Team #9822 Page 1 of 29

A solution to snowboard course

Abstract

The shape of a snowboard course has a significant impact on the performance of a snowboarder. Here in the paper, we focus on analyzing and modeling the impact of the course shape on the performance indicators such as vertical air.

Firstly, the movement of the snowboarder is divided into two parts, the movement in the cross section which is vertical with the central axisof the halfpipe, and the movement in the vertical section which parallels the central axisof the halfpipe.

Secondly, by analyzing the process of movement and the transformation of energy of the snowboarder in the cross section, we develop a kinetics equation of vertical air and dimensions which shows how the shape of course affects the vertical air. Then, we calculate the maximum vertical air by the theory of linear programming and get the corresponding dimensions of the course.

Thirdly, with the maximum vertical air and corresponding dimensions of cross section determined above, the movement in the vertical section is analyzed to determine dimensions in the vertical section.

Finally, for different shapes of halfpipe, we calculate the vertical air respectively, and explain how the shape impacts the vertical air thus forming a table. This table provides a practical instruction for halfpipes design and construction.

In addition, besides vertical air, we do the same analysis of other performance requirements such as the number of aloft stage that snowboarder can complete and twist in the air, thus concluding how the shape of the course impact these requirements.

The advantages and innovations of the model are as follows:

1. The movement of snowboarder is divided into two parts of two different directions, which may be slightly different from reality, but reasonable and acceptable under the background and simplifies the model by planarizing the space curve and force condition.

2. In the analysis of movement in cross section, we discuss how the friction coefficient between the snowboard and the halfpipe affects the vertical air, and develop an equation and a curve to show this relationship. Such analyses uncover the rule of the movement and meanwhile verify the veracity of our model.

Keywords: snowboarding halfpipe course, vertical air, kinetics analysis, linear programming, sensitivity analysis

Contents

Abstract1

1 Introduction3

1.1 Background3

1.2 Statement3

2 Analysis4

2.1 The structure of the course4

2.2 The process of movement5

2.2.1 Action analysis5

2.2.2 Movement track analysis6

3 variables and assumption8

3.1 Model assumption8

3.2 statement of basic variables9

4 Model10

4.1 The cross section model10

4.1.1 Model establishment10

4.1.2 The calculation of the cross section model14

4.2 The vertical section model17

4.2.1 Model establishment17

4.2.2 Calculation and analysis of model19

5 Model optimization21

5.1 Times of aloft stage21

5.2 maximum twist in the air22

5.3 Original velocity22

6 Practical application24

6.1 Practical constraints24

6.2 The table of dimensions for practical reference24

7 Conclusion26

8 Appendix28

MATLAB source code28

9 Reference29

1 Introduction1.1 Background

Snowboarding is a sport that involves descending a slope that is covered with snow on a snowboard attached to a rider's feet using a special boot set onto mounted binding [1].

Halfpipe was firstly used in 1970s by skateboarders as a way to create a perfect ride and the thrill of riding up and down the walls of a drainage ditch, and was adopted in later 1970s by snowboarders who wanted to upgrade the challenges of their sport, taking it to a new level of athletic excellence. Later throughout the 1990s and into the current decade the halfpipe as snowboard course gained increasing popularity thanks to Don McKay, Dave Rogers and Dough Waugh Who set a new standard for the sport and designed a machine which had the capacity to mechanize the construction of smooth pipe walls.

The sport has become a Winter Olympic Sport in 1998 and performance in a halfpipe has been rapidly increasing over recent years. The current limit performed by a top level athlete for a rotational trick in a halfpipe is 1440 degrees (4 full 360 degree rotations). In top level competitions rotation is generally limited to improve 'style and flow'. [2]

1.2 Statement

As we know, the performance of a snowboarder depends on several factors, among which are the snowboarders physical and technical conditions, the equipments and the snowboard course. Here, we propose to discus impact of the shape of snowboard course on the performance. Firstly, the shape of the course should be determined to maximize the production of vertical air which means the vertical distance above the edge of the halfpipe. Then, the shape needs to be tailored considering other possible requirements, including numbers of aloft stage, maximum twist in the air, original velocity.

