2009年数学建模美国赛获奖优秀论文.doc

.For office use onlyT1_T2_T3_T4_Team Control Number7238 Problem ChosenAFor office use onlyF1_F2_F3_F4_SummaryThis 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 isnt 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 isnt 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 perfect bat.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 balanceContents1. Introduction31.1 The development of baseball31.2 Sweet spot41.3 The sweet spot vary from different bats52. The Description of the Problem52.1 Where is the sweet spot?52.2 Does “corking” a bat enhance the “sweet spot” effect?62.3 Does the material out of which the bat is constructed matter?63. Models63.1 Basic Model63.1.1 Terms, Definitions and Symbols73.1.2 Assumptions93.1.3 The Foundation of Model93.1.4 Analysis of the Result113.2 Improved Model133.2.1 The Foundation of Model133.2.2 Solution and Result143.2.3 Analysis of the Result173.3 “corking” a bat183.3.1 How to cork a bat183.3.2 Methods193.3.3 Model203.3.4 Conclusions214. Conclusions214.1 Conclusions of the problem214.2 Methods used in our models224.3 Applications of our models225. Future Work226. References231. Introduction1.1 The development of baseballBaseball 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 teammates 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 bat1.2 Sweet spotEvery 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 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 spot2: (1) The location which produces least vibrational sensation in the batters 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 sweet 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 batsSweet 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 batters 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 bat3.The sweet spots 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-masscausing 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 zerosomething that can be measured by scientists.2. The Description of the Problem2.1 Where is the sweet spot?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. The first question require empirical finding.Modal analysis4 represents a reliable and important technique to study a structures dynamic characteristics, including its natural vibration frequencies and mode shapes. The intention of conducting this project was to carry out a modal analysis of a wooden baseball bat as part of a larger effort to find the principal modal parameters of the bat structure, such as the center of percussion (COP), the peak frequencies, main nodes, and the vibrational mode shapes along the bat as well as their relation to the so-called “sweet spot”, which will be shown to be more of a “sweet zone”.Fig. 2. A test for sweet spot2.2 Does “corking” a bat enhance the “sweet spot” effect? 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.2.3 Does the material out of which the bat is constructed matter?Today playing baseball no longer means you will experience the wonderful crack of the bat sound that brings back countless memories. In fact, wood bats are rare at most levels other than the pros. Below is the types of the different baseball bat materials available today: White Ash, Maple, Aluminum, Hickory, and Bamboo.3. Models3.1 Basic ModelAn explicit dynamic finite element model (FEM) has been constructed to simulate the impact between a bat and ball as shown in Fig. 3. A unique aspect of the model involves an approach used to accommodate energy losses associated with elastic colliding bodies. This was accomplished by modeling the ball as a viscoelastic material with high time dependence. The viscoelastic response of the ball was characterized through quasi-static tests and high-speed rigid-wall impacts. This approach found excellent agreement with experiment for comparisons involving rebound speed, contact force and contact duration. The model also captured the balls speed-dependent coefficient of restitution, which decreased with increasing impact speed.The model was verified by simulating a dynamic bat-testing machine involving a swinging bat and a pitched ball. The good agreement between the model and experiment for numerous bat and ball types, impact locations and speeds, as well as bat strain response indicate the model has broad application in accurately predicting bat performance5.Fig. 3. Diagram of finite element ball-bat impact modelIn the current FEM bat vibration also hindered the determination of its after impact speed. The problem was established by consideration of a momentum balance of the ball-bat system.3.1.1 Terms, Definitions and SymbolsIn the following comparisons, commercially available solid-wood and hollow-aluminium bats are considered. Each bat had a length of 860 mm(34 in). Their mass properties were measured and may be found in Table 1. The wood bat is slightly heavier and consequently exhibits a larger MOI. While this is typical, it should not be considered a rule. The hollow structure of metal bats allows manipulation of their inertia in non-obvious ways. The profiles of the two bats were similar, but not identical. The effect of bat profile for the normal and planar impacts considered here was not significant. The properties used for the ball were found from dynamic tests of a typical collegiate certified baseball. Table 1 Mass properties of a solid wood and hollow metal