A Chair for Rube Goldberg

Table of Contents 1...

Introduction

6...

Step 1: Collecting Junk

8...

Step 2: Designing

16...

Step 3: Calculating

44...

Step 4: Winging It

52...

How It Works

60...

Final Presentation

68...

Bibliography

1930 AMERICAN

CARTOONIST RUBE GOLDBERG

HONORED BY

WAT E R L O O

ARCHITECTURE STUDENTS Fourth year students CJ Yang and Amy Bao design Rube Goldberg a ridiculous chair inspired by his “Rube Goldberg Machines” which completed simple tasks through convoluted, indirect, and humorous ways.

Though Goldberg never built the machines himself, his legacy has led to working machines being created for film, television, music videos, and even engineering competitions. For a sample of his cartoons, see right.

1

Precedents

“This Too Shall Pass” music video by OK GO 2

2011 World Record Breaking Rube Goldberg Machine by Purdue Society of Professional Engineers team 3

4

5

Step 1

Collecting Junk Librarians, shop owners, professors, security guards, and students alike were surprised and amused as they were approached by Amy and CJ asking the unexpected question, “Do you have any random cool junk you don’t want that we can take for a project?” Special thanks goes out to Victoria from Brick and Mortar who was renovating a bar and had myriad scraps of wood, buckets, cups, etc; as well as the professor, Elizabeth, who donated her washing machine agitator and bought a cowbell from New Jersey.

6

7

Step 2 Designing

The chair machine had to be designed backwards, starting with the desired end (a chair popping up) and ending with the catalyzing input from the user. The basic concept of pieces of the chair pulled by springs on top of a two piece base emerged. The surrounding contraptions would have to be left until later.

8

4’

4’ 3’

Exploded Axo Main Structure

The two piece base allows for easier transportation. The two pieces of the chair are connected by base strips and dowels.

To move between floors, the smaller piece can fit through the UWASA elevator while the larger piece must be carried through the fire escape stairs. This also had to be factored in building the other components as everything that crossed in between the base pieces had to be detachable.

9

Dimensions

start position 48"

36" 3"

16.72"

42"

17.22"

8"

17.72"

17.72"

18"

3"

1"

10

1" 1" 1" 1"

end position 48"

36" 3"

17.72"

42"

18"

8"

3"

31.43"

17.72" 4" 1" 1"

11

12

A 1:5 sketch model of the basic chair mechanism and possible surrounding elements

Testing the sketch model

Testing the strength of the springs

The humble beginnings of something huge

After hours of chiseling and sanding, a smooth butt shape finally emerges from the seat

The underside of the seat is hollowed out with a router to reduce weight

Weighing down blocks of wood to glue them on as triangle reinforcements

The screws of the hinge pierce the seat back and wood block. Hinges are nestled in so the piece can lay flat

13

14

Attaching the pieces of the base

Assembling all pieces

Testing out the structural stability of the chair without springs

Cantilevering seat

Attaching the springs

Testing the power of 4 springs working in conjunction

Realizing that an additional force is needed to push the seat back up 45 degrees for the springs to activate

The beginnings of a catapult

15

Step 3

Calculations Table of Contents 17...

18...

26...

28...

34...

36...

40...

42...

16

Price and Weight Center of Gravity Reactions Body on Chair Overturn Pinned Frame Beam / Column Stress Combined Loading

4‘

4‘

3‘

Bamboo Bartop

x1

$0

1”x3”x6’ plank

x2

$4.99 x 2 = $9.98

3/4”x20”x36” Pine

x1

$23.93

1‘

4” Tubingx 10ft Springs Hinges

$8.78

x5 x5 sets

$5.13 x 5 = $25.65 $4.92 x 5 = $24.6

Steel Angles

many

$54.00

Zip Ties

x1 pack

$2.00

x1

$2.00

x 2 packs

$6.00

Yarn

Marbles

Total = $156.94

Total Weight of Chair = 61kg = 134.5 lbs

17

1. Center of Gravity start position BLOB 1

DWL 7

DWL 6

DWL 5

BAC

KV

DWL 4

IEW

DWL 3

TRI 1

2

DWL 2

TRI 2

3

DWL 1

BLOB 2

TRI 3

y

TRI 5

z

DWL 6

1

0

x

DWL 7

TRI 4

DWL 5

REC 1 DWL 4

FRO

NT

VIE

DWL 3

W DWL 2

REC 2 DWL 1

18

EW

E VI

SID

end position BLOB 1

3

BAC

KV

TRI 1

IEW

BLOB 2 TRI 2

2

1

y

REC 1

TRI 3

z

TRI 4

0

x

TRI 5

IEW

EV

SID

FRO

NT

VIE

W

REC 2

19

CoG CoG (start(start position) position)unit

in 40.36 inBLOBBLOB unit 40.36 2 2in

in in

40.00 40.00 DWL 4, DWL DWL 4, 11 DWL 11

in in

lbs*inlbs*in in lbs*inlbs*inlbs*inlbs*in

Dowels Dowels

use rhino use rhino END POSITION POSITION 32.00 32.00 DWL DWL 10 DWL 10 xi Goldberg berg i i vi (=b*h*t) vi DWL (=b*h*t) Di3, (lbs/in3zi Di3,(lbs/in3zi xi yi END yi Vi*Zi*Di Vi*Zi*Di Vi*Xi*Di Vi*Xi*Di Vi*Yi*Di Vi*Yi*Di 18.00 18.00 DWL 2, DWL DWL 2, 9DWL 9 PIECE PIECE 0 0 3024 30240.02 0.02 42.00 42.00 24.00 24.000.38 0.38 2540.16 2540.16 1451.52 1451.52 22.98 22.98 TOP TOP VIEW VIEW 1:10 1:10 Gravity r of Gravity / Reaction / Reaction Assignment Assignment 4.00 4.00 DWL 1, DWL DWL 1, 8DWL 8 PIECE PIECE 1 1 133.73133.730.02 0.02 19.42 19.42 24.00 24.001.25 1.25 51.94 51.94 64.19 64.193.34 3.34 44.51 44.51 3

3

PIECE PIECE 2 43.32 2 285.55 285.55 0.02 0.02 71.43 24.00 137.06137.06 13.36 13.36 43.32 TRI 1,TRI TRI 1,2,TRI TRI 2, 3,TRI TRI71.43 3,4 TRI 4 24.002.34 2.34407.94407.94 PIECE PIECE 3 35.27 3 596.43 24.00 24.001.23 1.23740.77740.77 286.29286.29 14.67 14.67 35.27 2 596.43 2 0.02 0.0262.1 62.1 TRI 56.00 TRI0.02 5 0.02 TRI 1 TRI 129.44 29.44 6.00 63.37 63.37 18.85 18.854.42 4.42 7.60 7.602.26 2.260.53 0.53 1 1 0.02 0.02 TRI 2 TRI 228.27 28.27 6.00 6.00 63.37 63.37 29.15 29.154.42 4.42 7.60 7.603.50 3.500.53 0.53 42.00 42.00 REC6.00 1, REC REC 1, 2, REC 0 45.66 2, 0 45.66 TRI 3 TRI 3 6.00 0.02 0.02 18.85 18.852.42 2.42 5.48 5.482.26 2.260.29 0.29 44.78 TOTAL 44.78 TOTAL TRI14 4 7,TRI 4 14 6.00 6.000.02 0.02 45.66 45.66 29.15 29.152.42 2.42 5.48 5.483.50 3.500.29 0.29 82.00 DWL 82.00 7, DWL DWL DWL TRI13 5 6,TRI 5 13 6.00 6.000.02 0.02 27.11 27.11 24.00 24.002.42 2.42 3.25 3.252.88 2.880.29 0.29 68.00 DWL 68.00 6, DWL DWL DWL Z 0.02Z 0.0242.00 42.001.50 1.50 61.03 BLOB 61.031 BLOB Rec 1 1 Rec 1 252 252 -0.50 -0.50211.68211.687.56 7.56 -2.52 -2.52 54.00 54.00 DWL 5, DWL DWL 12 5, DWL 12 Rec 2 Rec 2 252 252 0.02 0.02 42.00 42.00 46.50 -0.502DWL -0.50211.68 211.68 234.36234.36 -2.52 -2.52 DWL 1 DWL 1 DWL 2 DWL 3DWL 3 4 4 4 46.50 in3 in3 2 BLOB lbs*in lbs*in lbs*in44 lbs*in4Dowels Dowels lbs*in n)position) unit unit in total in 13.44in13.44 in 0.02lbs*in in0.02 40.36 40.36 BLOB 2 DWL DWL total 42.57 42.57 24.00 24.001.45 1.45 11.44 11.446.45 6.450.39 0.39 unit 40.00 40.00 DWL DWL DWL 114, DWL 11 use rhino use4,rhino BLOB 1BLOB 1 30 30 1 61.03 1 61.038.40 8.40 30.79 30.79 1830.90 1830.90 252.00252.00 923.70923.70 32.00 DWL 3,(lbs/in3zi DWL DWL 10 DWL 10 i i32.00 vi (=b*h*t) viDi (=b*h*t) Di3,2(lbs/in3zi yi 29.29 Vi*Xi*Di Vi*Yi*Di Vi*Xi*Di Vi*Yi*Di BLOB BLOB 2xi 12yixi 12 1Vi*Zi*Di 1 Vi*Zi*Di 29.29 41.54 41.54 10.69 10.69 351.48351.48 498.48498.48 128.28128.28 i 18.00 DWL 18.00 2, DWL DWL 9 2, DWL 9 X X PIECE 0 PIECE 03024 3024 0.02 42.00 0.02 24.00 42.00 24.00 0.38 2540.16 0.38 1451.52 2540.16 1451.52 22.98 22.98 SUM 4629.15 6387.41 6387.41 2952.31 2952.31 1103.62 1103.62 DWL 1 4.00 4.001, DWL DWL DWL 8 1,SUM DWL 84629.15 PIECE 1 PIECE 1 3 44.51133.73 0.02 19.42 0.02 24.00 19.42 24.00 1.25 1.25 51.94 64.19 51.94 64.19 3.34 3.34 DWL 2 44.51133.73 3 PIECE 2 PIECE 2 TRI 0.02 71.43 71.43 24.00 2.34 133.74 407.94 2.34 137.06 407.94 137.06 13.36 13.36 DWL 3 43.32285.55 43.32 1, 285.55 TRI TRI 2, TRI 1, TRI 3,0.02 TRI 2,Vi*Di TRI 4 24.00 TRI=4133.74 =3, Vi*Di PIECE 3 PIECE 596.43 3 596.43 0.02 62.1 0.02 24.00 62.1 24.00 1.23 740.77 1.23 286.29 740.77 286.29 14.67 14.67 DWL 4 35.27 2 35.27 2 29.44 29.44 TRI 5 0.02 TRI 5 TRI 1 TRI 1 6.00 6.00 63.37 0.02 18.85 63.37 18.85 4.42 4.42 7.60 2.26 7.60 0.53 2.26 0.53 DWL 5 z̄ =sumz̄ (ViZiDi)/sumVi*Di =sum (ViZiDi)/sumVi*Di 28.27 1 28.27 0.02 1 TRI 2 TRI 2 6.00 6.00 63.37 0.02 63.37 29.15 4.42divided 4.42 7.60 3.50 7.60 0.53 3.50 0.53 DWL 6 z̄ = z̄6387.41 =29.15 6387.41 divided by133.74 by133.74 42.00 42.00 REC 1, REC REC 2, 1, 0 45.66 REC 2, 018.85 TRI 3 TRI 3 6.00 0.02 6.00 0.02 18.85 2.42 2.42 5.48 2.26 5.48 0.29 2.26 0.29 DWL 7 z̄ = z̄ = 45.66 47.76 47.76 DWL 1 DWL 1 DWL 2 DWL 2 DWL 3 DWL DWL 3 4 DWL 4 DWL 5 DWL 5 DWL 6 DWL 6 DWL 7 DWL 7 TRI 4 TRI 4 6.00 0.02 6.00 45.66 0.02 29.15 45.66 29.15 2.42 2.42 5.48 3.50 5.48 0.29 3.50 0.29 DWL 8 TRI 5 TRI 5 6.00 0.02 6.00 27.11 0.02 24.00 27.11 24.00 2.42 2.42 3.25 2.88 3.25 0.29 2.88 0.29 DWL 9 REC 1REC 1 x̄=sumx̄(ViXiDi)/sumVi*Di =sum (ViXiDi)/sumVi*Di Z Z Rec 1 Rec 1 252 Z 0.02 252 Z42.00 0.02 42.00 1.50 -0.50 1.50 211.68 -0.50 211.68 7.56 -2.52 7.56 -2.52 DWL 10 x̄= x2952.31 ̄= 2952.31 divided divided by133.74 by133.74 Rec 2 Rec 2DWL 252 0.02 252 42.00 0.02 42.00 -0.50 46.50 211.68 -0.502 234.36 211.68 234.36 -2.52 DWL-2.52 1 DWL 1 DWL 2 DWL 3 DWL 3 DWL 4 REC DWL1 4 DWL REC11 1 x̄= x̄=46.50 22.07 22.07 X X DWL total DWL total 13.44 13.44 0.02 42.57 0.02 24.00 42.57 24.00 1.45 1.45 11.44 11.44 6.45 BLOB 0.39 6.45 DWL 12 BLOB 1 1 0.39 BLOB 1 BLOB 1 30 30 1 61.03 1 (ViZiDi)/sumVi*Di 61.03 8.40 30.79 8.40 1830.90 30.79 1830.90 252.00 923.70 252.00 923.70 DWL 13 ȳ=sum ȳ=sum (ViZiDi)/sumVi*Di BLOB 2 BLOB 2 12 12 1 29.29 29.29 10.69 41.54 10.69 351.48 351.48 128.28 498.48 128.28 DWL 14 ȳ= 11103.62 ȳ=41.54 1103.62 divided divided by133.74 by133.74498.48 X X 4629.15 SUM SUM4629.15 6387.41 2952.31 6387.41 1103.62 2952.31 1103.62 SUM ȳ= ȳ= 8.25 8.25