2 Analysis2.1 The structure of the course

We know from the background that snowboarding is a sport game involving big mountain (or free-ride), half-pipe, boarder-cross, rail jam, slope style, big air and racing [3]. The halfpipe originated from skateboarding, and has already become an indispensable style of snowboarding and a normal sport game in Winter Olympic.

The half-pipe is a semi-circular ditch or purpose built ramp (that is usually on a downward slope), between 8 and 22feet (6.7 m) in depth. Snowboarders perform tricks while going from one side to the other and in the air above the sides of the pipe.

Figure2.1: The front version of halfpipe

Figure2.2: The side version of halfpipe

We see from the figure above that the halfpipe resembles a half section of a large pipe, and the followings are the elements of a halfpipe [4]:

(1) Flat: the center flat floor of the Half-pipe

(2) Transitions/Trannies: the curved transition between the horizontal flat and the vertical walls

(3) Verticals/Verts: the vertical parts of the walls between the Lip and the Transitions

(4) Platform/Deck: the horizontal flat platform on top of the wall

(5) Entry Ramp: the beginning of the half-pipe where you start your run

Although in regular contests, dimensions of the halfpipe is limited to a strict criterion, which usually involves 120m in length, 15m in width, 3.5m in depth and an average slope angle of 18, fluctuations inside a specific extension are allowed according the terrain in practical construction. The specifications of dimensions are listed in the following table.

Table 2.1: The specifications of the half-pipe course [5]

Description

Recommended

1

Length of halfpipe

100 165m

2

Slope angle

16 18.5

3

Width of halfpipe

17.5 19.5 m

4

Width of decks

6 - 7.5 m

5

Depth of halfpipe

5-7m

6

Height of vertical

0.2m

7

Entry ramp length

15 m

8

Entry ramp width

10 m

9

Entry ramp height

at least 5.5 m

10

Distance from ramp to pipe

at least 9 m

2.2 The process of movement2.2.1 Action analysis

In the competition, a snowboarder rides from one wall to the other with the music while skiing and performing snowboard tricks on each transition as well as in the vertical air above the edge of the halfpipe. The performance usually contains 5-8 tricks. The judgments grade the performance according to the difficulty and aesthetic feeling. The score of every snowboarder should be less than 10, and the summation of the scores from the 5 judgments is the final score of the snowboarder in this round.

Basic tricks in the riding are listed as follows [5]:

(1) Traversing

It is traversing the transitions and the flat of the half-pipe, and this is the basic of snowboard.

(2) Slide Turns

It is making turns up in the transitions or the flat of the half-pipe. This is the basic of snowboard.

(3) Jump Turns

It is the take-off into the air and leaving the lip of the wall, when the snowboarder gets higher and higher up the wall.

While in the air, the tricks involve handstand, jumping handstand, twisting and grasping the board, etc.

2.2.2 Movement track analysis

According to the background and the action analysis above, the track of the snowboarder in the halfpipe can be drawn as the following figure:

Figure2.3: The track of movement

Considering the conversation of energy, we know that while riding from flat (the bottom part) with a maximum speed, through transition and vertical part, to the highest point of vertical air where the speed comes to zero, the snowboarders kinetic energy, which transform into gravitational potential energy and heat energy by friction, decreases to zero. Meanwhile, the gravitational potential energy comes to maximum as the vertical height comes to its maximum. Things comes reversely when snowboarder rides down the halfpipe from vertical air to the flat part of the course.

In order to observe and analyze the track conveniently, we assume that the crooked surface spreads out into a plane surface. Then, we propose to analysis the track on the plane.

Figure2.4: The plane version of movement track

As we see from the figure of the plane, the track of the ride from the midline of flat to the verge of vertical part on the plane must be a beeline to minimize the friction work, in turn to maximum the gravitational potential energy, thus leading to the maximum vertical air. The track in the vertical air (above the vertical part of the halfpipe) is a part of parabola.

3 variables and assumption3.1 Model assumption

To simplify the question, we make a series of appropriate assumptions as follows:

(1)The course in this issue is designed for skilled snowboarders, which means that when the shape of the halfpipe is determined, the snowboarder is able to control himself/herself to reach the maximum vertical air within his/her ability.

(2)In a halfpipe course, there are quite a lot possible styles of transition curve .However, we only consider the circular arc according to the reasons below. Firstly, in reality, the transition of practical halfpipe is all circular arc. Secondly, circular arc renders the calculation of the model more simple and convenient, without decreasing the fact significance of the model.

