Physics of the Trampoline Effect baseball golf tennis

Physics of the Trampoline Effect baseball, 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 – 1-e2 = fraction of initial ball KE lost to ball • e  COR for ball on flexible surface The Essential Physics: Toy Model • Cross (tennis, M=0) • Cochran (golf) • Naruo & Sato (baseball) kball kbat M m • Numerically solve ODE to get e = vf/vi – Energy lost (e<1) due to... • Dissipation in ball • Vibrations in bat ball bat • Essentially a 3-parameter problem: – e0 – rk  kbat/kball = PEball/PEbat – rm  m/M • Controls dissipation of energy stored in ball • Controls energy fraction stored in bat • f  (rk/rm) ( depends mainly on ball) • Controls energy transferred to bat (vibrations) Energy Flow Energy Fraction 1 0.8 0.6 Energy Fraction 1 PE-Ball r =25 r =1 k m 0.8 PE-Ball 0.6 0.4 0.2 KE-Ball E-Bat 0.4 0.2 r =10 r =1 k m KE-Ball E-Bat 1 0 0 0.2 0.4 0.6 t (ms) 0.8 1 0 0 0.2 0.4 0.6 t (ms) 0.8 wood-like: rk=75 (very stiff bat) aluminum-like: rk=10 (less stiff bat) e 0.60 f 2 kball kbat e e = 0.50 0 1.5 M 0.40 f 1 0.5 m ball bat 0.30 1. 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) 3. Intermediate coupling rk>1, f>1 e > e0 rm= m/M=0.25 0 energy fraction 0.80 0.60 0.40 0.20 25 50 r 75 0 100 k dissipation in ball ball 0.20 vibrations in bat 0.0 0 25 50 r k 75 100 Dependence on rm = m/M e f 5 0.70 0.60 solid: r =1/4 m dashed: r =1 m 4 e e = 0.50 0 f 3 2 0.40 1 f=1.1 0.30 0 25 50 r 75 0 100 • M  f max @ smaller rk k • Conclude: e depends on both rk and 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: e ball increases with vi k  e/e0 increases with vi 0 0.60 f 2 1.5 1 0.5 0 100 e = 0.50 1.5 • e/e0 (“BPF”) decreases as e0 increases 1.4 f 0.40 • Collision time increases as rk decreases e =0.45 e USGA pendulum test 1.2 1.1 1.0 5 10 e/e 1.3 0 0 0.30 0.20 r e =0.50 0 0 15 25 20 50 r 75 k k Realizing the Trampoline Effect in Baseball/Softball Bats Bending Modes vs. Hoop Modes kbat  R4: large in barrel  little energy stored kbat  (t/R)3: small in barrel  more energy stored f (170 Hz, etc)  < 1 f (1-2 kHz)  > 1  stored energyvibrations  energy mostly restored Net effect: e  e0 on sweet spot Net Effect: e<