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寻找甜蜜点的建模


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 s

tudied 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. 利用能量分析方法,提出了一个简单的模型,寻找“甜蜜点” ,并分析其参数属性。为了简化问题, 研究了一个高速球的蝙蝠是一个悬臂。通过建立动态模型和能量计算公式,发现甜点减少,使球 与球之间的能量传递最小化。数值计算方法是解决的动力学方程和计算的总能量的蝙蝠收到。利 用我们的模型和算法,我们可以很容易地计算出“甜点”的位置,并分析其变化的参数,如几何 参数和物理常数。我们的计算表明,能量转移到蝙蝠的位置 22 英寸从手柄的蝙蝠,这意味着最大 的力量转移到球。该模型是在良好的协议与实验数据。 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. 通过改变参数,如横截面的半径和转动惯量的模型中,我们也分析了很好的球拍,并分析了很好 的影响。我们发现很小可以忽略。 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. 由于不同材料的密度和杨氏模量的不同,将能量转移到蝙蝠的能量会发生改变。最后,通过制作 一个用来比较我们得到的解决方案的数字,你可以很容易地发现,铝蝙蝠比木头蝙蝠好。也许这 就是为什么大联盟棒球禁止金属蝙蝠。 Keywords: Sweet spot, energy, numeric method 关键词:甜点,能量,数值方法 Contents 高兴的 1. Introduction ……………………………………………………………………3 1。介绍.............................................................................. 3 1.1 Problem with the sweet spot and model’s goal ………….............3 1.1 Problem with the 甜蜜点与目标模型能够 S 3! ! ! 1.2 Model Assumption……………………………………………………...…3 1.2 模型,3? 2. Model for the Ball-Bat Collision………………………………………………4 2。模型的碰撞球跳动??4

2.1 ball-bat collision……………………………………………….…………..4 2.1 球打..................................................................... 4 碰撞 2.2 Energy convert ……………………………………………………………4 2.2 能源转换..................................................................... 4 2.3 Vibration of the bat……………………………………...…………….…..5 2.3 振动 of the 跳动?5 2.4 The numerical result and analysis……..………………………...……..6 数值结果与分析九龙山 2.4 the 6 3. The Corking effect…………………………………………………………….7 3。the corking 效应...................................................................... 7 4. Material influence……………………………………………………………..9 4。材料的影响....................................................................... 9 5.Summary………………………………………………………………………10 5.summary ................................................................................. 10 6. References……………………………………………………..…………….10 6。引用.............................................................................. 10 7. Appendix……………………………………………………………………...11 7。附录................................................................................. 11 1. Introduction 1 引言。 1.1 Problem with the sweet spot and our model’s goal 1.1 Problem with the 甜蜜点与我们的模型能够的目标 There 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. There is a part of a 点 on the fat 棒球打在最大功率是 transferred to the 球时打。反对 to the 解释基于 扭矩,it is not at the end of the 跳动知道 from experience。从我们的模型,我们可以解释为什么“甜 蜜点 is not at the end of the 跳动和准确地确定位置 of the 甜蜜点。 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. 一些人相信打凹成形敲缸 in the Head of the 跳动和填充它与软木橡胶或提高其“甜蜜点”效应。 我们增加了模型,它可以证明,corking 是无用的。 At last, the material (wood or aluminum) out of which the bat is constructed really matter that our model can predict. 最后,the 材料(木材或铝)out of which the 打了我们的模型是人工真的可以预测。 1.2 Model Assumption 1.2 模型假设 1. There is no friction between ball and bat generated by the collision. 1。没有摩擦球和蝙蝠 Generated by the 之间的碰撞。

