Evaluating baseball bat performancebaseball.physics.illinois.edu/evaluating_bat_performance.pdf · Evaluating baseball bat performance L. V. Smith School of Mechanical and Materials

Evaluating baseball bat performance

L. V. Smith

School of Mechanical and Materials Engineering, Washington State University, USA

AbstractSelected methodologies currently used to assess baseball bat performance wereevaluated through a series of ®nite element simulations. Results of the comparisonshow that current test methods contradict one another and do not describe theperformance advantage of modern hollow bats over solid wood bats. The discrepancywas related to the way performance was quanti®ed and the way the bat was tested.Performance metrics that do not consider a bat's mass moment of inertia (MOI) wereobserved to underestimate the hitting performance of light weight bats. A bat's centre ofpercussion was observed to be an unreliable indicator of its sweet spot (i.e. impactlocation providing the maximum hit ball speed). Bat performance was found to besensitive to the relative impact speed between the bat and ball. From these observationsthree recommendations concerning bat performance were made: (1) performanceshould be measured at relative speeds between the bat and ball that are representative ofplay conditions; (2) the ball should impact the bat at its experimentally determined sweetspot; and (3) performance should be quanti®ed from ball and bat speeds before and afterimpact. Using current test methods, an aluminium bat had a 0.9% higher performanceover wood (maximum), while using the proposed recommendations the difference was3.8% (average). The variation in the relative performance over three test conditionsreduced from � 4% using current test methods to � 0.3% when the above recommen-dations were followed.

Keywords: baseball bat, hitting performance, bat test methods

Introduction

The acceptance of nonwood bats in amateurbaseball has presented some challenges to thegame. The interest in alternative materials initiallyconcerned the expense associated with frequentwood bat failure, Crisco (1997). It was soondiscovered, however, that hollow bats hit the balldifferently than solid wood bats. This observationmotivated many bat producers to optimize theirproducts for performance rather than durability.

The increased performance has had many side-effects on the game, including safety, cost, andcompetitive balance.

There is a strong desire among amateur leaguesto limit the performance of modern bats. Whilethis idea may appear unique to baseball, it isachieved to some degree in many sports throughcontrol over ball performance, USGA (1998), ITF(2001). Since restrictions have not placed limits onthe composition the bat, the focus has been toensure a uniform performance among bats, inde-pendent of their material.

Assessing bat performance has proved to be a non-trivial task. The motion of the bat involves complexthree dimensional translation and rotation. Giventhe complexities of the bat motion and interaction

Correspondence address:L. V. Smith, School of Mechanical and Materials Engineering,Washington State University, Pullman, WA 99164±2920, USA.Tel.: 509 3353221. Fax: 509 3354662.E-mail: [email protected]

Ó 2001 Blackwell Science Ltd · Sports Engineering (2001) 4, 205±214 205

with the ball, it has been unclear how performancemight be assessed in a controlled laboratory envi-ronment that would represent response in play.

Many amateur leagues place controls on batweight since it is easily measured, understood andmost hollow bats are lighter than their solid woodcounterparts. It has been observed, however, thatthe rotational inertia of the bat is important to itsresponse, Crisco (1997). Test methods have beendeveloped to simulate bat motion and representprogress toward assessing bat performance. Thecurrent study will show, however, that current teststandards may not correctly describe the hittingperformance that may be observed in play. Thecomparison is made using a computational model,that has been veri®ed experimentally, Shenoy et al.(2001). This approach allows control over bat andball position, as well as the properties whosevariation can hinder experimental investigations.

Background

An experimental comparison of the test methodscurrently used to assess bat performance is outsidethe scope of this study. Recent advances in com-putational modelling of the ball-bat impact indicatethat such a comparison may not be necessary. It hasbeen observed, for instance, that the hitting per-formance of a bat is largely independent of itsconstraint, Nathan (2000), Smith et al. (2000). Thismay be qualitatively explained by considering therelatively short contact duration between the batand ball (�1 ms) and the load that may betransmitted to the bat from its constraints duringthis time. Constraint conditions represent a primarydifference between testing machines. If perform-ance may be considered independent of constraint,test results between machines may be compareddirectly. Bat constraint does have a large affect onthe loads transmitted to the bat after the impact iscomplete, however, and should be carefully con-sidered in bat durability studies, Axtell et al. (2000).

