沈阳航空航天大学国际数学建模论文18.pdf

2011201120112011 MathematicalMathematicalMathematicalMathematical ContestContestContestContest inininin ModelingModelingModelingModeling MCM MCM MCM MCM ControlControlControlControl SheetSheetSheetSheet Please review this page before subnmitting your solution to ensure that all of the information is correct Do not make changes by hand to the information on this control sheet If you need to change any of the information on this sheet login via the AdvisorAdvisorAdvisorAdvisor LoginLoginLoginLogin link on the MCM web site at make the changes online and print a new sheet You may NOT photocopy this control sheet to give to a new team nor may you assign any team a control number Each team must have its own control number obtained by registering via the MCM web site at Advisor Name Yanjie Li Department College of Science Institution Shenyang Aerospace University Address Shenyang Aerospace University Liaoning Shenyang 110136 ChinaBox 010 Daoyi South Street No 37 Daoyi Economical Developing 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ModelingModelingModelingModeling MCM MCM MCM MCM SummarySummarySummarySummary SheetSheetSheetSheet SummarySummarySummarySummary In recent years with snowboarding exercise rage among the young more and more snowboarding fans want to make more difficult movements this paper establish two models to design and optimize the ski trails to ensure that skiers can fly higher and have a longer time in the air 1 1 1 1 PracticalPracticalPracticalPractical momomomodeldeldeldel We assumed a set of ski movements slide rise soar slump of the athlete as a specific process if the player in field that designed by us has a maximum perpendicular height namely that the athlete in other fields do this particular process action the vertical height is lower than in the site designed by us then we can think that the site designed by us is relative ideally 1 We can provide the magnitude and direction of initial velocity directly and this have no effect to choose the site 2 Through the above assumptions we regard the field s parameters as constraints through the multiple objective programming to work out the maximum of vertical distance at this time the corresponding field parameters will be the optimal solution 2 2 2 2 TheoreticalTheoreticalTheoreticalTheoretical modelmodelmodelmodel Based on the practical model we analyzed the factors that influence the vertical distance more carefully such as the veneer skier s personal qualities proficiency and the loss of speed when pass in and out of the rabbet will have influence to the vertical distance Finally we designed out a more optimized practical venues through sensitivity analysis this site needs to be used as the special field of international snowboarding game Keywords Keywords Keywords Keywords Multi objective optimization vertical distance theory analysis SnowboardSnowboardSnowboardSnowboardCourseCourseCourseCoursetotototodesigndesigndesigndesign optimizationoptimizationoptimizationoptimization compromisecompromisecompromisecompromise TableTableTableTable ofof ofof ContentsContentsContentsContents 1 Introduction 2 1 1Problem Restatement 2 1 2 Problem Analysis 2 2 Model Establishment and Solution 3 2 1 Apply the method of physical mechanis 3 2 1 1 Variable hypothsis 3 2 1 2 Modeling 3 2 1 3 Model Solution 7 2 1 4 Conclusion 8 2 2Theoretical analysis 9 2 2 1The influence of the pose of body to the flying height and hang time when go into the groove 9 2 2 2 The influence of the posture in the air on the flight altitude and hang time 10 2 2 3The discuss among loss of velocity direction of blade and the speed after going into slot 11 2 2 4The restriction of the speed of slipping out of on the flying height 12 3 Sensitivity Analysis 13 4 Strengths and Weakness 14 4 1Strengths 14 4 2 Weakness 15 5 References 15 6 Appendix 16 1 1 1 1 IntroductionIntroductionIntroductionIntroduction 1 11 11 11 1 ProblemProblemProblemProblem RestatementRestatementRestatementRestatement Snowboarding is originated in the US in the 1960s though the time of the development of Snowboarding is short its widely admired by the young and it became popular in the world very fast Nowadays the ski field in developed country like Europe andAsia Snowboarding is tend to take the place ofdouble plate alpine skiing more than 90 percent of young people all stepped the veneer in white snow mountain freedomly romanticly and handsomely The giant slalom and veneer U shaped site veneer snow skills become the official event at the18th Olympic Winter Games in 1998 However in order to let more people both make more graceful movements and guarantee its safety Therefore we should design a ski trails shape to make the skier s vertical distance hang time is the longest So that they can have a plenty of time to make more actions Meanwhile some necessary factors should be taken in consideration whenwe design a practical site 1 21 21 21 2 ProblemProblemProblemProblemAnalysisAnalysisAnalysisAnalysis Afteraserious of discussion we analysis the problem from the veneer of the slope slide rink length width height and unilateral arc length of geometric factor and application of physical and mechanical and we need to consider the conditions and veneer skiers itself and some outside environmental factors We consider the above factors comprehensively