The Design of Snowboard HalfpipeAbstract: Based on the snowboard movement theory, the flight height depends on the out- velocity. We take the technical parameters of four sites and five excellent snowboarders for statistical analysis. As results show that the size of halfpipe (length, width and depth, halfpipe slope) influence the in- velocity and out- velocity. Help ramp, the angle between the snowboard’s direction and speed affect velocity ’s loss.For the halfpipe, we established the differential equation model, based on weight, friction, air density, resistance coefficient, the area of resistance, and other factors and the law of energy conservation. the model’s results show that the snowboarders’ energy lose from four aspects(1) the angle between the direction of snowboard and the speed, which formed because of the existing halfpipe(2) The friction between snowboard and the surface(3) the air barrier(4) the collision with the wall for getting vertical speed before sliping out of halfpipe.Therefore, we put forward an improving model called L-halfpipe，so as to eliminate or reduce the angle between the snowboard and the speed .Smaller radius can also reduce the energy absorption by the wall.At last, we put forward some conception to optimize the design of the halfpipe in the perspective of safety and producing torsion.Key words:snowboard; halfpipe; differential equation model;L-halfpipeContents1. Introduction (3)简介1.1the origin of the snowboard course problems (3)滑雪课程的起源问题。

1.2 the background (3)背景2. The Description of Problem (3)问题的描述2.1Practical halfpipe’s requirements (3)实用halfpipe的需求2.1.1 the maximum vertical and the largest body twist (3)最大垂直和最大的身体扭曲2.1.2 Speed analysis (3)速度分析2.2 Halfpipe’s own conditions (4)Halfpipe自身的条件2.2.1 Friction (4)摩擦2.2.2 the size of halfpipe (4)halfpipe的大小3. Model (4)模型3.1 Definitions and Symbols (4)定义和符号3.2 Assumptions (5)假设3.3 the simple analysis of gravity and friction when sliding in the halfpipe (5)简单的分析重力和摩擦力的halfpipe时滑动3.4 in-velocity of factors (6)速度的因素3.4.1 the snowboarder’ angle when in and the speed loss (6)滑雪在角和速度上的损失3.5 out-velocity of factors (8)初速度的因素3.5.1 Help ramp (8)帮助坡道3.5.2 the force point and the plate angle when out (9)力的点和板角3.5.3 the snowboarder’ angle when out and the speed loss (9)滑雪在角和速度上的损失3.5.4 H alfpipe’s Radius (11)Halfpipe的半径3.6 the in-velocity comparison with the out- velocity (14)速度与速率的对比3.7Snowboarder’s position impact on the speed (14)滑雪的位置影响速度3.8 the entire movement of the energychange in the halfpipe (15)在halfpipe中整个运动的能量变化3.9 the balance of speed after considering the air resistance (18)后速度的平衡考虑空气阻力3.10 L-halfpipe (19)左halfpipe3.11 Solution and Result (20)解决方案和结果4. Conclusions (21)总结4.1 Conclusions of the problem (21)结论的问题5. Future Work (21)工作展望5.1 other models (21)其他模型5.1.1 H alfpipe’s location outdoor (22)Halfpipe位置的户外5.1.2 H alfpipe’s material (22)Halfpipe的材料6. References (22)参考文献1. IntroductionIn order to indicate the origin of the snowboard course problems, the following background is worth mentioning.1.1 The origin of the snowboard course problemsIn the past, a significant amount of half pipe anxiety was due to the learning curve of a new sport, and educating resorts and pipe construction person nelson how to prepare the best shapes with basic resort equipment. This mode of operation is changing with the advent of new snowboard specific technology both in machine and hand tools. As technology has made half pipes better, the standards have also been proved. Most half pipe riders have a vision of what an ideal pipe should look like, but shifting that vision into reality seems to be a quantum leap.1.2 The backgroundThe problem lies in the fact that too many people who control the decision making process view of the half pipe as a fixed and static feature, and that once built, a pipe is left to the forces of nature. A severe change of opinions needed, as the half pipe needs to be thought of as an elastic form (almost lifelike) that changes daily and which needs continual maintenance. Another huge factor in developing consistent half pipes is a set of standards. Over the years, the NASBA, OP, USASA, USSA, ISF, and FIS have given differing pipe dimensions to resorts. All this help from various organizations has left pipe building more of an art than a science. Both the ISF and the FIS are now promoting similar versions of half pipe dimensions. So we need to redesign the shape of a snowboard course to maximize