Final Report of Design of

Experiment Course --A study on Factors Affecting Flying Distance of

Arrows Projected From a Crossbow

Group Leader: 侯燕铭

Group Members: 吕臣，胡欣然，吴梦春

Contents

1. Motivation ............................................................................................................... 3

2. Literature Review.................................................................................................... 3

2.1 The history of crossbow .................................................................................... 3

2.2 Principle of crossbow ........................................................................................ 3

2.3 Physical principle of firing arrows .................................................................... 4

2.4 Methods of firing arrows using crossbow ......................................................... 5

3. Factor Decomposition ............................................................................................. 6

3.1 Assumption ....................................................................................................... 6

3.2 Variable definition ............................................................................................. 6

3.2.1 Response variable ................................................................................... 6

3.2.2 Control variable ...................................................................................... 6

3.2.3 Noise factor ............................................................................................. 7

3.2.4 Constant factor ........................................................................................ 7

4. Design of Experiment ............................................................................................. 7

4.1 Sample size analysis ......................................................................................... 8

4.2 Factor level definition ....................................................................................... 8

4.3 Implementation of experiment .......................................................................... 8

4.4 The results of the experiment .......................................................................... 10

5. Data Analysis ........................................................................................................ 11

5.1 Basic analysis .................................................................................................. 11

5.1.1 Main effects .......................................................................................... 11

5.1.2 Interaction plot ...................................................................................... 12

5.1.3 Half normal plot .................................................................................... 13

5.1.4 Pareto plot ............................................................................................. 14

5.1.5 ANOVA table ........................................................................................ 14

5.1.6 Regression model .................................................................................. 15

5.1.7 Contour plots & Response surface plot ................................................ 17

5.2 Optimization design ........................................................................................ 19

6. Conclusion ............................................................................................................ 20

7. Possible Improvement for Our Experiment .......................................................... 21

1. Motivation

Crossbow originates from ancient China, which was widely used in the war.

Though today it is no longer used in military, many toy factories sell lots of toy

crossbows with little antipersonnel force. Especially, in recently years, with the

popularity of a card game named “Killers of the Three Kingdoms”, crossbow regains

its popularity among young people.

We decided to choose wooden crossbow for the prototype of the Design of

Experiment project because of the simple theorem, as well as the easy handing of the

crossbow. Besides, the noise factors are relatively fewer and we have the experience

of playing crossbow, which ensures the control stability. With the experiment, we try

to find the vital factors that affect the range of the flight, thus give good advice to the

users about how to shoot the array farther.

2. Literature Review

2.1 The history of crossbow

According to linguistic evidence, crossbow has originated among the cultures

neighboring ancient China where it first evolved in the form of unattended traps.

Bronze crossbow bolts used during as early as mid 5th century BC, were found at a

State of Chu burial site in Hubei province.

The earliest Chinese document mentioning a crossbow is in scripts from about 3rd

century BC attributed to the followers of Mozi(墨子). Sun Tzu's influential book, The

Art of War, has also mentioned the use of crossbows. We can easily catch the point

that crossbow was created and widely used in ancient China.

2.2 Principle of crossbow

A crossbow is mounted on a stick with a mechanism in it which holds the drawn

bow string. To project the array, a vertical rod is thrust up through a hole in the bottom

of the notch, thus forcing the string out. The rod is usually called a trigger.

When you are firing an array, you just need to draw the string and fix it to the

trigger. Then you would place the array in accurate position. Pull the trigger to let out

the array, which is quite easy and interesting.

The crossbow used in our experiment is in the follow picture.

Figure 1 The crossbow

2.3 Physical principle of firing arrows

According to the theory of our physics course, one similar conclusion of firing

arrows might be generated. If you want to throw something far, you need to make a

higher starting position, as well as the most appropriate degree of initial velocity

angle.

A theoretical calculation process can be obtained, and several variables need to be

stated first. Let t1 be the upward flying time, while t2 the downward flying time,

and t be the total time. Donate H as the starting point, α as the angle at which the

arrow is projected, and R as the flying horizontal distance.

