2011年美赛a题.doc

17Team #9262 Page of 17 Perfect Half-pipe: The Think of Snowboard Course AbstractWith the continuous progress and development, People are actively involved in sports and exploring in it continually. Skiing is popular with the majority of sports fans gradually under this condition. Especially，Snowboarding with good view, challenge and the basis of the masses develops rapidly and has become a major Olympic projects. In this paper, how to design and optimize the snowboard course of half pipe is discussed in detail. We strive to get the perfect course so that snowboarders can achieve the best motion state in the established physical conditions. Whats more, it may promote the development of the sport.This problem can be divided into three modules to discuss and solve. For the first problem of the design of half pipe, it can be based on the point of the energy conservation law. The method of functional analysis (Variation principle and Euler differential equation) is used to set up equations, when the secondary cause is ignored and the boundary conditions are taken into consideration. The curve equation is obtained by the above equation, that is, a skilled snowboarder can make the maximum production of “vertical air”. For the second question, athletes maximum twist in the air and some other factors need to be taken into account when to optimize the previous model, so that curve can meet the actual game conditions and appraisal requirements as much as possible. Ultimately, a satisfying curve will be got. The third problem is a problem relatively close contact with the actual, which is to setting down a series of tradeoffs that may be required to develop a “practical” course. In this paper, for the formulation of these factors, the main discussions are the thickness of snow on half pipe and the aspect of economy for the construction.After discussing these three aspects, the paper finally summarizes a construction program and evaluation criteria of the course in current conditions. Finally, by evaluating the advantages and disadvantages of the whole model，we put forward the advanced nature of the model, but also point out some limitations of the model.Key words: Snowboard course, Half-pipe,Functional, Euler equation, Fitted curve, Numerical differentiationTable of contentsTable of contents1I. Introduction.21.1 Half pipe structure.21.2 Background problem.31.3 Athletes aerials.31.4 Assume.3II. Models52.1 problem one.52.2 Problem two.112.3Problem three.13III. Conclusions.14IV. Future Work15V. Model evaluation.155.1ModelAdvantages.155.2Modeldisadvantages.16VI. References.17I. IntroductionSnowboarding is a popular pool game with the world of sports. The U-Snowboard length is generally 100-140m , U-type with a width of 14-18m,U-type Depth of 3-4.5m.the slope is 14-18 . In competition U-athletes Skate within the taxi ramp edge making the use of slide to do all sorts of spins and jumps action. The referees score according to the athletes performance as the Vertical air and the difficulty and effectiveness of action. The actions Consist mainly of the leaping grab the board, leaping catch of non-board , rotating leaping upside down and so on.1.1 Half pipe structure Half pipe structure contains: steel body frame, slide board, steps to help slide and rails.1.2 Background problemIn order to improve the movement of the watch, it can be improved from two aspects: orbit and the athletes themselves. Now according to the problem the orbit can be designed as a curve. on the curve the athletes can get a maximum speed. The design of orbit includes a wide range of content, such as the shape of U-groove design, track gradient, width and length designed to help the design of sliding section, and so on. The rational design of half pipe can be achieved to transform the energy to efficiency power, make the athletes achieve the best performance in the initial state of the air. This paper discusses the rational design of half pipe to these issues.1.3 Athletes aerialsAthletes on the hillside covering with thick snow skill down with the inertia of the platform, jump into the air, and complete a variety of twists or somersault. Rating criteria: vacated, takeoff, height and distance accounted for 20%; body posture and the level of skill accounted for50%; landing 30%. According to the provisions the difficulty of movements are ranged into small, medium and large. The athletes option the actions. However, the ground must have a slope of about 37 and 60 cm above the soft snow layer.1.4 AssumeIn order to simplify the model and can come to a feasible solution, making the following assumptions: 1 the shape of a snowboard course has a lowest point, the wide and the length of the snowboard course. 2 air resistance can be negligible. 3 it is assumed that the athletes themselves have no influence.II. The Description of the ProblemThis problem is a typical engineering design, involving a lot of disciplines, such as advanced mathematics, engineering mathematics, mechanical dynamics and biological dynamics, as well as the relevant provisions of sports competition and judging standards, and so on. According to the requirement of the problem, determine the shape of a snowboard course to maximize the production of “vertical air” by a skilled snowboarder. For this problem, we can use the boundary conditions and site properties (e.g. symmetry) and other requirements to establish functional combining with the variation principle Euler equations. The original equation can be changed into a functional extremum problem. Secondly, we optimize the model boundary and determine the appropriate snowboard courses slope toe to make the athletes perform maximum twist or do more difficult action.Finally a practical model should meet the