2025 Volume 66 Issue 2 Pages 180-185
Recently, metal bats have been widely used in high school and other baseball games, but owing to their improved performance, these bats have led to increased amount of accidents. Pitchers are often untimely hit the ball, resulting in serious injuries. However, the National Collegiate Athletic Association evaluates the repulsion performance of bats and balls using a standardized value called BBCOR (Bat-Ball Coefficient of Restitution). Thus, only bats with a value of 0.50 or less can be used. The BBCOR can quantitatively indicate the restitution performance of a bat, and several designs for controlling the BBCOR of metal bats have been proposed in previous studies. However, most of these require the bat to be actually manufactured; therefore, a simple method for predicting the BBCOR using dimensions and other factors during bat design is needed. In this study, the compressive load and BBCOR was first determined using simple compression and ball impact tests on aluminum alloy baseball bats with varying outside diameters and plate thicknesses. Subsequently, the relationship between the BBCOR and spring constant of the bats was examined, and the effects of the outside diameter and plate thickness on the spring constant were investigated. Based on these results, the effects of the outer diameter and plate thickness on the BBCOR were examined, and a simple equation that can determine the BBCOR from the shape (outer diameter and plate thickness) was proposed.
This Paper was Originally Published in Japanese in J. Soc. Mater. Sci., Japan 73 (2024) 597–602.
Fig. 6 Effect of thickness per diameter t/D on BBCOR of bats (I: t/D = 0.056, II: t/D = 0.060).
Players have been using wooden baseball bats since the introduction of baseball in Japan. However, wooden baseball bats are highly consumable and expensive because they can be damaged due to mishitting, etc., thereby imposing an economic burden on players and teams to which the players belong [1]. Under these circumstances, metal baseball bats, which are resistant to damage and relatively inexpensive, have been adopted in high school baseball from the 56th All Japan High School Baseball Championship held in the summer of 1974 since Mizuno released them in 1971. Since then, metal baseball bats have been predominantly used until the present [1–3].
Currently, the Special Rules for High School Baseball [4] stipulate the use of wooden, wood chip spliced, bamboo spliced, and metal baseball bats. No specific regulations are available regarding the material of metal baseball bats, but aluminum alloys of the 7,000 series are generally used. The tensile strength of the 7,000 series aluminum alloys varies based on their composition, processing, and heat treatment, but it is generally 550–675 MPa. Additionally, the Special Rules for High School Baseball [4] recommended that only metal bats that conform to SG (Safe Goods) standards can be used. The SG standard for nonwooden baseball bats [5], including metal baseball bats, stipulated the safety quality of baseball bats, which will not be destroyed by compression or flat tests, i.e., bats will not be deformed within the elastic range and will not be plastically deformed. Additionally, the modulus of longitudinal elasticity of aluminum alloys is almost constant regardless of its composition.
Metal baseball bats are hollow; thus, the striking part (barrel part) is deformed when it collides with the ball. The force generated when this deformation is restored repels the ball. This is called the trampoline effect, and it is generally one of the characteristics of metal baseball bats [6–9].
As time progressed, the performance of baseball bats improved. Additionally, the batted ball speed increased accordingly. Hence, fielders were unable to avoid batted balls, causing serious injuries. In particular, at the 101st All Japan High School Baseball Championship held in the summer of 2019, a player was hit in the head by a batted ball and suffered serious injury. Head injury is not only a sequela but also a life-threatening event in serious cases. Therefore, the board of directors of the Japan High School Baseball Federation approved a new standard for metal baseball bats in February 2022. The maximum diameter of the baseball bat was 64 mm, which was 3 mm smaller than the current 67 mm. Additionally, the thickness of the batted ball increased by approximately ≥1 mm from the current 3 mm. This aimed to suppress the rebound of the bat. Only baseball bats that meet the new standards will be available from the spring 2024 selection tournament in high school baseball [10].
