Team Control Number
For office use only T1 ________________ T2 ________________ T3 ________________ T4 ________________
9665
Problem Chosen
For office use only
F1 ________________ F2 ________________ F3 ________________ F4 ________________
A
Summary
As is known to all, in half-pipe games, the shape of the half-pipe has some influences on the performances of the snowboarders in many aspects, such as the “vertical air” and the average degree of rotation. It’s a big challenge for the snowboard designers to determine the shape of the half-pipe so as to achieve the desired performances. This problem can be attributed to an optimization problem.
For Question One, an interesting optimization model is constructed for the “vertical air” based on the theory of Dynamics. In particular, in order to achieve the maximum vertical distance above the edge of the half-pipe, we have derived the best height of the platform, which is 6.241 meters. On the other hand, by the relationship between the arc angle of the transitions and the vertical component of velocity, the best value for the arc angle of the transitions is determined to be 51.66°. Furthermore, according to the analytic expression of the curve of velocity recovery, we calculate the width of the flat is 19.18 meters. Finally, by the four steps method, the length of the U-shape bowl is calculated as 159.95 meters.
For Question Two, the mechanical characteristics model is constructed based on the theory of Sport Biomechanics. Particularly, we firstly derived the relationship between the torque and the horizontal twist angle. Then, the function relationship between the horizontal twist angle and the arc angle of the transitions is examined. Also, the situation of twist in the vertical direction can be discussed similarly and finally, in order to maximize the twist in the air, the best values of the arc angle of the transitions, the length and width of the U-shape bowl and the height of the platform are respectively calculated as 45°, 165.96 meters, 19.18 meters and 6.241 meters.
For Question Three, the weighted tradeoff model based on the “vertical air” and the twist angle is developed which ensures that the athlete would get the best scores. Based on the relationship among the “vertical air”, the twist angle and the corresponding scores obtained, we derived the appropriate weight coefficient and develop a practical course.
Moreover, the influences of other two different shapes of half-pipes on the “vertical air” and twist angle are also discussed. The model is tested by a numerical example and the sensitivity analysis about the initial velocity is performed. Finally, based on the analysis about the strengths and weaknesses of the proposed model, we have also considered the refinement of our model, taking the safety of the athlete and the air resistance into account.
Keywords:
Snowboard Half-Pipe Theory of Dynamics Optimization Mechanical Characteristics
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Fly Higher with More Twist: The Optimization of a Half-Pipe Content
I Introduction ................................................................................................................. 2 II Symbol Definitions .................................................................................................... 2 III Solution to Question One .......................................................................................... 3
3.1 Half Pipes ......................................................................................................... 3 3.2 The Description of Half-Pipe We Focused ...................................................... 4 3.3 Basic Model ..................................................................................................... 4
3.3.1 Assumptions and Notations ................................................................... 5 3.3.2 The Analysis of Snowboarder’s Movement .......................................... 6 3.3.3 The Calculation of Wf ......................................................................... 6 3.3.4 The Discussion of Wp .......................................................................... 7 3.3.5 The Function of Wp ............................................................................. 8 3.3.6 Approach Parameters of Half-Pipe ....................................................... 9
IVMechanical Characteristics Model of Question Two .............................................. 16
4.1 Assumptions ................................................................................................... 16 4.2 Torque ............................................................................................................ 16 4.3Mechanical characteristics .............................................................................. 16 4.4 The Determination of Parameters of Half-Pipe ............................................. 17 V Tradeoffs Model to Question Three ......................................................................... 18
5.1 The Determination of Parameters .................................................................. 18 5.2 The Inclined Angle of Half-Pipe’s Placement ................................................ 19 VI The Discussion of other Kinds of Half Pipe ........................................................... 20
6.1 The Half Pipe whose Transitions are Ellipse ................................................. 20 6.2 The Half-Pipe whose Flat is a Arc ................................................................. 21 VII The Verification of Our Models............................................................................. 21
7.1 The Verification by Living Example .............................................................. 21 7.2 Sensitivity Analysis ........................................................................................ 22 VIII Strengths and Weaknesses .................................................................................... 23 8.1 Strengths......................................................................................................... 23
8.2 Weaknesses .................................................................................................... 24 IX The Refinement of our Model ................................................................................ 24
9.1 The Refined Model Considering the Air Resistance ...................................... 24 9.2 The Refined Model Considering the Safety of Athlete .................................. 25 Reference ..................................................................................................................... 26 Appendix ...................................................................................................................... 27
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I Introduction
The snowboarding was firstly introduced in the early 1970s and since then, it has become more and more popular all over the world.
However, with the rapid development of modern science and technology, there is no doubt that the sport equipment has played an important role for an athlete to improve the performance.
