University of Iowa Colloquium, October 12, 2000 Page 1 How Does a Baseball Bat Work? The Dynamics of the Ball-Bat Collision Alan M. Nathan University of Illinois University of Iowa Colloquium October 12, 2000 l Introduction l Rigid body treatment of collision l Dynamic model of collision l Wood vs. aluminum l Summary

University of Iowa Colloquium, October 12, 2000 Page 2 REFERENCES l The Physics of Baseball, Robert K. Adair (Harper Collins, New York, 1990), ISBN l The Physics of Sports, Angelo Armenti (American Institute of Physics, New York, 1992), ISBN l l H. Brody, AJP 54, 640 (1986); AJP 58, 756 (1990) l P. Kirkpatrick, AJP 31, 606 (1963) l L. Van Zandt, AJP 60, 172 (1992) l R. Cross, AJP 66, 772 (1998); AJP 67, 692 (1999) l AMN, AJP 68, to appear in November, 2000

University of Iowa Colloquium, October 12, 2000 Page 3 Is This Heaven?

University of Iowa Colloquium, October 12, 2000 Page 4 Baseball and Physics: Murderers Rows 1927 N. Y. Yankees 1927 Solvay Conf.

University of Iowa Colloquium, October 12, 2000 Page 5 A Philosophical Note: “…the physics of baseball is not the clean, well-defined physics of fundamental matters but the ill-defined physics of the complex world in which we live, where elements are not ideally simple and the physicist must make best judgments on matters that are not simply calculable…Hence conclusions about the physics of baseball must depend on approximations and estimates….But estimates are part of the physicist’s repertoire…a competent physicist should be able to estimate anything...” “The physicist’s model of the game must fit the game.” “Our aim is not to reform baseball but to understand it.” --- Bob Adair in “The Physics of Baseball”, May, 1995 issue of Physics Today

University of Iowa Colloquium, October 12, 2000 Page 6 Hitting the Baseball “...the most difficult thing to do in sports” --Ted Williams BA:.344 SA:.634 OBP:.483 HR: 521 #521, September 28, 1960

University of Iowa Colloquium, October 12, 2000 Page 7 Here’s Why….. (Courtesy of Robert K. Adair)

University of Iowa Colloquium, October 12, 2000 Page 8 Description of Ball-Bat Collision l forces large (>8000 lbs!) l time is short (<1/1000 sec!) l ball compresses, stops, expands l kinetic energy  potential energy l bat affects ball….ball affects bat l hands don’t matter! l GOAL: maximize ball exit speed v f v f  105 mph  x  400 ft  x/  v f = 4-5 ft/mph What aspects of collision lead to large v f ?

University of Iowa Colloquium, October 12, 2000 Page 9 l What happens when ball and bat collide? YThe simple stuff: kinematics c frames of reference c conservation of momentum c conservation of angular momentum YThe really interesting stuff: energy dissipation c compression/expansion of ball c vibrations of the bat How to maximize v f ?

University of Iowa Colloquium, October 12, 2000 Page 10 Expect A weakly dependent on impact speed NCAA: * Bat Exit Speed Ratio (BESR) = A+0.5 * BESR <  A < For typical bat… v ball,f = 0.2 v ball,i v bat,i Kinematics: Frames of Reference Conclusion: v bat much more important than v ball Question: what bat/ball properties make BESR large?

University of Iowa Colloquium, October 12, 2000 Page 11 Kinematics: Conservation Laws v ball,f = 0.2 v ball,i v bat,i e  Coefficient of Restitution  0.5 r  recoil factor  0.24

University of Iowa Colloquium, October 12, 2000 Page 12 Energy in Bat Recoil. Translation. Rotation CM. z Important Bat Parameters: m bat, x CM, I CM =m bat k 2 CM Conclusion: All things being equal, want m bat, I bat large 0.17 ( ) = 0.24 Want r small to mimimize recoil energy

University of Iowa Colloquium, October 12, 2000 Page 13 But… l All things are not equal l Mass & Mass Distribution affect bat speed Conclusion: mass of bat matters….but probably not a lot see Watts & Bahill, Keep Your Eye on the Ball, 2nd edition, ISBN bat speed vs mass ball speed vs mass

University of Iowa Colloquium, October 12, 2000 Page 14 in CM frame: E f /E i = e 2 massive rigid surface: e 2 = h f /h i typically e  0.5 ~3/4 CM energy dissipated! depends on ball, surface, speed,... is the ball “juiced”? Energy Dissipated: Coefficient of Restitution (e): “bounciness” of ball

University of Iowa Colloquium, October 12, 2000 Page 15 Major League Baseball receives report on quality of baseballs Study finds no significant performance differences between 1999 and 2000 baseballs Posted on June 28, 2000 Major League Baseball has received the results of a study conducted by the UMass-Lowell Baseball Research Center regarding the performance of the baseballs used in Major League games, it was announced today. The study, in which 1999 and 2000 Major League baseballs and 2000 Minor League baseballs were tested for performance comparisons and specification compliance, revealed no significant performance differences and verified that the baseballs used in Major League games meet performance specifications. In all, Rawlings and Major League Baseball provided 192 baseballs to the research center for testing.

