REFERENCES
The Physics of Baseball, Robert K. Adair (Harper Collins, New York, 1990), ISBN 0-06-096461-8
The Physics of Sports, Angelo Armenti (American Institute of Physics, New York, 1992), ISBN 0-88318-946-1
www.npl.uiuc.edu/~a-nathan/pob H. Brody, AJP 54, 640 (1986); AJP 58, 756 (1990)
P. Kirkpatrick, AJP 31, 606 (1963) L. Van Zandt, AJP 60, 172 (1992) R. Cross, AJP 66, 772 (1998); AJP 67, 692 (1999)
AMN, AJP 68, to appear in November, 2000
University of Iowa Colloquium, October 12, 2000 Page 2
Is This Heaven?
University of Iowa Colloquium, October 12, 2000 Page 3
1927
N. Y. Yankees
Baseball and Physics: Murderers Rows
1927 Solvay Conf.
University of Iowa Colloquium, October 12, 2000 Page 4
A Philosophical Note:
“…the physics of baseball is not the clean, well-defined physics of fundamental matters but the ill-defined physics of the complex world in which we live, where elements are not ideally simple and the physicist must make best judgments on matters that are not simply calculable…Hence conclusions about the physics of baseball must depend on approximations and estimates….But estimates are part of the physicist’s repertoire…a competent physicist should be able to estimate anything ...”
“The physicist’s model of the game must fit the game.” “Our aim is not to reform baseball but to understand it.”
---Bob Adair
in “The Physics of Baseball”, May, 1995 issue of Physics Today
University of Iowa Colloquium, October 12, 2000 Page 5
Hitting the Baseball
“...the most difficult thing to do in sports”
--Ted Williams BA: SA: OBP: HR: .344 .634 .483 521
#521, September 28, 1960
University of Iowa Colloquium, October 12, 2000 Page 6
Here’s Why…..
(Courtesy of Robert K. Adair)
University of Iowa Colloquium, October 12, 2000 Page 7
Description of Ball-Bat Collision
forces large (>8000 lbs!) time is short (<1/1000 sec!) ball compresses, stops, expands kinetic energy potential energy bat affects ball….ball affects bat hands don’t matter! GOAL: maximize ball exit speed vf
vf 105 mph x 400 ft x/vf = 4-5 ft/mph
What aspects of collision lead to large v f ?
University of Iowa Colloquium, October 12, 2000 Page 8
How to maximize vf ?
What happens when ball and bat collide? The simple stuff: kinematics frames of reference conservation of momentum conservation of angular momentum The really interesting stuff: energy dissipation compression/expansion of ball vibrations of the bat
University of Iowa Colloquium, October 12, 2000 Page 9
Kinematics: Frames of Reference
vball,f Avball,i (1 A)vbat,i
• Expect A weakly dependent on impact speed • NCAA: * Bat Exit Speed Ratio (BESR) = A+0.5 * BESR < 0.728 A < 0.228 • For typical bat… vball,f = 0.2 vball,i + 1.2 vbat,i
Conclusion: vbat much more important than vball
Question: what bat/ball properties make BESR large?
University of Iowa Colloquium, October 12, 2000 Page 10
Kinematics: Conservation Laws
vball,f
e-r 1 e vball,i 1 r vbat,i 1 r
r recoil factor 0.24
e Coefficient of Restitution 0.5 vball,f = 0.2 vball,i + 1.2 vbat,i
University of Iowa Colloquium, October 12, 2000 Page 11
Energy in Bat Recoil
• Important Bat Parameters:
mbat, xCM, ICM=mbatk2CM
CM
. .
z
Translation Rotation
mball z2 r (1 2 ) mbat k CM
0.17 (1 + 0.41) = 0.24
Want r small to mimimize recoil energy
Conclusion: All things being equal, want mbat, Ibat large
University of Iowa Colloquium, October 12, 2000 Page 12
But… All things are not equal Mass & Mass Distribution affect bat speed
bat speed vs mass
ball speed vs mass
Conclusion: mass of bat matters….but probably not a lot
see Watts & Bahill, Keep Your Eye on the Ball, 2nd edition, ISBN 0-7167-3717-5
University of Iowa Colloquium, October 12, 2000 Page 13
Energy Dissipated: Coefficient of Restitution (e):
• in CM frame: Ef/Ei = e2
vrel,f “bounciness” of ball e vrel,i
• massive rigid surface: e2 = hf/hi
• typically e 0.5 ~3/4 CM energy dissipated!
• depends on ball, surface, speed,...
• is the ball “juiced”?
University of Iowa Colloquium, October 12, 2000 Page 14
Major League Baseball receives report on quality of baseballs Study finds no significant performance differences between 1999 and 2000 baseballs Posted on June 28, 2000 Major League Baseball has received the results of a study conducted by the UMass-Lowell Baseball Research Center regarding the performance of the baseballs used in Major League games, it was announced today. The study, in which 1999 and 2000 Major League baseballs and 2000 Minor League baseballs were tested for performance comparisons and specification compliance, revealed no significant performance differences and verified that the baseballs used in Major League games meet performance specifications. In all, Rawlings and Major League Baseball provided 192 baseballs to the research center for testing.