2. The time of the ball-bat collision varies little while the collision point is changing. 2。the time of the 球打碰撞碰撞点是各种各样的小而改变。 3. The baseball bat is an evenly proportioned cylinder 3。棒球蝙蝠是一 evenly proportioned 缸 4. In the article, the velocity we discuss is relative velocity and thus we can consider the bat to be static. 4。在我们讨论的文章,the 速度是相对速度和,因此我们可以认为蝙蝠是静态的。 (Assumption 3 and 4 give us enough reasonableness to consider the problem as a problem of cantilever.) (Assumption 3 和 4 给我们足够的合理性 to consider the problem as a problem of 悬臂。 ) 2. Mechanic Model For the Ball-Bat Collision 2。机械模型碰撞球打 Our 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. 我们的模型是 developed for Analyzing and the 碰撞 between the 打棒球。通过分析能源转换之间的 球和蝙蝠,我们找到的“甜蜜点”can be computed by Energy Analysis and the reason why the“甜蜜 点”is not at the Far End of the 跳动 can be explained by 能源转换。 2.1 ball-bat collision 2.1 球碰撞跳动 In this paper, the bat is modeled as a cantilever. 本文是 modeled 击败了悬臂。 Figure 1 图1 The 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’. 固定位置 is the 跳动控股 by the handle 的蝙蝠。 球打节拍 at the 位置 x0, 转移一些能源 to the 跳动。 我们的问题是: 这一点会弹回球打 along the most?according to the profound study in the last 20 年的 物理模型,包括蝙蝠复杂因子,which is difficult to handle。by using 节能法和分析能源转换过程在 打棒球的碰撞,简单的模型 is set up in this paper to the 打棒球的碰撞分析,找到有效的方法来计算 年和位置 of“甜蜜点” 。 2.2 Energy convert 2.2 能源转换 When 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

当高速棒球打了打,其动能 changed into three 配件:rebounding 动能驱动球飞了,变形能 of the 球,和能源转变 to the 跳动。the last one which makes the includes the 振动使难受,扭曲和变形能蝙 蝠。According to the following 能源节能理论,身份是抢劫 (1) (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. 在 K is the sum of the Energy transferred to the 跳动,is the potential energy of the 球。由于各种各样 的很小而碰撞点的变化,it can be regarded as the function of and independent with the structure of the 跳动。 So: 所以: (2) (2) According to the equality (2), transfer maximum power to the ball means make the energy K as small as possible. according to the Equality(2) ,传输最大功率 to the 球让你小 K 均值的能源是可能的。 According to mechanical knowledge, 根据力学知识, (3) (3) Where is the density, A is the cross section, E is the Young modulus and I is the moment of inertia. Where is the 密度,is the 截面,杨氏模量和 E 是 I is the 惯性矩。 2.3 Vibration of the bat 2.3 蝙蝠的振动 The mechanic model of the cantilever is 悬臂的力学模型 (4) (4) With initial conditions: 初始条件: (5) (5) And the boundary conditions: 和边界条件: , , , (6) , (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

实验表明,在接触周期的中间,球蝙蝠碰撞的最强的影响。因此,作为以下简单函数 (7) (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. 我们的研究目的是找 K 和 x0 的关系,所以我们可以找到“甜蜜点”通过能量分析。 To solve partial differential equation (4)-(6), Galerkin’s method is applied. For any given function , we have concluded from the equation (4) that 为解决偏微分方程(4)-(6) ,应用伽辽金法。对于任何给定的函数,我们得出结论,从方程(4) , (8) (8) Let: (9) 让: (9) Which fit the boundary conditions. B and C are unknown functions. Substitute function (9) and 适合边界条件。二、三是未知函数。替代功能(9)和 , , in turn into equation (8), we obtain the following ordinary differential equations 在方程(8)中,我们得到了以下常微分方程 (10) (10) (11) (11) With initial values: 初始值: , , 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. 然后, 我们用龙格-格-格-格-格方法求解方程组, 然后将溶液的乙、 丙三个函数 (9) , 得到模型 (4) (6)的数值解。 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 appendices 下一步是计算能量(3) 。用微分方程(3)代替数值解,并用数值微分和数值积分方法计算出任意 给定参数的能量。的算法和程序可以在附录中找到 2.4 Thenumerical results and analysis 2.4 计算结果与分析 Using 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: 使用我们的模型和上述的数值方法, 我们现在分析的能量转移到蝙蝠和球命中的位置之间的关系。 我们计算出对应于蝙蝠的每一点的能量,并画一个图: Figure 3 图3 By 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. 通过观察图 3 我们可以得出这样的结论:在 22 英寸到手柄的能量转移到蝙蝠是最小的。因此,能 量转移到棒球是最大。甜蜜点不在蝙蝠的末端。 3. The Corking effect 3。了很好的效果 A 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. 蝙蝠镂空圆筒在球棒头部用软木或橡胶填充它可以被看作是一个空心的蝙蝠在某种程度上。 所以, 横截面(一个)和惯性矩(我)将不同于实心木蝙蝠。 In the following discussion, we have made some reality-based assumptions that: 在下面的讨论中,我们已经提出了一些基于现实的假设: 1. We suppose that the radius of the baseball bat is R = 2.8(cm). Thus the square of cross section A = 25 1,我们认为棒球棒的半径是 2.8(厘米) 。因此,正方形的横截面为= 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: 2。在中空蝙蝠的情况下,外半径为 2.8(厘米) ,和内半径的蝙蝠被认为是 2.4(厘米) 。因此,横 截面的平方: A = 6.5 = 6.5 3. For solid bats moment of inertia I= while for hollow bats moment of inertia I = 3。对于固体蝙蝠的惯性矩,我=当中空蝙蝠的惯性矩我= Substitute above data into the equation (3) (4), namely: 将上述数据转换为(3) (4) ,即: Energy transferred to the bat: 能量转移到蝙蝠: The mechanic model of the cantilever: 悬臂梁的力学模型: 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.