batBatMass(g)C.G. (mm)MOI* ()Wood9064290.209Aluminium8634180.198*MOI and centre of gravity (C.G.) and measured from the bat bats centre of rotationThe motion of a swinging bat, as observed in play, may be described by an axis of rotation (not fixed relative to the bat), its rotational speed and location in space. The axis of rotation and its orientation move in space as the bat is swung and its rotational velocity increases. Thus, three-dimensional translation and rotation are required to describe the motion of a bat swung in play. Determining a bats hitting performance requires only a description of its motion during the instant of contact with the ball. The motion over this short time period has been observed as nearly pure rotation, with the fixed centre of rotation located near the hands gripping the bat. The exact motion of the bat will obviously vary from player to player. For the study at hand, a fixed centre of rotation, located 150 mm from the knob end of the bat was used, and is shown in Fig. 4. Fig. 4. Schematic of assumed bat motion during impact with a baseballThe impact location with the ball, is measured from the centre of rotation. The bats rotational speed before impact is designated , while the balls pitch speed is (Pitch speed is taken here as a negative quantity to maintain a consistent coordinate system.),which are shown as follows.: The mass moment of inertia of the bat: The mass of the ball: The bats rotational speed before impact: The bats rotational speed after impact:The balls pitch speed before impact: The distance from the impact location to the centre of rotation:The hit speed of the ball after impact: The short of “Ball Exit Speed Ratio”: The coefficient of restitution: The coefficient of restitution of the ball used for testing: the bats centre of percussion: The bats radius of gyration: The location of the bats centre of gravity: The sweet spot:Modulus of elasticity: Area moment of inertia: mass per unit length: Modulus of elasticity: The mass per unit length: Eigenfunctions belonging to : The roots of the beam equation3.1.2 AssumptionsTo test a bat one must assume a representative motion (typically rotation about a fixed centre), a bat and ball speed, a performance measure, and an impact location. From observations of amateur and professional players, typical bat swing speeds have been observed to range from 34 to 48 rad/s. Some practitioners of the game believe this number should be higher, but experimental measurements have not shown this. Pitch speed is more easily measured (often occurring live during a game) and may range from 20 m/s to 40 m/s. Thus, in a typical game, the relative speed between the point of contact on the bat and ball may vary by a factor of three. Of primary interest in bat performance studies is the maximum hit ball speed. To this end, tests are usually conducted toward the higher end of these relative speed ranges.3.1.3 The Foundation of ModelThe method involves pitching a ball toward a swinging bat, NCAA. This is the most difficult test to perform of the three methods and requires accurate positioning of the bat and ball, timing of their release and control of their speed. The problem was avoided by consideration of a momentum balance of the ball-bat system as(Eq. 1)where is the mass moment of inertia of the bat about its centre of rotation, is the bat rotational velocity after impact, and and are the mass and hit speed of the ball, respectively.The NCAA method uses what is termed a Ball Exit Speed Ratio (or ) to quantify bat performance at its experimentally determined sweet spot, . It is a ratio of the ball and bat speeds and is defined as(Eq. 2)Where is the impact location on the bat, and is the hit ball speed. It is used to normalize the hit ball speed with small variations that inevitably occur in controlling the nominal pitch and swing speeds. It may be found from the coefficient of restitution, , as (with ) .where for the ball-bat system (Eq. 3)The assumption of constant swing speed can lead to erroneous results if bats with different MOIs are being compared. A lighter bat will have a slower swing speed after impact with a ball than a heavy bat. Since would appear in the numerator of Eq. 2 as a negative contribution, the produces a lower measure of bat performance for light bats than would occur if . The is nevertheless popular because it avoids the experimentally difficult task of determining .The performance metric used by the ASTM method is termed the Bat Performance Factor (or BPF) and is found at the bats centre of percussion, , defined as(Eq. 4)where is the bats radius of gyration and is the location of its centre of gravity, both in relation to the centre of rotation.As observed by any player of the game of baseball, the hit-ball speed is dependent on its impact location with the bat. Many believe that a bats sweet spot coincides with its centre of percussion, , defined as the impact location that minimizes the reaction forces at its fixed centre of rotation. As will be shown below, they are offset slightly. The difference between the centre of percussion and the sweet spot may be partially explained by considering the momentum balance of the bat-ball impact. Combining Eqs 1 and 3 to eliminate produces an expression that may be solved for . The sweet spot, , can be obtained by minimizing this result with respect to , equating to