Nonwooden baseball bats, such as metal baseball bats, have been widely used since the National Collegiate Athletic Association (NCAA) approved their use in 1970 in the United States [8]. The NCAA has its rules for baseball bats, just like Japan. However, the difference between the NCAA and Japan is that it uses a value called the Bat–Ball Coefficient of Restitution (BBCOR) [11, 12], which standardizes the performance standard for bats. NCAA has stipulated that only baseball bats with a BBCOR of ≤0.50 can be used since the 2011 season [8, 12]. It is a quantitative indicator for determining the usage of a baseball bat. The BBCOR standard is 0.50 because the maximum BBCOR value of wooden baseball bats is 0.50 so nonwooden baseball bats can perform as well as wooden baseball bats [9].
As mentioned above, BBCOR quantitatively showed the rebound performance of a baseball bat, but spending money, such as the introduction of testing equipment and measuring equipment is necessary because impact tests are required to calculate BBCOR [13]. Therefore, simple methods of estimating the BBCOR of a baseball bat were studied for controlling BBCOR from the design stage of a baseball bat. Takashima et al. investigated the effects of the impact velocity of balls and the natural frequency (Hoop frequency) of baseball bats on the BBCOR of metal baseball bats [8]. They revealed that BBCOR is expressed by the power law of impact velocity and Hoop frequency. Additionally, Shinoyama et al. conducted compression tests on the barrel of a metal baseball bat [14]. They revealed that bats with larger compressive loads exhibited lower BBCOR, and proposed the BBCOR estimation equation expressed by the power law of compressive load. The compressive load of a baseball bat indicates its stiffness; the results of the study by Sasayama et al. revealed that the higher the stiffness of a baseball bat, the lower its BBCOR. Thus, several formulas for estimating the BBCOR of a metal baseball bat have been proposed. However, to obtain the Hoop frequency and compressive load, fabricating a baseball bat is necessary. Therefore, establishing a method to predict BBCOR more easily from the shape at the initial stage of designing a baseball bat is warranted.
Therefore, this study first conducted simple compression and ball impact tests on aluminum alloy baseball bats with various outer diameters and thicknesses, measuring their compressive load and BBCOR. The relationship between the BBCOR and the spring constant of baseball bats was then examined as well as the effect of their outer diameter and thickness on the spring constant. Hence, the effect of the outer diameter and thickness on the BBCOR of baseball bats was examined, and a simpler estimation formula that estimated the BBCOR from their shape (outer diameter and thickness) was examined.
A simple compression tester (Bat Testing Solutions, LLC, G4 SSL Baseball Bat Testers) [15] was used to perform compression tests on aluminum alloy baseball bats. Figure 1 shows a schematic diagram of a simple compression test [14]. First, a baseball bat was placed so that it was compressed at a position approximately 150 mm from the top of its barrel. A preload (375 lbs ≈ 4.45 N) was then applied. The preload was set to 0, and a certain amount of deformation (0.03 in ≈ 0.76 mm) was added by pulling the lever of the tester. The amount of increase in the load value was measured and used as the compressive load. The baseball bat was compressed once in the circumferential direction at 0°, 45°, −45° and 90°. The average value of each compressive load was the compressive load of the baseball bat.
The pressure experimental apparatus.
A light gas gun (Koatsu System Co., Ltd.) [7, 8, 13] was used as a ball launcher under test conditions for impact tests, conforming to ASTM F2219 [11]. Figure 2 shows a schematic diagram of the impact test equipment [8]. A high-density polyethylene sabot was utilized to launch balls, and the sabot and the ball were launched as one. The sabot stopper attached to the tip of the launch tube stopped the sabot and launched only the ball [7, 8]. The balls used match balls of baseball. To consider the effects of temperature and humidity [16], it was held for ≥48 hours at a temperature of 22°C ± 0.1°C and a humidity of 60% ± 1% using a constant temperature and humidity tank (Orion Machinery Co., Ltd., PAP01B-KJ-SP) before being subjected to impact tests [8]. The impact position of the balls was 150 mm from the top of the barrel of the baseball bat, and three impact tests were conducted for each bat. The impact surface of the balls was seamless. Two high-speed video cameras (Photron Limited, FASTCAM-APX RSX, and FASTCAM SA5) were used to measure the impact and repulsion velocity of the balls in three dimensions. The impact velocity was calculated from the difference between the position just before the ball hit the baseball bat and the time it took to pass through the position 30 frames before the impact. Additionally, the rebound velocity was calculated from the difference in the time it took to pass through the baseball bat at 6 inches (152.4 mm) and 12 inches (304.8 mm) from the baseball bat [11]. The BBCOR of baseball bats was calculated using the following equations (1) [7, 8, 11].
