1. kinematics of the ball-bat collision
How Does a Baseball Bat Work? The Physics of the Ball-Bat Collision Nuclear Chemistry Gordon Conference June 19, Alan M. Nathan University of Illinois at Urbana-Champaign
introduction
kinematics of the ball-bat collision
dynamic model for the ball-bat collision
applications: wood, aluminum, corked
summary/conclusions 1

2. Greatest baseball team
Baseball and Physics
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3. Introduction to the Ball-Bat Collision
forces large (>8000 lbs!)
time short (<1/1000 sec!)
ball compresses, stops, expands
kinetic energy  potential energy
lots of energy dissipated
bat is flexible
bat bends, compresses
the goals...
large hit ball speed
good “contact”

4. high-speed video of collision
These movies are owned by CE Composites Baseball (combatbaseball.com), designers and manufacturers of composite baseball bats, Ottawa, Ontario, Canada, and are shown here with their permission.

5. Kinematics of Ball-Bat Collision
vball
vbat vf eA
1+eA
r: bat recoil factor = mball/mbat,eff
(momentum and angular momentum conservation)
e: coefficient of restitution
(energy dissipation)
Primary dependence of vf on ball, bat speeds is in the relationship…weaker dependence through e,
which depends on rel ball-bat speed.
Inefficient collision (1 for superball on rigid wall)
Bat speed matters much more than ball speed
typical numbers: vf = 0.2 vball vbat

6. Kinematics: the recoil factorb
r = mball/mbat,eff mbat,eff = Ip/b2
typically pivot point is ~6” from knob
r ~ 0.25 for collision ~6” from barrel end
mass in handle doesn’t help
larger Ip better but ...

7. Ideal bat weight/MOI not easy to determine
Recent ASA Slow-Pitch Softball Field Tests (L. V. Smith, J. Broker, AMN)
fixed M
fixed MOI
Conclusions:
bat speed depends more on I6 than M:
vbat ~ (1/I6)1/4
rotation point close to knob
Ideal bat weight/MOI not easy to determine

8. Aside: Wood-Aluminum Differences
Inertial differences
CM closer to hands, further from barrel for aluminum
 Mbat,eff smaller 
larger recoil factor r, smaller eA
effectively, less mass near impact location
 MOIknob smaller  swing speed higher
 ~cancels  for many bats  
…but definite advantage for contact hitter 
Dynamic differences
Ball-Bat COR significantly larger for aluminum 
more on this later

9. Dynamics of Ball-Bat Collision COR and Energy Dissipation
e  COR  vrel,after/vrel,before
in CM frame: (final KE/initial KE) = e2
baseball on hard floor: e2 = hf/hi  0.25
typically e  0.5
~3/4 CM energy dissipated!
depends (weakly) on v
the bat matters too!
vibrations 
“trampoline” effect 

10. Accounting for Energy Dissipation:
Dynamic Model for Ball-Bat Colllision
Bat is flexible on short time scale
Collision excites vibrations
Vibrations reduce COR
Energy going to vibrations depends on
Impact location relative to nodes
Collision time (~0.6 ms) relative to 1/fvib
So far, just kinematic, plus a phenomonlogoical treatment of energy losses via COR. Now we want to dissect the collision 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)

11. The Details: A Dynamic Model20 y z
Step 1: Solve eigenvalue problem for free vibrations
Step 2: Ball-bat interaction (F) modeled as nonlinear lossy spring
Step 3: Expand in normal modes and solve
Free BC will be justified later

12. Normal Modes of the Bat:
Modal Analysis
demo
frequency domain
time domain
FFT
f2 = 582 Hz
f1 = 179 Hz
f3 = 1181 Hz
This can be skipped
frequencies and shapes

13.  Ball-Bat Force Details not important
--as long as e(v), (v) about right
Measureable with load cell
F vs. time
F vs. CM displacement 
This can be skipped

14. COR depends strongly on impact location
Effect of Bat on COR: Vibrations
nodes
f1 = 179 Hz
f2 = 582 Hz
the “sweet spot”
So, what role does the bat play in the COR of the ball-bat collision?
The collision can excite bending vibrations in the bat. These vibrations can be felt…it is the sting that is felt for off-sweet-spot hits. Sometimes the vibrational amplitude is so large that the bat even breaks. There are characteristic vibrational modes (frequencies and shapes), much like a guitar string. Lowest mode (shown here) is about Hz with a node about 6-7” from barrel end. Next higher mode is about 580 Hz, with a node a little further out. The possibility of exciting vibrations in the bat means that the ball-bat COR is not uniform across the length of the bat but looks something like this, peaked 4-7” from the end, where the nodes of the lowest few modes of vibration are clustered together (so that very little energy goes into vibrations), and dropping fairly rapidly on either side as the vibrations take more and more energy from the ball.
COR depends strongly on impact location

