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 axis of the halfpipe, and the
movement in the vertical section which parallels the central axis of 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 halfpipe’s 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
Team #9822
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Contents
Abstract ......................................................................................................1
1 Introduction ............................................................................................3
1.1 Background ...................................................................................................... 3
1.2 Statement.......................................................................................................... 3
2 Analysis ...................................................................................................4
2.1 The structure of the course ............................................................................... 4
2.2 The process of movement ................................................................................ 5
2.2.1 Action analysis ...................................................................................... 5
2.2.2 Movement track analysis ...................................................................... 6
3 variables and assumption ......................................................................8
3.1 Model assumption ............................................................................................ 8
3.2 statement of basic variables ............................................................................. 9
4 Model .....................................................................................................10
4.1 The cross section model ................................................................................. 10
4.1.1 Model establishment ........................................................................... 10
4.1.2 The calculation of the cross section model ......................................... 14
4.2 The vertical section model ............................................................................. 17
4.2.1 Model establishment ........................................................................... 17
4.2.2 Calculation and analysis of model ...................................................... 19
5 Model optimization ..............................................................................21
5.1 Times of aloft stage ........................................................................................ 21
5.2 maximum twist in the air ............................................................................... 22
5.3 Original velocity ............................................................................................ 22
6 Practical application ............................................................................24
6.1 Practical constraints ....................................................................................... 24
6.2 The table of dimensions for practical reference ............................................. 24
7 Conclusion ............................................................................................26
8 Appendix ...............................................................................................28
MATLAB source code ......................................................................................... 28
9 Reference...............................................................................................29
Team #9822
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1 Introduction
1.1 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 snowboarder’s 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.
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2 Analysis
2.1 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 22 feet (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]:
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(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 movement
2.2.1 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
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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 snowboarder’s 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.
Team #9822
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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.
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3 variables and assumption
3.1 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 snowboarder’s 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 snowboarder’s 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 n
-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 - downmidpoint 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.
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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
the vertical air of snowboarder in a stage of the
h
circulation
Units
m
wp
width of halfpipe, namely the distance from lip to lip
m
dp
depth of halfpipe, namely the distance from lip to
bottom of halfpipe
m
hv
height of the vertical part of the halfpipe
m
R
the radius of halfpipe’s transition section
m
wf
the width of halfpipe’s flat section
m
L
the total length of halfpipe
angle between central axis of halfpipe and horizontal
plane
the mass of the snowboarder
the speed of the snowboarder at the very beginning of
a stage, namely the speed when the snowboarder rides
cross the central axis
the angle between the speed vector and the cross
section of halfpipe at the very beginning of a stage
times of aloft stage that snowboarder can complete in
the circulation
friction coefficient between the snowboard and the
ground
gravity acceleration
m

m
v0

n

g
degree
kg
m/s
degree
dimensionless
dimensionless
m/s2
Team #9822
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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 axis of the halfpipe and the
movement in the vertical section which parallels the central axis of 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 axis of 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:
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1
2
mv0  0  mg R  hv  h   Q
2
(4-1)
In the equation above, h refers to the vertical air, which is unknown; v0 refers to the
velocity of snowboarder when passing the point A, which is a certain value according to
assumption (4); Q 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
Q  mg
wf
2
 Q1
(4-2)
Q1 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,
dv

mg sin   N  m


dt

2
 N  mg cos   mv

R

(4-3)
(4-4)
N refers to the support from the transition, and v refers to the linear velocity in a certain
time.
The derivative of the equation (3-4) with respect to t is,
dN
d m
dv
 mg sin 
  2v 
(4-5)
dt
dt
R
dt
The movement of object in transition can be described as,
d
vR
dt
d
After the substitution of v  R
and equation (4-3) into the equation (4-5), the
dt
result is,
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dN
d
d
 mg sin 
 (mg sin   N )  2 
dt
dt
dt
That is
dN
 3N  mg sin 
d
(4-6)
Then we calculate the equation (4-6) to get the expression of N . With the method of
variation of constant, this first-order linear inhomogeneous differential equation above
can be calculated as,
2 d 
  2 d
N  e
d  c   e 2   mg sin   e 2  d  C
(4-7)
  mg sin  e


where with methods of integration by parts, we get the following expression,
1  2 
 2 
 2 
 sin  e d   2 e sin    cos  e d