A comparison of test methods will typicallyconsider each test system and its interaction witha specimen. Given the small effect of constraint forthe case of bat performance, the focus of this

comparison will be directed toward the motion ofthe bat and ball, rather than entire systems (i.e. thedevice used to create the bat motion).

An explicit dynamic ®nite element model (FEM)has been constructed to simulate the impactbetween a bat and ball (using LS Dyna, version950) as shown in Fig. 1. Details of the model andits veri®cation with experiment may be foundelsewhere, Smith et al. (2000), Shenoy et al.(2001). A unique aspect of the model involves anapproach used to accommodate energy lossesassociated with elastic colliding bodies. This wasaccomplished by modelling the ball as a viscoelasticmaterial with high time dependence. The visco-elastic response of the ball was characterizedthrough quasi-static tests and high-speed rigid-wallimpacts. This approach found excellent agreementwith experiment for comparisons involving reboundspeed, contact force and contact duration. Themodel also captured the ball's speed-dependentcoef®cient of restitution, which decreased withincreasing impact speed.

The model was veri®ed by simulating a dynamicbat-testing machine involving a swinging bat anda pitched ball, Shenoy et al. (2001). The goodagreement between the model and experiment fornumerous bat and ball types, impact locations andspeeds, as well as bat strain response indicate themodel has broad application in accurately predict-ing bat performance.

With a model in place, comparisons betweencurrently used bat-performance test methods maybe conducted. Virtual testing of this type has theadvantage of complete control over the bat andball properties. These properties are dif®cult to

Figure 1 Diagram of ®nite element ball-bat impact model.

Evaluating baseball bat performance · L. V. Smith

206 Sports Engineering (2001) 4, 205±214 · Ó 2001 Blackwell Science Ltd

monitor experimentally, and can affect experimentalresults given the subtle, but signi®cant, differencesamong balls and bats. Thus, performance measures(experimental or theoretical) are most useful whenprovided on a relative basis with constant testconditions and using an accepted benchmark.

Bat properties

In the following comparisons, commercially avail-able solid-wood and hollow-aluminium bats areconsidered. Each bat had a length of 860 mm (34 in).Their mass properties were measured and may befound in Table 1. The wood bat is slightly heavierand consequently exhibits a larger MOI. While thisis typical, it should not be considered a rule. Thehollow structure of metal bats allows manipulationof their inertia in nonobvious ways. The pro®les ofthe two bats were similar, but not identical. Theeffect of bat pro®le for the normal and planarimpacts considered here was not signi®cant. Theproperties used for the ball were found fromdynamic tests of a typical collegiate certi®ed baseball.Detailsof the materialpropertiesused for the ball andbats may be found elsewhere, Shenoy et al. (2001).

Bat motion

The motion of a swinging bat, as observed in play,may be described by an axis of rotation (not ®xedrelative to the bat), its rotational speed and locationin space. The axis of rotation and its orientationmove in space as the bat is swung and its rotationalvelocity increases. Thus, three-dimensional trans-lation and rotation are required to describe themotion of a bat swung in play.

Determining a bat's hitting performance requiresonly a description of its motion during the instant

of contact with the ball. The motion over this shorttime period has been observed as nearly purerotation, with the ®xed centre of rotation locatednear the hands gripping the bat, Eggeman & Noble(1982). The exact motion of the bat will obviouslyvary from player to player. For the study at hand, a®xed centre of rotation, located 150 mm from theknob end of the bat was used, and is shown inFig. 2. The impact location with the ball, ri, ismeasured from the centre of rotation. The bat'srotational speed before impact is designated x1,while the ball's pitch speed is mp. (Pitch speed istaken here as a negative quantity to maintain aconsistent coordinate system.)

