How Does a Baseball Bat Work The Physics of the Ball and Bat

Baseball and Physics 1927 Yankees: Greatest baseball team ever assembled 1927 Solvay Conference: Greatest physics team ever assembled MVP’s Nuclear Chemistry Gordon Conference June 19, 2003 Page 2 Introduction to the 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 goals...  large hit ball speed  good “contact” Nuclear Chemistry Gordon Conference June 19, 2003 Page 3 high-speed video of collision These movies are owned by CE Composites Baseball (combatbaseball.com), designers and manufacturers of composite baseball bats, Ottawa, Ontario, Canada, and are shown here with their permission. Nuclear Chemistry Gordon Conference June 19, 2003 Page 4 Kinematics of Ball-Bat Collision  e-r  1+e  vf =   v ball   1+r  v bat 1+r    eA 1+eA r: bat recoil factor = mball/mbat,eff (momentum and angular momentum conservation) vball vbat vf e: coefficient of restitution (energy dissipation) typical numbers: vf = 0.2 vball + 1.2 vbat Nuclear Chemistry Gordon Conference June 19, 2003 Page 5 Kinematics: the recoil factor b e-r eA  1 r • r = mball/mbat,eff mbat,eff = Ip/b2  typically pivot point is ~6” from knob • r ~ 0.25 for collision ~6” from barrel end • mass in handle doesn’t help • larger Ip better but ... Nuclear Chemistry Gordon Conference June 19, 2003 Page 6 Recent ASA Slow-Pitch Softball Field Tests (L. V. Smith, J. Broker, AMN) Bat Speed at 6" Point vs. MOI 1.06 1.04 1.02 1 0.98 0.96 Bat Speed at 6" Point vs. W 1.04 fixed M fixed MOI ~(1/M) 0.25 1.02 1 dashed: n=0.25 solid: n=0.23 0.98 0.96 0.94 6000 7000 8000 9000 2 10000 11000 24 25 26 27 28 W (oz) 29 30 31 32 MOI (oz-in ) Conclusions: • bat speed depends more on I6 than M: Ideal vbat ~ weight/MOI not easy to determine • bat (1/I6)1/4 • rotation point close to knob Nuclear Chemistry Gordon Conference June 19, 2003 Page 7 Aside: Wood-Aluminum Differences  Inertial differences  CM closer to hands, further from barrel for aluminum  Mbat,eff smaller  * larger recoil factor r, smaller eA * effectively, less mass near impact location  MOIknob smaller  swing speed higher  ~cancels  for many bats    …but definite advantage for contact hitter   Dynamic differences  Ball-Bat COR significantly larger for aluminum  more on this later Nuclear Chemistry Gordon Conference June 19, 2003 Page 8 Dynamics of Ball-Bat Collision COR and Energy Dissipation e  COR  vrel,after/vrel,before  in CM frame: (final KE/initial KE) = e2   baseball on hard floor: e2 = hf/hi  0.25    typically e  0.5  ~3/4 CM energy dissipated! depends (weakly) on v the bat matters too!  vibrations  “trampoline” effect  Nuclear Chemistry Gordon Conference June 19, 2003 Page 9 Accounting for Energy Dissipation: Dynamic Model for Ball-Bat Colllision  Bat is flexible on short time scale  Collision excites vibrations  Vibrations reduce COR  Energy going to vibrations depends on  Impact location relative to nodes  Collision time (~0.6 ms) relative to 1/fvib see AMN, Am. J. Phys, 68, 979 (2000) Nuclear Chemistry Gordon Conference June 19, 2003 Page 10 The Details: A Dynamic Model 2y 2  2y  A 2  F - 2  EI 2  t x  x    0 2  Step 1: Solve eigenvalue problem for free vibrations 5 1 y 20 0 1  2   2 yn  2  EI   A n y n x 2  x 2      5 0 Step 2: Ball-bat interaction (F) modeled as nonlinear lossy spring Step 3: