1. 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
Introduction
The simple stuff (kinematics)
The interesting stuff (dynamics)
Wood vs. aluminum
Summary & Conclusions 1

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

3. 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 sec in timing, ball moves 1’.
Compare with golf, where ball is not even moving!
BA: .344
SA: .634
OBP: .483‡
HR:
‡career record

4. 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?

5. 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 vbat
Conclusion: vbat much more important than vball

6. 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”

7. 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….
 x 1.49 
All things equal, want r small
But….

8. 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

9. 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.

10. COR: Is the Ball “Juiced”?
MLB: e =  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:

11. 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.

12. More Realistic AnalysisCM
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.

13. 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)

14. The Essential Physics: A Toy Model
ball
bat
Mass=
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

15. A Dynamic Model of the Bat-Ball Collision20
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

16. 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)

17. FFT Measurements via Modal Analysis
Louisville Slugger R161 (33”, 31 oz)
FFT
frequency barrel node
Expt Calc Expt Calc
Conclusion: free vibrations
of bat can be well characterized

18. 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

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

20. 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

21. 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

23. 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

24. 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?

25. 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!

26. 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

27. 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

28. 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 =
Aluminum Bat
kball/kbat ~ 0.10
eeff =
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!

29. 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 

30. 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

31. 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

32. 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!