1 Restatement of the Problem. 2 Terms and Ambiguous Words. 3 Ambiguous Words. 4 Assumptions. # 6654 Page 2 / 19

Transcription

1 Contents 1 Restatement of the Problem... Terms and Ambiguous Words... 3 Ambiguous Words... 4 Assumptions... 5 The COP Model Introduction Development Examinations Strengths and Weaknesses The Plane-parallel Motion Model Introduction Development Numerical Analysis and Examination Extension: Oblique Collision The Plane-parallel Motion Model of a Corked Bat Development Analysis and Examination Weakness and Strengths Dynamics of Swinging a Bat Introduction Development The Comprehensive Consideration Introduction Analysis References... 19

2 # 6654 Page / 19 1 Restatement of the Problem Sweet spot is a spot on the fat part of a baseball bat where maximum power is transferred to the ball when hit. It is said that hollowing out a cylinder in the head of the bat and filling it with cork or rubber, called corking can enhance the sweet spot s effect. Any useful model must: Locate the sweet spot of a given bat. Analyze the effect of corking and explain why Major League Baseball prohibits corking. Judge whether the material matters or not, and then explain that why Major League Baseball prohibits metal bats. Terms and Ambiguous Words COM: the center of mass. COR: the coefficient of restitution, also denoted by e in the followed equations. COP: the center of percussion. ICOM: the moment of inertia about the axis through the COM. 3 Ambiguous Words The batted velocity of the ball: the velocity of the ball right after the batting. The axis of the bat: the longitudinal axis of the bat. The top of the bat: the bat s ending at the side of the barrel. The holding point: the point on the bat where the batter holds it. The collision point: the point on the bat where the bat collides with the ball. 4 Assumptions 4.1 The bat is swung in a plane. 4. The bat can be regarded as a rigid body. 4.3 The impulsive force between the bat and the ball is large and instantaneous when collide. So, the force acting on the bat by the batter s hand and the gravity of the bat and the ball are negligible. 4.4 The COR equals the ratio of the relative velocities before and after collision, and is essentially determined by the relative elastic properties of the bat and ball. [] When a bat is corked, the COR does not change responsibly.

3 # 6654 Page 3 / 19 5 The COP Model 5.1 Introduction In this model we consider the second definition of sweet spot. When a rigid body rotates about a fixed axis, collision at the COP results in zero recoil impulse at the pivot point [3]. That is to say, if we regard the swinging bat as a rigid body in the motion of fix-axis rotation about the axis through the holding point, batting a ball at the COP will bring no recoil force at the batter s hands. The distance between the pivot and the COP, denoted by s, can be calculated with the formula [3] s = J O md (5.1.1) J O is the moment of inertia of the bat about the pivot, m is the mass of the bat and d is the distance between the COM and the pivot. 5. Development If the COP is the sweet spot, then the location can be determined with the Equation J O can be calculated by using the Parallel-axis theorem [3] as follows. J O = J + md. J is the moment of inertia of the bat about the axis through the COM. So, s = J md + d s d = J md (5..1) (5..) The Equation (5..) shows that the relative location of the COP to the COM can be simply calculated with the ICOM, the mass of the bat and the distance between the holding point and the COM.

4 # 6654 Page 4 / Examinations The parameters of two bats are listed in Table The holding point is considered to be at the distance of 0.1 m to the knob end. Type TPX C000 Easton Pro Stix Maple-71 Length(m) Material Aluminum Maple m(kg) d(m) J(kg m ) Table.5.3.1: the parameters of two baseball bats. Data from The Impacts on Softball Coefficients of Restitution of Different Material Qualities Under the Different Environments, Chi-Chuan Lin, 1997 With the Equation (5..), the distances between the COM and the COP for the two bats are calculated to be 0.055m and 0.057m. So, the locations of the COPs are respectively at the distances of 0.576m and 0.68 to the knob. According to the data of the bats length, neither of the COPs is at the end of the bat. That is to say, the sweet spot is not the end of a bat. 5.4 Strengths and Weaknesses This Equation (5..) can give the location of sweet spot defined as the COP of the bat. This model is simple and it requires much fewer parameters than the Plane-parallel Model. Moreover, in the model the force acting on the bat by the hand is not ignored, so, the model can be used in some situations where the preconditions of Assumption 4.3 are not available. However, the model has obvious weaknesses. First, according to the COP model, the sweet spot is related to the location of the holding point. Second, this model is helpless to analysis a corked bat. Finally, the model cannot provide any information of the batting process. In sum, this model can only provide an approximate description of the sweet spot. A complete discussion of this problem is followed in the next model. 6 The Plane-parallel Motion Model 6.1 Introduction First we consider a simplification of batting process shown in Fig The ball moves on the plane on which the bat is swung. The colliding happens in the Right before the colliding, the ball

