数学美国数学建模写作培训.pptx

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1、Tailortheshapetooptimizeotherpossiblerequirements,suchasmaximumtwistintheair.Whattradeoffsmayberequiredtodevelopa“practical”course?第1页/共147页三个任务:1、确定滑道的形状，使腾空的垂直距离最大；2、调整滑道形状，使其它的参数，如空中扭转幅度，得到优化；3、设计一个实用的滑道会有哪些损失。第2页/共147页首先，思考一下运动员表演的精彩程度和哪些因素有关？1、滑道的形状，2、滑道的坡度3、运动员所选的滑行线路4、运动员本身的身体素质和滑行技术第3页/共147页

2、1、Shape of Course：It is usually accepted that the half-pipe is 100 to 150 meters long,17 to 19.5 meters wide and has a height of 5.4 to 6.5 meters(from floor to crown).The slope angle is 16 to 18.5 degrees.第4页/共147页In addition,the Federation International de Ski(FIS)recommended that the Width(W),Hei

3、ght(H),Transition(T)and the Bottom Flat(B)could be 15m,3.5m,5m,5m,respectively(as Shown in Figure1).第5页/共147页第6页/共147页 2、滑道的倾斜程度可以用滑道底部平面与水平线交角 表示 3、运动员滑行的路线是由滑道边缘某点从高处到低处滑向另一边缘某点，然后起跳到同侧边缘一点再进行下一轮滑行。第7页/共147页 滑行线路可以看成空间一条曲线，坐标用（x,y,z）表示。z轴平行于滑道的方向；y轴垂直于滑道；x轴、y轴、z轴构成右手直角坐标系。4、运动员的技术体现在对滑行路线的掌控和跳跃前体位

4、的转变获得的能量的大小两个方面第8页/共147页Figure 2 shows the definition of the coordinate variables x,y,z.x is the free variable,while y and z are functions of x.is dependent on the shape of the half-pipe.is dependent on the path of the snowboarder第9页/共147页第10页/共147页 The shape of the half-pipe can be represented by t

5、he function The path player chooses can be represented by the function z=z(x).The geometric meanings of are follows第11页/共147页第12页/共147页第13页/共147页第14页/共147页In a simply condition with constant friction,the energy loss will be minimizedif z is proportional to ,the length of the projection curvature of

6、the three dimensional curve to the x-z plane.(Figure 4)第15页/共147页第16页/共147页 Thus,we define the coordinate along z-axis as .Practically,we control z(x)using thelocation of the point right before the fly.第17页/共147页第18页/共147页第19页/共147页第20页/共147页第21页/共147页第22页/共147页 “A cycle”can be divided into two part

7、s.The first part is the movement on the half-pipe,and the second part is the air bourn performance.第23页/共147页第24页/共147页Flying distance():thedisplacementalongzdirectionduringtheflyFlying time():thedurationoftheflyCycle distance(Sc):thedisplacementalongzdirectionduringacycle第25页/共147页Procedures:A:find

8、 the relationships of main variables such as the energy loss resulted by friction,the velocity before flying,the duration of flying,flying height,the distance alone z-axis in each performance,etc.第26页/共147页 B:determine the shape of the course by fixing the width,height,thelength of bottom flat,the v

9、ert height,theshape of the transition.The objective is the maximum vertical height.第27页/共147页 C:analyze the factors affecting twist performance,particularly,the angularmomentum at the moment before flying.D:discuss the trade-off when developa practical course 第28页/共147页Analysis of the forces There a

10、re three kinds of force acted on the snowboarder,which are gravity mg,normal force N and resistance f.Resistance includes air drag and the friction between the snow and snow-board.第29页/共147页 N balances with the gravitys and the pseudo force centrifugal forces projection in the normal direction of th

11、e half-pipe surface at this point.The friction of the snow is propor-tional to the normal force,while hasnothing to do with velocity.第30页/共147页 It required that the angle between the direction of the snowboard and that of snowboarders velocity is constant,and the coefficient is.Air drag is proportio

