2011美国大学生数学建模竞赛特等奖提名

1、2011 Mathematical Contest in Ming (MCM) Summary SheetTeam Control Number9665Problem ChosenASummaryFor office use onlyT1 T2 T3 T4 For office use onlyF1 F2 F3 F4 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 aspec

2、ts, such as the “vertical air” and the average degree of rotation. Its 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 anoptimization problem.For Question One, an interesting optimization

3、 mis constructed for the “vertical air”based on the theory of Dynamics. In particular, in order to achieve theum vertical distanceabove 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 t

4、he 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 st

5、eps method, the length ofthe U-shape bowl is calculated as 159.95 meters.For Question Two, the mechanical characteristics mis constructed based on the theoryof Sport Biomechanics. Particularly, we firstly derived the relationship between the torque and the horizontal twist angle. Then, the function

6、relationship between the horizontal twist angle and thearc angle of the transitions is examined. Also, the situation of twist in the vertical direction can bediscussed similarly and finally, in order toize the twist in the air, the best values of the arcangle of the transitions, the length and width

7、 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 m based on the “vertical air” and the twist angle is developed which ensures that the athlete would get the best scores.

8、 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” andtwist angle are also

9、discussed. The mis tested by a numerical example and the sensitivityanalysis about the initial velocity is performed. Finally, based on the analysis about the strengthsand weaknesses of the proposed m, we have also considered the refinement of our m,taking the safety of the athlete and the airinto.K

10、eywords:SnowboardHalf-PipeTheory of DynamicsOptimizationMechanical CharacteristicsTeam # 9665Page 1 of 30Fly Higher with More Twist: The Optimization of a Half-PipeContentI Introduction2II Symbol Definitions2III Solution to Question One33.1 Half Pipes33.2 The Description of Half-Pipe We Focused43.3

11、Basic M. 43.3.1 Assumptions and Notations53.3.2 The Analysis of Snowboarders Movement63.3.3 The Calculation of Wf63.3.4 The Discussion of Wp73.3.5 The Function of Wp83.3.6 Approach Parameters of Half-Pipe9IVMechanical Characteristics Mof Question Two164.1 Assumptions164.2 Torque164.3 Mechanical char

12、acteristics164.4 The Determination of Parameters of Half-Pipe17V Tradeoffs Mto Question Three185.1 The Determination of Parameters185.2 The Inclined Angle of Half-Pipes Placement19VI The Discussion of other Kinds of Half Pipe206.1 The Half Pipe whose Transitions are Ellipse206.2 The Half-Pipe whose

13、Flat is a Arc21VII The Verification of Our Ms217.1 The Verification by Living Example217.2 Sensitivity Analysis22VIII Strengths and Weaknesses238.1 Strengths238.2 Weaknesses24IX The Refinement of our M. 249.1 The Refined M9.2 The Refined MConsidering the Air. 24Considering the Safety of Athlete25Ref

14、erence26Appendix27Team # 9665Page 2 of 30I IntroductionThe 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 ha

15、s 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)

16、. In order toachieve the desired performance, the key point in determining the shape of a half-pipeis to balance the possible requirements, such as theum production of “verticalair” andum twist in the air.The performance of an athlete heavily depends on the total air time, average degree of rotation

17、 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 todesign the shape of a h

18、alf-pipe for the purpose ofthe twist in the air.izing the vertical distance andII Symbol DefinitionsSymbolDefinitionhxrxdwxqaxDhmFNlSthe height of platform the radius of transitionsthe arc angle of transitions the width of flatthe twist angle of athlete in the sky The length of platformvertical airt

19、he 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 planethe slide distance from backing to flat to completing accelerationTeam # 9665Page 3 of 30rxvs vmax Smax fac ttwM xt

20、he radius of transitionsthe velocity when athlete making flight partthe biggest velocity before sliding on the transitions the slid distance during acceleration on the flatthe friction applied to snowboard the acceleration of athlete on the flatthe time in the air for athlete angular velocity of rot

21、ationthe bending moment about x axisIII Solution to Question One3.1 Half PipesA half-pipe is basically a U-shaped bowl that allows riders to move from one wallto the other by making jumps and perfor(Half-pipe snowboarding).Snowboarding Tricks on each transitionThere is a variety of different kinds o

