Transcript Baseball: It's Not Nuclear Physics (or is it?!)

Baseball and Physics
1927 Yankees:
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ever assembled
1927
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Greatest physics team
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How Does a Baseball Bat Work: The Physics of the Ball-Bat Collision
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Introduction: 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
bat is very flexible!

hands don’t matter!

GOAL: maximize ball exit speed vf
1 mph  4-5 ft

Question: What does vf depend on?
How Does a Baseball Bat Work: The Physics of the Ball-Bat Collision
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Kinematics: Reference Frames
“Lab” Frame
Bat Rest Frame
vrel
vball vbat
vf
eAvrel
vf = eA vball + (1+eA) vbat
eA  “Collision Efficiency” = “Ball Exit Speed Ratio” - 0.5
BESR
• property of ball & bat
• weakly dependent on vrel
• near “sweet spot” eA  0.2  vf  0.2 vball + 1.2 vbat
Conclusion: vbat much more important than vball
How Does a Baseball Bat Work: The Physics of the Ball-Bat Collision
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Conservation Laws and eA



Momentum Conservation
vball,f = eAvball vbat,f = r(1+eA)vball
r = mball/mbat
vball
Coefficient of Restitution
e  (vball,f+vbat,f)/vball  e A  e - r
1 r
Vball,f
Vbat,f
Energy Conservation
 e-r 
fball = 

1 r 
2
1 e 

1

r


frecoil = r 
2
1  e2
fdis =
1 r
How Does a Baseball Bat Work: The Physics of the Ball-Bat Collision
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Kinematics:
recoil factor r and coefficient of restitution e
e-r
eA 
1 r
=
.
.
CM
+
b
m ball
m ball m ballb 2
r 


m bat,eff m bat I bat,CM
 0.16 + 0.08

0.24
•
•
•
•
•
•
•
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
• All things equal, want r small
• But….
How Does a Baseball Bat Work: The Physics of the Ball-Bat Collision
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Kinematics:
Where Does the Energy Go?
=
.
.
CM
+
• Hit ball
• Recoil of Bat (r)
• Dissipation in ball and bat (e)
b
m ball
m ball m ballb 2
r 


m bat,eff m bat I bat,CM
 0.16 + 0.08

0.24
e-r
eA 
1 r
• All things equal, want r small
• But….
How Does a Baseball Bat Work: The Physics of the Ball-Bat Collision
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What is the Ideal Bat Weight?
(½ mv2)
Conclusion from Players: Lighter seems to be better!
How Does a Baseball Bat Work: The Physics of the Ball-Bat Collision
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Crisco/Greenwald Batting Cage Study
50
48

knob
46
(rad/s)
vbat  I-0.3
44
42
vbat  I-0.5
40
1.5 1.55 1.6 1.65 1.7 1.75 1.8 1.85 1.9
I
knob
4
2
(10 oz-in )
•vBAT(6”) = 1.2 mph/(1000 oz-in2) (vf=1.5  0.3 mph)
How Does a Baseball Bat Work: The Physics of the Ball-Bat Collision
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Kinematics: Coefficient of Restitution (e):
(Energy Dissipation)
“bounciness” of ball e 
• in CM frame: Ef/Ei =
e2
v rel, f
v rel,i
• 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!
How Does a Baseball Bat Work: The Physics of the Ball-Bat Collision
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COR: Is the Ball “Juiced”?
MLB: e = 0.546  0.032 @ 58 mph on massive rigid surface
R (ft)
COR Measurements
0.60
440
Lansmont
MLB/UML
COR
0.50
UML/BHM
MLB specs
0.45
0.40
Distance vs. COR
"90+70" collision
Lansmont/CPD
0.55
360
Briggs, 1945
60
80
100
120
equivalent impact speed (mph)
400
140
320
0.4
*
0.45
*
~ 35 '
0.5
cor
How Does a Baseball Bat Work: The Physics of the Ball-Bat Collision
0.55
0.6
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Putting it all together (rigid bat)...
12
10
vf = eA vball + (1+eA) vbat
8
6
4
e
CM
v (mph)
2
A
110
0
0.3
-2
0
5
100
10
90
15
20
25
30
eA
0.25
80
70
0.2
vf
60
vbat
50
0.15
40
30
16
18
20
22
24
26
28
30
0.1
32
z (inches)
How Does a Baseball Bat Work: The Physics of the Ball-Bat Collision
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Putting it all together (realistic bat)...
vf = eA vball + (1+eA) vbat
e
CM
v (mph)
A
0.3
110
100
0.25
90
80
vf
70
eA
0.2
60
0.15
50
40
30
16
18
20
22
24
26
28
30
0.1
32
z (inches)
How Does a Baseball Bat Work: The Physics of the Ball-Bat Collision
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III. Dynamics Model for Ball-Bat Colllision:
Accounting for Energy Dissipation