| \begin{equation} \textit{BBCOR} = \frac{v_{i} + v_{r}}{v_{i}}\left(\frac{m}{M_{e}} + 1 \right) - 1 \end{equation} | (1) |
where m indicates the mass of the ball (kg), vi and vr indicate the absolute values of the ball’s impact and rebound velocity (m/s), and Me indicates the effective mass of the baseball bat (kg). Me is the mass that considers the rotational motion when the baseball bat is supported at a position of 6 inches (152.4 mm) from the grip end, and is expressed by the following equations [7, 8, 11].
| \begin{equation} M_{e} = \frac{I + I_{\textit{pivot}}}{Q^{2}} \end{equation} | (2) |
where I indicates the moment of inertia around the support point of the baseball bat (kg·m2), Ipivot indicates the moment of inertia of the jig supporting the baseball bat (kg·m2), and Q indicates the distance from the support point to the impact point of the baseball ball (m).
Schematic diagram of the impact experimental apparatus [8].
Twenty-one aluminum alloy baseball bats were used, including commercially available and prototype baseball bats from Japan and overseas. Figure 3 shows the used bats. In general, the thickness of the barrel of the baseball bat is almost uniform and varies locally. This study only evaluated baseball bats with a thickness that is almost uniform in the length direction of the barrel. The thickness of the metal baseball bats was measured using an ultrasonic thickness gauge (A&D Co., Ltd., AD-3255). The thickness of each baseball bat in the 0°, 45° and 90° directions was measured at 150 mm from the top of the barrel, respectively, and the average value was used as the thickness of the baseball bat. Additionally, one wooden baseball bat was subjected to impact tests for comparison. Table 1 shows the length, mass, outer diameter, and thickness of the baseball bat at 150 mm from the top of the barrel, and the moment of inertia of the baseball bat. Especially, the moment of inertia of the baseball bat is the value when supported at a position of 6 inches (152.4 mm) from the grip end. Furthermore, the ID number of the baseball bat was set in ascending order of t/D, which is the value obtained by dividing the thickness t by the outer diameter D.
Samples of the variety aluminum alloy bats and the wood bat.
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No dents, etc., indicating that the baseball bats were deformed in the elastic range during compression when the surface of the baseball bats was checked after a simple compression test. Therefore, the value represents the radial spring constant of the barrel k if the compressive load of the baseball bat is divided by the amount of deformation due to compression. Therefore, this study evaluated the compression characteristics of the baseball bats by k. Figure 4 shows the association between BBCOR and k of the baseball bats as a log-log graph. Additionally, the three baseball bats with large outer diameters (B-1–3) could not be used in the simple compression tester and compression test data could not be obtained. Therefore, the relationship was shown in eighteen baseball bats in this study. A previous study [14] lower BBCOR for baseball bats with large compressive loads, i.e., large k and high rigidity.
Relationship between BBCOR and spring constant k of metal bats.
Furthermore, BBCOR can be expressed by the power law of k, as in the previous study [14], and it can be expressed as the following relation.
| \begin{equation} \textit{BBCOR} \propto k^{-\frac{1}{6}} \end{equation} | (3) |
Previous studies on sports equipment [8, 13] often expressed power as a single-digit fraction in the power law. Therefore, this study determined the power in eq. (3) based on the result of the power approximation of all the plots in the figure.
The relationship between t/D, which is the value obtained by dividing the k and the thickness t by the outer diameter D, was evaluated. Figure 5 shows the relationship between k and t/D of the baseball bats, wherein the larger the t/D, the larger the k as a log-log graph. k, which can be expressed by the power law of t/D can be expressed in the following relationship:
| \begin{equation} k \propto \left(\frac{t}{D}\right)^{\frac{6}{5}} \end{equation} | (4) |
This may be because the baseball bat deformation as the ball hits is within the elastic range, and the longitudinal elastic modulus of the baseball bats exhibited almost no difference; thus, the k of the baseball bats depends on the shape of the baseball bat, considering the thickness and the outer diameter. The method for determining the power in eq. (4) is similar to eq. (3).
