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1、MATH MODELING FOR SWEET SPOTAbstractBaseball 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 batted spe
2、ed during the hitting process. Firstly, this paper locates the 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 optimiz
3、ed 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 s
4、pot” 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 facto
5、rs 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 the viberational node reflect different form of energy conversion, the “space-distance” concept is introduced and the “Tec
6、hnique 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
7、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 coll
8、ision 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 ad
9、vantageous 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 bat
10、s 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 mode
11、l 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 advantages s and limitations, and providing suggestions on measuring the performance of a bat.Key w
12、ords: sweet spot, moment-of-inertia, Center-of-Percussion, Bat-Ball Coefficient-of-Restitution, Batted-Ball Speed ContentsSummary1Contents31.Restatement of the Problem42.Analysis of the Problem42.1 Analysis of Problem I42.2 Analysis of Problem II52.3 Analysis of Problem III53.Model Assumptions and S
13、ymbols63.1 Model Assumptions63.2 Symbols64.Modeling and Solution64.1 Modeling and Solution to Problem I64.1.1 Model Preparation64.1.2 Solutions to the two “sweet spot” regions84.1.3 Optimization Model Based on TOPSIS Method114.1.4 Verifying the “sweet spot” is not at the end of the bat124.2 Modeling
14、 and Solution to Problem II134.2.1 Model Preparation134.2.2 Controlling variable method analysis144.2.3 Analysis of corked bat and wood bat 56154.2.4 Reason for prohibiting corking4164.3 Modeling and Solution to Problem III174.3.1 Analysis of metal bat and wood bat 89174.3.2 Reason for prohibiting t
15、he metal bat 4185.Strengths and Weaknesses of the Model195.1.Strengths195.2 Weaknesses196.References201.Restatement of the ProblemExplain 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
16、hit. Why isnt 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 (hollow
17、ing 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 con
18、structed 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.Analysis of the Problem2.1 Analysis of Problem IFirst explain the “sweet spot” on a baseball bat, and then develop a
19、 model that helps explain why this spot isnt at the end of the bat.1There are a multitude of definitions of the sweet spot:1) the location which produces least vibrational sensation (sting) in the batters hands2) the location which produces maximum batted ball speed3) the location where maximum ener
20、gy is transferred to the ball4) the location where coefficient of restitution is maximum5) the center of percussionFor 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”
21、 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-p
22、ercussion (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 discu
23、ssed 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
24、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
25、bat.2.2 Analysis of Problem IIProblem II is to explain whether “corking” a bat enhances the “sweet spot” effect and why Major League Baseball prohibits “corking”.4In order to find out what changes will occur after corking the bat, the changes of the bats parameters should be analyzed first:1) The ma
26、ss 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 momen
27、t of inertia becomes smaller.56By 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.2.3 Analysis of Problem IIIFirst, explain whether the bat ma
28、terial 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?14The mass (M) and the center of mass (CM) of the bat are different because of the material out of which the
29、bat is constructed. The changes of the location of COP and moment of inertia () could be inferred.23Above physical attributes influence not only the swing speed of the player (the less the moment of inertia- is, the faster the swing speed is) but also the sweet spot effect of the ball which can be r
30、eflected 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.Model Assumptions and Symbols3.1 Model Assumption
31、s1) 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 disc
32、ussed are under common conditions.3.2 SymbolsTable 3-1SymbolsInstructionsa kinematic factorthe rotational inertia of the object about its pivot pointthe mass of the physical pendulumthe location of the center-of-mass relative to the pivot pointthe distance between the undetermined COP and the pivott
33、he gravitational field strengththe moment-of-inertia of the bat as measured about the pivot point on the handlethe swing period of the bat on its axis round the pivotthe length of the batthe distance from the pivot point where the ball hits the batvibration frequencythe mass of the ball4.Modeling an
34、d Solution4.1 Modeling and Solution to Problem I4.1.1 Model Preparation1) Analysis of the pushing force or pressure exerted on hands1Fig. 4-1As showed in Fig. 4-1:l If an impact forcewere to strike the bat at the center-of-mass (CM) then pointwould experience a translational acceleration - the entir
35、e 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.l If the impact forcestrikes the bat below the center-of-mass,
36、 but above the center-of-percussion, pointwould 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
37、the rotational acceleration to the right and a player would still feel an impulsive force in the hands. l If the impact force strikes the bat below the center-of-percussion, then pointwould still experience oppositely directed translational and rotational accelerations, but now the rotational accele
38、ration would be greater.l 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 fel
39、t by a player holding the bat in his/her hands.l Define point as the center-of-percussion（COP）1）Locating the COPAccording 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 pendulumbe concentrat
40、ed at a single point located at a distance L from 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 distancewas （4-1）This locationis known as the center-of-oscillation.A solid
41、 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 resto
42、ring 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 object, the location of the center-of-mass relative to the pivot point, the rotational inertia of the object about its pivot
43、 pointand the gravitational field strengthaccording to (4-2)2) Analysis of the vibration:1 Fig. 4-2As 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
44、. Define the position of the vibration source as the vibrational node. Now this vibrational node is one of the optional “sweet spots”.4.1.2 Solutions to the two “sweet spot” regions1) Locating the COP14l Determining the parameters：a. mass of the bat ;b. length of the bat (the distance between Block
45、1 and Block 5 in Fig 4-3);c. distance between the pivot and the center-of-mass ( the distance between Block 2 and Block 3 in Fig. 4-3);d. swing period of the bat on its axis round the pivot (take an adult male as an example: the distance between the pivot and the knob of the bat is 16.8cm (the dista
46、nce between Block 1 and Block 2 in Fig. 4-3);e. distance between the undetermined COP and the pivot (the distance between Block 2 and Block 4 in Fig. 4-3, that is the turning radius).Fig. 4-3Table 4-1Block 1knobBlock 2pivotBlock 3the center-of-mass（CM）Block 4the center of percussion (COP)Block 5the
47、end of the batl Calculation method of COP14:distance between the undetermined COP and the pivot: (is the gravity acceleration） （4-3） moment of inertia: (is the turning radius,is the mass) （4-4）l Results:The reaction force on the pivot is less than 10% of the bat-and-ball collision force. When the ba
48、ll falls on any point in the “sweet spot” region, the area where the collision force reduction is less than 10% is cm, which is called “Sweet Zone 1”.2) Determining the vibrational nodeThe 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 f