RAFT Friction Tester Design Report

MEGR 3156-002

December 10, 2020

Brendan Welch

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Table of Contents

1. Introduction ........................................................................................................................................... 3

1.1 Problem Statement ........................................................................................................................ 3

1.2 Design Overview .......................................................................................................................... 3

2. Test Plate ............................................................................................................................................... 4

2.1 Subsystem Introduction................................................................................................................. 4

2.2 Calculations ................................................................................................................................... 5

3. Controls ................................................................................................................................................. 8

3.1 Motor............................................................................................................................................. 8

3.2 Controller ...................................................................................................................................... 8

3.3 Code .............................................................................................................................................. 9

4. Electronics Tray .................................................................................................................................. 10

5. Trial Results ........................................................................................................................................ 11

6. Parts and Budget ................................................................................................................................. 12

7. References ........................................................................................................................................... 13

8. Appendix ............................................................................................................................................. 14

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1. Introduction

1.1 Problem Statement

This project was conducted to measure the coefficient of static friction of small samples on ABS plastic.

Finding the coefficient of static friction is done using a friction tester device that increases an angle of

incline until the sample starts to slip. Using this incline angle, the coefficient of static friction between the

two surfaces can be calculated. The derivation for this is shown below (See Figure 1)

Figure 1: Free body diagram of friction testing sample.

Summing the forces in the direction perpendicular to the test plate (shown in blue) gives the following

equation, (1).

𝑚𝑔 cos 𝛼 = 𝑁 (1)

Summing the forces in the direction parallel to the test plate gives the following equation, (2).

𝐹𝑓 = 𝑚𝑔 sin 𝛼 (2)

Given that frictional force is equal to the coefficient of static friction times the normal force, a new

equation, (3), is obtained, and solving for the coefficient of static friction yields (4).

µ(𝑚𝑔 cos 𝛼) = 𝑚𝑔 sin 𝛼 (3)

µ = tan 𝛼 (4)

Thus, in order to test for the coefficient of static friction, the design must be able to find the incline angle.

1.2 Design Overview

To find the incline angle, a stepper motor will be used to raise an incline until the sample starts to slip.

This motor will be controlled by an Arduino board that will count the number of steps as the incline

increases. The maximum sample mass will be 0.25 kg (or 0.55 lb). To detect the sample slipping, a

photodiode will be covered by the sample, and light readings will be continuously taken as the motor

increases the incline. Once the sample slips and the detected light level increases, the signal into the

Arduino board will tell the system to stop raising the incline and report the number of steps taken. This

number of steps can be plugged into (5) to calculate the coefficient of static friction.

µ = tan[(𝑠𝑡𝑒𝑝𝑠 𝑡𝑎𝑘𝑒𝑛)2𝜋

𝑚𝑜𝑡𝑜𝑟′𝑠 𝑠𝑡𝑒𝑝𝑠 𝑝𝑒𝑟 𝑟𝑒𝑣𝑜𝑙𝑢𝑡𝑖𝑜𝑛] (5)

An isometric view of the fully assembled design is shown below (See Figure 2).

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Figure 2: Assembled design isometric view.

The figure above shows the 3 main subsystems of the project, the test plate, the electronics tray, and the

controls. Each of these individual subsystems are discussed in detail below. One unique element shown in

the figure above is the clay mound housing a 28BYJ-48 stepper motor. The inclusion of this makeshift

motor frame is due to complications that occurred in the selection and implementation of a motor. This is

explained in Section 4.1.

2. Test Plate

2.1 Subsystem Introduction

The test plate was designed in two stages, the first was an initial design to accommodate the necessary

samples and sensors, and the second was in response to failure analysis. A detailed part drawing for the

test plate is shown below (See Figure 3).

Figure 3: Test plate drawing.

Test Plate

Electronics Tray

Stepper

Motor

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The hole containing the photodiode is located near the top of the testing plate face and was sized to fit a

5mm diameter photodiode with room for adjustment. The extrusion at the bottom of the plate was

designed to couple in the 8mm side of a 5mm to 8mm shaft coupler (5mm side to motor shaft). This was

accomplished with a 7.5mm diameter extrusion with a flattened section to allow a set screw to apply

pressure and hold the test plate in place. The height and width of the test plate face was sized to allow a

small sample to rest on it with room to slide once the test starts.

This part was 3D printed out of ABS Plastic using the rapid prototyping available from UNCC. The

material properties for ABS plastic are outlined in the table below (See Table 1).

Table 1: ABS plastic material properties1.