ANG (CJ) YANG BAO (AMY) BAO

Dowels

1, TRI 5, 2, 3 24.00 8.41 TRI BLOB 1, TRI 13 18.85 1.50 DWL 1, DWL 2, DWL 3, DWL 4, DWL 5, DWL 6, DWL 7 46.50 REC 2 29.15 TRI 2, TRI 0 4 24.00 1, TRIREC 5, 2, 13 24.00 1.50 TRI 1, TRI 3 18.85 TOTAL 46.50 2 10, DWL 8, DWLREC 9, DWL DWL 11, DWL 12, DWL 13, DWL 14 0 24.00 BLOB 2 TOTAL 1.50 REC 1 1 DWL 8, DWL 9, DWL 10, DWL 11, DWL 12, DWL 13, BLOB DWL 14 DWL 1, DWL 2, DWL 3, DWL 4, DWL 5, DWL 6,BLOB DWL 2 7 TRI BLOB 2, TRI 14 1, TRI 5, 2, 3 DWL 1, DWL 2, DWL 3, DWL 4, DWL 5, DWL 6, DWL 7 TRI 2, 1, TRI TRI 4 3 TRI 2 1, TRI REC 5, 2, 3 TRI 1, TRI 0 3 REC REC 12 0 REC 1

22.07 46.50 41.54 22.07 8.41 46.50 1.50 41.54 29.15 8.41 24.00 1.50 18.85 29.15 46.50 24.00 24.00 18.85 1.50 46.50 24.00 1.50

CoG (start position)

Vi*Di =

Vi*Di 133.74 =

TRI 1 TRI 1

133.74 1

1

2

2 0

in in lbs*inlbs*in lbs*inlbs*in unit unit in in in in in in lbs*inlbs*in use rhino use rhino DIMENSIONS DIMENSIONS IN INCHES IN INCHES i i vi vi zi zi xi xi yi yi Vi*Zi Vi*Zi Vi*Xi Vi*Xi Vi*Yi Vi*Yi DWL 1DWL 1 0.96 0.96 4.00 4.001.50 1.501.45 1.45 3.84 3.84 1.44 1.44 1.39 1.39 BASE DWL 2DWL 2 0.96 0.9618.00 18.001.50 1.501.45 BASE 1.4517.28 17.28 1.44 1.44 1.39 1.39 DWL 3DWL 3 0.96 0.9632.00 32.001.50 1.501.45 CHAIR 1.4530.72 30.72 1.44 1.44 1.39 1.39 CHAIR DWL 4DWL 4 0.96 0.9640.00 40.001.50 1.501.45 ATTACHMENTS 1.4538.40 38.40 1.44 1.44 1.39 1.39 ATTACHMENTS DWL 5DWL 5 0.96 0.9654.00 54.001.50 1.501.45 TOTAL 1.4551.84 51.84 1.44 1.44 1.39 1.39 TOTAL DWL 6DWL 6 0.96 0.9668.00 68.001.50 1.501.45 1.4565.28 65.28 1.44 1.44 1.39 1.39 DWL 7DWL 7 0.96 0.9682.00 82.001.50 1.501.45 1.4578.72 78.72 1.44 1.44 1.39 1.39 DWL 8DWL 8 0.96 0.96 4.00 4.00 46.50 46.501.45 1.45 3.84 3.8444.64 44.64 1.39 1.39 DWL 9DWL 9 0.96 0.9618.00 18.00 46.50 46.501.45 1.4517.28 17.2844.64 44.64 1.39 1.39 DWL 10 DWL 10 0.96 0.9632.00 32.00 46.50 46.501.45 1.4530.72 30.7244.64 44.64 1.39 1.39 DWL 11 0.96 0.96 40.00 40.00 1.4538.40DWL 38.4044.64 1.39 1.39 DWL 4DWL 4 1RECDWL DWL DWL 5 46.50 46.501.45 6DWL 44.64 6 REC 1 11 45 in3 inin3 lbs*in lbs*in44 lbs*in lbs*in44 lbs*in4 unit inin 12 in in 0.96 lbs*in in DWL DWL 12 0.9654.00 54.00 46.50 46.501.45 1.4551.84 51.8444.64 44.64 1.39 1.39 use rhino use rhinoDWL 13 DWL 13 0.96 0.9668.00 68.00 46.50 46.501.45 1.4565.28 65.2844.64 44.64 1.39 1.39 i vi vizi zixi 14 yixi 0.96 Vi*Zi yi Vi*Xi Vi*Xi Vi*Yi Vi*Yi DWL DWL 14 0.9682.00Vi*Zi 82.00 46.50 46.50 1.45 1.45 78.72 78.7244.64 44.64 1.39 1.39 DWL 1 0.96 0.96 4.00 1.50 4.00 1.45 1.50 1.45 3.84 3.84 1.44 1.44 1.39 572.16 1.39 SUM SUM 13.44 13.44 572.16 322.56322.5619.49 19.49 DWL 2 0.96 18.00 0.96 18.00 1.50 1.45 1.50 17.28 1.45 17.28 1.44 1.44 1.39 1.39 BLOBBLOB 1 1 DWL 3 0.96 32.00 0.96 32.00 1.50 1.45 1.50 30.72 1.45 30.72 1.44 1.44 1.39 1.39 DWL 4 0.96 40.00 0.96 40.00 1.50 1.45 1.50 38.40 1.45 38.40 1.44 1.44 1.39 1.39 DWL 5 0.96 54.00 0.96 54.00 1.50z̄ dwl=sum 1.45 51.84 1.45 51.84 1.44 1.44 1.39 1.39 z̄1.50 dwl=sum (ViZi)/sumVi (ViZi)/sumVi DWL 6 0.96 68.00 0.96 68.00 1.50 z̄ 1.45 1.50z̄ dwl= 65.28 1.45 572.16 65.28 1.44 1.44 1.39 1.39 dwl= 572.16 divided divided by13.44 by13.44 DWL 7 0.96 82.00 0.96 82.00 1.50 z̄ 1.45 1.50z̄ dwl=78.72 1.45 42.57 78.72 1.44 1.44 1.39 1.39 dwl= 42.57 DWL 8 0.96 0.96 4.00 46.50 4.00 46.50 1.45 1.45 3.84 44.64 3.84 44.64 1.39 1.39 DWL 9 0.96 18.00 0.96 46.50 18.00x̄ dwl=sum 46.50 1.45 17.28 1.45 17.28 44.64 44.64 1.39 1.39 x̄ dwl=sum (ViXi)/sumVi (ViXi)/sumVi DWL 100.96 32.00 0.96 46.50 32.00 x̄ 46.50 1.45 30.72 1.45 30.72 44.64 44.64 1.39 1.39 dwl=x̄ dwl= 322.56322.56 divided divided by13.44 by13.44 DWL 110.96 DWL 40.00 0.96 46.50 40.00 46.50 1.45 38.40 1.45 DWL38.40 44.64 1.39 5 DWL 5 6 DWL 44.64 6 1.39 DWL 7 DWL 7 TRI 1 TRI 1 x̄ dwl=x̄ dwl=24.00 24.00 DWL 120.96 54.00 0.96 TRI46.50 54.00 1.45 51.84 1.45 51.84 44.64 44.64 1.39 1.39 3 TRI 3 46.50 DWL 130.96 68.00 0.96 46.50 68.00ȳ dwl=sum 46.50 1.45 65.28 1.45 65.28 44.64 44.64 1.39 1.39 ȳ dwl=sum (ViZi)/sumVi (ViZi)/sumVi DWL 140.96 82.00 0.96 46.50 82.00TOTAL 46.50 1.45 1.45 19.49 78.72 44.64 44.64 1.39 1.39 TOTAL ȳ= ȳ=78.72 19.49 divided divided by13.44 by13.44 SUM 13.44 13.44 572.16 572.16 322.56 322.56 19.49 19.49 ȳ= ȳ= 1.45 1.45