We can use the equations:

t1 =v sinα

g

1

2gt1

2 + H =1

2gt2

2

From which we can figure out the results:

t = t1 + t2 =v sin α

g+ (

v sinα

g)2 +

2H

g

R = v cosα [v sinα

g+

v sinα

g 2

+2H

g]

Also, we can use a simple schematic diagram to indicate the flying process of an

arrow.

Figure 2 Physical principle

2.4 Methods of firing arrows using crossbow

The methods of firing an arrow with crossbow can be various. You can fire it at the

height of your stomach, or at the height of your neck. You can also firing the array

with full draw length or half draw length. Also, you can choose any angle to release

the arrow from horizontal to vertical.

Other factors may also affect the range of flying distance. For instance, if you fire it

following the wind, you can get a farther distance. In addition, different people may

have different levels of skill, which also affects the final result.

3. Factor Decomposition

3.1 Assumption

To simplify the experiment, some assumptions are necessary: the crossbow is as

common as other toy crossbow, not specifically designed; the experimenters are all

skillful enough to guarantee the stability of each firing; environmental factors which

are difficult to deal with are regarded as having no influence on the experiment result,

such as the temperature.

3.2 Variable definition

3.2.1 Response variable

The horizontal flying distance of the arrow projected from the crossbow

We use ribbon tape to measure the horizontal distance of each firing and the

accuracy is 0.01 meter, which can directly reflect the influence of different setting of

parameters. Note that the range is from the starting position to the first point the arrow

hits the floor, since it usually glides on the floor after it stops its flying.

3.2.2 Control variable

The drawing length

When we fire an arrow, we can let the firing force small or large, which mean the

drawing length is short or long, respectively. According to common sense, the longer

the drawing length, the stronger firing force will be, and the larger initial speed the

arrow possesses.

Initial angle

This factor concerns the angle of the horizon and the direction of the arrowhead.

According to our experience, people rarely fire the arrow at a high angle. Thus, in

order to simplify, the range of angle is from 0 to 45 degree.

Initial height

Considering the height of human body, we set the height range at 1.0 to 1.6 meters

for our experiment.

Whether with the arrowhead

We think that the arrowhead will influence the air resistance, since its material is

different from the body of the arrow. Thus, the flying distance will decrease if the air

resistance increases.

3.2.3 Noise factor

Noise factors cannot be handled and hard to measure, however, we would like to

minimize their effects in our experiments as possible. Thus, we would use

randomization for these factors, and the sequence of different treatments is random.

The noise factors that we considered to exist in the experiment are as follows.

Variation of the arrows

Variation of the experimenters

Random error of measurement of distance

3.2.4 Constant factor

Factors that are not of interest to us can be set to distant factors. The experiment

can be simplified in this way. The constant factors involved in the experiment are as

follows:

The speed of wind

The roughness of the floor

Room temperature

4. Design of Experiment

After defining the factors, we continue to come up with a feasible and effective

experiment design. We thought that a full factorial design is good based on the

following reasons:

(a) There are four factors totally, not too much.

(b) Relationships among all factors are to be identified.

(c) All main effects and all interactions are to be estimated.

We can measure the responses by using various combinations of factors and levels

in order to determine the large-effect factors and distinct interactions between factors.

4.1 Sample size analysis

Sample size analysis is important in the design of experiments. We use a replicated

24full factorial design with 3 replicates for each treatment for the following reasons:

(a) The experiment is easy and takes short time for a single trial.

(b) More degrees of freedom is provided thus makes it easier for main factors

analysis.

4.2 Factor level definition

In order to take a full factorial design, we need to set all the four factors at two

levels—low and high. Considering the noise which has the potential risk to

overwhelm the signal in data, the distance between two levels of factors should be

appropriately increased while at the same time we keep the levels in the range of

practical use. The results of factor level we chose are shown in the following table:

Factor -1 1

Height(meter) 1.3 1.5

Angle(degree) 0 45

Whether with arrowhead No Yes

Drawing length Short Long

4.3 Implementation of experiment

With the help of Minitab, we create a factorial design. The combination of factors

and orders, the random run order are given by Minitab. Since the drawing length

indicates the energy available for one firing, we use the name energy to represent the

factor drawing length.