Test methods

To test a bat one must assume a representativemotion (typically rotation about a ®xed centre), abat and ball speed, a performance measure, and animpact location. From observations of amateur andprofessional players, typical bat swing speeds havebeen observed to range from 34 to 48 rad s)1,Crisco et al. (2000). Some practitioners of the gamebelieve this number should be higher, but experi-mental measurements have not shown this, Fleisiget al. (1997) and Koenig et al. (1997). Pitch speed ismore easily measured (often occurring live during agame) and may range from 20 m s)1 to 40 m s)1.Thus, in a typical game, the relative speed betweenthe point of contact on the bat and ball may vary bya factor of three. Of primary interest in batperformance studies is the maximum hit ball speed.To this end, tests are usually conducted toward thehigher end of these relative speed ranges.

Table 1 Mass properties of a solid wood and hollow metal bat

Bat Mass (g) C.G. (mm) MOI (kg m2)

Ash 906 429 0.209

Aluminium 863 418 0.198

MOI and centre of gravity (C.G.) are measured from the bat'scentre of rotation.

Figure 2 Schematic of assumed bat motion during impact with abaseball.

L. V. Smith · Evaluating baseball bat performance


Three test methods are commonly used toevaluate bat performance. The ®rst method involvespitching a ball toward a swinging bat, NCAA(1999). This is the most dif®cult test to performof the three methods and requires accuratepositioning of the bat and ball, timing of theirrelease and control of their speed. The NCAAcurrently uses this type of test to certify bats forcollegiate play.

A second method involves pitching a ball towardan initially stationary bat. This has been accepted asan ASTM standard, ASTM (2000). This test ismuch simpler to perform than the NCAA test,since bat speed and timing do not need to becontrolled. The method requires measurement ofthe bat speed after impact, however, which can bedif®cult to discern from its vibrating response.

In the current FEM bat vibration also hinderedthe determination of its after impact speed. Theproblem was avoided by consideration of amomentum balance of the ball-bat system as

Ix1 �mmpri � Ix2 �mmhri; �1�where I is the mass moment of inertia of the batabout its centre of rotation, x2 is the bat rotationalvelocity after impact, and m and mh are the mass andhit speed of the ball, respectively.

A third method for testing bats involves swingingit toward an initially stationary ball. This methodhas not yet been incorporated into a test standard,but it is often used to study bat performanceunof®cially. It shares similar advantages of simpli-city with the ASTM method over the NCAAmethod. It does require fabricating a device toswing a bat, which may be more costly than the batsupport ®xture and pitching machine used with theASTM method.

Current performance metrics

A bat's performance should be determined in a waythat allows comparison with other bats and testmethods. Three metrics are commonly used toquantify bat performance. The ®rst simply uses themeasured hit-ball speed obtained from a test

directly. While this value is of primary interest inplay, its use as a performance metric in thelaboratory has several limitations. First, it is sen-sitive to variations in the pitched ball and bat swingspeeds, so that this variation will cause scatter inthe performance metric. The second limitationconcerns the momentum of the swinging bat. If allbats are tested at the same pitch and swing speed,then the bats with greater inertia will generallyproduce higher hit-ball speeds. The opposite trendis typcially observed in play, however, where thelighter, low inertia, hollow bats typically hit the ballfurther.