2.2 Calculations

Once the basic geometry of the part was decided on, failure criteria were established and calculated for.

Firstly, the part must not deflect more than 0.5mm to maintain precision in measurement. Secondly, the

part must not yield.

The figure below shows the free body diagram for the test plate (See Figure 4).

Figure 4: Free body diagram of test plate. Modeled as a cantilever beam.

For sake of simplicity, the point load was assumed at the end of the test plate instead of above the hole for

the photoresistor, and the distributed load was considered constant throughout the entirety of the part.

Using the maximum deflection of 1mm, a MatLab code was constructed to analyze the thickness required

for the part. The equations in put into MatLab are listed below and each variable’s value is tabulated (See

Table 2).

𝐼 =𝑏𝑡3

12 (6)

𝐷𝑒𝑓𝑙𝑒𝑐𝑡𝑖𝑜𝑛 𝑏𝑦 𝐹 =−𝐹𝐿3

3𝐸𝐼 (7)2

𝑤 = 𝜌𝐴𝑐𝑔 = 𝜌𝑏𝑡𝑔 (8)

𝐷𝑒𝑓𝑙𝑒𝑐𝑡𝑖𝑜𝑛 𝑏𝑦 𝑤 = −𝑤𝐿4

6𝐸𝐼 (9)2

Density E Yield Stress

1530 kg/m3

1.379 GPa 20.68 MPa

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Table 2: Deflection variables & values.

Because the maximum deflection for both deflection equations above, (7) and (9), their values were able

to be added together for each variable thickness. After plugging in the values above to the deflection

equations along with a thickness range of 2mm to 10mm, the following deflection versus thickness plot

was generated (See Figure 5).

Figure 5: Deflection versus thickness plot. Maximum deflection of 1mm marked in black.

Based on the plot, a thickness of roughly 4mm is required to adhere to the maximum deflection of 1mm.

With a safety factor of 1.5 applied, the thickness chosen for the plate was 6mm.

With the thickness determined, the distributed load ‘w’ and moment of inertia ‘I’ were calculated. The

table below shows the updated set of variable values.

Table 3: Updated variable values with known thickness.

Symbol Title Value

I Moment of Inertia (dependent on t)

b Width 50mm

t Thickness (solving for)

F Point Load 2.45 N

w Distributed Load (dependent on t)

E Young's Modulus 1.379 GPa

Symbol Title Value

I Moment of Inertia 9E-10 m4

b Width 50mm

t Thickness 6mm

F Point Load 2.45 N

w Distributed Load 4.5 N/m

E Young's Modulus 1.379 GPa

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Using the values from the table above and the free body diagram of the part, a stress analysis was

conducted across the plate. First, shear and moment diagrams were generated. A figure showing the shear

and moment diagrams is shown below (See Figure 6).

Figure 6: Shear and moment diagrams across length of test plate.

The equations for the reactionary force ‘Ry’ and the reactionary moment ‘M0’ are given below.

𝑅𝑦 = 𝑤𝐿 + 𝐹 (10)

𝑀0 =𝑤𝐿2

2+ 𝐹𝐿 (11)

Using the calculus method, equations for the shear and moment in terms of distance were found.

𝑉(𝑥) = 𝑅 − 𝑤𝑥 (12)

𝑀(𝑥) = −𝑤𝑥2

2+ 𝑅𝑥 − 𝑀0 (13)

A few points of interest for stress analysis along the part were the fixed joint (at x = 0mm), the joint

between the thinner cross-section and plate (x = 10mm), and any arbitrary point on the plate itself (x =

11mm). Using (14) where ‘𝝈’ is the bending stress, ‘M’ is the moment at that location, ‘t’ is the thickness of the part, and ‘I’ is the moment of inertia, the bending stresses were calculated. A table showing the location, moment, moment of inertia, and bending stress at each of the three locations specified is shown below (See Table 4).

𝜎 =𝑀𝑡

2𝐼 (14)

Table 4: Stress analysis results.

x [mm] M [Nm] I [m4] Stress [MPa]

0 0.211 3.6E-10 2.34

10 0.239 3.6E-10 2.034

11 0.242 9E-10 0.601

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The moments of inertia for the first two locations were lower because of the decreased cross-sectional

area for a small portion of the part from 0mm to 11mm from the center of the round extrusion. All of the

stresses were much lower than the Yield Stress of ABS plastic, 20.68 MPa, thus this part has passed all

failure criteria.

