美国数学建模优秀论文0.doc

Team #6336 Page 13 of 13Model for seeking sweet spotAbstract Using an energy analyzing approach, a simple model is raised for finding the “sweet spot” and analyzing its parametric property. To simplify the problem, the bat hit by a high speed ball is studied as a cantilever. By set up its dynamic model and energy formula, finding the sweet spot reduced to finding the spot which minimize the energy transferring between the ball and the bat. Numerical methods are given to solve the dynamical equation and compute the total energy the bat received. Using our model and algorithms, we can easily calculate the position of the “sweet spot” and analyze its variation with the parameters such as the geometrical parameters and physics constants. Our calculation shows that the energy transferred into the bat is minimized at the position 22 inches from the handle of the bat, which means the maximum power transferred to the ball. The model is in excellent agreement with experimental data.By changing the parameters such as cross section, radius and moment of inertia in the model, we also analyzed the corking bat, and analyzed the influence of corking. We find that corking is slight enough to ignore.Because of the difference of density and Young modulus in different materials, the energy transferred to the bat will be changed. Finally by making a figure which is used to compare the solutions we get, you can easily find that aluminum bats are better than the wood bats. Maybe this is why Major League Baseball prohibits metal bats.Keywords: Sweet spot, energy, numeric methodContents1. Introduction 31.1 Problem with the sweet spot and our models goal .31.2 Model Assumption.32. Model for the Ball-Bat Collision42.1 ball-bat collision.42.2 Energy convert 42.3 Vibration of the bat.52.4 Thenumerical results and analysis.63. The Corking effect.74. Material influence.95. Summary106. References.107. Appendix.111. Introduction1.1 Problem with the sweet spot and our models goalThere is a spot on the fat part of a baseball bat where maximum power is transferred to the ball when hit. Oppose to the explanation based on torque, it is not at the end of the bat knowing from experience. From our model, we can explain why the sweet spot is not at the end of the bat and accurately determine the location of the sweet spot. Some players believe that hollowing out a cylinder in the head of the bat and filling it with cork or rubber enhances the “sweet spot” effect. We augment out model so it can prove that corking is useless. At last, the material (wood or aluminum) out of which the bat is constructed really matter that our model can predict.1.2 Model Assumption1. There is no friction between ball and bat generated by the collision.2. The time of the ball-bat collision varies little while the collision point is changing.3. The baseball bat is an evenly proportioned cylinder4. In the article, the velocity we discuss is relative velocity and thus we can consider the bat to be static.(Assumption 3 and 4 give us enough reasonableness to consider the problem as a problem of cantilever.)2. Mechanic Model For the Ball-Bat CollisionOur model is developed for analyzing the collision between the baseball and bat. By analyzing the energy transformation between the ball and the bat, we find the “sweet spot” can be computed by energy analysis and the reason why the “sweet spot” is not at the far end of the bat can be explained by energy transformation.2.1 ball-bat collisionIn this paper, the bat is modeled as a cantilever. Figure 1The fixed position is the bat handle holding by the batter. The ball hit the bat at the position x0, transferring some energy to the bat. Our problem is: which point along the bat will rebound the ball most? According to the profound study in the last 20 years, the physics model of the bat includes complicated factors, which is difficult to handle. By using energy conservation Law and analyze the energy transformation process during bat-ball collision, a simple model is set up in this paper to analyze the bat-ball collision, and find an efficient method to compute the position of “sweet spot. 2.2 Energy convertWhen a high speed baseball hit a bat, its kinetic energy changed into three parts: the rebounding kinetic energy driving the ball flying away, the deformation energy of the ball, and energy transformed to the bat. The last one includes the vibration which makes the batter uncomfortable, and the deformation energy distorted the bat. According to the energy conservation theory, following energy identity is hold (1) Where K is the sum of the energy transferred to the bat, is the potential energy of the ball. Since varies very little while the collision point change, it can be regarded as the function of and independent with the structure of the bat. So: (2)According to the equality (2), transfer maximum power to the ball means make the energy K as small as possible.According to mechanical knowledge, (3)Where is the density, A is the cross section, E is the Young modulus and I is the moment of inertia.yWhere is the best place to receive the ball?x Figure 22.3 Vibration of the batThe mechanic model of the cantilever is (4) With initial conditions: (5)And the boundary conditions: ，, (6) Experiments show that the strongest impact of ball-bat collision is in the middle of the contact period. Therefore, f is taken as the following