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 ©20032011 Daniel A. Russell 
Right up front I want to make something clear: The NCAA BESR bat performance standard does NOT regulate the maximum battedball speed for a bat. In fact, the BESR standard does not even measure battedball speed at all. And, for that matter, it doesn't measure the ball exit speed either. What the standard does measure is the ratio of the ball exit speed to the combined speeds of the pitched ball and swung bat. Knowledge of the BallExitSpeedRatio along with the MomentOfInertia of the bat can be used to calculate the battedball speed. In fact, the 97mph hit ball speed usually associated with the BESR standard is a calculation based on a measured BESR for a specific 34inch wood bat, an assumed bat swing speed of 66mph and an assumed pitchedball speed of 70mph. It may be a representative BBS value for wood bats, but this 97mph value does not represent the maximum hitball speed for a wood bat. In order to leave most MLB ballparks, a baseball has to be hit at speeds around 110mph, something that MLB players do with wood bats rather frequently. In the only field study of wood and aluminum bats to date, a wood bat was found to hit balls with a maximum battedball speed of 101mph, and an average battedball speed of 91.4mph. The calculated value of the battedball speed, calculated from the laboratory measurement of BESR and MOI, for this same bat was 95.0mph, which is 6mph lower than the maximum battedball speed measured in the field study.
The interpretation of the BattedBall Speed equation (3) means that a larger BESR will produce a larger BBS, assuming ball and bat speeds don't change. Faster bat speeds also result in faster battedball speeds, with a change in bat speed having a greater effect on the BBS than a similar change in pitchedball speed. This last fact is very important. Consider two bats that have the same BESR as measured in the lab, but which can be swung with significantly different speeds. The bat which can be swung faster will produce a higher battedball speed in the field if the BESR value is the same for both bats. The one important factor that BESR cannot measure is the manner in which bat weight and momentofinertia affect bat speed.
As an example, a typical 34" wood bat has a moment of inertia of 11,500 ozin^{2} measured with respect to a pivot at the 6inch point on the handle. The linear bat speed at the sweet spot is then calculated to be: CriscoGreenwald v_{bat} = 68.6 mph and Fleisig v_{bat} = 59.3 mph. For the calculations I describe below, I used an average of the two batswing equations.
The graphic shows a plot of BESR versus MOI which are the two quantities the NCAA regulates for high school and college baseball bats. The solid red dot represents a particular bat with a certain BESR and MOI. First, let's see what happens of the BESR is held constant, but the MOI is lowered. Since the BESR is the same, the collision efficiency is also the same. But, since the MOI is lower, the bat can be swung faster which means that the BattedBall Speed will increase. Data points on the left of our graph will have higher BattedBall Speeds than data points on the right side off the graph. As a second case, let's keep the MOI constant, but lower the BESR. Now the swing speed will remain constant (same MOI) but the collision efficiency will decrease due to the lower BESR. This will result in a lower BBS. Data points at the bottom of the graph will have lower BattedBall Speeds than data points at the top of the graph. And finally, if we increase the BESR and the MOI at the same time, then the collision efficiency will increase, the but bat speed will decrease. If the increase in BESR and the decrease in bat speed offset each other, then the BattedBall Speed will not change.
 
This result implies that we could draw some diagonal lines that represent constant BBS contour lines on our graph of BESR versus MOI. If the BESR and MOI change together so as to offset each other, then the location of our red dot simply moves back and forth diagonally along a constant BBS contour line. The BattedBall Speed would be the same for any combination of BESR and MOI that fall on the same contour line. But, if we change the BESR and MOI values in different amounts (or keep one the same while we change the other) then we jump to another contour line which could have a higher or lower value for BattedBall Speed.
The reason that a constant BBS contour lines exist is due to a subtle fact that the BESR itself depends on the MOI of the bat. A lower MOI bat has less effective collision with a ball, resulting in a lower value of the BESR. A bat with a larger MOI has a more effective collision with the ball and the resulting BESR is higher. So, while a lower MOI bat can be swung faster, the lower MOI renders the collision less effective. It is a relatively easy calculation to show that the two effects (higher bat speed versus lower BESR) almost exactly cancel each other so that if the elastic properties of the barrel are kept constant, changing the MOI ends up having very little effect on the final BattedBall Speed. This is precisely the reason that corked wood bats don't actually offer any real advantage to a player trying to hit a home run  the increased swing speed to due the lower MOI is offset by a less effective collision between bat and ball (corking a wood bat changes the elastic properties of the bat by a negligible amount). Experimental data for the six bats in the Brown University batting cage study^{[7]} show the same result. The variation in measured batswing speeds due variations in bat MOI was almost completely offset by a decrease in the measured value of the BESR so that the differences in performance can be entirely explained by differences in the elastic properties of the barrel (i.e., the trampoline effect).  