the production of vertical air by a skilled snowboarder.2. The Description of the Problem2.1 Practical halfpipe’s requirements2.1.1 the maximum vertical and the largest body twistSnowboarders’ greatest height, the number of rotations (the larges t body twist) and the beautiful action will affect the athlete's score. the longer the spare time left, the more rotations to do for snowboarders. The basic physics principle at work here is the conservation of angular momentum. The angular momentum of the snowboarder is determined at takeoff, and cannot be changed once the snowboarder is airborne. So to make turns in the air the snowboarder must give himself initial rotation upon takeoff. In order to reach the maximum height, the maximum out-velocity would be required.so we analyzed the in- velocity and the out-velocity, and the shape of space (length, width, depth, field gradient) affect the in- velocity and the out-velocity obviously.But the height can not be too high, because too high speed would be a big threat to the safety of snowboarders. Therefore, in order to control the maximum speed, we need to redesign the halfpipe.2.1.2 Speed analysisWhether to reach the maximum vertical height or to produce the largest body twist speed is is a reflection of practical indicators to the halfpipe design.The composition of the factors in the action.Including the fly height, difficulty, diversity, qualitycompletion of the action, Site use and landing conditions and so on because the height have an limit effect on difficulty, diversity, quality of action completement, so the fly height is the core elements of many factors.To conclude,no height,no no flight time and no flight time,no difficult action.As the free fall shows:V=.The height snowboarders can reach have a veryyclose relationship with the speed.2.2 Half pipe’s own conditions2.2.1 FrictionFriction, including friction between the board and the snow as well as air friction.The dynamic friction coefficient between Snow and the board changes from 0.03 to 0.2.Take 0.2 for example, the maximum friction coefficient and the full effect of body weight to calculate the vertical friction0.2Wf=, that the acceleration less due to friction is generated to accelerate the role of body weight 0.2 times, much smaller than resulting in the acceleration of gravity effect. Air friction 2f C Av, in our model, we do not consider the influence of air friction.0.5a a d2.2.2 the size of halfpipeUnder certain circumstances,as the length, depth, tilt angle increases, the speed will be. In view of snowboard safety, speed can not be infinite, which has some of the value of the constraints.3. Models3.1 Definitions and SymbolsFlat：the bottom ground of U grooveTransitions：the transition zone between Horizontal and vertical groove bottom wall Verticals：the vertical parts of the walls between the Lip and the Transitions Platform：the level platform on the snow wall surfaceEntry Ramp：the slippery position of U-shaped slotm：Athlete's qualityg：Gravity accelerationV：Athletes’ speed when first enter u-shaped slot1V：Athletes’ speed when last sliding out u-shaped slottl：under side rectangular width of U-shaped slot1l：the length of U-shaped slot2R：the deep of U-shaped slotn：Athletes emptied timesβ：Angle between Athletes’ speed and slot edge horizontal when first enteru-shaped slotu ： the frictional factor between Skateboarding and snowf A ：how much work friction do when Athletes vertically into a u-shaped slot in arc d C ：Air resistance coefficienta ρ：Air densityν：Athletes’ speed relative air movementA ：Corresponding to the projective area of v3.2 Assumptions1.Assuming frictional factor is a constant when athletes are in taxiing process2.Assuming no melting snow when athletes are in taxiing process)3.Assuming the maximizing friction is gravity, frictional factor as the biggest 0.2, when compared friction work and gravity work4.Assuming the loss of speed is 2 meters per second because of the Angle between the speed and direction of existence with blade when athletes come into (out) the slots every time3.3 the simple analysis of gravity and friction when sliding in the halfpipeIf the athlete slip into the half pipe with a certain speed. Athletes in motion of constantly falling in vertical direction Increasing gravitational potential energy. The process in motion need to overcome the frictional resistance acting between the skate and snow acting must also overcome the air resistance