Hunting for the Sweet Spot by a Seesaw Model

Hunting for the Sweet Spot by a Seesaw Model Haiyan Li 1, Jianling Li, Shijun Li 1, and Zhaotian Liu 1 1 School of Information Science and Engineering, Yunnan University, Kunming, Yunnan, 650091, China School of Humanities and Social Sciences, Jinling Institute of Technology, Nanjing, Jiangshu,10038, China jiallko@sina.cn Abstract. A seesaw model was proposed to hunt for the sweet spot on a baseball bat. After analyzing the vibration of the bat and the ball, a basic vibration model was abstracted to simulate the interaction of the bat and the ball, which simplifies the vibration of the bat and the ball into two components, one is a seesaw part, whose pivot is the sweet spot of the bat, the other is the source of the vibrator (SOC). The model demonstrate that the sweet spot is not at the end of the bat while is a sweet zone close to the end of the bat and also illustrates that corking in the head of the bat will decrease that area of the sweet zone while corking in the end of the bat will increase the area of the sweet zone. The model shows that the area of the sweet zone is increased if the material of the bat is metal compared to solid wooden bat. Keywords: baseball bat, sweet spot, seesaw model, vibration model. 1 Introduction Baseball, the national ball in USA and Japan, is popular and is thought as the combination of competition, wisdom, braveness and cooperation. Batters know from experience that there is a sweet spot [1] on the baseball bat, about 17 cm from the end of the barrel, where the shock of the impact, felt by the hands, is reduced to such an extent that the batter is almost unaware of the collision. At other impact points, the impact is usually felt as a painful sting or jarring of the hands and forearm, particularly if the impact occurs at a point which is not sweet spot. Therefore, many researches have been done on bat and ball. Most of these studies are based on the physical phenomena of ball-bat collision. In this paper we proposed a seesaw model by just taking the essence of physical phenomena into consideration rather than the specific process of the physical phenomenon. The model defines the sweet spot as the minimum energy loss area, maximum energy transfer area []. After analyzing the vibration of the bat and the ball, a basic vibration model was abstracted to simulate the interaction of the bat and the ball, which simplifies the vibration of the bat and the ball into two components, one is Y. Wu (Ed.): ICCIC 011, Part IV, CCIS 34, pp. 33 41, 011. Springer-Verlag Berlin Heidelberg 011

34 H. Li et al. a seesaw part, whose pivot is the sweet spot of the bat, the other is the source of the vibrator (SOC). The model demonstrates that the sweet spot is not at the end of the bat while is a sweet zone close to the end of the bat and also illustrates that corking in the head of the bat will decrease that area of the sweet zone while corking in the end of the bat will increase that area of the sweet zone. The model shows that the area of the sweet zone is increased if the material of the bat is metal compared to solid wooden bat. Approach At first, the vibration mode of the baseball bat is summarized through relevant literatures [3, 4], and then we take the vibration of the basic mode caused by the ball colliding with the bat as a simple model, shown in Fig. 1, which the vibration curve is enlarged in Fig. 1, actually the real vibration amplitude is not as large as that shown in fig. 1. Subsequently, we improve the model to enable them to be used in practice based on leverage theory. Fig. 1. Schematic diagram of fundamental vibration mode A. The Model without Considering Torque The baseball bat vibrates after colliding with a ball. A Vibration Modes of a Baseball Bat was proposed in [4], which implicated that an impact at the node will not cause the bat to vibrate, and thus none of the initial energy of the ball would be lost to the bat. To illustrate the theory, we have done an experiment called "The effect of the ball rebound height when the ball falls on the seesaw at different locations, shown in fig.. It is clear that the rebound height of the ball falling on the pivot O is the highest. The loss energy of the ball is the minimum at the pivot O. Fig. shows the difference of the rebound height of a ball, whose initial height is the same but the collision point on the seesaw is different. H 1 is the rebound height which the ball falls at the pivot; H is the rebound height which the ball falls at the non-pivot. H1 > H.

Hunting for the Sweet Spot by a Seesaw Model 35 Fig.. The effect of the ball rebound height when the ball falls at different locations B. The Model Considering the Torque According to the abovementioned conclusion, the node is the smallest energy loss point, shown in fig. 3. Then, there will be three best sweet spots in fig.3, which is contradicted to the actual situation. Therefore, in order to further determine the sweet spot of the baseball bat, we must consider the torque of the baseball bat. Fig. 3. The second bending mode From the perspective of torque, the sweet spot should be close to the end of baseball bat, where the energy transmitted to the baseball is the maximum, whether the tangential speed or the torque of the bat is the maximum when the player hits the ball with the same speed. In order to find the real sweet spot out, we divide a baseball bat into two parts based on the above analysis on the vibration and torque of the bat. One part called "SEESAW" is AB segment shown in Fig. 4 whose pivot is near the end of node (O), that is, a sweet spot. The other part called "SOV" (the Source of Vibrator) is BC segment shown in Fig. 4. "SEESAW" part and "SOV" parts have no vibrations when there is no impact between the bat and the ball. Once the collision occurred, "SOV" starts to vibrate and affects "SEESAW" swing up and down. In general, bat has such a Fig. 4. Basic SEESAW Model

