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  
v bat


1+r 
 1+r 
eA
vball vbat
vf
1+eA
r: bat recoil factor = mball/mbat,eff
(momentum and angular momentum conservation)
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. W
Bat Speed at 6" Point vs. MOI
1.06
fixed M
1.04
1.02
1.02
1
1
0.98
fixed MOI
1.04
dashed: n=0.25
solid: n=0.23
~(1/M)
0.25
0.98
0.96
0.96
0.94
6000
7000
8000
9000
10000
11000
24
2
MOI (oz-in )
25
26
27
28
29
30
31
32
W (oz)
Conclusions:
• bat speed depends more on I6 than M:
Ideal• vbat
weight/MOI
not easy to determine
1/4
bat ~ (1/I6)
• 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 
02

Step 1: Solve eigenvalue problem for
free vibrations
 2   2 yn 
2


EI


A

n yn
2 
2 
x  x 


Step 2: Ball-bat interaction (F)
modeled as nonlinear lossy spring
Step 3: Expand in normal modes and
solve
y(x,t )   q n (t )yn ( x)
n
Nuclear Chemistry Gordon Conference
2
51
y
20
01
5
0
5-
y
-10
z
-15
-20
d qn
F(t) yn ( z )
2
 n q n 
2
dt
A
June 19, 2003
0
5
01
51
02
52
03
53
Page 11
Normal Modes of the Bat:
demo
Modal Analysis
frequency domain
time domain
FFT(R)
0.15
1
582
0.5
R
FFT
0
1181
0.1
-0.5
1830
179
0.05
-1
-1.5
2400
0
5
10
t (ms)
15
20
f1 = 179 Hz
0
0
500
f2 = 582 Hz
1000
1500
frequency (Hz)
2000
2500
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
nodes
0.5
CM
f1 = 179 Hz
0.4
0.3
the “sweet spot”
0.2
0.1
0
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
CM
initial
final
node
/v
initial
nodes
0.35
0.4 rigid bat
0.3
rigid bat
0.25
0.3
0.2
flexible bat
0.2
0.15
data from Rod Cross
freely suspended bat
v = 2.2 mph
0.1
flexible bat
0.1
0.05
data from Lansmont BBVC
bat pivoted about 5-3/4"
=100 mph
v
initial
i
0
16
20
24
28
distance from knob (inches)
32
0
23
only lowest mode excited
24
25 26 27 28 29 30
distance from knob (inches)
31
lowest 4 modes excited
Conclusion: essential physics under control
Nuclear Chemistry Gordon Conference
June 19, 2003
Page 15
displacement (mm)
time evolution
10
0 - 1 ms
0.1 ms intervals
8
6
• rigid-body motion develops
only after few ms
4
2
0
• far end of bat has no effect on
-2
ball
-4
 knob moves after 0.6 ms
impact point
200
150
1-10 ms
1 ms intervals
 collision over after 0.6 ms
 nothing on knob end
matters
• size, shape
• boundary conditions
• hands
Nuclear Chemistry Gordon Conference
100
50
0
impact point
-50
0
June 19, 2003
5
10 15 20 25 30
distance from knob (inches)
Page 16
Vf independent of end support
Vf (mph)
120
110
100
90
4.75'' pivot
80
6.75'' pivot
free
70
swing/hit
60
50
60
70
80
90
100
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
% Energy Dissipated
Nodes
0.55
80
0.5
70
COR
0.45
60
Ball
0.4
50
0.35
40
0.3
30
0.25
20
0.2
10
Vibrations
0.15
0
2
4
6
8
10
12
Losses in ball
anti-correlated
with vibrations
in bat
0
14
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, ...
Nuclear Chemistry Gordon Conference
June 19, 2003
demo
Page 19
The “Trampoline” Effect: A Closer Look
e
ball-bat
1
0.9
e
= 0.5
0.8
e
= 1.0
ball
bat
0.7
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
100
k /k
bat
Ideal
Situation:
like
 For
wood batbat
For
aluminum
ball
person on trampoline
k

k7k
: most
of energy
stored
in bat:
e  ebat
k
50k
: ~2%
energy
stored
bat
kbat
~15%
ofofenergy
stored
ininbat
ball
batbat
ball:ball
e

1:
stored
ee
doesn’t
1: energy
energy
stored in
in bat
bat returned
returned
bat
matter
batbat

ee e

1,
of
e


1.2
e
“BPF”
=
1.20
eindependent
ball
ball
ball
Nuclear Chemistry Gordon Conference
June 19, 2003
Page 20
The “Trampoline” Effect:
A Closer Look
Bending Modes
kbat  R4: large in barrel
 little energy stored
vs.
Hoop Modes
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
COR-model
COR-expt
0.60
0.55
0.50
0.45
0.40
500
1000
1500
f
hoop
Nuclear Chemistry Gordon Conference
2000
(Hz)
June 19, 2003
Page 23
Where Does the Energy Go?
Energy (J)
Energy (J)
400
400
Ball KE
350
Wood Bat
300
Ball KE
350
Aluminum Bat
300
250
250
Ball PE
200
200
150
Ball PE
150
Bat Recoil KE
100
Bat Recoil KE
100
50
Bat Vibrational E
0
0
0.2
0.4
0.6
t (ms)
Nuclear Chemistry Gordon Conference
0.8
1
Bat Vibrational E
50
0
0
June 19, 2003
0.2
0.4
0.6
t (ms)
0.8
1
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
• possible advantage for
“contact” hitter
Nuclear Chemistry Gordon Conference
uncorked
90
corked
80
calculation
70
2
3
4
5
6
7
8
distance from knob (inches)
June 19, 2003
9
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
Nuclear Chemistry Gordon Conference
June 19, 2003
Page 29
And in conclusion...
 Thanks
for inviting me here
I
love talking about this stuff, so ask
me lots of questions!
Nuclear Chemistry Gordon Conference
June 19, 2003
Page 30