EFFECT OF BALL PROPERTIES ON THE BALL-BAT COEFFICIENT OF RESTITUTION

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1 EFFECT OF BALL PROPERTIES ON THE BALL-BAT COEFFICIENT OF RESTITUTION A. M. NATHAN 1 AND L. V. SMITH 2 1 Univrsity of Illinois, 1110 W. Grn Strt, Urbana, IL 61801, USA, a-nathan@illinois.du 2 Washington Stat Univrsity, P.O Box , Pullman, WA 99164, USA, lvsmith@wsu.du A thortical modl is prsntd that rlats th ball-bat cofficint of rstitution to th ball cofficint of rstitution 0 and dynamic stiffnss k. Th modl is usd to dvlop a tchniqu to normaliz to valus of 0 and k for a standard ball. Th fficacy of this normalization tchniqu is dmonstratd by comparison with xprimntal data. It is shown to b vastly suprior to a widly usd tchniqu that is basd on th physically unjustifid assumption that th ratio / 0, commonly rfrrd to as th Bat Prformanc Factor or BPF, is indpndnt of both 0 and k. 1 Introduction In rcnt yars, an ffort has bn undr way to masur and rgulat th prformanc of nonwood basball and softball bats. Th masurmnt tchniqu involvs projcting a ball from a high-spd cannon onto a stationary bat and masuring th spd of th ball both bfor and aftr th collision. From ths masurmnts, a valu can b drivd for th ball-bat cofficint of rstitution (COR), which is a masur of nrgy dissipation in th ball-bat systm. If is to b a maningful mtric of bat prformanc, it is ncssary to control th proprtis of th balls usd to masur it. On such ball proprty is 0, th COR of th ball whn colliding with a rigid objct, which dtrmins th fraction of comprssional nrgy stord in th ball that is rturnd as kintic nrgy. A scond ball proprty is k, th ffctiv spring constant or dynamic stiffnss of th ball. For a givn bat, th ball stiffnss controls how th initial nrgy is partitiond btwn comprssional nrgy stord in th ball and that stord in th bat. Th largr th ball stiffnss, th lss comprssional nrgy is stord in th ball, lading to lss ovrall nrgy dissipation and largr. This phnomnon is popularly known as th trampolin ffct. Basd on ths gnral idas, a highly-simplifid thortical modl is constructd that dscribs th dpndnc of on 0 and k. This modl is usd to dvlop a tchniqu to normaliz to valus of 0S and k S for a standard ball. Th normalization tchniqu is tstd by applying it to xprimntal data takn at th bat tsting facility at th Sports Scincs Laboratory at Washington Stat Univrsity. Whil not prfct, th tchniqu is shown to b vastly suprior to anothr widly usd tchniqu. 1

2 2 2 Thortical Considratons 2.1. Toy Modl for th Ball-Bat Collision Th starting point is a two-spring modl for th ball-bat collision, Fig. 1, which was prviously dvlopd by Cross as a modl for th trampolin ffct in th intraction of tnnis balls with th rackt strings (Cross, 2000). In this modl, th ball and bat ar ach rprsntd as masss on linar lossy springs, with forc constants k 0 and k 1, rspctivly. W hraftr rfr to k 0 as th dynamic stiffnss" of th ball. Th two springs mutually comprss ach othr, convrting th initial cntr-of-mass (CM) kintic nrgy ntirly into comprssional potntial nrgy. Figur 1. Simplifid physical modl for th ball-bat collision. Th fundamntal quation for th nrgy dissipatd in th collision is as follows: 1-2 = (1-0 2 )f 0 + (1-1 2 )f 1, (1) whr f 0 and f 1 ar th fraction of th initial CM nrgy stord in th ball and bat, rspctivly; th quantitis (1-0 2 ) and (1-1 2 ) ar th fraction of stord nrgy that is dissipatd in th ball or bat; and (1-2 ) is th fraction of total CM nrgy that is dissipatd in th collision. For linar springs, f 0 =k 1 /(k 1 +k 0 ) and f 1 =k 0 /(k 1 +k 0 ). Dfining r k 1 /k 0, which is th ratio of nrgy stord in th ball to that stord in th bat, Eq. 1 can b rarrangd to obtain: r + = r. (2) Assuming no losss in th bat (i.., 1 =1), a rasonabl assumption for impacts nar th swt spot of th bat, thn Eq. 2 can b rwrittn to obtain Cross s rsult (Cross, 2000):

