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1、Half pipe-snow boardSummaryThis essay mainly discusses two problems. One is to determine the shape of a snowboard course to maximize the production of “vertical air”; the other is to modify the shape of the course to allow for the performance of certain snowboarding movements and decide what tradeof
2、fs may be required to develop a “practical”course.First, the following assumptions are made: no wind in the course; the snowboarder being a skilled one; the transitional zone being arc-shaped, and the entry starting from the leading edge of the course at a fixed initial velocity. Second, to deal wit
3、h the first problem, the course is taken as an arc. The optimization mathematical model of the course is build with vertical air as the target function. Given the multiple variables involved, certain variables are artificially fixed to study the influence of some variable has on the vertical air. Th
4、e major conclusion is: width of the flat bottom is in inverse proportion to the vertical air. In theory, to get the largest vertical air, the width should be minimized. When the width reduced to zero, the maximum vertical air is 7.152080m. When conducting sensitivity analysis, we find the minor chan
5、ge in width of the flat bottom has little impact on the final results. However, in reality, the flat bottom is necessary as it provides the snowboarder a chance to adjust his movement before entering the arc transitional zone. The second problem focuses on the maximization of the twist in the air. T
6、he initial velocity is disintegrated into two directions to ensure the remaining potential energy conversed from the off-slot kinetic energy must reach its maximum amount. An optimization model with maximum twist and vertical air as the target function is similarly set. The conclusion is that when a
7、 skilled snowboarder enters the course at an initial velocity of 11.96m/s, the ideal maximum vertical air he achieves is 5.367006m, and under this condition, the maximum twist in the air is also achieved.The strengths of this essay lie in its exploration in building optimization model based on the e
8、nergy conservation process during snowboarding and controlling certain variable to study other variables in solving multiple variable problems. The weakness is that our sensitivity analysis is only conducted on the impact of width of the flat bottom on the final results while neglecting the influenc
9、e of other variables. In addition, we assume the transitional zone as an arc-shaped one, which makes the analysis of other shaped zone such as over and double-curve impossible.Keywords: half pipe-snow board twist optimisticIntroductionSnowboarding is a popular winter sport discipline in Europe and A
10、merica. It combines skiing, twisting and other skills together and the competition has two events: half-pipe and super G. The paper discusses the design of u-shaped pool to ensure the success of a skilled snowboarder. A u-shaped half-pipe is consisted of verticals, Arc transition zone, and bottom pl
11、atform.The issue will be discussed in the following two parts.According to the contest rules, the snowboarders routine comprises a series of separate movements. The diversity of these actions fully demonstrates a players proficiency and has a great impact on his final scores. Among these, vertical a
12、ir is a key measurement.“Vertical air” here refers to the maximum vertical distance above the edge of the halfpipe. The larger the vertical air, the longer the air time; the longer the air time, the more time a snowboarder has to finish the aerial acrobatic maneuvers. Thus, it is believed that verti
13、cal air is critical for any complex and difficult maneuver and is the specific requirement of the u-shaped snowboarding event. Then, how to produce the maximum vertical air in a u-shaped course? In this essay, all the elements affecting this distance is taken into account to solve this problem, incl
14、uding the sloped, wall, leveling degree, snow, takeoff zone of the course and the snowboarder himself.In an ideal condition, a skilled snowboard is capable of minimize his errors and complete the whole routine in a smooth way and therefore is desirable for the model setting. To simplify the matter,
15、the course, in our design, is consisted of two symmetrical circular arc cambers which are connected by a flat bottom platform. Our analysis will focus on the structure of this u-shaped course and the snowboarders mechanical process. When designing the structure, the radius, central angel of the arc,
16、 width of the bottom platform, the height of the wall are all important factors need to consider. As for the mechanical analysis, law of energy conservation is applied to study the arc friction. To maximize the vertical air, a snowboarder must manage to get as great geopotential as possible while pe