University of Iowa Colloquium, October 12, 2000 Page 16 COR and the “Juiced Ball” Issue MLB: e =  58 mph on massive rigid surface

University of Iowa Colloquium, October 12, 2000 Page 17 l CM energy shared between ball and bat l Ball is inefficient:  75% dissipated l Wood Bat Yk ball /k bat ~ 0.02 Y  80% restored Ye eff = l Aluminum Bat Yk ball /k bat ~ 0.10 Y  80% restored Ye eff = c“trampoline effect” l Bat Proficiency Factor  e eff /e Effect of Bat on COR: Local Compression E bat /E ball  k ball /k bat   x bat /  x ball >10% larger! tennis ball/racket Recent BPF data: (Lansmont BBVC/Trey Crisco)  0.99 wood  1.12 aluminum More later on wood vs. aluminum

University of Iowa Colloquium, October 12, 2000 Page 18 l Collision excites bending vibrations in bat YOuch!! Thud!! YSometimes broken bat YEnergy lost  lower v f l Bat not rigid on time scale of collision l What are the relevant degrees of freedom? Beyond the Rigid Approximation: A Dynamic Model for the Bat-Ball collision see AMN, Am. J. Phys, 68, in press (2000)

University of Iowa Colloquium, October 12, 2000 Page 19  >> 1 m on M a +M b (1 on 6) ball bat  << 1 m on M a (1 on 2) Bat not rigid on time scale of collision rigid bat The Essential Physics: A Toy Model Mass= 1 2 4

University of Iowa Colloquium, October 12, 2000 Page y z y A Dynamic Model of the Bat-Ball Collision Solve eigenvalue problem for free oscillations (F=0)  normal modes (y n,  n ) Model ball-bat force F Expand y in normal modes Solve coupled equations of motion for ball, bat ‡ Note for experts: full Timoshenko (nonuniform) beam theory used Euler-Bernoulli Beam Theory ‡

University of Iowa Colloquium, October 12, 2000 Page 21 Normal Modes of the Bat Louisville Slugger R161 (33”, 31 oz) Can easily be measured (modal analysis) f 1 = 177 Hz f 2 = 583 Hz f 3 = 1179 Hz f 4 = 1821 Hz nodes

University of Iowa Colloquium, October 12, 2000 Page 22 frequency barrel node Expt Calc Measurements via Modal Analysis Louisville Slugger R161 (33”, 31 oz) Conclusion: free vibrations of bat can be well characterized

University of Iowa Colloquium, October 12, 2000 Page 23 Model for the Ball 3-parameter problem: k   n  v-dependence of  m  COR F=kx n F=kx m

University of Iowa Colloquium, October 12, 2000 Page 24 Putting it all together…. impact pointball compression Procedure: specify initial conditions numerically integrate coupled equations find v f = ball speed after ball and bat separate

University of Iowa Colloquium, October 12, 2000 Page 25 Conclusion: only modes with f n  < 1 strongly excited General Result force normalized to unit impulse energy in n th mode Fourier transform

University of Iowa Colloquium, October 12, 2000 Page 26 collision time  2.2 ms  only lowest mode excited Comparison with Experiment 1. Low-speed collision

University of Iowa Colloquium, October 12, 2000 Page 27 Comparison with Experiment 2. High-speed collisions collision time  0.65 ms e eff /e distance from barrel (m) Batting Cage Data: Crisco/Greenwald calculation Conclusion: essential physics under control

University of Iowa Colloquium, October 12, 2000 Page 28 CMnodes Application to realistic conditions: 90 mph ball; 70 mph bat at 28”

University of Iowa Colloquium, October 12, 2000 Page Maximum v f (~28”) 2. Minimum vibrational energy (~28”) 3. Node of fundamental (~27”) 4. Center of Percussion (~27”) 5. “don’t feel a thing” Insights into collision process: 1. The “sweet spot”

University of Iowa Colloquium, October 12, 2000 Page 30 nodes Conclusions: size, shape, boundary conditions at far end don’t matter hands don’ t matter! Insights into collision process: 2. The effect of hands

University of Iowa Colloquium, October 12, 2000 Page 31 Insights into collision process: 3. Time evolution of bat

University of Iowa Colloquium, October 12, 2000 Page 32 Wood versus Aluminum: 1. General Considerations Length and weight “decoupled” * Can adjust shell thickness * Fatter barrel, thinner handle Weight distribution more uniform * Easier to swing * Less rotational recoil * More forgiving on inside pitches Stiffer for bending * Less energy lost due to vibrations More compressible * COR larger

University of Iowa Colloquium, October 12, 2000 Page 33 Wood versus Aluminum: 2. More Realistic Comparisons a. direct comparision b. 9% larger COR c. 8% higher bat speed

University of Iowa Colloquium, October 12, 2000 Page 34 Wood versus Aluminum: 3. Dynamics of “Trampoline” Effect “bell” modes: “ping” of bat Want k small to maximize stored energy Want  >>1 to minimize retained energy Conclusion: there is an optimum 

University of Iowa Colloquium, October 12, 2000 Page 35 Things I would like to understand better l Relationship between bat speed and bat weight and weight distribution l Location of “physiological” sweet spot l Better model for the ball l Better understanding of trampoline effect for aluminum bat l Why is softball bat different from baseball bat? l Effect of “corking” the bat

University of Iowa Colloquium, October 12, 2000 Page 36 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