Figure 3 Experimental design in Minitab

After the design is finished, some detailed methods of operation are then taken into

account.

1. Measurement methods.

For good accuracy and experiment result, a set of fine measurement methods are

critical. Some of the factors can be well measured by specific instruments, but some

others cannot.

a. We borrowed a tape measure with the help of which we marked the position

of the right height in the 2 levels on our clothes. Thus we can make sure that

the arrow is launched at the right initial height.

b. When the launch angle is at its high level (45°), one of our team members

will help the one who is responsible for launching the arrow to confirm the

angle with a protractor.

c. The factor of with or without arrowhead is simple in terms of controlling.

d. The drawing length has 2 levels, one is sure to be accurate because the

facility we used has a slot for measuring, which can be regarded as the higher

level. We marked on the crossbow to help us locate the distance for the low

level.

e. We made smart estimate on the flying distance. There are marble grids on the

floor and we measured the length of each grid. During the experiment, we

just check how many grids the flying distance has covered and we only need

to measure the distance in the last grid which is covered partly.

2. The experiment environment

a. We performed the experiment in the hall of Shunde Building, and the floor is

considered in the same smooth condition everywhere.

b. Since we performed the experiment indoor, the influence of wind is

minimized and can be neglected. Other related temperature factors are also

considered stable.

c. To control the influence of individual characteristics, all the arrows are

launched by the same person.

4.4 The results of the experiment

After constructing the experiment design, we carried out the experiment and get the

results in the table below, and the results are listed in experimenting sequence. Note

that Rep 1, 2, 3 means the result of replication 1, 2, 3, respectively.

Height angle Arrowhead Energy Rep 1 Rep 2 Rep 3

-1 -1 -1 -1 3.85m 4.20m 4.15m

1 -1 -1 -1 4.35m 4.80m 4.35m

-1 1 -1 -1 5.20m 5.25m 4.90m

1 1 -1 -1 4.80m 4.95m 5.05m

-1 -1 1 -1 2.85m 3.05m 2.80m

1 -1 1 -1 2.95m 3.00m 2.95m

-1 1 1 -1 3.75m 3.60m 3.40m

1 1 1 -1 4.20m 4.25m 4.35m

-1 -1 -1 1 6.50m 6.05m 6.30m

1 -1 -1 1 6.80m 6.50m 6.45m

-1 1 -1 1 9.65m 9.30m 9.70m

1 1 -1 1 9.50m 9.35m 9.75m

-1 -1 1 1 7.80m 7.50m 7.55m

1 -1 1 1 7.95m 8.05m 8.20m

-1 1 1 1 8.15m 8.50m 8.00m

1 1 1 1 8.20m 8.60m 7.90m

5. Data Analysis

5.1 Basic analysis

5.1.1 Main effects

With the help of Minitab, we can get the main effect plot, which is presented in

Figure 3 below.

Figure 4 Main effect plot

We can learn from the plot that both energy and angle are the factors that influence

the distance of projected arrows significantly. In addition, the influence is positive.

However, the factor height and arrowhead have a relatively weak effect on the

distance. Next, let us take a look at the interaction effect plot.

5.1.2 Interaction plot

Figure 5 Interaction plot

Similarly, we can obtain the plot indicating interaction effects in Minitab. From

the plot above, we can find that factors angle, energy and arrowhead show relatively

significant interaction effect, which need to be further checked in the later analysis. In

order to determine the key factors, we formed half normal plot of effects in the

following part.

5.1.3 Half normal plot

Figure 6 Half normal plot

We can see from the plot that there are altogether 9 main effects and/or interaction

effects are relatively significant. But, which are the most significant ones? We can

refer to the Pareto Plot.

5.1.4 Pareto plot

Figure 7 Pareto plot

From the plot, we can omit insignificant factors to get the parameters involved in

our model. We can list them in order according to the plots: D, B, CD, C, BCD, BC, A.

Then we can use the ANOVA table to confirm our view.