(3) In the model, we do not consider the impact of air resistance throughout the movement for the following reasons. Firstly, the area of the windward is an important factor in air resistance, which is, however, extremely difficult to determine, for the reasons that movement of snowboarder is quite complex, with the snowboarders own personality. For example, skilled snowboarders can change the air resistance by changing the windward area. Such change of air resistance is difficult to evaluate. Secondly, the formula for air resistance is very complicated. There is no simple and practicable formula for its calculation yet. Otherwise, if we use a simplified formula, the result would be of no practical significance. Thirdly, the surface of professional clothing in snowboarding is quite smooth and tight enough, so as to reduce the air resistance. Fourthly, during a snowboarders movement, air resistance is nearly unchanged in each action, which means the air resistance can be regarded as a constant in a specific course. In one word, in the study of shape, we can ignore the impact of air resistance.

(4) According to the assumption (1), skilled snowboarders can control the speed and the process of his/her movement effectively, so we suppose that snowboarder could control his/her movement, thus reaching the same speed and angle of the speed when passing the middle line of the course. Above all, all the process could be simplified into a -time circulation of a stage which is a process including entering the halfpipe and the existing. We only study a stage, a process of midpoint of halfpipe - rise air - down- midpoint of halfpipe ", in the circulation.

(5) According to the analysis of course in chapter2.1, the halfpipe consists of curve and line, and there is a certain angle between the central axis and horizontal plane, which is the same with the slope of the hill. Consequently, the movement track of the snowboarder is a complicated space curve. For the purpose of simplicity, we divide the movement of the snowboarder into two parts: the movement in the cross section which is vertical with the central axis of the halfpipe and the movement in the vertical section which parallels the perpendicular bisector of the halfpipe. Then, the model in this article considers the movement of snowboarder from these two parts separately.

3.2 statement of basic variables

Basic variables involved in the model are stated in the following table:

Table3.1: Statement of variables

Variables

Explanation

Units

the vertical air of snowboarder in a stage of the circulation

m

width of halfpipe, namely the distance from lip to lip

m

depth of halfpipe, namely the distance from lip to bottom of halfpipe

m

height of the vertical part of the halfpipe

m

the radius of halfpipes transition section

m

the width of halfpipes flat section

m

the total length of halfpipe

m

angle between central axis of halfpipe and horizontal plane

degree

the mass of the snowboarder

kg

the speed of the snowboarder at the very beginning of a stage, namely the speed when the snowboarder rides cross the central axis

m/s

the angle between the speed vector and the cross section of halfpipe at the very beginning of a stage

degree

times of aloft stage that snowboarder can complete in the circulation

dimensionless

friction coefficient between the snowboard and the ground

dimensionless

gravity acceleration

m/s2

4 Model

The major steps of this model are:

Firstly, the movement of the snowboarder is divided into two parts, the movement in the cross section which is vertical with the central axisof the halfpipe and the movement in the vertical section which parallels the central axisof the halfpipe.

Secondly, the movement in the cross section is analyzed to get the mathematic expression of vertical air, which is associated with the dimension in the cross section. Then, the dimension in the cross section is determined when they make the vertical air get the largest value.

Thirdly, after the dimension of cross section and the maximum vertical air are determined, the movement in the vertical section is analyzed and the dimension in the vertical section is also determined.

Finally, the sensitive analysis is executed in order to determine the influence of the dimension change to the maximum value of vertical air.

4.1 The cross section model

In the following part, the movement in the cross section is analyzed in order to get the equation of vertical air, which is associated with the dimensions in the cross section. Then, dimensions in the cross section are determined when they maximize the vertical air.

4.1.1 Model establishment

The figure of the cross section, which is vertical with the central axisof the halfpipe, is showed as the following figure.

Figure 4.1: The cross section of the halfpipe

The energy transformation, load condition and the movement track in the cross section are analyzed in the following parts.

(1) The equation of energy

When a snowboarder starts from point A, and get the peak of the cross section, point E, the energy transformation of the whole process can be described as the following formula:

(4-1)

In the equation above, refers to the vertical air, which is unknown; refers to the velocity of snowboarder when passing the point A, which is a certain value according to assumption (4); refers to the friction work in the process from point A to point E.