Now let's take the concepts of the BESR, MOI, and constant BBS contour lines and see if we can understand how the NCAA uses them to regulate the performance of nonwood baseball bats. In 1999 the NCAA tested a 34inch prostock wood bat and found that it produced a BESR of 0.728 in a test apparatus that used a ball speed of 70mph and a bat swing speed of 66mph. This BESR value was set as the upper limit for any 34inch bat, wood, aluminum, or composite. However, since the BESR standard does not account for swing speed (which depends on MOI), the NCAA also regulates the weight and MOI for nonwood bats. The NCAA has decided that a nonwood 34inch baseball bat must have a weight of at least 31oz, and must have a MOI greater than 9530 ozin^{2}, as indicated by the blue shaded region on the graph. The NCAA also tested the BESR for a 33inch and 32inch wood bat and used the results to extrapolate a sliding scale (shown at left) of BESR and MOI limits for bats of any length.
It should be noted, again, that the NCAA bat performance standard does not regulate battedball speed. Instead, it requires that nonwood bats have a ballexitspeedratio (BESR) equal to or less than the value for a high quality wood bat of the same length. And, the momentofinertia of the nonwood bat must be greater than some lower limit.  
Now, If I take the BESR and MOI values for the 34inch reference wood bat, assume a pitchedball speed of 70mph and use the batswing formulas to calculate bat speed for an impact at the sweet spot of a 34inch bat of a given MOI, then I obtain a calculated value of the battedball speed to be 97mph for this 34" wood bat. This 97mph BBS does not represent the maximum possible BBS, however. Players who can swing this bat faster will generate higher BBS, as can easily be verified with the BBS equation (3) above. In the hands of a good MLB player, this bat could easily hit balls in excess of 115mph. I can use the BESR and MOI for this reference bat to determine the BatBallCoefficientofRestitution, or BBCOR which describes the elastic properties of the batball collision. Keeping this elastic property constant, I can change the MOI, recalculate the BESR and construct a constant 97mph contour line as shown on the graph at right. Notice that there is a triangular section above and to the left of the 97mph contour line representing allowable nonwood bats that will have BattedBall Speeds higher than 97mph.
If I use the batswing formulas to calculate batswing speed for a bat with BESR=0.728 and MOI9530 ozin^{2} (at the upper left hand corner of the blue shaded region), I find that it is possible for a bat to produce a battedball speed of 102mph and still fall within the NCAA BESR and MOI limits. This does not mean that every NCAA approved nonwood bat hits balls 5mph faster than wood bats. I do not know how many (if any) NCAA approved nonwood bats come close to this BBS value  that data is not in the public domain.(See update at right) However, the performance standard currently used by the NCAA and NFHS to regulate baseball bat performance clearly allows for nonwood bats to hit balls up to 5mph faster than the high quality wood reference bat. This 5mph potential advantage for nonwood bats would appear to contradict the claims of some groups (like the Don't Take My Bat Away! Coalition) who have repeatedly gone on record claiming that today's legal metal bats generate battedball speeds exactly the same as wood. However, the allowance for a 5mph advantage for nonwood bats is fact  built into the BESR+MOI standard, and backed up by experimental data. Moreover, the NCAA is aware of this possible 5mph metal bat advantage, and they are quite happy with the current state of college baseball. For the sake of argument, though, if the NCAA wanted to regulate bat performance so that nonwood bats would hit balls with exactly the same speed as wood bats using the BESR+MOI performance standard then they would have to either^{[8]}

Update: July 8, 2008 I have just seen some proprietary experimental data that is not yet in the public domain, but at the risk of "getting in trouble" I will briefly summarize because it proves that the above charts and interpretations are not just theoretical ideas. Sixteen wood and 72 nonwood (metal and composite) 34inch baseball bats were recently certified as passing the NCAA BESR+MOI performance standard. The 16 wood bats had laboratory experimental battedball speeds ranging from 94mph to 97.5 mph. The 72 nonwood bats had laboratory experimental battedball speeds ranging from 93mph to 102.5mph. Both ranges are exactly where the charts above predict them to fall. From this I can draw two conclusions: (1) Some current nonwood bats produce the same battedball speeds (or even a little less) than wood. (2) Many current nonwood bats can hit balls up to 5mph faster than the best wood bats. Both conclusions show that the BESR+MOI standard does exactly what it was designed to do. Furthermore, this range of laboratory data also agrees with the limited experimental data from field tests as I discuss below. 