5.1.5 ANOVA table

拟合因子: length 与 Height, Angle, arrowhead, energy

length 的效应和系数的估计（已编码单位）

项 效应 系数 系数标准误 T P

常量 6.0260 0.02832 212.80 0.000

Height 0.2187 0.1094 0.02832 3.86 0.001

Angle 1.3062 0.6531 0.02832 23.06 0.000

Arrowhead -0.5896 -0.2948 0.02832 -10.41 0.000

Energy 3.9687 1.9844 0.02832 70.08 0.000

Height*Angle -0.0938 -0.0469 0.02832 -1.66 0.108

Height*Arrowhead 0.0854 0.0427 0.02832 1.51 0.141

Height*Energy -0.0312 -0.0156 0.02832 -0.55 0.585

Angle*Arrowhead -0.6187 -0.3094 0.02832 -10.93 0.000

Angle*Energy 0.4396 0.2198 0.02832 7.76 0.000

Arrowhead*Energy 0.6354 0.3177 0.02832 11.22 0.000

Height*Angle*Arrowhead 0.1396 0.0698 0.02832 2.46 0.019

Height*Angle*Energy -0.0937 -0.0469 0.02832 -1.66 0.108

Height*Arrowhead*Energy -0.0396 -0.0198 0.02832 -0.70 0.490

Angle*Arrowhead*Energy -0.7438 -0.3719 0.02832 -13.13 0.000

Height*Angle*Arrowhead*Energy -0.1688 -0.0844 0.02832 -2.98 0.005

S = 0.196188 PRESS = 2.77125

R-Sq = 99.48% R-Sq（预测） = 98.82% R-Sq（调整） = 99.23%

对于 length 方差分析（已编码单位）

来源 自由度 Seq SS Adj SS Adj MS F P

主效应 4 214.233 214.233 53.5582 1391.50 0.000

2因子交互作用 6 11.963 11.963 1.9938 51.80 0.000

3因子交互作用 4 6.996 6.996 1.7490 45.44 0.000

4因子交互作用 1 0.342 0.342 0.3417 8.88 0.005

残差误差 32 1.232 1.232 0.0385

纯误差 32 1.232 1.232 0.0385

合计 47 234.765

In the ANOVA table obtained above, the significant factors (P-value close to 0) are

marked by yellow color. According to these values, we can formulate our model of the

experiment.

5.1.6 Regression model

拟合因子: length 与 Height, Angle, arrowhead, energy

length 的效应和系数的估计（已编码单位）

项 效应 系数 系数标准误 T P

常量 6.0260 0.03378 178.39 0.000

Height 0.2187 0.1094 0.03378 3.24 0.002

Angle 1.3062 0.6531 0.03378 19.33 0.000

Arrowhead -0.5896 -0.2948 0.03378 -8.73 0.000

Energy 3.9687 1.9844 0.03378 58.74 0.000

Angle*Arrowhead -0.6187 -0.3094 0.03378 -9.16 0.000

Angle*Energy 0.4396 0.2198 0.03378 6.51 0.000

Arrowhead*Energy 0.6354 0.3177 0.03378 9.41 0.000

Angle*arrowhead*Energy -0.7438 -0.3719 0.03378 -11.01 0.000

S = 0.234039 PRESS = 3.23590

R-Sq = 99.09% R-Sq（预测） = 98.62% R-Sq（调整） = 98.90%

对于 length 方差分析（已编码单位）

来源 自由度 Seq SS Adj SS Adj MS F P

主效应 4 214.233 214.233 53.5582 977.80 0.000

2因子交互作用 3 11.758 11.758 3.9194 71.55 0.000

3因子交互作用 1 6.638 6.638 6.6380 121.19 0.000

残差误差 39 2.136 2.136 0.0548

失拟 7 0.905 0.905 0.1292 3.36 0.008

纯误差 32 1.232 1.232 0.0385

合计 47 234.765

According to the results of coefficients in the table above, we can formulate the

regression model with binary variables:

