Physics and Acoustics of Baseball & Softball Bats
Daniel A. Russell, Ph.D.
Graduate Program in Acoustics
The Pennsylvania State University

The contents of this page are ©2003-2011 Daniel A. Russell

Today is The contents of this page were last modified on April 6, 2007

How is Bat Performance Regulated?

The Ultimate Performance Metric: Batted-Ball Speed

The only performance metric for baseball and softball bats that has any real meaning in the field of play is the Batted-Ball Speed (BBS), the speed with which a pitched ball is hit by a bat swung by an actual player. Once the collision between the bat and ball is over the flight path of a baseball or softball is determined by: The ball dimensions will be the same for every batter, though they are different for baseball and softball. The atmospheric conditions depend on the weather and the location of the ball park. The launch angle and ball spin are things that a batter can control - and which can greatly influence how far a hit ball travels - but which cannot be regulated. The Batted-Ball Speed, the speed with which the ball leaves the bat, depends on the interaction between ball and bat, and this is a quantity that can be regulated.

Field Measurements of Batted-Ball Speed

It might seem like the best way to measure batted-ball speed is to conduct field trials where actual players swing bats at pitched balls, and the batted-ball speed is measured directly in the field of play. However, while field testing does provide information about how well a particular bat performs when used for actual play, it is a very costly and difficult process to conduct a meaningful field study of bat performance. The photograph and graphs at right are from the 1997 Brown University field study of baseball bats[1,2] in which 19 players swung one of 6 bats at pitched balls from a pitching machine. The field study was conducted at an indoor batting cate to eliminate the influence of wind and weather. The study took three days and required several computers to control four infrared camera to track the motion of the bats and balls in 3-D. Apprixately a dozen people helped to set things up and to collect the data over the three day period. The actual analysis of the data required several months of computational analysis of the film footage to extract bat and ball trajectories and speeds. Such a study provided useful results for those six bats tested, but the cost (and time) needed to generate performance data for every single bat on the market would make this method of assessing bat performance completely unrealistic as a means of regulating bat performance.

Besides the cost and time required to conduct an meaningful field measurement of bat performance there is the problem of repeatability. The plots below right show data for two of the baseball bats tested in this batting cage field study[1,2]. Each data point represents one good impact between bat and ball. Immediately, one can see that not every ball is hit with the same speed. Actual field measurements of hit balls produce a wide range of hit ball speeds. There is a maximum batted-ball speed for each bat, but only a very small number of hits actually produce this maximum value. So, how do we compare bats? Do we use the maximum hit-ball speed from the field trial, the average value, or the range of values? And how many data points are necessary to have statistical confidence in the data?

Batted-ball speeds for the wood bat (orange dots) range from 70 to 101-mph, with an average of 91.4-mph. Metal bat M1 (red dots) had batted-ball speeds between 70 and 100-mph with a single hit producing 108-mph. If you ignore this single outlying point, the maximum BBS for the wood bat is actually faster than the maximum BBS for this metal bat. However, the average of all batted-ball speeds for bat M1 was 94.6-mph, shich is 3.2-mph faster than the wood bat. The blue dots for bat M2 are for a very high performance baseball bat* that produced batted-ball speeds between 70 and 107-mph with an average of 101.5-mph. Even though the maximum BBS was around 107-mph, there are many hits producing batted-ball speeds much lower than the maximum BBS for wood. Not every single hit produces the maximum possible batted-ball speed.

The point I wish to make by showing this data is that in order to accurately measure the performance of a bat in a field study, a rather large amount of data must be collected to ensure that the performance quantity (maximum BBS or average BBS) is measured with statistical confidence. The scatter in the data and the number of data points necessary to detect a clear trend is a considerable limitation of field tests as a means of measuring bat performance for regulation purposes.

*This particular metal bat is not legal under the current NCAA BESR and MOI performance standards, which were adopted in 1999. The barrel diameter is too large (2.75" instead of 2.625"), the bat is 33" long and weighs 29oz so it fails to meet the "minus three" rule, and it exceeds the maximum BESR limit for 33" bats.

Laboratory Measurements of Batted-Ball Speed

So it would appear that in order to measure (and regulate) the performance of every baseball and softball on the market we need a method of testing bats in a laboratory setting that provides repeatable and reliable data without the high cost and limitations of a field test, and without the wide range in the spread of batted-ball speeds that result from real players swinging real bats. The only thing one must require is that the results from the laboratory test must accurately represent results measured in the field.