Relationship between spring constant k and thickness per diameter t/D of metal bats.
Therefore, the BBCOR of the baseball bat can be expressed by the power law of t/D from eqs. (3) and (4) as follows:
| \begin{equation} \textit{BBCOR} \propto \left(\frac{t}{D}\right)^{-\frac{1}{5}} \end{equation} | (5) |
Figure 6 shows the relationship between BBCOR and t/D. The dot-dash line in the figure indicates the line with a BBCOR of 0.50, the two-dotted line indicates the BBCOR measurement result of a wooden baseball bat. Larger the t/D, lower the BBCOR. This may be because the trampoline effect has been reduced due to the larger t/D, considering the outer diameter of the baseball bat has become thinner, or the thickness has been thickened. Therefore, BBCOR can be expressed by the power law of t/D, as shown in eq. (6) below.
| \begin{equation} \textit{BBCOR} = 0.275\left(\frac{t}{D}\right)^{-\frac{1}{5}} \end{equation} | (6) |
Effect of thickness per diameter t/D on BBCOR of bats (I: t/D = 0.056, II: t/D = 0.060).
Figure 6 shows the curve according to eq. (6) as a solid line, and the error range is presented as dashed lines. Figure 7 shows the estimated value of BBCOR obtained from eq. (6) and the measured value obtained from impact tests. The estimated value is consistent with the measured value with an error of ±2.0%, and the BBCOR of the bat can be estimated within this error range. Here, the dashed lines in Figs. 6 and 7 present not only the margin of error but also the test data variation. Therefore, the t/D with a BBCOR of ≤0.5 is ≥0.056 of point I in Fig. 6 when the impact velocity is 67 m/s.
Comparison of estimated and measured BBCOR.
The results include impact tests conducted at an impact speed of 67 m/s following ASTM. However, the NCAA, which oversees high school and college baseball in the United States, stipulates an impact velocity of 136 mph (60.8 m/s). Additionally, the results of the study of the relationship between BBCOR and the impact velocity v indicate that a relationship between BBCOR and v was presented as the following equation (7) [8].
| \begin{equation} \textit{BBCOR} \propto v^{-\frac{1}{6}} \end{equation} | (7) |
Considering these factors, correcting the BBCOR measured and calculated by the impact velocity of this study to the impact velocity of the NCAA baseball team is necessary. Therefore, consider the BBCOR at the impact velocity of 60.8 m/s from eq. (7), which is specified in the NCAA at approximately 1.6% higher than the BBCOR at 67 m/s. Therefore, the prediction line can be shown by the long dashed line in Fig. 6 if a +2.0% line of 60.8 m/s from a +2.0% line of 67 m/s is predicted. In this case, the t/D, which has a BBCOR of ≤0.5, is predicted to be ≥0.060, which is the II point in Fig. 6. Based on these predicted values, the thickness can be calculated to be approximately 3.9 mm if the diameter of the baseball bat is 64 mm following the new standard of the Japan High School Baseball Federation. Additionally, the assumed value of the thickness of the new standard can be reasonable. BBCOR can be predicted from the shape of the baseball bat by following the above methods and steps.
This study focused on a baseball bat with an almost uniform barrel thickness. Detailed investigation of the application of this method to a baseball bat with a locally changing thickness will be planned.
This study conducted simple compression and ball impact tests on baseball bats made of aluminum alloy. The effect of the outer diameter and the thickness of the barrel on the BBCOR of metal baseball bats and an estimation formula that predicts BBCOR more simply were evaluated. The results of this study include the following.
Part of the content of this paper was given at the 71st Annual Meeting held in May of 2022. In addition, various measurements and tests were conducted at the Composite Materials Laboratory, Department of Mechanical Engineering, Faculty of Science and Engineering, Hosei University, with the cooperation of students. We express our gratitude to them here.
