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1、For office use only T1_ T2_ T3_ T4_ Team Control Number7474Problem ChosenAFor office use only F1_ F2_ F3_ F4_2010 Mathematical Contest in Modeling (MCM Summary Sheet(Attach a copy of this page to each copy of your solution paper.Type a summary of your results on this page. Do not include the name of

2、 your school, advisor, or team members on this page.The simple harmonic model of baseball collision system There is a tremendous amount of physics and engineering that goes into the design of a baseball or softball bat and an amazing amount of physics involved in the bat-ball collision, and in the p

3、erformance and behavior of the bat itself. Sweet spot plays an important role in baseball for it can give the ball maximum of energy. Many studies showed that the bat vibrate violently when meets with the baseball, and cost a lot of energy. Through the careful studying, we abstract the collision sys

4、tem into a simple harmonic motion system which is constituted by 4 parts of objects and elastic force between them. By analyzing the Simple harmonic system, we build differential equations model for the collision, and gain the ball leave speed at all locations of the hitting point. Then, we can get

5、the location of sweet point by finding the maximum return speed of the ball. For problem 2 and 3, we change some coefficient of the model used in problem1 according to the situation, and concluded that corking a bat will slow the speed of ball, while using a bat made of aluminum will increase the ba

6、ll speed.Keywords: simple harmonic motion system , differential equations model , collision systemThe simple harmonic model of baseball collision systemAbstractThere is a tremendous amount of physics and engineering that goes into the design of a baseball or softball bat and an amazing amount of phy

7、sics involved in the bat-ball collision, and in the performance and behavior of the bat itself. Sweet spot plays an important role in baseball for it can give the ball maximum of energy. Many studies showed that the bat vibrate violently when meets with the baseball, and cost a lot of energy. Throug

8、h the careful studying, we abstract the collision system into a simple harmonic motion system which is constituted by 4 parts of objects and elastic force between them. By analyzing the Simple harmonic system, we build differential equations model for the collision, and gain the ball leave speed at

9、all locations of the hitting point. Then, we can get the location of sweet point by finding the maximum return speed of the ball. For problem 2 and 3, we change some coefficient of the model used in problem1 according to the situation, and concluded that corking a bat will slow the speed of ball, wh

10、ile using a bat made of aluminum will increase the ball speed.Keywords: simple harmonic motion system , differential equations model , collision systemI IntroductionThere are many definitions for what constitutes the "Sweet Spot" on a baseball bat. From a player's viewpoint, the sweet

11、spot is the place on the bat barrel where the contact between bat and ball results in the best hit - the ball leaves the bat with the greatest speed and the player's hands feel very little vibration from the impact. But the scientific definition is unclear, because different locations can have d

12、iffering effects. One possible sweet spot is the center of percussion, which is the location where the ball may impact the bat without causing a reaction force on the hand. An impact at any other location can cause the handle to feel like it is jumping in the hand. Another possible sweet spot is a l

13、ocation called a vibrational node. The impact of the ball causes the bat to vibrate in waves that have dead spots, or nodes. A third location of interest is the bat's center of mass, which is located by balancing the bat horizontally. The sweet spot is usually located not at the center of mass,

14、but somewhere between it and the end of the bat. There are some other definitions too. Some players believe that “corking” a bat enhances the “sweet spot” effect. There are some arguments about that .Such asa corked bat has (slightly less mass., less mass (lower inertia means faster swing speed and

15、less mass means a less effective collision. These are just some peoples views, other people may have different opinions. Whethercorking is helpful in the baseball game has not been strongly confirmed yet. Experiments seem to have inconsistent results.The material difference is also viewed as an infl

16、uence factor. Some people had done research on the performance of wood, metal and other materials.In our model, we try to make the outgoing speed of the baseball maximum .This is our definition of the sweet spot. In the process of solving the problem and reach our goal, we will consider the differen

17、ce of the parameters of different materials. In the finally, we can find which position is the sweet spot.II Assumptions of Our ModelOur assumptions are as the following:(1The bat is a homogeneous rod.(2The heat generated by the collision of the baseball and bat is ignored.(3The collision process oc

18、curs in the horizontal plane,so we do not take gravity into consideration.(4The bats rotary fulcrum is at the handle position.(5The bat for the problem 1 is standard bat and is make of wood.III Notation and Definitions IV Our Model4.1 Problem AnalysisFrom the experiment of baseball players, we know

19、that there is a sweet spot (or region on the bat, 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 minimized1. Many studies and experiments have showed that there is strong vibration when the ball collides with the bats

20、. A great deal of energy is lost due to the vibration of the bat.2 Therefore, the ball can hardly get the maximum energy, bounce back with a relatively low speed.There are many definitions about the “sweet spot”. Some thinks that the “sweet spot” is the location where the sensation in hands is minim