Dr. Alan Nathan has spent a considerable amount of time analyzing the raw data from this batting cage study and was able to extract BatBallCoefficientsofRestitution for each of the 538 impacts recorded. Knowing the BBCOR and MOI of a bat, it is possible to calculate the BESR and make a prediction of BattedBall Speed. In the Fall of 2005 Dr. Nathan and I arranged to have the six bats from the Brown University study tested using the ASTM F2219 high speed BESR testing protocol, and the laboratory BESR measurements agreed with the data from the field study almost perfectly. Armed with the BBCOR, BESR and MOI values for the bats we can locate the bats on the BESRMOI scale, predict the BattedBall Speed and compare to the field measurements.
Two of the bats (one wood and one metal) in the Brown University batting cage study were 34" in length, and the other four metal bats were 33" long. The blue shaded graphic at right shows the BESR and MOI limits for 34" bats along with constant BBS contours for 93mph, 97mph and 102mph. The white circle and solid black square data points represent BESR and MOI values for the 34" wood bat and the 34" metal bat M5, respectively, from the Brown University batting cage study. The batted ball speed given in parenthesis is calculated from the BBS equation (3) above using the BESR, MOI, along with the batswing speed calculated from the formulas with the bat's MOI, and an assumed pitchedball speed of 70mph.
The predicted BBS of 93.8mph for the wood bat from the Brown University study is significantly lower than the 97mph NCAA reference wood bat used to set the BESR standard, as should be expected since its BESR is much lower. It is interesting to note that the predicted BBS for the 34" metal bat from the Brown University study is only 0.7mph faster than the predicted 97 mph BBS for the NCAA 34" reference wood bat. This 34" metal bat would not be legal today, however, because it weighs 30oz and thus fails the "minus3" rule.
So, how well does the BESR standard predict the field performance of bats? Pretty well actually  as long as you remember that the BESR standard assumes a pitchedball speed of 70mph and a batswig speed of 66mph. The scatter plots at right show the measured BattedBall Speeds as a function of batswing speed (impact speed on the plots) for the 34" wood bat and 34" metal bat M5 from the Brown University study.^{[10]} Using the batswing speed formulas (discussed above) and the MOI values of the bats, the swing speeds are calculated to be 66mph and 69mph for the wood and metal bats respectively. Looking at the left scatter plot for the wood bat, a batswing speed of 66mph corresponds to a maximum BBS of 94mph. Using the measured BESR and MOI for this wood bat, I calculated a BBS of 93.8mph. Looking at the right scatter plot for metal bat M5, a batswing speed of 69mph corresponds to a maximum BBS of 98mph. Using the the measured BESR and MOI for this metal bat, I calculated a BBS of 97.7mph. I would say the BESR and MOI values and the BBS equation allows fora pretty accurate prediction of the maximum battedball speed in the field for a given swing speed. After comparing the BESRMOI graphic and the BBSvsBat Speed scatter plots, two important observations stand out to me:


The other four metal bats from the Brown University field study were 33" in length. The green shaded area in the graphic at right shows the BESR and MOI limits for 33" bats along with constant BBS contours corresponding to 97mph and 102mph. I've also plotted the BESR and MOI data point for a high quality 33" wood bat, which has the maximum allowed BESR and has a swing speed of 66mph so that the BattedBall speed (assuming a pitchedball speed of 70mph) is right about 97mph. Notice that the maximum allowed BBS for a nonwood bat that falls within the BESR and MOI limits is 102mph, exactly the same as was the case for 34" bats. The black square and circle data points represent the predicted BattedBall Speeds (calculated from experimental values of BESR and MOI) for the four 33" metal bats from the Brown University study. Notice that the highest performing bat from the Brown University study exceeds the BESR limit and so would not be legal today. Three of these 33" metal bats (square data points) also have a barrel diameter of 2.75" which is larger than the 2.625" diameter limit adopted by the NCAA in 1999, and they also fail to meet the "minus 3" rule (one is 4 and two are 5), and so none of these three bats would be legal under the current NCAA performance standards. In fact, of the five metal bats field tested in the Brown University study only one would be legal under the current NCAA performance standards  the bat indicated by the solid black circle. This legal metal bat has a predicted BBS only 0.5higher than the "woodbat" standard. How well do these BBS predictions match field measurements? The scatter plots below right show the