36 H. Li et al. movement model,that is, the impact of the ball and the bat prompts bat vibration, while the vibration which in turn affects the total energy of the ball when it is bounced back. In Fig.4 AB is the "Seesaw" part, whose pivot is O, BC is the vibration source. It is concluded that there is a point of minimum energy loss in the seesaw part of the bat, which is the pivot of seesaw, called sweet spot. 3 Simulation and Result Analysis A. Why Isn t the Sweet Spot at the End of the Bat? The question why the sweet spot is not at the end of the bat is now translated into why the pivot of "Seesaw" is not at the end. Based on the torque balance theory, it can be concluded by the Seesaw shown in Fig. 5: F1 L1 = F L (1) Fig. 5. "SEESAW" model If the pivot is at the end, then the L 1 (or L ) tends to 0, so F 1 (or F ) tends to, while it is impossible to achieve F 1 (or F ). Therefore, the pivot of "seesaw" cannot be at the end, that is, the sweet spot cannot be at the end. C. Corking a Bat Affects the Sweet Spot Some players believe that "corking" a bat enhances the "sweet spot" effect. Here we use the proposed model to discuss this issue. The previous conclusion only considers the sweet spot of the bat in a vibration mode, but the vibration of a baseball bat is very complex based on many previous studies[3,4,6], which has a lot of bending modes, as shown in Fig. 6. However, the first and second bending modes have significant impact on ball energy, therefore, according to the convention used by Rod Cross [6], the sweet zone is defined as the region located between the nodes of the first and second modes of vibration (about 4-7 inches from the barrel end of a 30-inch Little League bat), shown in Fig. 7. Since the vibration motion of the bat is small in this region, the impact in this region results in little vibration of the bat but a solid hit results in maximum energy transferred to the ball.

Hunting for the Sweet Spot by a Seesaw Model 37 Fig. 6. Three bending modes of a freely- supported bat Fig. 7. The sweet zone In this way, the model is further improved if the sweet spot become the sweet zone, which the vibration amplitude of SOV change will affect the SEESAW vibration amplitude. Particularly, the larger SOV vibration amplitude is, the larger SEESAW vibration amplitude will be, and vice versa. Now we analyze the sweet zone change based on the amplitude of SEESAW changes. In order to make the analysis more reliable, we assume: (1) The relationship between the collision energy loss of ball, bat and their position before the collision is that node has no loss of energy, at other location, the greater the amplitude of the bat is, the greater the energy of ball loss is, and vice versa. () The relationship between the oscillation amplitude of vibration and its quality, for a fixed frequency of the vibration source and fixed external factors, can be defined as that the larger of the quality the smaller of the amplitude; and vice verse. Based on the above analysis, it is clear that "corking" the bat decreases the quality of the SOV, so that its amplitude increases. The increased amplitude of SOV results in amplitude increase of SEESAW accordingly. When SEESAW amplitude increases,

38 H. Li et al. the location where the bat loss the same energy, moves closer to the pivot of the SEESAW, and so the sweet zone becomes smaller. With the above analysis and conclusions, we know that "corking" a bat in the head will not enhance the "sweet spot" effect. Therefore, why Major League Baseball (MLB) prohibits "corking" does not depend on this effect. D. Is the Material Which the Bat is Constructed Matters? We make some more assumptions to predict how different material impacts the performance of a bat: (1) The bat has the same shape and the same volume () Only the density of the material affects the sweet spot of bat made of different materials. (3) The baseball bat is solid and the density is uniformly distributed (4) The mass of a bat changes because of the material density, that is, the mass increases as the density increases and vice verse. With the above assumptions, we make a more detailed analysis on the model in the basic vibration model diagram shown in Fig. 1, the basic vibration model. If the material of the baseball bat is changed, then the mass of the bat will be changed. The change finally leads to change the pivot of the "seesaw" part. Now we analyze the specific change. With the "seesaw" shown in Fig. 8, we have the following assumptions: (1) The balance of the see-saw is defined as the status when the seesaw does not swing. The seesaw is always in a state of equilibrium as long as the material remains unchanged. () The transition from one equilibrium state to another is achieved by shifting the pivot. (3) We only consider the final result of the transition that the see-saw varies from one equilibrium status to another while ignoring the specific process. (4) Left is defined as the positive shift direction. (5) When material changes, Δm denotes the difference between the variation of the mass increment on the left and the mass increment on the right. Fig. 8. A special "seesaw" model

