1. Introduction: Description of 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 goal...
large hit ball speed
Define COR of ball
Usual definition
Energy terms
Effect of bat on COR
Bending vibrations
Shell vibrations
Model for the collision

2. Kinematics of the Ball-Bat Collision
vball
vbat vf
r  bat recoil factor = mball/Mbat,effective
e  Coefficient of Restitution (COR)
typical numbers:
r  e  0.50  vf = 0.2 vball vbat
Relationship between eA and COR,r uses only conservation of momentum (for a free bat), conservation of angular momentum (for free or pivoted bat), and the definition of COR. For the experts, the definition is ratio of relative ball-bat speed after to before the collision, where only the rigid body motion of the bat is implied. There are no approximations in arriving at this expression. Given COR and inertial properties of bat, eA can be predicted. Conversely, given measurements of eA, one can get COR (and BPF). In fact, this is the basis of the Gilman proposal to measure the BPF. The same physics is contained in the Brandt technique, but the algebra is different because the recoil speed of the bat is measured rather than the post-impact speed of the ball.
Keep in mind the following: angular momentum and/or momentum conservation implies that if the initial ball or bat speed is known, then a measurment of either the exit speed of the ball or the recoil speed of the bat uniquely determines the other. You do not need to measure both, unless you desire some redundancy. Lansmont has the ability to measure both.
Note: this talk focuses entirely on COR

3. COR and Energy Dissipation
e  COR  vrel,after/vrel,before
in CM frame: (final KE/initial KE) = e2
e.g., drop ball on hard floor: COR2 = hf/hi  0.25
typically COR  0.5
~3/4 CM energy dissipated!
depends on impact speed
mostly a property of ball but…
the bat matters too!
vibrations  , “trampoline” effect 
Figure out way to combine r with COR into one slide

4. Accounting for Energy Dissipation:
Dynamic Model for Ball-Bat Colllision
Collision excites bending vibrations
Ouch!! Thud!! Sometimes broken bat
Energy lost  lower COR, vf
Find lowest mode by tapping
Reduced considerably if
Impact is at a node
Collision time (~0.6 ms) >> Tvib
So far, just kinematic, plus a phenomonlogoical treatment of energy losses via COR. Now we want to dissect th ecollision 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)

5. The Essential Physics: A Toy Model
ball
bat
Mass= 
rigid limit
  1
1 on 
Reductin of COR due to vibratins equivalent to reduction of bat mass
flexible limit
 1
1 on 2

6. The Details: A Dynamic Model20 y z
Step 1: Solve eigenvalue problem for free vibrations
Step 2: Nonlinear lossy spring for ball-bat interaction
Step 3: Expand in normal modes and solve

7. Normal Modes of the Bat Louisville Slugger R161 (34”, 31 oz)
f1 = 177 Hz
f2 = 583 Hz
f3 = 1179 Hz
f4 = 1821 Hz
 prop to k2 for lowest frequencies, imples considerable dispersion
Can easily be measured: Modal Analysis

8.  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 

9. COR maximum near 2nd node
Vibrations and the COR
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 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 maximum near 2nd node

10. Some interesting insights:
24”
27”
30”
Center of Percussion close to lowest 27”
Coincides neither with max 29”
…nor with max. vf
Far end of bat doesn’t matter
mass, grip, …
Velocity at handle…..
27” near node of mode 1
30” near node of mode 3
hands don’t matter….
1. Near cop …. Collision transmits no forct to hands, vice versa
2. Pulse propagation time… no clamping effect
3. Forcethat hands could exert << ball-bat force
Where is batter’s sweet spot?

11. Time evolution of the bat T= 0-1 ms Ball leaves bat T= 1-10 ms
Conclusions:
Knob end doesn’t matter
Batter’s grip doesn’t matter
vibrations and rigid motion
indistinguishable on
short time scale

12. Bounce superballs from beam (Rod Cross)
Conclusion:
Nothing on far end
of beam matters

13. 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 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.

14. The “Trampoline” Effect:
A Closer Look
Compressional energy shared between ball and bat
PEbat/PEball = kball/kbat (= s)
PEball mostly dissipated (75%)
BPF = Bat Proficiency Factor  e/e0
Ideal Situation: like person on trampoline
kball >>kbat: most of energy stored in bat
f >>1: stored energy returned
e2  (s+e02)/(s+1)
 1 for s >>1
 eo2 for s <<1
Tennis racket like Al bat.
Al bat: possibly 10% in COR==>7 mph==>35’ or more
technology of Al bats: thinner wall==>increase r
bat: “tennis racket”-like efficient even for dead ball
measurement techniques:
Brndt-technique: measure incident ball speed and recoil bat rotational speed for ball on stationary bat
BHM: measure incident and rebound ball speeds using BHM to swing bat
BBVC: measure incident and rebound ball speed plus recoil bat angular speed for ball on stationary pivoted bat
New NCAA restrictions

15. Trampoline Effect: Toy Model, revisited
ball
bat
Mass= 

16. The “Trampoline” Effect: A Closer Look
Bending Modes vs Shell Modes
k  R4: large in barrel
 little energy stored
f (170 Hz, etc) > 1/
 energy lost to vibrations
Net effect: BPF  1
k  (t/R)3: small in barrel
 more energy stored
f (1-2 kHz) < 1/ 
 energy mostly restored
Net Effect: BPF > 1

17. Where Does the Energy Go?

18. Some Interesting Consequences (work in progress)
BPF increases with …
Ball stiffness
Impact velocity
Decreasing wall thickness
Decreasing ball COR
Note: effects larger for “high-s” than for “low-s” bats
“Tuning a bat”
Tuning due to balance between storing energy (k small) and returning it (f large)
Tuning not related to phase of vibration at time of ball-bat separation
s  kball/kbat
e2  (s+e02 )/(s+1)
BPF = e/e0

22. 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
should be tested experimentally