2011MCM美賽A題論文snowboard course.pdf

Team 8038 February 23 2010 SummarySummarySummarySummary Baseball is a popular bat and ball game involving both athletics and wisdom There are strict restrictions on the material size and manufacture of the bat It is vital important to transfer the maximum energy to the ball in order to give it the fastest battedspeedduringthehittingprocess Firstly thispaperlocatesthe center of percussion COP and the viberational node based on the single pendulum theory and the analysis of bat vibration With the help of the synthesizing optimization approach a mathematical model is developed to execute the optimized positioning for the sweet spot and the best hitting spot turns out not to be at the end of the bat Secondly based on the basic model hypothesis taking the physical and material attributes of the bat as parameters the moment of inertia and the highest batted ball speed BBS of the sweet spot are evaluated using different parameter values which enables a quantified comparison to be made on the performance of different bats Thus finally explained why Major League Baseball prohibits corking and metal bats In problem I taking the COP and the viberational node as two decisive factors of the sweet zone models are developed respectively to study the hitting effect from the angle of energy conversion Because the different sweet spots decided by COP and theviberationalnode reflectdifferentformof energy conversion the space distance concept is introduced and the Technique for Order Preferenceby Similarity to Ideal Solution TOPSIS is used to locate the sweet zone step by step And thus it is proved that the sweet spot is not at the end of the bat from the two angles of specific quantitative relationship of the hitting effects and the inference of energy conversion In problem II applying new physical parameters of a corked bat into the model developed in Problem I the moment of inertia and the BBS of the corked bat and the original wood bat under the same conditions are calculated The result shows that the corking bat reduces the BBS and the collision performance rather than enhancing the sweet spot effect On the other hand the corking bat reduces the moment of inertia of the bat which makes the bat can be controlled easier By comparing the two conflicting impacts comprehensively the conclusion is drawn that the corked bat will be advantageous to the same player in the game for which Major League Baseball prohibits corking In problem III adopting the similar method used in Problem II that is applying different physical parameters into the model developed in Problem I calculate the moment of inertia and the BBS of the bats constructed by different material to analyze the impact of the bat material on the hitting effect The data simulation of metal bats performance and wood bats performance shows that the performance of the metal bat is improved for the moment of inertia is reduced and the BBS is increased Our model and method successfully explain why Major League Baseball for the sake of fair competition prohibits metal bats In the end an evaluation of the model developed in this paper is given listing its advantagessandlimitations andprovidingsuggestionsonmeasuringthe performance of a bat KeyKeyKeyKey words words words words sweetspot moment of inertia Center of Percussion Bat Ball Coefficient of Restitution Batted Ball Speed ContentsContentsContentsContents Summary 1 Contents 3 1 Restatement of the Problem 4 2 Analysis of the Problem 4 2 1Analysis of Problem I 4 2 2Analysis of Problem II 5 2 3Analysis of Problem III 5 3 ModelAssumptions and Symbols 6 3 1 Model Assumptions 6 3 2 Symbols 6 4 Modeling and Solution 6 4 1 Modeling and Solution to Problem I 6 4 1 1 Model Preparation 6 4 1 2 Solutions to the two sweet spot regions 8 4 1 3 Optimization Model Based on TOPSIS Method 11 4 1 4 Verifying the sweet spot is not at the end of the bat 12 4 2 Modeling and Solution to Problem II 13 4 2 1 Model Preparation 13 4 2 2 Controlling variable method analysis 14 4 2 3 Analysis of corked bat and wood bat 5 6 15 4 2 4Reason for prohibiting corking 4 16 4 3 Modeling and Solution to Problem III 17 4 3 1Analysis of metal bat and wood bat 8 9 17 4 3 2 Reason for prohibiting the metal bat 4 18 5 Strengths and Weaknesses of the Model 19 5 1 Strengths 19 5 2 Weaknesses 19 6 References 20 1 1 1 1 RestatementRestatementRestatementRestatement ofofofof thethethethe ProblemProblemProblemProblem Explain the sweet spot on a baseball bat