21、ized or cannot feel any sensation. There are also tremendous of people who think that the “sweet spot” is the location where the ball can get the maximum speed after collision. However, some studies have indicated that these two points are not exactly the same. In order to be accord with the title,

22、we consider the latter one.Our objective is to find the sweet spot where the velocity of baseball is maximum after collision. Since the process of the collision and vibration is quite complex, we should have an intuitive understanding of the collision process before the modeling.Large numbers of exp

23、eriments have showed that the time of collision is extremely short (about 0.7ms when the baseball moves at the regular speed of 40.2m/s. 3The peak force between bat and ball is an extremely violent one, in which the bat imparts a huge force on the ball thereby causing it to change directions and gai

24、n speed. Fig.1 illustrates the relationship between force and time. Fig.1Through the close analysis into the collision, we know that the ball experiences a significant amount of deformation during the collision, much more so than the bat. And the speed of the ball is considerably less after the coll

25、ision than it was before. The ratio of incoming velocity to outgoing velocity is called the coefficient of restitution (COR.The bat and ball vibrate in the process of collision, which is the main point we will analysis below.In the process of collision, both the baseball and bat has their own deform

26、ation, but they all have their internal force to overcome the deformation. The force is called stress in physics. Therefore, they will recover their shapes after a period of time. And the ball will leave very quickly before the bat fully recovered and the bat will remain vibrate for a short time. Th

27、e stress for both the bat and ball is determined by the deformation of bat and ball. The bigger the deformation, the greater the stress will be. Its in accord with the SHM4 (Simple Harmonic Motion during the contact period. For the study convenience, some scientists use a SHM system (showed in Fig.2

28、 to model the vibration of ball and bat during the collision. Considering the vibration is the damped vibration, two dampers are added, and their damped coefficient is closely related to the material of ball and bat. is also determined by the material of baseball. Fig.2The differential equations des

29、cribing this mass-spring model are:11212( a s x x x x - is the elasticity inside the ball. This is non-linearbecause the linear model of the elasticity doesnt fit well with the reality. To modulate the linear model, we use this power function. So does the damping force 11212( b c x x x x - inside th

30、e ball.This model is intuitive. It simplifies the collision process between ball and bat and provides a relatively theoretical physics model to simulate thex x x m x s x c xs x x x x c x x x x =-=-+-+-collision process. It is a convenient way to study the process for many people are familiar with th

31、e mass-spring model. Simple harmonic model applies to Hookes Law, the equations above make some modifications , power function take the place of simple linear function, it will have better applicability. It is a efficient model. But there is a little problem with the analysis above, because it just

32、applies to the situation when the ball hits the bat at its center-of-mass. This very rarely happens - hits at the sweet spot are several inches from the center-of-mass. And the parameter which depends on the hit location and material so cannot be assumed easily. It can hardly be expressed as a funct

33、ion of distance explicitly. It does not suit the baseball and bat model well. Therefore, we make some improvements based on the simple mass-spring model.4.2 The Improved ModelLets take a closer look at the model:When the bat was hit by a ball with tremendous speed, the part of the bat right below th

34、e ball was hit at the location of center-of mass, and was pulled by the parts nearby, as shown in Fig.3: Fig.3The shadow part stands for the region which was right under the baseball and have the center-of mass hit with the ball.3F is the press caused by the baseball1F is the traction caused by the

35、right side of the shadow part.2F is the traction caused by the left side of the shadow part.Since 12, F F is caused by the deformation, and according to the Hooke law,F k x = which is similar with the spring, so we build an physics model below to simulate the bat-ball collision system. Fig.4We divid

36、e the bat into three parts. The middle part is the one contacting with the baseball, and the other two parts of the bat give the traction to the middle part to recover its location during the collision. Since the bat is held by people, we ignore the movement of .That is to say, was clamped firmly. C

37、ertainly, the vibration is also the damped vibration, so dampers are added.We can see that the , the coefficient of elasticity, is determined by the material of bat. So do the , is determined by the location of the hit point. By using the 2nd Newtons law, we can construct a series of differential eq

38、uations to describe this model.The following is the analysis of each object in this simple harmonicsystem. The object labled represents the baseball, and represents the quality of the baseball. is the elasticity coefficient of the ball. The objects labled represent the three part of the bat, represe

39、nt the quality of each part respectively.Each object subjects to two kinds of forces, that is damping force and elastic force. The elastic force is stress generated from the collision process indeed. We assume that is fixed for in the realistic model, object is near the handle. Next, we will make me

40、chanical analysis of objects . According to Newtons Second Law, we write out the following equations. Before the ball hits the bat, the three parts of the bat are relatively static. In the process of the collision, the four objects have the following relations. What we must explain here and what is

