How Does a Baseball Bat Work: Dynamics of the Ball-Bat Collision Page 2
Baseball and Physics
1927 Yankees:
Greatest baseball team
ever assembled
1927
Solvay Conference:
Greatest physics team
ever assembled
MVP’s
How Does a Baseball Bat Work: Dynamics of the Ball-Bat Collision Page 3
Hitting the BaseballHitting the Baseball
“...the most difficult thing to do in sports”
--Ted Williams
BA: .344SA: .634OBP: .483‡
HR: 521#521, September 28, 1960
‡career record
How Does a Baseball Bat Work: Dynamics of the Ball-Bat Collision Page 4
Introduction: Description of Ball-Bat CollisionIntroduction: Description of Ball-Bat Collision
forces large (>8000 lbs!) time is short (<1/1000 sec!) ball compresses, stops, expands kinetic energy potential energy bat compresses ball….ball bends bat hands don’t matter!
GOAL: maximize ball exit speed vf
vf 105 mph x 400 ft x/vf = 4-5 ft/mph
How to predict vf?
How Does a Baseball Bat Work: Dynamics of the Ball-Bat Collision Page 5
Kinematics: Reference Frames
vf = eA vball + (1+eA) vbat
Conclusion: vbat much more important than vball
vball vbat
vf
“Lab” Framevrel
eAvrel
Bat Rest Frame
eA “Apparent Coefficient of Restitution” = “BESR” - 0.5
• property of ball & bat• weakly dependent on vrel
0.2 vf 0.2 vball + 1.2 vbat
How Does a Baseball Bat Work: Dynamics of the Ball-Bat Collision Page 6
Kinematics: Conservation Laws
(Accounting for eA)
r1
r-e eA
v
eAvm2
m1
m1
r bat recoil factor = mball/mbat,eff
e “Coefficient of Restitution”
How Does a Baseball Bat Work: Dynamics of the Ball-Bat Collision Page 7
Kinematics: bat recoil factor
typical numbers • mball = 5.1 oz• mbat = 31.5 oz• k = 9.0 in• b = 6.3 in• r = .24• e = 0.5• eA = 0.21
. .CM .
b
k
b 1
m
m r
2
2
bat
ball
= +
0.16 x 1.49
0.24
r1
r-e eA
• All things equal, want r small• But….
How Does a Baseball Bat Work: Dynamics of the Ball-Bat Collision Page 8
All things are not equal Mass & Mass Distribution affect bat speed
Conclusion:mass of bat matters….but probably not a lot
60
70
80
90
100
110
120
20 30 40 50 60
vbat
(mph)
Mbat
(oz)
Mbat
vbat
2 constant
(Mbat
+M0)v
bat
2 constant
vbat
constant
vf vs. M
bat
40
50
60
70
80
90
100
20 30 40 50 60
vf (mph)
Mbat
(oz)
Mbat
vbat
2 constant
(Mbat
+M0)v
bat
2 constant
vbat
constant
vbat
vs. Mbat
How Does a Baseball Bat Work: Dynamics of the Ball-Bat Collision Page 9
• in CM frame: Ef/Ei = e2
• massive rigid surface: e2 = hf/hi
• typically e 0.5~3/4 CM energy dissipated!
• probably depends on impact speed
• depends on ball and bat!
Kinematics: Coefficient of Restitution (e):
(Energy Dissipation)“bounciness” of ball
i rel,
f rel,
v
v e
How Does a Baseball Bat Work: Dynamics of the Ball-Bat Collision Page 10
COR: Is the Ball “Juiced”?
MLB: e = 0.546 0.032 @ 58 mph on massive rigid surface
320
360
400
440
0.4 0.45 0.5 0.55 0.6
R (ft)
cor
*
*~ 35 '
Distance vs. COR "90+70" collision
0.40
0.45
0.50
0.55
0.60
60 80 100 120 140equivalent impact speed (mph)
COR
Briggs, 1945
UML/BHM
Lansmont
MLB specs
MLB/UML
COR Measurements
Lansmont/CPD
r r)1(v
v e
iball,
fball, For ball on stationary bat:
How Does a Baseball Bat Work: Dynamics of the Ball-Bat Collision Page 11
Putting it all together…..
vf = eA vball + (1+eA) vbat
30
40
50
60
70
80
90
100
110
0.1
0.15
0.2
0.25
0.3
16 18 20 22 24 26 28 30 32
eAv (mph)
z (inches)
vbat
vf
eA
CM
-2
0
2
4
6
8
10
12
0 5 10 15 20 25 30
How Does a Baseball Bat Work: Dynamics of the Ball-Bat Collision Page 12
More Realistic Analysis
30
40
50
60
70
80
90
100
110
0.1
0.15
0.2
0.25
0.3
16 18 20 22 24 26 28 30 32
eAv (mph)
z (inches)
vf e
A
CM
vf = eA vball + (1+eA) vbat
How Does a Baseball Bat Work: Dynamics of the Ball-Bat Collision Page 13
Collision excites bending vibrations in bat
Ouch!! Thud!!
