2011年美赛a题

Team #9262
Perfect Half-pipe: The Think of
Snowboard Course



Abstract
With the continuous progress and development, People are actively involved in
sports and exploring in it continually. Skiing is popular with the majority of sports
fans gradually under this condition. Especially，Snowboarding with good view,
challenge and the basis of the masses develops rapidly and has become a major
Olympic projects. In this paper, how to design and optimize the snowboard course of
half pipe is discussed in detail. We strive to get the perfect course so that
snowboarders can achieve the best motion state in the established physical conditions.
What’s more, it may promote the development of the sport.
This problem can be divided into three modules to discuss and solve. For the first
problem of the design of half pipe, it can be based on the point of the energy
conservation law. The method of functional analysis (Variation principle and Euler
differential equation) is used to set up equations, when the secondary cause is ignored
and the boundary conditions are taken into consideration. The curve equation is
obtained by the above equation, that is, a skilled snowboarder can make the maximum
production of “vertical air”. For the second question, athletes’ maximum twist in the
air and some other factors need to be taken into account when to optimize the
previous model, so that curve can meet the actual game conditions and appraisal
requirements as much as possible. Ultimately, a satisfying curve will be got. The third
problem is a problem relatively close contact with the actual, which is to setting down
a series of tradeoffs that may be required to develop a “practical” course. In this paper,
for the formulation of these factors, the main discussions are the thickness of snow on
half pipe and the aspect of economy for the construction.
After discussing these three aspects, the paper finally summarizes a construction
program and evaluation criteria of the course in current conditions. Finally, by
evaluating the advantages and disadvantages of the whole model，we put forward the
advanced nature of the model, but also point out some limitations of the model.


Key words: Snowboard course, Half-pipe,Functional, Euler equation,

Fitted curve,

Numerical differentiation

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Table of contents

Table of contents……………………………………………1
I. Introduction……………………………….…….………..2
1.1 Half pipe structure…………………….……………..2
1.2 Background problem……..………………………….3
1.3 Athletes aerials….………………………..…………..3
1.4 Assume………………………………………………..3
II. Models……………………………………………………5
2.1 problem one……………………………….………….5
2.2 Problem two…………..……………………………11
2.3 Problem three……………………………………….13
III. Conclusions…………………………………………….14
IV. Future Work……………………………………………15
V. Model evaluation……………………………….……….15
5.1 Model Advantages………………...…………………15
5.2 Model disadvantages…………………..…………….16
VI. References……………………………………..………..17

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I. Introduction
Snowboarding is a popular pool game with the world of sports. The
U-Snowboard’ length is generally 100-140m , U-type with a width of
14-18m,U-type Depth of slope is 14°-18 °. In competition
U-athletes Skate within the taxi ramp edge making the use of slide to do
all sorts of spins and jumps action. The referees score according to the
athletes’ performance as the Vertical air and the difficulty and
effectiveness of action. The actions Consist mainly of the leaping grab the
board, leaping catch of non-board , rotating leaping upside down and so
on.
1.1 Half pipe structure
Half pipe structure contains: steel body frame, slide board, steps
to help slide and rails.

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1.2 Background problem
In order to improve the movement of the watch, it can be improved
from two aspects: orbit and the athletes themselves. Now according to the
problem the orbit can be designed as a curve. on the curve the athletes can
get a maximum speed. The design of orbit includes a wide range of
content, such as the shape of U-groove design, track gradient, width and
length designed to help the design of sliding section, and so on. The
rational design of half pipe can be achieved to transform the energy to
efficiency power, make the athletes achieve the best performance in the
initial state of the air. This paper discusses the rational design of half pipe
to these issues.
1.3 Athletes aerials
Athletes on the hillside covering with thick snow skill down with the
inertia of the platform, jump into the air, and complete a variety of twists
or somersault. Rating criteria: vacated, takeoff, height and distance
accounted for 20%; body posture and the level of skill accounted for50%;
landing 30%. According to the provisions the difficulty of movements
are ranged into small, medium and large. The athletes option the actions.
However, the ground must have a slope of about 37 ° and 60 cm above
the soft snow layer.
1.4 Assume
In order to simplify the model and can come to a feasible solution,

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making the following assumptions:
1? the shape of a snowboard course has a lowest point, the wide and the
length of the snowboard course.
2? air resistance can be negligible.
3? it is assumed that the athletes themselves have no influence.
II. The Description of the Problem
This problem is a typical engineering design, involving a lot of
disciplines, such as advanced mathematics, engineering mathematics,
mechanical dynamics and biological dynamics, as well as the relevant
provisions of sports competition and judging standards, and so on.
According to the requirement of the problem, determine the shape of a
snowboard course to maximize the production of “vertical air” by a
skilled snowboarder. For this problem, we can use the boundary
conditions and site properties (e.g. symmetry) and other requirements to
establish functional combining with the variation principle Euler
equations. The original equation can be changed into a functional
extremum problem.
Secondly, we optimize the model boundary and determine the
appropriate snowboard course’s slope toe to make the athletes perform
maximum twist or do more difficult action.
Finally a practical model should meet the requirements of safety,
sustainability and economic. According to the high degree of human

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security, the source of the snow and the topography the model will be
optimized more reasonable.