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)
How Does a Baseball Bat Work: The Physics of the Ball-Bat Collision
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The Essential Physics: A Toy Model
bat
ball
Mass= 1
2
4
e
A
rigid
0.7
 << 1
0.6
0.5
m on Ma
 >> 1
(1 on 2)
m on Ma+Mb
0.4
(1 on 6)
flexible
0.3
0
2
4

6
8
10
How Does a Baseball Bat Work: The Physics of the Ball-Bat Collision
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02
A Dynamic Model of the Bat-Ball Collision
51
y
20
01
Euler-Bernoulli Beam Theory‡
5
2  2y 
2y
 EI 2   A 2  F(z, t)
2 
z  z 
t
0
5-
y
-10
z
• Solve eigenvalue problem for free oscillations (F=0)
-15
-20
0
 normal modes (yn, n)
5
01
51
02
52
03
53
• 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
How Does a Baseball Bat Work: The Physics of the Ball-Bat Collision
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Normal Modes of the Bat
Louisville Slugger R161 (33”, 31 oz)
f1 = 177 Hz
f3 = 1179 Hz
f2 = 583 Hz
nodes
f4 = 1821 Hz
Can easily be measured (modal analysis)
0
5
10
15
20
25
30
35
How Does a Baseball Bat Work: The Physics of the Ball-Bat Collision
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Measurements via Modal Analysis
Louisville Slugger R161 (33”, 31 oz)
FFT(R)
1
0.15
0.5
FFT
582
R
0
1181
0.1
-0.5
-1
-1.5
0
5
10
t (ms)
15
20
2400
frequency
Expt Calc
barrel node
Expt Calc
179
582
1181
1830
26.5
27.8
29.0
30.0
177
583
1179
1821
1830
179
0.05
26.6
28.2
29.2
29.9
0
0
500
1000
1500
frequency (Hz)
2000
2500
Conclusion: free vibrations
of bat can be well characterized
How Does a Baseball Bat Work: The Physics of the Ball-Bat Collision
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Model for the Ball
force (pounds)
4
1 10
8000
2
approx quadratic
6000
F=kxn
4000
F=kxm
2000
1.6
 (ms)
1.2
collision time versus impact speed
0.8
0
0
Force (lb)
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.4
compression (inches)
0
20
40 60 80 100 120 140
impact speed (mph)
10000
8000
3-parameter problem:
160 mph
k
6000
4000
n  v-dependence of 
80 mph
2000
m  COR of ball with rigid surface
0
0
0.2
0.4
0.6
0.8
time (ms)
How Does a Baseball Bat Work: The Physics of the Ball-Bat Collision
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Putting it all together….
y 2n (x 0 )F(s, t)
d 2q n
2
 n qn 
2
dt
A
d 2 y ball
m ball
 - F(s, t)
2
dt
s   q n ( t ) y n ( x 0 ) - y ball ( t)
n
ball compression
Procedure:
• specify initial conditions
• numerically integrate coupled equations
• find vf = ball speed after ball and bat separate
How Does a Baseball Bat Work: The Physics of the Ball-Bat Collision
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General Result
 I y (x ) 
2if t
0
En  
  F (t )e n dt
 2 A  0
2