Sometimes broken bat
Energy lost lower vf (lower e)
Bat not rigid on time scale of collision
What are the relevant degrees of freedom?
see AMN, Am. J. Phys, 68, 979 (2000)
III. Dynamics Model for Ball-Bat Colllision:
Accounting for Energy Dissipation
How Does a Baseball Bat Work: Dynamics of the Ball-Bat Collision Page 14
0.3
0.4
0.5
0.6
0.7
0 2 4 6 8 10
eA
>> 1
m on Ma+Mb
(1 on 6)
ball bat
<< 1
m on Ma
(1 on 2)
The Essential Physics: A Toy Model
Mass= 1 2 4
rigid
flexible
How Does a Baseball Bat Work: Dynamics of the Ball-Bat Collision Page 15
20
-2 0
-1 5
-1 0
-5
0
5
10
15
20
0 5 10 15 20 25 30 35
y
z
y
A Dynamic Model of the Bat-Ball Collision
• Solve eigenvalue problem for free oscillations (F=0)
normal modes (yn, n)
• Model ball-bat force F
• Expand y in normal modes
• Solve coupled equations of motion for ball, bat‡ Note for experts: full Timoshenko (nonuniform) beam theory used
Euler-Bernoulli Beam Theory‡
t)F(z, t
yA
z
yEI
z 2
2
2
2
2
2
How Does a Baseball Bat Work: Dynamics of the Ball-Bat Collision Page 16
Normal Modes of the Bat
Louisville Slugger R161 (33”, 31 oz)
Can easily be measured (modal analysis)0 5 10 15 20 25 30 35
f1 = 177 Hz
f2 = 583 Hz
f3 = 1179 Hz
f4 = 1821 Hznodes
How Does a Baseball Bat Work: Dynamics of the Ball-Bat Collision Page 17
-1.5
-1
-0.5
0
0.5
1
0 5 10 15 20
R
t (ms)
0
0.05
0.1
0.15
0 500 1000 1500 2000 2500
FFT(R)
frequency (Hz)
179
582
1181
1830
2400
frequency barrel nodeExpt Calc Expt Calc 179 177 26.5 26.6 582 583 27.8 28.21181 1179 29.0 29.21830 1821 30.0 29.9
Measurements via Modal Analysis
Louisville Slugger R161 (33”, 31 oz)
Conclusion: free vibrationsof bat can be well characterized
FFT
How Does a Baseball Bat Work: Dynamics of the Ball-Bat Collision Page 18
0.4
0.8
1.2
1.6
2
0 20 40 60 80 100 120 140
(ms)
impact speed (mph)
collision time versus impact speed
Model for the Ball
3-parameter problem:
k
n v-dependence of
m COR of ball with rigid surface
0
2000
4000
6000
8000
1 104
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
force (pounds)
compression (inches)
approx quadratic
F=kxn
F=kxm
0
2000
4000
6000
8000
10000
0 0.2 0.4 0.6 0.8
Force (lb)
time (ms)
160 mph
80 mph
How Does a Baseball Bat Work: Dynamics of the Ball-Bat Collision Page 19
Putting it all together….
t)F(s,- dt
ydm
A
t))F(s,(xyq
dt
qd
2ball
2
ball
02n
n2n2
n2
ball compression
Procedure: • specify initial conditions• numerically integrate coupled equations• find vf = ball speed after ball and bat separate
t)(y- )x(y)t(qs ball0nn
n
How Does a Baseball Bat Work: Dynamics of the Ball-Bat Collision Page 20
Conclusion: only modes with fn < 1 strongly excited
General Result2
2
0
02n
2
n dt)( A2
)x(yI E tifnetF
0
2000
4000
6000
8000
10000
0 0.2 0.4 0.6 0.8
Force (lb)
time (ms)
160 mph
80 mph
0
0.2
0.4
0.6
0.8
1
0 0.5 1 1.5 2f
energy in nth mode
Fourier transform
How Does a Baseball Bat Work: Dynamics of the Ball-Bat Collision Page 21
Results: Ball Exit SpeedLouisville Slugger R16133-inch/31-oz. wood bat
Conclusion: essential physics under control
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
23 24 25 26 27 28 29 30 31
vfinal
/vinitial
distance from knob (inches)
data from Lansmont BBVCbat pivoted about 5-3/4"
vinitial
=100 mph
rigid bat
flexible bat
nodes
only lowest mode excited lowest 4 modes excited
0
0.1
0.2
0.3
0.4
16 20 24 28 32
vfinal
/vinitial
distance from knob (inches)
rigid bat
flexible bat
CM node
data from Rod Crossfreely suspended bat
vi = 2.2 mph
How Does a Baseball Bat Work: Dynamics of the Ball-Bat Collision Page 22
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0 0.05 0.1 0.15 0.2 0.25 0.3
eeff
/e
distance from barrel (m)
Trey Crisco's Batting Cage Data(wood)
calculation
How Does a Baseball Bat Work: Dynamics of the Ball-Bat Collision Page 23
0
10
20
30
40
50
60
70
16 20 24 28 32
% Energy
rigid recoil
ball
vibra tions
losses inball
(a)
0
5
10
15
20
25
30
16 20 24 28 32
distance from knob (cm)
Total
1
3
>3
2