To meet the needs of skilled snowboarders, manufactures have introduced lots of different designs utilizing different materials and shapes for snowboarder courses, which are also currently known as half-pipes (Brennan, et al, 2003). In order to achieve the desired performance, the key point in determining the shape of a half-pipe is to balance the possible requirements, such as the maximum production of “vertical air” and maximum twist in the air.
The performance of an athlete heavily depends on the total air time, average degree of rotation and the “vertical air”. In fact, these elements are also related to the shape of a half-pipe, especially the transition radius, the height and the length and width of the flat bottom of the half-pipe.
In this paper, the theory of Dynamics and Sport Biomechanics are employed to design the shape of a half-pipe for the purpose of maximizing the vertical distance and the twist in the air.
II Symbol Definitions
Symbol
hxDefinition the height of platform the radius of transitions the arc angle of transitions
the width of flat
the twist angle of athlete in the sky
The length of platform
vertical air
the coefficient of sliding friction of the surface of half-pipe the sustain force given by half pipe to snowboard an athlete the angle between gravity and vertical direction of arc plane
the slide distance from backing to flat to completing acceleration
rx wxaxh
FN STeam # 9665 Page 3 of 30
rxvsthe radius of transitions
the velocity when athlete making flight part the biggest velocity before sliding on the transitions the slid distance during acceleration on the flat
the friction applied to snowboard the acceleration of athlete on the flat
vmaxSmaxfactt
ˆMx
the time in the air for athlete angular velocity of rotation the bending moment about x axis
III Solution to Question One
3.1 Half Pipes
A half-pipe is basically a U-shaped bowl that allows riders to move from one wall
to the other by making jumps and performing Snowboarding Tricks on each transition (Half-pipe snowboarding).
There is a variety of different kinds of half-pipes utilized internationally and the main differences between them are:
the bottom region is flat or arched,
the two side arches of the half-pipe are circular or other arches.
Considering the possible differences mentioned above, we mainly divide the following three kinds of half-pipes:
(1) The bottom region and its two sides are respectively flat and circular. This kind of
half-pipe is used in Winter Olympics.
(2) The bottom region and its two sides are respectively arched and circular. This
kind of half-pipe is used in previous games.
(3) The bottom region and its two sides are respectively flat and elliptical. This kind
of half-pipe is used in some local games. For a skilled snowboarder, it’s obvious that the shape of a half-pipe will affect his achievement. In the following discussions, we shall develop a mathematical model and focus on analyzing the first kind of half-pipe. In addition, a few designs of some other kinds of half-pipes are also discussed and compared with respect to the influences on the athlete’s performances in games.
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3.2 The Description of Half-Pipe We Focused
The Halfpipe Schematic drawn in Figure 1:
Figure 1. Snowboarding Half-pipe Schematic (Half-pipe snowboarding) Sources: http://www.abc-of-snowboarding.com/snowboardinghalfpipe.asp To illustrate the half-pipe more specific, we throw light upon its main elements.
Flat
Is the center flat floor of the Half-pipe Transitions
The curved transition between the horizontal flat and the vertical walls Verticals
The vertical parts of the walls between the Lip and the Transitions Platform/Deck
The horizontal flat platform on top of the wall Entry Ramp
The beginning of the half-pipe where athlete start their run
3.3 Basic Model
In this section, a basic model is developed in order to model the design of a half-pipe and analyze its influence on the performance of an athlete.
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3.3.1 Assumptions and Notations
For the sake of convenience of the following discussions, we firstly assume that: (1) The “ vertical air “ for an specific athlete is a constant.
(2) The snowboard and snowboarder are an entity and the internal action of them can
be ignored.
(3) The air resistance is ignored.
(4) The coefficient of sliding friction of the half pipe surface is fixed.
Now, we are to introduce some notations which will be used in the subsequent Sections.
hx: denotes the height of platform; rx: denotes the radius of transitions; : denotes the arc angle of transitions; wx: denotes the width of the flat; tx: denotes the thickness of flat;
ax: denotes the length of U-shape bowl; bx: denotes the width of the U-shape bowl.
In particular, a scheme of a half-pipe with the notions introduced above is drawn and presented in Figure 2.
Figure 2. A scheme of a half-pipe with notations.
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Based on the discussions above, we have
hx(rxrxcos)tx,bxwx2rxsin2tx.
Obviously, in order to complete the design of a half-pipe, we should find the numerical solutions of each variable defined in Figure 2. 3.3.2 The Analysis of Snowboarder’s Movement
As shown in Figure 2, when an athlete starts from point O to a random extreme point A, there only exists the gravity and friction which are in work in the whole process.
According to the law of conservation of energy, we can get
WpWfmgh, hhhx,where W p denotes the work done by the snowboarder on the flat and it is used for
acceleration, Wf denotes the work done by friction.
