1. 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

2. 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  e 0 : the trampoline effect e 0  COR for ball on rigid surface –1-e 0 2 = fraction of ball PE dissipated e  COR for ball on flexible surface –1-e 2 = fraction of initial ball KE lost to ball

3. m k ball ball M k bat bat The Essential Physics: Toy Model Numerically solve ODE to get e = v f /v i –Energy lost (e<1) due to... Dissipation in ball Vibrations in bat Essentially a 3-parameter problem: –e0–e0 Controls dissipation of energy stored in ball –r k  k bat /k ball = PE ball /PE bat Controls energy fraction stored in bat –r m  m/M f    (r k /r m ) (  depends mainly on ball) Controls energy transferred to bat (vibrations) Cross (tennis, M=0) Cochran (golf) Naruo & Sato (baseball)

4. Energy Flow wood-like: r k =75 (very stiff bat) aluminum-like: r k =10 (less stiff bat)

5. m k ball ball M k bat bat 1.Strong coupling limit: r k >>1, f  >1 E bat /E ball <<1 e = e 0 2.Weak coupling limit: r k <<1, f  <<1 m on M e=(e 0 -m/M)/(1+m/M) 3.Intermediate coupling r k >1, f  >1 e > e 0 r m = m/M=0.25

6. Dependence on r m = m/M M   f  max @ smaller r k Conclude: e depends on both r k and r M Not unique function of f Limiting case: r k >1 (r m  0) (thin flexible membrane) e  1, independent of e 0 f  =1.1

7. Important Results (all confirmed experimentally) Harder ball or softer bat decreases r k, increases e Nonlinear baseball: k ball increases with v i  e/e 0 increases with v i e/e 0 (“BPF”) decreases as e 0 increases Collision time increases as r k decreases USGA pendulum test

8. Bending Modes vs. Hoop Modes k bat  R 4 : large in barrel  little energy stored f (170 Hz, etc)  < 1  stored energy  vibrations Net effect: e  e 0 on sweet spot e<<e 0 off sweet spot no trampoline effect k bat  (t/R) 3 : small in barrel  more energy stored f (1-2 kHz)  > 1  energy mostly restored Net Effect: e/e 0 = 1.20-1.35 trampoline effect Realizing the Trampoline Effect in Baseball/Softball Bats

9.  bb <  sb  curve “stretches” to higher f Trampoline Effect: Softball vs. Baseball Net result: ordering reversed should be tested experimentally

10. Summary Simple physical model developed for trampoline effect Model qualitatively accounts for observed phenomena with baseball/softball bats –Both r k and r M are important –e/e 0 not a bat property independent of e 0 Relative performance of bats depends on the ball! –But this needs to be tested