Expand in normal modes and solve 2 5 - y 1 -0 z 1 -5 2 -0 0 5 0 1 5 1 0 2 5 2 0 3 5 3 y(x,t )   q n (t )y n ( x) n d qn F(t) y n ( z ) 2  n q n  2 dt A June 19, 2003 Nuclear Chemistry Gordon Conference Page 11 1 0.5 0 Normal Modes of the Bat: demo Modal Analysis frequency domain time domain FFT(R) 0.15 582 R FFT 0.1 1181 -0.5 0.05 -1 -1.5 179 1830 2400 0 5 10 t (ms) 15 20 0 0 500 1000 1500 frequency (Hz) 2000 2500 f1 = 179 Hz f2 = 582 Hz f3 = 1181 Hz frequencies and shapes Nuclear Chemistry Gordon Conference June 19, 2003 Page 12 Ball-Bat Force • Details not important --as long as e(v), (v) about right • Measureable with load cell F vs. CM displacement force (pounds) F vs. time 1 10 4 8000  approx quadratic 6000 4000 2000 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 compression (inches) Nuclear Chemistry Gordon Conference June 19, 2003 Page 13 Effect of Bat on COR: Vibrations COR 0.5 0.4 0.3 0.2 0.1 0 nodes CM f1 = 179 Hz the “sweet spot” f2 = 582 Hz 0 2 4 6 8 10 12 distance from tip (inches) 14 0 5 10 15 20 25 30 35 COR depends strongly on impact location Nuclear Chemistry Gordon Conference June 19, 2003 Page 14 Comparison with Data: Ball Exit Speed Louisville Slugger R161, 33/31 v final /v v initial CM node final /v initial 0.35 0.3 0.25 0.2 rigid bat nodes 0.4 rigid bat 0.3 0.2 0.1 0 16 20 24 28 distance from knob (inches) 32 flexible bat flexible bat data from Rod Cross freely suspended bat v = 2.2 mph i 0.15 0.1 0.05 0 23 data from Lansmont BBVC bat pivoted about 5-3/4" v =100 mph initial 24 25 26 27 28 29 30 distance from knob (inches) 31 only lowest mode excited lowest 4 modes excited Conclusion: essential physics under control Nuclear Chemistry Gordon Conference June 19, 2003 Page 15 time evolution • rigid-body motion develops only after few ms displacement (mm) 10 8 6 4 2 0 0 - 1 ms 0.1 ms intervals • far end of bat has no effect on -2 ball -4  knob moves after 0.6 ms  nothing on knob end matters • size, shape • boundary conditions • hands Nuclear Chemistry Gordon Conference impact point 200 150 100 50 0 impact point -50 0 5 10 15 20 25 30 distance from knob (inches) Page 16  collision over after 0.6 ms 1-10 ms 1 ms intervals June 19, 2003 Vf independent of end support Vf (mph) 120 110 100 90 4.75'' pivot 80 70 60 50 60 70 80 90 100 6.75'' pivot free swing/hit Vi or Swing (mph) Data courtesy of Keith Koenig Nuclear Chemistry Gordon Conference June 19, 2003 Page 17 Flexible Bat and the “Trampoline Effect” COR 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0 2 Nodes % Energy Dissipated 80 70 60 COR Ball 50 40 30 20 Vibrations 4 6 8 10 12 10 0 14 Losses in ball anti-correlated with vibrations in bat inches from barrel Nuclear Chemistry Gordon Conference June 19, 2003 Page 18 The “Trampoline” Effect:  Compressional energy shared between ball and bat  PEbat/PEball = kball/kbat  ~75% of PEball dissipated  If some energy stored in bat and if PEbat effectively returned to ball, then COR larger Effect occurs in tennis, golf, aluminum bats, ... June 19, 2003  demo Page 19 Nuclear Chemistry Gordon Conference The “Trampoline” Effect: A Closer Look e ball-bat 1 0.9 0.8 0.7 e e ball bat = 0.5 = 1.0 e k bat  e k ball e  k bat +k ball 2 2 ball 2 bat 0.6 0.5 0.01 0.1 1 10 ball 100 k bat /k Ideal Situation:bat  For wood bat For aluminum like person