simple function (7)Our purpose of our study is to find the relationship between K and x0, so we can find the “sweet spot” through energy analysis.To solve partial differential equation (4)-(6), Galerkins method is applied. For any given function, we have concluded from the equation (4) that (8)Let: (9)Which fit the boundary conditions. B and C are unknown functions. Substitute function (9) and , in turn into equation (8), we obtain the following ordinary differential equations (10) (11)With initial values: , We then use Runge-Kutta method to solve the equation system and then substitute the solution B, C into function (9), the numerical solution of model (4)-(6) is obtained.The next step is computing the energy (3). Substitute the numerical solution of y into equation (3), and using numerical differentiation and numerical integral methods, the energy can be computed for any given parameters. The algorithms and the programs can be found in the appendices2.4 Thenumerical results and analysisUsing our model and above numerical methods, we now analyze the relationship between the energy transferred to the bat and the location the ball hit. We calculate energy corresponding to every point on the bat and draw a figure below:Figure 3By observing the figure 3 we can draw the conclusion that at the 22 inches to the handle the energy transferred to the bat is the minimum. Thus the energy transferred to the baseball is the maximum. The sweet spot is not at the end of the bat.3. The Corking effectA bat hollowing out a cylinder in the head of the bat and filling it with cork or rubber can be viewed as a hollow bat to some extent. So, the cross section (A) and moment of inertia (I) will be different with the solid wood bat.In the following discussion, we have made some reality-based assumptions that:1. We suppose that the radius of the baseball bat is R = 2.8(cm). Thus the square of cross section A = 25 2. In the case of the hollow bat, the outer radius is R = 2.8(cm), and the inner radius of the bat is considered to be r = 2.4(cm). Thus the square of cross section: A = 6.5 3. For solid bats moment of inertia I= while for hollow bats moment of inertia I = Substitute above data into the equation (3) (4), namely:Energy transferred to the bat:The mechanic model of the cantilever:Use the same method we do in the above, we can get the energy transferred to the bar of both corking bat and solid bat. Comparing these two K value, we will find that the two functional images are almost the same, as is shown in figure 4. This means that corking bat has little effect on the energy loss of the collision. Figure 44. Material influenceIn order to discuss the influence of the material out of which the bat is constructed, we use some parameter info based on real case which is given in the following:The comparison of the wood bat and the aluminum bat: MaterialDensity(g/)Young modulus(psi)Wood0.61500000Aluminum2.710000000By calculating the sum of the energy transferred to the bat in both cases, we have concluded that the impact of the material of the bat is significant: the aluminum bat can reduce the loss of the energy during the collision more effectively than the wood bat as can be observed in figure 5.Figure 55. SummaryThe confusing problem that why isnt the sweet spot at the end of the bat can be explained clearly by the model: we find that vibrations play major role in energy loss, and thus we build our model to find the sweet spot using the knowledge of the mechanic model of the cantilever. And by using the conservation theory and difference equation, we are able to find the energy loss in every locations of baseball bat and find the least energy loss locationsweet spot. In the case of the corking bat, we find the corking effect is tiny. However, predicted by our model the change of the material can affect the energy loss during the collision. Our findings are limited by the assumptions of our model. We ignored some factors which may also have impacts on the selection of the sweet spot.6. References:The sweet spot of a baseball bat.1 H.Brody. “The sweet spot of a baseball bat”. Physics Department, University of Pennsylvania, Philadelphia, Pennsylvania 19104.2 Alan M. Nathan. “Dynamics of the baseball-bat collision”. Department of Physics, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801.3 R.Cross. “The bounce of a ball”. Am.J.Phys.67.222-227 4 L.L.Van Zandt. “The dynamical theory of the baseball bat”. Am.J.Phys.60,172-1815 Robert K. Adair. “The physics of Baseball (HarperCollins, New York, 1994)”, 2nd,pp 71-77. Appendix:We order two Matlab-Program which is used to calculate the energy transferred to the bat.1: Main function:global rhou length1 cross1 EI x0 f0 dt dx%-% Parameter input% rhou: mass density% cross1: area of the cross section% Length1: length% EI: Young mod% x0: position of the collision % f0: force% dt: duration of the collision% dx: interval of the collision%-rhou=0.56;length1=34;cross1=38.47;f0=100000;dt=0.001;dx=1;m=100;n=100;mm=20;%solving partial differential equations t1=linspace(0,dt,m);x1=linspace(0,length1,n);batenergy=zeros(1,mm);EI=11*108;for i=1:mm x0=length1*i/mm;tout,yout=ode23(batfun,0,dt,zeros(4,1);y1=spline(tout,yout(:,1),t1);y2=spline(tout,yout(:,3),t1);t,x=meshgrid(t1,x1);y1,x=meshgrid(y1,x1);y2,x=meshgrid(y2,x1);yy=x.3.*(y1.*(x-4*length1/3)+y2.*(x-length1).2);%energy on the batyt=yy;yt(1,:)=zeros(1,n);yt(2:end