Distance = 6.0260 + 0.1094 × Height + 0.6531 × Angle− 0.2948 × Arrowhead

+ 1.9844 × Energy− 0.3094 × Angle × Arrowhead + 0.2198

× Angle × Energy + 0.3177 × Arrowhead × Energy− 0.3719

× Angle × Arrowhead × Energy

To replace the binary variable with the actual values, we can get the following

model:

Distance = 3.8431 + 1.094 × Height + 0.029 × Angle + 0.0146

× Arrowhead + 1.7646 × Energy− 0.0138 × Angle

× Arrowhead + 0.0098 × Angle × Energy + 0.6896

× Arrowhead × Energy− 0.0165 × Angle × Energy

× Arrowhead

The range of these variables involved in the final model is shown in the following

table.

Factor -1 1

Height(meter) From 1.0 to 1.6, continuous

Angle(degree) From 0 to 45, continuous

Whether with arrowhead No Yes

Drawing length Short Long

Next step is to analysis the residual of our model, in order to test whether the model

is sufficiently good. The residual plot obtained in Minitab is as follow:

Figure 8 Residual plot

The residual plot shows that the distribution of residuals is good, thus the model is

a reasonable one.

5.1.7 Contour plots & Response surface plot

In order to explore the interactions between factors, we can use the contour plots

and the response surface plot to test whether interactions exist.

Figure 9 Contour plots

If a contour plot is curved, we can conclude that there exists interaction between

the two factors. Thus, from the plot above, we can see that the interactions of angle

and arrowhead, energy and angle, energy and arrowhead are significant. The result

agrees with the results obtained in 5.1.6.

Also, we can use response surface plot to achieve the same goal. If the surface is

plane, we can think that there is no interaction between the two factors. Here we take

factor angle and energy as an example. The plot can be obtained.

Figure 10 Response surface plot

From the plot, we can find that the surface is not a plane. Thus, the interaction of

the factor energy and angle is significant.

5.2 Optimization design

Our experiment is intended for an optimization research, which attempts to find the

setting of parameters that let the arrow fly further from the crossbow. The data

collected can serve for the purpose of the experiment. The result obtained by response

optimization in Minitab is as follow:

Figure 11 Response optimization

From the response optimization, we can find that the optimal treatment should be

lowest height, largest angle and energy, and without arrowhead as well. The

theoretical longest flying distance of the arrow is 9.55 meters, and the composite

desirability is 1, though there are several observations that are larger than the optimal

value.

6. Conclusion

Through the whole process of our experiment, several useful conclusions can be

drawn.

The factors Angle and Energy have significant positive effects on the distance of

the projected arrows. Higher energy means larger drawing length of the rubber,

which indicating larger initial speed of the arrow. According to related physical

theory, the higher initial speed the arrow has, the longer it will fly. As for angle’s

effect, in the range from 0 to 45 degree, the corresponding flying distance

increases when the angle becomes larger.

Note that though the main effects of Height and Arrowhead are relatively weak,

some deductions can be obtained. The increase in height leads to increase in

distance, which agrees with physical theory. However, arrows with arrowhead fly

shorter than those without arrowhead. Perhaps it is because the material of

arrowhead is plastic, not metal. Thus, using arrowhead will bring more air

resistance for the arrow, which decreases its flying distance.

When the angle is 0, the influence of arrowhead is not significant. However, if the

angle becomes large, the arrow would fly for more time, thus the effect of air

resistance becomes obvious. This is why the arrow with arrowhead flies much

shorter if the angle is 45 degree.

Though the arrowhead can decrease the flying distance of the arrow, when energy

becomes large, the decrease becomes less and less significant. If we use the

largest drawing length to shoot, the distances will be the same whether the

arrowhead is used or not.

One of the best methods to project an arrow which flies long is using the long

drawing length at a position of low height, and firing the arrow at a large angle

without the arrowhead.

7. Possible Improvement for Our Experiment

Our experiment followed the basic design method of full factorial. Since there are

many limitations, especially time, in the process, there may be several improvements

for our experiment.

Initial design with center points

Center points will help us to test the significance of curvature, and whether we need

to modify our final model.

Use response surface to search for optimal point

Using RSM can help us move towards the point of highest distance gradually.