1

TOTALTOTAL 03 3

TRI 1 5 TRI 5

2

0

2

0

3

BLOB 1

3

BLOB 1

BLOB BLOB 1 1

z̄ =sum (ViZiDi)/sumVi*Di z̄=sum (ViZiDi)/sumVi*Di z̄ dwl=sumz̄ (ViZi)/sumVi dwl=sum (ViZi)/sumVi TRI 5 TRI 5 2 2 4 4 4 4 4 4 3 3 by13.44 4 4 3 4 4 4 4 z̄= 6387.41 z̄ = divided 6387.41 by133.74 divided by133.74 z̄ dwl= z̄572.16 dwl= divided 572.16 by13.44 divided in3 lbs*in lbs*in lbs*in lbs*in lbs*in lbs*in CoG CoG (end (end position) position) Dowels Dowels (same (same position) position) unit unit unit in in 2 TRI in 2 in in in in lbs*inlbs*inlbs*inlbs*in in3 in3 in in in lbs*inlbs*in TRI unit 2 TRIin2 in in TRI z̄= 47.76 z̄ = 47.76 z̄ dwl= z̄ 42.57 dwl= TRI 442.57 TRIuse 4 rhino use use rhino use DWL 2 DWL 2 DWL 3 DWL 3 4 DWL 4 DWL 5 rhino DWL 5 DWL 6 DWL 6 DWL 7 DWL 7 TRI rhino 2 TRI 2 i i vi (=b*h*t) vi (=b*h*t) Di (lbs/in3zi Di (lbs/in3zi xi xi yi yi Vi*Zi*Di Vi*Zi*Di Vi*Xi*Di Vi*Xi*Di Vi*Yi*Di Vi*Yi*Di i i vi vi zi zi xi xi yi yi Vi*Zi Vi*Zi Vi*Xi Vi*Xi Vi*Yi Vi*Yi REC 1 REC 1 x̄=sum (ViXiDi)/sumVi*Di x̄=sum (ViXiDi)/sumVi*Di x̄ dwl=sumx̄(ViXi)/sumVi dwl=sum (ViXi)/sumVi PIECE PIECE 0 0 3024 30240.02 0.02 42.00 42.00 24.00 24.000.38 0.38 2540.16 2540.16 1451.52 1451.52 22.98 22.98 DWL 1DWL 1BLOB 0.96 BLOB 2 0.96 2 4.00 4.001.50 1.501.45 1.45 3.84 3.84 1.44 1.44 1.39 1.39 x̄= 2952.31 x̄= divided 2952.31 by133.74 divided by133.74 x̄ dwl= x322.56 ̄ dwl= divided 322.56 by13.44 divided by13.44 TOTALTOTAL 1 1 PIECE PIECE 1 1 133.73133.730.02 0.02 28.27 28.27 24.00 24.009.11 9.11 75.61 75.61 64.19 64.19 24.37 24.37 DWL 2DWL 2 0.96 0.9618.00 18.001.50 1.501.45 1.4517.28 17.28 1.44 1.44 1.39 1.39 x̄= 22.07 x̄= 22.07 x̄ dwl= x̄ 24.00 dwl= 24.00 DWL 2 DWL 3DWLDWL 3 4DWL 4 DWL 5DWL 5 DWL 6DWL 6 DWL 7DWL 7 TRI 1Y 1DWL 1TRI 1 DWL 2DWL PIECE PIECE 2 2 285.55285.55 0.02 0.02 35.27 24.00 24.00 17.88 17.88201.43201.43 137.06137.06 102.11102.11 YDWL DWL10 3DWL 10 0.96 0.96 32.00 32.00 1.50DWL 1.50 1.45 1.45 30.72 30.72 1.44 1.44 1.39 1.39 BLOB 2 21 35.27 BLOB 1 BLOB BLOB TRI 83DWL 8TRI 3 DWL 9DWL 9DWL 3DWL DWL 11 DWL 11 DWL 12 12 DWL 13 DWL 13 DWL 14 DWL 14 PIECE PIECE 3 3 596.43596.430.02 0.02 44.51 44.51 24.00 24.00 17.77 17.77530.94530.94 286.29286.29 211.97211.97 DWL 4DWL 4 0.96 0.9640.00 40.001.50 1.501.45 1.4538.40 38.40 1.44 1.44 1.39 1.39 ȳ=sum (ViZiDi)/sumVi*Di ȳ=sum (ViZiDi)/sumVi*Di ȳ dwl=sumȳ(ViZi)/sumVi dwl=sum (ViZi)/sumVi 0 0 4 TRI 4 TRI TRI 5 TRI 5 TRI 1 TRI 1 6.00 6.000.02 43.32 18.85 18.85 16.80 16.80 5.20 5.202.26 2.262.02 2.02 DWL 5DWL 5 0.96 0.9654.00 54.001.50 1.501.45 1.4551.84 51.84 1.44 1.44 1.39 1.39 Z ZTOTAL REC 0.02 2REC 2 43.32 TOTAL ȳ= 1103.62ȳ= 1103.62 divided by133.74 divided by133.74 ȳ= 19.49 ȳ= divided 19.49 by13.44 divided by13.44 TRI 2 TRI 2 6.00 6.000.02 0.02 43.32 43.32 29.15 29.15 16.80 16.80 5.20 5.203.50 3.502.02 2.02 DWL 6DWL 6 0.96 0.9668.00 68.001.50 1.501.45 1.4565.28 65.28 1.44 1.44 1.39 1.39 REC 11REC 1 ȳ= DWL 9DWL 8.25 9 DWL 10 ȳ= 1.45 ȳ= 1.45 BLOB 1 BLOB DWL 10 DWL 11 DWL 11 DWL 12 DWL 12 DWL 13 DWL 13 DWL 14 DWL 14 DWLȳ=8DWL 8 8.25 TRI 3 TRI 3 6.00 6.000.02 0.02 43.32 43.32 18.85 18.851.42 1.42 5.20 5.202.26 2.260.17 0.17 DWL 7DWL 7 0.96 0.96 82.00 REC 2REC 2 82.001.50 1.501.45 1.4578.72 78.72 1.44 1.44 1.39 1.39 3 3 TRI 1 TRI 1 TRI 4 TRI 4 6.00 6.000.02 0.02 43.32 43.32 29.15 29.151.42 1.42 5.20 5.203.50 3.500.17 0.17 DWL 8DWL 8 0.96 0.96 4.00 4.00 46.50 46.501.45 1.45 3.84 3.8444.64 44.64 1.39 1.39 0 0 1 TRI 5 1 TRI 52 2 5 TRI 5 TOTAL6.00 6.000.02 0.02 22.94 22.94 24.00 24.001.42 1.42 2.75 2.752.88 2.880.17 0.17 DWL 9DWL 9 0.96 0.9618.00 18.00 46.50 46.501.45 1.4517.28 17.2844.64 44.64 1.39 1.39 2 2 TRITOTAL BLOBBLOB 2 2 1 1 0 0 3 3 Rec 1 Rec 1 252 2520.02 0.02 42.00 42.001.50 1.50 -0.50 -0.50211.68211.687.56 7.56 -2.52 -2.52 DWL 10 DWL 10 0.96 0.9632.00 32.00 46.50 46.501.45 1.4530.72 30.7244.64 44.64 1.39 1.39 TRI 5 TRI 5 2 2 Rec 2 Rec 2 252 2520.02 0.02 42.00 42.00 46.50 -0.50 -0.50211.68211.68 234.36234.36 -2.52 -2.52 DWL 11 DWL 11 0.96 0.9640.00 40.00 46.50 46.501.45 1.4538.40 38.4044.64 44.64 1.39 1.39 4 4 4 46.50 3 4 lbs*in lbs*in lbs*in44 lbs*in4Dowels (same inTRI inin3 in3 in3 lbs*in lbs*in lbs*in44 lbs*in lbs*in44 lbs*in4 lbs*in lbs*in position) ) Dowels position) (same position) unit unit unit unit in in in in in in in in in in TRI 2 2 3 3 TRI 2 TRI 2 DWL total DWL total 13.44 13.440.02 0.02 42.57 42.57 24.00 24.001.45 1.45 11.44 11.446.45 6.450.39 0.39 DWL 12 DWL 12 0.96 0.9654.00 54.00 46.50 46.501.45 1.4551.84 51.8444.64 44.64 1.39 1.39 TRI 4 use TRIrhino 4 use rhino use rhino use rhinoBLOB 1BLOB 1 TRI 2 TRIDWL 2 13 30 30 1 61.03 1 61.038.40 8.40 30.79 30.79 1830.90 1830.90 252.00252.00 923.70923.70 DWL 13 0.96 0.9668.00 68.00 46.50 46.501.45 1.4565.28 65.2844.64 44.64 1.39 1.39 i i vi (=b*h*t)viDi (=b*h*t) (lbs/in3zi Di 2(lbs/in3zi yi 29.29 Vi*Xi*Di Vi*Yi*Di Vi*Xi*Di Vi*Yi*Di i vi vizi RECDWL xi 14 yixi 0.96 Vi*Zi yi Vi*Xi Vi*Xi Vi*Yi Vi*Yi 2zi REC 2 14 BLOB BLOB 2xi 12yixi 12 1Vi*Zi*Di 1 Vi*Zi*Di 29.29 41.54 41.54 10.69 10.69 351.48351.48 498.48498.48 128.28128.28 i DWL 0.9682.00Vi*Zi 82.00 46.50 46.50 1.45 1.45 78.72 78.7244.64 44.64 1.39 1.39 PIECE 0 PIECE 0Z 3024 3024 0.02 42.00 0.02 4629.15 24.00 42.00 24.00 0.38 2540.16 0.38 1451.52 2540.16 1451.52 22.98 5988.87 22.98 DWL 1 0.96 0.96 4.00 1.50 4.00 1.45 1.5013.441.45 3.84 3.84 1.44 1.44 1.39 572.16 1.39 BLOB 2 BLOB 2 TOTAL TOTAL13.44 SUM SUM 4629.15 5988.87 2952.31 2952.31 1413.30 1413.30 DWL 1 SUM SUM 572.16 322.56322.5619.49 19.49 Z REC 1REC 1 1 2 1 2 0.96 1DWL DWL 2DWL 228.27 3 DWL DWL 3 4 DWL 4 DWL 5 DWL 5 DWL 6 DWL 6 DWL 7 DWL 7 PIECE 1DWL PIECE 133.73 11 133.73 0.02 0.02 DWL 24.00 28.27 24.00 9.11 9.11 75.61 64.19 75.61 24.37 64.19 24.37 DWL DWL 18.00 0.96 18.00 1.50 1.45 1.50 17.28 1.45 17.28 1.44 1.44 1.39 1.39 DWL 8DWL 8 DWL 9DWL 9 DWL DWL DWL6 12 DWL 12 6 DWL 7 DWL 13 DWL 13 11 DWL DWL 1 DWL 1 DWL 2 10 DWL10 2 DWL 3 DWL DWL 3 4 11 DWL 4 DWL 5 DWL 5 DWL DWL DWL 7 Y Y PIECE 2 PIECE285.55 2 285.55 0.02 2 35.27 0.02Vi*Di 24.00 35.27 17.88 24.00 17.88 201.43 137.06 201.43 102.11 137.06 102.11 DWL 3 DWL 3 0.96 32.00 0.96 32.00 1.50 1.45 1.50 30.72 1.45 30.72 1.44 1.44 1.39 1.39 BLOB BLOB 2 =Vi*Di = 133.74133.74 DWL1011 DWL 11DWL 12 DWL 12 DWL 13 DWL 13 DWL 14 DWL 14 DWL 8 DWL 8 DWL 9 DWL 9DWL 10 DWL X X PIECE 3 PIECE596.43 3 596.43 0.02 44.51 0.02 24.00 44.51 17.77 24.00 17.77 530.94 286.29 530.94 BLOB 211.97 286.29 DWL 4 DWL 4 0.96 40.00 0.96 40.00 1.50 1.45 1.50 38.40 1.45 38.40 1.44 1.44 1.39 1.39 BLOB 1 1211.97 TRI 4 0 54.00 TRI 4 TRI 5 5 TRI 5 5 0.96 0 54.00 TRI 1 TRI 1 6.00 0.02 6.00 0.02 43.32 16.80 18.85 16.80 5.20 2.26 5.20 2.02 2.26 2.02 DWL DWL 0.96 1.50 1.45 51.84 1.45 51.84 1.44 1.44 1.39 1.39 Z Z REC 2 z̄43.32 REC(ViZiDi)/sumVi*Di 2 18.85 z̄ =sum (ViZiDi)/sumVi*Di =sum z̄ dwl=sum z̄1.50 dwl=sum (ViZi)/sumVi (ViZi)/sumVi TRI 2 TRI 2 6.00 0.02 6.00 43.32 0.02 43.32 16.80 29.15 16.80 5.20 3.50 5.20 2.02 3.50 2.02 DWL 6 DWL 6 0.96 68.00 0.96 68.00 1.50 z̄ 1.45 1.50z̄ dwl= 65.28 1.45 572.16 65.28 1.44 1.44 1.39 1.39 z̄ = 12 z̄5988.87 =29.15 5988.87 divided divided by133.74 by133.74 dwl= 572.16 divided divided by13.44 by13.44 REC 1 REC 1 DWL 9 DWL 9DWL 10 DWL DWL 10 11 DWL 11 DWL DWL 12 DWL 13 DWL 13 DWL 14 DWL 14 TRI 3 TRI 3 6.00 0.02 6.00 43.32 0.02 18.85 43.32 18.85 1.42 1.42 5.20 2.26 5.20 0.17 2.26 0.17 DWL 7 DWL 7 0.96 82.00 0.96 82.00 1.50 1.45 1.50 78.72 1.45 78.72 1.44 1.44 1.39 1.39 REC 2 REC 2 z̄ = z̄ = 44.78 44.78 z̄ dwl=z̄ dwl=42.57 42.57 BLOB BLOB 1 1 TRI 4 TRI 4 6.00 0.02 6.00 43.32 0.02 29.15 43.32 29.15 1.42 1.42 5.20 3.50 5.20 0.17 3.50 0.17 DWL 8 DWL 8 0.96 0.96 4.00 46.50 4.00 46.50 1.45 1.45 3.84 44.64 3.84 44.64 1.39 1.39 TRI 1 TRI 1 TRI 3 TRI 3 TRI 5 TRI 5 6.00 0.02 6.00 0.02 (ViXiDi)/sumVi*Di 24.00 22.94 24.00 1.42 1.42 2.75 2.88 2.75 0.17 2.88 0.17 DWL 9 DWL 9 0.96 18.00 0.96 46.50 18.00x̄ dwl=sum 46.50 1.45 17.28 1.45 17.28 44.64 44.64 1.39 1.39 x̄=sumx22.94 ̄ (ViXiDi)/sumVi*Di =sum x ̄ dwl=sum (ViXi)/sumVi (ViXi)/sumVi BLOB 2 BLOB 2 TOTALTOTAL Rec 1 Rec 1 252 0.02 252 42.00 0.02 1.502952.31 -0.50 1.50 211.68 -0.50 7.56 -2.52 7.56 -2.52 DWL 10 DWL 100.96 32.00 0.96 46.50 32.00 x̄ 46.50 1.45 30.72 1.45 322.56 30.72 44.64 44.64 1.39 1.39 x̄= x2952.31 ̄ = 42.00 divided divided by133.74 by133.74211.68 dwl=x̄ dwl= 322.56 divided divided by13.44 by13.44 3 3 2 Rec 2 Rec 2 252 0.02 0.02 42.00 -0.50 46.50 -0.50 234.36 211.68 234.36 -2.52 2 -2.52 DWL 11 DWL 110.96 40.00 0.96 46.50 40.00 x̄ 46.50 1.45 1.45 24.00 38.40 44.64 44.64 1.39 1.39 5 22.07 0 0 211.68 1252 1 42.00 x̄TRI = 5x̄TRI =46.50 22.07 dwl=x̄ dwl=38.40 24.00 DWL total DWL total 13.44 13.44 0.02 42.57 0.02 24.00 42.57 24.00 1.45 1.45 11.44 11.44 6.45 0.39 6.45 0.39 DWL 12 DWL 120.96 54.00 0.96 46.50 54.00 46.50 1.45 51.84 1.45 51.84 44.64 44.64 1.39 1.39 TRI 2 TRI923.70 2 4 TRI 41830.90 BLOB 1 BLOB 1 30 30 1 61.03 1 (ViZiDi)/sumVi*Di 61.03 8.40 30.79 8.40 TRI 1830.90 30.79 252.00 923.70 252.00 DWL 13 DWL 130.96 68.00 0.96 46.50 68.00ȳ dwl=sum 46.50 1.45 65.28 1.45 65.28 44.64 44.64 1.39 1.39 ȳ=sum ȳ=sum (ViZiDi)/sumVi*Di ȳ dwl=sum (ViZi)/sumVi (ViZi)/sumVi REC 2 REC 2 BLOB 2 BLOB 2 12 12 1 29.29 29.29 10.69 41.54 10.69 351.48 351.48 128.28 498.48 128.28 DWL 14 DWL 140.96 82.00 0.96 46.50 82.00 46.50 1.45 1.45 19.49 78.72 44.64 44.64 1.39 1.39 ȳ= 11413.30 ȳ=41.54 1413.30 divided divided by133.74 by133.74498.48 ȳ= ȳ=78.72 19.49 divided divided by13.44 by13.44 BLOB BLOB 2 2 SUM SUM4629.15 4629.15 5988.87 2952.31 5988.87 1413.30 2952.31 1413.30 SUM SUM 13.44 13.44 572.16 572.16 322.56 322.56 19.49 19.49 REC 1 REC ȳ= 1ȳ= 10.57 10.57 ȳ= ȳ= 1.45 1.45 DWL 2

DWL 2 DWL 3 DWL 4 3 DWL 4 DWL 8DWL DWL 8

Vi*Di =

Vi*Di 133.74 =

DWL 5

DWLDWL 5 DWL 9 6DWLDWL 9 6 DWL 7

DWL 7DWL 10 DWL 10

Y

BLOB 2 BLOB 1 BLOB BLOB 21

z̄ =sum (ViZiDi)/sumVi*Di z̄=sum (ViZiDi)/sumVi*Di REC 2REC 2 z̄= 5988.87 z̄ = divided 5988.87 by133.74 divided by133.74 DWL 12 DWL 12 DWL 8z̄ = DWL 9DWL 9 DWL 10 DWLDWL 10 11 DWL 11 z̄= 8DWL44.78 44.78 TRI 3

TRI 3

x̄=sum (ViXiDi)/sumVi*Di x̄=sum (ViXiDi)/sumVi*Di TOTAL TOTAL x̄= 2952.31 x̄= divided 2952.31 by133.74 divided by133.74 3 TRI 1 1x̄= 5 22.07 x̄TRI = 5 0 22.07 0 TRI 4

TRI 4

ȳ=sum (ViZiDi)/sumVi*Di ȳ=sum (ViZiDi)/sumVi*Di ȳ= 1413.30ȳ= 1413.30 divided by133.74 divided by133.74 ȳ= 10.57 ȳ= 10.57

20

BLOB 2

BLOB 2

TRI 1 32 TRI 2

DWL 13 DWL 13

DWL 14 DWL 14

TRI 1

TOTAL TOTAL DWL 13 DWL 13

DWL 12 DWL 12

DWL 11 DWL 11

133.74

DWL 14 DWL 14

TRI 1 TRI 1 DWL 3DWLDWL 3 4DWL DWL 2DWL 2 TRI43 TRI 3DWL 5DWL 5 TRI 2 DWL TRI 26DWL 6 DWL5 10 DWLDWL 10 11 DWL 9DWL 9TRI 5 TRI DWL DWL 12 DWL 12 DWL 13 DWL 13 TRI11 4 TRI 4 z̄ dwl=sumz̄ (ViZi)/sumVi dwl=sum (ViZi)/sumVi 2 2 0 0 1 1 3 3 Z Zz̄ dwl= z̄572.16 dwl= divided 572.16 by13.44 divided by13.44

