Fermilab Colloquium, July 12, 2000 Page 1 Baseball: It’s Not Nuclear Physics (or is it?!) Alan M. Nathan University of Illinois FNAL Colloquium, July 12,

Fermilab Colloquium, July 12, 2000 Page 2

REFERENCESREFERENCES

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

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 Sept. 2000 L. Briggs, AJP 27, 589 (1959) R. Mehta, Ann. Ref. Fluid Mech. 17, 151 (1985) www.npl.uiuc.edu/~a-nathan/pob


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


Hitting the BaseballHitting the Baseball

“...the most difficult thing to do in sports”

--Ted Williams

BA: .344SA: .634OBP: .483HR: 521

#521, September 28, 1960


Here’s Why…..

(Courtesy of Robert K. Adair)


Description of Ball-Bat CollisionDescription 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 more later

What aspects of collision lead to large vf?


What happens when ball and bat collide?

The simple stuff: kinematics conservation of momentum conservation of angular momentum

The really interesting stuff: energy dissipation compression/expansion of ball vibrations of the bat

How to maximize vf?


The Simple Stuff: Kinematics

ibat,iball,fball, vr1

e1 v

r1

r-e v

vball,f = 0.2 vball,i + 1.2 vbat,i

Conclusion: vbat much more important than vball

Question: what bat/ball properties make vball,f large?

e Coefficient of Restitution 0.5

r recoil factor 0.25


Sosa’s 500’ Blast(s) in Home Run Derby:

A Numerical Analysis

• D = 500

* vf 127 mph

• vball,i 60 mph

* vbat,i 96 mph!

• if vball,i were 90 mph

* D = 530


Energy in Bat Recoil

.

Translation

.Rotation

CM .

z

• Important Bat Parameters:

mbat, xCM, ICM

• wood vs. aluminum

bat

2ball

bat

ball

I

zm

m

m r

Conclusion: All things being equal, want mbat, Ibat large

0.17 + 0.07 = 0.24

Want r small to mimimizerecoil energy


But… All things are not equal Mass & Mass Distribution affect bat speed

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

40

50

60

70

80

90

100

20 30 40 50 60

mass of bat (oz)

constant bat energy

constant bat+batter energy

60

70

80

90

100

110

120

20 30 40 50 60

mass of bat (oz)

constant bat energy

constant bat speed

constant bat+batter energy

bat speed vs mass

ball speed vs mass


• in CM frame: Ef/Ei = e2

• massive rigid surface: e2 = hf/hi

• typically e 0.5~3/4 CM energy dissipated!

• depends on ball, surface, speed,...

• is the ball “juiced”?

Energy Dissipated:

Coefficient of Restitution (e):“bounciness” of ball

i rel,

f rel,

v

v e


COR and the “Juiced Ball” Issue

MLB: e = 0.546 0.032 @ 58 mph on massive rigid surface

Conclusion: more systematic studies needed

0.40

0.45

0.50

0.55

0.60

40 60 80 100 120 140equivalent impact speed (mph)

COR

Briggs, 1945

UML/BHM

Lansmont BBVC

MLB specs

MLB/UML

COR Measurements

320

360

400

440

0.4 0.45 0.5 0.55 0.6

R (ft)

cor

*

*~ 35 '

Distance vs. COR "90+70" collision


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

Aluminum Bat

kball/kbat ~ 0.10

80% restored eeff = 0.55-0.58

“trampoline effect”

Bat Proficiency Factor eeff/e

Effect of Bat on COR: Local CompressionEffect of Bat on COR: Local Compression

Ebat/Eball kball/kbat xbat/ xball

>10% larger!

tennis ball/racket

Recent BPF data:(Lansmont BBVC/Trey Crisco)

0.99 wood 1.12 aluminum

More later on wood vs. aluminum


Collision excites bending vibrations in bat

Ouch!! Thud!!

