1. How Does a Baseball Bat Work: The Physics of the Ball-Bat Collision Page 1 SIU Colloquium, October 12, 2001 How Does a Baseball Bat Work? The Physics of the Ball-Bat Collision SIU Colloquium, October 12, 2001 Alan M. Nathan University of Illinois at Urbana-Champaign a-nathan@uiuc.edu l Introduction l The simple stuff (kinematics) l The interesting stuff (dynamics) l Wood vs. aluminum l Summary & Conclusions http://www.npl.uiuc.edu/~a-nathan/pob

2. How Does a Baseball Bat Work: The Physics of the Ball-Bat Collision Page 2 Baseball and Physics 1927 Yankees: Greatest baseball team ever assembled 1927 Solvay Conference: Greatest physics team ever assembled MVP’s

3. How Does a Baseball Bat Work: The Physics of the Ball-Bat Collision Page 3 Introduction: 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 compresses ball….ball bends bat Ybat is very flexible! l hands don’t matter! l GOAL: maximize ball exit speed v f Y1 mph  4-5 ft l Question: What does v f depend on?

4. How Does a Baseball Bat Work: The Physics of the Ball-Bat Collision Page 4 Kinematics: Reference Frames v f = e A v ball + (1+e A ) v bat Conclusion: v bat much more important than v ball v ball v bat vfvf “Lab” Frame v rel e A v rel Bat Rest Frame e A  “Collision Efficiency” = “Ball Exit Speed Ratio” - 0.5 property of ball & bat weakly dependent on v rel near “sweet spot” e A  0.2  v f  0.2 v ball + 1.2 v bat BESR

5. How Does a Baseball Bat Work: The Physics of the Ball-Bat Collision Page 5 Conservation Laws and e A l Momentum Conservation v ball,f = e A v ball v bat,f = r(1+e A )v ball r = m ball /m bat l Coefficient of Restitution e  (v ball,f +v bat,f )/v ball  l Energy Conservation f ball = f recoil = f dis = V ball,f v ball V bat,f

6. How Does a Baseball Bat Work: The Physics of the Ball-Bat Collision Page 6 Kinematics: recoil factor r and coefficient of restitution e typical numbers m ball = 5.1 oz m bat = 31.5 oz k  = 9.0 in b = 6.3 in r =.24 e = 0.5 e A = 0.21.. CM. b = +  0.16 + 0.08  0.24 All things equal, want r small But….

7. How Does a Baseball Bat Work: The Physics of the Ball-Bat Collision Page 7 Kinematics: Where Does the Energy Go?.. CM. b = +  0.16 + 0.08  0.24 All things equal, want r small But…. Hit ball Recoil of Bat (r) Dissipation in ball and bat (e)

8. How Does a Baseball Bat Work: The Physics of the Ball-Bat Collision Page 8 What is the Ideal Bat Weight? (½ mv 2 ) Conclusion from Players: Lighter seems to be better!

9. How Does a Baseball Bat Work: The Physics of the Ball-Bat Collision Page 9  v BAT (6”) = 1.2 mph/(1000 oz-in 2) (  v f =1.5  0.3 mph) v bat  I -0.3 v bat  I -0.5

10. How Does a Baseball Bat Work: The Physics of the Ball-Bat Collision Page 10 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! probably depends on impact speed depends on ball and bat! Kinematics: Coefficient of Restitution (e): (Energy Dissipation) “bounciness” of ball

11. How Does a Baseball Bat Work: The Physics of the Ball-Bat Collision Page 11 COR: Is the Ball “Juiced”? MLB: e = 0.546  0.032 @ 58 mph on massive rigid surface

12. How Does a Baseball Bat Work: The Physics of the Ball-Bat Collision Page 12 Putting it all together (rigid bat)... v f = e A v ball + (1+e A ) v bat CM -2 0 2 4 6 8 10 12 051015202530

13. How Does a Baseball Bat Work: The Physics of the Ball-Bat Collision Page 13 v f = e A v ball + (1+e A ) v bat CM Putting it all together (realistic bat)...