Obviously, in order to maximize the “vertical air”, we should maximize Wp and minimize Wf, respectively. Note that the resistant function of friction for the snowboard is much smaller than the work done by the snowboarder, the achievement of maximizing the “vertical air” can approximately be obtained only by maximizing
Wp.
3.3.3 The Calculation of Wf
Firstly, it’s known that the work done by friction is given by WfflFNl. In order to compute the force FN, we consider the following two different situations: When the snowboarder is on the flat, then we have FNmg, where m is
the total weight including both the snowboard and the athlete.
When the snowboarder is on the transition, then the centripetal force is
provided by the support force of arch and the gravity. Thus, we have
mv2FNmgcos,
2where v is the velocity, (0) is the angle between the gravity and the vertical direction of arch. We can also get that dlrxd and the force analysis of the snowboarder is presented in Figure 3.
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Figure 3. Force analysis of the snowboarder.
Assume that the snowboarder has completed n times flight actions, we can approximately calculate the distance l by the following equation:
ln(wx2r)/cos.
Thus, we can get
nmgwxmv2Wfn(mgcos)2rxd.
0cosrxNote that it is difficult to deal with the friction changes of the transitions and the
work done by the friction is much less than that done by the athlete used for acceleration, we approximate the value of the friction of the transitions as mg and it will not induce a big difference. So, we have
Wfnmg(wx2r)/cos.
3.3.4 The Discussion of Wp
Snowboarders get higher velocity by stomping or rising leg on the flat. However, as to friction of half-pipe, the velocity has a maximum value vmax. When the velocity achieves vmax, snowboarder can’t improve it by snowboarding skills. We assume the work done by snowboarder when he accelerate for ith on the flat as Wpi. However, in the process of design the half-pipe, we don’t care about the numerical solution of
Wpi. The reason of discussing it is to ensure the integrity and accuracy of our model. Combing with the equations derived above, we obtain
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hWi1npinmg(wx2r)/cosmg.
Obviously, the value of h is decided by Wp directly. For the process of every slid motion done by athlete is same basically, we can calculate Wp by
1nWpWpi
ni1By analyzing the above equations, we can get that the “vertical air” is maximum if the velocity when athlete make flight action is limit value. 3.3.5 The Function of Wp
Wp is the work done by athlete used for accelerating and it is used to compensate the lose caused by the work done by friction.
Figure 4 shows the specific situation we investigate.
Figure 4. The scheme of slid motion done by athlete
As is shown in Figure 4, from point A to B, the velocity of athlete decreases because of the friction. In this process, the following equations are obeyed.
1212mvmvBWf1max22. Wff1(AA'BB')1From point B to C, the velocity recoveries to the maximum value with the help of
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Wp. So we have,
1212mvmaxmvBWf2Wp, 22Wff2BC2where, f1 is the friction applied to snowboard of AA' and BB', f2 is that of
BC and Wf1, Wf2 are the corresponding work done by f1 and f2. Thus, we have
WpWf1Wf2
3.3.6 Approach Parameters of Half-Pipe
The height of the platform and the radius of transition
The height of platform is related to the original velocity of athletes. In the process of sliding down from the entrance, there is almost no loss of energy, namely, the gravitational potential energy translates into kinetic energy totally. So we have
12mghxmv02
The original velocity decides the achievement of athletes in competition and is related to the “vertical air”. Thus , it is very important to consider it in the design of the height of platform.
According the study by Huang and his team, in order to get a relative good score, athlete should ensure the original velocity when he or she enters the half-pipe to be 11.06m/s at least.
Since the abusive high of platform will result in the big loss of velocity before the flight part made, the relative suitable original velocity for design platform is the least one. Thus,
2v0hx6.241 m.
2g
The arc angle of transitions
Athletes make flight part on the top of transitions and his velocity of flight can be resolute in the three directions-x, y and z. Namely,
vxvscosx, vyvscosy and vzvscosz,
where, vx, vy and vz are the component velocities of vs in the directions of x,
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y and z respectively and x,y and z are the angle of vx,vy and vz. Figure 5 shows the specific situation.
Figure 5. The scheme of resolution of vs
What is more, in vertical plane, just like what is shown in Figure 6 , the relationship of , x and y obeys the following equations approximately.
x and y90 .
Thus,
vxvscos and vyvssin.
Figure 6. The scheme of component velocities in vertical plane.
The height of flight h increase with the vertical component velocity vy. As is shown in Figure 4, from point A to A', the relationship of energy obeys
mvs212mvmaxf1AAmghx. 22Namely,
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vsThen, we can get
wr212(mvmaxf1(xx)mgrx(1cos)) m2cos.wr212(mvmaxf1(xx)mgrx(1cos))sin m2cosvyWhere f1 is the friction force in the inclined plane, which decrease with . For the variation of it is not significant, we regard it as a constant.
