2011年美赛a题

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 3-4.5m.the 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,

1212mv0WmvmghAf22 (1)

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Where

v0 is the initial velocity （m/h）,

v is the velocity towards destination （m/h）, 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：

v2NmgcosmrWhere

（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：

v2fN mgcosmr （3）

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

svt21212mv0Wmvmghmgcosmdl （4） 22r0Friction acting Af：

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vt2vt2mgBsAf mgcosmrdl2mrdl （5）

00s

Use higher mathematics：

r1y'y''322 （6）

2 dl1y'dx （7）

According to the nature of the curve, the speed can be assumed to satisfy this expression：

1vv0kxe （8）

Put all these formulas in order and suppose the expression for the functional：

2vt2mv0y''mdldx （9） 2kx2r1y'00esB2Set F2mv0y''e2kx1y'2 （10）

Reference Euler equation：

FdFd22ydxy'dxF 11） y''0 （Obtained：

F0 （ 12） yTeam #9262 Page 8 of 17

2mv0y''y'F 3y'2kx22e1y' （13）

Fy''2mv01 （14）

e2kx1y'22To（12）-（14）into equation（11）， Obtained：

dmv20y''y'd22dx3e2kx1y'22mv0 dx2e2kx1y'212Integrate it：

mv20y''y'dmv20e2kx1y'232dxe2kx1y'212C Simplified：

mv20y''y'2kmv20''y'C e2kx1y23'2e2kx3mv20y1y23'2e2kx251y'2y''2k1y'2y'32y' Suppose： dydxpy So：

d2ydpdydpdx2dydxpdy Substituted into the above equation：

dp2k1p2pdyp32p 15）

16）

17）

18）

19）

（ （ （ （ （Team #9262 Page 9 of 17

pIntegrate it：

42p2dp2kdy （20） 21pp42p2dp2kdy （21） 21pObtained the final results：

13p3p3arctanp2kyc0 （22） 3For 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 [xi, xi +1],

the boundary conditions： y (0) = 0, y '(0) = 0. Into the formula（22） for the boundary conditions： C = 0

yi1yiy'hPut into formula（22）：

1yi1yiyi1yiyi1yi33arctan2kyi0 3hhh

3

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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 = 0.0003x6 - 0.0063x5 + 0.0435x4 - 0.12x3 + 0.1594x2 - 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'Vx （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.

12121112122mvx1my1WpAafJ1w12J2w2mvxmvy22222222

（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.

2kgmBy checking the literature, moment of inertia J1 is 1.1() and J2 is

2kgm84.3( ). Combining with the known data, we get the relationship

of w1, w2,

Vy. According to the value of V, the relationship of

Vx and

Vy will be got. The boundary angle is

.

From

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tan∠ =

VyVx/

∠ =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 = 0.0003x6 - 0.0063x5 + 0.0435x4 - 0.12x3 + 0.1594x2 - 0.0571x -0.0008

h=4(m) x0=8(m) ∠=83.30

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 . 6/2008. 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 Snowboarding.China.

[4]

Building

A

Zaugg

Half-Pipe.America.

http://www.zauggamerica.com/resort/pipegroomers/pipe.shtml

[5] Olympic Half Pipe Snowboarding Rules.US.

http://www.ehow.com/way_5150384_olympic-half-pipe-snowboarding-rules.html

[6]The Physics Of Snowboarding.

http://www.real-world-physics-problems.com/physics-of-snowboarding.html