使用我们在做上述同样的方法,我们可以转移到蝙蝠和蝙蝠都很坚实的酒吧的能量。比较这两者 的值,我们会发现,这 2 个函数的图像几乎是相同的,如图 4 所示。这意味着,塞住蝙蝠对碰撞 能量损失的影响不大。 Figure 4 图4 4. Material influence 4。材料影响 In 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: 为了讨论材料的影响,该材料的结构,我们使用了一些参数信息的基础上真实的情况下,这是在 以下: The comparison of the wood bat and the aluminum bat: 木材蝙蝠与铝蝙蝠的比较: Material Density(g/ ) 材料密度(克/) Young modulus(psi) 杨氏模量(PSI) Wood 0.6 1500000 木材 1500000 0.6 Aluminum 2.7 10000000 铝 10000000 2.7 By 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. 通过计算在这两种情况下的能量转移到蝙蝠的总和,我们得出的结论是,蝙蝠的材料的影响是显 着的:铝蝙蝠可以减少能量的损失,在碰撞过程中更有效地比木材蝙蝠可以观察到在图 5。 Figure 5 图5 5. Summary 5。总结 The confusing problem that why isn’t 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 location—sweet 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. 模型中的振动对能量损失的重要作用,我们发现,在能量损失的振动中发挥重要作用,因此,我 们建立我们的模型,以找到使用的机械模型的悬臂梁的振动的问题,为什么不是在结束的问题,

这一混乱的问题。通过利用守恒原理和差分方程,我们能够找到棒的每一个位置的能量损失,找 到最小能量损失位置的甜蜜点。在塞上蝙蝠的情况下,我们找到了很好的效果是很小的。然而, 我们的模型预测的材料的变化可以影响在碰撞过程中的能量损失。我们的研究结果是有限的,我 们的模型的假设。我们忽略了一些因素,这些因素也可能会影响到甜斑的选择。 6. References: 6。参考文献: The sweet spot of a baseball bat. 棒球棒的甜蜜点。 [1] H.Brody. “The sweet spot of a baseball bat”. Physics Department, University of Pennsylvania, Philadelphia, Pennsylvania 19104. [ 1 ] h.brody。 “一个棒球棒的甜蜜点” 。宾夕法尼亚大学物理系,宾夕法尼亚 19104。 [2] Alan M. Nathan. “Dynamics of the baseball 弥敦·M·艾伦·M· · · · · · · 。 “棒球的动力学 -bat collision”. Department of Physics, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801. -蝙蝠碰撞。物理系,伊利诺伊大学 Urbana-Champaign 分校,伊利诺斯,61801。 [3] R.Cross. “The bounce of a ball”. Am.J.Phys.67.222-227 [ 3 ] r.cross。 “球反弹” 。am.j.phys.67.222-227 [4] L.L.Van Zandt. “The dynamical theory of the baseball bat”. Am.J.Phys.60,172-181 [ 4 ] l.l.van Zandt。 “棒球棍的动力学理论” 。am.j.phys.60172-181 [5] Robert K. Adair. “The physics of Baseball (HarperCollins, New York, 1994)”, 2nd, [ 5 ]罗伯特 K.阿代尔。 “棒球的物理(HarperCollins 出版社,纽约,1994) ” ,第二, pp 71-7 PP 71-7 7. Appendix: 7。附录: We order two Matlab-Program which is used to calculate the energy transferred to the bat. 我们以两个 MATLAB 程序,用于计算转移到蝙蝠的能量。 1: Main function: 1:主要功能: global rhou length1 cross1 EI x0 f0 dt dx 全球 RhoU length1 cross1 EI x0 F0 DT DX %-----------------------------------------------------------% -----------------------------------------------------------% Parameter input %参数输入 % rhou: mass density % RhoU:质量密度 % cross1: area of the cross section % cross1:横截面积的