  2 

1  2 
sin  
e
cos    e  2  sin d 
e
2


sin   2  cos   2 
sin   2 

e

e

e
d
2
2
4
4 2

1
2
That is,
cos   2  sin   2 
e

e
e  2  cos   2 sin  
2
4 2
 2 
sin

e
d




1
4 2  1
1
4 2
(4-8)
After the substitution of equation (4-8) to equation (4-7), the result is,
N  mg
cos   2 sin 
 Ce 2 
2
4  1
(4-9)
Then the constant C can be calculated with the initial condition, which is listed as
follows,
when t  0 ，  0 ，N  mg 
mv12
R
In the initial condition above, v1 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,
v0 2  v1 2  2as

wf

s 
2

a  g

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They can be calculated to,
v1  v0  gw f
2
2
(4-10)
After the substitution of equation (4-9), the result is,
mv12
mg
cos  2 sin    Ce 2
mg 

R
4 2  1
Then the value of constant C is,
C  mg 
mv12
mg
4 2 mg mv12



R
R
4 2  1 4 2  1
After the substitution of const value, the expression of N is,
N
2
mv12 
mg
2   4 mg




cos


2

sin


e

 4 2  1
R 
4 2  1

(4-11)
In conclusion, the friction work in the transition is,


 mg
 4 2 mg mv12

Q1   Nds   2 NRd  R  2  2
cos   2 sin    e 2   2

0
0
R
4


1
 4  1


2


mg 
 R  4 2 mg mv1  2  2
2
2


 R 2

e
  sin  0  2 cos  0  
 2
0
R 
4  1 
 2  4  1

 d

mg
R  4 2 mg mv12  

e  1



1

2



2  4 2  1
R 
4 2  1
2  1  2 2 mg 1 2  
 mgR 2

 mv1 e  1
4  1  4 2  1 2

 R
So the total friction work in the whole process is,
Q
 2 2 mg 1
1
2  1
2
mg  w f  mg 2
 R   2
 mv1  e   1
2
4  1
 4  1 2



(4-12)
(3) The calculation of vertical air
The equation of energy, that is equation (4-1), is,
1
mv02  0  mg R  hv  h   Q
2
After the substitution of equation (4-12) and equation (4-10), the final result, the
expression of vertical air, is,
1
2
mv0  Q
h 2
 R  hv
mg


v0 2  e  8 2    1  2 2 e 
e 


 R  hv 
 wf
2g
2
4 2  1
2
(4-13)
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(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 R 、hv 、w f , and the coefficient of
those variables are all minus, that is with the increase of R 、hv 、w f , the value of
vertical air will decrease.
Actually, the increase of R 、hv 、w f 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 w p , 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 w p . Since w p  w f  2 R , so it comes
to
17.5m  w f  2 R  19m
.
When it comes to d p , 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 d p vary between 5.4
and 6.5. Since d p  hv  R , while the recommendation for hv in FIS is 0.2m. Thus, we
get R  d p  hv  d p  0.2 , 5.2m  R  6.3m
Additionally, we have got restrictions on both w f and R , that is:
17.5  w f  2 R  19
5.2  R  6.3
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In conclusion, considering the expression of h in equation (4-13), we get a linear
programming problem with an objective function of h , that is,


v 2  e  8 2    1  2 2 e 
e 
max h  0


R

h

 wf
v
2g
2
4 2  1
2
s.t. 17.5  w f  2 R  19
5.2  R  6.3
(2) Calculation of the linear programming problem
Moreover, from the statistical data [6] we know that if the velocity exceeds 15 m/ s , the
snowboarder will be unable to control the movement. Consequently we determine 15
m/ s as the maximum value of v and 9.8 m / s 2 as g 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   0.07 as an
example to calculate the maximum h and the corresponding dimensions of halfpipe.
When  =0.07, h  0.9386 R  0.0436w f  8.465 .
Considering R as the x axis and w f as the y axis, the feasible zone indicated in
17.5  w f  2 R  19 , 5.2  R  6.3 can be shown in the coordinate system as follows:
Figure 4.3: Linear programming
With the objective function transferred into y  21.53x  22.9h  194.1 , it can be
calculated according to the graph, that on the dot  x, y   R, w f   5.2 ,7.1 , h comes to
its maximum value, that is: max h  3.27m .
(3) Effects of friction coefficient on vertical air
Equation (4-13) is the function of vertical air associated with the dimension variables R 、
hv 、w f . Actually, the vertical air is also related to the factor v0 and  . According to
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the assumption, the influence analysis of v0 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 R , hv , w f and the initial velocity v0 are constant, which is equal to the value
calculated in the linear programming. Then we analyze the influence of  on the
vertical air.
R  5.2m, w f  7.1m, hv  0.2m, v0  15m / s
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
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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
ag
(4-14)
So the acceleration parallel with the speed v is
a x  g cos
(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
vertical air of snowboarder in a stage of the circulation,
of axis when snowboarder moves in halfpipe,
lv
l0
h
is the
is the distance in direction
is the distance in direction of axis
when snowboarder moves in air. It is shown in the following figure4.3.
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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, n should be controlled
between 5 and 8.So from figure 4.3, we know,
n2l0  lv   L
It is
so:

(4-16)
that represents the angle between the line AB and the cross section of halfpipe ,
1


l 0   R  hv  w f   tan 
2
2

(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 course’s 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 course’s lip, so
v 2  2 g cos  h
That is
v  2 g cos  h
(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
t
v
g cos 
Considering the symmetry of movement, the whole time is
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t '  2t 
2v
g cos 
During the process, the acceleration in horizontal direction is constant, that is,
a y  g sin  , so
2
 v 
1
  4
lv  g sin  
2
 g cos 
That is,
lv  4h tan 
(4-19)
Take equation (4-16), (4-18) into (4-15),so
2l0  4h tan  
L
n
(4-20)
1


where l 0   R  hv  w f   tan 
2
2

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 n reaching its maximum ( n  8 ), so does the
speed( v0  15 ). According to the rule and recommendations of FIS (International Ski
Federation) snowboard world cup, for the 18 feet halfpipe, the recommendation of L ,
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 L [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   25 in normal conditions.
In the calculation, we take the result of 4.1.2:   0.07 , R  5.2 , hv  0.2 , w f  7.1 ,
h  3.27   25 and L  120 into equation (4-20). It comes to:
  16.58
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  ,  , L ,
shown in the follows,
(4-21)
190.624 tan   104.64 tan   L
The relationship that  is in inverse ratio to  to some degree is shown above, that is
to say, with a constant L ,  decreases with the increase of  . In addition,  is in a
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direct ratio with L to some degree. Namely, with a constant  and a large  , L should
be large enough to enable the snowboarder to complete the performance, which is
accord with the common sense.
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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:
L
2l 0  4h tan  
(5-1)
n
1


where l 0   R  hv  w f   tan 
2
2

after simplification, it comes to:
n
L
R  2hv  w f tan   4h tan 
(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 n 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 n times of aloft stage. Consequently, the equation
(5-2) turns into,
n
L
R  2hv  w f tan   4h tan 
(5-3)
From the equation above, we can draw the conclusion that n is related to several
dimensions:
(1) n 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) n 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 L .
(3) n decreases with the increase of R 、hv 、w f . In reality, with the increase of R 、hv 、
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w f the distance the snowboarder need to complete a suspended aloft will increase. As
,
a result, the times of the stage will decrease.
(4) n increases with the increase of L . 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
t '  2t 
2v
g cos 
and
v  2 g cos   h , we can get the equation below,
t'  2 2 
h
g cos 
(5-4)
It shows that t ' is in a direct ratio with h . t ' would be maximum co-responding with
the maximum h . Also, in chapter 4.1,the calculation of h has no relationship with  .
Equation (5-4) indicates that the larger the  is, the smaller the cos would be. What’s
more, the hang time t ' would be longer, the more twist in the air in turn. So we take the
maximum h and the largest  in Chapter 4.2 into the equation (5-4) so as to get the
maximum t ' , that is,   16.58 .
5.3 Original velocity
The original velocity v0 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 v0 .
The force condition of the snowboarder on the slope is as the following graph, where l
refers to the distance between ramp and halfpipe, while  refers to the angle between
central axis of halfpipe and horizontal plane:
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N
f
m
g
Figure5.1: Load condition
The energy equation of the snowboarder is:
1 2
mv  mgl sin   mg cos   l
2
From which we get:
v 2  2l g sin   g cos  
From the formula above, we can indicate that v is in a direct ratio with both l and  , that
is to say, the original velocity increases with the increase of either l or  . Consequently,
we should make the angle  and l larger 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.
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6 Practical application
6.1 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  , the  and 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 R , hv and w f , 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
wp
h
dp
5.4
5.5
5.6
5.7
5.8
5.9
6
6.1
6.2
6.3
6.4
6.5
wp
17.5
3.27
3.18
3.10
3.01
2.93
2.84
2.76
2.67
2.59
2.50
2.42
2.33
18
3.25
3.16
3.08
2.99
2.90
2.82
2.73
2.65
2.56
2.48
2.39
2.31
18.5
3.22
3.14
3.05
2.97
2.88
2.80
2.71
2.63
2.54
2.46
2.37
2.29
19
3.20
3.12
3.03
2.95
2.86
2.78
2.69
2.61
2.52
2.44
2.35
2.27
19.5
3.18
3.09
3.01
2.92
2.84
2.75
2.67
2.58
2.50
2.41
2.33
2.24
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It is necessary to explain that the reference of the table above should be based on
certain condition, that is v0  15m / s ,   0.07 and hv  0.2m .
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7 Conclusion
In this paper, the movement of the snowboarder is divided into two parts of two
different directions. We analyze the shape’s 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
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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 snowboarder’s 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.
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8 Appendix
MATLAB 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
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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 Eggiwil “Snowboard 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