We assume that m75kg, 0.1 and 5, then the solution of is transformed as a question which calculating the maximum of function of one variable.
Utilizing the MATLAB software, we can get the value of corresponding to the max component velocity in vertical direction is 51.66°. The relationship of and vy is shown in Figure 7
14component speed in the vertical direction12108642000.20.40.60.811.21.41.6the arc angle of transitions
Figure 7 .The relationship of and vy
Just as shown in Figure7, the relationship of two variables obeys parabola approximately. When 51.66, the vy reaches the maximum value. According the equation
vy22gh,
we can derive the “vertical air” is maximal when 51.66. So, the radius of transitions can be calculated as
hxrx16.44 m.
1cosTeam # 9665 Page 12 of 30
The width of flat
The process of athlete doing work for acceleration can’t be completed in a very short distance. It must reach the minimum distance Smax for accelerating velocity to
vmax. Base on this, we promote the curve of velocity recovery.
Just as shown in Figure 8, the curve has following characteristics:
(1) Horizontal axis presents the slid distance to the start point, and vertical axis
presents the maximum velocity can be achieved corresponding to a fixed slid distance.
(2) The recovery function to velocity decreases with the increment of slid
distance S, namely the slope in figure decreases with S. (3) In order to obtain vmax, Smaxshould be slid at least.
(4) If the slid distance is longer than Smax, the achieved velocity at last can be
regarded as a constant approximately.
Figure 8. The curve of velocity recovery
Fortunately, Harding, et al. (2007) investigate the relationship between the acceleration and the time used for sliding by utilizing a sliding Fast Fourier Transform (FFT) window and subsequent power analysis and characterize it in Figure 9 .
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Figure 9. A unfiltered accelerometer trance associated with a snowboarding run
In Figure 9, the point Mcharacterizes the time when athlete attach flat after leaving transitions. The acceleration is very big at the moment of M. The point N presents the slid distance of an athlete reaches Smax and the velocity is max. The point Pcharacterizes the time when athlete attach the other transition after completing the slide on flat.
We can get, by analyzing, the conclusion that the velocity of athlete will not change after acceleration for some time in the condition of the length of flat is long enough. To investigate it more deeply, we distill the relationship in the figure into a approximate broken line which shown in Figure 10.
Figure 10 .The distilled relationship between acceleration and time
Combing the data shown in Figure 9, The relationship between the acceleration and time when athlete accelerate in flat can be characterized by the following equation.
2.5g2.5gt, t1s. ac0 t>1sTeam # 9665 Page 14 of 30
Thus, the velocity of athlete at random can be calculated by
vv1(2.5g2.5gt)dtv12.5gt1.25gt2.
01The slid distance when the acceleration decreased to 0 at the time of t1s is
Smax, which can be calculated by
Smax(v12.5gt1.25gt2)dt
0.
Note that v111.06 m/s, we get Smax19.27m.
In addition, combing the discussion above and the assumption that 5, we get
1wxSmaxcos19.18m
The solution of the length of a half-pipe
The length of a half-pipe is decided by the slid distance between two continuous fight part and the times of flight parts made by athlete simultaneously. Thus,
.
Figure 11. The scheme of process of sliding
From Figure 11, we can get that
a0DDsin2rxsinvxtt.
In order to get the numerical solution of ax, the following 4 steps should be applied.
Step1: Calculating vs From Figure 4, we have
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1212mvsmvmaxf1AA’mghx. 22Thus,
vs212(mvmaxmgAAmghx). m2In order to calculate vs, the parameters m, vmax, AA' and vs should be known.
We assume that m is 75kg which is the average weight of a snowboarder. Then,
vmax(v12.5gt1.25gt2)|t123.31 m/s.
Note that the range of coefficient of sliding friction is from 0.03 to 0.2, we assume that it is 0.1. From Figure 4, we can also get that
r. AAxcosThus,vs can be calculated.
Step2: The solution of vx and tt by utilizing vs
vxvscosxvscos, vyvssin and tt2
Step3: The solution of DD
vyg
DDStep4: Getting ax by calculation of a0
wx cosCombing the above derivation, we can get that
a022.85 m
According the current investigations, the times of flight part for a athlete is about 7. Therefore, we have
ax159.95m
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IVMechanical Characteristics Model of Question Two
4.1 Assumptions
Athlete can transform the horizontal velocity to the one twist with the central
axis of snowboarder.
The path of sliding and the “ vertical air” are unchangeable
vmax The velocity of athlete can reach
4.2 Torque
In order to maxim twist in the air, athlete must bend his body. A torque, which is perpendicular to snowboard’s axis of rotation, is generated simultaneously when snowboarder bending his body. Figure 12 describe the specific situation.
Figure 12. Torque applied to the snowboard and the angle of twist
Source: Brennan, et al. Modelling the mechanical characteristics and on-snow performance of snow boards.