12、nal to the square of the velocity,and the coefficient is.第31页/共147页第32页/共147页第33页/共147页A:vertical air without considering the rotation第34页/共147页 在管内滑行的第一阶段,运动员以一定的初始能量跳入滑道,并保持身体始终垂直于立足点的切平面方向,由于阻力作用,损失一定能量后到达管道另一端,此时运动员改变姿势成与水平垂直位置,进入第二阶段 起跳,完成空中直体旋转或翻转之后落到管壁边缘,从而完成一个动作循环.两个过程就是能量转换的过程.从三个方面分析这个过程.第

13、35页/共147页第36页/共147页 以管道最高处U形横截面对称中心为原点O，过O点的水平线为零势能点，运动员在任意一点（x,y,z）的势能为管道宽度的一半长度为 。Step1 calculation of第37页/共147页第38页/共147页第39页/共147页第40页/共147页第41页/共147页第42页/共147页第43页/共147页第44页/共147页The work done by the snowboarder is第45页/共147页Step 3.The calculation of“vertical air”At the edge of the half-pipe bef

14、ore thefly,is 0.Consider the potential energy at the start point as 0,From the Energy Conservation Principle,we have 第46页/共147页Since第47页/共147页a.If we neglect the air drag during the fly The flying track from the start point toend point is that in figure.第48页/共147页第49页/共147页From the second Newton law

15、 in the y-axis,第50页/共147页第51页/共147页第52页/共147页b.If we take the air drag during the flyinto account The air drag is with the same direction as the tangent line The component force on the y-axis is 第53页/共147页According to the Second Newton Law:=Let Sincethen 第54页/共147页Separating the variables yieldssoTh

16、us 第55页/共147页Similarly,from the second Newtonlaw in z-axis,obtain flying distanceThe decrease of gravitational potential is第56页/共147页Fromand 第57页/共147页The energy conversion and conser-vation relationship at the beginning and the end of“a cycle”is 第58页/共147页B:Discussing the Rotation “Vertical air”is

17、not the only judgingcriteria of a snowboarders performance.The total degree of rotation and the levelof difficulty are also essential.第59页/共147页Analyzing the problem As soon as the snowboarder flies into the air,if we neglect the air drag,hisangular momentum cannot be changed,so his rotation only de

18、pends on thesnowboarder himself.第60页/共147页 By changing his own posture,thesnowboarder can do twist,flip or whatever he wants.Thus,since we are discussing the influence of the shape of the half-pipe,we only need to considerthe initial angular momentum of the fly.To maximize the twist,we have to maxim

19、ize the initial angular momentum.第61页/共147页 The angular momentum can bedisintegrated into three directions-x,y,and z.Lets represent them with Lx0,Ly0,and Lz0.The shape of the half-pipe canonly affect Lz0 while Lx0 and Ly0 are determined by the snowboarder,with thehelp of the friction of the snow.第62

20、页/共147页第63页/共147页 From the definition of angularmomentum,we have(Regard the snowboarder as a stick.is the angular velocity about the axispassing through and perpendicular to oneend of the stick.m is the mass of the snowboarder;is the height of snowboarder)第64页/共147页 is the radius of curvature of the

21、 cross-section of the half-pipe at the edge and the part of the curve that close to the edge.Plusthen 第65页/共147页 From equation above we can see that the snowboarder should stretch his body as much as possible to increase .And to maximize should be as small as possible and should be as large as possi

22、ble.Here we still use numerical method to find the optimal half-pipeshape.第66页/共147页 When the angular momentum is alongx and y direction.This kind of twist isbeginners level and doesnt have muchto do with the shape of the half-pipe.Instead,it depends on the snowboardersmovement right before the fly.

