When Ash Meets Cowhide: The Physics of the Ball-Bat Collision Alan M When Ash Meets Cowhide: The Physics of the Ball-Bat Collision Alan M. Nathan University of Illinois at Urbana-Champaign a-nathan@uiuc.edu Introduction The simple stuff (kinematics) The interesting stuff (dynamics) Wood vs. aluminum Summary & Conclusions http://www.npl.uiuc.edu/~a-nathan/pob 1

Greatest baseball team Baseball and Physics 1927 Yankees: Greatest baseball team ever assembled 1927 Solvay Conference: Greatest physics team ever assembled MVP’s

Hitting the Baseball “...the most difficult thing to do in sports” #521, September 28, 1960 “...the most difficult thing to do in sports” --Ted Williams 90 mph ball…<0.5 s to evaluate, guess at trajectory, go through a complicated physiological process to transfer energy from your muscles to the bat, and put the bat in the right place at the right time. Very easy to fail! E.g., if off 0.01 sec in timing, ball moves 1’. Compare with golf, where ball is not even moving! BA: .344 SA: .634 OBP: .483‡ HR: 521 ‡career record

Introduction: Description of Ball-Bat Collision forces large (>8000 lbs!) time is short (<1/1000 sec!) ball compresses, stops, expands kinetic energy  potential energy bat compresses ball….ball bends bat hands don’t matter! GOAL: maximize ball exit speed vf vf  105 mph  x  400 ft x/vf = 4-5 ft/mph How to predict vf?

Kinematics: Reference Frames “Lab” Frame Bat Rest Frame vball vbat vf “Lab” Frame vrel eAvrel Bat Rest Frame vf = eA vball + (1+eA) vbat Two reasons why: 1. The “1”, which simply comes because bat is already moving in the “right” direction 2. The small value of eA All of the interesting physics is in eA. However, one should not lose sight of the fact that the single most important factor is bat speed. Still, baseball is a “”game of inches”, so even seemingly marginal things can matter in the long run. Collision is very … How to account for eA? eA  “Apparent Coefficient of Restitution” = “BESR” - 0.5 property of ball & bat weakly dependent on vrel  0.2  vf  0.2 vball + 1.2 vbat Conclusion: vbat much more important than vball

Kinematics: Conservation Laws (Accounting for eA) eAv m1 Recoil factor: related to energy loss due to rigid recoil of bat (conservation of momentum and angular momentum) COR: dissipation of energy in ball or bat Note: for massive bat, r=0 for elastic collision, e=1 for both, eA = 1 e.g. superball on massive rigid surface (demo) Generally want r small and e large to maximize eA r  bat recoil factor = mball/mbat,eff e  “Coefficient of Restitution”

Kinematics: bat recoil factor . CM b = typical numbers mball = 5.1 oz mbat = 31.5 oz k = 9.0 in b = 6.3 in r = .24 e = 0.5 eA = 0.21 + Cute trick: Can replace linear + rotational problem with purely rotational if vbat refers to impact location and mbat is b-dependent effective mass To make r small… mbat large Ibat,CM large b small but….  0.16 x 1.49  0.24 All things equal, want r small But….

mass of bat matters….but probably not a lot All things are not equal Mass & Mass Distribution affect bat speed 1. Kinematics: For given bat speed, heavy bat give higher ball speed than light bat 2. Kinematics: For given bat KE, light bat gives higher ball speed than heavy bat 3. These establish limiting curves 4. Model: fixed energy (force x distance), shared by bat and body (about equally for “normal” bat). Result: as mbat increases , Ebat increases, Ebody decreases, and vbat decreases (approx as 1/m, in agreement with “experiment”) 5. Tendency towards lighter bats. NCAA rules, etc.. Seems to contradict this analysis. Mass distribution not considered here (impt for wood-Al differences). Also control, timing, and reaction time. 6. Could do similar analysis for MOI about handle Conclusion: mass of bat matters….but probably not a lot

Kinematics: Coefficient of Restitution (e): (Energy Dissipation) “bounciness” of ball in CM frame: Ef/Ei = e2 massive rigid surface: e2 = hf/hi typically e  0.5 ~3/4 CM energy dissipated! probably depends on impact speed depends on ball and bat! Conservation of momentum not full story. Need to say something about inelasticity of collision…after all, a super ball will react differently than a baseball.