acting. We use all ski areas in China to analyze the data[1] as follows in Table1:Table 2 National snowboard half pipe skiing skill to the situation Championship Series17 ° slope of more than 100 meters along the length of glide in the groove The competition is in the17 ° slope and along the length of more than 100 meters slide in the grooves and do all kinds of flip, twist, grasp the difficulty of board action, the action is completed in a certain vertical height of drop. The standards of international competition venues, can be obtained by calculating the U-groove vertical drop 150*sin17h =.Those athletes complete the maneuver in the vertical direction to produce the height of 40 meters gap. A gap of more than 40 meters in the vertical direction athletes can have a very substantial increase in the rate. A gap of more than 40 meters in the vertical direction athletes can have a very substantial increase in the rate. In terms of free fall calculations 22104020y V hg =≈⨯⨯≈m / s, However the snow and the board’s dynamic friction coefficient between 0.03 to 0.2, the maximum friction coefficient and the full body weight to calculate the friction force acting perpendicular 0.2W of t =.That the speed less is due to the friction resistance, it is weight generated to accelerate the role of body weight 0.2 times, far less than the acceleration of gravity produces results. Therefore, venue’s height of fall is an important way for athletes obtained the vertical velocity. Athletes can complete the vertical velocity and level velocity conversion with a reasonable technology, So that Athletes most likely to get to the maximum vacate height at the last vacate.3.4 in-velocity of factors[1]3.4.1 the snowboarder’ angle when in and the speed lossPlayers control the skis taxiing around the edge of the board into the slot ，both the before and the after of snowboard have the effect of braking, so in order to reduce the loss of speed, so that ，the speed of the body center of gravity in the same direction with the board's longitudinal axis as far as possible ，to reduce the braking effect when the snowboard have instant contact with the snow, and homeopathic slide, taking fulladvantage of wall height difference obtained acceleration. It can be seen the speed of full contact is less than the speed of front panel from Table 3, indicating that the human body has a loss of speed when completely into the slot, Since the existence of wall resistance, the speed loss is normal. However, if the speed of body center of gravity has the same direction with the blade, the speed of the losses will be reduced. As can be seen from Table 3, the athlete’ gravity speed direction has an angle with direction of blade center, the minimum is 1.2, and the maximum is 5.4, the speed ofdirection and the direction with the blade did not reach exactly the same. Decrease the maximum rate reached 27.5%, a minimum rate of 6.8%.Figure 1 the angle between the rate of speed loss and direction with the blade when into the slotIt can be seen that the speed loss rate and direction with the blade angle has not exactly the same trend from Figure 1, there may be several reasons as follows:(1)players is not very skilled when sliding into the slot, the ability of controlling board is not strong(2) It may require different sliding speed for the different air movement in the next time, resulting in players want to control taxi speed on purpose (3)the center of gravity is too forward, the gravity torque is too large, have Side effect, So the technology will have a major impact in speed.3.5 out-velocity of factors [1]3.5.1 help rampAthletes for the first time into the slot before sliding into the slot with help, Athletes should be actively obtained the speed of access to controlled, If the snowboarder into the slot before , after slide a certain distance at the edge of the slot, Obtain a certain speed. and before leaping into the slot and in a certain height 0E , you'll get some initial energy reserves 000E E E 动势(0E Representative athlete ofthe initial energy, 0动E representing athletes initial kinetic energy,0势E .Representing athletes Initial potential) With the completion of the action into the groove, getting smaller and smaller potential energy athletes to complete, in the case of gravity does positive work, the potential energy of the players is correspondingincrease, that the athletes will get the vertical speed by energy transfer. After get some