23、第67页/共147页C:Numerical Computation Firstly,determine the values of some parameters,such as :friction coefficient between the snow and snowboard,:the drag coefficient of air,m:themass of the snowboarder,:the initial mechanical energy at thebeginning of a cycle.第68页/共147页 According to H.Yan,P.Liu,and F

24、.Guo8,the coefficient of kinetic friction between snow and snowboard is generally about 0.03 to 0.2.In addition,J.Chen,R.Li and Z.Guan 9 discoveredthe friction coefficient to be 0.0312.Thus,we generally assume to be 0.03 in our discussion about shape of half-pipe.=0.03第69页/共147页 As N.Yan stated in r

25、eference 10,theair drag coefficient is about 0.15.=0.15 Assume the mass of a snowboarder is typically 60kg which is the mass of Shaun White,one of the most outstand-ing snowboarders in the world.m=60kg第70页/共147页According to reference 11,the drop-in ramp height should be at least 5.5mand distance fro

26、m ramp to pipe shouldbe at least 9m.Since the slope angle isabout 18,the potential energy of gravity is:mg(5.5+9 cos(18)8267 J.第71页/共147页 As may not exceed 0.2(in which case E0 would be 6613 J),we conser-vatively assume the initial mechanicalenergy at the beginning of a cycle to be 7000 J.To sum up,

27、we assume that:0.03,0.15,m 60kg,E0 7000 J.第72页/共147页 We assume the default value of these variables as the table shows and analyze the effect of each variable on thevertical air respectively while maintainingother variables.第73页/共147页第74页/共147页 W(width)第75页/共147页 第76页/共147页Conclusion:With W(width of

28、 ramp)varying from10 to 25,the wider the ramp is,the faster the snowboarder can speed up before thefly and,consequently,the larger“Vertical Air”is.第77页/共147页 H(height)第78页/共147页第79页/共147页Conclusion:Within about 15 meter(possibly the same value of the width),higher ramp can results in higher velocity

29、 and larger“Vertical Air”.But when the ramp is higher than 15 meter,the velocity and the “Vertical Air”will decrease with height.第80页/共147页 B(length of flat bottom)第81页/共147页第82页/共147页Conclusion:With other parameters fixed to defaultvalues,the effects of flat is complex.Longer flat provides larger“V

30、ertical Air”.But the decreasing value meansthe potential maximum“Vertical Air”is decreased,the“Vertical Air”is affectedby the direction the player flies to.第83页/共147页(gradient angle of the venue)第84页/共147页第85页/共147页(location of start point of fly)第86页/共147页第87页/共147页Conclusion:It is obvious that the

31、 steeper the venue is,the faster players can slide.The larger is,the steeper the pathis and more momentum before the fly.Butif the path is too steep(5 meter)theplayer will fly more down the pipe insteadof higher above the pipe.第88页/共147页Global Optimal:We enumerate width from 13m to18m,height from 2

32、to 6,flat length from 4to 6,from 0 to 15 and from 15to 20.Use the kinetic energy before thefly and“Vertical Air”as criteria:第89页/共147页 Best parameters for“Vertical Air”when the snowboarder does not changehis velocity direction right before the flyare：The optimum shape of the course isdetermined by G

33、ene Algorithm.第90页/共147页 Solutions to the problems Question1:Determine the shape of a snowboard course(currently known as a“half-pipe”)to maximize the production of“vertical air”by a skilled snowboarder.“Vertical air”is the maximum vertical distance above the edge of the half-pipe.第91页/共147页 The goa

34、l of this question is to optimizethe shape of the course which provides maximum vertical distance above the edgeof the half-pipe during the fly.Mathematically,we define followingobjective function:第92页/共147页 The function is determined by thefollowing parameters:W(width),H(depth),B(length of flat bot

35、tom),(fly point)andtransition shape.:z coordinate along z-axis right before the fly,which determine sliding path the snowboarder chooses.第93页/共147页 Transition Shape:the y-x relationship,indicating the shape of transitionpart connecting Vert and flat bottom.We follow the following procedures,includin

36、g random Generation of convexcurves,controlling Splines and GeneAlgorithm,to find the optimal transitionshape.第94页/共147页1.Generate sampling points of convex curve as transition part Given:h=0.01,W=第95页/共147页Method:defineso第96页/共147页为了求y(i)，我们来构造k(i):1)随机产生N-1个正数，设2）令满足第97页/共147页 Therefore the sampli