Let's start with the assumption that Batted-Ball Speed is the desired end quantity to regulate, and see what it depends on from a physics perspective. If one starts from basic physics conservation laws (conservation of linear and angular momentum, and the conservation of energy), it is relatively simple to derive the equation for batted-ball speed to be:[3,4]

where vball is the pitched speed of the ball, vbat is the linear speed of the bat at the impact location, and eA is a term called the collision efficiency (sometimes referred to as the "apparent coefficient of restitution"). The collision efficiency eA depends on the elastic properties of both the bat and ball, as well as on the moment-of-inertia of the bat, and the location of the impact on the bat barrel. It is the most important quantity we can measure regarding bat performance.

The Collision Efficiency eA

The collision effeciency may be easily measured in a laboratory test. Basic physics theory and experimental results confirm that the collision efficiency eA has the following properties So, how is the collision effeciency measured in the laboratory? Depending on the type of laboratory test we may have balls and bats moving both before and after a collision. If we rearrange equation (1) to solve for eA we get
where I have replaced the term "BBS" with vf to represent the final velocity of the ball after the bat-ball collision, so as not to confuse it with a value of the Batted-Ball Speed in the field. Some bat performance standards measure a quantity called the Ball-Exit-Speed-Ratio (BESR) which is the ratio of the final velocity of the ball to the combined initial velocities of the bat and ball. Mathematically, the BESR is related to the collision efficiency by
so that the two are essentially the same.

In all four of the laboratory test standard protocols that are currently being used (ASA BBS, NCAA BESR, USSSA BPF1.20 and Little League BPF1.15) a ball is fired with a speed vball from a cannon towards a bat which is clamped in a pivot so that it is free to rotate after the ball hits it. The bat is initially at rest so vbat=0. This means that the collision efficienty may be simply measured as the ratio of ball speeds after and before the collision with the bat,

Once we have measured the value of eA in the lab, we can use equation (1) to predict the Batted-Ball Speed in the field. All we need is a value for the pitched-ball speed vball and a value for the bat swing speed vbat based on the bat's moment-of-inertia.

Swing Speed Formulas

In order to predict Batted-Ball Speed for a particular bat, we need to know the speed with which a typical player would swing this bat. Several comprehensive field studies for for players swinging baseball bats[5,6] and softball bats[7,8] at pitched balls have concluded that the speed with which a bat can be swung is directly related to the moment-of-inertia of the bat. The field test experimental data for bat swing speed can be fit rather well to an equation, so that if we know the moment-of-inertia of the bat we can predict the speed with which that bat can be swung. The moments-of-inertia and the resulting swing speed equations are different for baseball and softball. Rather than lay them all out here, I'll introduce the equations relative to each sport when I discuss each specific test method below.

Comparing the Three Commonly Used Performance Metrics

Considering that Batted-Ball Speed is the ultimate performance metric for baseball and softball bats, the following three methods are currently used by various governing bodies and organizations to regulate BBS in the field. Follow the links below to much more detailed descriptions of the three common bat performance standards.


[1] R.M. Greenwald R.M., L.H. Penna , and J.J. Crisco,"Differences in Batted Ball Speed with Wood and Aluminum Baseball Bats: A Batting Cage Study," J. Appl. Biomech., 17, 241-252 (2001).
[2] J.J. Crisco, R.M. Greenwald, J.D. Blume, and L.H. Penna, "Batting performance of wood and metal baseball bats," Med. Sci. Sports Exerc., 34(10), 1675-1684 (2002)
[3] A.M. Nathan, "Dynamics of the bat-ball collision," Am. J. Phys. 68, 979-990 (2000)
[4] A.M. Nathan, "Characterizing the performance of baseball bats," Am. J. Phys. 71(2), 134-143 (2003)
[5] G. Fleisig, N. Zheng, D. Stodden, and J. Andrews, "Relationship between bat mass properties and bat velocity," Sports Engineering, 5(1), 1-8 (2002).
[6] J. Clutter and K. Koenig, "The effect of baseball bat properties on swing speed," in: Proceedings of the 4th meeting of the American Society of Mechanical Engineers; Bioengineering Division, Edited by V. Goel, R. Spilker, G. Atheshian, and L. Soslowsky, Big Sky, MT, June (1999), 629-630.
[7] L. Smith, J. Broker, and A. Nathan, "A Study of Softball Player Swing Speed," in: Sports Dynamics Discovery and Application, Edited by A. Subic, P. Trivailo, and F. Alam, (RMIT University, Melbourne Australia, 2003), 12-17.
[8] K. Koenig, N. Mitchel, T. Hannigan, and J. Clutter, "The influence of moment of inertia on baseball/softball bat swing speed," Sports Engineering, 7(2), 105-118 (2004).

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