2
n
2
energy in nth mode
1
Force (lb)
Fourier transform
10000
0.8
8000
160 mph
0.6
6000
0.4
4000
0.2
80 mph
2000
0
0
0
0
0.2
0.4
time (ms)
0.6
0.8
0.5
1
1.5
2
f
Conclusion: only modes with fn  < 1 strongly excited
How Does a Baseball Bat Work: The Physics of the Ball-Bat Collision
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Results: Ball Exit Speed
v
final
Louisville Slugger R161
33-inch/31-oz.
wood bat
v /v
/v
CM
initial
final
node
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)
only lowest mode excited
32
0
23
24
25 26 27 28 29 30
distance from knob (inches)
31
lowest 4 modes excited
Conclusion: essential physics under control
How Does a Baseball Bat Work: The Physics of the Ball-Bat Collision
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1.4
Trey Crisco's Batting Cage Data
(wood)
1.2
1
calculation
0.8
eeff/e
0.6
0.4
0.2
0
0
0.05
0.1
0.15
0.2
0.25
0.3
distance from barrel (m)
How Does a Baseball Bat Work: The Physics of the Ball-Bat Collision
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Application to realistic conditions:
(90 mph ball; 70 mph bat at 28”)
% Energy
70
(a)
60
rigid recoil
losses in
ball
50
CM
40
nodes
30
100
20
rigid bat
80
v (mph)
10
ball
30
0
flexible bat
f
16
25
60
Louisville Slugger
R161 (33", 31 oz)
40
20
vibra tions
20
24
28
distance from knob (inches)
32
24
28
32
(b)
Total
20
1
15
10
16
20
2
3
5
>3
0
16
20
24
28
distance from kno b (cm)
How Does a Baseball Bat Work: The Physics of the Ball-Bat Collision
32
Page 24
The “sweet spot”
vibrational velocity at handle
displacement at handle
80
impact @ 24.8"
40
24.8"
26.8"
y 0
26.8"
28.8"
-40
28.8"
-80
0
2
4
6
8
t (ms)
1. Maximum vf (~28”)
0
10
% Energy
70
rigid recoil
60
2. Minimum vibrational energy (~28”)
3. Node of fundamental (~27”)
4. Center of Percussion (~27”)
5. “don’t feel a thing”
2
4
6
8
10
t (ms)
losses in ball
50
40
30
vibrations
20
10
0
ball
16
20
24
28
distance from knob (inches)
How Does a Baseball Bat Work: The Physics of the Ball-Bat Collision
32
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Boundary conditions
3
0.5
Displacement at 5”
2
R161A: free vs. pivoted
pivoted-rigid
0.4
1
0.3
free-rigid
y (mm)
0
e
A
-1
impact at 27"
0.2
0.1
flexible:
free or rigid
0
-2
-0.1
-3
0
0.5
1
1.5
t (ms)
Conclusions:
2
-0.2
16
20
24
28
32
z (inches from knob)
• size, shape, boundary conditions at far end don’t matter
• hands don’ t matter!
How Does a Baseball Bat Work: The Physics of the Ball-Bat Collision
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displacement (mm)
10
8
0.1 ms intervals
6
4
T= 0-1 ms
pivot point
2
0
-2
-4
Time evolution200
of the bat
impact point
150
1 ms intervals
100
T= 1-10 ms
pivot point
50
0
impact point
-50
0
5
10
15
20
25
30
distance from knob (inches)
How Does a Baseball Bat Work: The Physics of the Ball-Bat Collision
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20
Time evolution
15
20
of the bat
10
15
20
5
10
1.0 ms 150
205
0.8 ms
0.6 ms
10
-5
150
5
-10
10
-5
20
0
-15
5
-10
15
-5
-20
0
10
0.4 ms -15
0
5
10
15
20
25
30
35
-15 0
5
10
15
20
25
30
35
5
10
15
30
35
-10
-55
-20
0.2 ms -100
-20
-15
-5 0
20
25
impact
location
-20
-10 Does a Baseball Bat Work: The Physics of the Ball-Bat Collision
How
Page 28
Wood versus Aluminum
Kinematics
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
Dynamics
xCM
k0
kh
f1
f2
kball/k
eeff/e
Stiffer for bending
* Less energy lost due to vibrations
More compressible
* COReff larger
Wood
Aluminum
22.7
20.9
9
9.4
24.4
22.9
156
222
548
721
0.02
0.1
1
1.1-1.2
How Does a Baseball Bat Work: The Physics of the Ball-Bat Collision
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Effect of Bat on COR: Local Compression

CM energy shared between ball and bat
Ebat/Eball  kball/kbat  xbat/ xball

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
tennis ball/racket
>10%
larger!
How Does a Baseball Bat Work: The Physics of the Ball-Bat Collision
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Wood versus Aluminum:
Dynamics of “Trampoline” Effect
bending modes
bell modes
“bell” modes:
0
ω
t
R2
 t
k  
R
1000
3
2000
3000
frequency (Hz)
4000
“ping” of bat
• Want k small to maximize stored energy
• Want >>1 to minimize retained energy
• Conclusion: there is an optimum 
How Does a Baseball Bat Work: The Physics of the Ball-Bat Collision
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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)
0.8
1
Bat Vibrational E
50
0
0
0.2
0.4
0.6
t (ms)
How Does a Baseball Bat Work: The Physics of the Ball-Bat Collision
0.8
1
Page 32
Which Performance Metric?
eA
0.2
0.15
eA or COR or vf
0.1
e 0.05
A
0
(my most important slide) -0.05
wood: BPF=0.99
aluminum: BPF=1.07
-0.1
-0.15
0
5
10
15
distance from tip (inches)
100
0.5
vf
90
COR
0.4
80
v (mph)
f
70
e
0.3
0.2
60
wood: BPF=0.99
aluminum: BPF=1.07
50
40
0
5
10
distance from tip (inches)
wood: BPF=0.99
aluminum: BPF=1.07
0.1
15
0
0
5
10
15
distance from tip (inches)
How Does a Baseball Bat Work: The Physics of the Ball-Bat Collision
Page 33
Things 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
 velocity dependence
 wall thickness dependence

Effect of “corking” the bat
How Does a Baseball Bat Work: The Physics of the Ball-Bat Collision
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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 probably outperforms wood!
How Does a Baseball Bat Work: The Physics of the Ball-Bat Collision
Page 35