The NCAA method uses what is termed a BallExit Speed Ratio (or BESR) to quantify batperformance at its experimentally determined sweetspot, r � rs. It is a ratio of the ball and bat speedsand is de®ned as

BESR � mh ÿ 1=2�x1ri � mp�x1r ÿ mp

; �2�

where ri is the impact location on the bat, and mh isthe hit ball speed. It is used to normalize the hit-ball speed with small variations that inevitablyoccur in controlling the nominal pitch and swingspeeds. It may be found from the coef®cient ofrestitution, e, as BESR � e + � (with x1 � x2)where for the ball-bat system

e � x2ri ÿ mh

mp ÿ x1ri: �3�

The assumption of constant swing speed can leadto erroneous results if bats with different MOI's arebeing compared. A lighter bat will have a slowerswing speed after impact with a ball than a heavybat. Since x2 would appear in the numerator ofEq. (2) as a negative contribution, the BESRproduces a lower measure of bat performance forlight bats than would occur if x1 ¹ x2. The BESRis nevertheless popular because it avoids theexperimentally dif®cult task of determining x2.

The performance metric used by the ASTMmethod is termed the Bat Performance Factor(or BPF) and is found at the bat's centre of percus-sion, r � q, de®ned as



q � k2o

�r�4�

where k2o is the bat's radius of gyration and �r is the

location of its centre of gravity, both in relation tothe centre of rotation, Meriam & Kraige (1997).The BPF considers initial variation in speed, themomentum of the bat and ball, as well as variationsthat may occur between balls. It is de®ned as theratio of the ball-bat coef®cient of restitution, e, andthe coef®cient of restitution of the ball used fortesting, eb, as

BPF � e

eb: �5�

While this metric accounts for the primary factorsaffecting bat performance, it should not be consid-ered as a quantity independent of test conditions.The response of the bat and ball are known to berate dependent, for instance. It would be inappro-priate therefore to compare the performance of twobats under different impact speeds with any of theseperformance metrics.

The quantitative values of the BESR and BPF formany bats are similar. This coincidence should notimply that they should be compared directly,however. In the following, bat performance willbe discussed on a relative basis as

P � Pa ÿ Pw

Pw�6�

where P is the performance measure (BESR or BPF)and the subscript a and w represent a wood oraluminium bat, respectively.

Wood and metal bats are compared in Fig. 3using the BPF for three circumstances involving thefollowing initial conditions: stationary bat (ASTM),stationary ball, and moving ball and bat. The BPFwas found at each bat's centre of percussion, q, assuggested by the ASTM standard. The speeds usedhere are near the range observed for play, and arethe same that the NCAA method uses for certi®-cation. The pitch speed is 3 m s)1 higher than theASTM recommended speed, although the ASTMmethod does have provision for `elevated speeds'.The relative performance of the wood and metal

bats is observed to vary greatly between the threetest conditions. The metal bat is expected to have ahigher performance, but this is only observed forthe moving bat and ball case. This does not implythat the wood bat has a higher hitting performance,but illustrates the effect that test conditions canhave on bat performance.

The BESR of each bat was found at its sweet spot(as recommended by the NCAA) and is shown inFig. 4. A large difference in the relative perform-ance of the wood and metal bats is again apparent,and is extreme for the initially stationary bat. Thisis a result of using a constant bat speed in theBESR, which is a poor approximation for theinitially stationary bat case. Considering the ASTMand NCAA tests from Figs 3 and 4, we haveP � )7% and P � 0.9%, respectively. This studywill show that both of these test methods under-estimate the relative hitting performance of themetal bat.

Another example illustrating the differencesbetween the BESR and BPF is given in Fig. 5 fora hypothetical metal bat as a function of wallthickness (percentage of nominal). Of signi®cancein the ®gure is the opposite effect that wallthickness (or MOI) has on the two metrics for

Figure 3 The predicted bat performance factor found at thebats' respective centre of percussions for three test initialconditions.



thin walled bats. The BESR and BPF begin toconverge for heavy bats. This may be expectedsince the assumption of x1 � x2 used in the BESRimproves as the bat MOI increases. The change inwall thickness considered here is unrealisticallylarge, primarily for demonstration purposes. Thelarge BPF with decreasing wall thickness shown in

Fig. 5, for instance, would be limited in practice bybarrel yielding, which has not been included in themodel.

