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Introduction: Description of Ball-Bat Collision

Introduction: Description of 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 goal... large hit ball speed. v ball. v bat.

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Introduction: Description of Ball-Bat Collision

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  1. Introduction: Description of 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 goal... • large hit ball speed

  2. vball vbat vf Kinematics of the Ball-Bat Collision r bat recoil factor = mball/Mbat,effective e Coefficient of Restitution (COR) typical numbers: r  0.25 e  0.50  vf = 0.2 vball + 1.2 vbat Note: this talk focuses entirely on COR

  3. COR and Energy Dissipation • e  COR  vrel,after/vrel,before • in CM frame: (final KE/initial KE) = e2 • e.g., drop ball on hard floor:COR2 = hf/hi 0.25 • typicallyCOR  0.5 • ~3/4 CM energy dissipated! • depends on impact speed • mostly a property of ball but… • the bat matters too! • vibrations  , “trampoline” effect 

  4. Accounting for Energy Dissipation: Dynamic Model for Ball-Bat Colllision • Collision excites bending vibrations • Ouch!! Thud!! Sometimes broken bat • Energy lost  lower COR, vf • Find lowest mode by tapping • Reduced considerably if • Impact is at a node • Collision time (~0.6 ms) >> Tvib see AMN, Am. J. Phys, 68, 979 (2000)

  5. bat ball Mass= 1 2  The Essential Physics: A Toy Model rigid limit   1 1 on  flexible limit  1 1 on 2

  6. y 20 y z The Details: A Dynamic Model • Step 1: Solve eigenvalue problem for free vibrations • Step 2: Nonlinear lossy spring for ball-bat interaction • Step 3: Expand in normal modes and solve

  7. f1 = 177 Hz f3 = 1179 Hz f2 = 583 Hz f4 = 1821 Hz Normal Modesof the Bat Louisville Slugger R161 (34”, 31 oz) Can easily be measured: Modal Analysis

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

  9. Vibrations and the COR the “sweet spot” COR maximum near 2nd node

  10. 24” 27” 30” Some interesting insights: • Center of Percussion close to lowest node @ 27” • Coincides neither with max COR @ 29” • …nor with max. vf • Far end of bat doesn’t matter • mass, grip, …

  11. T= 0-1 ms Ball leaves bat T= 1-10 ms Time evolution of the bat • Conclusions: • Knob end doesn’t matter • Batter’s grip doesn’t matter • vibrations and rigid motion • indistinguishable on • short time scale

  12. Bounce superballs from beam (Rod Cross) Conclusion: Nothing on far end of beam matters

  13. Flexible Bat and the “Trampoline Effect” Losses in ball anti-correlated with vibrations in bat

  14. The “Trampoline” Effect: A Closer Look • Compressional energy shared between ball and bat • PEbat/PEball = kball/kbat (= s) • PEball mostly dissipated (75%) • BPF = Bat Proficiency Factor  e/e0 • Ideal Situation: like person on trampoline • kball >>kbat: most of energy stored in bat • f >>1:stored energy returned • e2 (s+e02)/(s+1)  1 for s >>1  eo2 for s <<1

  15. Trampoline Effect: Toy Model, revisited Mass= 1 2  bat ball

  16. The “Trampoline” Effect:A Closer Look Bending Modes vs. Shell Modes • k  (t/R)3: small in barrel •  more energy stored • f (1-2 kHz) < 1/  •  energy mostly restored • Net Effect: BPF > 1 • k  R4: large in barrel •  little energy stored • f (170 Hz, etc) > 1/ •  energy lost to vibrations • Net effect: BPF  1

  17. Where Does the Energy Go?

  18. Some Interesting Consequences(work in progress) s  kball/kbat e2 (s+e02 )/(s+1) BPF = e/e0 • BPF increases with … • Ball stiffness • Impact velocity • Decreasing wall thickness • Decreasing ball COR • Note: effects larger for “high-s” than for “low-s” bats • “Tuning a bat” • Tuning due to balance between storing energy (k small) and returning it (f large) • Tuning not related to phase of vibration at time of ball-bat separation

  19. 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 • should be tested experimentally

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