5 # 6654 Page 5 / 19 of mass m is at the velocity of v 0 and the bat is moving in a plane. According to the Assumption 4.1 and Assumption 4., the motion of the bat can be regarded as Plane-parallel Motion of Rigid Bodies. The motion of any point of the bat can be resolved to two independent motions: the translation along with the COM and the rotation about the axis through the COM. In this scenario, right before the colliding, the translation velocity of the COM of the bat is u 0, and the angular velocity relative to the COM is ω 0. The colliding point is at the distance of s the COM at the side of the barrel end. After the colliding the velocity of the ball changes to v 1, the translation velocity of the bat COM changes to u 1 and the angular velocity changes to ω 1. All the velocities are perpendicular to the contact interface and the directions are shown in Fig According to the Angular Momentum Principle, the moment of impulse equals to the change of the bat s angular momentum m v 0 + v 1 s = J(ω 0 ω 1 ) (6.1.1) J is the ICOM of the bat. According to Assumption 4.3, the total momentum of the bat and the ball is conserved. That is to say, mv 0 + Mu 0 = mv 1 + Mu 1 (6.1.) Fig A simplification of batting process. According to Assumption 4.4, at the collision point the equation can be established. e = v 1 (u 1 + ω 1 s) v 0 + (u 0 + ω 0 s) (6.1.3) e is the COR. u 0 + ω 0 s and u 1 + ω 1 s are the velocities of the collision point before and after the collision.

6 # 6654 Page 6 / 19 Then we established the equations which describe the simple colliding: m v 0 + v 1 s = J(ω 0 ω 1 ) mv 0 + Mu 0 = mv 1 + Mu 1 e = v 1 (u 1 + ω 1 s) v 0 + (u 0 + ω 0 s) (6.1.4) 6. Development Equations is a linear equation set. By solving the equations, v 1 can be given as a function of s when other variables are regarded as given parameters: v 1 = v 0 + JM(1 + e)(u 0 + v 0 ) + JMω 0 (1 + e)s Jm + JM + mms (6..1) Set We find s = dv 1 ds = 0 0, wen ω 0 = 0 mm(u 0 + v 0 ) ± (mm(u 0 + v 0 )) + 4mMJω 0 (m + M) mmω 0, oters The values of s is the extreme point of the function v 1 (s). On the other hand, the graph of any function in the type of is alike to what is shown in Fig 6..1: f x = A + Bx C + Dx, BD > Fig The graph of the function f x = A+Bx C+Dx

7 # 6654 Page 7 / 19 That is to say, the positive one is the maximum point of v 1 (s). So, the value of s below maximizes v 1. 0, wen ω 0 = 0 s = mm(u 0 + v 0 ) + (mm(u 0 + v 0 )) + 4mMJω 0 (m + M) mmω 0, oters (6..) Set v r = v 0 + u 0 as the relative velocity of the ball to the bat. Then equation # can be simplified as s = mmv r + mmv r + 4mMJω 0 (m + M) mmω 0, ω 0 0 (6..3) If the sweet spot is defined as the point which maximizes the batted velocity, then the Formula (6..3) just provides the location of it. From this result we can find: If ω 0 = 0, which means that the bat is in complete translation, the Formula (6..) indicates that the sweet spot lies right at the COM. Common sense can support this conclusion, because only central collision results no rotation which wastes energy. For a bat, the M and J is fixed. Then s, the distance between the sweet spot and the COM, varies when v r or ω 0 changes. But, according to the actual situations, the values of v r and ω 0 must lie in certain intervals. Then, the values of s calculated with the Formula (6..3) are also lie in a certain interval. That is to say, the sweet spot is rather a region on the bat than a single line. The sweet spot is not at the top of the bat unless the calculated value of s is large than the distance between the fatter ending and the COM. Moreover, the further numerical analysis shows that with the increasing of v r, the value of s decreases. That means the sweet spot for a high speed ball is near the bat s COM. Further analysis is conducted with numerical methods as follows. 6.3 Numerical Analysis and Examination A particular bat, an 871g, 840mm long Louisville Slugger R 161 wood bat and a particular baseball, a 0.14 kg Rawlings Official League Baseball are used for further analysis. The center of mass of the bat was located 560mm from the knob end, and J was measured to be 0.039kg m. [1] So M = kg, J = 0.039kg m, m = 0.14 kg. With different values of v r and ω 0, values of s are given by the Formula (6..3). Fair curves are