22、f half-pipes utilized internationally and themain differences between them are:llthe 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 thefollowing three kinds of half-pipes:(1)The

23、bottom region and its two sides are respectively flat and circular. This kind of half-pipe is used in Winter Olympics.The bottom region and its two sides are respectively arched and circular. This kind of half-pipe is used in previous games.The bottom region and its two sides are respectively flat a

24、nd elliptical. This kind of half-pipe is used in some local games.For a skilled snowboarder, its obvious that the shape of a half-pipe will affect(2)(3)his achievement. In the following discussions, we shall develop a mathematical mand focus on analyzing the first kind of half-pipe. In addition, a f

25、ew designs of some other kinds of half-pipes are also discussed and compared with respect to the influences on the athletes performances in games.Team # 9665Page 4 of 303.2 The Description of Half-Pipe We FocusedThe Halfpipe Schematic drawn in Figure 1:Figure 1. Snowboarding Half-pipe Schematic (Hal

26、f-pipe snowboarding) Sources:To illustrate the half-pipe more specific, we throw light upon its main elements.§ FlatIs the center flat floor of the Half-pipe§ TransitionsThe curved transition between the horizontal flat and the vertical walls§ VerticalsThe vertical parts of the walls

27、between the Lip and the Transitions§ Platform/DeckThe horizontal flat platform on top of the wall§ Entry RampThe beginning of the half-pipe where athlete start their run3.3 Basic MIn this section, a basic mis developed in order to mthe design ofahalf-pipe and analyze its influence on the p

28、erformance of an athlete.Team # 9665Page 5 of 303.3.1 Assumptions and NotationsFor the sake of convenience of the following discussions, we firstly assume that: The “ vertical air “ for an specific athlete is a constant.The snowboard and snowboarder are an entity and the internal action of them canb

29、e ignored.(1)(2)(3)(4)The airis ignored.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 subsequentSections.hx : denotes the height of platform;rx : denotes the radius of transitions;d : denotes the arc angle of t

30、ransitions;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 drawnand presented in Figure 2.Figure 2. A scheme of a half-

31、pipe with notations.Team # 9665Page 6 of 30Based on the discussions above, we have×cosd ) = tx ,×sin d + 2tx .hbObviously, in order to complete the design of a half-pipe, we should find thenumerical solutions of each variable defined in Figure 2.3.3.2 The Analysis of Snowboarders MovementA

32、s 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- Wf = mg × Dh,ìïWpíDh = h - h ,ïîxwhere Wp

33、 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 toize the “vertical air”, we shouldizeWpandminimize Wf , respectively. Note that the resistant function offriction for thesnowboard is much smaller than the

34、 work done by the snowboarder, the achievementofWp .izing the “vertical air” can approximately be obtained only byizing3.3.3 The Calculation of WfFirstly, its known that the work done by friction is given by Wf = f ×l = m × FN ×l . In order to compute the force FN , we consider the fo

35、llowing two different situations:u When the snowboarder is on the flat, then we have FN = mg , where m isthe total weight including both the snowboard and the athlete.u When the snowboarder is on the transition, then the centripetal force is provided by the support force of arch and the gravity. Thu

36、s, we havemv2FN - mg × cos l =,2where v is the velocity, l ( 0 < l < d ) is the angle between the gravity and the vertical direction of arch. We can also get that dl = rx × dl and the force analysis of the snowboarder is presented in Figure 3.Team # 9665Page 7 of 30Figure 3. Force an

37、alysis of the snowboarder.Assume thatthe snowboarder has completed n times flight actions, we canapproximately calculate the distance l by the following equation:l = n × (wx + 2r ×s ) / cosa .Thus, we can getm × nmg × wmv2dò0W =+ mn(mg cos l +) × 2r × dl .xfcosaxrx

38、Note that it is difficult to deal with the friction changes of the transitions and thework done by the friction is much less than that done by the athlete used formmgacceleration, we approximate the value of the friction of the transitions as it will not induce a big difference. So, we haveWf = m &#

39、215; nmg × (wx + 2r ×d ) / cosa .and3.3.4 The Discussion of WpSnowboarders get higher velocity by stomas to friction of half-pipe, the velocity has aor rising leg on the flat. However,um value vmax . When the velocityachieves vmax , snowboarder cmprove it by snowboarding skills. We assume