Sometimes broken bat

Energy lost lower vf

Lowest modes easy to find by tapping

Location of nodes important

Beyond the Rigid Approximation:

A Dynamic Model for the Bat-Ball collision

see AMN, Am. J. Phys, 68, in press (2000)


20

-2 0

-1 5

-1 0

-5

0

5

10

15

20

0 5 10 15 20 25 30 35

y

z

y

t)F(z, t

yA

z

yEI

z 2

2

2

2

2

2

A Dynamic Model of the Bat-Ball Collision

• Solve eigenvalue problem for 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‡


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


-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


0.4

0.8

1.2

1.6

2

0 20 40 60 80 100 120 140

t (ms)

impact speed (mph)

collision time versus impact speed

Model for the Ball

3-parameter problem:

k t

n v-dependence of t

m COR

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


t)(y-t),y(xs t)F(s,- dt

ydm

A

t))F(s,(xyq

dt

qd

)x(y)t(qt)y(x,

ball02ball

2

ball

02n

n2n2

n2

nn

n

impact pointball compression

Putting it all together….

Expectation: only modes with fn t < 1 strongly excited


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


rigid bat

flexible bat

CM node

data from Rod Crossfreely suspended bat

vi = 2.2 mph


• Under realistic conditions…

• 90 mph, 70 mph at 28”

0

10

20

30

40

50

60

70

16 20 24 28 32

% Energy

rigid recoil

ball

vibrations

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)


flexible bat

rigid bat

Louisville SluggerR161 (33", 31 oz)

CM nodes


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


Results:The “sweet spot”

30

40

50

60

70

80

90

100

110

20 22 24 26 28 30 32

vf (mph)

x (inches)

flexible (free or pivoted)

rigid pivoted

rigid free

nodes

-20

0

20

0 2 4 6 8 10

v (m/s)

t (ms)

Motion of Handle

24”

27”

30”

Possible “sweet spots”

1. Maximum of vf (~28”)

2. Node of fundamental (~27”)

3. Center of Percussion (~27”)

-3

-2

-1

0

1

2

3

0 0.5 1 1.5 2

y (mm)

t (ms)

impact at 27"

13 cm

Hands don’t matter!


Wood versus Aluminum

• Length and weight “decoupled”* Can adjust shell thickness* Fatter barrel, thinner handle

• More compressible* COR larger

• Weight distribution more uniform* Easier to swing* Less rotational recoil* More forgiving on inside pitches* Less mass concentrated at impact point

• Stiffer for bending* Less energy lost due to vibrations

0

20

40

60

80

100

16 20 24 28 32

vf (mph)


wood

aluminum-1

aluminum-2

wood versus aluminum


How Would a Physicist Design a Bat?How Would a Physicist Design a Bat?

Wood Bat already optimally designed

highly constrained by rules! a marvel of evolution!

Aluminum Bat lots of possibilities exist but not much scientific research a great opportunity for ...

fame fortune


Things I would like to understand betterThings I would like to understand better

Relationship between bat speed and bat weight and weight distribution

Effect of “corking” the bat Location of “physiological” sweet spot Better model for the ball FEA analysis of aluminum bat Why is softball bat different from baseball bat?


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


Aerodynamics of a BaseballAerodynamics of a Baseball

Forces on Moving Baseball

No Spin Boundary layer separation DRAG! FD=½CDAv2

With Spin

Ball deflects wake ==>Magnus

force FMRdFD/dv Force in direction front of ball

is turning

Pop

Pbottom

Drawing courtesty of Peter Brancazio


How Large are the Forces?How Large are the Forces?