8 # 6654 Page 8 / 19 plotted to fit the data as shown in Fig Fig.6.3.1: The graphs of s as the function of v r with different ω 0 as parameters. ω 0 = ω i = 5i + 7 (i = 1,,3,4,5). The v r is determined by v 0 and u 0, both of which has a upper and lower limit according to baseball rules and actual situations. According to the MLB s records, v 0 is usually between 40.m/s and 46.9m/s. And under the normal circumstance, the angular velocity of the bat ω 0 is below 3rad/s and the linear speed of the bat center of mass is below 17.87m/s for the given baseball bat. So, we choose v r = v 0 + u 0 from 40.m/s to 64.77m/s(that is 17.87m/s m/s) as the value range. v r ω 1 ω ω 3 ω 4 ω Table Values of s when the relative velocity v r is 40.m/s and 64.77m/s. ω 0 = ω i = 5i + 7 (i = 1,,3,4,5). We can find the maximum of s from the Fig.3. is 0.118m. So the sweet spot is meters far from the end of the bat. That is to say, to identify the end of the bat as the sweet spot is incorrect. Fig.6.3.: the batted velocity v 1 and s, the distance between the center of mass and the collision position when

9 # 6654 Page 9 / 19 v0 = 46.9m/s, u0 = 17.87m/s, and ω 0 = ω i = 5i + 7 (i = 1,,3,4,5). Fig shows the relation between v 1 and s with each ω 0 when v 0 = 46.9m/s and u 0 = 17.87m/s. According to Fig.6.3.1, the curves reach their peak values where the values of s are respectively , , , , and And those are the values we can get from the carves in Fig where v r = 64.77m/s. The figure is used to prove our assumption of the sweet spot is at the exact position that can maximize v Extension: Oblique Collision The Formula (6..) can be extended to describe the process of oblique collision. Consider the situation shown in Fig The only difference is that the direction of initial velocity of the ball is not perpendicular to the contact interface. The angel made with the normal line of the contact interface and the direction of ball s velocity is θ before the colliding. Fig.6.4.1: a simplification of the oblique collision process Because no impulse acts on the ball in the direction paralleling the contact interface, the angel remains θ after the colliding. In this scenario, according to Assumption # equation # needs to be adapted to m v 0 + v 1 s cos θ + m v 0 + v 1 r sin θ = J(ω 0 ω 1 ) mv 0 cos θ + Mu 0 = mv 1 sin θ + Mu 1 e = v 1 cos θ (u 1 + ω 1 s) v 0 cos θ + (u 0 + ω 0 s) r is the distance between the centre of the ball and the bat s axis. Then we can find, (6.4.1)

10 # 6654 Page 10 / 19 JM u 0 + v 0 cos θ 1 + e + JMsω 0 (1 + e) v 1 = v 0 + Jm cos θ + JM cos θ + mmrs sin θ + mms cos θ (6.4.) In the situation of a minor angle oblique collision, which means θ is not large, rs sin θ is small enough to be neglected. Common sense tells us that an efficient batting always requires a small angle between the directions of the bat and the ball s velocities, so the assumption of minor angel is rational. Then we find v 1 = v 0 + JM u 0 + v 0 cos θ 1 + e + JMsω 0 (1 + e) Jm + JM + mms It is very alike to the Formula (6..3.) Set 1 cos θ (6.4.3) The analysis in 6. shows that when u 0 + v 0 cos θ = V r mmv r + mmv r + 4mMJω 0 (m + M) s = mmω 0 (6.4.4) v 1 reaches its maximum. So the value of s given by the Formula (6.4.4) locates the sweet spot of the Oblique Colliding. From the formula (6.4.4) we can find: Comparing Formula (6.4.4) with Formula (6..3), we find that the location of sweet spot, or say sweet region, changes little when the batting is oblique. Because the forms of Formula (6.4.4) and Formula (6..3) are almost the same, the result we find in 6.1,6. and 6.3 can be applied to the scenario of Oblique Collision. Consider a more complex situation: the velocity of the bat before the colliding is not perpendicular to the contact interface either. The angel between the normal line and the direction of this velocity is φ. Fig a more complex collision In the same way, resolve the bat s velocity to the two components perpendicular to and parallel to the contact interface. Then the Equation can be adapted to:

11 # 6654 Page 11 / 19 Set m v 0 + v 1 s cos θ + m v 0 + v 1 r sin θ = J(ω 0 ω 1 ) mv 0 cos θ + Mu 0 cos φ = mv 1 sin θ + Mu 1 cos φ e = v 1 cos θ (u 1 cos φ + ω 1 s) v 0 cos θ + (u 0 cos φ + ω 0 s) u 0 cos φ + v 0 cos θ = V r The similar formula as formula 6..1 is given as follows, v 1 = v 0 + JM u 0 cos φ + v 0 cos θ 1 + e + JMsω 0 (1 + e) Jm + JM + mms mmv r + mmv r + 4mMJω 0 (m + M) s = mmω 0 1 cos θ The following analysis is the same as 6.1 and The Plane-parallel Motion Model of a Corked Bat 7.1 Development Hollowing out a cylinder in the head of a bat and filling with other material, called corking a bat, cause several effects on the bat. The location of COM, the mass and the moment of inertia about the axis through the COM all changes. According to the formula # and #, the sweet spot and the batted velocity changes responsibly. Consider the corked bat shown in Fig Fig.7.1.1:the corked bat The original density of the bat is ρ 0 and that of the material corked is ρ. Set Δρ = ρ ρ 0 Δρ can be negative. The radius of the hollowed cylinder is r. The distance between the bottom of

12 # 6654 Page 1 / 19 the hollow for corking and the original location of the COM is L. L is positive when the bottom is above the COM and is negative when below. The distance between the COM and the bat s top is L. So the volume of the corked part The change of the mass can be given as V = πr (L 0 L) ΔM = Δρπr (L 0 L) (7.1.1) The mass of the hollowed part of the bat is m 1, and the COM of this part is at the distance of r 1 to the original COM of the whole bat. The mass of the remained part of the bat is m, and the COM of this part is at the distance of r to the original COM of the whole bat. The COM of the corked bat is at the distance of R to the original COM of the whole bat. According to definition of the COM, we can establish the equations: 0 = m 1r 1 + m r m 1 + m ρ ρ m 1 r 1 + m r R = 0 ρ ρ m 1 + m 0 Solve the equation set we can find Because R = m 1r 1 (ρ ρ 0 ) m 1 ρ + m ρ 0 m 1 = ρπr L 0 L m = M m 1 r 1 = L 0 + L So, R = Δρπr L L 0 m + (L L 0 )Δρπr (7.1.) It is also the change of the location of the COM. The moment of inertia has the property of additivety. The moment of inertia of the hollowed part about the axis through the original COM of the bat is And that of the corked part is 1 3 ρ 0πr L 0 3 L ρπr L 0 3 L 3 Then the change of the moment of inertia about the axis through the original COM can be given as ΔJ 0 = 1 3 Δρπr L 0 3 L 3 (7.1.3)

13 # 6654 Page 13 / 19 According to the Parallel-axis theorem [3], the change the moment of inertia about the axis through the COM of the corked bat is ΔJ = M + ΔM R + ΔJ 0 (7.1.4) The three equations can be employed to locate the sweet spot of the corked bat with Formula (7.1.1) ~ (7.1.4): s = m(m + ΔM)v r + m(m + ΔM)v r + 4m(M + ΔM)(J + ΔJ)ω 0 (m + M + ΔM) m(m + ΔM)ω 0 (7.1.5) According to the Assumption 4.4, the COR remains the same after corking. Also the batted velocity can be calculated with Formula (6..1) as follows: v 1 = v 0 + M(J + ΔJ)(1 + e)(u 0 + v 0 ) + ω 0 (J + ΔJ)(M + ΔM)(1 + e)s J + ΔJ (m + (M + ΔM)) + m(m + ΔM)s (7.1.6) 7. Analysis and Examination Does corking enhance the sweet spot effect of a bat? The problem depends on how the sweet spot effect is interpreted. Two different understandings on the sweet spot effect are listed as follows: If corking makes the region of sweet spot larger, we say that the corking enhances the sweet spot effect. If corking makes the batted velocity of the ball larger when the swinging velocity of the bat remains the same, then we consider that the sweet spot effect is enhanced. The two understandings are both discussed in the following sections. We still use long Louisville Slugger R 161 in 6.3 for analysis. Commonly, the diameter of the corked cylinder is about 1 inch, and the height is from 6 to 10 inches. So r = 0.017m

