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NUMERICAL METHODS FOR ENGINEERS (COEB223)

FINAL REPORT

TOOTHPICK TOWER PROJECT

GROUP MEMBERS:

AHMAD HAZIM BIN AMINUDDIN (ME089463)

MUHAMAD IQBAL BIN MUHAMMAD SIDIK (ME089385)

CHIN FWU QUAN (ME089478)

MUHAMAD HAZIQ BIN MUHAMAD JAMAL (ME089454)

SECTION : 1A

LECTURER : MISS SHANGARI K. RAVEENDRAN

DATE : 15/1/2014


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INTRODUCTION

This progress report is regarding the group project assigned to us in this course. We are

required to build tower using toothpicks that can support a heavy load. The toothpicks can only

be connected using any type of glue. The tower needs to be at least 15 cm tall and cannot

exceed the area of 15x15 cm. When the tower has finished, we are required to present it to our

lecturer. We are required to explain the building processes, problems encountered, and

solutions. Finally, our towers strength and quality will be tested.

PROBLEM STATEMENT

1. How do we make the tower to be able to support as much load as possible?2. How do we make the tower to be as tall and as light as possible?3. How do we build the tower with quality, and at the same time minimizing the amount of

toothpicks and glues used?

OBJECTIVE

The objective of the project is to enable students to apply the numerical methods learned in

the lectures in analyzing an engineering related problem.


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LITERATURE REVIEW

For this project, we are required to build a toothpicks tower that will be able to support heavy

loads. Firstly, the most important part of the tower is the design. We did some research in

order to find the most stable and strong tower. We also need to consider the size of the base

and the height of the tower carefully in order to withstand the maximum load.

Toothpick tower comes in many shapes such as rectangular, triangular, square and others.

From our research, we agreed to conclude that we will build hexagonal shaped tower.

Hexagonal tower has six edges, thus the load will be distributed more to each edge instead of a

triangular or rectangular shaped tower. We will also be using triangular links between each

level instead of using only a vertical link to make it stronger.

For the materials, we will use only toothpick as the building materials. We use OK 3 second

glue and a construction adhesive bonding nails as the connectors between the toothpicks.

Firstly, we started building the hexagonal base, then we proceed to the first level, until finish.

METHODOLOGY

Find a few model of the toothpick to get an idea and create our own model of thetoothpick.

Discuss and decide which model to use that included the requirement of the project. Make a sketch of the tower to ease our work. Decide the amount of the toothpick and the glue that we have to use to build the tower. We make a prototype to satisfy the requirement of the project about the height, weight

and the maximum load that the tower can hold.

Identify the problem and discuss the problem and do an adjustment to make a bettertower.


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TECHNICAL DESCRIPTION

1. 3 toothpicks are used as a link instead of 1 toothpick because it will be much stronger.

2. Hexagonal shape tower has 6 edges, thus the load will be distributed more to each edge

instead of a triangular shape or rectangular shape tower.

3. Triangular links are formed between each level instead of only a vertical link because it will

be stronger.

4. Horizontal links are added at the side to hold each of the vertical links so that it will be

more stable and it won't collapse so easily.

5. The center is the main part of the tower. In order to have a strong center, 16 toothpicks are

used as a link at the center.


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NUMERICAL METHOD CALCULATIONS

For the calculation, we used LU decomposition to calculate the number of toothpick, time and

droplet number of glue used. First, there are few types of links, which are 3 center links, 36

slanting links, and a total of 72 horizontal link and vertical link.

For each center link, 16 toothpicks, 60 seconds and 8 droplets of glue are used.

For each slanting link, 3 toothpicks, 8 seconds and 2 droplets of glue are used.

For each horizontal or vertical link, 3 toothpicks, 5 seconds and 2 droplets of glues are used.