Hunting for the Sweet Spot by a Seesaw Model 39 When the "seesaw" is balanced, if M denotes the mass of the right, then the mass of the left is M +Δ M and Δ M > 0. With the above assumptions and theory about torque, we obtain: ( M +Δ M) gl = MgL () Equation (3) minus (),we get: 1 ( M +Δ M +Δm) g( L Δ L) = Mg( L +Δ L) (3) 1 Δ ml1 =ΔL( Δ M +Δ m) (4) As shown in Fig. 8, L 1 < L and when the seesaw is in equilibrium state, according to () we obtain ρvgl 1 1 = ρvgl, therefore V1 > V. So if the density of the seesaw increases, we get Δ m > 0.And because L 1 > 0 and Δ M > 0 so the condition that makes the (3) correct is Δ L > 0. Then, according to the assumption about the shift direction of fulcrum, it is concluded that the fulcrum should be shifted from right to left (positive direction) as the density of see-saw increases or the fulcrum should be shift from left to right (negative direction) as the density of see-saw decreases. With above conclusions, we can predict different behavior of wood (usually ash) or metal (usually aluminum) bats. It is clear that the metal density is larger than that of wood. So the fulcrum of a wood seesaw shifts from right to left when the material has been replaced by metal. From the two bending modes (the fundamental bending mode and second bending mode as shown in Fig. 6), it is concluded that the sweep spot [3,4] is greatly impacted because the shift movement are not the same. Now we will figure out the difference of the movement. From (4), we obtain: ΔmL Δ L = 1 (5) Δ M +Δ m To compare Δ L in the fundamental bending mode and the second bending mode, we set that L1 in (5) is corresponding to L 11 and L 1 in the fundamental bending mode and the second bending mode, respectively. In the same way, ΔM is corresponding to ΔM 1 and Δ M, ΔL is corresponding to ΔL 1 and Δ L, Δm is corresponding to Δm 1 and Δ m.therefore, ΔmL 1 11 Δ L1 = (6) Δ M1 +Δ m1 ΔmL 1 Δ L = (7) Δ M +Δ m Then (5) is expressed as (6) and (7) in the fundamental bending mode and the second bending mode.according to our "seesaw" model, we know: L 11 > L 1 Δ M <Δ M, Δm Δm 1 So ml 1 11 ml 1 Δ >Δ and Δ M1 +Δ m1 <Δ M +Δ m, then, it is achieved that:, 1

40 H. Li et al. Δ L >Δ L (8) 1 Equation (6) shows that the shift movement of the fulcrum in the fundamental bending mode is larger than that in the second bending mode when the material varies from wood to metal. Then with Fig. 7 we can conclude, the sweet zone of a solid metal bat is smaller than that of a solid wood bat. From the above analysis we can conclude that the greater the density of the material, the smaller the sweet spot area when the bat have the same shape and volume. On the contrary, the smaller the density of the material the greater the sweet spot area. Therefore, it is not the reason why MLB prohibit the use of metal bats. However, we must consider that metal (usually aluminum) baseball bat is hollow and its mass is even smaller than the wood bat. From this point of view, although the density of metal (usually aluminum) is larger than the density of wood, but the average density of a hollow metal bat is smaller than that of a solid wood bat. Therefore, the use of metal enlarges the sweet spot area. When the sweet spot area increases, in terms of the competition, to get the same batting effect needs lower batter's batting skill. In other words, when the sweet spot area increases, the fairness and competitiveness of the competition decrease. Therefore, in order to ensure the competitiveness and fairness of the baseball tournament, MLB should prohibit metal bats. 4 Conclusions A vibration model was proposed to simulate the interaction of bat and ball, which simplifies the vibration of the bat and the ball into two components, a seesaw part, whose pivot is the sweet spot of the bat; and the source of the vibrator (SOC). Based on the SEESAW model and the leverage theory, the model drew conclusions: (1) the sweet spot is a sweet zone close to the end of the bat. () corking a bat in the head decreases the area of the sweet zone while corking a bat at the end increases the area of the sweet zone; and (3)the sweet zone is increased if the material of the bat is metal compared to the solid wooden bat. Acknowledgment. This work was supported by the Grant (008YB009) from the Science and Engineering Fund of Yunnan University, the Grant (113014) from the Young and Middle-aged Backbone Teacher s Supporting Programs of Yunnan University and the Grant (113014) from on-the-job training of PHD of Yunnan University. References [1] Russell, D.A.: The sweet spot of a hollow baseball or softball bat invited paper at the 148th meeting of the Acoustical Society of America, San Diego, CA, November 15-19 (004); Abstract published in J. Acoust. Soc. Am., 116(4), Pt., pg. 60 (004) [] http://www.exploratorium.edu/baseball/sweetspot.html

Hunting for the Sweet Spot by a Seesaw Model 41 [3] Russell, D.: Vibrational Modes of a Baseball Bat. Science & Mathematics Department, Kettering University (003) [4] Russell, D.A.: Vibrational Bending Modes of a Baseball Bat. Science & Mathematics Department, Kettering University (003) [5] Cross, R.: The sweet spot of a baseball bat. American Journal of Physics 66(9), 771 779 (1998)