their category number. As shown in Figure 34, the drill bits have the
diameters of ″, ″, 1″, ″, ″ (1.27cm, 2.22cm, 2.54cm, 2.86cm, 3.18cm). Drill bits larger than 21
87 1 8
1 1 41 1 4
1
″ (3.18cm) did not drill through and hence are not used. A dot is labelled at half of the width and 6.00 ±
0.05 cm from the top. The metal strip is then being drilled with a drill press with the drill bit according to
the number associated with the strip aiming at the dot, as shown in Figure 35. T1 and T2 are drilled with
1″ (2.54cm) and ″(,3.18cm) drill bits and T3 is not drilled. While drilling, safety goggles and oven1 41
gloves are worn for protection. See Figure 13 for the completed metal strips.
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Figure 34. (Left) ″, ″, 1″, ″, ″ drill bits from left to right.21
87 1 8
1 1 41
Figure 35. (Right) Drilling through the metal strip using a drill press under teacher supervision.
Before being used in experiments, as shown in Figure 36 and 37, the metal strips are flattened using an
iron vice. This ensures consistency in shape and decreases deformation during the experiment.
Figure 36. (Left) Figure 37. (Right) Iron vice flatten the aluminium strip in different directions.
The eccentric gear system is made from VEX robotics parts. As shown in Figure 38, a ″ (0.64cm) ×832
1.750″ (4.445cm) button head screw is installed through the outer hole on the 60T high strength gear and
secured by a ″ (0.64cm) keps nut using a ″ (0.24cm) hex screwdriver. Between the nut and gear, a832
332
pillow block lock plate is put in place with a 2" (5.08cm) square bar drive shaft to keep the motion of the
gear and the screw together. The shaft is secured by two 4mm shaft collars, one on each side of the gear,
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secured by ″ (0.20cm) hex key. Majority of the shaft is left on the front face of the gear, as shown in564
Figure 39.
Figure 38. (Left) Back view of the eccentric gear. Figure 39. (Right) Front view of the eccentric gear.
The metal that is tested is located on a wooden desk with 18" (45.72cm) trigger clamp with a napkin
underneath to increase friction and decrease damage to the desk surface. The center of the hole is right at
the edge of the table, as shown in Figure 40 to ensure consistency of the distance to the applied stress. The
eccentric gear is then installed into a drill, and the screw goes in the drill bit holding position, as shown in
Figure 41. The drill’s trigger is taped tightly to the drill itself so it is always on maximum power.
Figure 40. (Left) Metal strip clamped on the table where half of the circle is on the desk.
Figure 41. (Right) Eccentric gear installed into the drill.
The drill is being put in a woodwork vice with some newspaper in between to regulate location. The
woodwork vice is secured on the table so that the eccentric wheel is right on top of the metal strip as
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shown in Figure 42. A 4" (10.16cm) iron G-clamp is secured on the trigger clamp, tilting down for later
use. The drill is then connected a variac transformer as shown in Figure 43 for speed control. In the
experiment, the potential difference is maintained at 50 ± 5 V.
Figure 42. (Left) Drill clamped on a woodwork vice. Figure 43. (Right) Variac transformer turned on.
Appendix D - Stopwatch Time to Seconds Python 2.7 Code
#Read in stopwatch data in format hours:minutes'seconds"centiseconds
import re
stopwatchData = str(raw_input("Please input stop watch reading: "))
re.sub("[^0-9]", "", stopwatchData)
totalSecondsInt = 3600*int(stopwatchData[0])
totalSecondsInt += 60*int(stopwatchData[2:4])
totalSecondsInt += int(stopwatchData[7:9])
print("The results in seconds is " + str(totalSecondsInt)+"."+str(stopwatchData[-2:]))
Appendix E - Video Footage of Three Stages Used to Calculate Revolutions Per Second
The videos are recorded on a Pixel XL phone using 240 frames per second, using the slow motion
function. In the video, at first the footage is normal speed but after about 2 seconds become slow motion.
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When calculating the revolution per second, only the normal speed part is looked at. There might be
uncertainty while counting the normal speed part’s length because of loss during video compression.
https://photos.app.goo.gl/E838CQ9uHAqMNSfe8
Appendix F - Seconds to Stage Cycles and Fatigue Life Python 2.7 Code
#turn time for each stage into cycles based on calculated revolutions per second and calculate total life #include uncertainty calculations rps = [12, 11, 12] rpsUncertainty = [0.34, 0.29, 0.21] timeUncertainty= [2, 4, 4] stageT1 = float(raw_input("Please input stage 1 time: ")) stageT2 = float(raw_input("Please input stage 2 time: ")) stageT3 = float(raw_input("Please input stage 3 time: ")) fatigueLife1 = rps[0]*stageT1 fatigueLife2 = rps[1]*stageT2 fatigueLife3 = rps[2]*stageT3 fatigueLifeTotal = fatigueLife1 + fatigueLife2 + fatigueLife3 if stageT1 == 0:
percentUncertaintiy1 = rpsUncertainty[0]/rps[0] else:
percentUncertaintiy1 = rpsUncertainty[0]/rps[0] + timeUncertainty[0]/stageT1 if stageT2 == 0:
percentUncertaintiy2 = rpsUncertainty[1]/rps[1] else:
percentUncertaintiy2 = rpsUncertainty[1]/rps[1] + timeUncertainty[1]/stageT2 if stageT3 == 0:
percentUncertaintiy3 = rpsUncertainty[2]/rps[2] else:
percentUncertaintiy3 = rpsUncertainty[2]/rps[2] + timeUncertainty[2]/stageT3 fatigueLifeUncertainty1 = fatigueLife1*percentUncertaintiy1 fatigueLifeUncertainty2 = fatigueLife2*percentUncertaintiy2 fatigueLifeUncertainty3 = fatigueLife3*percentUncertaintiy3
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fatigueLifeUncertaintyTotal = fatigueLifeUncertainty1 + fatigueLifeUncertainty2 + fatigueLifeUncertainty3 print("Stage 1 fatigue life is: " + str(int(fatigueLife1)) + " +- " + str(int(fatigueLifeUncertainty1))) print("Stage 2 fatigue life is: " + str(int(fatigueLife2)) + " +- " + str(int(fatigueLifeUncertainty2))) print("Stage 3 fatigue life is: " + str(int(fatigueLife3)) + " +- " + str(int(fatigueLifeUncertainty3))) print("Total fatigue life is: " + str(int(fatigueLifeTotal)) + " +- " + str(int(fatigueLifeUncertaintyTotal))) Appendix G - Picture of Aluminium Strips after Fatigue Failure
Pictures of all aluminium strips after fatigue failure. https://photos.app.goo.gl/sKYPcukndtzdzz767
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