University of Iowa Colloquium, October 12, 2000 Page 15
COR and the “Juiced Ball” Issue MLB: e = 0.546 0.032 @ 58 mph on massive rigid surface
0.60
COR Measurements
Lansmont
R (ft) 440
0.55
Lansmont/CPD
MLB/UML UML/BHM MLB specs
COR
0.50
Distance vs. COR "90+70" collision
400
0.45
Briggs, 1945
360
*
0.45
* ~ 35 '
0.5 cor 0.55 0.6
0.40
60
80
100
120
140
320 0.4
equivalent impact speed (mph)
University of Iowa Colloquium, October 12, 2000 Page 16
Effect of Bat on COR: Local Compression
tennis ball/racket
CM energy shared between ball and bat
Ball is inefficient: 75% dissipated Wood Bat kball/kbat ~ 0.02 80% restored eeff = 0.50-0.51
Ebat/Eball kball/kbat xbat/ xball
Aluminum Bat kball/kbat ~ 0.10 80% restored eeff = 0.55-0.58 “trampoline effect”
>10% larger!
Recent BPF data:
(Lansmont BBVC/Trey Crisco)
0.99 wood 1.12 aluminum
Bat Proficiency Factor eeff/e
More later on wood vs. aluminum
University of Iowa Colloquium, October 12, 2000 Page 17
Beyond the Rigid Approximation: A Dynamic Model for the Bat-Ball collision
Collision excites bending vibrations in bat Ouch!! Thud!!
Sometimes broken bat Energy lost lower vf
Bat not rigid on time scale of collision What are the relevant degrees of freedom?
see AMN, Am. J. Phys, 68, in press (2000)
University of Iowa Colloquium, October 12, 2000 Page 18
The Essential Physics: A Toy Model ball Mass= 1 bat 2
|v /v |
f i
0. 8
4
0. 7
0. 6
>> 1 m on Ma+Mb (1 on 6)
0 2 4 vibration 6 8 10
<< 1
m on Ma
0. 5
0. 4
0. 3
(1 on 2)
Bat not rigid on time scale of collision
Energy Fraction 0. 6
rigid bat
0. 5
0. 4
recoil
0. 3
0. 2
ball
0. 1
0
0
2
4
6
8
10
University of Iowa Colloquium, October 12, 2000 Page 19
0 2
5 1
y
0 1 5
A Dynamic Model of the Bat-Ball Collision
20
Euler-Bernoulli Beam Theory‡
2 2 y 2 y EI 2 A 2 F(z,t) 2 z z t
0
5 -
y
-10 -15
z • Solve eigenvalue problem for free oscillations (F=0)
-20 0
normal modes (yn, n)
5 0 1 5 1 0 2 5 2 0 3 5 3
• 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
University of Iowa Colloquium, October 12, 2000 Page 20
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
University of Iowa Colloquium, October 12, 2000 Page 21
Measurements via Modal Analysis Louisville Slugger R161 (33”, 31 oz)
1 0.5
FFT(R) 0.15 582
R
0
0.1
1181
-0.5 -1
0.05
-1.5 0 5 10 t (ms) 15 20
179
1830
2400 0 0 500 1000 1500 frequency (Hz) 2000 2500
frequency Expt Calc
179 582 1181 1830 177 583 1179 1821
barrel node Expt Calc
26.5 27.8 29.0 30.0 26.6 28.2 29.2 29.9
Conclusion: free vibrations of bat can be well characterized
University of Iowa Colloquium, October 12, 2000 Page 22
Model for the Ball
force (pounds) 4 1 10
8000 approx quadratic 6000
2 1.6 (ms) 1.2 0.8
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
collision time versus impact speed
4000
F=kxn
F=kxm
2000
0
0.4
Force (lb)
10000
0
20
compression (inches)
40 60 80 100 120 140 impact speed (mph)
8000
160 mph
3-parameter problem:
k n v-dependence of m COR
0.8
6000
4000
80 mph
2000
0 0 0.2 0.4 0.6
time (ms)
University of Iowa Colloquium, October 12, 2000 Page 23
Putting it all together….