15. Comparison with Data: Ball Exit Speed Louisville Slugger R161, 33/31
For free rigid bat, 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

16. time evolution rigid-body motion 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
nothing on knob end matters
size, shape
boundary conditions
hands

17. Vf independent of end support
Vf (mph)
Data courtesy of Keith Koenig

18. Flexible Bat and the “Trampoline Effect”
Losses in ball anti-correlated with vibrations in bat
So, what role does the bat play in the COR of the ball-bat collision?
The collision can excite bending vibrations in the bat. These vibrations can be felt…it is the sting that is felt for off-sweet-spot hits. Sometimes the vibrational amplitude is so large that the bat even breaks. There are characteristic vibrational modes (frequencies and shapes), much like a guitar string. Lowest mode (shown here) is about Hz with a node about 6-7” from barrel end. Next higher mode is about 580 Hz, with a node a little further out. The possibility of exciting vibrations in the bat means that the ball-bat COR is not uniform across the length of the bat but looks something like this, peaked 4-7” from the end, where the nodes of the lowest few modes of vibration are clustered together (so that very little energy goes into vibrations), and dropping fairly rapidly on either side as the vibrations take more and more energy from the ball.

19. The “Trampoline” Effect:
Compressional energy shared between ball and bat
PEbat/PEball = kball/kbat
~75% of PEball dissipated
If some energy stored in bat and if PEbat effectively returned to ball, then COR larger
Effect occurs in tennis, golf, aluminum bats, ...
Ask question:
Which give more “power”: tighter strings or looser strings on tennis racket?
Then ask (if they get it wrong): can a person bounce higher from a hardwood floor or from a trampoline?
demo

20. The “Trampoline” Effect: A Closer Look
Ideal Situation: like person on trampoline
kbat kball: most of energy stored in bat: e  ebat
ebat  1: energy stored in bat returned
 e  1, independent of eball
For aluminum bat
kbat  7kball: ~15% of energy stored in bat
ebat  1: energy stored in bat returned
 e  1.2 eball “BPF” = 1.20
For wood bat
kbat  50kball: ~2% of energy stored in bat
ebat doesn’t matter
 e  eball
Ask question:
Which give more “power”: tighter strings or looser strings on tennis racket?
Then ask (if they get it wrong): can a person bounce higher from a hardwood floor or from a trampoline?

21. The “Trampoline” Effect: A Closer Look
Bending Modes vs Hoop Modes
kbat  R4: large in barrel
 little energy stored
f (170 Hz, etc) > 1/
 stored energyvibrations
Net effect: e  e0 on sweet spot
ee0 off sweet spot
kbat  (t/R)3: small in barrel
 more energy stored
f (1-2 kHz) < 1/ 
 energy mostly restored
Net Effect: e > e0
“BPF”  e/e0 = !
Little effect of bending modes at sweet spot.

22. Modal analysis: Dan Russell and AMN
bending modes
hoop modes
hoop modes

23. COR vs. Hoop Mode Frequency
Energy left in hoop vibrations

24. Where Does the Energy Go?

25. Some Interesting Consequences (work in progress)
e/e0 increases with …
Ball stiffness
Impact velocity
Decreasing wall thickness
Decreasing ball COR
Note: effects larger for “low-s” (high-performance) than for “high-s” (low-performance) bats
“Tuning a bat”
Tune by balancing between storing energy (k small) and returning it (f large)
Tuning not simply related to phase of vibration at time of ball-bat separation
s  kbat/kball
e2  (1+se02 )/(s+1)
e  1 for s << 1

26. Some Interesting Consequences (work in progress)
USGA “pendulum” test---(Wed. NYT)
4 parameters
mball, mclub, kball, kclub
make mball >> mclub and kball >> kclub
heavy, stiff steel ball on clubhead
collision time determined by mball (known) and kclub
measure collision time to determine kclub
kclub determines trampoline effect
implementation expected Jan. 2004

27. So What’s the Deal with Corked Bats?
~1” diameter hole ~10” deep; fill with whatever
similar to aluminum bat
easier to swing and control 
but less effective at transferring energy 
Is there a “trampoline” effect from hole or filler?
probably not 
Net result:
little or no effect for home run hitter 
possible advantage for “contact” hitter 

28. Bat Research Center, UML, Sherwood & amn, Aug. 2001
Not Corked DATA Corked
COR:   0.005
Conclusions:
no trampoline effect!
no advantage to corked
for home run hitter
possible advantage for
“contact” hitter
calculation

29. Summary Dynamic model developed for ball-bat collision
flexible nature of bat included
simple model for ball-bat force
Vibrations play major role in COR for collisions off sweet spot
Far end of bat does not matter in collision
Physics of trampoline effect mostly understood and interesting consequences predicted
Corking bat has little effect on home run

30. And in conclusion... Thanks for inviting me here
I love talking about this stuff, so ask me lots of questions!