37、ng points of a random convex curve is acquired.2.Fitting the curve Use B-Spline,two-point method or three-point method to fit the sampling points into a curve .第98页/共147页3.Use Gene Algorithm to optimize curve Consider(x(i),y(i)as the 1st generationin Gene Algorithm.Define the position of controlling

38、 pointsas Gene”.Define Fitness-Function as the Vertical Air determined by the shape of course and the default parameters.第99页/共147页 After roulette wheel selection,uniformcross-over,random mutation,optimizingthe Fitness of Gene”,new population derived.Repeat the process.This optimized shape providing

39、 largest“Vertical Air”is close to an ellipse.第100页/共147页第101页/共147页Conclusion 1:Wider and higher ramp,larger gradient angle,all contribute to“Vertical Air”.While the width and height of the course is fixed,the shape of the course is close to an ellipse.Sometimes,a short Vert and Flat is also preferr

40、ed.第102页/共147页 Question2:Tailor the shape to optimizeother possible requirements,such as maximum twist in the air.Tailored the shape of the course byoptimizing degree of rotation which correlates to following requirements.第103页/共147页1)Optimize total duration of fly 2)Optimize initial angular momentu

41、m along z-direction 3)Optimize total degree of rotation 第104页/共147页 1)Optimize total duration of fly also known as“air time”.The duration of fly is Analyze the effect of each factor onthe“air time”when maintaining other Variables as default values 第105页/共147页第106页/共147页Relationship between width and

42、 fly time第107页/共147页Relationship between flat and fly time第108页/共147页Relationship between and fly time第109页/共147页Relationship between height and fly time第110页/共147页Relationship between theta and fly time第111页/共147页Conclusion for 1):Here we enumerate width from 13 to 18,height from 2 to 6,flat length

43、 from 4 to 6,from 0 to 15 and from 15to20to optimize the fly time and obtain the optimal shape of course:Width=18m,Height=5m,Flat=6m,=4m,=20，=2.749s 第112页/共147页2)Optimize initial angular momentum along z-axis As we have computed before,第113页/共147页Conclusion for 2):We use the parameter recommendedby

44、FIS,第114页/共147页 Using Gene Algorithm,we obtain the optimal shape as follows:第115页/共147页 After fitting with second order curve,we found it is close to ellipse.And this curve is tailored from the optimal curve for“Vertical Air”in question1.The“Vert”and“Flat”is much shorter.This result makes sense sinc

45、e“Vert”and a long“Flat”will result in large(actually infinite)radius of curvature.第116页/共147页3)Optimize total degree of rotation Plus,the total degree of rotation also is affect by duration of fly.第117页/共147页Tailor the course As the perception model9 and automated scoring system10,developedby Hardin

46、g JW,etc,has shown,the overallperformance of half-pipe snowboard can be treated as a complex combination of“Vertical Air”,Average Air Time(AAT)and Average Degree of Rotation(ADR).第118页/共147页 They also put forward an equation topredict score of snowboarder automat-ically.Predicted Score=11.424AAT+0.0

47、13ADR 2.223 第119页/共147页 Air time(flying time)is almostproportional to the square root of“VerticalAir”in the motion of Projectile,so theycan be regarded as a single criterion.We set objective function in Gene Algorithm as the“Predicted Score”tofind the optimal shape of course.第120页/共147页 Fitting with

48、 2nd order curves,we found this curve is perfectly fitting bytwo ellipse which fit optimal curves for“vertical air”and total degree of rotation(respectively)第121页/共147页第122页/共147页Question3:What tradeoffs may be required to develop a“practical”course?a.Radius of Curvature The radius of curvature of t

49、he half-pipecross-section cannot be too small,or the snow may fall off12第123页/共147页 And due to the fact that thesnowboard is more than one meter long,it is obvious that a too small radius of curvature is dangerous and infeasible.第124页/共147页b.Flat bottom Since the 1980s,the flat ground givesthe athle

50、te time to regain balance after landing and more time to prepare for thenext trick1 第125页/共147页 Moreover,according to our numericalcomputation,when a half-pipe has widerflat bottom,and decreases,whereas the duration of fly,and vertical air increases.第126页/共147页c.represents the final kineticenergy of