Y DWL 1DWL 1 DWL 8DWL 8

z̄ dwl=

z̄ 42.57 dwl=

REC 1,REC REC1,2REC 2 BLOB 1 BLOB 1

42.57

x̄ dwl=sumx̄(ViXi)/sumVi dwl=sum (ViXi)/sumVi x̄ dwl= x322.56 ̄ dwl= divided 322.56 by13.44 divided by13.44 x̄ dwl= x̄ 24.00 dwl= 24.00

2 TRI 2

ȳ dwl=sumȳ(ViZi)/sumVi dwl=sum (ViZi)/sumVi ȳ= 19.49 ȳ= divided 19.49 by13.44 divided by13.44 BLOB 2 BLOB 2 ȳ= 1.45 ȳ= 1.45 TOTAL

Y

DWL 1 DWL 8

Y

DWL 1 DWL 2 DWL 8 DWL 9

TOTAL

DWL 43 TRI DWL DWL DWL 2 DWL 3 3 4 DWL TRI 35 TRI 95 DWLTRI 10 5 DWL DWL 11 10 TRI DWL DWL 4 11 DWL TRI 412

TRI 1 TRI 1 DWL 6 2 DWL 6 DWL 7 TRI52 DWL TRI DWL 12DWL 13 DWL 13DWL 14

DWL 7 DWL 14

DWL 7DWL 7 DWL 14 DWL 14

(CJ) YANG YANG Y) (AMY) BAOBAO

ȳ=sum ȳ=sum (ViZiDi)/sumVi*Di (ViZiDi)/sumVi*Di 54.00 DWL DWL 5,divided DWL 5, DWL 12 12 ȳ=54.00 ȳ= 1103.62 1103.62 divided by133.74 by133.74 40.36 40.36 BLOB BLOB 2 2 ȳ= ȳ= 8.25 8.25 40.00 40.00 DWL DWL 4, DWL 4, DWL 11 11 x̄=sum x(ViXiDi)/sumVi*Di ̄ =sum (ViXiDi)/sumVi*Di 32.00 32.00 DWL DWL 3, DWL 3, DWL 10 10 x̄=44.78 2952.31 x̄=44.78 2952.31 divided divided by133.74 by133.74 TOTAL TOTAL 18.00 18.00 DWL DWL 2, DWL 2, DWL 9 9 x̄=82.00 x̄=22.07 82.00 DWL22.07 7,DWL DWL7,14 DWL 14 z̄ =

z̄ =47.76 47.76

ȳ dwl=sum ȳ dwl=sum (ViZi)/sumVi (ViZi)/sumVi ȳ= ȳ= 19.49 19.49 divided divided by13.44 by13.44 ȳ= ȳ= 1.45 1.45

z̄ dwl= z̄ dwl= 42.57 42.57

x̄ dwl=sum x̄ dwl=sum (ViXi)/sumVi (ViXi)/sumVi x̄ dwl= x̄ dwl= 322.56322.56 divideddivided by13.44by13.44 x̄ dwl= x̄ dwl= 24.00 24.00

4.00 4.00 DWL DWL 1, DWL 1, DWL 8 8 68.00 68.00 DWL 6,DWL DWL6,13 DWL 13 44.51 44.51 3 3

ATTACHMENTS ATTACHMENTS TOTAL TOTAL

61.03 61.03 BLOB BLOB 1 1 ȳ=sum ȳ=sum (ViZiDi)/sumVi*Di (ViZiDi)/sumVi*Di ȳ dwl=sum ȳ dwl=sum (ViZi)/sumVi (ViZi)/sumVi 3 3 4 4 4 4 4 4 4 4 3 3 TRI 4 4 4 4 43.32 43.32 TRI 1, TRI 1, TRI 2, TRI 2, TRI 3, TRI lbs*in lbs*in lbs*in lbs*in in in lbs*in lbs*in lbs*in lbs*in lbs*in lbs*in lbs*in lbs*in CoG CoG (end (end position) position) Dowels Dowels (same (same position) position) unitunit unit in in in in in in in in3, TRI in4 in4 in in DWLdivided 5,DWL DWL 5,unit 12 DWL 12in in ȳ=54.00 1103.62 ȳ=54.00 1103.62 divided by133.74 by133.74 ȳ= ȳ= 19.49 divided 19.49 divided by13.44by13.44 35.27 35.27 2 2 40.36 40.36 BLOB BLOB 2 2 use use rhino rhino 5 5 useuse rhino rhino 29.44 29.44 TRITRI ȳ= ȳ= 8.25 8.25 ȳ= ȳ=1.45 1.45 40.00 40.00 DWL 4,DWL DWL DWL 11vi (=b*h*t) i 4,i11 vi (=b*h*t) Di (lbs/in3zi xi xi yi yi Vi*Zi*Di Vi*Zi*Di Vi*Xi*Di Vi*Xi*DiVi*Yi*Di Vi*Yi*Di i i vi vi zi zi xi xi yi yi Vi*Zi Vi*Zi Vi*Xi Vi*Xi Vi*Yi Vi*Yi 28.27 28.27 1 1Di (lbs/in3zi 32.00 32.00 DWL 3,DWL DWL 3,PIECE 10 DWL PIECE 0 010 3024 3024 0.02 0.02 24.00 0.38 0.38 2540.16 2540.161451.52 1451.52 22.98 22.98 DWL DWL 1 1 0.96 0.96 4.00 4.00 1.50 1.50 1.45 1.45 3.84 3.84 1.44 1.44 1.39 1.39 42.00 42.00 REC REC 1, REC 1, REC 2, 02,42.00 0 42.00 24.00 18.00 18.00 DWL 2,DWL DWL2,9DWL 9 PIECE PIECE 1 1 133.73 133.73 0.02 0.02 28.27 28.27 24.00 24.00 9.11 9.11 75.61 75.61 64.19 64.19 24.37 24.37 DWL DWL 2 2 0.96 0.96 18.00 18.00 1.50 1.50 1.45 1.45 17.28 17.28 1.44 1.44 1.39 1.39 4.00 4.00 DWL 1,DWL DWL1,8DWL 8 PIECE 2 2 285.55 285.55 0.02 0.02 35.27 35.27 24.00 24.00 17.88 17.88 201.43 201.43 137.06 137.06102.11 102.11 DWL DWL 3 3 0.96 0.96 32.00 32.00 1.50 1.50 1.45 1.45 30.72 30.72 1.44 1.44 1.39 1.39 44.51 44.51 3 3 PIECE PIECE PIECE 33,3TRI 596.43 0.02 0.02 44.51 24.00 17.77 286.29211.97 211.97 DWL DWL 4 4 0.96 0.96 40.00 1.50lbs*in 1.45 1.44 1.44 1.39 1.39 4 44.51 4 24.00 4 4 17.77 4 4 530.94 286.29 3 4 4 38.40 3 4 1.45 4 40.004 1.50 1, TRI TRI 1, 2, TRI TRI TRIin3, 4 596.43 TRI lbs*in lbs*in530.94 in3 in in3 43.32 inTRI lbs*in4lbs*in lbs*in38.40 position) on) Dowels Dowels (same (same position) position) unit unit in unit unit43.32 in in in in in lbs*in lbs*in in 2, in in4 in Z Z in lbs*in lbs*inlbs*in lbs*in 1 1 6.00 6.00 0.02 0.02 43.32 43.32 18.85 18.85 16.80 16.80 5.20 5.20 2.26 2.26 2.02 2.02 DWL DWL 5 5 0.96 0.96 54.00 54.00 1.50 1.50 1.45 1.45 51.84 51.84 1.44 1.44 1.39 1.39 35.27 35.27 2 2 TRITRI DWL DWL 1 1 DWL DWL 2 2 DWL DWL 3 3 DWL DWL 4use 4REC DWL DWL 5 5 DWL DWL 6 6 REC 1 1 use rhino use rhino use rhino rhino 29.44 29.44 TRI 5 TRI 5 TRITRI 2 2 6.00 6.00 0.02 0.02 43.32 43.32 29.15 29.15 16.80 16.80 5.20 5.20 3.50 3.50 2.02 2.02 DWL DWL 6 6 0.96 0.96 68.00 68.00 1.50 1.50 1.45 1.45 65.28 65.28 1.44 1.44 1.39 1.39 i i 28.27 vi (=b*h*t) vi1(=b*h*t) Di (lbs/in3zi (lbs/in3zi xi xi yi yi Vi*Zi*Di Vi*Zi*Di Vi*Xi*Di Vi*Xi*Di Vi*Yi*Di Vi*Yi*Di i i vi vi zi zi xi xi yi yi Vi*Zi Vi*ZiVi*Xi Vi*XiVi*Yi Vi*Yi 28.27 1 Di TRITRI 3 3 6.00 6.00 0.02 0.02 43.32 43.32 18.85 18.85 1.42 1.42 5.20 5.20 2.26 2.26 0.17 0.17 DWL DWL 7 7 0.96 0.96 82.00 82.00 1.50 1.50 1.45 1.45 78.72 78.72 1.44 1.44 1.39 1.39 PIECE 0PIECE 0 42.00 3024 0.02 0.02 24.00 24.00 0.38 0.38 2540.16 2540.16 1451.52 1451.52 22.98 DWL 1 DWL 1 0.96 0.964.00 4.00 1.50 1.45 1.44 1.39 42.00 REC3024 1,REC REC 1,2, 0 442.00 2, 0 42.00 TRI TRI 4REC 6.00 6.00 0.02 0.02 43.32 43.32 29.15 29.15 1.42 1.4222.985.20 5.20 3.50 3.50 0.17 0.17 DWL DWL 8 1.50 8 0.96 0.961.453.84 4.00 4.003.84 46.50 46.50 1.44 1.45 1.45 1.39 3.84 3.84 44.64 44.64 1.39 1.39 PIECE 1PIECE 1 133.73133.730.02 0.02 28.27 28.27 24.00 24.00 9.11 9.11 75.61 75.61 64.19 64.19 24.37 24.37 DWL 2 DWL 2 0.96 0.96 18.00 18.00 1.50 1.50 1.45 1.45 17.28 17.28 1.44 1.44 1.39 1.39 TRITRI 5 5 X X 6.00 6.00 0.02 0.02 22.94 22.94 24.00 24.00 1.42 1.42 2.75 2.75 2.88 2.88 0.17 0.17 DWL DWL 9 9 0.96 0.96 18.00 18.00 46.50 46.50 1.45 1.45 17.28 17.28 44.64 44.64 1.39 1.39 PIECE 2PIECE 2 285.55285.550.02 35.27 35.27 24.00 24.00 17.88 17.88 201.43 201.43 137.06 137.06 102.11 102.11 DWL 3 DWL 3 0.96 0.9632.00 32.00 1.45 30.72 1.44 1.39 RecRec 10.02 1 252252 0.02 0.02 42.00 42.00 1.50 1.50 -0.50 -0.50 211.68 211.68 7.56 7.56 -2.52 -2.52 DWL DWL 101.50 10 1.50 0.96 0.961.4530.72 32.00 32.00 46.50 46.50 1.44 1.45 1.45 1.39 30.72 30.72 44.64 44.64 1.39 1.39 BLOB BLOB 1 1 PIECE 3PIECE 3 596.43596.43 44.51 44.51 24.00 24.00 17.77 17.77 530.94 530.94 286.29 286.29 211.97 211.97 DWL 4 DWL 4 0.96 0.9640.00 40.00 1.45 38.40 1.44 1.39 Z 0.02 ZRec Rec 20.02 2 252252 0.02 0.02 42.00 42.00 46.50 46.50 -0.50 -0.50 211.68 211.68 234.36 234.36 -2.52 -2.52 DWL DWL 111.50 11 1.50 0.96 0.961.4538.40 40.00 40.00 46.50 46.50 1.44 1.45 1.45 1.39 38.40 38.40 44.64 44.64 1.39 1.39 TRI 1 TRI 1 6.00 6.000.02 0.02 43.32 43.32 18.85 18.85 16.80 16.80 5.20 5.20 2.26 2.26 2.02 2.02 DWL 5 DWL 5 0.96 0.96 54.00 54.00 1.50 1.50 1.45 1.45 51.84 51.84 1.44 1.44 1.39 1.39 DWL total total 13.44 13.44DWL 0.02 42.57 24.00 24.00 1.45 1.45DWL11.44 11.443 6.45 6.45 0.39 DWL DWL 12 12 0.96 0.96 54.00 54.00 46.506 1.45 1.45 51.84 51.84 44.64 44.64 1.39 1.39 DWL 1DWL 1 DWL 20.02 DWL 242.57 3DWL DWL 0.39 4DWL 4 1REC 1 DWL 5DWL 5 DWL46.50 6DWL DWL 7DWL 7 REC TRI 2 TRI 2 6.00 6.000.02 0.02 29.15 5.20 3.50 2.02 DWL 6 DWL 6 0.96 0.9668.00 68.00 1.45 65.28 1.44 1.39 BLOB BLOB 143.32 1 43.32 3029.15 3016.80 116.80 1 61.03 61.035.20 8.40 8.40 3.50 30.79 30.79 2.02 1830.90 1830.90 252.00 252.00923.70 923.70 DWL DWL 131.50 13 1.50 0.96 0.961.4565.28 68.00 68.00 46.50 46.50 1.44 1.45 1.45 1.39 65.28 65.28 44.64 44.64 1.39 1.39 TRI 3 TRI 3 6.00 6.000.02 0.02 18.85 5.20 2.26 0.17 DWL 7 DWL 7 0.96 0.9682.00 82.00 1.45 78.72 1.44 1.39 BLOB BLOB 243.32 2 43.32 1218.85 12 1.42 1 1.42 1 29.29 29.295.20 41.54 41.54 2.26 10.69 10.69 0.17 351.48 351.48 498.48 498.48128.28 128.28 DWL DWL 141.50 14 1.50 0.96 0.961.4578.72 82.00 82.00 46.50 46.50 1.44 1.45 1.45 1.39 78.72 78.72 44.64 44.64 1.39 1.39 TRI 4 TRI 4 6.00 6.000.02 0.02 43.32 43.32 29.15 29.15 1.42 1.42 5.20 5.203.50 3.50 0.17 0.17 DWL 8 DWL 8 0.96 0.964.00 4.00 46.50 46.50 1.45 1.39 SUM SUM 4629.15 4629.15 5988.87 5988.872952.31 2952.311413.30 1413.30 SUM SUM 13.44 13.441.453.84 3.8444.64 44.641.39 572.16 572.16 322.56 322.56 19.49 19.49 TRI 5 TRI X 5 22.94 22.94 24.00 24.00 1.42 1.42 2.75 2.752.88 2.88 0.17 0.17 DWL 9 DWL 9 0.96 0.9618.00 18.00 46.50 46.50 1.45 1.4517.28 17.2844.64 44.641.39 1.39 X6.00 6.000.02 0.02 Rec 1 Rec 1 252 2520.02 0.02 42.00 42.00 1.50 1.50 -0.50 -0.50 211.68211.687.56 7.56 -2.52 -2.52 DWL 10DWL 10 0.96 0.9632.00 32.00 46.50 46.50 1.45 1.4530.72 30.7244.64 44.641.39 1.39 Vi*Di Vi*Di = = 133.74 133.74 BLOB BLOB 1 1 Rec 2 Rec 2 252 2520.02 0.02 42.00 42.00 46.50 46.50 -0.50 -0.50 211.68211.68 234.36234.36 -2.52 -2.52 DWL 11DWL 11 0.96 0.9640.00 40.00 46.50 46.50 1.45 1.4538.40 38.4044.64 44.641.39 1.39 DWL total DWL total 13.44 13.440.02 0.02 42.57 42.57 24.00 24.00 1.45 1.4511.44 11.446.45 6.45 0.39 0.39 DWL 12DWL 12 0.96 0.9654.00 54.00 46.50 46.50 1.45 1.4551.84 51.8444.64 44.641.39 1.39 z̄ =sum z̄ =sum (ViZiDi)/sumVi*Di (ViZiDi)/sumVi*Di z̄ dwl=sum z̄ dwl=sum (ViZi)/sumVi (ViZi)/sumVi TRI 1 1 46.50 BLOB 1BLOB 1 30 30 1 61.03 1 61.03 8.40 8.40 30.79 30.79 1830.90 1830.90 252.00252.00 923.70923.70 DWL 13DWL 13 0.96 0.9668.00 TRI 68.00 46.50 1.45 1.4565.28 65.2844.64 44.641.39 1.39 z̄ = z̄ = 5988.87 5988.87 divided divided by133.74 by133.74 z̄ dwl= z̄ dwl= 572.16 572.16 divided divided by13.44 by13.44 TRI 3 3 46.50 BLOB 2BLOB 2 12 12 1 29.29 1 29.29 41.54 41.54 10.69 10.69 351.48351.48 498.48498.48 128.28128.28 DWL 14DWL 14 0.96 0.9682.00 TRI 82.00 46.50 1.45 1.4578.72 78.7244.64 44.641.39 1.39 z̄ = z̄ = 44.78 44.78 z̄ dwl= z̄ dwl= 42.57 42.57 SUM SUM 4629.15 4629.15 5988.87 5988.87 2952.31 2952.31 1413.30 1413.30 SUM SUM 13.44 13.44 572.16572.16 322.56322.5619.49 19.49