• Drag is comparable to weight• Magnus force < 1/4 weight)

0

0.5

1

1.5

2

0 25 50 75 100 125 150Dra

g/W

eig

ht

or

Mag

nu

s/W

eig

ht

Speed in mph

Drag/Weight

Magnus/Weight =1800 RPM


The Flight of the Ball:The Flight of the Ball:Real Baseball vs. Physics 101 BaseballReal Baseball vs. Physics 101 Baseball

Role of Drag

Role of Spin

Atmospheric conditions Temperature Humidity Altitude Air pressure Wind

approx linear

Max @ 350

-100

0

100

200

300

400

0 20 40 60 80 100

Range (ft)

q (deg)

Range vs. q

0

20

40

60

80

100

120

0 100 200 300 400 500 600 700

y (ft)

x (ft)

no drag

with drag

100

200

300

400

500

50 60 70 80 90 100 110 120

Range (ft)

vi (mph)

Range vs. v

0

50

100

150

200

250

-100 0 100 200 300 400

horizontal distance in feet

200

350

500

750900


The Role of FrictionThe Role of Friction

Friction induces spin for oblique collisions

Spin Magnus force

Results

Balls hit to left/right break toward foul line

Backspin keeps fly ball in air longer

Topspin gives tricky bounces in infield

Pop fouls behind the plate curve back toward field

batball

topspin ==>F down backspin==>F up

sidespin ==> hook

bat

ball


The Home Run SwingThe Home Run Swing

• Ball arrives on 100 downward trajectory

• Big Mac swings up at 250

• Ball takes off at 350

•The optimum home run angle!


Pitching the BaseballPitching the Baseball

“Hitting is timing. Pitching isupsetting timing”

---Warren Spahn

vary speeds manipulate air flow orient stitches


Let’s Get Quantitative!Let’s Get Quantitative!How Much Does the Ball Break?How Much Does the Ball Break?

Kinematics z=vT x=½(F/M)T2

Calibration 90 mph fastball drops 3.5’ due to

gravity alone Ball reaches home plate in ~0.45

seconds Half of deflection occurs in last 15’ Drag: v -8 mph Examples:

“Hop” of 90 mph fastball ~4” Break of 75 mph curveball ~14”

slower more rpm force larger

3

4

5

6

7

0 10 20 30 40 50 60Ve

rtic

al

Po

sit

ion

of

Ba

ll (

fee

t)

Distance from Pitcher (feet)

90 mph Fastball

0

0.2

0.4

0.6

0.8

1

1.2

0 10 20 30 40 50 60

Ho

rizo

nta

l Def

lect

ion

of

Bal

l (fe

et)

Distance from Pitcher (feet)

75 mph Curveball


Examples of PitchesExamples of Pitches

Pitch V(MPH) (RPM) T M/W

fastball 85-95 1600 0.46 0.10

slider 75-85 1700 0.51 0.15

curveball 70-80 1900 0.55 0.25

What about split finger fastball?


Effect of the StitchesEffect of the Stitches

Obstructions cause turbulance

Turbulance reduces dragDimples on golf ballStitches on baseball

Asymmetric obstructions

Knuckleball

Two-seam vs. four-seam delivery

Scuffball and “juiced” ball


Example 1: FastballExample 1: Fastball

85-95 mph1600 rpm (back)12 revolutions0.46 secM/W~0.1


Example 2: Split-Finger FastballExample 2: Split-Finger Fastball

85-90 mph1300 rpm (top)12 revolutions0.46 secM/W~0.1


Example 3: CurveballExample 3: Curveball

70-80 mph1900 rpm

(top and side)17 revolutions0.55 secM/W~0.25


Example 4: SliderExample 4: Slider

75-85 mph1700 rpm (side)14 revolutions0.51 secM/W~0.15


SummarySummary

Much of baseball can be understood with basic principles of physics

Conservation of momentum, angular momentum, energy

Dynamics of collisions

Excitation of normal modes

Trajectories under influence of forces

gravity, drag, Magnus,….

There is probably much more that we don’t understand

Don’t let either of these interfere with your enjoyment of the game!

Documents

Fermilab Colloquium, July 12, 2000 Page 1 Baseball: It’s Not Nuclear Physics (or is it?!) Alan M. Nathan University of Illinois FNAL Colloquium, July 12,