of the vertical velocity into the tank, the athletes have a certain amount of kinetic energy reserves; athletes using the kinetic energy reserves, transformation to the potential when out the half pipe, it can achieve the purpose of improving flight altitude; flight altitude do reserve for potential of the next action into the half pipe for the next action to provide time and space to ensure the successful completion However, athletes in the kinetic and potential energy conversion, to achieve the speed must be controllable. If the speed is not contro llable, it will affect the athlete’s performance, Otherwise it will lead to serious accidents. From Table 4, it can be seen that the athletes Lei Pan rear positive blade rate of 540movement into the tank thelargest; is s.14, the minimum Shi wan Cheng's anti-blade rate of om93720front foot movement, is s11. The actions are successful action, but also a national athlete,.m06so you can give a preliminary conclusion: the speed of athletes in the following speed control 15 meters per second.3.5.2 the force point and the plate angle when outIn the trench wall of the moment, because of losing the support of the front skis, then, the stress point should be to leave the center of board, and gradually transition back to the board, so that the stress point is always forcing plate wall, front foot homeopathic slide, back foot should be gradually forced pedal. When reaction force in sufficient, maintain parabolic path smooth, increasing the speed, and maintain a reasonable angle of the slot. At the same time of achieving the goal of increasing height highly effective, also get into the appropriate slot speed and angle of twist. Reaching movements while floating high, reducing the level of speed and the effect of resistance into the half pipe, reasonably come into the groove; do energy reserves for the next the action.3.5.3 the snowboarder’ angle when out and the speed lossCheng（20）Sun ZhiFeng front o72010.20 8.24 4.0 19.2Huang Shi Ying Anti-fronto72013.73 12.09 3.9 11.9ZenXiao Hua front o72011.65 11．55 0.3 0.9Liu JiaYu behind o54011.209.82 4.1 12.3Pan Lei behind o54012.00 9.11 3.0 24.0 Table 4 is part of the e lite athlete’s slotting board kinematic parameters. By comparing the data in Table 4, we can find what the speed of completely clear out the slot is less than the speed of the front panel instantaneous slip out the slot. It can be seen that five players’ speed and the direction of blade angle have positively correlated with the loss rate in Figure 2, indicating that the greater of angle between speed and direction with the blade, the greater of loss speed, so you need to control the sliding board direction, letting the long axis have the same direction with the speed of human body.Figure 2 the angle between the rate of speed loss and direction with the snowboardwhen out of the halfpipe3.5.4 H alfpipe’s RadiusAppropriate reduced orbit radius can increase the speed when athletes slip out half pipe, and favor the athletes to make various actions in the air. Sides rail identifiable by two 1/4arcs, we can deduces the formulaof tfti o i oi f I dt M w I w ∑⎰+=)(Then taking orbit design into consideration, the optimal speedup method is to reduce the rail depth (by our hypothesis know depth and arc radius is equal), namely decreases of r , and so can reduce of I , effectively increase f w . But, taking the athlete's safety into consideration, the athletes' speed may not excessive, namely orbit radius cannot be too small. General provisions half pipe orbit radius scope for 3-4.5m, guarantee the slot speed are not more than 15, also ensures the athlete's safety.The basic snowboarding physics behind this phenomenon can be understood by applying the principle of angular impulse and momentum.The schematic of the physics of snowboarding in this analysis is given below.Figure 3 the analysis of forceWhere:i w is the initial angular velocity of the body (consisting of snowboarder plusboard), at position (1)f w is the final angular velocity of the body, at position (2), which is the point at which the snowboarder exits the half-pipei V is the initial velocity of the center of mass G of the body, at position (1)f V is the final velocity of the center of mass G of the body, at position (2)i r is the initial distance from the center of rotation o to the body's center of massG, at position (1)f r is the final distance from the center of rotation o to the body's