4.3Mechanical characteristics
We treat the snowboard as a thin beam and relate the applied moment and applied torque to the radius of curvature and the rate of twist by the relationship (Kollár & Springer 2003):
1W WMˆx2224xˆ, W W44 T 24
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ˆ is the bending moment about the x axis, 1where Mxx is the curvature of the
ˆ is the applied torque, is the rate of twist longitudinal y axis (Figure 4), T
which is related to the angle of twist by the expression:d and W22 W24 dyand W44 are the elements of the compliance matrix. The origin of the local xyz co-ordinate system is at the centroid（Brennan, et al，2003）. When only a bending moment is applied we have:
1xˆ and WMˆ. W22Mx24xSince, W24 is decided by the intrinsic property of half pipe. we only need to calculate
ˆ. Note that Mˆ is the product of M(the mass of half pipe) and the projection of Mxxutransition arc at the direction of x. Namely,
ˆMrsin, Mux where rx is determined by . Thus, in order to maxim the angle of twist, we should maximize and can get that
ˆdy W24Mx4.4 The Determination of Parameters of Half-Pipe
ˆ should be In order to maximize the twist angle, the bending moment Mxmaximized. The method of determination of parameters of half-pipe is similar to that of question 1.
Athlete accelerates his velocity to vmax on the flat, slides to transition at this velocity and makes flight part. The key of ensuring the maximum of twist angle of athlete is design of height and arc of transition.
We resolute the velocity vs when athlete begins his flight part, and get:
vxvscosx, vyvscosy and vzvscosz.
After the flight part, vx and vz are unchangeable, and athlete can convert vx
into the rotational velocity (vx) whose central axis is that of snowboard. So,
vxvx, where is the efficiency of converting and only related to the
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competence of athlete. In addition, with the consideration of benefitting athlete to return back snowboarding run, z should be relative big. Since the radius of rotation after athlete making fight part is
l0, we can get the 2angular velocity of rotation ( vx/(l0/2)). In addition, the total time for athlete flight tt( tt2vy/g) can be calculated. Thus, the angle of twist could be calculated as
tt
For x and y90 , we can get
vxvscos and vyvssin.
Thus, we can derive that
ttvsincos2svs2sin22.
Based on the above investigation, we can get the following conclusion. When the arc angle of transition equals 45 degree, the twist angle is maximal. Regarding twist angle as an optimizing object and by the method used in question one, we obtain
hx6.241 m.
And the radius of transitions can be calculated as
rxhx/(1cos)21.3 m
The method used in calculating the width of half pipe is similar to the one used in question one. However, the arch angles of transitions in two methods are different. The numerical solution is ax165.96m.
V Tradeoffs Model to Question Three
5.1 The Determination of Parameters
Base on analysis we have done, we can conclude that the parameters calculated with different objectives- “vertical air” or twist in the air-are different. In order to design the half-pipe which is most suitable to athletes, we tradeoff the two kinds of
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objectives and get the corresponding parameters.
Objective of optimization: the best achievement of athlete in the competition. Harding and his team (2008) provide the relationship among the “vertical air”, degree of rotation and the predicted scores of athlete obeys the following equation.
PS3.42AVA0.011ADR1.794,
where PS is the predicted scores of athletes in the competition, AVA is the average “vertical air” of athletes, and ADR is the average degree of rotation. Basing on the above equation, we can find that the “vertical air” is much more important than the degree rotation for the scores of athletes.
For the objective of optimization is the scores given by judges, we regard the coefficients of AVA and ADR as weight when designing half-pipe. The weight of “vertical air” is
3.4210.997.
3.4210.011And , the weight of degree of rotation is
0.011k20.003.
3.4210.011
k1As the width of flat and the height of platform are equal in the calculations with different objective, there is no need to weight them. Namely,
hx6.241m and wx19.18m
To the remain parameters, ax and rx can be decided by , hx and wx. Thus, in order to complete the design of half-pipe in the tradeoff model, the only things to do is to determine the value of .
According to the principle of weighting, the most suitable arc angle of transitions
t should be calculated by
tk11k2251.63,
where the 1 and 2 are arc angle of transitions in question one and two respectively.
It is easy to get the following parameters
rx16.37m and ax160.19m.
5.2 The Inclined Angle of Half-Pipe’s Placement
The inclined angle of half-pipe’s placement refers to the angle of flat plane and
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horizontal one, which is unrelated to the size of half-pipe and only decided the method of placement.
The angle is used for benefitting to capture the high velocity and reducing the friction force. And a multitude of researchers have done the investigation of inclined angle. Combing them, we get a recommended value which is 18°. If the size of half-pipe is relative small, the angle should decrease slightly, however, it should in the range of 15~18°.
