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Physics of the Trampoline Effect baseball, golf, tennis, . Alan M. Nathan a , Daniel Russell b , Lloyd Smith c a University of Illinois at Urbana-Champaign b Kettering University c Washington State University. The “Trampoline” Effect: A Simple Physical Picture.

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Physics of the trampoline effect baseball golf tennis l.jpg

Physics of the Trampoline Effectbaseball, golf, tennis, ...

Alan M. Nathana, Daniel Russellb, Lloyd Smithc

aUniversity of Illinois at Urbana-Champaign

bKettering University

cWashington State University


The trampoline effect a simple physical picture l.jpg
The “Trampoline” Effect:A Simple Physical Picture

  • Two springs mutually compress each other

    • KE  PE  KE

  • PE shared between “ball spring” and “bat spring”

    • PE stored in ball mostly dissipated

    • PE stored in bat mostly restored

  • Net effect: less overall energy dissipated

    • e  e0: the trampoline effect

      • e0 COR for ball on rigid surface

        • 1-e02 = fraction of ball PE dissipated

      • e  COR for ball on flexible surface

        • 1-e2 = fraction of initial ball KE lost to ball


The essential physics toy model l.jpg

kbat

kball

M

m

ball

bat

The Essential Physics: Toy Model

  • Cross (tennis, M=0)

  • Cochran (golf)

  • Naruo & Sato (baseball)

  • Numerically solve ODE to get e = vf/vi

    • Energy lost (e<1) due to...

      • Dissipation in ball

      • Vibrations in bat

  • Essentially a 3-parameter problem:

    • e0

      • Controls dissipation of energy stored in ball

    • rk kbat/kball = PEball/PEbat

      • Controls energy fraction stored in bat

    • rm  m/M

      • f  (rk/rm) ( depends mainly on ball)

      • Controls energy transferred to bat (vibrations)


Slide4 l.jpg

Energy Flow

wood-like: rk=75

(very stiff bat)

aluminum-like: rk=10

(less stiff bat)


Slide5 l.jpg

kball

kbat

M

m

ball

bat

rm= m/M=0.25

  • Strong coupling limit:

  • rk>>1, f>1 Ebat/Eball<<1

  • e = e0

  • 2. Weak coupling limit:

  • rk<<1, f<<1

  • m on M

  • e=(e0-m/M)/(1+m/M)

  • Intermediate coupling

  • rk>1, f>1

  • e > e0


Slide6 l.jpg

Dependence on rm = m/M

f=1.1

  • M  f max @ smaller rk

  • Conclude: e depends on bothrkand rM

    • Not unique function of f

  • Limiting case: rk<<1 and f>>1 (rm0) (thin flexible membrane)

    • e1, independent of e0


Important results all confirmed experimentally l.jpg
Important Results(all confirmed experimentally)

  • Harder ball or softer bat decreases rk, increases e

  • Nonlinear baseball: kball increases with vi

     e/e0 increases with vi

  • e/e0 (“BPF”) decreases as e0 increases

  • Collision time increases as rk decreases

    USGA pendulum test


Realizing the trampoline effect in baseball softball bats l.jpg

  • kbat (t/R)3: small in barrel

  •  more energy stored

  • f (1-2 kHz)  > 1

  •  energy mostly restored

  • Net Effect:

  • e/e0 = 1.20-1.35

  • trampoline effect

  • 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

  • no trampoline effect

Realizing the Trampoline Effect in Baseball/Softball Bats

Bending Modes vs. Hoop Modes


Trampoline effect softball vs baseball l.jpg

  • bb< sb  curve “stretches” to higher f

Trampoline Effect:Softball vs. Baseball

  • Net result:

  • ordering reversed

  • should be tested experimentally


Summary l.jpg
Summary

  • Simple physical model developed for trampoline effect

  • Model qualitatively accounts for observed phenomena with baseball/softball bats

    • Both rk and rM are important

    • e/e0 not a bat property independent of e0

  • Relative performance of bats depends on the ball!

    • But this needs to be tested