COR: Is the Ball “Juiced”? MLB: e = 0.546  0.032 @ 58 mph on massive rigid surface Variety of techniqutes used: Briggs: fired slab of wood at stationay ball; measured both rebound speeds with dual ballistic pendulum methond Lansmont BBVC: fire ball into statinoary wall up to 140 mph. Can also fire into stationary bat constrained to pivot about point on handle. UML: fire ball into stationary wall or use BHM to swing bat at projected ball (up to 70+70). Latter technique used to test bats for NCAA. 90+70 equivalent to 144 for ball on wall For ball on stationary bat:

Putting it all together….. -2 2 4 6 8 10 12 CM vf = eA vball + (1+eA) vbat 5 10 15 20 25 30 NOTE: this is just kinematics and treats the bat as a rigid body. This as far as we can go without considering the bat as a flexible dynamic object.

More Realistic Analysis CM vf = eA vball + (1+eA) vbat A preview of what is to come..and the most important result of my calculations from the point of view of the game. This the outcome of a dynamic model of the collision. I will tell you how the model is constructed and then use that model to gain some interesting insights into the collision process. One of the goals is to account for and understand the differences between the full and dashed curves.

III. Dynamics Model for Ball-Bat Colllision: Accounting for Energy Dissipation Collision excites bending vibrations in bat Ouch!! Thud!! Sometimes broken bat Energy lost  lower vf (lower e) Bat not rigid on time scale of collision What are the relevant degrees of freedom? So far, just kinematic, plus a phenomonlogoical treatment of energy losses via COR. Now we want to dissect th ecollision process, time slice by time slice, to see what is really going on during the time the ball and bat are in contact. In doing so, we want to try to do a strict accounting for where the energy goes in the collision. So, we want to go beyond kinematics and talk about dynamics. We know that a purely rigid body treatment cannot be right…for example, we know that the collision can excite vibrations in the bat. see AMN, Am. J. Phys, 68, 979 (2000)

The Essential Physics: A Toy Model ball bat Mass= 1 2 4 rigid  << 1 m on Ma (1 on 2)  >> 1 m on Ma+Mb (1 on 6) Tau determined mainly by compressibility of ball w is resonant freq of bat elastic vs. quasi-free: elastic: energy conserving 1 on 6 quasifree: energy conserving 1 on 2 The essential point: ball does not see full mass of bat…it sees only the local mass … the reduced mass increases the local recoil, with less energy going to the ball. Control parameter is w tau. So, we need to understand the frequency spectrum of vibrational modes as well as the time scale of the collision. In addition, we need to know the spatial structure of the vibrations (where they are strongly or weakly excited). flexible

A Dynamic Model of the Bat-Ball Collision 20 Euler-Bernoulli Beam Theory‡ y z Solve eigenvalue problem for free oscillations (F=0)  normal modes (yn, n) Model ball-bat force F Expand y in normal modes Solve coupled equations of motion for ball, bat Two “free” parameters…Young’s and shear modulus ‡ Note for experts: full Timoshenko (nonuniform) beam theory used

Louisville Slugger R161 (33”, 31 oz) Normal Modes of the Bat Louisville Slugger R161 (33”, 31 oz) f1 = 177 Hz f2 = 583 Hz f3 = 1179 Hz f4 = 1821 Hz nodes  prop to k2 for lowest frequencies, imples considerable dispersion Can easily be measured (modal analysis)

FFT Measurements via Modal Analysis Louisville Slugger R161 (33”, 31 oz) FFT frequency barrel node Expt Calc Expt Calc 179 177 26.5 26.6 582 583 27.8 28.2 1181 1179 29.0 29.2 1830 1821 30.0 29.9 Conclusion: free vibrations of bat can be well characterized

Model for the Ball 3-parameter problem: F=kxn F=kxm k   More complicated model not necessary. 3-parameter problem: k   n  v-dependence of  m  COR of ball with rigid surface

Putting it all together…. ball compression Procedure: specify initial conditions numerically integrate coupled equations find vf = ball speed after ball and bat separate

General Result energy in nth mode Fourier transform Only lowest 4 or so modes important for tau 0.5 ms Nodes 5-6” from end….little effect of vibrations there Conclusion: only modes with fn  < 1 strongly excited