VI The Discussion of other Kinds of Half Pipe
6.1 The Half Pipe whose Transitions are Ellipse
The design parameters of this kind half pipe can be applied by two methods, as is shown in Figure 13.
Figure 13 a. The first method for designing half-pipe
Figure 13 b. The second method for designing half-pipe
To the fist method, for the slope of transition is too steep, it is very hard for athlete sliding to the top which used for making flight part. Even though he or she can reach the top of half pipe, the huge loss on velocity can be ignored too. Thus, the difficulty of snowboarding on this kind of half-pipe is relative big, and it is suit for the snowboarder who pursuit high difficulty.
To the second one, it is equivalent to extend a distance at each sides of flat. In the condition that the width of flat is wide enough, this method will waste the resource of half-pipe. What is more, the arc angle of transitions is difficult to control and it is easy to result in athlete flying out the half-pipe for the reason of the slope of transition is
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too gentle.
6.2 The Half-Pipe whose Flat is a Arc
The transition of this kind of half-pipe is the same to that of the one we mainly discussed, but, the flat of it is arced. For a variety of reasons such as athlete is more vulnerable when he or she sliding on it for it is difficult to stop on the arc run, the effect of acceleration is not obvious for it rely on the static friction, this kind of half-pipe have been abolished.
Figure 14 shows the shape of this kind of the half-pipe.
Figure 14. The shape of half-pipe whose flat is arced.
VII The Verification of Our Models
7.1 The Verification by Living Example
We design three kinds of U-shape half-pipes, and their parameters are shown in Table 1. In order to compare the difference between the current half-pipes used in the competition and the one we design, we collect 6 sets of data which character the parameters of 6 different kinds of practical half-pipe, see Table 2.
Table 1. The parameters of half-pipes we design unit (m)
Parameters Half-pipe in Half-pipe in Half-pipe in Unit
question 1 question 2 question 3
The arc angle of transitions 51.66 45 51.63 degree The length of half-pipe 159.95 165.96 160.19 m The width of flat 19.18 19.18 19.18 m The height of platform 6.241 6.241 6.241 m
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Table 2. The parameters of current half-pipes unit (m)
Parameters
The length of half-pipe The width of flat The height of platform
By Table 1, we can find, obviously, our half-pipe belongs to the relative big one. This is because of the original velocity used for designing the height of platform is that of athlete who competing in big games. Thus, the half-pipe decided by the parameters has a relative big size and can be used for big competitions, such as Winter Olympics.
Combing the analysis of the two tables, the parameters of our half-pipes are close to that of current big one. And this phenomenon can verify the correctness of our model.
Data appeared in different reference
160-220 160 80 100-110 160-200 18 18 18 13-15 18 7~8 6 5 3-3.5 5-6
7.2 Sensitivity Analysis
By changing the original velocity used for designing hx and utilizing our model,
we calculating a set of parameters of half-pipe. The range of original velocity is from 7m/s to 12m/s, and we take 11 velocities totally with the interval of 0.5 m/s. The data of parameters with different velocity is drawn in Figure 15.
the height of platform864210.90.80.76810120.6681012the arc angle of transtionsthe length of halfpipe18016014012022201816681012146the width of flat81012
Figure 15 The trend of half-pipe parameters in sensitivity analysis
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As Figure 15 showing, we can get that The size of half-pipe increase with v0
hx, axand wx are linear with v0 approximately
The arc angle of transitions has the broken line relationship with v0, and the
linearity is better when v0 is relatively big.
The specific data of them are shown in Table 3
Table 3. The data calculated in sensitivity analysis
v0 m/s
7 7.5 8 8.5 9 9.5 10 10.5 11 159.1 6.17 19.24 51.58
11.5 12 ax m124.31 128.72 135.12 139.78 144.25 148.47 152.45 156.15 hx m2.50 2.89 3.27 3.69 4.13 4.60 5.10 5.63 wx m15.22 15.73 16.23 16.73 17.23 17.73 18.24 18.74 degree40.69 40.79 44.67 46.03 47.17 48.29 49.39 50.49
162.94 165.94 6.75 7.35 19.74 20.24 52.64 53.70
From Table 3, when v0 takes the value of 8, 9, 10 and 11m/s, the theoretical results match the current half-pipe very well, which agrees with our model.
Base on the data in Table 3, we divide the half pipes into three kinds-small, medium and big and recommend the relative classic parameters of them. The big half-pipe:
ax159.71, hx6.17, wx19.24 and 51.58.
The medium half-pipe:
ax139.78, hx3.69, wx16.73 and 46.03.
The small half-pipe:
ax128.72, hx2.89, wx15.73 and 40.79.
On the condition of other requirement required, the designer can look up Table 3 to get the corresponding parameters of half-pipe. Obviously, by sensitivity analysis, the range available is extended largely.
