1. University of Kentucky April 25, 2003 Page 1 When Ash Meets Cowhide: The Physics of the Baseball-Bat Collision Colloquium, U. of Kentucky, April 25, 2003 Alan M. Nathan University of Illinois at Urbana-Champaign a-nathan@uiuc.edu http://www.npl.uiuc.edu/~a-nathan/pob “...the most difficult thing in sports” --Ted Williams: 1918-2002 introduction kinematics dynamics

2. University of Kentucky April 25, 2003 Page 2 1927 Solvay Conference: Greatest physics team ever assembled Baseball and Physics 1927 Yankees: Greatest baseball team ever assembled MVP’s

3. University of Kentucky April 25, 2003 Page 3 Description of Ball-Bat Collision l forces large (>8000 lbs!) l time short (<1/1000 sec!) l ball compresses, stops, expands  kinetic energy  potential energy  lots of energy dissipated l bat is flexible  bat bends, compresses l the goal...  large hit ball speed

4. University of Kentucky April 25, 2003 Page 4 ©Champaign News-Gazette, April 19, 2003

5. University of Kentucky April 25, 2003 Page 5 e A  “Collision Efficiency”: -1  e A  +1 property of ball & bat: frame-independent can measure, then use to predict v f depends weakly on v rel, strongly on impact location near “sweet spot”: e A  0.2 v f  0.2 v ball + 1.2 v bat The Ball-Bat Collision: Kinematics v f = e A v ball + (1+e A ) v bat v ball v bat vfvf “Lab” Frame v rel e A v rel Bat Rest Frame

6. University of Kentucky April 25, 2003 Page 6 What Does e A Depend On? Kinematics: recoil of bat (r) Dynamics: energy dissipation (e) Small r is best r  0.25 typical…depends on …. mass of bat mass distribution of bat impact location.. CM. b = + Heavier bat is better but….

7. University of Kentucky April 25, 2003 Page 7 Recent ASA Slow-Pitch Softball Field Tests ( L. V. Smith, J. Broker, AMN) Conclusion: bat speed more a function of mass distribution than mass fixed M fixed MOI

8. University of Kentucky April 25, 2003 Page 8 COR and Energy Dissipation (primary focus of this talk) l e  COR  v rel,after /v rel,before l in CM frame: (final KE/initial KE) = e 2  baseball on hard floor: e 2 = h f /h i  0.25 l typically e  0.5  ~3/4 CM energy dissipated! l depends (weakly) on v l the bat matters too!  vibrations   “trampoline” effect

9. University of Kentucky April 25, 2003 Page 9 Aside: Wood-Aluminum Differences l Inertial differences  CM closer to hands, further from barrel for aluminum  M bat,eff smaller  *larger recoil factor r, smaller e A *effectively, less mass near impact location  MOI knob smaller  swing speed higher  cancels for many bats l Dynamic differences  Ball-Bat COR significantly larger for aluminum

10. University of Kentucky April 25, 2003 Page 10 Aside: Regulating Bat Performance l ASTM:  Regulate ball-bat COR  Recoil factor approximately cancelled by v bat l NCAA:  Regulate e A to be “wood-like”  But aluminum can be swung faster * Supplemental W and MOI lower limits l ASA:  Regulate v f based on e A measurements and model for v bat

11. University of Kentucky April 25, 2003 Page 11 l Bat is flexible on short time scale  Collision excites vibrations  Vibrations reduce COR l Vibrations reduced if  Impact is at a node (“sweet spot”)  Collision time (~0.6 ms) >> T vib see AMN, Am. J. Phys, 68, 979 (2000) Accounting for Energy Dissipation: Dynamic Model for Ball-Bat Colllision

12. University of Kentucky April 25, 2003 Page 12 ball bat Mass= 5 10 20 The Essential Physics: A Toy Model   1: flexible limit ball “sees” M a (5 on 10)   1: rigid limit ball “sees” M a +M b (5 on 30) On short time scale, ball sees reduced bat mass:  COR reduced, vibrations excited

13. University of Kentucky April 25, 2003 Page 13 The Details: A Dynamic Model 20 y z y l Step 1: Solve eigenvalue problem for free vibrations l Step 2: Nonlinear lossy spring for F l Step 3: Expand in normal modes and solve

14. University of Kentucky April 25, 2003 Page 14 Normal Modes of the Bat: Modal Analysis time domain frequency domain FFT frequencies and shapes f 2 = 582 Hz f 1 = 179 Hz f 3 = 1181 Hz

15. University of Kentucky April 25, 2003 Page 15 Mode1 Mode 2 Mode 3 Bending Modes of Bat

16. University of Kentucky April 25, 2003 Page 16 Ball-Bat Force F vs. time  F vs. CM displacement Details not important --as long as e(v),  (v) about right Measureable with load cell