center of mass G , at position (2)g is the acceleration due to gravityN is the normal force acting on the snowboard, as shownF is the friction force acting on the snowboard, as shownIt is assumed that the half-pipe is a perfect circle with center at o. The physics of snowboarding in this analysis can be treated as a two-dimensional problem. Now, apply the equation for angular impulse and momentum to the system (consisting of snowboarder plus board):tfoi i o of fti I w M dt I w +=∑⎰Where:oi I is the initial moment of inertia of the body (consisting of snowboarder plus board) about an axis passing through point o and pointing out of the page, at position(1)of I is the final moment of inertia of the body (consisting of snowboarder plus board) about an axis passing through point o and pointing out of the page, at position(2)o M ∑ is the sum of the moments about point o. These moments are integrated between an initial time i t (at position 1) and a final time f t (at position 2)Here we are assuming that the body can be treated as rigid at positions (1) and (2), even though the snowboarder does in fact change his moment of inertia between thesetwo positions. But as it turns out, when using this equation we only need to know the initial and final values of the moment of inertia of the body.The line of action of the normal force N passes through point o, so it does not exert a moment on the body about point o. The friction force F is small so it can be neglected in terms of its moment contribution. This leaves only the gravitational force which exerts a moment on the body about point o. (Note that the gravitational force acts through the center of mass of the body, consisting of snowboarder plus board). In the above equation isolate f w . Thus,tfoi i o tif of I w M dtw I +=∑⎰Now,22oi Gi i of Gf f I I mr I I mr =+=+Where:Gi I is the initial moment of inertia of the body about an axis passing through point G and pointing out of the page, at position (1)Gf I is the final moment of inertia of the body about an axis passing through point G and pointing out of the page, at position (2)m is the mass of the bodyIn the above equation for f w , if we decrease of I the angular velocity f w will increase beyond the value it would be if we did not decrease of I . In practice this can be accomplished by sufficiently reducing the distance from the center of mass of the body G to the point o. In other words, make f r small enough and f w will increase. Note also that the terms Gf I and o M ∑ may also change somewhat. But the dominant effect will be that of reducing f r .At positions (1) and (2), the velocity of the center of mass G is given byi i if f fV w r V w r ==These two velocities are parallel to the half-pipe since the body is rigid at positions (1) and (2).r small appropriate, Looking at the above equations for velocity, if we makesfw. This in turn will result in his velocity the snowboarder will significantly increasefV) being greater than otherwise.exiting the pipe (f3.6 The in-velocity comparison with the out- velocity [1]It can be seen that the speed of athletes when athletes slip out half pipe is less than the speed of athletes when athletes slip out half pipe from Figure 4. The biggest difference between the two is the Shi wan Cheng, the smallest difference between the two is that Zen Xiao Ye. The average speed is 11.69m s when slip into half pipe, the average down is1.94m s,the speed decline will lead to altitude declining when slip out half pipe, having effect on the speed of slipping into half pipe next time, which restricts movements of athletes and sports techniques to improve the difficulty level of play, but also make the action quality greatly reduced, so the players should pay attention to the completion of a continuous action of the hair lower limb muscle strength.Figure 4 the chart of comparison about speed change when into (out of)half pipe 3.7 Snowb oarder’s position impact on the speedPumping on a half-pipe is used by snowboarders to increase their vertical take-off speed when they exit the pipe. This enables them to reach greater height and performmore aerial tricks, while airborne. The principle is exactly the same as for skateboarders pumping on a half-pipe.The snowboarder is able to increase his speed on the half-pipe with his feet remaining firmly on the board. This begs the question, what is the physics of snowboarding taking place that enables the snowboarder to increase