II. Models
2.1 problem one
As shown（3.1）A is the lowest point of the snowboard course.
From A to B we want to find a curve to make the athletes get the
maximum vertical distance above the edge of the snowboard course.

Figure 1 Half-pipe

Set A as origin of coordinate.

Awing of conservation energy and neglecting air resistance, the
mathematical function is,
1
2
mv
2
0
?W?
1
2
mv
2
?mgh?A
f
(1)

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Where
v
0
is the initial velocity （mh）,
v is the velocity towards destination （mh）,
m is the mass of an athlete (kg),
w is the

energy which is made by the athlete (J),
h is the Vertical height (m),
Af is the friction work （J），

According to mechanical analysis：

N?mgcos
?
?m
Where
v
2
r

（2）
θ is the angle the angle between the tangent and the horizontal line,
N is the pressure on the object,
r is the radius of curvature,
According to friction formula：
2
?
v
?
?
f?
?
N?
?
?

?
mgcos
?
?m
r
?

（3）
??
To (3) into equation (1), combined with calculus：

1
2
mv
0< br>?W?
2
1
2
mv
2
2
?
v
t
?
?
dl

（4）
?mgh?
?
?
?
mgcos
?
?m

??
r
??
0
s
Friction acting A
f
：

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2
?
v
t
?
?
mgB
??
A
f
?
?
?
?< br>mgcos
?
?mdl??
?
r
?
2
?0
ss
?
?
m
0
v
t
r
2< br>dl

（5）


Use higher mathematics：

r?
?
1?y'
?2
3
2
y''
2

（6）

dl?1?y'dx

（7）

According to the nature of the curve, the speed can be assumed to
satisfy this expression
：
v?v
0
1
e
kx

（8）

Put all these formulas in order and suppose the expression for the
functional：
B
s
??
?
?
m
0
v
t
r
2
2
dl?
?
0
?
mv
0
y' '
e
2kx
2
1?y'
2
dx

（9）

Set

F?
?
mv0
y''
e
2kx
2
1?y'
2

（10）

Reference Euler equation：
?F d
?
?F
?
d
??
??
??
?ydx?
?y'
?
dx
2
2
?
?F
?
??

11）

??
?0


（
?y''
??
Obtained：
?F
?y
?0


（

12）

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?F
?y'

??
e
?
mv
0
y''y'
2kx
2?
1?y'
2
?
3
2


（13）

?F

?y''
?
?
mv
2
0
1

（14）

e
2kx
?
1?y'
2
?2
To（12）-（14）into equation（11），
Obtained：

d
?
?
?
mv
2< br>?
0
y''y'
?
2
??
dx
?
? ?
d
?
?
mv
2
?
?
??
3?
1
?
e
2kx
1?y'
2
2
?dx
2
?
0
?
?
e
2kx
??
?
2
?

?
1?y'
2
?
Integrate it：
2
??
?
?
mv
0
y''y'
?
d
?
?
mv
2
0
?
3
?
?C

e
2kx
?
1?y'
2
?
2
dx
?
?
?
e
2kx
?
1?y'
2
?
1
2
?
?
Simplified：
?
mv
2
2k
?
mv
2

0
y''y'
0
?
mv
2
0
y''y'< br>3
?
e
2
?
3
?
3
5
?C

kx
?
1?y'
2
?
2
e< br>2kx
1?y'
2
?
2
e
2kx
?
1?y'
2
?
2
y''?
2k
?
1?y'
2
?
y'
3
?2y'


dy
Suppose：

dx
?p
?
y
?

2
So：

dy
dx
2
?
dp
dy
?
dy
d x
?p
dp
dy

Substituted into the above equation：
p
dp2k
?
1?p
2
?
dy
?
p
3
?2p

15）

16）
17）

18）
19）

（

（

（

（

（

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?
p
Integrate it：
4
?2p
1?p
2
2
?
dp
?2 kdy


（20）
?< br>?
p
4
?2p
1?p
2
2
?
dp< br>?
?
2kdy

（21）

Obtained the final results：
1
3
p?3p?3arctanp?2ky?c?0

（22）

3
For the difficult equation, we obtain numerical solutions by numerical
differentiation, and then obtained function equation by numerical fitting

method：
Discrete interval [0,8],
Where
take steps ：h = 1.
Each point xi, i = 0,1, …… 8.
Every interval [x
i
, x
i +1
],
the boundary conditions： y (0) = 0, y '(0) = 0.
Into the formula（22） for the boundary conditions：
C = 0
y'?
y
i?1
?y
i
h
3
Put into formula（22）：
1
?
y
i?1
?y
i
??
y
i?1
?y
i
??
y
i?1?y
i
?
??
?3
??
?3arctan
??
?2ky
i
?0

3
?
hhh
?????

