DOE

Free Fall Time of Paper Cones

Jiankun SUN, Nan CHEN, Donghui LI, Quan YUAN,Tianyang XU

DOE Contents

Pre-experiment2

Conclusion4

Introduction31

Formal Experiment33

IntroductionLiterature Review

《浅探纸锥下落快慢与纸锥大小的关系》 ,陆文彬 Only study free fall time of paper cones with

same radius and different angles Derive a complicated equation to describe the

relationship between acceleration and velocity and the impacts from paper cone properties

IntroductionFactor Decomposition

Response variable:free fall time

Input variables:paper thickness, cone radius, central angle,

Control variable:height

Nuisance factor:operator performance, recording error

Pre-experimentPre-experiment Design

Purpose Check the repeatability and reproducibility Get a general idea of the degree of factors’ influence

Design 3 replications for each level and each operator

  VariablesCode Radius Thickness Angle of cone

-1 5 cm Thin 150°

1 10 cm Thick 270°

Formal Experiment

Experiment Process

At platform of stairs in the hall of Shunde Building

The operator stretch out the arm horizontally and hold the paper cone with two fingers, and then release the cone without initial velocity

A camera is used to record the process for time analysis

Pre-experimentAnalysis: Reproducibility

Check whether there is significant difference between results of different operators

Cannot claim significant difference

Pre-experimentAnalysis: Influence Degree

Check which variables have influence and how much the influence is

Influence from certain variables and interaction

Formal Experiment

Formal Experiment Design

Full Factorial Design Level setting

radius thickness angle-1 1 -1 1 continuous variable

5cm 10cm 70g/m2 200g/m2 90+20k(k=0,1,2…9)

Formal Experiment

Data Collection

大小 厚薄 锥角 　 时间 　1 -1 270 4.73 4.40 4.531 -1 250 4.20 4.17 3.901 -1 230 3.90 4.00 3.731 -1 210 3.57 3.47 3.471 -1 190 3.27 3.20 3.101 -1 170 3.00 2.90 2.931 -1 150 2.70 2.60 2.601 -1 130 2.43 2.30 2.471 -1 110 2.67 2.47 2.371 -1 90 2.00 2.00 2.70

Formal Experiment

Data analysis

Descriptive analysis:

Thickness may have influence while radius may not

Formal Experiment

Data Analysis

Scatter Plot:

There seems to be a linear relationship between time and angel.

280260240220200180160140120

5. 0

4. 5

4. 0

3. 5

3. 0

2. 5

2. 0

Angl e

Time

-1 -1-1 11 -11 1

Radi us Thi ckness

Scatterpl ot of Ti me vs Angl e

Formal Experiment

Data Analysis

Main effects plot

1-1

3. 63. 22. 82. 42. 0

1-1

270250230210190170150130

3. 63. 22. 82. 42. 0

321

Radi us

Mean

Thi ckness

Angl e Repl i cati on

Mai n Eff ects Pl ot for Ti meData Means

Formal Experiment

Data Analysis

Interaction plot

No significant interaction between radius and thickness or between angle and radius.

Interaction between thickness and angle.

1- 1 270250230210190170150130

4

3

2

4

3

2

Radi us

Thi ckness

Angl e

-11

Radi us

-11

Thi ckness

I nteracti on Pl ot f or Ti meData Means

Formal Experiment

General Linear Model (GLM)

Formal Experiment

Residual Plots

The residual versus is a bowl-like curve May be due to interaction

0. 500. 250. 00-0. 25-0. 50

99. 999

90

50

10

10. 1

Resi dual

Perc

ent

432

0. 50

0. 25

0. 00

-0. 25

-0. 50

Fi tted Val ue

Resi

dual

0. 40. 20. 0-0. 2-0. 4

24

18

12

6

0

Resi dual

Freq

uenc

y

9080706050403020101

0. 50

0. 25

0. 00

-0. 25

-0. 50

Observati on Order

Resi

dual

Normal Probabi l i ty Pl ot Versus Fi ts

Hi stogram Versus Order

Resi dual Pl ots for Ti me

Formal Experiment

Conclusion from GLM

Free fall time of paper cones have strong linear relationship with central angle and paper thickness

At 95% confidence level, the correlation between free fall time and cone radius is not significant

Formal Experiment

GLM adding interaction

Formal Experiment

Residual Plots

0. 40. 20. 0-0. 2-0. 4

99. 999

90

50

10

10. 1

Resi dual

Perc

ent

432

0. 4

0. 2

0. 0

-0. 2

-0. 4

Fi tted Val ue

Resi

dual

0. 30. 20. 10. 0-0. 1-0. 2-0. 3

20

15

10

5

0

Resi dual

Freq

uenc

y

9080706050403020101

0. 4

0. 2

0. 0

-0. 2

-0. 4

Observati on Order

Resi

dual

Normal Probabi l i ty Pl ot Versus Fi ts

Hi stogram Versus Order

Resi dual Pl ots for Ti me

Formal Experiment

Conclusion from 2nd GLM

A relatively higher, and no obvious pattern in the residual versus fits

At 95% confidence level, The effect of the interaction between paper thickness and central angle is significant.

Formal Experiment

Conclusion from 2nd GLM

A relatively higher, and no obvious pattern in the residual versus fits

At 95% confidence level, The effect of the interaction between paper thickness and central angle is significant.

ConclusionConclusion & Explanation

Equations thin paper: Time = 0.461408 + 0.0143581 Angle thick paper: Time = 0.461408 + 0.00927134 Angle

Explanation Air resistance increases with central angle. For different paper thickness, different mass of paper

contributes to different change in fall time as the central angle changes.

Wrong sign of coefficient of thickness: not significant

DOE