14 # 6654 Page 14 / 19 We only consider the situation that the corked material is lighter than the original. The batted velocity is respectively calculated with Formula (7.1.6) when equals 6, 8, and 10 inches and the graph is shown in Fig7..1. Fig The velocity of batted ball v 1 and the change of the density of the baseball bat ρ when u 0 = 17.87m/s. Fig.7..1 reveals that the more the density changed, the lower the velocity of batted ball is if other conditions keep still. That means, corking the wood bat with other material like corks or rubbers will decrease the efficiency of the collision and then decrease the velocity of batted ball. All these conclusions depend on that u 0 does not change. 7.3 Weakness and Strengths In our models, the COR is a parameter determined by the elastic properties of the bat and ball, and it does not only vary from bat to bat but also changes when a bat is corked. But in the models, we fix the value of e for a bat regardless of whether it is corked. The coefficient of restitution is too complex for us to find the relationship between its value and its factors. And this difficulty is another reason that we use fixed COR. In addition, the Equation 6..1 is independent of e, so our conclusion on the location of the sweet spot is still right. Normally, the coefficient of restitution is got by tests, and is fathomable. Our model s main weakness lies in that the COR we use is fixed according to Assumptions 4.4. In order to strengthen our model s reliability, enough statistics of COR for different bats, or the quantitative relationship between the COR and corking is required. 8 Dynamics of Swinging a Bat 8.1 Introduction A complete analysis of the mechanics of the ball and the bat would be complex. An alternative semi-empirical method based on curve fitting is introduced to simplify the problem. Common sense tells us that although the swinging velocities of different bat may differ, the trajectory of

15 # 6654 Page 15 / 19 swinging bat is almost the same for a batter. An experiment was conducted to record the position of a swinging bat s two ends every ms [1]. The record is shown in Fig [1] Fig 8.1.1:the trace of COM of the bat when swing. Because the COM is fixed at certain point of a bat, the trajectory of a bat s COM is also the same regardless of the bat s weight, material and so on. A good fit to the experimental points of the COM is the logarithmic spiral shown in Fig.8.1. [1], defined by R = Ae λθ (8.1.1) Fig.8.1. the logarithmic spiral This is the expression of polar coordinate. The polar axis is the bat s axis when swinging begins,

16 # 6654 Page 16 / 19 and the origin point is at the initial location of the bat s ending of handle. 8. Development Convert the polar coordinates R, θ into rectangle coordinates, x = Ae λθ cos θ y = Ae λθ sin θ (8.1.) According to the Newton Second Law, equations are established as follows involving F, the force acting on the bat by the hand and M, the mass of the bat, d x dt = F cos α M d y dy = F sin α M α is the angel made with the x-axis and the direction of F. So, And d x dt + d y dt = F M d x dt = Aeλθ ( 1 + λ cos θ λ sin θ) dθ dt + (Aeλθ λ cos θ Ae lamθ sin θ) d θ dt d y dt = Aeλθ (λ cos θ + ( 1 + λ ) sin θ) dθ dt + (Aeλθ cos θ + Ae λθ λ sin θ) d θ dt d x dt + d y dt = A e λθ 1 + λ d θ dt. + dθ d θ dθ λ + dt dt dt 1 + λ dθ dt stands for the angular velocity of COM and d θ dt stands for the angular acceleration. Suppose the colliding happened at the point of P. The angular velocity of the COM is expected to reach its maximum at P. That indicates the angular acceleration at P is zero. If not, the angular velocity will keep increasing and cause energy wastage. So, set Then we find On the other hand, d θ dt = 0 F M = Aeλθ 1 + λ dθ dt (8.1.3) v = dx dt + dy dt = d dt (Aeλθ cos θ) + d dt (Aeλθ sin θ) = Ae λθ 1 + λ dθ dt