17. University of Kentucky April 25, 2003 Page 17 Vibrations and the COR COR maximum near 2 nd node the “sweet spot”

18. University of Kentucky April 25, 2003 Page 18 Comparison with Data: Ball Exit Speed Louisville Slugger R161, 33/31 Conclusion: essential physics under control only lowest mode excited lowest 4 modes excited

19. University of Kentucky April 25, 2003 Page 19 Conclusion: ideal ball-bat collision can be simulated simulation Batting cage data

20. University of Kentucky April 25, 2003 Page 20 Time Evolution rigid-recoil develops only after few ms far end of bat has no effect on ball --knob moves after 0.6 ms --collision over after 0.6 ms --hands don’t matter!

21. University of Kentucky April 25, 2003 Page 21 superballs bounced from beam (Rod Cross) free clamped 30 cm 60 cm 110 cm 30 cm 60 cm 110 cm

22. University of Kentucky April 25, 2003 Page 22 Flexible Bat and the “Trampoline Effect” Losses in ball anti-correlated with vibrations in bat

23. University of Kentucky April 25, 2003 Page 23 The “Trampoline” Effect: l Compressional energy shared between ball and bat  PE bat /PE ball = k ball /k bat  ~75% of PE ball dissipated l If some energy stored in bat and if PE bat effectively returned to ball, then COR larger demo

24. University of Kentucky April 25, 2003 Page 24 l Ideal Situation: like person on trampoline  k bat  k ball : most of energy stored in bat: e  e bat  e bat  1: energy stored in bat returned  e  1, independent of e ball The “Trampoline” Effect: A Closer Look l For wood bat  k bat  50k ball : ~2% of energy stored in bat  e bat doesn’t matter  e  e ball l For aluminum bat  k bat  7k ball : ~15% of energy stored in bat  e bat  1: energy stored in bat returned  e  1.2 e ball “BPF” = 1.20

25. University of Kentucky April 25, 2003 Page 25 Bending Modes vs. Hoop Modes k bat  R 4 : large in barrel  little energy stored f (170 Hz, etc) > 1/   stored energy  vibrations Net effect: e  e 0 on sweet spot e  e 0 off sweet spot k bat  (t/R) 3 : small in barrel  more energy stored f (1-2 kHz) < 1/   energy mostly restored Net Effect: e > e 0 “BPF”  e/e 0 = 1.20-1.35! The “Trampoline” Effect: A Closer Look

26. University of Kentucky April 25, 2003 Page 26 Modal analysis: Dan Russell and AMN hoop modes bending modes

27. University of Kentucky April 25, 2003 Page 27 Thanks to Dan Russell Hoop Modes and the “ping”

28. University of Kentucky April 25, 2003 Page 28 Where Does the Energy Go?

29. University of Kentucky April 25, 2003 Page 29 Some Interesting Consequences (work in progress) l e/e 0 increases with …  Ball stiffness  Impact velocity  Decreasing wall thickness  Decreasing ball COR  Note: effects larger for “low-s” than for “high-s” bats l “Tuning a bat”  Tune by balancing between storing energy (k small) and returning it (f large)  Tuning not related to phase of vibration at time of ball-bat separation l Does “corking” a bat produce trampline effect? s  k bat /k ball e 2  (1+se 0 2 )/(s+1) e  1 for s << 1

30. University of Kentucky April 25, 2003 Page 30 Some Interesting Consequences (work in progress) l Simple measurements to predict BPF  Measure static compression of bat  Measure frequency of shell modes  Measure collision time with massive steel ball  m ball >> m bat  k ball >> k bat  Collision time =    (m ball /k bat ) * Similar to USGA method for metal drivers

31. University of Kentucky April 25, 2003 Page 31 Summary l Dynamic model developed for ball-bat collision  flexible nature of bat included  simple model for ball-bat force l Vibrations play major role in COR for collisions off sweet spot l Far end of bat does not matter in collision l Physics of trampoline effect mostly understood and interesting consequences predicted  should be tested experimentally

32. University of Kentucky April 25, 2003 Page 32 l More typically for aluminum bat  k bat  7k ball : ~15% of energy stored in bat  e bat  1: energy stored in bat returned  e  1.2 e 0 “BPF” = 1.20 l For wood bat  k bat  98k ball : ~2% of energy stored in bat  e bat  1: energy stored in bat returned  e  e 0

33. University of Kentucky April 25, 2003 Page 33 Trampoline Effect: toy model with dissipation in ball ball bat Mass= 1 2  k ball /k bat determines energy stored f  detemines e bat