to make the skiers do more actions in the air 2 2 2 2 ModelModelModelModel EstablishmentEstablishmentEstablishmentEstablishment andandandand SolutionSolutionSolutionSolution 2 12 12 12 1 ApplyApplyApplyApply thethethethe methodmethodmethodmethod ofofofof physicalphysicalphysicalphysical mechanismechanismechanismechanis 2 1 12 1 12 1 12 1 1VariableVariableVariableVariablehypothsishypothsishypothsishypothsis Slope Angle Initial velocity direction and site longitudinal Angle The half of the width of U shaped slot The distance of the initial position to enter slope segment The mass of Veneer skier Gravity acceleration The depth of U shaped slot Friction between the skis and snow The velocity The speed of Exposing to arc Vertical distance Friction factor The speed of arriving at the highest edge of arc 2 1 22 1 22 1 22 1 2 ModelingModelingModelingModeling As is shown in figure 1 research the part of OA figure 1 we can get that According to the law of conservation of energy obtain figure 2 because we get Then we discuss the condition move along the arc figure 3 figure 4 Accordingtothetheoremof kineticenergy obtain decomposed the work done by frictioninto the work done by upward force along the arcand the work done by friction along the direction of slide In this process we regard the athlete as particle therefore 1 2 For 2 seek a linear derivative of time then 3 Bringand 1 into the upper formula then Use the method of variation of constant we can be solution that Among them bring this formula into 4 get Take the initial conditions into ConsiderationWhen 0 and 0 then therefore Known by definition of work And because of the work done by frictionalong the slide direction So at this moment the work of friction is Athletes fly from the edge of arc to the highest point because the speed is lesser at this moment and the air resistancefrom this is lesser too At this moment the air resistance can be ignored after the athlete s suspension we can regard it as a stand cast movement with a beginning speed Asisshowninfigure 5 figure 5 Because during the course of ascent only gravity should be taken into consideration therefore when have a maximum height 2 1 32 1 32 1 32 1 3 ModelModelModelModel SolutionSolutionSolutionSolution According to the international snowboarding slide s parameters See table 1 table 1 Snowboarding slide parameters minmax Slope angle 17 5 18 5 Mass 50kg75kg Height floor to crown 18ft22ft Note Other parameters 0 03 0 22 3 25m 4 35m 10m s 16m s 45 90 5 4m 6 5m According to the above parameters use matlab to get the solution show as the followings 2 1 42 1 42 1 42 1 4ConclusionConclusionConclusionConclusion Through this model we can predict the maximum height 5 3447 m then theother corresponding parameters are respectively 0 3230rad 50 0000kg 16 0000m s 5 4000m 0 0936 5 5210m 3 2500m 1 5710rad Therefore the design of snowboarding slide show as the picture below figure 6 2 22 22 22 2TheoreticalTheoreticalTheoreticalTheoreticalanalysisanalysisanalysisanalysis 2 2 12 2 12 2 12 2 1TheTheTheThe influenceinfluenceinfluenceinfluence ofof ofof thethethethe poseposeposepose ofof ofof bodybodybodybody totototo thethethethe flyingflyingflyingflying heightheightheightheight andandandandhanghanghanghang timetimetimetime whenwhenwhenwhengogogogo intointointointo thethethethe groovegroovegroovegroove The braking phenomenon will have different influences to the increase of speed when go into the groove It Caused the loss of energy and influence the height of next action In order to reduce the lossed of speed when landing the hip knee ankle of leg should have an appropriate buffer increase the action time of cell wall and plate absorb the counterforce when landing on the ground F is ground counterforce H is the force distance FH is the work done by counterforce Reduce the lossed of speed making the potential energy of body fully into the kinetic energy When veneer skiers slide into the slot at the same time body should lean properly towards slot base the center of gravityshoulelayin the first half of plate See figure 7 We make full use of gravity and heavy torque role to make human body accelerated to obtain the best kinetic energy so it can make athletes are flying at an altitude of ascension and rise the hangtimer figure 7 Into the slot action 2 2 2 2 2 2 2 2 2 2 2 2 TheTheTheThe influenceinfluenceinfluenceinfluence ofof ofof thethethethe postureposturepostureposture inininin thethethethe airairairair onononon thethethethe flightflightflightflight altitudealtitudealtitudealtitude andandandandhanghanghanghang timetimetimetime Athletes will fly up along the parabolic after flighting If want to increasehangtime athletes should shrink thedirectiontowards chest direction as far as possible and prevent body or arms from shaking and swaying Because the body s shaking rocking will break the balance of human body system making self control of athletesweaken reducing the movement quality in the air Athletes in flight had better keep the body s natural and relaxed and reduce redundant actions to ensure the smooth of movement in the air In addition the action of shrinking thigh toward chest is favorable to complete catching plate movements Angular momentum is conservative I1W1 I2W2 In the condition of momentum conservation the action of shrinking thigh towards chest can drivethe upper