2010年美国数学建模竞赛获奖论文,英文版

2010年美国数学建模竞赛获奖论文

For office use only

T1 ________________

T2 ________________

T3 ________________

T4 ________________ Team Control Number 8038A For office use only F1 ________________ F2 ________________ F3 ________________ F4 ________________ Problem Chosen

Team #8038

February 23, 2010

Summary

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 batted speed 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 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 the viberational node reflect different form of 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

2010年美国数学建模竞赛获奖论文

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 advantages s and limitations, and providing suggestions on measuring the performance of a bat.

Key words: sweet spot, moment-of-inertia, Center-of-Percussion, Bat-Ball Coefficient-of-Restitution, Batted-Ball Speed

Contents

Summary .......................................................................................................................... 1

Contents ........................................................................................................................... 2

1.Restatement of the Problem .............................................................................................. 3

2.Analysis of the Problem ................................................................................................... 3

2.1 Analysis of Problem I............................................................................................. 3

2.2 Analysis of Problem II............................................................................................ 4

2.3 Analysis of Problem III .......................................................................................... 5

3.Model Assumptions and Symbols ...................................................................................... 5

3.1 Model Assumptions ............................................................................................... 5

3.2 Symbols ............................................................................................................... 5

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......................................................... …… 此处隐藏：25608字，全部文档内容请下载后查看。喜欢就下载吧 ……