on trampoline kk  50kball: ~2% of energy stored in bat ebat kbatbat k7kballmost of energy stored in bat: e  bat  ball: : ~15% of energy stored in bat eebat doesn’t matter ebat  1: energy stored in bat returned bat  1: energy stored in bat returned Nuclear Chemistry Gordon Conference June 19, 2003  e  eballball “BPF”of eball  e  1, independent = 1.20 e  1.2 e Page 20 The “Trampoline” Effect: A Closer Look 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 > e0 ee0 off sweet spot “BPF”  e/e0 = 1.20-1.35! Nuclear Chemistry Gordon Conference June 19, 2003 Page 21 Modal analysis: Dan Russell and AMN bending modes hoop modes hoop modes Nuclear Chemistry Gordon Conference June 19, 2003 Page 22 COR vs. Hoop Mode Frequency COR 0.70 Energy left in hoop vibrations 0.65 0.60 0.55 0.50 0.45 0.40 500 1000 f hoop COR-model COR-expt 1500 (Hz) 2000 Nuclear Chemistry Gordon Conference June 19, 2003 Page 23 Where Does the Energy Go? Energy (J) 400 350 300 250 200 150 100 50 0 0 0.2 0.4 0.6 t (ms) 0.8 1 Energy (J) 400 Ball KE 350 Ball KE Wood Bat 300 250 Aluminum Bat Ball PE 200 Ball PE Bat Recoil KE 150 100 Bat Recoil KE Bat Vibrational E 0 0.2 0.4 0.6 t (ms) 0.8 1 Bat Vibrational E 50 0 Nuclear Chemistry Gordon Conference June 19, 2003 Page 24 Some Interesting Consequences (work in progress)  e/e0 increases with … s  k /k  Ball stiffness  Impact velocity  Decreasing wall thickness  Decreasing ball COR bat ball e2  (1+se02 )/(s+1) e  1 for s << 1  Note: effects larger for “low-s” (high-performance) than for “high-s” (low-performance) bats  “Tuning a bat”  Tune by balancing between storing energy (k small) and returning it (f large)  Tuning not simply related to phase of vibration at time of ball-bat separation Nuclear Chemistry Gordon Conference June 19, 2003 Page 25 Some Interesting Consequences (work in progress)  USGA “pendulum” test---(Wed. NYT)  4 parameters mball, mclub, kball, kclub  make mball >> mclub and kball >> kclub  heavy, stiff steel ball on clubhead  collision time determined by mball (known) and kclub  measure collision time to determine kclub  kclub determines trampoline effect  implementation expected Jan. 2004 Nuclear Chemistry Gordon Conference June 19, 2003 Page 26 So What’s the Deal with Corked Bats?  ~1” diameter hole ~10” deep; fill with whatever similar to aluminum bat * easier to swing and control  * but less effective at transferring energy   Is there a “trampoline” effect from hole or filler? probably not   Net result: little or no effect for home run hitter  possible advantage for “contact” hitter  Nuclear Chemistry Gordon Conference June 19, 2003 Page 27 Bat Research Center, UML, Sherwood & amn, Aug. 2001 COR: Not Corked DATA Corked 0.445  0.005 0.444  0.005 v (mph) f Conclusions: • no trampoline effect! • no advantage to corked for home run hitter 90 uncorked corked 80 calculation 70 2 3 4 5 6 7 8 distance from knob (inches) 9 • possible advantage for “contact” hitter Nuclear Chemistry Gordon Conference June 19, 2003 Page 28 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 Corking bat has little effect on home run June 19, 2003 Page 29  Nuclear Chemistry Gordon Conference And in conclusion...  Thanks I for inviting me here love talking about this stuff, so ask me lots of questions! 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