Dowels (no change)

TRI BLOB 2, TRI 14 29.15 8.41 24.00 1, TRI 5, 2, 3 1.50 DWL 1, DWL 2, DWL 3, DWL 4, DWL 5, DWL 6, DWL 7 TRI 1, TRI TRI 4 3 18.85 29.15 TRI 2, 2 46.50 24.00 1, TRI REC 5, 2, 3 24.00 TRI 1, TRI 0 3 18.85 1.50 REC 1 REC 2 46.50 TOTAL 0 24.00 DWL 8, DWL 9, DWL 10, DWL 11, REC DWL112, DWL 14 TOTAL 1.50 13, DWL BLOB14 2 DWL 8, DWL 9, DWL 10, DWL 11, DWL 12, DWL 13, DWL BLOB BLOB 12 DWL 1, DWL 2, DWL 3, DWL 4, DWL 5, DWL 6,BLOB DWL 17 TRI6,2,DWL TRI 7 4 DWL 1, DWL 2, DWL 3, DWL 4, DWL 5, DWL 1,TRI TRI2,5,TRI 2, 34 1,TRI TRI1,5,TRI 2, 3 TRI 1,REC TRI 23 0 REC 2 REC 10 REC 1

22.07 46.50 22.07 41.54 46.50 8.41 41.54 1.50 8.41 29.15 1.50 24.00 29.15 18.85 24.00 46.50 18.85 24.00 46.50 1.50 24.00 1.50

CoG (end position)

TOTAL TOTAL

x̄=sum x̄=sum (ViXiDi)/sumVi*Di (ViXiDi)/sumVi*Di Vi*Di =Vi*Di = 133.74133.74 x̄= x̄= 2952.31 2952.31 divided divided by133.74 by133.74 x̄= x̄= 22.07 22.07 z̄=sum z̄(ViZiDi)/sumVi*Di =sum (ViZiDi)/sumVi*Di z̄ = 5988.87 z̄ = 5988.87 divideddivided by133.74 by133.74 ȳ=sum ȳ=sum (ViZiDi)/sumVi*Di (ViZiDi)/sumVi*Di z̄ = z̄ =44.78 44.78 ȳ= ȳ= 1413.30 1413.30 divided divided by133.74 by133.74 ȳ= ȳ=

x̄=sum x(ViXiDi)/sumVi*Di ̄ =sum (ViXiDi)/sumVi*Di x̄= 2952.31 x̄= 2952.31 divideddivided by133.74 by133.74 x̄= x̄=22.07 22.07

x̄ dwl=sum x̄ dwl=sum (ViXi)/sumVi (ViXi)/sumVi x̄ dwl= x̄ dwl= 322.56 322.56 divided divided by13.44 by13.44 x̄ dwl= x̄ dwl= 24.00 24.00

3 3

0(ViZi)/sumVi 0 2 2 z̄ dwl=sum z̄ dwl=sum (ViZi)/sumVi TRI 1 TRI 1 572.16572.16 divideddivided by13.44 by13.44 TRI 3 TRI 3 z̄ dwl= z̄ dwl= ȳ dwl=sum ȳ dwl=sum (ViZi)/sumVi (ViZi)/sumVi

5 5 1 1 TRITRI

z̄ dwl= z̄ dwl= 42.57 42.57

10.57 10.57

ȳ= ȳ= 19.49 19.49 divided divided by13.44 by13.44 TRITRI 2 2 ȳ= ȳ= 1.45 1.45 TRITRI 4 4 x̄ dwl=sum x̄ dwl=sum (ViXi)/sumVi (ViXi)/sumVi x̄ dwl= x̄ dwl= 322.56322.56 divideddivided by13.44by13.44 3 x̄ dwl= x̄ dwl= 24.00 24.00

TOTALTOTAL

1

TRI 1 5 TRI 5

2

0

2

3

0

ȳ=sum ȳ=sum (ViZiDi)/sumVi*Di (ViZiDi)/sumVi*Di ȳ= 1413.30 ȳ= 1413.30 divideddivided by133.74 by133.74 ȳ= ȳ=10.57 10.57

ȳ dwl=sum ȳ dwl=sum (ViZi)/sumVi (ViZi)/sumVi ȳ= ȳ= 19.49 divided 19.49 divided by13.44by13.44 ȳ= ȳ=1.45 1.45

TRI 2 TRI 2 TRI 4 TRI 4

BLOB BLOB 2 2 DWL DWL 1 1

DWL DWL 2 2

DWL DWL 3 3 DWL DWL 4 4

X X

DWL 2 DWL 2

DWL DWL 5 5

DWL DWL 6 6

DWL DWL 7 7

REC REC 1 1

Z Z

REC REC 2 2

DWL DWL 8 8

DWL 3 DWLDWL 3 4 DWL 4

DWL 5 DWL 5

DWL 6 DWL 6

REC 1 REC 1

DWL 8DWL 8

DWL DWL 9 9

DWL DWL 10 BLOB BLOB 2 10 2

BLOB BLOB 1 1

TRITRI 5 5

BLOB BLOB 1 1

REC 2REC 2

TOTAL TOTAL 9DWL 0DWL 0 1BLOB 3 3 19 BLOB

2 2

DWL DWL 13 13

DWL 7 DWL 7

TRITRI 1 1

1 1

DWL DWL 12 12

DWL DWL 11 11

DWL 10 DWL 10

DWL 12 DWL 12

DWL 11 DWL 11

DWL 13 DWL 13

DWL 14 DWL 14

2 2 3 3

TRITRI 2 2 TRITRI 21 2 BLOB 1BLOB TRI 1 TRI 1 1

1

2

2 0

TOTALTOTAL 03 3

BLOB BLOB 2 2

TRI 5 TRI 5 TRI 2 TRI 2 DWL DWL 8 8

DWL DWL 9 9

DWL DWL 10 10

DWL DWL 1 1 Y Y DWL DWL 8 8 Z Z

REC REC 2 2 DWL DWL 11 11

DWL DWL 12 12

DWL DWL 13 13

DWL 1 DWL 1 Y Y DWL 8 DWL 8

BLOB 2BLOB 2

Z

REC 2 REC 2 DWL 9 DWL 9

DWL 10DWL 10DWL 11DWL 11 Z Z DWL DWL 1 1

X X

DWL DWL 2 2

DWL DWL 14 14

DWL 12DWL 12

DWL 13DWL 13

REC REC 1 1 DWL DWL 3 3DWL DWL 4 4

DWL 2 DWL 2 DWL 9 DWL 9 Z

3 TRI 3 TRI 5 5 TRI 2 TRI 2

BLOB 2BLOB 2 TOTALTOTAL 1 DWL 3 DWLDWL 3 4 DWL 4 DWL 5 DWL 5 DWL 10DWL DWL 10 11DWL 11 DWL 12DWL 12

TRI 5 TRI 5

0

DWL DWL 6 6 DWL DWL 13 13

DWL DWL 7 7 DWL DWL 14 14

0 0 TRITRI 4 4 REC REC 1 1 REC REC 2 2 DWL 6 DWL 6 DWL 13DWL 13

DWL 7 DWL 7 DWL 14DWL 14

0 4 TRI 4 TRI

REC 1 REC 1 REC 2 REC 2

DWL 14DWL 14 DWL DWL 5 5

1

2

BLOB BLOB 2 2 TOTAL TOTAL 1 1 DWL DWL 3 3 DWL DWL 4 4 DWL DWL 5 5 DWL DWL 10 10 DWL DWL 11 11 DWL DWL 12 12

DWL DWL 2 2 DWL DWL 9 9 2

DWL DWL 6 6 BLOB BLOB 1 1

DWL DWL 7 7

21

START POSITION TOP VIEW 1:10

Rube Goldberg Center of Gravity / Reaction Assignment

DIMENSIONS IN INCHES BASE CHAIR ATTACHMENTS TOTAL

ZI QING (CJ) YANG YUN RU (AMY) BAO

Start Position Top View (1:20)

47.76 82.00 68.00 61.03 54.00 40.36 40.00 32.00 18.00 4.00 71.43 63.37 62.01 45.66 27.11 19.42 42.00

TOTAL DWL 7, DWL 14 DWL 6, DWL 13 BLOB 1 DWL 5, DWL 12 BLOB 2 DWL 4, DWL 11 DWL 3, DWL 10 DWL 2, DWL 9 DWL 1, DWL 8 2 TRI 1, TRI 2 3 TRI 3, TRI 4 TRI 5 1 REC 1, REC 2, 0

Z 22.07 46.50 41.54 8.41 1.50 29.15 24.00 18.85 46.50 24.00 1.50

DWL 1

DWL 2

DWL 3

DWL 5

DWL 4 REC 1

DWL 6

DWL 7

TOTAL DWL 8, DWL 9, DWL 10, DWL 11, DWL 12, DWL 13, DWL 14 BLOB 2 BLOB 1 DWL 1, DWL 2, DWL 3, DWL 4, DWL 5, DWL 6, DWL 7 TRI 2, TRI 4 1, TRI 5, 2, 3 TRI 1, TRI 3 REC 2 0 REC 1

X BLOB 1

TRI 1

TRI 3 TOTAL TRI 5

1

3

0

2

TRI 2

TRI 4

BLOB 2

REC 2 DWL 8

DWL 9

DWL 10

DWL 11

DWL 12

DWL 13

DWL 14

Start Position Side View (1:20)

BLOB 1

2 3 TRI 1 TRI 2

10.57 17.88 17.77 16.80 9.11 1.42 30.79 10.69 1.45 0.50 0.38

TOTAL 2 3 TRI 1, TRI 2 1 TRI 3, TRI 4, TRI 5 BLOB 1 BLOB 2 DWL (ALL) 0 REC 1, REC 2

BLOB 2 TOTAL 1

Y DWL 1 DWL 8

DWL 2 DWL 9

Z

DWL 3 DWL 10 TRI 5

DWL 4 DWL 11 TRI 3 TRI 4 0 REC 1, REC 2

DIMENSIONS IN INCHES BASE CHAIR ATTACHMENTS TOTAL

22

42.00 28.27 29.44 35.27 43.32 44.51 4.00 18.00 32.00 40.00 40.36 54.00 61.03 68.00 82.00 44.78

REC 1, REC 2, 0 1 TRI 5 2 TRI 1, TRI 2, TRI 3, TRI 4 3 DWL 1, DWL 8 DWL 2, DWL 9 DWL 3, DWL 10 DWL 4, DWL 11 BLOB 2 DWL 5, DWL 12 BLOB 1 DWL 6, DWL 13 DWL 7, DWL 14 TOTAL

DWL 5 DWL 12

DWL 6 DWL 13

DWL 7 DWL 14

END POSITION TOP VIEW 1:10

Rube Goldberg Center of Gravity / Reaction Assignment

DIMENSIONS IN INCHES BASE CHAIR ATTACHMENTS TOTAL

ZI QING (CJ) YANG YUN RU (AMY) BAO

End Position Top View (1:20)