his speed on the half-pipe?To increase his speed, the snowboarder crouches down in the straight part of the half-pipe. Then when he enters the curved portion of the half-pipe he lifts his body and arms up, which results in him exiting the pipe at greater speed than he would otherwise.r small Looking at the above equations for velocity, if the snowboarder makesfw. This in enough (by lifting his body and arms up), he will significantly increasefV being greater than if he did not lift turn will result in his velocity exiting the pipe (fhis body and arms up.By continually pumping his body (by crouching down and lifting his body and arms up in the curved portion of the half-pipe), the snowboarder is able to continually increase his velocity, eventually allowing sufficient height to be reached (upon exiting the half-pipe) to perform a variety of mid-air tricks.A more intuitive (non-mathematical) explanation of the physics of snowboarding taking place here is that pumping adds energy to the system in the same way that a child pumping on a swing adds energy, and results in him swinging higher. Therefore, the physics of snowboarding related to pumping on a half-pipe is similar to pumping on a swing.As a snowboarder lifts his arms and body up he feels resistance due to the force of centripetal acceleration which tends to push his body away from the center of rotation o. This resistance is proof that work is being done, and therefore energy is being added to the system.3.8 the entire movement of the energy change in the halfpipeHow the energy change during the Athletes’ entire movement in the half pipe.Figure 5 3-D half pipeFigure 6 halfpipe’s cr oss-sectionFrom Figure 6, we can know both sides of the curved part is the 1 / 4 cylinder in the side, the middle is rectangle.As shown, we assume that the depth of half pipe is R, the middle part length is 1l , the width of half pipe is 12l R , the half pipe ’s length is 2l , half pipe ’s inclination angle is α.When the athletes straight down into the tank by the vertical speed, we analysis the friction’s work in this process.When the athletes straight down into the tank where has friction, the friction’ work can be applied to functional principle, considering the given state can find out friction ’s work, But this does not consider the specific forms of friction force. By the analysis of analytical solution, we can describe its distribution characteristics.[2] α1l R 2l βAs shown in Fig 7, Objects satisfied Newton equations, the tangent of the form and normal directions form is (considered f uN =),dt dv m uN mg =-θcos ，————————————————（1）R v m mg N 2sin =-θ，————————————————（2）Figure 7 objects in circular orbit forcePray for (2) a derivative timedt dv v R m dt d mg dt dN 2cos =-θθ， dtd R v θ= and (1) David into the type dtd uN mg dt d mg dt dN θθθθ)2cos 2(cos -=-， θθcos 32mg uN d dN =+—————————————————— (3) Solving (3) type is the key to solve the solution of friction()()[]()[]⎰⎰⎰⎰+-=+-=c d u mg u c d ud mg ud N θθθθθθθθ*2exp *cos 3)2exp(*2exp *cos 32exp ——————————（4） Among them;()()()()()θθθθθθθθθθθθθθθd u u u u u u d u u u u d u *2exp *cos 412exp 4sin 2exp 2cos 2exp 2sin )2exp(2cos 2exp *cos 22⎰⎰⎰-+=+=namely ())sin cos 2(14)2exp(*2exp *cos 2θθθθθθ++=⎰u u u d u ， So:)2exp()sin cos 2(4132θθθu C u umg N -+++=， fBecause of: 0,0==N θ，will,24132U mg uC +-=. So:))2exp(2sin cos 2(4132θθθu u u umg N --++=———————————（5） And:))2exp(*2sin cos 2(4132θθθu u u umg uN f --++== ())cos sin 2(4132sin cos 241322202θθθθθθθθθu u f e u u umgR d ue u u umgR uNRd uNds A --+-+-=-++-=-=-=⎰⎰⎰3.9 the balance of speed after considering the air resistance [3]If in the process of straight downhill snow is flat and snowboard does not leave the ground can be approximately described by plane hinged to the relationship between ski and snow we watch skis and skiers as a whole force people ski and snowboard in the force of both concentration a nd reduced to a couple Torques’s’. At this point slide in the snow is equivalent to a single degree of freedom motion system as Figure 8When the system is in static equilibrium with⎩⎨⎧=-=+-0cos 0)(sin ααmg f f f mg sr a Which 20.5a a d f C Av ρ=,r s f uf =.Joint Solution available ：cos r f umg α=Can be seen, friction and gravity components is balance at the balance.Figure8 single degree of freedom motion systemInto the above equation can be obtained:0)cos 5.0(sin 2=+-αραumg Av C mg d aAC u mg v d a ραα)cos (sin 2-= suppose 12sin (1)u u β-=+，。