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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 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

• 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

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)

wood-like: rk=75

(very stiff bat)

aluminum-like: rk=10

(less stiff bat)

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

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)

• 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

• 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

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

Trampoline Effect:Softball vs. Baseball

• Net result:

• ordering reversed

• should be tested experimentally

• 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