41、very important for the following analysis is that the hitting position is related to the mass distribution of the three parts. That is to say, we first assume the distance from the pivot to the hitting position is sl and we have got the data of the bats size and density. So we can get the relationsh

42、ip of the quality of the three parts, which is described by sl.In order to better fit the process, we can use the rudimentary model described above to analysis the subject :For object :=(-(And we let , be the damping force between object and .The same with the symbol and=-.-.-.+(-+(For object :=.+.I

43、n addition,=(23223232( s F c xc x x =- Since is relatively small, 3223232sin tan x x l -=By the same method, =24224224( s F c xc x x =-The meanings of the symbols are:the distance from the mass center of object to the axis of the bat damping factorcoefficient of elasticity of the spring connecting o

44、bjectandcoefficient of elasticity of the springdamping force between object anddamping force between object andelastic force between objectandelastic force between objectandthe angle between the axis andthe line of object andthe angle between the axis andthe line of object andthe relative displaceme

45、nt between object andthe relative displacement between object andthe distance between and s center of massthe distance between and s center of massand is closely related with the hit location. The following diagram of a standard baseball bat (Fig.5 shows the relationship between the hit location and

46、 ,The mathematic describe model is: =(-(=-.-.-.+( -+( (3=.+. Fig.5From the diagram above, we can model the shape of a standard baseball bat is(cm:212( ( r x xdx dm r x dx =Certainly, the result of is a function related to the variable ls. can be easily get from the simple geometry, which is also a f

47、unction related to the variable ls.Since 1123l ls dm l dm ls =-=- , we correlate with the function(3, and give theappropriate initial condition, we can simulate the vibration function of ball and bat for any certain ls .For example, when ls=0.95,we can solve the function(3 by matlab, and get the wav

48、e form of the bat below: Fig.6This is the wave form of the battle, we can see that it vibrate strongly atthe 0.12m away from the tip of the bat. But it not the real wave of the bat. Since the collision time(less than 1ms is extremely short for 5s. That is to say, the ball will leave the bat after ab

49、out 1ms. So we amplify thepicture and take a close look at the ball and the bat. The following diagram is the wave form in 1ms: Fig.7In order to know the velocity of baseball after the collision, suppose we take downward as the opposite direction,we can see their velocity figure below: Fig.8From the

50、 velocity simulation above, we can find the balls leaving speed. From the basic knowledge of the physics, we know that its the second point that the two curves come across. Simulation loses its effect after that point because the ball leaves the collision system. Although the picture shows the bats

51、velocity will be quick than the ball, its unreliable because the simulation loses its effect after the point. So the collision time is about 0.65ms, which its similar to the experimental data.So the velocity of the ball after the collision can be found at every ls .We use matlab to calculate the eve

52、ry velocity at each ls, and gain the picture below: Fig.9It can be easily seen that the sweet spot is located at about 0.92m, 0.15m from the tip.V Corking a bat5.1 Brief introductionBatters swinging at baseballs have to swing the big ol' bat fast, if they want good things to happen. Heavy bats m

53、ake for hard work. For the last 100 years, players have tried to quicken their bats by breaking the rules.A cork-lined bat might also be a lighter weight, which would make its swing speed a lot faster than that of a heavy, old, wooden shaft. Fig.105.2 Model and AnalysisWe still use the model describ

54、ed in the problem1 to illustrate the difference. The model in the problem 1 just needs few changes and it can also be capable to deal with this.The changes in the model was just few things described below:1. Since the figure 10 shows that there is a little cork at the end of the bat, we can regard t

55、he quality of the became lighter than before, that is too say, the density of becomes lower. Then,(2 is changed since it contain the density .2. The material of has changed. Then, some properties like elasticity coefficient and damped coefficient of changed. The elasticity of wood is about 0.12*107

56、N/cm, and the rubber is about 0.118*105 N/cm. For simulation, we can simply choose a number between them to stand for the elasticity of the composition, so does the damped coefficient;3. The center of mass has moved. Its movement is too small to be ignored.We change them in the model, and simulate,

57、the result is in accord withmany scientists study: Fig.11The picture clearly shows that when the ball hit at or near the sweet spot, standard one perform better than the corked one. So, many people have the illusion that corking a bat can have a better performance. We can see that Major League Baseb

58、all prohibits “corking” for the reason that they want to keep the race brillient.VI Aluminum bat vs wood bat6.1 Model and analysisAs described above, when the material is made of metal, we should change some coefficient in the model we used in the problem1:1. The density of the material has changed, so the mass of the objects will change.2. Some properties like elasticity coefficient and damped coefficient of the bat has changed.3. The center of mass has moved due to the h