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Numerical Solution of each point is obtained in turn：

x
y
0
0
1 2 3 4 5 6 7 8
0.0069 0.0094 0.0336 0.1329 0.3669 0.7872 1.4127 4.0382
Functions images and function equation are obtained by numerical
fitting on Excel：


Figure 2 The results of the numerical solution of the fitting image

After fitting the equation：
y = 0.0003x
6
- 0.0063x
5
+ 0.0435x
4
- 0.1289x
3
+ 0.1594x
2
- 0.0571x -0.0008 （23）
Then the entire image can be got by symmetry along the y-axis. This
models of problem one can be solved.
2.2 Problem two
On question 2, its main purpose is to improve the model in problem
one under the condition of meeting the requirements of other possible
cases. Analyze other possible requirements which include a number of

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aspects, such as the maximum twist in the air, players’ safety when they
leave the ground and the stability of athletes when they land. Among
them, we mainly consider the maximum twist of snowboarder in the air.
When players leave the ground, they are only affected by gravity and
air resistance. We ignore the players’ adjustment in the air. After the
project flying out of the ground, in order to analyzing simply and thinking
clearly, the velocity of the object is divided into lines velocity and angular
velocity. Velocity contains components of three different directions:
horizontal, vertical and longitudinal. Angular velocity consists of
somersault angular velocity and twist angular velocity.

Figure 3 Flip velocity analysis
After athletes flying out of the course, the velocity of longitudinal
depends on a rational allocation of their own energy when they ski, so the
design of course can not be considered. Vertical speed determines the
maximum height with which athletes fly out of the course. So it is the

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requirement for design of the maximum “vertical air”. For the horizontal
velocity, players’ reaction force when they leave the course should be
taken into account. And the horizontal velocity generated by reaction
force must satisfy the equation below.

V
'
?V
x
（24）
Thus it can ensure that athletes fall back to ground safely after flying out,
as the same time, it also meet appreciation, technical and safety
requirements. On the problem that athletes reverse in the air,
Conservation of energy can be used in the cross section.

1
2
mv
2
x1
?
1
2
m
y1?W
p
?A
af
?
2
1
2
J
1
w
1
?
2
1
2
J
2
w
2
?
2
1
2
mv
2
x2
?
12
mv
2
y2

（25）

Where
V is the velocity of each state,
Wf is the effective bio-energy an athlete release,
J is the moment of inertia under different rotations,
Aaf is the energy dissipated by air resistance.
By checking the literature, moment of inertia J1 is 1.1(
84.3(
kg?m
2
kg?m
2
) and J2 is
). Combining with the known data, we get the relationship
. According to the value of V, the relationship of
V
x
of w1, w2,
V
y
V
y
and
will be got. The boundary angle is
?
.
From

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tan∠
?
=
V

V
x

y
∠
?
=83.30
So the boundary angle is 83.30
2.3 Problem three
In practice, there are many factors to consider, for example, the
thickness of snow covering and the construction of the economy, in
addition to shape. The topography should be made the best use of to save
project cost. Climate also is a constraint. Snow can be smoother and be
used longer when the weather is cold.

III. Conclusions
The basis of this model is snowboarding skilled players can
generate the maximum vertical air. Awing of numsolve and fitted the
mathematical function is,
y = 0.0003x
6
- 0.0063x
5
+ 0.0435x
4
- 0.1289x
3
+ 0.1594x
2
-
0.0571x -0.0008

h=4(m) x
0
=8(m)

∠
?
=83.3
0

slope angle ∠
?
= 180 （International recommended values）

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Figure 4

Half-pipe

The ultimate resolution of model takes various factors into account.
The model can be applied to other similar improvements similar problems,
such as the design of emergency chute.

IV. Future Work
Although this paper considered a wide variety, but only one purpose
getting the best track shape. However, in the actual construction process
the aims to be achieved are complex and the design aspects are various. If
you want to continue the track design, the following areas to be
discussed，
1. The run-up route’s height and inclination.
2. Design of the best athletes’ running track. In the process you need
to consider artistic, challenging and security.

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3. Design the length of the orbit to make athletes can efficiently
complete the 5-8 vacated performances.

V. Model evaluation
5.1 Model Advantages
(1). this model is infusion and the result is intuitive.
(2). this paper has
Strong theory
with calculating the best shape theory.
(3). This Problem is close to the real life situation, because of
considered comprehensive.
5.2 Model Disadvantages
Solving the model is complicated and some factors only have the
qualitative analysis and not quantitative discussion.

VI. References

[1] Jason W. Harding , Kristine Toohey, David T. Martin1, Allan G.
Hahn, Daniel A. James . 62008. TECHNOLOGY AND HALF-PIPE
SNOWBOARD COMPETITION –INSIGHT FROM ELITE-LEVEL
JUDGES. ISEA.

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[2] Wu Wei,Xia Xiujun. 2006. Half-pipe snow-board skiing skill
training field in summer Explore and Design. China.
[3]Xiao Ningning,Gao Jun.2009. Research of the Technical
Characteristics of Half-pipe .
[4] Building A Zaugg a.
http:
[5] Olympic Half Pipe Snowboarding .
http:_5150384_olympic-half-pipe-snowboardi

[6]The Physics Of Snowboarding.
http:sics-of-snowboard