part of the bodydescendit can make for completing catching skids and increase hanq time under the condition that the course of core is stable 2 2 2 2 2 2 2 2 3The3The3The3The discussdiscussdiscussdiscuss amongamongamongamong losslosslossloss ofof ofof velocity directionvelocity directionvelocity directionvelocity direction ofof ofof bladebladebladeblade andandandand thethethethespeedspeedspeedspeedafterafterafteraftergoinggoinggoinggoing intointointointo slotslotslotslot Athletes control skis taxiing agains paries after going into slote because the blade all have the role of brake in order to reduce the loss of speed athletes should adjust the speed direction of core to be equal to skids verticalvertical axisaxis direction as far as possible it can decrease the braking efficiency decending from touching with snow coasting down the pariesandtake full advantage of interior wall altitude intercept to gain greater acceleration In this way it can make speed becomes larger therefore athletes also can fly higher and produce more hangtime See figure 8 figure 8 A slot speed when losing with blade direction Angle relation graph 2 2 2 2 2 4The2 4The2 4The2 4The restrictionrestrictionrestrictionrestriction ofofofof thethethethe speedspeedspeedspeed ofofofof slippingslippingslippingslipping outoutoutout ofofofof onononon thethethethe flyingflyingflyingflying heightheightheightheight Snowboardingexercisesetsintaxiing leapskill soithasboth competitivenesscompetitiveness and entertainment In the constitutive factors it contains many factors such as flying height difficulty diversity the quality of finishing movements utilization of area and landing situation and so on Because the height will have a restriction on difficulty diversity and the quality of finishing movement flying height is the core in many factors and it is the foundation in air erformancehe No height is no longer flight time no flight time safeguard was unable to complete difficult moves fail to get good grades and the main factors that restrict the flying height is a slot speed 1 1 1 1 TheTheTheThe theoreticaltheoreticaltheoreticaltheoretical analysisanalysisanalysisanalysis thatthatthatthat firstfirstfirstfirst helpinghelpinghelpinghelping skiigskiigskiigskiig enteringenteringenteringentering intointointointo slotslotslotslot hashashashas a a a a influenceinfluenceinfluenceinfluence onononon rollingrollingrollingrolling soltsoltsoltsolt Athletes should chose helping skiing to enter slot athletes should be proactive to obtain controlled speed If the athlete slipe a certain distance around the border he wii obtain a certain speed and if the athlete caprioles a certain height he wii obtain a certain initial energy reservation E0 E1 E2 E0 represent initial energy of athlete E1 represent initial energy of motion of athlete E2 represent initial potential energy of athlete With the completion of the slot action the E2 that athlete commplete fewer and fewer Under the circumstances that gravity do it swork E1 of the athlete wii increase accordingly namely athlete wii gain a certain vertical speed Athletes had certain energy of motion Reservation after gaining a certain vertical speed It gains transformation to potential energy after rolling slot so it can achieve the purpose of increasing flying height The achievement of flying height also make a potential energy preparation for entering into slot next time also offer space time guard for finishing next motion smoothly but under the course of transformation between the energy of motion and potential energy the speed that wii be achieved must be controlled if the speed is uncontrollable it wii affect their technical movement of play even cause accidental seriously so we should advocate the active and controlled speed 2 2 2 2 thethethethe pointpointpointpoint ofofofof thethethethe boardboardboardboard andandandandlongitudinallongitudinallongitudinallongitudinal angleangleangleangle ofofofof boradboradboradborad havehavehavehave anananan effecteffecteffecteffect onononon speedspeedspeedspeed ofofofof rollingrollingrollingrolling speedspeedspeedspeed The stress point and longitudinal angle of borad also affect the speed of rollingoutof slot and height At the moment when athlete rolling out of the slot the stress point should leave the center of borad due to the front edge losing it s support Gradually to board hind transition make plate stress point always in Taiwan in front wall slide out foot pedal force should be gradual After obtaining enough reaction force at the same time keep track of smooth increase parabola out slot speed and keep the reasonable out groove Angle In the purpose of effectively increase height at the same time athletes still can obtain appropriate swivel speed and the angle of out of groove When achievingfloating action at the same timereduce the effect of the slot resistance achieve reasonable into the slot for the next action make preparation for reserving energy 3 3 3 3 SensitivitySensitivitySensitivitySensitivity AnalysisAnalysisAnalysisAnalysis 1 1 1 1 TheTheTheThe influenceinfluenceinfluenceinfluence ofof ofof airairairair resistanceresistanceresistanceresistance totototo thethethethe modelmodelmodelmodel The objects moved in the air will be influenced by the resistance of the air if the speed in the air lower than 2 5 M we can think that the