Every hitter knows that there is a spot on the fat part of a baseball bat where maximum power is transferred to the ball when hit Why isn t this spot at the end of the bat A simple explanation based on torque might seem to identify the end of the bat as the sweet spot but this is known to be empirically incorrect Develop a model that helps explain this empirical finding Some players believe that corking a bat hollowing out a cylinder in the head of the bat and filling it with cork or rubber then replacing a wood cap enhances the sweet spot effect Augment your model to confirm or deny this effect Does this explain why Major League Baseball prohibits corking Does the material out of which the bat is constructed matter That is does this model predict different behavior for wood usually ash or metal usually aluminum bats Is this why Major League Baseball prohibits metal bats 2 Analysis2 Analysis2 Analysis2 Analysis ofofofof thethethethe ProblemProblemProblemProblem 2 12 12 12 1 AnalysisAnalysisAnalysisAnalysis ofofofof ProblemProblemProblemProblem I I I I First explain the sweet spot on a baseball bat and then develop a model that helps explain why this spot isn t at the end of the bat 1 There are a multitude of definitions of the sweet spot 1 the location which produces least vibrational sensation sting in the batter s hands 2 the location which produces maximum batted ball speed 3 the location where maximum energy is transferred to the ball 4 the location where coefficient of restitution is maximum 5 the center of percussion For most bats all of these sweet spots are at different locations on the bat so one is often forced to define the sweet spot as a region If explained based on torque this sweet spot might be at the end of the bat which is known to be empirically incorrect This paper is going to explain this empirical paradox by exploring the location of the sweet spot from a reasonable angle Based on necessary analysis it can be known that the sweet zone which is decided by the center of percussion COP and the vibrational node produces the hitting effect abiding by the law of energy conversion The two different sweet spots respectively decided by the COP and the viberational node reflect different energy conversions which forms a two factor influence This situation can be discussed from the angle of space distance concept and the Technique for Order Preference by Similarity to Ideal Solution TOPSIS could be used 2 The process is as follows first let the sweet spots decided by the COP and the viberational node be optional sweet spots second define the regions that these optional sweet spots may appear as the sweet zones and the length of each sweet zone as distance then the sweet spot could be located by sequencing the sweet zones of the two kinds on the bat Finally compare the maximum hitting effect of this sweet spot with that of the end of the bat 2 22 22 22 2AnalysisAnalysisAnalysisAnalysis ofofofof ProblemProblemProblemProblem II II II II Problem II is to explain whether corking a bat enhances the sweet spot effect and why Major League Baseball prohibits corking 4 In order to find out what changes will occur after corking the bat the changes of the bat s parameters should be analyzed first 1 The mass of the corked bat reduces slightly than before 2 Less mass lower moment of inertia means faster swing speed 3 The mass center of the bat moves towards the handle 4 The coefficient of restitution of the bat becomes smaller than before 5 Less mass means a less effective collision 6 The moment of inertia becomes smaller 5 6 By analyzing the changes of the above parameters of a corked bat whether the hitting effect of the sweet spot has been changed could be identified and then the reason for prohibiting corking might be clear 2 32 32 32 3AnalysisAnalysisAnalysisAnalysis ofofofof ProblemProblemProblemProblem IIIIIIIIIIII First explain whether the bat material imposes impacts on the hitting effect then develop a model to predict different behavior for wood or metal bats to find out the reason why Major League Baseball prohibits metal bats 1 4 The mass M and the center of mass CM of the bat are different because of the material out of which the bat is constructed The changes of the location of COP and moment of inertia bat I could be inferred 2 3 Above physical attributes influence not only the swing speed of the player the less the moment of inertia bat Iis the faster the swing speed is but also the sweet spot effect of the ball