ybat (x, t) q n (t) yn (x)
n
y2 (x0 )F(s,t) d 2q n 2 n q n n dt 2 A d 2 yball mball - F(s, t) 2 dt s ybat (x0 , t) - yball (t)
impact point
ball compression
Procedure: • specify initial conditions • numerically integrate coupled equations • find vf = ball speed after ball and bat separate
University of Iowa Colloquium, October 12, 2000 Page 24
General Result
I y (x 0 ) En F (t )e2ifnt dt 2A 0
2 2 n
2
energy in nth mode
Force (lb)
10000
force normalized to unit impulse
1
8000
160 mph
0.8 0.6 0.4
Fourier transform
6000
4000
80 mph
2000
0.2
0 0.2 0.4 0.6 0.8
0
0 0 0.5 1 1.5 2
time (ms)
f
Conclusion: only modes with fn < 1 strongly excited
University of Iowa Colloquium, October 12, 2000 Page 25
Comparison with Experiment 1. Low-speed collision
v /v
50 losses in ball 40
final
initial
CM
node
30
rigid recoil
V =1 m/s
i
COR=0.66 20 vibrations in bat
0.4 rigid bat 0.3
ball
10
0 18
20
22
24
26
28
30
32
0.2 0.1
flexible bat data from Rod Cross freely suspended bat vi = 2.2 mph
16 20 24 28 distance from knob (inches) 32
35 V =1 m/s 30 total 25 20 15 10 5 0 Modes >2 -5 18 20 22 24 26 28 30 32 Mode 2 Mode 1
i
COR=0.66
0
collision time 2.2 ms
only lowest mode excited
University of Iowa Colloquium, October 12, 2000 Page 26
Comparison with Experiment 2. High-speed collisions collision time 0.65 ms
v
final
/v
initial
0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 23 rigid bat
nodes
1.4 1.2 1 0.8
Batting Cage Data: Crisco/Greenwald
calculation
flexible bat
e /e
eff
0.6 0.4 0.2
data from Lansmont BBVC bat pivoted about 5-3/4" v =100 mph
initial
24
25 26 27 28 29 30 distance from knob (inches)
31
0 0 0.05 0.1 0.15 0.2 0.25
0.3
distance from barrel (m)
Conclusion: essential physics under control
University of Iowa Colloquium, October 12, 2000 Page 27
Application to realistic conditions: 90 mph ball; 70 mph bat at 28” CM
100 rigid bat 80 v (mph)
f
% Energy 70
(a)
60 50 40 30 20 vibra tions rigid recoil losses in ball
nodes
flexible bat
10 30 0
60 40 20
ball
16 20 24 28 32
Louisville Slugger R161 (33", 31 oz)
16 20 24 28 distance from knob (inches) 32
(b)
25 20
Total
1
15 10 5
2 3 >3
0 16 20 24 28 distance from kno b (cm) 32
University of Iowa Colloquium, October 12, 2000 Page 28
Insights into collision process: 1. The “sweet spot”
80 40 y 0 -40 -80
displacement at handle
impact @ 24.8"
24.8"
vibrational velocity at handle
26.8"
26.8"
28.8"
28.8"
0
2
4 t (ms)
6
8
10
0
2
4 t (ms)
6
8
10
1. Maximum vf (~28”)
% Energy 70 rigid recoil 60 50 40 30 20 10 0 16 ball vibrations
2. Minimum vibrational energy (~28”) 3. Node of fundamental (~27”) 4. Center of Percussion (~27”)
losses in ball
5. “don’t feel a thing”
20 24 28 distance from knob (inches)
32
University of Iowa Colloquium, October 12, 2000 Page 29
Insights into collision process:
2. The effect of hands
3
v (mph) f 110
nodes
2
1 y (mm) 0
Displacement at 5”
100 90 80 70
rigid pivoted
rigid free
-1
impact at 27"
-2 -3 0 0.5 1 t (ms) 1.5 2
60 50 40 30 20
flexible (free or pivoted)
22
24
Conclusions:
• hands don’ t matter!
26 28 x (inches)
30
32
• size, shape, boundary conditions at far end don’t matter
University of Iowa Colloquium, October 12, 2000 Page 30
Insights into collision process: 3. Time evolution of bat
displacement (mm) 10 8 6 4 2 0 -2 -4 0 5 10 15 20 impact point 25 30 pivot point
0.1 ms intervals
distance from knob (inches)
University of Iowa Colloquium, October 12, 2000 Page 31
Wood versus Aluminum:
1. General Considerations
• Length and weight “decoupled” * Can adjust shell thickness v (mph) f wood versus aluminum * Fatter barrel, thinner handle wood 30 • Weight distribution more uniform aluminum * Easier to swing 20 * Less rotational recoil * More forgiving on inside pitches 10 150 mph ball • Stiffer for bending stationary bat * Less energy lost due to vibrations 0 20 25 30 • More compressible distance from knob (inches) * COR larger
University of Iowa Colloquium, October 12, 2000 Page 32
Wood versus Aluminum:
2. More Realistic Comparisons
v (mph)
f
wood versus aluminum
100
aluminum
80
a. direct comparision
b. 9% larger COR
70 mph ball pivoted bat wood
60
c. 8% higher bat speed
40 20 25 30 distance from knob (inches)
University of Iowa Colloquium, October 12, 2000 Page 33
Wood versus Aluminum:
3. Dynamics of “Trampoline” Effect
bending modes bell modes
“bell” modes:
0 1000
k t 2 meff R
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
University of Iowa Colloquium, October 12, 2000 Page 34
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
Why is softball bat different from baseball bat? Effect of “corking” the bat
University of Iowa Colloquium, October 12, 2000 Page 35
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
University of Iowa Colloquium, October 12, 2000 Page 36