% Length1: length 长度:长度为 % EI: Young mod % EI:年轻的 MOD % x0: position of the collision % x0:碰撞的位置 % f0: force % F0:力 % dt: duration of the collision DT:%的碰撞时间 % dx: interval of the collision DX:%的碰撞时间 %-----------------------------------------------------------% -----------------------------------------------------------rhou=0.56;length1=34;cross1=38.47;f0=100000;dt=0.001;dx=1; RhoU = 0.56;长度= 34;cross1 = 38.47;F0 = 100000;DT = 0.001;dx = 1; m=100;n=100;mm=20; 米= 100;100;毫米= 20; %solving partial differential equations %解偏微分方程 t1=linspace(0,dt,m);x1=linspace(0,length1,n); T1 = linspace(0,DT,m) ;X1 = linspace(0,长度 1,n) ; batenergy=zeros(1,mm); batenergy =零(1 毫米) ; EI=11*10^8; EI = 11 * 10 ^ 8; for i=1:mm 对于我= 1:毫米 x0=length1*i/mm; x0 =长度*我/毫米; [tout,yout]=ode23('batfun',[0,dt],zeros(4,1)); [你] = ode23 吹捧, ('batfun” ,[ 0 ],DT,零点(4,1) ) ; y1=spline(tout,yout(:,1),t1); Y1 =样条(吹捧,你的(: ,1) ,T1) ; y2=spline(tout,yout(:,3),t1); Y2 =样条(吹捧,你的(: ,3) ,T1) ; [t,x]=meshgrid(t1,x1); 【T、X ] = meshgrid(T1,X1) ; [y1,x]=meshgrid(y1,x1); 【Y1,X ] = meshgrid(Y1,X1) ;

[y2,x]=meshgrid(y2,x1); 【Y2,X ] = meshgrid(Y2,X1) ; yy=x.^3.*(y1.*(x-4*length1/3)+y2.*(x-length1).^2); YY = X(Y1 ^ 3。*。*(X-4×长度/ 3)+ y2。*(x-length1) 。^ 2) ; %energy on the bat %的能量在蝙蝠 yt=yy;yt(1,:)=zeros(1,n);yt(2:end,:)=diff(yy)/(t1(2)-t1(1)); YT = YY;YT(1, : )= 0(1,N) ;YT(2:最后, : )= diff(YY)/(T1(2)- T1(1) ) ; y2x=yy;y2x(:,1)=zeros(m,1);y2x(:,end)=zeros(m,1); y2x = YY;y2x(: ,1)= 0(m,1) ;y2x(: ,结束)= 0(m,1) ; y2x(:,2:n-1)=diff(yy,2,2)/(x1(2)-x1(1))^2; y2x(: ,2:1)= diff(YY,2,2)/(X1(2)-(1)^ X1)2; ff=rhou*cross1*yt.^2/2+EI*y2x.^2/2; FF = RhoU * cross1 * YT。^ 2 / 2 + Ei×y2x ^ 2 / 2; batenergy(i)=trapz(x1,ff(end,:)); batenergy(我)= trapz(X1,FF(结束, : ) ) ; end 结束 xx=length1*[4:mm-4]/mm; XX =长度* [ 4 ] /毫米 4; plot(xx,batenergy(4:mm-4),'*-') 情节(XX,batenergy(4:4) , “*” ) %text(xx,batenergy(4:mm-4),'wood') %的文本(XX,batenergy(4:4) ,'wood” ) hold on 坚持 EI=69*10^8; EI = 69 * 10 ^ 8; rhou = 2.7 RhoU = 2.7 for i=1:mm 对于我= 1:毫米 x0=length1*i/mm; x0 =长度*我/毫米; [tout,yout]=ode23('batfun',[0,dt],zeros(4,1)); [你] = ode23 吹捧, ('batfun” ,[ 0 ],DT,零点(4,1) ) ; y1=spline(tout,yout(:,1),t1); Y1 =样条(吹捧,你的(: ,1) ,T1) ; y2=spline(tout,yout(:,3),t1); Y2 =样条(吹捧,你的(: ,3) ,T1) ;