Impact location

As observed by any player of the game of baseball,the hit-ball speed is dependent on its impactlocation with the bat. Many believe that a bat'ssweet spot coincides with its centre of percussion,r � q, de®ned as the impact location that minimizesthe reaction forces at its ®xed centre of rotation. Aswill be shown below, they are offset slightly.

The difference between the centre of percussionand the sweet spot may be partially explained byconsidering the momentum balance of the bat-ballimpact. Combining Eqs (1) and (3) to eliminate x2

produces an expression that may be solved for vh.

The sweet spot, r � rs, can be obtained by minim-izing this result with respect to rs, equating to zero,then solving for rs, as

rs �mp

x1�

��I

m� mp

x1

� �2s

�7�

In the above expression, mp < 0, so that rs � 0when x1 � 0. Thus, under rigid-body-motionassumptions, an initially stationary bat will impartthe highest hit-ball speed when impacted at its®xed constraint. This may be expected since aninitially stationary bat has no rotational energy toimpart to the ball, and any other impact locationwould transfer energy from the ball to the bat.Comparison of Eqs (4) and (7) demonstrates thatrs ¹ q. While rigid body motion is clearly an oversimpli®cation of the bat-ball impact, Eq. (7)suggests that the sweet spot of a bat may not be®xed but depend on the initial bat and ball speed.The computational model described above wasused to investigate the effect of a nonrigid bat'srotational speed on its sweet spot by simulatingimpacts along its length. This was done at 50-mmintervals over the range of 350 mm > r > 650 mm.This produced a curve of hit-ball speed vs. impactlocation, as shown in Fig. 6. The location of themaximum of this curve yielded rs.

Figure 4. The predicted ball exit speed ratio found at the bats'respective sweet spots for three test initial conditions.

Figure 5. Bat performance measures as a function of wallthickness of an aluminium bat.



A comparison of the sweet spots, as a function ofswing speed, was obtained from the momentumbalance, Eq. (7) and the FEM for the two bat typesdescribed in Table 1, and may be found in Fig. 7.The momentum balance clearly overestimates themagnitude of the dependence of rs on x1. Adependence is observed in the ®nite elementresults, where the sweet spot is observed to moveup to 50 mm between the stationary and swingingconditions. A similar dependence was observedwith pitch speed (but not shown here for brevity),where the sweet spot was observed to move up to35 mm between stationary and pitched ball condi-tions. The sweet spot locations from the FEM arecontrasted with the centre of percussion foundfrom Eq. (4) for the metal and wood bats, in Fig. 8.A notable observation from Fig. 8 concerns therelative locations of rs and q for the wood and metalbats. The centre of percussion for the metal bat is25 mm inside its sweet spot, while the centre ofpercussion for the wood bat lies only 2 mm insideof its sweet spot. (The centre of percussion of themetal bat is observed to be 10 mm inside that ofwood, a trend that was consistent from a group of12 bats including bats of solid wood, and hollowmetal and composite construction.) Thus, tests

which consider impacts of bats at r � q mayprovide an unfair comparison, and do not representtheir relative hit-ball speed potential. The utility ofusing rigid body dynamics to determine an appro-priate impact location for dynamic bat testingappears dubious. Similar observations could be

Figure 6 Hit ball speed as a function of impact location for awood and aluminium bat.

Figure 7 Effect of bat rotational speed on the location of itsswet spot.

Figure 8 Comparison of the location of the sweet spot andcentre of percussion for a wood and aluminium bat. (Sweet spotfound using mp � 31 m s)1 and x � 53 rad s)1.)



made for tests that consider a ®xed impact location,independent of bat composition.

Improved performance measure

While the contradictions between current testmethodologies are unfortunate, the following willshow that they may be improved and simpli®ed toprovide a more accurate description of bat per-formance. In selecting a numerical quantity todescribe bat performance, the results of Fig. 5 showthat the effect of bat MOI can be important and isdescribed by the BPF.