(b)
20
40
60
80
100
16 20 24 28 32
vf (mph)
distance from knob (inches)
flexible bat
rigid bat
Louisville SluggerR161 (33", 31 oz)
CM nodes
Application to realistic conditions:
(90 mph ball; 70 mph bat at 28”)
How Does a Baseball Bat Work: Dynamics of the Ball-Bat Collision Page 24
1. Maximum vf (~28”)
2. Minimum vibrational energy (~28”)
3. Node of fundamental (~27”)
4. Center of Percussion (~27”)
5. “don’t feel a thing”
The “sweet spot”
0
10
20
30
40
50
60
70
16 20 24 28 32
% Energy
distance from knob (inches)
rigid recoil
ball
vibrations
losses in ball
-80
-40
0
40
80
0 2 4 6 8 10
y
t (ms)
impact @ 24.8"
26.8"
28.8"
displacement at handle
0 2 4 6 8 10
vibrational velocity at handle
t (ms)
24.8"
26.8"
28.8"
How Does a Baseball Bat Work: Dynamics of the Ball-Bat Collision Page 25
-3
-2
-1
0
1
2
3
0 0.5 1 1.5 2
y (mm)
t (ms)
impact at 27"
Displacement at 5”
Conclusions:
• size, shape, boundary conditions at far end don’t matter
• hands don’ t matter!
Boundary conditions
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
16 20 24 28 32
eA
z (inches from knob)
R161A: free vs. pivoted
pivoted-rigid
free-rigid
flexible: free or rigid
How Does a Baseball Bat Work: Dynamics of the Ball-Bat Collision Page 26
Time evolution
of the bat
-4
-2
0
2
4
6
8
10
displacement (mm)
0.1 ms intervals
impact point
pivot point
-50
0
50
100
150
200
0 5 10 15 20 25 30distance from knob (inches)
1 ms intervals
impact point
pivot point
T= 0-1 ms
T= 1-10 ms
How Does a Baseball Bat Work: Dynamics of the Ball-Bat Collision Page 27
Wood versus AluminumKinematics
Length, weight, MOI “decoupled”
* shell thickness, added weight* fatter barrel, thinner handle
Weight distribution more uniform
* ICM larger (less rot. recoil)
* Ihandle smaller (easier to swing)
* less mass at contact point
DynamicsStiffer for bending
* Less energy lost due to vibrations
More compressible
* COReff larger
Wood AluminumxCM 22.7 20.9
k0 9 9.4
kh 24.4 22.9
f1 156 222
f2 548 721
kball/k 0.02 0.1
eeff/e 1 1.1-1.2
How Does a Baseball Bat Work: Dynamics of the Ball-Bat Collision Page 28
CM energy shared between ball and bat
Ball inefficient: 75% dissipated
Wood Bat kball/kbat ~ 0.02 80% restored eeff = 0.50-0.51
Aluminum Bat
kball/kbat ~ 0.10
80% restored eeff = 0.55-0.58
Effect of Bat on COR: Local CompressionEffect of Bat on COR: Local Compression
Ebat/Eball kball/kbat xbat/ xball
>10% larger!
tennis ball/racket
How Does a Baseball Bat Work: Dynamics of the Ball-Bat Collision Page 29
Wood versus Aluminum:
Dynamics of “Trampoline” Effect
“bell” modes: 3
2 R
t k
R
t
“ping” of bat
• Want k small to maximize stored energy
• Want >>1 to minimize retained energy
• Conclusion: there is an optimum
0 1000 2000 3000 4000frequency (Hz)
bending modes
bell modes
How Does a Baseball Bat Work: Dynamics of the Ball-Bat Collision Page 30
-0.2
-0.1
0
0.1
0.2
20 25 30
vf (mph)
distance from knob (inches)
wood versus aluminum
woodaluminum
150 mph ballstationary bat
Performance Comparison
40
60
80
100
20 25 30
vf (mph)
distance from knob (inches)
wood versus aluminum
wood
aluminum
70 mph ballpivoted bat
How Does a Baseball Bat Work: Dynamics of the Ball-Bat Collision Page 31
Things I would like to understand betterThings I would like to understand better
Relationship between bat speed and bat weight and weight distribution
Location of “physiological” sweet spot
Better model for the ball
Better understanding of trampoline effect for aluminum bat
Why is softball bat different from baseball bat?
Effect of “corking” the bat
How Does a Baseball Bat Work: Dynamics of the Ball-Bat Collision Page 32
Summary & Conclusions
• The essential physics of ball-bat collision understood
* bat can be well characterized
* ball is less well understood
* the “hands don’t matter” approximation is good
• Vibrations play important role
• Size, shape of bat far from impact point does not matter
• Sweet spot has many definitions
• Aluminum outperforms wood!