measured field data for the highest performing metal bat M2 and the legal metal bat M1. Using MOI for bat M2, the swingspeed formulas give a typical swing speed of 69mph. The scatter plot shows that a batswing speed of 69mph corresponds to a maximum measured BBS of about 102mph. This agrees rather well with the 101.2mph prediction. For the other metal bat, M1, the MOI and swing speed formulas give a swing speed of about 67mph which corresponds to a predicted BBS of 98mph. This agrees with the predicted value of 97.5mph. As was the case for the 34" bats, the predicted BBS values  which are based on a typical swing speed  do not represent the maximum possible BattedBall Speeds (107mph for M2 and 99mph for M1). Players who can swing the bats faster can generate faster BattedBall Speeds. I can draw several conclusions from this analysis of the NCAA BESR and MOI bat performance standard:


The BESR standard, and the definition of the BESR itself, were defined with reference to a specific test apparatus that rotates a bat gripped at the handle to impact a moving ball. The rotational speed of the bat and the speed of the ball can be controlled by the operator. According to the original NCAA Certification Protocol^{[12]} established in 1999, this method was used to test bats between 32" and 35" in length. The bat is mounted onto a rotating axis by a grip centered at 57/16" from the end of the knob. The bat is rotated so that a point on the barrel 6inches from the end of the bat is traveling at a linear speed of 66±1 mph when it impacts the ball which is moving towards the bat with a linear speed of 70±2 mph. This means that the combined relative speed between bat and ball is 136 mph at the moment of impact. The ball exit speed is measured with a set of light speedgates and the ball is required to pass through the small diamond shaped hole in the wall (shown in the photo) in order to be counted as a valid hit. The BESR is first measured at the 6inch point on the barrel, then also at the 5" and 7" points to ensure that the maximum BESR value is found. An average of 5 valid measurements is recorded as the BESR result. The measured BESR value is compared to the maximum value allowed for the specific bat length, and if the measured BESR is below the accepted value the bat is considered to be BESR Certified.
All BESR testing of bats for NCAA college and high school use is conducted in the laboratory at the Baseball Research Center at the University of Massachusetts Lowell. There are only two existing copies of the bat hitting machine used to test bats to the BESR standard  the second copy is in the possession of the guy who invented it (but I'm pretty sure I heard that his hasn't worked in some time). Due to licensing restrictions imposed by the machine's inventor, this device has not been allowed to be used for research purposes. Unfortunately, this means that very little research has been conducted to verify the success of the BESR standard for shorter bats. 
Bat hitting machine used to conduct the moving bat moving ball version of the BESR test protocol. Photo courtesy of the UMassLowell Baseball Research Center. 
In 2006 the inventor of the batball hitting machine formerly used to conduct the BESR test filed a multimillion dollar lawsuit against the Baseball Research Center and the NCAA for supposedly violating the terms of their license agreement. The NCAA has since abandoned the use of this machine in favor of a highspeed cannon test identical to that used by the Amateur Softball Association to regulate the performance of softball bats.
For this new test standard protocol^{[13]} (which follows the ASTM standard F2219^{[14]}) a baseball is fired from a cannon at a speed of 138mph towards a bat which is initially at rest (v_{bat}=0), gripped in pivot that is free to rotate after the ball hits the bat. The speeds of the ball as it approaches the bat and as it rebounds from the bat are measured with a series of light gates. The BESR is calculated from equation (5) above with v_{bat}=0 so that This test method and equipment has no restrictions on its use for research purposes. Current testing is underway at the Baseball Research Center in an attempt to investigate the extension of the BESR standard to shorter length youth baseball bats. Unfortunately, one of the problems with this "ballin, ballout" method is that bats momentsofinertia below 7000 ozin^{2} tend to have extremely low collision efficiency (or BESR) so that they often don't rebound with enough speed to pass through the ball speed gates. A possible fix to this problem is to measure the bat rebound speed after the collision and to use this to determine the BESR. 
Ball cannon, light gates, and bat pivot used to conduct the highspeed "ballin ballout" version of the BESR test protocol. Photo courtesy of the WSU Sports Science Laboratory..

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