(2) The calculation of friction work

The friction work can be divided into two parts: the friction work in the flat and the friction work in the transition, which is

(4-2)

refers to the friction work in the transition.

Then we propose to calculate the friction work in the transition, and the load condition in the transition is as follows:

Figure 4.2: The load condition in the transition

The load equation in the transition is,

(4-3)

(4-4)

refers to the support from the transition, andrefers to the linear velocity in a certain time.

The derivative of the equation (3-4) with respect to is,

(4-5)

The movement of object in transition can be described as,

After the substitution of and equation (4-3) into the equation (4-5), the result is,

That is

(4-6)

Then we calculate the equation (4-6) to get the expression of. With the method of variation of constant, this first-order linear inhomogeneous differential equation above can be calculated as,

(4-7)

where with methods of integration by parts, we get the following expression,

That is,

(4-8)

After the substitution of equation (4-8) to equation (4-7), the result is,

(4-9)

Then the constant can be calculated with the initial condition, which is listed as follows,

when

In the initial condition above, refers to the velocity of snowboarder when passing by the lowest point of the circle. With the effect of friction, the movement of snowboarder is a process of decelerating from point A to point B. So the equations of movement are,

They can be calculated to,

(4-10)

After the substitution of equation (4-9), the result is,

Then the value of constant is,

After the substitution of const value, the expression of is,

(4-11)

In conclusion, the friction work in the transition is,

So the total friction work in the whole process is,

(4-12)

(3) The calculation of vertical air

The equation of energy, that is equation (4-1), is,

After the substitution of equation (4-12) and equation (4-10), the final result, the expression of vertical air, is,

(4-13)

(4) Analysis of the result

Until now, the expression of vertical air has been calculated. After the analysis of the expression above, the following conclusion can be received: there is a linear connection between the vertical air and the variables, such as , and the coefficient of those variables are all minus, that is with the increase of , the value of vertical air will decrease.

Actually, the increase of leads to the increase of friction work, and as a result, the kinetic energy of snowboarder will also decrease when leaving the halfpipe, so as the vertical air.

We explain and confirm the analysis above once more: although the division of snowboarder movement into two parts is not accordant with the reality, it is logical to suppose the shape corresponding to the maximum vertical air will not change even with the combination of the two parts. Besides, the problem will be too difficult to solve when analyzing the space curve directly. So the division of movement is also an important way to simplify the problem.

4.1.2 The calculation of the cross section model

(1) Establishment of linear programming

According to the rule and recommendation of FIS (International Ski Federation) snowboard world cup, the halfpipe dimensions should be within a specific extent, and then we determine the limitation of dimensions as following [5]:

For the 18feet halfpipe, the recommendation of, which means the width of halfpipe, is between 17.5m and 18m, while for the 22 feet ones the recommendation is 19.5m. Consequently, we take 17.5-19m as the extent of. Since, so it comes to.

When it comes to, which means the depth of halfpipe, the recommendation for 18feet is 5.4m, and that for 22 feet ones is 6.5m. Then, we make vary between 5.4 and 6.5. Since, while the recommendation for in FIS is 0.2m. Thus, we get,

Additionally, we have got restrictions on bothand, that is:

In conclusion, considering the expression of in equation (4-13), we get a linear programming problem with an objective function of, that is,

s.t.

(2) Calculation of the linear programming problem

Moreover, from the statistical data [6] we know that if the velocity exceeds 15, the snowboarder will be unable to control the movement. Consequently we determine 15 as the maximum value of and 9.8 as in the objective function above.

As it is stated above, the kinetic friction coefficient between the snowboard and the halfpipe is between 0.03 and 0.2. We propose to make the coefficient as an example to calculate the maximumand the corresponding dimensions of halfpipe.

When =0.07, .

Consideringas theaxis andas theaxis, the feasible zone indicated in, can be shown in the coordinate system as follows:

Figure 4.3: Linear programming

With the objective function transferred into, it can be calculated according to the graph, that on the dot, comes to its maximum value, that is: .

(3) Effects of friction coefficient on vertical air

Equation (4-13) is the function of vertical air associated with the dimension variables. Actually, the vertical air is also related to the factor and. According to the assumption, the influence analysis of to the vertical air is unnecessary. But it is still necessary to analysis the influence of the friction coefficient on the vertical air.