which can be reflected by the maximum batted ball speed BBS The BBS of different material can be got by analyzing the material parameters that affect the moment of inertia Then it can be proved that the hitting effects of different bat material are different 3 3 3 3 ModelModelModelModelAssumptionsAssumptionsAssumptionsAssumptions andandandand SymbolsSymbolsSymbolsSymbols 3 13 13 13 1 ModelModelModelModelAssumptionsAssumptionsAssumptionsAssumptions 1 The collision discussed in this paper refers to the vertical collision on the sweet spot 2 The process discussed refers to the whole continuous momentary process starting from the moment the bat contacts the ball until the moment the ball departs from the bat 3 Both the bat and the ball discussed are under common conditions 3 23 23 23 2 SymbolsSymbolsSymbolsSymbols Table 3 1 SymbolsInstructions ka kinematic factor 0 I the rotational inertia of the object about its pivot point Mthe mass of the physical pendulum dthe location of the center of mass relative to the pivot point Lthe distance between the undetermined COP and the pivot g the gravitational field strength bat I the moment of inertia of the bat as measured about the pivot point on the handle Tthe swing period of the bat on its axis round the pivot Sthe length of the bat z the distance from the pivot point where the ball hits the bat f vibration frequency ball m the mass of the ball 4 4 4 4 ModelingModelingModelingModeling andandandand SolutionSolutionSolutionSolution 4 14 14 14 1 ModelingModelingModelingModeling andandandand SolutionSolutionSolutionSolution totototo ProblemProblemProblemProblem I I I I 4 4 4 4 1 1 1 1 1 1 1 1 ModelModelModelModel PreparationPreparationPreparationPreparation 1 Analysis of the pushing force or pressure exerted on hands 1 Fig 4 1 As showed in Fig 4 1 If an impact forceFwere to strike the bat at the center of mass CM then pointPwould experience a translational acceleration the entire bat would attempt to accelerate to the left in the same direction as the applied force without rotating about the pivot point If a player was holding the bat in his her hands this would result in an impulsive force felt in the hands If the impact forceFstrikes the bat below the center of mass but above the center of percussion pointPwould experience both a translational acceleration in the direction of the force and a rotational acceleration in the opposite direction as the bat attempts to rotate about its center of mass The translational acceleration to the left would be greater than the rotational acceleration to the right and a player would still feel an impulsive force in the hands If the impact force strikes the bat below the center of percussion then pointPwould still experience oppositely directed translational and rotational accelerations but now the rotational acceleration would be greater If the impact force strikes the bat precisely at the center of percussion then the translational acceleration and the rotational acceleration in the opposite direction exactly cancel each other The bat would rotate about the pivot point but there would be no net force felt by a player holding the bat in his her hands Define pointOas the center of percussion COP 1 Locating the COP According to physical knowledge it can be determined by the following method Instead of being distributed throughout the entire object let the mass of the physical pendulumMbe concentrated at a single point located at a distance Lfrom the pivot point This point mass swinging from the end of a string is now a simple pendulum and its period would be the same as that of the original physical pendulum if the distanceLwas Md I L bat 4 1 This locationLis known as the center of oscillation Asolid object which oscillates about a fixed pivot point is called a physical pendulum When displaced from its equilibrium position the force of gravity will attempt to return the object to its equilibrium position while its inertia will cause it to overshoot As a result of this interplay between restoring force and inertia the object will swing back and forth repeating its cyclic motion in a constant amount of time This time called the period depends on the mass of the objectM the location of the center of mass relative to the pivot pointd the rotational inertia of the object about its pivot point 0 Iand the gravitational field strengthgaccording to Mgd I T 0 2 4 2 2 Analysis of the vibration 1 Fig 4 2 As showed in Fig 4 2 mechanical vibration occurs when the bat