VIII Strengths and Weaknesses
8.1 Strengths
Simplicity: The mathematical model we construct is simple enough to
understand with a small amount of mathematical skill. And, the calculation of it can solved by simple mathematical software such as Matlab.
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Flexibility: Models in this paper consider influence on the performance of
athletes given by the parameters-height, width, angle and so on –of half pipe to ensure the flexibility and entity of the design. By doing so, it can design different shape of half pipes with different requirements.
Operability: Our models take into account the criteria of snowboarding and
promote the design plan, which helps athlete perform better in the competitions.
Developed from history data: All the data we used are from the records of
Olympic Games.
Refinement: Base on considering other factors such as injures of athlete, we
refine our model.
8.2 Weaknesses
Assumptions: Simplifying assumptions such as ignoring air resistance had to
be made in order to create a calculable model. Inputs：This verification of model requires a multitude data, some of which is
difficult to obtain.
IX The Refinement of our Model
9.1 The Refined Model Considering the Air Resistance
As the equation (Answer.com) of calculating the air resistance is
1facSwv2
2where, fa is the air resistance, cis the coefficient of air resistance, is the density of air. Sw is the frontal area of object and v is relative velocity of object and air. In addition, c and are constants. And Sw is variable, however, the variation is very small.
Based on the analysis above, we can conclude that
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In the process of sliding, the air resistance is proportional to the square of velocity. In horizontal direction,
dSvdt.
And the work done by air resistance is
SmaxWfa 0SmaxfadS(cSw2)vdS2.
0The equation of velocity on the flat investigated in question is applicable and
Smax is reached when t1s. Thus,
1cSwcSw3Wfa()vdt()(v12.5gt1.25gt2)3dt22001 cSw24253952(gg5v12gv13)22248
For the given c, Sw, and v1, the numerical solution of Wfa is a constant. Thus, just by substituting the sum of Wf and Wfainto the models we construct, the parameters of half-pipe can be calculated.
9.2 The Refined Model Considering the Safety of Athlete
Snowboarders are vulnerable in the competition, and combing the safety factor into the design of half-pipe is necessary. By analyzing the relationship of safety and the twist angle, we can get the function between them. Then, by substituting the factor of safety by the twist angle into the model we constructed, the parameters of half-pipe are also be calculated.
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Reference
[1] Brennan,S.M. et al. 2003. Modelling the mechanical characteristics and on-snow
performance of snowboards. Sports Engineering 6, 193-206.
[2] ABC-OF-SNOWBOARDING. Halfpipe Snowboarding/ Half Pipes Halfpipes.
http://www.abc-of-snowboarding.com/snowboardinghalfpipe.asp.
[3] Hongguang,Yuan.et al.2009.Factors Influencing Velocity Away from Decks in
Snowboard Half-pipe.Journal of Shenyang Sport University 28(3),16-28.
[4] Harding,J.W. et al. 2007. Feature Extraction of Performance Variables in Elite
Half-Pipe Snowboarding Using Body Mounted Inertial Sensors. http://www.anarchistathlete.com/wp-content/uploads/2009/05/harding-et-al-_2007_-feature-extraction-hp-kpvs-inertial-sesnors.pdf. [5] Kollár, L.P. and Springer, G.S. 2003. Mechanics of Composite Structures, p. 209.
Cambridge University Press, New York, NY, USA.
[6] Answers.com. What is the formula for air resistance please state what each
symbol or letter stands for? http://wiki.answers.com/Q/What_is_the_formula_for _air_resistance_please_state_what_each_symbol_or_letter_stands_for.
[7] Harding, J.W and James,D.A.2010. Performance Assessment Innovations for Elite
Snowboarding. 8th Conference of the International Sports Engineering Association.
[8] Harding,J.W. et al. 2008. Automated scoring for elite half-pipe snowboard
competition: important sporting development or techno distraction? Sports Technology Journal. No.6, 277-290.
[9] Snowboarding. Halfpipe Construction: The Shape of Things to Come.
http://snowboarding.transworld.net/1000022552/how-to/halfpipe-construction-the-shape-of-things-to-come/
[10] Wikipedia, the free encyclopedia. Half-pipe, http://en.wikipedia.org/wiki/Half
pipe.
[11] Subic, Aleksandar,et al.2010. Experimental installation for evaluation of
snowboard simulated on snow dynamic performance.8th Conference of the International Sports Engineering Association
[12] Buffinton, Keith W.2003.Laboratory, computational and field studies of
snowboard dynamics, Sports Engineering 6,129-137.