14. How Does a Baseball Bat Work: The Physics of the Ball-Bat Collision Page 14 l Collision excites bending vibrations in bat YOuch!! Thud!! YSometimes broken bat YEnergy lost  lower v f (lower e) l Bat not rigid on time scale of collision l What are the relevant degrees of freedom? see AMN, Am. J. Phys, 68, 979 (2000) III. Dynamics Model for Ball-Bat Colllision: Accounting for Energy Dissipation

15. How Does a Baseball Bat Work: The Physics of the Ball-Bat Collision Page 15  >> 1 m on M a +M b (1 on 6) ball bat  << 1 m on M a (1 on 2) The Essential Physics: A Toy Model Mass= 1 2 4 rigid flexible

16. How Does a Baseball Bat Work: The Physics of the Ball-Bat Collision Page 16 20 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 ‡

17. How Does a Baseball Bat Work: The Physics of the Ball-Bat Collision Page 17 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

18. How Does a Baseball Bat Work: The Physics of the Ball-Bat Collision Page 18 frequency barrel node Expt Calc 179 17726.526.6 582 58327.828.2 1181117929.029.2 1830182130.029.9 Measurements via Modal Analysis Louisville Slugger R161 (33”, 31 oz) Conclusion: free vibrations of bat can be well characterized FFT

19. How Does a Baseball Bat Work: The Physics of the Ball-Bat Collision Page 19 Model for the Ball 3-parameter problem: k   n  v-dependence of  m  COR of ball with rigid surface F=kx n F=kx m

20. How Does a Baseball Bat Work: The Physics of the Ball-Bat Collision Page 20 Putting it all together…. ball compression Procedure: specify initial conditions numerically integrate coupled equations find v f = ball speed after ball and bat separate

21. How Does a Baseball Bat Work: The Physics of the Ball-Bat Collision Page 21 Conclusion: only modes with f n  < 1 strongly excited General Result energy in n th mode Fourier transform

22. How Does a Baseball Bat Work: The Physics of the Ball-Bat Collision Page 22 Results: Ball Exit Speed Louisville Slugger R161 33-inch/31-oz. wood bat Conclusion: essential physics under control only lowest mode excited lowest 4 modes excited

23. How Does a Baseball Bat Work: The Physics of the Ball-Bat Collision Page 23

24. How Does a Baseball Bat Work: The Physics of the Ball-Bat Collision Page 24 CMnodes Application to realistic conditions: (90 mph ball; 70 mph bat at 28”)

25. How Does a Baseball Bat Work: The Physics of the Ball-Bat Collision Page 25 1. 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” The “sweet spot”

26. How Does a Baseball Bat Work: The Physics of the Ball-Bat Collision Page 26 Conclusions: size, shape, boundary conditions at far end don’t matter hands don’ t matter! Boundary conditions

27. How Does a Baseball Bat Work: The Physics of the Ball-Bat Collision Page 27 Time evolution of the bat T= 0-1 ms T= 1-10 ms

28. How Does a Baseball Bat Work: The Physics of the Ball-Bat Collision Page 28 0.2 ms 0.4 ms 0.6 ms 0.8 ms 1.0 ms Time evolution of the bat impact location

29. How Does a Baseball Bat Work: The Physics of the Ball-Bat Collision Page 29 Wood versus Aluminum Kinematics Length, weight, MOI “decoupled” * shell thickness, added weight * fatter barrel, thinner handle Weight distribution more uniform * I CM larger (less rot. recoil) * I handle smaller (easier to swing) * less mass at contact point Dynamics Stiffer for bending * Less energy lost due to vibrations More compressible * COR eff larger

30. How Does a Baseball Bat Work: The Physics of the Ball-Bat Collision Page 30 l CM energy shared between ball and bat l Ball inefficient:  75% dissipated l Wood Bat Yk ball /k bat ~ 0.02 Y  80% restored Ye eff = 0.50-0.51 l Aluminum Bat Yk ball /k bat ~ 0.10 Y  80% restored Ye eff = 0.55-0.58 Effect of Bat on COR: Local Compression E bat /E ball  k ball /k bat   x bat /  x ball >10% larger! tennis ball/racket

31. How Does a Baseball Bat Work: The Physics of the Ball-Bat Collision Page 31 Wood versus Aluminum: 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 

32. How Does a Baseball Bat Work: The Physics of the Ball-Bat Collision Page 32 Where Does the Energy Go?

33. How Does a Baseball Bat Work: The Physics of the Ball-Bat Collision Page 33 Which Performance Metric? e A or COR or v f vfvf COR eAeA (my most important slide)

34. How Does a Baseball Bat Work: The Physics of the Ball-Bat Collision Page 34 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 Yvelocity dependence Ywall thickness dependence l Effect of “corking” the bat

35. How Does a Baseball Bat Work: The Physics of the Ball-Bat Collision Page 35 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 probably outperforms wood!