hits the ball Hands feel comfortable only when the holding position lies in the balance point The batting point is the vibration source Define the position of the vibration source as the vibrational node Now this vibrational node is one of the optional sweet spots 4 1 24 1 24 1 24 1 2 SolutionsSolutionsSolutionsSolutions totototo thethethethe twotwotwotwo sweetsweetsweetsweet spotspotspotspot regionsregionsregionsregions 1 Locating the COP 1 4 Determining the parameters a mass of the batM b length of the batS the distance between Block 1 and Block 5 in Fig 4 3 c distance between the pivot and the center of massd the distance between Block 2 and Block 3 in Fig 4 3 d swing period of the bat on its axis round the pivot T take an adult male as an example the distance between the pivot and the knob of the bat is 16 8cm the distance between Block 1 and Block 2 in Fig 4 3 e distance between the undetermined COP and the pivotL the distance between Block 2 and Block 4 in Fig 4 3 that is the turning radius Fig 4 3 Table 4 1 Block 1knob Block 2pivot Block 3the center of mass CM Block 4the center of percussion COP Block 5the end of the bat Calculation method of COP 1 4 distance between the undetermined COP and the pivot 2 2 4 gT L gis the gravity acceleration 4 3 moment of inertia 2 2 0 4 MgLT I Lis the turning radius Mis the mass 4 4 Results The reaction force on the pivot is less than 10 of the bat and ball collision force When the ball falls on any point in the sweet spot region the area where the collision force reduction is less than 10 is 1 1 9 0 LLcm which is called Sweet Zone 1 2 Determining the vibrational node The contact between bat and ball we consider it a process of wave ransmission When the bat excited by a baseball of rapid flight all of these modes as well as some additional higher frequency modes are excited and the bat vibrates We depend on the frequency modes list the following two modes Thefundamentalbendingmodehastwonodes orpositionsofzero displacement One is about 6 1 2 inches from the barrel end close to the sweet spot of the bat The other at about 24 inches from the barrel end 6 inches from the handle at approximately the location of a right handed hitter s right hand Fig 4 4 Fundamental bending mode 1 215 Hz The second bending mode has three nodes about 4 5 inches from the barrel end a second near the middle of the bat and the third at about the location of a right handed hitter s left hand Fig4 5 Second bending mode 2 670 Hz The figures show the two bending modes of a freely supported baseball bat The handle end of the bat is at the right and the barrel end is at the left The numbers on the axis represent inches this data is for a 30 inch Little League wood baseball bat These figures were obtained from a modal analysis experiment In this opinion we prefer to follow the convention used by Rod Cross 2 who defines the sweet zone as the region located between the nodes of the first and second modes of vibration between about 4 7 inches from the barrel end of a 30 inch Little League bat Fig 4 6 The figure of Sweet Zone 2 The solving time in accordance with the searching times and backtrack times It is objective to consider the two indices together 4 1 34 1 34 1 34 1 3 OptimizationOptimizationOptimizationOptimization ModelModelModelModel BasedBasedBasedBased onononon TOPSISTOPSISTOPSISTOPSIS MethodMethodMethodMethod Table4 2 swing period T bat mass M bat length S CM position d coefficient of restitution BBCOR initial veloci ty in v swing speed bat v ball mass ball m wood bat ash 0 12s 876 0 15g 86 4 cm 41 62cm0 4892 27 7m s 15 3 m s 850 5g Adopting the parameters in the above table and based on the quantitative regions in sweet zone 1 and 2 in 4 1 2 the following can be drawn 2 Sweet zone 1 is 1 1 9 0 LL 8358 57 50 cmcm Sweet zone 2 is 2 1 LL 23 55 41 48 cmcm As shown in Fig 4 3 define the position of Block 2 which is the pivot as the origin of the number axis andxas a random point on the number axis 1 Optimization modeling 2 The TOPSIS method is a technique for order preference by similarity to ideal solution whose basic idea is to transform the integrated optimal region problem into seeking the difference among evaluation objects distance That is to determine the most ideal position and the acceptable most unsatisfactory position according to certain principals and then calculate the distance between each evaluation object and the most ideal position and the distance between each evaluation object and the acceptabl