kinematics of the ball-bat collision How Does a Baseball Bat Work? The Physics of the Ball-Bat Collision Nuclear Chemistry Gordon Conference June 19, 2003 Alan M. Nathan University of Illinois at Urbana-Champaign a-nathan@uiuc.edu http://www.npl.uiuc.edu/~a-nathan/pob introduction kinematics of the ball-bat collision dynamic model for the ball-bat collision applications: wood, aluminum, corked summary/conclusions 1

Greatest baseball team Baseball and Physics 1927 Yankees: Greatest baseball team ever assembled MVP’s 1927 Solvay Conference: Greatest physics team ever assembled

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”

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.

Kinematics of Ball-Bat Collision vball vbat vf eA 1+eA r: bat recoil factor = mball/mbat,eff (momentum and angular momentum conservation) e: coefficient of restitution (energy dissipation) Primary dependence of vf on ball, bat speeds is in the relationship…weaker dependence through e, which depends on rel ball-bat speed. Inefficient collision (1 for superball on rigid wall) Bat speed matters much more than ball speed typical numbers: vf = 0.2 vball + 1.2 vbat

Kinematics: the recoil factor b 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 ...

Ideal bat weight/MOI not easy to determine Recent ASA Slow-Pitch Softball Field Tests (L. V. Smith, J. Broker, AMN) fixed M fixed MOI Conclusions: bat speed depends more on I6 than M: vbat ~ (1/I6)1/4 rotation point close to knob Ideal bat weight/MOI not easy to determine

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

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 

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 So far, just kinematic, plus a phenomonlogoical treatment of energy losses via COR. Now we want to dissect the collision process, time slice by time slice, to see what is really going on during the time the ball and bat are in contact. In doing so, we want to try to do a strict accounting for where the energy goes in the collision. So, we want to go beyond kinematics and talk about dynamics. We know that a purely rigid body treatment cannot be right…for example, we know that the collision can excite vibrations in the bat. see AMN, Am. J. Phys, 68, 979 (2000)

The Details: A Dynamic Model 20 y z Step 1: Solve eigenvalue problem for free vibrations Step 2: Ball-bat interaction (F) modeled as nonlinear lossy spring Step 3: Expand in normal modes and solve Free BC will be justified later

Normal Modes of the Bat: Modal Analysis demo frequency domain time domain FFT f2 = 582 Hz f1 = 179 Hz f3 = 1181 Hz This can be skipped frequencies and shapes

 Ball-Bat Force Details not important --as long as e(v), (v) about right Measureable with load cell F vs. time F vs. CM displacement  This can be skipped

COR depends strongly on impact location Effect of Bat on COR: Vibrations nodes f1 = 179 Hz f2 = 582 Hz the “sweet spot” So, what role does the bat play in the COR of the ball-bat collision? The collision can excite bending vibrations in the bat. These vibrations can be felt…it is the sting that is felt for off-sweet-spot hits. Sometimes the vibrational amplitude is so large that the bat even breaks. There are characteristic vibrational modes (frequencies and shapes), much like a guitar string. Lowest mode (shown here) is about 160-180 Hz with a node about 6-7” from barrel end. Next higher mode is about 580 Hz, with a node a little further out. The possibility of exciting vibrations in the bat means that the ball-bat COR is not uniform across the length of the bat but looks something like this, peaked 4-7” from the end, where the nodes of the lowest few modes of vibration are clustered together (so that very little energy goes into vibrations), and dropping fairly rapidly on either side as the vibrations take more and more energy from the ball. COR depends strongly on impact location

Comparison with Data: Ball Exit Speed Louisville Slugger R161, 33/31 For free rigid bat, peak is at CM For full calculation, peak determined by interplay between rotational recoil and vibrational nodes. At lowest node, rigid-full only lowest mode excited lowest 4 modes excited Conclusion: essential physics under control

time evolution rigid-body motion develops only after few ms far end of bat has no effect on ball knob moves after 0.6 ms collision over after 0.6 ms nothing on knob end matters size, shape boundary conditions hands

Vf independent of end support Vf (mph) Data courtesy of Keith Koenig

Flexible Bat and the “Trampoline Effect” Losses in ball anti-correlated with vibrations in bat So, what role does the bat play in the COR of the ball-bat collision? The collision can excite bending vibrations in the bat. These vibrations can be felt…it is the sting that is felt for off-sweet-spot hits. Sometimes the vibrational amplitude is so large that the bat even breaks. There are characteristic vibrational modes (frequencies and shapes), much like a guitar string. Lowest mode (shown here) is about 160-180 Hz with a node about 6-7” from barrel end. Next higher mode is about 580 Hz, with a node a little further out. The possibility of exciting vibrations in the bat means that the ball-bat COR is not uniform across the length of the bat but looks something like this, peaked 4-7” from the end, where the nodes of the lowest few modes of vibration are clustered together (so that very little energy goes into vibrations), and dropping fairly rapidly on either side as the vibrations take more and more energy from the ball.

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, ... Ask question: Which give more “power”: tighter strings or looser strings on tennis racket? Then ask (if they get it wrong): can a person bounce higher from a hardwood floor or from a trampoline? demo

The “Trampoline” Effect: A Closer Look Ideal Situation: like person on trampoline kbat kball: most of energy stored in bat: e  ebat ebat  1: energy stored in bat returned  e  1, independent of eball For aluminum bat kbat  7kball: ~15% of energy stored in bat ebat  1: energy stored in bat returned  e  1.2 eball “BPF” = 1.20 For wood bat kbat  50kball: ~2% of energy stored in bat ebat doesn’t matter  e  eball Ask question: Which give more “power”: tighter strings or looser strings on tennis racket? Then ask (if they get it wrong): can a person bounce higher from a hardwood floor or from a trampoline?

The “Trampoline” Effect: A Closer Look Bending Modes vs. Hoop Modes 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 kbat  (t/R)3: small in barrel  more energy stored f (1-2 kHz) < 1/   energy mostly restored Net Effect: e > e0 “BPF”  e/e0 = 1.20-1.35! Little effect of bending modes at sweet spot.

Modal analysis: Dan Russell and AMN bending modes hoop modes hoop modes

COR vs. Hoop Mode Frequency Energy left in hoop vibrations

Where Does the Energy Go?

Some Interesting Consequences (work in progress) e/e0 increases with … Ball stiffness Impact velocity Decreasing wall thickness Decreasing ball COR 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 s  kbat/kball e2  (1+se02 )/(s+1) e  1 for s << 1

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

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 

Bat Research Center, UML, Sherwood & amn, Aug. 2001 Not Corked DATA Corked COR: 0.445  0.005 0.444  0.005 Conclusions: no trampoline effect! no advantage to corked for home run hitter possible advantage for “contact” hitter calculation

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

And in conclusion... Thanks for inviting me here I love talking about this stuff, so ask me lots of questions!