17、rforming a vertical air maneuver. Based on correlations among these factors, a function can be established and by adjusting each element, the largest vertical air will be achieved and by collecting these data, the shape of the u-shaped course can be finally determined. Then, the shape will be modifi
18、ed to enable the player to perform the maximum twist in the air. While skiing out of the U-shaped course, the snowboarder will conduct vertical air maneuver and twist following a parabolic path. In this process, the bigger the snowboarders vertical displacement, the farther his horizontal displaceme
19、nt, the larger the distance he finishes, the more air time he gets and the greater possibility he wins to perform difficult and complex actions. The players initial velocity can be disintegrated into two directions: vertical and horizontal. The former enables him to perform vertical air and the latt
20、er, twist. Assumptions1、Since the air resistance is relatively small in snowboarding, it is neglected in our model-construction.2、In this model, the snowboarder and his snowboard is taken as a whole system, and this system, a mass point.3、During snowboarding, the friction coefficient between snow an
21、d snowboard on the whole course is assumed as a constant value.4、The curve segment of u-shaped course is assumed as a circular arc and the bottom horizontal line, a line segment.Definitions and notations vertical air friction skiing downward along the arc work generated by the above movement Frictio
22、n skiing upward along the transition zone Work generated by the above movement Friction on the flat bottom Work resulted from the above friction Friction coefficient between the snow and snowboard Central angel of the arc transition zone An intersection angle between the radius of the transition zon
23、e and the flat bottom and Width between the two edges of the u-shaped course Radius of the transition zone Total quality of the snowboarder and snowboard Width of the flat bottom Initial energy when entering the course Initial velocity when entering the courseModelsBased on the above analysis, we di
24、vide the whole snowboarding process into three parts: snowboarding downward, horizontally, and upward. The model is designed according to this division.1 Course design to achieve maximum vertical air1.1 Construction of the energy-relation model（1）、Energy calculation before entering the courseIt is k
25、nown that an experienced snowboard will manage to get a controllable speed on starting his snowboarding by choosing the suitable though to ski for a while and jumping to certain height before entering the trough. In this way, he can get certain amount of initial energy reserves. In the conversion be
26、tween the snowboarders kinetic energy and potential energy, the velocity resulted from this must be controllable. If not so, the snowboarders performance will be compromised and in the worst case, an accident will occur. Therefore, a controllable initial velocity is strongly recommended.Here, the av
27、erage initial velocity is 11.96m/s. Based on the theorem of kinetic energy, the initial energy E is got based on the theorem of kinetic energy.（2）、Calculation of friction workWhile skiing downward along the arc transition zone, the snowboarder suffers a friction f. Based on the analysis; two equatio
28、ns of tangential and normal direction are listed as the following (with )figure1: the stress of Object in a circular orbit After solving the above equation, the following friction is got with initial value t=0 and when the object is on the top of the arc.Work done by is Friction while snowboarding u
29、pward along the arc isWork done by isWork done by friction in the horizontal snowboarding is（3）、Energy calculation after leaving the thoughThe moment the snowboarder entering the though, the potential energy begin to decrease. With gravity working normally, the kinetic energy increases accordingly,
30、that is, the snowboarder is gaining certain vertical velocity through energy conversion. After getting certain vertical velocity, the snowboarder has certain amount of kinetic energy reserves. By transforming these reserves into potential energy, the vertical air can be greatly increased when the sn
31、owboarder enters the though, and this further serves as the source of potential energy of next similar maneuver, which, in turn, makes next movement possible. Once entering the flat bottom platform, the friction and velocity here will fall and thus enable the snowboarder to adjust his movement to pr