[13] Thomas, H, et al.2010.Recognizing Turns and Other Snowboarding Activities with
a Gyroscope.http://ieeexplore.ieee.org/xpl/mostRecentIssue.jsp?punumber= 5648406
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Appendix
A 1. The solution of in question 1
f = @(x)-sqrt(2/75*(0.5*75*23.31^2-0.1*75*9.8*(19.81+25.*x)/cos(5*pi/180) -75*9.8*25*(1-cos(x)))).*sin(x); % The positive number of vy
x = fminbnd(f, 0, 2) y = -f(x)
A2. The Figure characterizes the relationship between arch angle and velocity syms o;
o=0.001:0.001:pi/2;
g=9.8;a=5*pi/180;rx=25;wx=19.81; m=75;vmax=23.31;f1=0.1*m*g;
vy=sqrt(2/m*(0.5*m*vmax^2-f1*(wx+rx.*o)/cos(a)-m*g*rx*(1-cos(o)))).*sin(o); plot(o,vy)
A3. The solution of width of half-pipe
a=5*pi/180;rx=16.44;o=51.66*pi/180;smax=19.27;rx=16.44; hx=6.241;u=0.1;m=75;g=9.8;vmax=23.31;wx=19.18;n=7; AB=rx*o/cos(a); CD=smax+2*AB;
vs=sqrt(2/m*(0.5*m*vmax^2-u*m*g*AB-m*g*hx)); vx=vs*cos(o); vy=vs*sin(o); tt=2*vy/g;
a0=CD*sin(a) +2*rx*o*sin(a) +vx*tt; % the solution for horizontal distance ax=a0*n
A4. Sensitivity Analysis
% The solution for parameters of half-pipe
thta=[40.6857 40.7889 44.6735 46.0314 47.1659...
48.2889 49.3947 50.4948 51.5777 52.6434 53.6976]; o=thta*pi/180;
vmax=[19.2500 19.7500 20.2500 20.7500 21.2500 21.7500... 22.2500 22.7500 23.2500 23.7500 24.2500];
wx=[15.2246 15.7265 16.2284 16.7303 17.2322 17.7342... 18.2361 18.7380 19.2399 19.7418 20.2437];
smax=[15.1667 15.6667 16.1667 16.6667 17.1667 17.6667...
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18.1667 18.6667 19.1667 19.6667 20.1667]; hx=[2.5 2.89 3.27 3.69 4.13 4.60 5.1 5.63 6.17 6.75 7.35]; % The parameter matrix corresponding to the chosen data a=5*pi/180;rx=16.37; u=0.1;m=75;g=9.8; AB=rx*o/cos(a); CD=smax+2*AB;
vs=sqrt(2/m*(0.5*m*vmax.^2-u*m*g*AB-m*g*hx)); vx=vs.*cos(o); vy=vs.*sin(o); tt=2*vy/g;
a0=(CD*sin(a) +2*rx*o*sin(a) +vx.*tt); a1=a0*7
% Plotting the relationship figures of vy and parameters of half-pipe v0=7:0.5:12;
subplot(231) plot(v0,hx,'r-o')
title('hx'),xlabel('v0')
subplot(232)
plot(v0,smax,'b-d') title('smax'),xlabel('v0')
subplot(233) plot(v0,o,'k-*') title('o'),xlabel('v0')
subplot(234) plot(v0,a1,'m-p') title('a1'),xlabel('v0')
subplot(235)
plot(v0,vmax,'K-+') title('vmax'),xlabel('v0')
subplot(236) plot(v0,wx,'c-<') title('wx'),xlabel('v0')
A5. The data appeared in references
Parameters Data appeared in different reference The length of half-pipe 160-220 160 80 100-110
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The width of flat The height of platform The inclined angle
Parameters
The length of half-pipe The width of flat The height of platform The inclined angle
18 7~8 18 18 6 18 18 5 15 13-15 3-3.5 15~18
Data appeared in different reference 135 135 120 130 17.5 17.5 17.0 16.0 4.8 4.8 4.5 4.6 16.5 16.5 16.0 16.5
A6. The results calculated in the solution
v 0 m/s ax m hx m wx m degree m Smax m vmaxvs m/s tt s
7 7.5 8 8.5 9 9.5 10 10.5 11 159.1 6.17 19.24 51.58 19.17 23.25 19.76 3.16
11.5 12
124.31 128.72 135.12 139.78 144.25 148.47 152.45 156.15 2.50 2.89 3.27 3.69 4.13 4.60 5.10 5.63 15.22 15.73 16.23 16.73 17.23 17.73 18.24 18.74 40.69 40.79 44.67 46.03 47.17 48.29 49.39 50.49 15.17 15.67 16.17 16.67 17.17 17.67 18.17 18.67 19.25 19.75 20.25 20.75 21.25 21.75 22.25 22.75 17.28 17.62 17.91 18.23 18.55 18.86 19.17 19.46 2.30 2.35 2.57 2.68 2.78 2.87 2.97 3.06 162.94 165.94
6.75 7.35 19.74 20.24 52.64 53.70 19.67 20.17 23.75 24.25 20.05 20.34 3.25 3.35
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