The BPF was modi®ed by computing it ateach bat's sweet spot as shown in Fig. 9. Thevariations between the test conditions are observedto decrease, where for two of the three conditionsP � 3%. For the third condition, however,P � )6%. The dependence of bat performance onthe test condition is obviously undesirable and isaddressed below.

As observed in Fig. 5, a hollow bat's hittingperformance increases with decreasing wall thick-ness. It is possible that this phenomenon is relatedto an increased radial barrel deformation thatwould occur with a thinner wall. This may beexplained in terms of ball deformation and impulse.

Impact studies of various types of baseballs haveshown its coef®cient of restitution to decreasewith increasing impact velocity (or deformation).Impacts involving a deforming hollow barrel wouldreduce the amount of ball deformation, and poten-tially lengthen the duration of contact between the

Figure 9 The predicted bat performance factor found at thebats' respective sweet spots for three test initial conditions.

Figure 10 The predicted BPF found at the bats' respective sweetspots using a constant relative impact speed between the balland the bat for three test initial conditions.

Figure 11 The predicted BESR found at the bats' respectivesweet spots using a constant relative impact speed between theball and the bat for three test initial conditions.



ball and bat. Thus, hollow bats have the potentialof reduced energy loss during ball impact andimparting greater impulse to the ball.

From the foregoing, and in the context of batevaluation, it is desirable to compare bats underimpact conditions that are similar to play condi-tions. In the comparisons of Figs 3, 4, and 9, therelative velocity of the bat and ball varied between30 m s)1 and 60 m s)1. A constant relative speed of60 m s)1 (comparable to play conditions) may beaccomplished by increasing the pitch and swingspeeds of the initially stationary bat and ball tests,respectively. This provides a similar impact forceon the bat for all three conditions, and reducesperformance variation due to barrel deformation.The results of impacts at the bat's sweet spot areshown in Figs 10 and 11, for the BPF and BESR,respectively. The magnitude of the BPF and BESRcontinue to depend on the test condition. Thevariation in the relative difference between thewood and metal bats for the three conditions isreduced considerably, however (P � 3.8 � 0.3%and P � 1.0 � 2.3% for the BPF and BESR,respectively). As was noted earlier, the performancedifference between the metal and wood bats islower and the variation is higher using the BESRthan that found with the BPF.

Conclusions

A computational model has been presented andused to evaluate test methods from industry, theNCAA and the ASTM in assessing bat perform-ance. A comparison between a representative woodand aluminium bat shows that current standard testmethodologies may not accurately represent a bat'shitting performance in play. Recommendations forimproving bat assessment include accounting forthe bat's inertia to quantify bat performance,evaluating a bat at its experimentally determinedsweet spot (as done by the NCAA) and usingrelative impact speeds between the bat and ball thatare representative of play conditions. By followingthese recommendations it was shown that therelative performance of a metal bat over woodwas nearly four times that found using current test

methods. The variation in performance over threetest conditions using the above recommendationswas 13% of that occurring from current testmethods. These results also suggest that batperformance can be accurately determined using asimpli®ed test involving either an initially station-ary bat or ball.

Acknowledgements

This work has been funded by the WashingtonState Technology Center and The Brett BrothersBat Company. Their support is gratefullyacknowledged.

References

ASTM F1881±98 (2000) Standard Test Method for MeasuringBaseball Bat Performance Factor, ASTM, WestConshohocken, PA.

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Crisco, J.J. (1997) NCAA Research Program on Bat andBall Performance, Final Report, Providence, R.I.,pp. 6±24, Brown University, Providence, RI.

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Koenig, K. et al. (1997) Inertial Effects on Baseball BatSwing Speed. In NCAA Research Program on Bat andBall Performance, Final Report, Providence, R.I,pp. 88±100.

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Nathan, A.M. (2000) Dynamics of the Baseball-Bat Colli-sion. American Journal of Physics, 68 (11), 979±990.

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Evaluating baseball bat performancebaseball.physics.illinois.edu/evaluating_bat_performance.pdf · Evaluating baseball bat performance L. V. Smith School of Mechanical and Materials