17 # 6654 Page 17 / 19 Compare two equations of Equation (8.1.3) and Equation (8.1.4), we find (8.1.4) v = 1 + λ F M (8.1.5) That is to say, when the swinging force acting on the bat is constant, the velocity of the bat at the collision point is in inverse proportion to the mass of the bat. With the Equation 8.1.3, we can calculate the swinging velocity of a bat from that of another once the ratio of the two bat s mass is known. 9 The Comprehensive Consideration 9.1 Introduction Since the process of batting consists of two stages--the swinging and the colliding, we need to combine the Plane-parallel motion model and the Dynamics of Swinging as a whole. The Dynamics of Swinging can provide the velocity of the bat before the collision, and using this result the Plane-parallel motion model gives the batted velocity of the ball and the location of the sweet spot. 9. Analysis According to the dynamics of swinging a bat, u 0 = 1 + λ F M the more the mass of the bat M decrease, the more u 0 will increase, and the mass of the bats is in inverse proportion to the velocity right before the collision. We take the increase of u 0 into consideration as follows. Fig.7... The velocity of batted ball v 1 and the change of the density of the baseball bat ρ when u 0 = 1 + λ F M is changing.

18 # 6654 Page 18 / 19 According to Fig.7.., there is an obvious increase of v 1, and the more the mass of the bat M decrease, the more v 1 will increase, which means corking a bat may enhance the sweet spot effect. Combine with the two figures above, we concluded that although corking a bat would decrease the efficiency of the collision, but the linear speed of the bat center of mass u 0 increases. After comprehensive consideration, our conclusion is that corking a bat usually enhances the sweet spot effect. Since that corking can enhances the sweet spot effect, for every competitor s sake, Major League Baseball prohibits corking. All above is the analysis on wood bats. Bats constructed of metal (usually aluminum) are analyzed as follows. To compare the collision efficiency of wood bats with metal bats, we use Easton Adult SV1-3 Baseball Bat-009 for further analysis. The bat s length is 31 inches, weighs 8 ounces, and its MOI is 70 lb inch. That is, M = kg, J = kg m, and other conditions kept still. The statistics of wood bats for comparing still accords to that of the Louisville Slugger R 161 wood bat. According to the dynamics of swinging a bat mentioned above, the mass of the bats is in inverse proportion to the velocity right before the collision. So v 0 = 19.60m/s. We calculate the answer in the condition that the angular velocity of the bat ω 0 = 3rad/s. Fig.9..1 The batted velocity v 1 with different materials out of which the bat are constructed (curve v standing for aluminum bat while curve v1 standing for wood bat in the figure) and s, the distance between the center of mass and the collision position. The World Baseball Association prescribes that the bats weight should not less than 0.9 kg for the youth league aiming at protecting the pitch. The analysis above shows that the lighter the bat is, the faster it is swung. Normally, the pitcher needs at least 0.4s to reflect when the ball is batted to him straightly. In the adult league, a professional pitcher can pitch a ball at the velocity reaching at 46.9 m/s. From Fig.9..1, the ball batted by the metal bat reaches the peak at 4.59m/s while ball batted by the wood bat reaches the peak at 38.57m/s.

19 # 6654 Page 19 / 19 The distance between the pitcher and the batter is about 18.6m. With these statistics, the time for the ball batted by the metal bat reaching at the pitcher is 0.433s, and the time for the ball batted by the wood bat reaching at the pitcher is 0.478s. That means the time for pitcher to reflect to the batted ball is 0.045s shortted when other conditions kept still. Because of 0.433s Is extremely close to the limit of reflecting time, which is 0.4s, if we choose another lighter aluminum bat, the time to reflect may break the limit. And if the limit is broken, the pitcher may get injured. Above all, we find that the metal bats behave better than the wood one. The metal bats are much lighter and easier to swing and make the velocity of the batted-ball larger than the wood ones, and that causes the pitcher more dangerous. In order to make the game fair and safe, the metal bats are prohibited by the Major League Baseball. References [1] R. Cross, Mechanics of swinging a bat, Am. J. Phys. 1, 36-43(009). [] D. A. Russell, The sweet spot of a hollow baseball or softball bat, Kettering University(004). [3] Q.L.Liu, Theoretical Mechanics, Science Press, Beijing, 005