In order to analyze the influence of the friction coefficient on the vertical air, the value of several variables should be const. Therefore, we hypnosis the values of dimension elements,,and the initial velocity are constant, which is equal to the value calculated in the linear programming. Then we analyze the influence ofon the vertical air.

With the help of software MATLAB, the curve of vertical air and the friction coefficient can be depicted as follows, while the source code of the MATLAB program is shown in the appendix.

Figure 4.4: The influence of friction coefficient to the vertical air

Based on the curve above, following conclusions can be received,

Firstly, the vertical air decreases, with the increase of friction coefficient, which is accord with the common sense. Actually, with the increase of friction coefficient, friction work increases, thus leading to a lower vertical air.

Secondly, when the friction coefficient comes above 0.13, the vertical air becomes negative, which, however, can be explained. On the one hand, the snowboarder may become unable to ride out of the halfpipe. On the other hand, the change of friction coefficient affects the ratios in the objective function, thus altering the whole linear programming. The best shape changes in turn. Actually, the analysis above merely aims to show the trend of vertical air with the change of friction coefficient.

In a word, the analysis above not only shows how friction coefficient impacts the vertical air, but also inspects the model established above.

4.2 The vertical section model

With the shape of the cross section determined, we begin to study the movement in vertical section so as to define the shape of vertical section which leads to the maximum vertical air.

4.2.1 Model establishment

From figure2.2 (the figure of the vertical section), we analyze the load condition, geometric condition and the process of movement. Results are shown as follows:

(1) Analysis of load condition

By analyzing the load condition in the direction of vertical axis, we can conclude that, the snowboarder is only affected by gravity when in the air, as the following figure shows.

Figure4.5: load condition in vertical axis

(4-14)

So the acceleration parallel with the speed is

(4-15)

(2) Analysis of geometric condition

According to the track of the snowboarder in the halfpipe mentioned in charpter2.2.2, we calculate the geometric condition in graph on the plane. We suppose that is the vertical air of snowboarder in a stage of the circulation, is the distance in direction of axis when snowboarder moves in halfpipe, is the distance in direction of axis when snowboarder moves in air. It is shown in the following figure4.3.

Figure4.6: Spread plane-vertical section

Considering the previous analysis of snowboarding action and rules, times of the aloft stage that snowboarder can complete in the circulation, should be controlled between 5 and 8.So from figure 4.3, we know,

(4-16)

It is that represents the angle between the line AB and the cross section of halfpipe , so:

(4-17)

(3) Analysis of movement process

From figure 4.3, when snowboarder moves from B to D, he/she would try his/her best to make himself/ herself vertical to the courses lip, in order to gain more vertical air. Suppose the most ideal condition, that is, before reaching the air, the direction of speed has already been vertical to the courses lip, so

That is

(4-18)

When snowboarder rides out of the course, his movement track is a parabola. Thus, the time in the air from B to D is

Considering the symmetry of movement, the whole time is

During the process, the acceleration in horizontal direction is constant, that is,, so

That is,

(4-19)

Take equation (4-16), (4-18) into (4-15),so

(4-20)

where

4.2.2 Calculation and analysis of model

The issue aims at a best halfpipe in theory. So we only calculate in the limiting case, that is, when the times of the aloft stage reaching its maximum (), so does the speed(). According to the rule and recommendations of FIS (International Ski Federation) snowboard world cup, for the 18 feet halfpipe, the recommendation of, which means the length of halfpipe, is between 100m and 150m, while for the 22 feet ones the recommendation is between 120m and 165m. Consequently, we take 120m as the value of[5].

Besides, when it comes to , from formula (4-20), is in inverse ratio to to some degree, that is to say, with a constant , it will be impossible for the snowboarder to complete the performance if comes too large, which is consistent with common sense. In the calculation, we take in normal conditions.

In the calculation, we take the result of 4.1.2:,,,,and into equation (4-20). It comes to:

Moreover, without considering the constant, we merely use other variables that are already defined about the shape of halfpipe, and comes to the relationship of,, , shown in the follows,

(4-21)

The relationship that is in inverse ratio to to some degree is shown above, that is to say, with a constant, decreases with the increase of. In addition, is in a direct ratio with to some degree. Namely, with a constantand a large, should be large enough to enable the snowboarder to complete the performance, which is accord with the common sense.