32、epare for the following upward skiing into the arc transition zone.While skiing upward, the snowboarders center of gravity gradually moves to the back of the snowboard. At the moment of vertical air maneuver, with the leading edge of the board losing support, the force bear point completely falls on
33、 the boards ending edge, which is heavily presses against the wall of the course. By doing so, the snowboarder gains sufficient counter-force, maintains a clean parabolic path, enhances the off-slot speed and attains a reasonable off-slot angle. While effectively enlarging the vertical air, the snow
34、boarder also get s proper twist velocity and in-slot angle. Moreover, during the vertical air movement, the snowboarders horizontal velocity is reduced, which further reduces the in-slot resistance, and prepares sufficient energy reserves for the next maneuver. Though the direction of the off-slot r
35、emains unknown, qualitative analysis shows that only in the vertical air maneuver can the snowboarder reach the maximum height in the air. According to the energy conservation theorem, That is, 1.2 Establishment of the structure-correlation modelThe following figure is made in accordance with the re
36、quirements of the contest. figure 2:the schematic drawing of arc structureThe equation is listed below after careful calculation Since the height of the u-shaped course is fixed,2. U-shaped course designing to achieve maximum twistThe vertical air got in the former model is larger than the real dist
37、ance with the effect of twist and rotation not taking into consideration. In this model, the two elements will be considered. Since the off-slot movement follows a parabolic path, the off-slot velocity will be disintegrated into vertical and horizontal directions. Among which, the vertical velocity
38、enables the snowboarder to movement vertically and the horizontal one, skillfully. Both the twist and rotation derive their energy from the horizontal velocity as showed below: figure 3: Post-flight speed analysis is the total energy used in twist and rotation; is the off-slot velocity, energy is 3.
39、 Model solution3.1. Model solution with the maximum vertical airWith regard to the correlation between energy and structure, the variation of s and l will affect R, and. With h reaches its maximum, such effect will also become the largest. （1）、Energy solution before entering the courseNormally, mg=7
40、00N. When g is approximately equal to 10kg/m3 , m=70kg. In common situation, the initial velocity is between 10m/s-15m/s, is between 3500J-7875J（2）、Friction work solutionThe friction coefficient varies with the changing temperature and pressure, but the variation is within 0.9-1.5. Neglecting the in
41、fluence of these two factors, friction coefficientTherefore:（3）、Off-slot energy solutionThe following equation is got based on the above conclusion and equation （4）、Determination of the structural parametersis between 13-18,，和， With being fixed and s changing, the results are listed as the following
42、: table 1: the parameters change in the determined 1600.1204606E+100.7920838E-080.99442107.15208030.5466712E+090.1522681E-070.89605526.85208060.4465042E+090.1615634E-070.76579296.55208090.4234920E+090.1480116E-07 0.59242746.25208012no feasible answerWith being fixed and s changing, the results are l
43、isted as the following: table 2: the parameters change in the determined 5130.4583634E+090.1431285E-070.65569576.652080140.3914006E+090.1756965E-070.71335816.652080150.4465042E+090.1615634E-070.76579296.652080160.8555937E+090.8846507E-080.81342836.652080170.5316193E+090.1493508E-070.85670586.6520801
44、80.5020764E+090.1657927E-070.89605536.652080Conclusion: As shown in Table 1, when remains fixed and s keeps increasing, the ideal vertical air h will diminish, which means to get the maximum vertical air, width of flat bottom should be reduced to its lowest level and even to be zero with other facto
45、rs neglected completely.As shown in Program1 and Table1, with width of flat bottom s increasing, arc radium R, central angle will change accordingly. With and width of flat bottom rising, the slope angle will decrease. It means the snowboarder consumes more energy on a relatively even slope.Model so
46、lution with the maximum rotationCorrelations among the different variables: As shown in Program2 and Table 2, in an ideal situation, the vertical air is only influenced by variation of width of flat bottom and has little to do with the change of. But variation of will affect the result of optimal so
47、lution. When rises,will rise subsequently. It is worth noting that change of h in line 6 is largely attributed to the change of width of flat bottom in the first line. Parts of the program debugging results are shown in Table 3: table 3: The optimal solution in the different 0130.5133642E+090.1621473E-070.89605544.3665800140.5793600E+090.1505125E-070.93188234.6142350150.57936