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.

24

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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