5 Model optimization

In the model above, we have analyzed how the shape of course impacts the vertical air. Then, in the following model optimization, we propose to discuss the effect of the shape on the other performance requirements to a certain degree, aiming to change the shape to optimize other possible requirements.

5.1 Times of aloft stage

From the analysis above, we know that the times of stage should be between 5 and 8 in norm conditions.

From the vertical section analysis in chapter 4.2, we know the relationship between the times and shape can be expressed as follows:

(5-1)

where

after simplification, it comes to:

(5-2)

In the equation (5-1), we take the inequality sign rather than equality sign to ensure there is a specific distance after the times of aloft stage. We take the critical condition into account to simplify the calculation, that is to say, assume we take the equality sign in equation (5-2) after times of aloft stage. Consequently, the equation (5-2) turns into,

(5-3)

From the equation above, we can draw the conclusion that is related to several dimensions:

(1) decreases with the increase of. In fact, with the increase of, the slope becomes steeper, and the speed of snowboarder becomes larger In turn, leading to a shorter time, and finally shorten the reaction time.

(2) decreases with the increase of . Actually, the increase of means the increase of angle between the velocity and the cross section. If direction of the velocity does not change, the distance of snowboarder falls in the direction of central axis during the time of one aloft stage will increase. Consequently, the times of the stage will decrease with a constant.

(3) decreases with the increase of . In reality, with the increase of , the distance the snowboarder need to complete a suspended aloft will increase. As a result, the times of the stage will decrease.

(4) increases with the increase of . It is understandable that with the constant distance in a aloft stage, the increase of the length of the course will definitely result in the increase of the times of the stage.

5.2 maximum twist in the air

Through the analysis of movement, we can draw the conclusion that, the factor affects the twist in the air most seriously is the hang time. The longer the hang time is, the more twist in the air. When combining the equation and , we can get the equation below,

(5-4)

It shows that is in a direct ratio with. would be maximum co-responding with the maximum . Also, in chapter 4.1,the calculation of has no relationship with . Equation (5-4) indicates that the larger theis, the smaller the would be. Whats more, the hang time would be longer, the more twist in the air in turn. So we take the maximum and the largest in Chapter 4.2 into the equation (5-4) so as to get the maximum, that is,.

5.3 Original velocity

The original velocity refers to the very velocity when snowboarder enters the halfpipe after a long acceleration by riding along the specific snow slope. Consequently, the original velocity is determined by the process of the movement on the slope before the halfpipe. We propose to find how shape of the slope impacts.

The force condition of the snowboarder on the slope is as the following graph, where refers to the distance between ramp and halfpipe, whilerefers to the angle between central axis of halfpipe and horizontal plane:

(Nmgf)

Figure5.1: Load condition

The energy equation of the snowboarder is:

From which we get:

From the formula above, we can indicate that is in a direct ratio with bothand, that is to say, the original velocity increases with the increase of eitheror. Consequently, we should make the angle andlarger to reach larger original velocity, thus leading to a high vertical air in turn. At the same time, however, the original velocity should not exceed a certain value to ensure the safety of snowboarder.

6 Practical application6.1 Practical constraints

In a practical issue, when building a halfpipe , people should consider the actual terrain from all respects. In one word, unlike the model, there are so many practical constraints. Now constrains are listed as below.

(1)In practical condition, the, that is the angle between central axis of halfpipe and horizon plane is usually a constant, equaling to the slope. Although the builder could intend to change the value of , theand the slope are mostly equated.

(2) According to the contents in the chapter4.1, the vertical air will increase with the decrease of friction coefficient, which will be beneficial to performance of the snowboarder. But actually, it may cost a lot to keep the friction coefficient. Besides, the friction coefficient will also change while the blade angle of the snowboard and the skill of snowboarder change.

(3)Besides, a series of dimension, including,and, are also restricted by the practical landform.

6.2 The table of dimensions for practical reference

In order to provide practical recommendation and support to the builder of the snowboard course, we calculate the dimensions and the corresponding vertical air, and list them in the table below. The table is designed for a practical and convenient reference.

