2009年数学建模美国赛获奖优秀论文

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7238

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A

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Summary

This paper describes model testing of baseball bats with the purpose of finding the so-called ―sweet spot‖. We establish two models and solve three problems. Basic model describes sweet spot which isn’t this spot at the end of the bat and helps explain this empirical finding. It predicts different behavior for wood (usually ash) or metal (usually aluminum) bats and explains Major League Baseball prohibits metal bats. Improved model proves that corking a bat enhances the sweet spot effect and explains Major League Baseball prohibits corking.

Selected methodologies currently used to assess baseball bat performance were evaluated through a series of finite element simulations. According to the momentum balance of the ball-bat system, basic model equation was established. The sweet spot can be found by the solution of the equation, when the baseball bat performance metrics were defined, considering initial variation in speed, the momentum of the bat and ball. Then, the improved model illustrates the vibrational behavior of a baseball bat and finds the peak frequencies and vibration modes and their relation to the ―sweet spot‖. From these observations two recommendations concerning bat performance were made:

(1) This spot isn’t at the end of the bat. The bat is related to materials out of which it is constructed. This model can predict different behavior for wood or metal bats. That is why Major League Baseball prohibits metal bats.

(2) In Improved model, a bat (hollowing out a cylinder in the head of the bat and filling it with cork or rubber, then replacing a wood cap) enhance the ―sweet spot‖ effect. This explains why Major League Baseball prohibits ―corking‖.

In some sense we have come full circle to the problem that there is no single definition of the sweet spot for a hollow baseball or softball bat. There are locations on the barrel which result in maximum performance and there are locations which result in minimal discomfort in the hands. These locations are not the same for a given bat, and there is considerable variation in locations between bats. Hopefully this conclusion will enhance the understanding of what the sweet spot is and what it is not, as well as encouraging further research into the quest for the \

To the second question, we used three methods to cork a bat. From the test we know the corked bat can improve performance metrics and enhance the sweet spot. That is why Major League Baseball prohibits corking.

To the third question, we used the first model and get that Aluminum bats can clearly out perform wood bats.

Finally, model testing analysis are made by simulation and conclusions are obtained. The strengths of our model are brief, clear and tested, which can be used to calculate and determined the sweet spot. The weaknesses of our model are need to further investigate, which is shown in the paper.

Key words: Sweet spot Finite element simulation Baseball bat performance

Ball-bat system Momentum balance

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Contents

1. Introduction ................................................................................................................... 4

1.1 The development of baseball ................................................................................... 4 1.2 Sweet spot ............................................................................................................ 4

1.3 The sweet spot vary from different bats .................................................................... 5 2. The Description of the Problem ........................................................................................ 6

2.1 Where is the sweet spot?......................................................................................... 6 2.2 Does ―corking‖ a bat enhance the ―sweet spot‖ effect? ............................................... 6 2.3 Does the material out of which the bat is constructed matter? ...................................... 6 3. Models.......................................................................................................................... 7

3.1 Basic Model.......................................................................................................... 7

3.1.1 Terms, Definitions and Symbols .................................................................... 7 3.1.2 Assumptions ............................................................................................... 9 3.1.3 The Foundation of Model............................................................................ 10 3.1.4 Analysis of the Result..................................................................................11 3.2 Improved Model.................................................................................................. 13

3.2.1 The Foundation of Model............................................................................ 14 3.2.2 Solution and Result .................................................................................... 15

3.2.3 Analysis of the Result................................................................................. 17 3.3 ―corking‖ a bat .................................................................................................... 18

3.3.1 How to cork a bat....................................................................................... 18 3.3.2 Methods ................................................................................................... 19 3.3.3 Model....................................................................................................... 20 3.3.4 Conclusions .............................................................................................. 21

4. Conclusions ................................................................................................................. 21

4.1 Conclusions of the problem................................................................................... 21 4.2 Methods used in our models.................................................................................. 22 4.3 Applications of our models ................................................................................... 22 5. Future Work ................................................................................................................ 22 6. References................................................................................................................... 23

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1. Introduction

1.1 The development of baseball

Baseball is a bat-and-ball sport played between two teams of nine players each. The goal is to score runs by hitting a thrown ball with a bat and touching a series of four bases arranged at the corners of a ninety-foot square, or diamond. Players on one team (the batting team) take turns hitting against the pitcher of the other team (the fielding team), which tries to stop them from scoring runs by getting hitters out in any of several ways. A player on the batting team can stop at any of the bases and later advance via a teammate's hit or other means. The teams switch between batting and fielding whenever the fielding team records three outs. One turn at bat for each team constitutes an inning; nine innings make up a professional game. The team with the most runs at the end of the game wins.

Evolving from older bat-and-ball games, an early form of baseball was being played in England by the mid-eighteenth century. This game and the related rounders were brought by British and Irish immigrants to North America, where the modern version of baseball developed. By the late nineteenth century, baseball was widely recognized as the national sport of the United States. Baseball on the professional, amateur, and youth levels is now popular in North America, parts of Central and South America and the Caribbean, and parts of East Asia. The game is sometimes referred to as hardball, in contrast to the derivative game of softball [1].

Fig. 1. The colliding of ball and bat

1.2 Sweet spot

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. Trying to locate the exact sweet

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spot on a baseball or softball bat is not as simple a task as it might seem, because there are a multitude of definitions of the sweet spot[2]:

(1) The location which produces least vibrational sensation 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

(6) The node of the fundamental vibrational mode

(7) The region between nodes of the first two vibrational modes

(8) The region between center of percussion and node of first vibrational mode For most bats all of these \spots\are at different locations on the bat, so one is often forced to define the sweet spot as a region, approximately 5-7 inches from the end of the barrel, where the batted-ball speed is the highest and the sensation in the hands if minimized. For the purposes of this paper, we will attempt to examine the sweet spot in terms of two separate criteria. One will be the location where the measured performance of the bat is maximized, and the other will be the location where the hand sensation, or sting, is minimized.

1.3 The sweet spot varies from different bats

Sweet spots on a baseball bat are the locations best suited for hitting pitched baseballs. At these points, the collision between the bat and the ball produces a minimal amount of vibrational sensation (sting) in the batter's hands and/or a maximum speed for the batted ball (and, thus, the maximum amount of energy transferred to the ball to make it travel further). On any given bat, the point of maximum performance and the point of minimal sting may be different. In addition, there are variations in their locations between bats, mostly depending on the type of bat and the specific manufacturer. Generally, there is a 1.57-2.0-in (3.8-5.1 cm) variation in the location of the sweet spot between different bat types. On average, the sweet spot occurs between 5 and 7 in (12.7 and 17.8 cm) from the barrel end of the bat[3].

The sweet spot's location for maximizing how far the batted ball travels after being hit can be calculated scientifically. When a batter hits a ball, the bat will rebound from the force of the collision. If the ball is hit closer to the handle end, a translational (straight-line) force will occur at the pivot point. If the ball is hit nearer to the barrel end, a rotational force will occur at the handle end near its center-of-mass—causing the handle to move away from the batter. This rotating motion causes a force in the opposite direction at the pivot point. However, impacts at the sweet spot results in these two opposite forces being balanced, causing a net force of zero—something that can be measured by scientists.

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