3. Controls

3.1 Motor

With the test plate geometry and sample weight selected, the torque that the selected motor must output

was calculated. The test plate was estimated as a rectangular paralleliped with a point mass (the sample)

at its end. The equation for the rotational moment of inertia, (14), is given below where ‘m’ is the overall

mass and ‘mL’ is the mass of the part. Once the rotational moment of inertia was calculated, it was

plugged into the equation for torque, (15), where ‘T’ is the torque required, ‘Jz’ is the rotational moment

of inertia, and ‘α’ is the angular acceleration.

𝐽𝑧 = 𝑚𝐿2 +𝑚𝐿𝑡2

12+

𝑚𝐿𝐿2

3 (14)

𝑇 = 𝐽𝑧𝛼 (15)

The tables below show the inputs and outputs of the torque calculations (See Tables 5 & 6).

Table 5: Rotational moment of inertia inputs and result.

Table 6: Torque inputs and result.

Based on this torque requirements, the Nema 17 would be an affordable and appropriately sized stepper

motor given its hodling torque of 28 Ncm3. The 28BYJ-48 is also an acceptable motor with a torque

rating of 8 Ncm (11.4 oz. in.)4. Because of the circular design and the low weight of the 28BYJ-48

stepper motor, the Nema 17’s was initially chosen for the design.

During the assembly and testing of the design, complications arose with the Nema 17 stepper motor.

Because of this, the 28BYJ-48 stepper motor was substituted in. Because the support extruding from the

top of the electronics tray was designed at a height such that it would support the test plate with the shaft

centered at the shaft height of the Nema 17 (20mm), a clay support frame was constructed for the 28BYJ-

48 stepper motor to raise the height and properly attach the motor to the plate.

3.2 Controller

This design operates off an Arduino Uno R3 controller connected to a ULN 2003 motor driver and a

photodiode. A circuit diagram of the setup used to control the friction tester is shown below (See Figure

7).

mp mL m Jz

0.2500 0.0370 0.2870 0.0019

Desired

Speed [rpm]

Time To

Accelerate [s]

Angular Acceleration

[rad/s2]

Torque

[Nm]

Torque

[Ncm]

5 0.5 1.047197551 0.002 0.2

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Figure 7: Final circuit diagram.

For this circuit, the only power input into the circuit is coming from the Arduino itself. Because of the

low speeds of the stepper motor and the low voltage requirements for analyzing the photodiode, the

power supply via USB was more than enough. Analog pin ‘A0’ was responsible for reading the voltage

from the photodiode. PWM pins 9, 10, 11, and 12 were wired to the motor driver to initiate steps.

3.3 Code

Figures 8-12 show the breakdown of the full Arduino code (full code can be found in the Appendix).

Before any of the code is initialized, the sample should be placed just covering the photodiode with its top

edge.

Figure 8: Arduino code defining stepper motor library and properties.

The code starts by including the Stepper library with predefined functions designed for easy operation of

stepper motors. Then it defines the step count for the 28BYJ-48 stepper motor, 2038 steps per revolution,

yielding a resolution of 0.18 degrees per step. Finally, the PWM pins are defined in the order required to

actuate the motor.

Figure 9: Code initializing variable names.

Two variables ‘phr’ and ‘count’ are established. ‘Phr’ corresponds to the analog pin A0 reading from the

photodiode. ‘Count’ refers to the step count of the motor that will increase as the incline is raised.

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Figure 10: Code setting up while loop.

The setup begins by setting the stepper motor speed to 3rpm. The system was initially designed for 5rpm

(as discussed during the torque calculations), but the speed was lowered to ensure an accurate

measurement. It continues by starting the serial monitor and printing the initial light level measurement

from the photodiode. Finally, the step count is set to zero before the while loop is initiated.

Figure 11: While loop.

The while loop starts by ensuring that the measured light level is lower than 13. From repeated tests, the

minimum ambient light level printed by the system was 13. If the light level is lower than 13, then the

sample must still be covering the photodiode over the hole, so the stepper motor steps 5 times. By

grouping steps in segments of 5, the resolution broadens from 0.18 degrees per step to 0.88 degrees per

step. After the steps have been taken, it takes another analog reading, increases the count by 5, and waits

10 ms. Once one cycle has been completed, it goes back to the top of the loop to check to see if the light

level is still lower than 13. This continues until the light level is measured at some value higher than 13,

implying that the sample has slipped past the photodiode hole.

Figure 12: Code to display steps and reset stepper motor.

Once the exit criteria for the while loop is met, the serial monitor prints the final step count. The system

then waits 3 seconds and steps the stepper motor back to its starting position.