3 2 2 r0 + 1 = (3) 1 + r Eq. 3 is th basis for our normalization procdur. A plot of vs. r is shown in Fig. 2(a) for svral diffrnt valus of 0. Th limiting cass hav simpl physical intrprtations. For r>>1, ssntially all th CM nrgy is stord in th ball, non in th bat, and approachs 0, th valu for th ball alon, ssntially indpndnt of r. This rgim is typical of wood bats and low-prforming hollow bats. In th opposit limit, r<<1, vry littl nrgy is stord in th ball, so that approachs 1 (or 1 ) indpndnt of 0. In th intrmdiat rang, is gnrally largr than 0, as som of th nrgy that might hav bn stord and mostly dissipatd in th ball is instad stord in th bat. For modrn hollow mtal or composit bats, r is gnrally in th rang 2-15, a rang in which dpnds on th two ball proprtis, 0 and k. 3 Figur 2. (a) Plot of vs. r (Eq. 3) for thr valus of 0. (b) Plot of th ratio / 0, commonly calld th BPF, vs. r for thr valus of 0, dmonstrating that th BPF is not indpndnt of ithr ball COR or dynamic stiffnss. For th non-wood bat studid xprimntally, 2.2 r Normalizing to a Standard Ball Suppos a ball of known COR 0 and dynamic stiffnss k is usd to masur th ball-bat COR for a particular bat, obtaining. Givn that information, a tchniqu is sought to prdict th ball-bat COR S whn th sam bat is tstd with a standard or normalizing ball S with COR 0S and dynamic stiffnss k S. In th contxt of th two-spring modl, an xact procdur can b obtaind via Eq. 3. Aftr som algbraic manipulation, our proposd normalization prscription is obtaind: two-spring modl: S = k(1 ) + k ( ) k(1 ) + k ( ) S S S 0. (4)

4 4 A diffrnt normalization procdur (Brandt, 1997) is widly usd and is basd on th assumption that th ratio / 0, commonly known as th Bat Prformanc Factor or BPF, is a proprty of th bat alon and indpndnt of both 0 and k. Th BPF normalization is givn by th formula BPF: S 0S =. (5) 0 Howvr, th BPF assumption is not in gnral consistnt with th two-spring modl. Indd, a carful inspction of Eq. 4 or Fig. 2(b) shows that that / 0 is indpndnt of 0 and k only in th limit r>>1, i.., only for wood or low-prforming hollow bats. 3 Exprimnt and Rsults Th bat and ball tsting facility at th Sports Scinc Laboratory at Washington Stat Univrsity (Smith & Cruz, 2008; Smith, 2008) was usd to study th dpndnc of on th ball proprtis 0 and k, with th spcific goal of tsting th normalization procdur of Eq. 4. Th masurmnts consistd of firing a softball from an air cannon at 110±1 mph onto a stationary bat and masuring th incoming and rbound spd of th ball, from which th ball-bat COR is drivd using standard formulas (Nathan, 2003). Th masurmnts utilizd 78 diffrnt standard softballs, whos COR and dynamic stiffnss wr dtrmind in supplmntal xprimnts (ASTM WK8910) and rangd from and ,000 lb/inch, rspctivly. Th balls wr dividd into groups of six, with balls in ach group having narly th sam valu of 0 and k. Th primary bat studid was a high-prforming non-wood bat (Louisvill Sluggr Catalyst, 34 inchs long, 26.5 oz). As w will discuss shortly, th r valus for this bat and th balls usd wr in th rang From Fig. 2, w s that in this rgim th ball-bat COR is a much strongr function of k than of 0, whras th BPF is a strong function of both k and 0. This bat should thrfor b particularly usful in distinguishing btwn th two normalization tchniqus. Th impact location was fixd at 6.5 inchs from th barrl tip. Additional data wr takn on a wood bat (Brtt Brothrs Pro-Modl 110, 33 inchs long, 29 oz), for which th ball-bat COR is xpctd to b indpndnt of k. Aftr normalizing th wood bat COR to 0 using Eq. 5, w confirm our xpctations by finding th COR to b indpndnt of k. Th root-man-squar (rms) scattr of th normalizd valus about th man is 0.005, which w tak as an indication of th ovrall prcision of our COR masurmnts. Th rsults of our study for th non-wood bat ar prsntd in Fig. 3, whr th plottd valus ar avrags ovr th six balls in ach group. Fig. 3(a) shows th dpndnc of and S on k for balls with 0.36< 0 <0.37, whr S is calculatd using Eq. 4, with normalizing valus 0S =0.36 and k S =6700 lb/inch. Th rsults show that has a narly linar dpndnc on k with a slop of pr 1000 lb/in. Th slop is