Here are the steps to calculate using LU decomposition:

1. [A][X] = [B]2. From [A], find [U] then [L]3. [L][D] = [B], find [D]4. [U][X] = [D], find [X]

Now, begin our steps:

From the problem statement, interpret the data into equations

72 = 3x5y2z ------------------------------------- Horizontal + Vertical links 36 = 3x8y2z ------------------------------------- Slanting links 3 = 16x60y8z ------------------------------------- Center links

X = number of toothpick

Y = time in second

Z = droplet number of glue


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Step 1

[A][X] = [B]

3 -5 -2 X 72

3 -8 -2 Y = 36

16 -60 -8 Z 3

Step 2

Find [U]

3 -5 -2 3 -5 -2

[U] = 0 -3 0 = 0 -3 0

0 -100/3 8/3 0 0 8/3

u21= a21f21a11

a21f21a11= 0

3f21(3) = 0

f21= 1

u22= a22f21a12= -8 + 1(5) = -3

u23= a23f21a13= -2 + 1(2) = 0


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u31= a31f31a11

a31f31a11= 0

16f31(3) = 0

f31= 16/3

u32= a32f31a12= -60(16/3)(-5) = -100/3

u32= a32f32a22

a32f32a22= 0

-100/3f32(-3) = 0

f32= 100/9

u33= a33f32a23= 8/3(100/9)(0) = 8/3

Find [L]

1 0 0 1 0 0

[L] = f21 1 0 = 1 1 0

f31 f32 1 16/3 100/9 1


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Step 3

[L][D] = [B]

1 0 0 D1 72

1 1 0 D2 = 36

16/3 100/9 1 D3 3

D1= 72 ; D2= -36 ; D3= 19 ;

Step 4

[U][X] = [D]

3 -5 -2 X 72

0 -3 0 Y = -36

0 -100/3 8/3 Z 19

X = 48.75 ; Y = 12 ; Z = 7.125 ;


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Thus,

Total number of toothpick = 3x + 3x + 16x

= 22x (substitute x = 48.75)

= 22 (48.75)

= 1072.5

= 1073 toothpicks

Total time = 5y + 8y + 60y

= 73y (substitute y = 12)

= 73 (12)

= 876 seconds

= 14.6 minutes (assuming non-stop work)

Total droplet number of glue = 2z + 2z + 8z

= 12z (substitute z = 7.125)

= 12 (7.125)

= 85.5 drops (assuming no wastage)

(Approximately 0.1ml per drop, thus approximately 9ml is used)


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RESULTS AND DISCUSSIONS

1. In this experiment, we used LU decomposition to calculate the number of toothpick, timeand glue we used to build a tower.

2. The steps involve: Interpret data into equations Write equations into [A][X] = [B] From [A], find [U] then [L] [L][D] = [B], find [D] [U][X] = [D], find [X]

3. Calculate the total by substituting the unknowns.4. We found that we used 1073 toothpicks, 876 seconds and 85.5 droplets of glue to build all

the links for a tower.

5. In other words, our tower weigh 107.3g (1073 toothpick x 0.1g/toothpick), 14.6 minutestime taken and approximately 9ml of glue is used.

6. Anyway, the values calculated are the values without considering the wastages.7. To complete a toothpick tower, approximately 2 weeks are used to come out with the ideas

and sketches, which are the shape of the tower, arrangement of links, type of glue used etc.


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CONCLUSIONS

As a conclusion, our tower is built. 3 toothpicks are used as a link to increase the

strength of the tower. Besides, hexagonal shape is chosen for the tower shape instead of

triangular shape or rectangular shape because hexagonal has 6 edges, thus load will be

distributed more to each edge. Vertical links are used between each level to decrease the

weight of the tower and the usage of toothpick, at the same time increase the height of the

tower. On the other hand, triangular links are added between each level instead of using only

vertical links to increase the strength of the tower.

To calculate all the material used, LU decomposition is used. To build all the links

required for a tower, 1073 toothpicks, 14.6 minutes and 9ml of OK 3 seconds glue are used.

From the number of toothpicks used too, we can calculate that our tower weigh approximately

107.3 gram.

With the orientation the links and the shape of the tower, we estimates that it can

withstands a load of 40kg. Anyway, only 30kg load is tested to avoid the tower collapses.


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REFERENCES

1) http://www.usd116.org/ums/projects/sciencenet/projects/tthints.html2) http://mat.gsia.cmu.edu/classes/QUANT/NOTES/chap7.pdf3) http://en.wikipedia.org/wiki/Simplex_algorithm4) http://www.isye.gatech.edu/~spyros/LP/LP.html5) http://www.math.cuhk.edu.hk/course/math3210/lpch3.pdf6) http://www.youtube.com/watch?v=JwapuC2-VlA


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APPENDIX

Bamboo toothpicks

3 seconds glue construction adhesive bonding nail


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base

1st

level

link


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2nd

level

Finished tower

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