requirements of safety, sustainability and economic. According to the high degree of human security, the source of the snow and the topography the model will be optimized more reasonable.II. Models2.1 problem oneAs shown（3.1）A is the lowest point of the snowboard course.From A to B we want to find a curve to make the athletes get the maximum vertical distance above the edge of the snowboard course.Figure 1 Half-pipeSet A as origin of coordinate.Awing of conservation energy and neglecting air resistance, the mathematical function is, (1)Where v0 is the initial velocity （m/h）,v is the velocity towards destination （m/h）,m is the mass of an athlete (kg),w is the energy which is made by the athlete (J),h is the Vertical height (m),Af is the friction work （J），According tomechanical analysis： （2）Where is the angle the angle between the tangent and the horizontal line, N is the pressure on the object, r is the radius of curvature, According to friction formula： （3）To(3)intoequation(1), combined with calculus： （4）FrictionactingAf： （5）Use highermathematics： （6） （7）According to the nature of the curve, the speed can be assumed to satisfy this expression： （8）Put all these formulas in order and suppose the expression for the functional： （9）Set （10）ReferenceEulerequation： （11）Obtained： （12） （13） （14）To（12）-（14）intoequation（11），Obtained： （15）Integrate it： （16）Simplified： （17） （18）Suppose： So： Substituted intothe above equation： （19） （20）Integrate it： （21）Obtained the final results： （22）For the difficult equation, we obtain numerical solutions by numerical differentiation, and then obtained function equation by numerical fittingmethod：Discrete interval 0,8,Where take steps ：h = 1.Each point xi, i = 0,1, 8.Every interval xi, xi +1,the boundary conditions： y (0) = 0, y (0) = 0.Into theformula（22） fortheboundary conditions：C = 0Put into formula（22）： Numerical Solution ofeach point isobtainedin turn：x012345678y00.00690.00940.03360.13290.36690.78721.41274.0382Functionsimages andfunction equation are obtained bynumerical fittingon Excel：Figure 2 The results of the numerical solution of the fitting imageAfterfittingtheequation：y = 0.0003x6 - 0.0063x5 + 0.0435x4 - 0.1289x3 + 0.1594x2 - 0.0571x -0.0008 （23）Then the entire image can be got by symmetry along the y-axis. This models of problem one can be solved.2.2 Problem two On question 2, its main purpose is to improve the model in problem one under the condition of meeting the requirements of other possible cases. Analyze other possible requirements which include a number of aspects, such as the maximum twist in the air, players safety when they leave the ground and the stability of athletes when they land. Among them, we mainly consider the maximum twist of snowboarder in the air. When players leave the ground, they are only affected by gravity and air resistance. We ignore the players adjustment in the air. After the project flying out of the ground, in order to analyzing simply and thinking clearly, the velocity of the object is divided into lines velocity and angular velocity. Velocity contains components of three different directions: horizontal, vertical and longitudinal. Angular velocity consists of somersault angular velocity and twist angular velocity.Figure 3 Flipvelocity analysisAfter athletes flying out of the course, the velocity of longitudinal depends on a rational allocation of their own energy when they ski, so the design of course can not be considered. Vertical speed determines the maximum height with which athletes fly out of the course. So it is the requirement for design of the maximum “vertical air”. For the horizontal velocity, players reaction force when they leave the course should be taken into account. And the horizontal velocity generated by reaction force must satisfy theequation below. （24）Thus it can ensure that athletes fall back to ground safely after flying out, as the same time, it also meet appreciation, technical and safety requirements. On the problem that athletes reverse in the air, Conservation of energy can be used in the cross section. （25）WhereV is the velocity of each state,Wf is the effective bio-energy an athlete release,J is the moment of inertia under different rotations,Aaf isthe energydissipated byair resistance.By checking the literature, moment of inertia J1 is 1.1() and J2 is 84.3( ). Combining with the known data, we get the relationship of w1, w2,. According to the value of V, the relationship of and will be got. The boundary angle is .Fromtan = / =83.30So the boundary angle is 83.302.3Problem threeIn practice, there are many factors to consider, for example, the thickness of snow covering and the construction of the economy, in addition to shape. The topography should be made the best use of to save project cost. Climate also is a constraint. Snow can be smoother and be used longer when the weather is cold.III. ConclusionsThe basis of this model is snowboarding skilled players can generate the maximum vertical air. Awing of numsolve and fitted the mathematical function is,y = 0.0003x6 - 0.0063x5 + 0.0435x4 - 0.1289x3 + 0.1594x2 - 0.0571x -0.0008h=4(m) x0=8(m) =83.30 slope angle = 180 （International recommended values） Figure 4 Half-pipeThe ultimate resolution of model takes various factors into account. The model can be applied to other similar improvements similar problems, such as the design of emergency chute. IV. Future WorkAlthough this paper considered a wide variety, but only one purpose getting the best track shape. However, in the actual construction process the aims to be achieved are complex and the design aspects are various. If you want to continue the track design, the following areas to be discussed，1. The run-up routes height and inclination.2. Design of the best athletes running track. In the