[t,x]=meshgrid(t1,x1); 【T、X ] = meshgrid(T1,X1) ; [y1,x]=meshgrid(y1,x1); 【Y1,X ] = meshgrid(Y1,X1) ; [y2,x]=meshgrid(y2,x1); 【Y2,X ] = meshgrid(Y2,X1) ; yy=x.^3.*(y1.*(x-4*length1/3)+y2.*(x-length1).^2); YY = X(Y1 ^ 3。*。*(X-4×长度/ 3)+ y2。*(x-length1) 。^ 2) ; %energy on the bat %的能量在蝙蝠 yt=yy;yt(1,:)=zeros(1,n);yt(2:end,:)=diff(yy)/(t1(2)-t1(1)); YT = YY;YT(1, : )= 0(1,N) ;YT(2:最后, : )= diff(YY)/(T1(2)- T1(1) ) ; y2x=yy;y2x(:,1)=zeros(m,1);y2x(:,end)=zeros(m,1); y2x = YY;y2x(: ,1)= 0(m,1) ;y2x(: ,结束)= 0(m,1) ; y2x(:,2:n-1)=diff(yy,2,2)/(x1(2)-x1(1))^2; y2x(: ,2:1)= diff(YY,2,2)/(X1(2)-(1)^ X1)2; ff=rhou*cross1*yt.^2/2+EI*y2x.^2/2; FF = RhoU * cross1 * YT。^ 2 / 2 + Ei×y2x ^ 2 / 2; batenergy(i)=trapz(x1,ff(end,:)); batenergy(我)= trapz(X1,FF(结束, : ) ) ; end 结束 plot(xx,batenergy(4:mm-4),'-r') 情节(XX,batenergy(4:4) , “R” ) xlabel('place of collision') xlabel('place 碰撞” ) ylabel('Energy') ylabel('energy” ) 2: Subprogram: 2:子程序: function f=func(t,y) 函数 f(t,y)功能 global rhou length1 cross1 EI x0 f0 dt dx 全球 RhoU length1 cross1 EI x0 F0 DT DX AA=[1 0 0 0 0001 0 7/90*rhou*cross1*length1^6 0 -4/315*rhou*cross1*length1^7 7 / 90 * 0 * 6 * cross1 RhoU length1 ^ 0 - 4 / 315 * * * 7 ^ length1 cross1 RhoU 0010 0100

0 -4/315*rhou*cross1*length1^7 0 rhou*cross1*length1^8/280]; 0 - 4 / 315 * * * cross1 RhoU length1 ^ RhoU * * 7 0 cross1 length1 ^ 8 / 280 ]; BB=[y(2) BB = [ Y(2) 20*EI*length1^2*y(1)+1/2*f0*dx*(2*x0+dx-8*length1/3) 20 * 2 * ^ EI * length1 Y(1)+ 1 / 2 *(2 * * * DX F0 x0 + dx-8 *长度 1/3) y(4) -EI*(8*y(1)*length1^3-6*y(3)*length1^4)+dx/3*f0*(3*(x0-length1)^2+3*dx*(x0-length1)+dx^2)]; Y(4)- EI *(8 * Y(1)*长度^ 3-6 *(3)*长度^ 4)+ DX / 3 * f0 *(3 *(2 + 3 * x0-length1)^ DX DX ^ *(x0-length1)+ 2)]; f=AA\BB; F = AA、BB; return 返回


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