5 considrably rducd to pr 1000 lb/in by normalization, an improvmnt by a factor of thr. Idally th normalizd slop would b zro, so thr is som additional dpndnc of on k that is not accountd for by th two-spring modl. Fig. 3(b) shows th dpndnc on 0 for balls with 6500<k<7000 lb/in. Th two-spring normalization rmovs ssntially all th dpndnc on 0, rducing th slop of a linar fit by a factor of fiv. On th othr hand, Eq. 5 ovrcorrcts for 0, rsulting in a slop largr in magnitud and opposit in sign compard to th uncorrctd data. Th scattr plot in Fig. 3(c) of all th data shows that th larg sprad in unnormalizd valus of is rducd considrably whn th prscription of Eq. 4 is usd to normaliz. By comparison, th BPF normalization tchniqu, Eq. 5, shows a sprad comparabl to th unnormalizd valus. Using Eq. 3, w stimat that th bat stiffnss is approximatly 22,000 lb/inch, so that r falls in th rang That th BPF tchniqu works so poorly can b asily undrstood from Fig. 2(b), givn th rang of r. Indd, th xprimntal BPF valus ar far from constant, ranging from and for th data in Fig. 3(a) and 3(b), rspctivly. 5 Figur 3. Rsults for th ball-bat COR, with unnormalizd valus in blu and normalizd valus using Eq. 4 or 5 in rd and black, rspctivly. Th dottd lins ar linar fits to th data. (a) vs. k for approximatly constant 0 ; (b) vs. 0, for approximatly constant k; (c) Scattr plot of all th data, whr th horizontal lin is th man valu for ach and th box contains all but th uppr and lowr 5% of th points. Closly spacd points hav bn displacd horizontally for clarity.

6 6 4 A Usful Approximation For low-prformanc bats, i.., thos with not much largr than 0, a usful approximation to Eq. 4 can b drivd, taking advantag of th fact that r>>1, so that should b narly indpndnt of k: + (6) S 0S To dmonstrat th ffctivnss of th approximation, considr a bat with =0.54 whn masurd with a ball of 0 =0.52. W normaliz to a ball of th sam k and 0S =0.50, obtaining and using Eqs. 4 and 6, rspctivly, a diffrnc of only 0.25%. 5 Summary W hav prsntd a modl of th ball-bat collision that xplicitly dmonstrats th dpndnc of th ball-bat COR on th COR 0 and dynamic stiffnss k of th ball. W hav usd this modl to dvlop a tchniqu for normalizing to proprtis of a standard ball. W hav tstd th modl with a high-prformanc softball bat for which thr is a strong narly linar dpndnc of on k and hav shown that th normalization tchniqu, whil not prfct, rducs that dpndnc by about a factor of thr. W hav furthr shown that th dpndnc of on 0 is rmovd by th normalization. W hav shown xprimntally that th ratio / 0, known as th BPF, dpnds on both 0 and k, as prdictd by th two-spring modl. Thrfor it is not surprising that th BPF normalization mthod fails for th non-wood bat tstd. Finally w hav drivd an approximat normalization xprssion which is valid for low-prforming bats. 0 Rfrncs Cross, R. (1993) Th cofficint of rstitution for collisions of happy balls, unhappy balls, and tnnis balls, Amrical Journal of Physics, 68, Brandt, R. (1997) Mthod and apparatus for dtrmining th prformanc of sports bats and similar quipmnt. Unitd Stats Patnt US , publishd 30 Spt. Smith, L. and Cruz C. (2008) Idntifying altrd softball bats and thir ffct on prformanc, Sports Tchnology, 1/4-5, Smith, L. (2008) Progrss in Masuring th Prformanc of Basball and Softball Bats, Sports Tchnology, 1/6, Nathan, A. (2003) Charactrizing th prformanc of basball bats, Amrical Journal of Physics, 71, ASTM WK8910. Nw Standard Tst Masurmnt for Masuring th Dynamic Stiffnss (DS) of Basballs and Softballs.

EXST Regression Techniques Page 1

EXST Regression Techniques Page 1 EXST704 - Rgrssion Tchniqus Pag 1 Masurmnt rrors in X W hav assumd that all variation is in Y. Masurmnt rror in this variabl will not ffct th rsults, as long as thy ar uncorrlatd and unbiasd, sinc thy