40、thework done by snowboarder when he accelerate for i th on the flat as Wpi . However,in the process of design the half-pipe, we dont care about the numerical solution ofWpi . The reason of discussing it is to ensure the integrity and accuracy of our mCombing with the equations derived above, we obta

41、in.lTeam # 9665Page 8 of 30nå- mnmg(wx + 2rd ) / cosaWpiDh = i=1mg.Obviously, the value of Dh is decided by Wp directly. For the process of everyslid motion done by athlete is same basically, we can calculate Wp bynW = 1 × åWppini=1By analyzing the above equations, we can get that the

42、 “vertical air” isthe velocity when athlete make flight action is limit value.um if3.3.5 The Function of WpWp 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

43、 scheme of slid motion done by athleteAs is shown in Figure 4, from point A to B, the velocity of athlete decreasesbecause of the friction. In this process, the following equations are obeyed.ì1 mv2= 1 mv2 + Wï 2maxBfíïîWf12.1= f1 × ( AA'+ BB ')From point B to C

44、, the velocity recoveries to theum value with the help ofTeam # 9665Page 9 of 30Wp . So we have,ì1 mv2= 1 mv2 -W +Wï 2maxBf2p ,í2ïîWf2= f2 × BCwhere,f1 is the friction applied to snowboard of AA' and BB ' ,f2 is that ofBC and Wf , Wfare the corresponding work do

45、ne byf1 andf2 . Thus, we have12Wp = W + Wff123.3.6 Approach Parameters of Half-PipeThe height of the platform and the radius of transitionThe 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, namel

46、y, thegravitational potential energy translates into kinetic energy totally. So we havemg × h = 1 mv2x02The 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 platfo

47、rm.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 theflight partone. Thus, th

48、e relative suitable original velocity for design platform is the leastv2h = 0 = 6.241 m .x2gThe arc angle of transitionsAthletes make flight part on the top of transitions and his velocity of flight can beresolute in the three directions-x, y and z. Namely,vx = vs ×cosax ,and vz = vs ×cosa

49、z ,vy = vs ×cosa ywhere, vx , vy andvzvsare the component velocities ofin the directions of x,Team # 9665Page 10 of 30a x ,a yand a z are the angle ofvx , vyand vz . Figure 5y and z respectively andshows the specific situation.Figure 5. The scheme of resolution of vsWhat is more, invertical pla

50、ne, just like what is shown in Figure 6 , theand a y obeys the following equations approximately.a xrelationship of d ,ax » d anda y » 90 - do.Thus,vx = vs ×cosdand vy = vs ×sin d .Figure 6. The scheme of component velocities in vertical plane.The height of flight Dh increase wit

51、h the vertical component velocity vy . As is shown in Figure 4, from point A to A' , the relationship of energy obeysm × v 212- f × AA¢ - mg × h . s =2× m × v2max1xNamely,Team # 9665Page 11 of 30wx + rxdcosa2 1v =- f1 () - mgrx (1 - cosd )2(m 2mvmaxs.Then, we can ge

52、twx + rxdcosa2 1v =( mv- f () - mgrx (1- cosd ) ×sin d2ymax1m 2is the friction force in the inclined plane, which decrease with d . Forf1Wherethe variation of it is not significant, we regard it as a constant.We assume that m = 75kg , m = 0.1 and a = 5o , then the solution of distransformed as

53、a question which calculating theum of function of one variable.dUtilizing thesoftware, we can get the value ofcorresponding to themax component velocity in vertical direction is 51.66°.dThe relationship ofandvy is shown in Figure 71412108642000 20.40.60.811.21.41.6the arc angle of transitionsFi

54、gure 7 .The relationship of dand vyJust as shown inFigure7, the relationship of two variables obeys parabolaapproximately. When d = 51.66o , the vreaches theyAccording the equationvy = 2g ×Dh ,2um value.al when d = 51.66o . So, the radius ofwe can derive the “vertical air” istransitions can be

55、calculated ascomponent speed in the vertical directionTeam # 9665Page 12 of 30hx1- cosdr = 16.44 m .xThe width of flatThe process of athlete doing work for acceleration cant be completed iySmaxshort distance. It must reach the minimum distancefor accelerating velocity tovmax . Base on this, we promo

56、te 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 axispresents thedistance.um velocity can be achieved corresponding to a fixed slid(2)The recovery function to velocity decreases with the increment of sli