44.78 82.00 68.00 61.03 54.00 40.36 40.00 32.00 18.00 4.00 44.51 43.32 35.27 29.44 28.27 42.00

TOTAL DWL 7, DWL 14 DWL 6, DWL 13 BLOB 1 DWL 5, DWL 12 BLOB 2 DWL 4, DWL 11 DWL 3, DWL 10 DWL 2, DWL 9 DWL 1, DWL 8 3 TRI 1, TRI 2, TRI 3, TRI 4 2 TRI 5 1 REC 1, REC 2, 0

Z

TOTAL DWL 8, DWL 9, DWL 10, DWL 11, DWL 12, DWL 13, DWL 14 BLOB 2 BLOB 1 DWL 1, DWL 2, DWL 3, DWL 4, DWL 5, DWL 6, DWL 7 TRI 2, TRI 4 1, TRI 5, 2, 3 TRI 1, TRI 3 REC 2 0 REC 1

22.07 46.50 41.54 8.41 1.50 29.15 24.00 18.85 46.50 24.00 1.50

DWL 1

DWL 2

DWL 3

DWL 4 REC 1

DWL 5

DWL 6

DWL 7

DWL 13

DWL 14

X BLOB 1

TRI 1 TRI 3 TOTAL

1

TRI 5

3

0

2

TRI 2 TRI 4

BLOB 2

REC 2 DWL 8

DWL 9

DWL 10

DWL 12

DWL 11

End Position Side View (1:20)

10.57 3.42 2.42 2.34 1.25 1.23 30.79 10.69 1.45 0.50 0.38

TOTAL TRI 1, TRI 2 TRI 3, TRI 4, TRI 5 2 1 3 BLOB 1 BLOB 2 DWL (ALL) 0 REC 1, REC 2

BLOB 1

BLOB 2 TOTAL

Y DWL 1 DWL 8

TRI 5

DWL 2 DWL 9

Z

1

DWL 3 DWL 10

TRI 3 TRI 4

DWL 4 DWL 11 0

DWL 5 DWL 12

TRI 1 TRI 2

3

DWL 7 DWL 14

DWL 6 DWL 13 2

REC 1, REC 2

DIMENSIONS IN INCHES BASE CHAIR ATTACHMENTS TOTAL

42.00 19.42 27.11 45.66 62.01 63.37 71.43 4.00 18.00 32.00 40.00 40.36 54.00 61.03 68.00 82.00 47.76

REC 1, REC 2, 0 1 TRI 5 TRI 3, TRI 4 3 TRI 1, TRI 2 2 DWL 1, DWL 8 DWL 2, DWL 9 DWL 3, DWL 10 DWL 4, DWL 11 BLOB 2 DWL 5, DWL 12 BLOB 1 DWL 6, DWL 13 DWL 7, DWL 14 TOTAL

23

oldberg of Gravity / Reaction Assignment END POSITION Start Position J) YANG TOP VIEW 1:10 n Assignment MY) BAO

START POSITION FRONT VIEW 1:10

D DIMENSIONS IN INCHES

Front View (1:10)

1

2

BASE CHAIR ATTACHMENTS TOTAL

DWL 14 DWL 13 DWL 12 DWL 11 DWL 10 DWL 9 DWL 8

TRI 2, TRI 3, TRI 4

REC 2, 0

Z DWL 3

DWL 2

DWL 41 REC 1 BLOB

DWL 5

DWL 6

DWL 7

BLOB 1

TOTAL BLOB 1 BLOB 2 DWL (ALL) TRI 1, TRI 2 TRI 3, TRI 4, TRI 5 2 1, 3 0 REC 1, REC 2

TRI 1 TRI 3

8.94 30.79 10.69 1.45 3.42 2.42 2.34 1.25 0.38 0.50

1

TOTAL

BLOB 2 TRI 5

0

2

Y

3

TOTAL TRI 2 TRI 4

DWL 1-7

DWL 9

24

DWL 10

X

DWL 8-14

TRI 5 TRI 3

1.50 24.00 46.50 18.85 24.00 29.15 1.50 8.41 41.54 46.50 22.07

TRI 2

TRI 1 2 1

3

TRI 4

0 REC 1 0 REC 2 TRI 1, TRI 3 1, TRI 5, 2, 3 TRI 2, TRI 4BLOB 2 DWL 1, DWL 2, DWL 3, DWL 4, DWL 5, DWL 6, DWL 7 BLOB 1 BLOB 2 DWL 8, DWL 9, DWL 10, DWL 11, DWL 12, DWL 13, DWL 14 TOTAL REC 2 DWL 11

DWL 12

DWL 13

DWL 14

oldberg of Gravity / Reaction Assignment END POSITION End Position TOP VIEW 1:10 n Assignment ) YANG

MY) BAO

1

2

END POSITION FRONT VIEW 1:10

DI DIMENSIONS IN INCHES

Front View (1:10)

BASE CHAIR ATTACHMENTS TOTAL

DWL 14 DWL 13 DWL 12 DWL 11 DWL 10 DWL 9 DWL 8

TRI 2, TRI 3, TRI 4

REC 2, 0

Z

BLOB41 REC 1 DWL

DWL 3

DWL 2

DWL 5

DWL 6

DWL 7

BLOB 1

2 TRI 1 TRI 3

3 TRI 1

TOTAL BLOB 1 BLOB 2 DWL (ALL) 2 3 TRI 1, TRI 2 1 TRI 3, TRI 4, TRI 5 0 REC 1, REC 2

TOTAL

TRI 5

2

0

Y

DWL 9

DWL 8-14 TRI 3

1.50 24.00 46.50 18.85 24.00 29.15 1.50 8.41 41.54 46.50 22.07 DWL 10

BLOB 2

TOTAL

3

TRI 2 TRI 4

DWL 1-7

10.57 30.79 10.69 1.45 17.88 17.77 16.80 9.11 1.42 0.38 0.50

1

TRI 2

TRI 5

TRI 4

0 X REC 1 0 REC 2 TRI 1, TRI 3 1, TRI 5, 2, 3 TRI 2, TRI 4 BLOB 2 DWL 1, DWL 2, DWL 3, DWL 4, DWL 5, DWL 6, DWL 7 BLOB 1 BLOB 2 DWL 8, DWL 9, DWL 10, DWL 11, DWL 12, DWL 13, DWL 14 TOTAL REC 2 DWL 11

DWL 12

DWL 13

DWL 14

25

2. Reactions

start position

TOP view There are 4 spacers at 4 coners of the base:

RF1=31.16#

STEP 2 - FRONT VIEW

We can assume the whole chair as a beam, and calcualte the CoG (center of gravity) as a point load on this beam:

47.76"

IEW

RB1=34.95#

NT

We visually “flatten” the chair, and look from the SIDE view to find the total reaction force action on the 2 spacers at the front and the 2 spacers at the back of the chair.

KV

CoG

FRO

STEP 1 - SIDE VIEW

BAC

RB2=41.2# 133.74#

Since we found out the whole chair’s center of gravity, we can now assume the whole hair as one object, and try to find the reaction force from the ground for both positions.

The answer from STEP 1 is a combination of reaction forces on the 2 spacers at the front/back.

VIE

STEP 3 - BACK VIEW Now, we do the same steps as STEP 2, but looking from the BACK view.

Looking at the chair from the FRONT view, we can find reaction forces on each spacer.

25.93"

22.07"

22.07" CoG

Now, we can assume the combined reaction forces as a point load acting on a beam. The 2 upwards reaction forces would add up to this point load.

36.24"

EW

E VI

SID

RF2=26.43#

W

25.43"

21.57"

25.93"

RB =76.15#

CoG CoG 47.26"

RB1

35.74"

Wchair

ΣMB2 = 0 RF =57.59#

RB

RF

RF1

Wchair = 133.74” ΣMB = 0 RF = ΣMF = 0 RB = 26

Wchair x 35.74” 35.74” + 47.26” Wchair x 47.26” 35.74” + 47.26”

RB2

21.57" 25.43"

ΣMF2 = 0 = 57.59#

RF1 = ΣMF1 = 0

= 76.15#

RF2=

RF x 25.43” 21.37” + 25.43” RF x 21.57” 21.37” + 25.43”

RF2

RB1 = ΣMB1 = 0 RB2=

= 31.16#

= 26.43#

RB x 21.57” 21.37” + 25.43” RB x 25.43” 21.37” + 25.43”

= 34.95#

= 41.20#

end position

TOP view There are 4 spacers at 4 BAC coners of the base: KV IEW

We follow the same steps to find the reaction forces of the chair at the END position.

FRO

NT

VIE

KV

RB2=38.69#

IEW

CoG

RB1=32.75# RF1=33.76#

EW

E VI

W

BAC

133.74#

SID

FRO

NT

VIE

EW

E VI

SID

W

RF2=28.63#

STEP 1 - SIDE VIEW

STEP 2 - FRONT VIEW

We visually “flatten” the chair, and look from the SIDE view to find the total reaction force action on the 2 spacers at the front and the 2 spacers at the back of the chair.

We can assume the whole chair as a beam, and calcualte the CoG (center of gravity) as a point load on this beam: 44.78"

39.22"

The answer from STEP 1 is a combination of reaction forces on the 2 spacers at the front/back.

STEP 3 - BACK VIEW Now, we do the same steps as STEP 2, but looking from the BACK view.

25.93"

Looking at the chair from the FRONT view, we can find reaction forces on each spacer.

CoG

Now, we can assume the combined reaction forces as a point load acting on a beam. The 2 upwards reaction forces would add up to this point load.

22.07"

25.43"

RB =71.35# CoG

38.72"

RB1

Wchair

RF

21.57"

25.93"

CoG

44.28"

22.07"

21.57" 25.43"

RF =62.39#

RB

Wchair = 133.74”

ΣMB2 = 0

RF1

RF2

ΣMB = 0 Wchair x 38.72” RF = = 62.39# 38.72” + 44.28”

ΣMF2 = 0 RF x 25.43” RF1 = 21.37” + 25.43”

ΣMF = 0 Wchair x 44.28” RB = 38.72" + 44.28” = 71.35#

ΣMF1 = 0 RF x 21.57” RF2= 21.37” + 25.43”

RB1 = ΣMB1 = 0 RB2=

RB2

RB x 21.57” 21.37” + 25.43”

RB x 25.43” 21.37” + 25.43”

= 38.60#

= 32.75#

= 33.76#

= 28.63# 27

3-1. Body on Chair

Posture 1 BAC

KV

We are looking to find out the reaction forces acting on our chair when a person sits on it.

IEW

BAC

KV

IEW

R4=84.16#

F1=135.45# F2=54# Wchair=133.74#

Depending on the person's sitting position, the center of gravity of the body is different.

We have assume 3 different sitting postures.

FRO

NT

VIE

W

* People will only be able to sit in the chair when it's in the END position.

Therefore, the calculations only looks at the chair in this position.

STEP 1 - BODY COMPONENTS We combine the body components as groups to simplify the calculations.

Then, we assume the locaiton of its center of gravity:

F2 F1 F3

R3=78.31#

EW

E VI

SID

FRO

NT

VIE

Now we distribute these weight on the chair, use superposition to find out the reaction force from each group of component. We add them together to find the overall reaction force at each corner caused by the body.

SID

R2=95.59#

F1 FRONT

23.50"

RF =62.39#

ΣV = 0 RF1 = RF2 =

F1

RF 2

BACK

73.55#

=

23.50"

F1

= 0.602kN x 225#/1kN

RF2

RF1

45.07"

= (90+420+92) N

23.50"

RF

For each component, we look at it from the SIDE view to find the combined reaction forced acting at the front and the back of the chair. Then we look at the FRONT/BACK view, and use the combined reaction force from above to find reactions at each spacer F1 SIDE

EW

E VI

W

STEP 2

37.93"

F1 = Head + Torso + Upp. Arms

F3=35.55#

R1=100.72#

2

= 36.78#

23.50"

RB

= 135.45# F1=135.45#

F2 = Low. Arms + Thigh = (40+200) N = 0.24kN x 225#/1kN = 54# F3 = Low. Legs + Feet = (130+28) N = 0.158kN x 225#/1kN = 35.55# 28

RF

RB

RB =61.90#

ΣMB = 0 F1 x 45.07” RF = = 73.55# 37.93” + 44.07” ΣV = 0 F1 = RB + RF RB = F1 - RF = 61.90#

RB2

RB1 ΣV = 0 RB1 = RB2 =

RB 2

=

61.90# 2

= 30.95#

30.00"

23.13"

54.00"

F2

F2

SIDE

59.87"

F3

STEP 3

SIDE

Now we add all the RF1 from F1, F2, F3, and the chair's weight together to find the total reaction force at that spacer. We repeat the same step for the other spacers.

F3

F2=54#

F3=35.55#

RB

RF ΣMB = 0 F2 x 54” RF = 30” + 54”

= 34.71#

ΣMF = 0 F2 x 30” RB = 30” + 54” F2 FRONT

23.50"

= 19.29#

RB

RF ΣMB = 0 F3 x 59.87” RF = 23.13" + 59.87"

IEW

F3 FRONT

RF

KV

IEW

S

23.50"

F1=135.45# F2=54#

= 25.64#

ΣMF = 0 F3 x 23.13” RB = = 9.91# 23.13" + 59.87" IDE VIEW FRO N TV

23.50"

BAC

KV

IEW

Wchair=133.74# R3

R1

F3=35.55# FRO

NT

23.50"

BAC

R4

VIE

E VIE

SID

W

W

R2

RF

R1 = 36.78+12.82+17.36+33.76 = 100.72# RF =34.71#

RF2

RF1

23.50"

F2 BACK

23.50"

F3 BACK

ΣV = 0 RB1 = RB2 =

2

=

19.29# 2

R4 = 30.95+4.96+9.65+38.60 = 84.16#

23.50"

RB =9.91#

RB2 RB

= 78.31#

RB

RB =19.29#

RB1

R3 = 30.95+4.96+9.65+32.75

ΣV = 0 25.64# RF = RF1 = RF2 = = 12.82# 2 2

23.50"

RB

= 95.59#

RF2

RF1

ΣV = 0 34.71# RF = RF1 = RF2 = = 17.36# 2 2

R2 = 36.78+12.82+17.36+28.63

RF =25.64#

RB2

RB1 ΣV = 0 = 9.65#

RB1 = RB2 =

RB 2

=

9.91# 2

= 4.96# 29

To find the reaction forces in this position, we follow the same procedure as Posture 1.

