How Does a Baseball Bat Work? The Physics of the Ball-Bat Collision Nuclear Chemistry Gordon Conference June 19, 2003 Alan M. Nathan University of Illinois at Urbana-Champaign [e-mail] http://www.npl.uiuc.edu/~a-nathan/pob

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How Does a Baseball Bat Work?The Physics of the
Ball-Bat Collision Nuclear Chemistry Gordon
ConferenceJune 19, 2003Alan M. Nathan
University of Illinois at Urbana-Champaigna-nath
an_at_uiuc.eduhttp//www.npl.uiuc.edu/a-nathan/pob

• introduction
• kinematics of the ball-bat collision
• dynamic model for the ball-bat collision
• applications wood, aluminum, corked
• summary/conclusions

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Baseball and Physics
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Introduction to the Ball-Bat Collision

• forces large (gt8000 lbs!)
• time short (lt1/1000 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

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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.
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Kinematics of Ball-Bat Collision
r bat recoil factor mball/mbat,eff (momentum
and angular momentum conservation) e
coefficient of restitution (energy
dissipation)
typical numbers vf 0.2 vball 1.2 vbat
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Kinematics the recoil factor

• 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 doesnt help
• larger Ip better but ...

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Recent ASA Slow-Pitch Softball Field Tests(L. V.
Smith, J. Broker, AMN)

• 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
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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

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Dynamics of Ball-Bat CollisionCOR 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 ?

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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

see AMN, Am. J. Phys, 68, 979
(2000)
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The Details A Dynamic Model

• 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

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Normal Modes of the Bat Modal Analysis
demo
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Ball-Bat Force

• Details not important
• --as long as e(v), ?(v) about right
• Measureable with load cell

?
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Effect of Bat on COR Vibrations
the sweet spot
COR depends strongly on impact location
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Comparison with Data Ball Exit
Speed Louisville Slugger R161, 33/31
only lowest mode excited
lowest 4 modes excited
Conclusion essential physics under control
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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

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Vf independent of end support
Data courtesy of Keith Koenig
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Flexible Bat and the Trampoline Effect
Losses in ball anti-correlated with vibrations in
bat
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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, ...

demo
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The Trampoline Effect A Closer Look
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The Trampoline EffectA Closer Look

• Bending Modes vs. Hoop Modes

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Modal analysis Dan Russell and AMN
hoop modes
hoop modes
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COR vs. Hoop Mode Frequency
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Where Does the Energy Go?
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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 ? (1se02 )/(s1) e ? 1 for s
ltlt 1
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Some Interesting Consequences(work in progress)

• USGA pendulum test---(Wed. NYT)
• 4 parameters
• mball, mclub, kball, kclub
• make mball gtgt mclub and kball gtgt 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

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So Whats 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 ?

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Bat Research Center, UML, Sherwood amn, Aug.
2001
Not Corked DATA Corked COR
0.445 ? 0.005 0.444 ? 0.005

• Conclusions
• no trampoline effect!
• no advantage to corked
• for home run hitter
• possible advantage for
• contact hitter

calculation
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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

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And in conclusion...

• Thanks for inviting me here
• I love talking about this stuff, so ask me lots
  of questions!