Table6.1: The table of dimensions for practical reference

17.5

18

18.5

19

19.5

5.4

3.27

3.25

3.22

3.20

3.18

5.5

3.18

3.16

3.14

3.12

3.09

5.6

3.10

3.08

3.05

3.03

3.01

5.7

3.01

2.99

2.97

2.95

2.92

5.8

2.93

2.90

2.88

2.86

2.84

5.9

2.84

2.82

2.80

2.78

2.75

6

2.76

2.73

2.71

2.69

2.67

6.1

2.67

2.65

2.63

2.61

2.58

6.2

2.59

2.56

2.54

2.52

2.50

6.3

2.50

2.48

2.46

2.44

2.41

6.4

2.42

2.39

2.37

2.35

2.33

6.5

2.33

2.31

2.29

2.27

2.24

It is necessary to explain that the reference of the table above should be based on certain condition, that is,and .

7 Conclusion

In this paper, the movement of the snowboarder is divided into two parts of two different directions. We analyze the shapes effect on vertical air in two processes, and come to the conclusion of how the shape influences the vertical air.

In cross section, we develop a kinetics equation of vertical air and dimensions which shows how the shape of course affects the vertical air. And we calculate the maximum vertical air by the theory of linear programming .Finally, we get the corresponding dimensions of the course. It is seen as follows:

Figure 7.1 The best dimensions of the cross section

In vertical section, we determine dimensions in vertical section on the base of the dimensions in cross section.

Figure 7.2 The best dimensions of the vertical section

In model optimization, more indicators are taken into considerations, such as times of the stage (suspended aloft), maximum twist in the air, original velocity. Of each indicator, we all calculate it in order to uncover the factors.

Finally, we develop a table, which shows the best dimensions with its maximum vertical air in different condition. It is of great convenience to design and build halfpipe course in reality.

The advantages and innovations of the model are as follows:

1. Physical theory and kinetics are used to go deep into analyzing the forces, the process of movement and the transformation of energy of the snowboarder, thus explaining the objective law in the movement clearly.

2. The movement of the snowboarder is divided into two parts of two different directions, which may be slightly different from the reality, but reasonable and acceptable under the background of our model. This division simplifies the model by planarizing the space curve and force condition.

3. The practical calculation on dimensions explains how the shape impacts those performance indicators, thus providing a credible theory to determine the dimensions of course to perfect snowboarders performance in practical construction.

4. In the analysis of movement in the cross section, we discuss how the friction coefficient between the snowboard and the halfpipe affects the vertical air, and develop a formula and curve to show this effect. Such analysis uncover the rule of the movement and at the same time verify the veracity of our model

5. Different indicators are taken into consideration to optimize our model.

6. The model shows its great advantage in practical part, giving people great convenience in designing and building process.

The disadvantages are as follows:

1.The division of the movement into two parts in planes fails to reflect the reality precisely, thus leading to slight error in the result.

2.Several forces which have little influence on the movement such as the air resistance are not taken into account, which may also bring about error in the result.

3.While analyzing effects of dimensions on the performance, we conduct qualitative analysis instead of quantitative one because of the complexity of those expressions of indicators, which maybe unconvincing to some degree.

8 AppendixMATLAB source code

The influence of friction coefficient to the vertical air

function y=friction(u,v0, R, hv, wf)

y=v0^2*(2-exp(3.14*u))/(2*9.8)-((8*(u.^2)-u+1-2*(u.^2).*exp(u*3.14))*R)./(4*u.^2+1)-hv-0.5*u.*exp(u*3.14)*wf;

end

>> u=[0.03:0.01:0.2];

>> y=friction(u,15,5.2,0.2,7.1)

>> plot(u,y),grid,pause

9 Reference

[1]http://en.wikipedia.org/wiki/Snowboarding Feb.2011

[2]http://en.wikipedia.org/wiki/Half-pipe origin of the Half-pipe Feb.2011

[3]http://en.wikipedia.org/wiki/Snowboarding#Half-pipe Feb.2011

[4]http://www.abc-of-snowboarding.com/snowboardinghalfpipe.asp Feb.2011

[5]Zaugg Zg EggiwilSnowboard resort information sheet for FIS world cup Switzerland 2010

[6]YAN Hongguan, LIU Pin, GUO Fen Journal of Shenyang Sport University :Factors Influencing Velocity Away from Decks in Snowboard Half-pipe Shenyang, China May.2009

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