Posture 2 BAC

KV

IEW

BAC

KV

F1=117#

STEP 1

Wchair=133.74#

- BODY COMPONENTS

We combine the body components as groups to simplify the calculations.

Then, we assume the locaiton of its center of gravity: FR ON

TV

IEW

R3=69.02# F2=108#

R1=109.99#

EW E VI

SID

FRO

NT

VIE

33.00"

50.00"

F1

SID

R2=104.86#

23.50"

F1

F1

SIDE

EW

E VI

W

F1 F2

IEW

R4=74.87#

BACK

23.50"

RB

F1 = Torso + half Upp. Leg = (420+100) N

F1=117#

= 0.52kN x 225#/1kN

RB =46.52#

= 117# F2 = Head + Upp.Arms + Low Arms+ half Upp.Leg + Low Leg+ Feet = (90+92+40+100+130+28) N = 0.48kN x 225#/1kN = 108#

STEP 2 Now we distribute these weight on the chair, use superposition to find out the reaction force from each group of component. We add them together to find the overall reaction force at each corner caused by the body.

For each component, we look at it from the SIDE view to find the combined reaction forced acting at the front and the back of the chair. Then we look at the FRONT/BACK view, and use the combined reaction force from above to find reactions at each spacer

30

RF

RB

ΣMB = 0 RF =

ΣV = 0

F1 x 50”

= 70.48#

33” + 50"

ΣV = 0 F1 = RB + RF RB = F1 - RF = 46.52# F1

23.50"

FRONT

RB2

RB1

RB1 = RB2 =

RB 2

20.00"

F2 SIDE

23.50"

=

46.52# 2

= 23.26#

63.00" F2

RF F2=108#

RF RF =70.48#

RF1

RF2

ΣV = 0 70.48# RF = RF1 = RF2 = = 35.24# 2 2

RB

ΣMB = 0 RF =

F2 x 63” 20” + 63"

= 81.98#

ΣV = 0 F2 = RB + RF RB = F2 - RF = 26.02#

23.50"

F2

FRONT

23.50"

R1 = 35.24+44.99+33.76

RF

= 109.99# R2 = 35.24+44.99+28.63 = 104.86# R3 = 23.26+13.01+32.75

RF =81.98#

= 69.02# R4 = 12.26+13.01+38.60

RF2

RF1

= 74.87#

ΣV = 0 81.98# RF = RF1 = RF2 = = 44.99# 2 2 23.50"

F2 BACK

23.50"

RB

RB =26.02#

RB2

RB1 ΣV = 0 RB1 = RB2 =

RB 2

=

26.02# 2

= 13.01#

STEP 3 Now we add all the RF1 from F1, F2, F3, and the chair's weight together to find the total reaction force at that spacer. We repeat the same step for the other spacers.

BAC

KV

F1=117#

IEW

R4 Wchair=133.74#

R3 F2=108#

R1 FRO

NT

VIE

E VIE

SID

W

W

R2

31

To find the reaction forces in this position, we follow the same procedure as Posture 1.

Posture 3 BAC

KV

IEW

ON

TV

IEW

IEW

12# R3=74.35# F3=58.05#

R1=104.68#

EW E VI

SID

FRO

NT

F2 F1 F3

KV

F1=144.45# R4=80.20# F2=22.5# Wchair=133.74#

STEP 1 - BODY COMPONENTS We combine the body components as groups to simplify the calculations.

Then, we assume the locaiton of its center of gravity: FR

BAC

12#

37.50"

VIE

46.50"

F1

SID

R2=99.55#

23.50"

F1

F1

SIDE

EW

E VI

W

BACK

23.50"

RB

F1 = Head&neck + Torso + Upp. Arm + Low. Arm = (90+240+92+40) N

F1=144.45#

= 0.642kN x 225#/1kN = 144.45# F2 = half Upp. Leg = 0.1kN x 225#/1kN = 22.5# F2 = half Upp. Leg + Low. Leg + Feet = (130+28+100)N x 225#/1kN = 58.05#

STEP 2

RF

RB

ΣMB = 0 RF =

37.5”+46.5"

23.50"

FRONT

32

RB1 = RB2 =

RB 2

=

64.49# 2

28.50"

F2 SIDE

= 32.25#

55.50" F2

23.50"

RF

Now we distribute these weight on the chair, use superposition to find out the reaction force from each group of component. We add them together to find the overall reaction force at each corner caused by the body.

For each component, we look at it from the SIDE view to find the combined reaction forced acting at the front and the back of the chair. Then we look at the FRONT/BACK view, and use the combined reaction force from above to find reactions at each spacer

= 79.96#

RB2

RB1 ΣV = 0

F1 x46.5”

ΣV = 0 F1 = RB + RF RB = F1 - RF = 64.49# F1

RB =64.49#

F2=22.5#

RF RF =79.96#

RF1

RF2

ΣV = 0 79.96# RF = RF1 = RF2 = = 39.98# 2 2

RB

ΣMB = 0 RF =

F2 x 55.5” 28.5"+55.5"

ΣV = 0 F2 = RB + RF RB = F2 - RF = 7.63#

= 14.87#

23.50"

F2

FRONT

23.50"

23.50"

F3 FRONT

RF

23.50"

R1 = 39.98+7.44+23.5+33.76 = 104.68#

RF

R2 = 39.98+7.44+23.5+28.63 = 99.55# R3 = 32.25+3.82+5.53+32.75

RF =14.87#

RF2

RF1

F2 BACK

R4 = 32.25+3.82+5.53+38.60 = 80.20#

RF2

RF1

ΣV = 0 14.87# RF = RF1 = RF2 = = 7.44# 2 2 23.50"

= 74.35#

RF =47.00#

ΣV = 0 47.00# RF = RF1 = RF2 = = 23.50# 2 2

23.50"

RB

23.50"

F3 BACK

23.50"

RB

RB =7.63#

ΣV = 0 RB1 = RB2 =

RB =11.06#

RB2

RB1 RB 2

16.00"

F3

=

7.63#

RB2

RB1

= 3.82#

2

ΣV = 0 RB1 = RB2 =

68.00"

RB 2

=

11.06# 2

= 5.53#

SIDE

F3

STEP 3 Now we add all the RF1 from F1, F2, F3, and the chair's weight together to find the total reaction force at that spacer.

We repeat the same step for the other spacers.

F3=58.05# BAC

KV

RF ΣMB = 0 RF = ΣMF = 0 FRO

NT

VIE

RWB =

BAC

12#

KV

F1=144.45# R4 F2=22.5# Wchair=133.74#

F3 x 68”

= 47.00#

16" + 68" F3 x16.00” 16" + 68"

RB

IEW

E VIE

SID

W

= 11.06#

IEW

12# R3 F3=58.05#

R1 FRO

NT

VIE

E VIE

W

SID

W

R2

33

3-2. Overturn

start position

PUSH!

highest point

BAC

KV

If we push the highest point on the chair, how much force does it take to tip our chair over?

CoG

FRO

NT

The smallest force happens when we are pushing at the highest point of the chair in the direction of its least resistance to overturn.

h=50"

26.24"

13.66"

F1 back view

side view

Wchair

Mresisting = Wchair x 13.66" Moverturning = F1 x h

This is the smallest moment arm - creating the least resisting moment

SIDE

F1

= F1 x 50"

ÎŁM = 0, Wchair = 133.74# therefore, Mresisting = Moverturning 133.74# x 13.66" = F1 x 50"

F1 = 36.54#

26.24" highest point

50.00" CoG

34

EW

E VI

SID

W

Moverturning = F1 x h

BACK

TOP

CoG

VIE

highest point

CoG

13.66"

IEW

In conclusion, it takes 36.54# of force to tip this chair in this view.

= F1 x 50"

= 36.54# x 50"

= 1827 lbf-in

end position

highest point

BAC

KV

We follow the same steps to find the overturn force for the chair in its END position.

IEW

BAC

KV

IEW

CoG

FRO

NT

The smallest force happens when we are pushing at the highest point of the chair in the direction of its least resistance to overturn.

VIE

EW

E VI

SID

W

FRO

NT

VIE

EW

E VI

SID

W

highest point

Moverturning = F1 x h

BACK

CoG

h=50"

TOP

24.22"

= F1 x 50"

= 36.54# x 50"

= 1827 lbf-in

13.66"

F2 13.66"

CoG

back view

39.22"

side view

Wchair

Mresisting = Wchair x 13.66" Moverturning = F2 x h

This is the smallest moment arm - creating the least resisting moment

SIDE

24.22" 15.00"

F2

= F2 x 50"

ÎŁM = 0, Wchair = 133.74# therefore, Mresisting = Moverturning 133.74# x 13.66" = F2 x 50"

F2 = 36.54#

highest point

50.00" CoG

Both the start position and end position requires the same amount of force to tip the chair over : 36.54#.

In conclusion, it takes 36.54# of force to tip this chair in this view.

35

4. Pinned Frame

Posture 1 BAC

KV

From the 3 positions, we will be taking down the load, and finding out which member has the most stress.

BAC

KV

IEW

R4=84.16#

F1=135.45# F2=54# Wchair=133.74#

We find the force on each member by taking the moment at point A&B to find its vertical forces. Then, we explode the members and find the forces acting FRO NT on each member VIE separately.

W

SIDE VIEW

IEW

R3=78.31#

EW

E VI

SID

F3=35.55#

R1=100.72# FRO

NT

VIE

Overall

F2 F1

SID

R2=95.59#

member DF

ΣMA = 0 A = A

F3

EW

E VI

W

ΣV = 0 VD = VF = 108.09#

36.55# x 26.27"+54# x 30.5"+135.34# x 38.43" = VB x 84"

VB = 93.00#

23.63" 6.37" 8.43" 45.57"

Σvertical = 0 VA+VB = (54+135.45+36.55) # VA = 133.00#

#

5 # .4 54 135 #

.55

36

E

C

A 26.27"

F D

5#

# .4 54 135 VE=81.36 E VE=81.36

VC=81.36

E C #

.55

36

HA

C

=0

VA=133.00

F

VE = 81.36#

In conclusion,

ΣMD = 0 VA x (26.27+18.24)" = 36.55# x (6.37 + 8.43 + 45.57-39.49)" + VC x 18.24" + VB x 39.49"

VF=108.09

VC = 81.36#

D VD=108.09

D VD=108.09

HB

B VB=93.00

36

VF + VE = 54# + 135.45#

member AB

VF=108.09

F

ΣV = 0

because, B is a roller, there is no horizontal force HA=HB =0 therefore, Hc=HD =HE=HF =0

39.49" B

18.24"

member EF

=0

Σvertical = 0 VA+VB = VC+VD + 36.55# VD = 108.09#

In Posture 1, member AB is going through most stress.

Posture 2 BAC

KV

IEW

BAC

KV

F1=117#

We follow the same procedure to find the member experiencing the worst stress

FRO

NT

VIE

F2=108#

R1=109.99#

SID

W

FRO

NT

VIE

ΣMA = 0 A = A

F2

108# x 20.5"+117# x (20.5+7.77)" = VB x (20.5 + 7.77 + 5.23 + 11 + 39.5)"

VB = 65.73# 39.5"

Σvertical = 0 VA+VB = (117+108) # VA = 159.27#

#

117 8#

10

E C

A 26.27"

F D

18.24"

EW

E VI

W

Overall

F1

5.23" 20.5" 7.77" 11"

Wchair=133.74#

R3=69.02#

EW E VI

SIDE VIEW

IEW

R4=74.87#

SID

R2=104.86#

member CE

ΣV = 0 VC = VE = 117.01#

member DF

ΣV = 0 VF = VD = 0.01#

because, B is a roller, there is no horizontal force HA=HB =0 therefore, Hc=HD =HE=HF =0

39.49" B

#

117 VE=117.01 E VE=117.01

E C

VC=117.01

#

8 10 HA =0

A VA=159.27

C

F

member AB

VF=0.01

ΣMD = 0

F

VF=0.01

D

VD=0.01

D

VC=117.01 VD=0.01

VB x 39.5"+108# x (11+5.23+7.77)" + VC x (5.23+11)" = VA x (20.5+7.77+5.23+11)"

VC = 117.01#

HB

B

=0

In conclusion, In Posture 2, member CE is going through the most stress.

Σvertical = 0 VA+VB = VC+VD VD = -0.01#

VB=65.73

37

Posture 3 BAC

KV

IEW

NT

VIE

R3=74.35#

E

5#

.0 58

F

C

A

.5# 44 1

E C

18.97"

D

# .45

22

18.22"

5#

B

HF=23.53

F VF=88.51 VE=78.44 HE=11.53

F D

VF=88.51 HF=23.53 HD=11.53 VD=88.51

HC=11.53 VC=78.44

HA =0

VA=139.22

C

D

VC=78.44 VD=88.51

B VB=85.78

38

=0

VB = 139.22#

member EF

ΣH = 0 HF = HE -12# = (23.53 - 12)# = 11.53# member EC

ΣH = 0

85.78# x 39.5" + 58.05# x (11.02+1.48+9+7)" = VC x (1.48+16)" + 139.2# x (84- 39.5)"

VC = -78.44# ΣMB = 0 139.22# x 84"+58.05# x (84-16)" + 65.92# x (1.48+9+7+39.5)"= VD x 39.5"

VD = 88.51#

VF = VD = 88.51#

HD = 11.53#

VC = VE = 78.44#

144.45# x (39.5+7)"+22.5# x (16+39.5)"+12# x 18.97"+58.05# x (84-16)" = 12# + (18.22 + 19.97)" + VB x 84"

ΣV = 0

12# x 18.22"= HD x18.97"

ΣV = 0

ΣMB = 0 B = B

member DF

HB

HF = 23.53#

member EC

ΣMD = 0

# .05 HC=11.53 HD=11.53

58

ΣMF = 0

member AB

12#

R2=99.55#

12# x (18.22+18.97)"= HF x18.97"

VB = 85.78#

.5# 4.4 14

22

SID

ΣMD = 0

ΣMA = 0 A = A

39.5"

EW

E VI

W

58.05# x 16"+22.5# x (16+12.5)" + 144.45# x (16+12.5+9)" + 12# x (18.22+18.97)" = 12# + 18.97" + VB x 84"

12#

12#

VIE

Overall

1.48" 11.02" 9" 7"

E HE=11.53 VE=78.44

FRO

NT

12# F2 F1

12#

F3=58.05#

R1=104.68#

SID

W

12# F3

16"

IEW

12#

EW E VI

SIDE VIEW

KV

F1=144.45# R4=80.20# F2=22.5# Wchair=133.74#

We follow the same procedure to find the member experiencing the worst stress

FRO

BAC

12#

HC = HE = 11.53# member AB

because, B is a roller, there is no horizontal force HB =0, HA=Hc - HD = 0

By comparing the members in Posture 1, 2, & 3, we can see, member CE is going through the most stress.