4. Electronics Tray

With all of the control and hardware elements selected and designed, the plate to hold all of them was

next to be designed. A detailed part drawing for the electronics tray is shown below (See Figure 13).

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Table 7: Friction testing results.

Figure 13: Electronics tray drawing.

The square and rectangular indents were designed to nest the Nema 17 motor and Arduino Uno R3

respectively. The vertical extraction was designed to support the test plate while at rest given that the

shaft was centered at the shaft height of the Nema 17. Because the Nema 17 was not ultimately chosen, a

clay frame was added onto the electronics tray and adjusted so that the shaft of the 28BYJ-48 was

centered at the same height as the Nema 17, 20mm.

5. Trial Results

After the design was assembled and tested, 3 trials were run on 3 separate specimens. The figure below

shows each of the three samples tested (See Figure 14).

Figure 14: 2x2 Lego brick (left), 1018 Steel Sample (center), and Pink Eraser (right) samples.

The coefficient of static friction was tested on each of the samples three times. The step count from serial

monitor was plugged into (5) to yield the coefficient of static friction. The table below details the test

results (See Table 7).

Trial # Step Count Angle [deg] Angle [rad] Coefficient Step Count Angle [deg] Angle [rad] Coefficient Step Count Angle [deg] Angle [rad] Coefficient

1 170 30.03 0.52 0.58 165 29.15 0.51 0.56 235 41.51 0.72 0.89

2 170 30.03 0.52 0.58 155 27.38 0.48 0.52 235 41.51 0.72 0.89

3 175 30.91 0.54 0.60 165 29.15 0.51 0.56 240 42.39 0.74 0.91

AVE 171.67 30.32 0.53 0.58 161.67 28.56 0.50 0.54 236.67 41.81 0.73 0.89

Lego Brick 1018 Steel Eraser

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Based on the results from the table above, 1018 Steel has the lowest coefficient of static friction with

ABS plastic among the selected samples at an average value of 0.54. It is followed by the Lego Brick at

0.58 and then the eraser at 0.89. The step count precision is impressive, considering each test has only one

difference of a single counting cycle (5 steps) with the exception of 1018 steel at 2 counting cycles.

6. Parts and Budget

The following is a comprehensive list of the parts used in the design, their quantity, and their price.

Table 8: Price and quantity of each part used in friction tester design.

A lot of the project was able to be completed using only elements from the Arduino Elegoo kit. The

exceptions to this being the 3D printed parts, the ‘Electronics Tray’ and the ‘Test Plate’, and a few

miscellaneous components needed to complete the design and allow it to function.

Part Name Qty. Price Part Name Qty. Price

Arduino Uno 1 Electronics Tray 1 $15.66

Small Breadboard 1 Test Plate 1 $13.36

28BYJ-48 1 5mm to 8mm shaft coupler 1 $2.20

Jumpers 15 Photodiode 1 $0.95

ULN2003 Driver 1 Clay 1 $0.82

Resistor (100 ohm) 1 Electrical Tape 1 $10.00

$79.98Total Cost

$36.99

(Elegoo

Kit)

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7. References

1. DM Dielectric Manufacturing. “ABS (Acrylonitrile-Butadlene-Styrene).”

https://dielectricmfg.com/knowledge-

base/abs/#:~:text=ABS%20offers%20a%20good%20balance,hardness%2C%20rigidity%20and%

20electrical%20characteristics.&text=ABS%20plastic%20remains%20hard%2C%20rigid,heat%

2Dresistant%20and%20platable%20grades.

2. Philpot, T. A. (2017). Mechanics of Materials: an Integrated Learning System. Hoboken, NJ:

John Wiley & Sons, Inc.

3. Data Sheets 4 U. “17HS4401 Datasheet 2 Phase Hybrid Stepper Motor.”

https://www.datasheet4u.com/datasheet-pdf/MotionKing/17HS4401/pdf.php?id=928661

4. Solarbotics. “28BYJ-48-64 64:1 Stepper Gearmotor.”

https://solarbotics.com/product/22310/#:~:text=Torque%20%405V%3A,in*oz)%20%40%20200

mA%20draw

5. Circuit Lab. “Circuit Lab Online Editor.” https://www.circuitlab.com/editor/

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8. Appendix

An additional set of figures showing the overall design, MatLab code, and the full Arduino code are

shown below.

Figure A1: Unlabeled assembled design isometric view.

Figure A2: Assembled design rear view.

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Figure A3: Assembled design rear isometric view.

Figure A4: MatLab code to establishing variables.

Figure A5: MatLab code calculating for maximum deflection with changing thickness.

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Figure A6: Full Arduino Code.