Results: Ball Exit Speed Louisville Slugger R161 33-inch/31-oz. wood bat For rigid, peak is at CM For full calculation, peak determined by interplay between rotational recoil and vibrational nodes. At lowest node, rigid-full only lowest mode excited lowest 4 modes excited Conclusion: essential physics under control

Application to realistic conditions: (90 mph ball; 70 mph bat at 28”) CM nodes Position of max: between CM (minimize rotational recoil) and end (maximize bat speed) max near 2nd node (not at first node as commonly believed) nearly coincides with rigid body value at max (an “accident” … nodes are stacked up in 27”-30” region and higher modes not important) Energy…tradeoff between vibrations and losses in ball

The “sweet spot” 1. Maximum vf (~28”) 2. Minimum vibrational energy (~28”) 3. Node of fundamental (~27”) 4. Center of Percussion (~27”) 5. “don’t feel a thing” Velocity at handle….. 27” near node of mode 1 30” near node of mode 3 hands don’t matter…. 1. Near cop …. Collision transmits no forct to hands, vice versa 2. Pulse propagation time… no clamping effect 3. Forcethat hands could exert << ball-bat force Where is batter’s sweet spot?

Boundary conditions Conclusions: -3 -2 -1 1 2 3 0.5 1.5 y (mm) t (ms) impact at 27" Displacement at 5” Group velocity about 700 m/s Dispersions implies high frequencies arrive first hands don’t matter…. 1. Near cop …. Collision transmits no forct to hands, vice versa 2. Pulse propagation time… no clamping effect 3. Forcethat hands could exert << ball-bat force Conclusions: size, shape, boundary conditions at far end don’t matter hands don’ t matter!

T= 0-1 ms Time evolution of the bat T= 1-10 ms Discuss the collision in this context elastic wave…..evolving into rigid body motion Rigid body motion evolves very slowly on time scale of collision T= 1-10 ms

Wood versus Aluminum Kinematics Length, weight, MOI “decoupled” shell thickness, added weight fatter barrel, thinner handle Weight distribution more uniform ICM larger (less rot. recoil) Ihandle smaller (easier to swing) less mass at contact point Dynamics Stiffer for bending Less energy lost due to vibrations More compressible COReff larger This summarizes the important properties of bats

Effect of Bat on COR: Local Compression CM energy shared between ball and bat Ball inefficient:  75% dissipated Wood Bat kball/kbat ~ 0.02 80% restored eeff = 0.50-0.51 Aluminum Bat kball/kbat ~ 0.10 eeff = 0.55-0.58 Ebat/Eball  kball/kbat  xbat/ xball tennis ball/racket Tennis racket like Al bat. Al bat: possibly 10% in COR==>7 mph==>35’ or more technology of Al bats: thinner wall==>increase r bat: “tennis racket”-like efficient even for dead ball measurement techniques: Brndt-technique: measure incident ball speed and recoil bat rotational speed for ball on stationary bat BHM: measure incident and rebound ball speeds using BHM to swing bat BBVC: measure incident and rebound ball speed plus recoil bat angular speed for ball on stationary pivoted bat New NCAA restrictions >10% larger!

Dynamics of “Trampoline” Effect Wood versus Aluminum: Dynamics of “Trampoline” Effect “bell” modes: Wood: Not much one can do, given constraints imposed by rules. Aluminum: NCAA! “ping” of bat Want k small to maximize stored energy Want >>1 to minimize retained energy Conclusion: there is an optimum 

Performance Comparison Direct comparison: since Al CM closer to handle, the peak of acor is closer to handle, where bat speed is not as high. Hence, Al is not as effective under such a direct comparison Increase COR Increase bat speed

Things I would like to understand better Relationship between bat speed and bat weight and weight distribution Location of “physiological” sweet spot Better model for the ball Better understanding of trampoline effect for aluminum bat Why is softball bat different from baseball bat? Effect of “corking” the bat

Summary & Conclusions The essential physics of ball-bat collision understood bat can be well characterized ball is less well understood the “hands don’t matter” approximation is good Vibrations play important role Size, shape of bat far from impact point does not matter Sweet spot has many definitions Aluminum outperforms wood!