39

5.Beam/Column Stress In this assignment, we are looking for the "beam" and "column" that we expect to experience the highest stress.

We will look at the load/shear/moment diagrams of the beam element to find the stress through axial & bending stress formulas.

For columns, we use the buckling stress formula.

BEAM

FRONT

It does not matter which body posture pre-assumptions we choose, since all the postures we did in the last chapter are all symmetrical in the front view. As the body's weight is always the same, and its CoG lcoation on the beam from the front view is the same, we only need to look at one body posture.

22.78" 24.22" P

CoG of body & chair together

R1

R2

P = 133.74# + 225# = 358.74#

ΣMR2 = 0 R1 = (358.72#x24.22")/ (22.78+24.22)" = 184.87# R2 = P - R1 = 173.87#

BEAM

24"

FRONT

BAC

KV

IEW

NT

VIE

E VIE

W

SID

184.87# SHEAR:

0

0

=CoG

-173.87#

=person's weight @ body CoG

FRO

24"

22.07" 25.93"

Mmax

y MOMENT:

x

fb = M/S

W

since S = (bh²)/6

0

0

Mmax = (P x 22.78"x24.22")/(22.78+24.22)" = 4211.24 lbs. in

= [84" x (1")²] / 6 = 14 in³ This is the BEAM that will have the highest stress, because it's taking most of the load from the seats of the chair and the body.

We need to first simplify the calculations by finding a center of gravity for the chair and the human body as one:

We will see it bending from the front view.

since we are looking at the FRONT view, we only need to look at x

Let's look at this piece closely:

i

weight (#)

xi (in)

chair

133.74

22.07

body

225

24

x = Vi Xi / sum Vi =

(133.74# x 22.07") + (225# x 24") (133.74+225)#

= 23.28" 40

Bending Stress Formula:

fb = M/S

since S = 14 in³ fb = 4211.24 lbs.in / 14 in³

= 300.80 psi Shear Stress Formula: fv = (V x Qmax)/Ib since R1 > R2, & since V = max reaction force = R1

I = (bh³)/12 = [84"x(1")³]/12

= 7 in⁴ Qmax = A x ry = bxhxry Fv = (184.87#x84"x1"x0.5")/(7 in⁴x84") = 13.16 psi

COLUMN - POSTURE 2 ACK VIE From the last chapter, W we find that Pose 2 was creating the most stress in member CE. B

FRO

NT

VIE

BAC

KV

Therefore, we will use this pose to find the buckling stress for this member EW E VI SID

W

FRO

NT

COLUMN

VIE

IEW

EW

E VI

W

SID

ry=0.5" Acr = 1"x8" = 8in² fa = P/Acr BAC

KV

IEW

= 117.01#/8in² = 14.63 psi 14.62psi is the critical buckling stress

FRO

NT

VIE

E VIE

W

SID

W

This is the COLUMN that unergoes the highest stress

Let's look at this panel closely:

P = 117.01# (from last chapter)

41

6.Combined Loading

member EF

In this assignment, we are trying to assess the stress in the member caused by both bending and axial stress together. Pose 1&2 have no members undergoing both stress at the same time. Therefore, we will be using Pose 3 for this analysis.

is in both axial & bending stress

F2 = 22.5#

12# HE=11.53#

Analysis of POSTURE 3 from PINNED-FRAME ANALYSIS:

F1 = 144.45#

HF=23.53# VE=78.44# 1.48"

9"

VF=88.51# 7"

BENDING STRESS:

SIDE VIEW

78.44#

12# F2 F1

SHEAR:

55.94#

0

0

12# F3

-88.5# BAC

K

VIE W Mmax = 619.55#"

1.48" 11.02" 9" 7"

16"

12# 12#

#

E HE=11.53 VE=78.44 E C

NT

18.97"

D

5 .5# 4.4 22 14

HA =0

VA=139.22

VE=78.44 HE=11.53

F

EV

W

SID

D

HD=11.53 VD=88.51

HC=11.53 VC=78.44

D

VC=78.44 VD=88.51

B

FRO

NT

Let's take a strip from the member, and find its bending stress: section:

VF=88.51 HF=23.53

VB=85.78

42

0 IEW

12#

HF=23.53

F VF=88.51

C

VIE

0

B

# .05 HC=11.53 HD=11.53

58

MOMENT:

FRO

F

C

A

12#

18.22"

5#

E

#

KV

116.09#"

39.5"

.5# .4 22 144

.05

58

BAC

HB =0

fb = M/S since Sy = (bh²)/6

= [1" x (2")²]/6

= 2/3 in³ fby = 619.55 lbs.in / (2/3)in³ = 929.33 psi

VIE

E VIE

W

SID

W

IEW

7.Actual vs. Allowable Stress We will be comparing the actual stress with the allowable stress to see if the member will fail under the load.

1

BEAM from Chapter 5 - in bending

Member EF from Chapter 6 - in bending & compression

3 BAC

KV

2

FRO

NT

VIE

IEW

BAC

KV

5 COLUMN from Chapter W E VIE SID - in compression

FRO

W

BAC BAKCVIEW KV IEW

BAC

KV

IEW

NT

VIE

E VIE

SID

W

IEW

W

we know from the calculation: fb = 929.33psi since material = Bambusa Bamboo, Fb = 5800psi, E=1.3x10⁶psi

W E VIE SID VIEW E SID

FRO FRNOT VIE NT W VIE W

we know from the calculation: fb = 300.80 psi, and the shear stress is fv = 13.16 psi since material = plywood, Fb = 2000 psi, Fv = 190psi max deflection (Δmax) happens at the point of load: Δmax = (Pl³)/48EI "I" from chapter 5 = 7in³ "E" from material chart = 1.8x10⁶ psi Δmax = [358.74# x (22.78" + 24.22")³] / (48x1.8x10⁶psi x 7in⁴)

= 0.062 in 300.80psi < 2000psi

Compare fb & Fb: fb < Fb 13.16psi < 190psi

Compare fv & Fv: fv < Fv Because the actual stress is much smaller than allowable, this member will not fail.

FRO

NT

VIE

E VIE

W

SID

W

we know from the calculation: fa = 14.63psi, fcr = 4134psi since material = plywood, Fc (allowable stress in comression) = 1640psi fa=14.63psi < Fc=1640psi < fcr=4134psi

ANALYSIS: The actual stress caused by the body and the chair weight is much smaller than the critical stress (buckling stress) and the allowable compression stress.

Since the allowable compression stress is smaller than the critial, it means that when the load is very high, the "column" will crush before it can start buckling.

fb/Fb = 929.33psi/5800psi = 0.16 we also know from the calculation: fa = 11.77psi

Fcr = (∏²E)/[(l/r)²]

ry = √(I/A) = √[(bh³)/12÷(b x h)

= √(h²/12) when h=2" when h=1" ry = √[(1)²/12] ry = √[(2)²/12] = 0.58" = 0.29"

Choose the smaller "r" to get a smaller Fcr. Fcr. = (∏²x 1.3x10⁶psi)/[(17.48"/0.29")²] = 3531.48# Fa = Fcr/A = 3531.48#/2in² = 1765.74psi since Fa<Fcr, it will buckle before crushing fa/Fa = (11.77/1765.72)psi = 0.0067 Now, we add the bending and axial stress ratio together to see if it exceeds the max stress limit this member can take: fa/Fa + fb/Fb = 0.16+0.0067 = 0.1667

since 0.1667 < 1, This member will not fail in either buckling or bending in this situation.

43

Step 4 Winging It

The complicated contraptions of the Rube Goldberg Machine do not lend themselves to scale models or digital design and planning. The process had to be spontaneous. Elements emerged one after another working backwards until the surrounding space around the chair on the base was filled to the brim. Repeated testing throughout the design informed further decisions. Processes had to be nondestructive, easily reset, and durable for the sake of this project.

44

The first base for the catapult, which ended up not working

Starting with the bare bones beginnings

CJâ&#x20AC;&#x2122;s complicated lever system grows

A collection of junk to be transformed into something spectacular 45

46

A spontaneous creative process also led to many dead ends

Testing each component as they were being built.

Using collected junk in creative ways

An incredibly messy work area

Maniac for scale

Moving to the student lounge for more space

Repairing the chair after moving it to the loft

The view from the other side

47

STARTING POSITION 48

ENDING POSITION 49

A HAPPY CUSTOMER AFTER THE FIRST SUCCESS 50

ALTERNATIVE POSITIONS FOR A RIDICULOUS CHAIR 51

How It Works Due to the nature of a Rube Goldberg Machine, the individual processes can be hard to follow and understand. Please use this section as a helpful breakdown of each step that leads to the final result. In real time, this machine takes less than half a minute to run its full course. Aside from the first action performed by the user, the rest is driven entirely by domino effects of gravity, kinetic energy transfer, and stored energy.

52

53

Right View I

P D

A

J E F

B

K

G L N

54

M

O

C

H

The user takes the marble from the holder and drops it through the target (A).

Marble hits a series of metal plates (B) on its way down ramps.

Marble falls into weighted bucket (C) hanging from string.

Bucket full of marbles (D) is pulled over by string weighed down by bucket (C).

Marbles bounce and roll down plank of nails (E).

Marble drop into funnel of nails (F).

Inside of funnel (F).

Marbles fall into half tube (G).

Marbles roll along into bucket (H).

55

Front View

S R

Q

P

Lever is pulled down by bucket full of marbles (H), which tilts up the platform (I).

Dominoes (L) land on top of taut string.

O

M N

The dominoes continue along the ramp, which turns and changes heights.

56

Platform (I) holding marble releases it into tube (J).

Marble exits tube (J) on to rail (K).

Marble exits tube and rolls down rail (K) until it hits cowbell (L) at the bottom which rings and knocks over dominoes (M).

Taut string pulls out pin (M) holding the foot of the chair (N) down.

Foot of chair (N) is pulled up quickly by spring. An attached ruler hooks onto a strong which connects to dominoes above (P).

The first domino (P) is pulled down by the string yanked down by the seat (O).

The last domino hits the small marble at a corner angle (Q), knocks it out its perch, and the marble begins rolling.

The small marble rolls down the stepped ramps (R).

The marble enters the bucket (S) to change directions, and it continues to roll down a ramp in the opposite direction.

57

Left View

Q Σ X

R Z

T

Y

Δ

U λ

W

V

58

β

S

The marble keeps rolling as the ramp transitions into a tube (T).

The small marble exiting the pink tube hits a medium marble which then hits a big marble on a small ramp (U).

The large marble hits wood piece (W). All marbles land in bag (V).

Wood piece (W) turns on its axis from the marble, and the bent piece hits the end of wood piece (X).

Wood piece (X) falls over and lands on woods piece (Y), which pulls down a connected string.

The pin (Z) is pulled out by the string weighed down by the fallen piece of wood (X), dropping one end of a plank.

The stack of books (Σ) slides down the now inclined plank and falls off to hit the pizza pan. The weight pushes down the fulcrum (β).

The chair back (Δ) is catapulted up 45 degrees, which is when the springs take action and pull it completely forward.

The seat (λ) which is hinged to the chair back, comes slamming down, and the chair is complete.

59

Final Presentation On Wednesday July 24th, 2019, students at the University of Waterloo Architecture campus gathered to witness the spectacle three months in the making. Against all odds, the chair was 100% successful during the show. Concerns about safety and functionality arose among the crowd as CJ and Amy began setting up the mechanisms of the chair, but were mostly forgotten as the presentation went on, giving way to joyous surprise and loud laughter.

When asked for a statement, Amy said “It’s stupid and I love it” while CJ claimed “I’m never doing this again”.

60

Set up begins

Amy replaces the marbles

Resetting the dominoes

CJ sets up the books

61

62

“You want to do the honors?”

“So if this fails, is it my fault?”

Amy recording the action

Following the action

Looking on proudly

Completely enraptured

â&#x20AC;&#x153;Whew! It worked!â&#x20AC;?

CJ - triumphant amidst applause.

63

64

65

66

67

Bibliography

68

Goldberg, Rube. “Simple Alarm Clock.” Cartoon. Rube Goldberg. Accessed August 10, 2019. https://www.rubegoldberg.com/artwork/simple-alarm-clock/.

Goldberg, Rube. “Umbrella Stealing Alarm.” Cartoon. Rube Goldberg. Accessed August 10, 2019. https://www.rubegoldberg.com/artwork/umbrella-stealing-alarm/.

OK Go. YouTube. March 01, 2010. Accessed August 10, 2019. https://www.youtube.com/watch?v=qybUFnY7Y8w.

Schenke, Jim, and Joan Myers. “Rube Goldberg Machine Shatters Guinness World Record, Destroys Planet.” Phys.org. May 30, 2011. Accessed August 10, 2019. https://phys.org/news/2011-05-rube-goldberg-machine-shatters-guinness.html.

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