1

Experimental Baseball PhysicsAlan M. Nathan

a-nathan@uiuc.eduwebusers.npl.uiuc.edu/~a-nathan/pob

Department of PhysicsUniversity of Illinois

Courtesy, Trey Crisco

Courtesy, Dan Russell

2

1927

Solvay Conference:

Greatest physics team

ever assembled

The Baseball/Physics Connection

1927 Yankees:

Greatest baseball team

ever assembled

MVP’s

3

Some Topics I Will Cover

• The ball-bat collision

• The flight of the baseball

4

The Ball-Bat Collision

ball bat

e-rq =

1+re-r 1+e

BBS = v v1+r 1+r

e: “coefficient of restitution” 0.50 (energy dissipation—mainly in ball, some in bat)

r mballz2/I6 : bat recoil factor = 0.25(momentum and angular momentum conservation)

---heavier is better but…

q=0.20

BBS = q vball + (1+q) vbatvball vbat

BBS

z

5

Studies of the Collision Efficiency q

• Independent of reference frame– Measure in bat rest frame: q=vf/vi

– Use q to predict field performance

Sports Sciences Laboratory, Washington State University

6

• strike bat in barrel—look at response in handle

• handle moves only after ~0.6 ms delay

• collision nearly over by then

• nothing on knob end matters• size, shape• boundary conditions• hands!

• confirmed experimentally

-30.00

-20.00

-10.00

0.00

10.00

20.00

30.00

0 1 2 3 4 5

v (m/s)

t (ms)

Independence of End Conditions

7

Studies of the Collision Efficiency q

• Independent of end conditions

60

70

80

90

100

110

120

50 60 70 80 90 100

Vi or Swing (mph)

4.75'' pivot

6.75'' pivot

free

swing/hit

Vf (mph)

Courtesy, Keith Koenig

8

q: Field Study vs. Laboratory

Crisco, Smith, AMN

9

Modal Analysis of a Baseball Batwww.kettering.edu/~drussell/bats.html

0

0.05

0.1

0.15

0 500 1000 1500 2000 2500

FFT(R)

frequency (Hz)

179

582

1181

1830

2400

frequency

-1.5

-1

-0.5

0

0.5

1

0 5 10 15 20

R

t (ms)

time

0 5 10 15 20 25 30 35

f1 = 179 Hz

f2 = 582 Hz

f3 = 1181 Hz

f4 = 1830 Hz

10

Vibrations, COR, and the “Sweet Spot”

Evib

vf

e

+

0.1

0.2

0.2

0.3

0.3

0.4

0.4

0.5

0

20

40

60

80

100

120

0 5 10 15

e

vf (mph)

distance from tip (inches)

nodes4 3 2 1

Strike bat here

at ~ node 2

vibrations minimized

COR maximized

BBS maximized

best “feel”

Note: COP is irrelevant to feel and performance

11

•Two springs mutually compress each other KE PE KE

• PE shared between “ball spring” and “bat spring”

…sharing depends on “kball/kbat”

• PE in ball mostly dissipated (~80%!)

• PE in bat mostly restored

• Net effect: less overall energy dissipated...and therefore higher ball-bat COR

…more “bounce”—confirmed by experiment

…and higher BBS

• Also seen in golf, tennis, …

Aluminum Bats and the “Trampoline” Effect:A Simple Physical Picture

demo

12

0.40

0.45

0.50

0.55

0.60

0.65

0.70

500 1000 1500 2000

COR-modelCOR-expt

COR

fhoop

(Hz)

Softball Data and Model

Conclusions: • COR of Al bat can be significantly higher• essential physics is understood

Russell, Smith, AMN

change kballchange kbat

Wood

13

Regulating Performance of Non-Wood Bats:A Science-Based Approach Used by NCAA

• Specify maximum q – approx. same as for wood bats of similar wt.– implies bats swung alike will perform alike

• Specify minimum MOI to limit bat speed– smaller than wood

• Together, these determine a maximum BBS– gap between wood and aluminum 5%– does that mean aluminum should be banned?

• an issue many are struggling with

BBS = q vball + (1+q) vbat

14

40

42

44

46

48

50

8.5 9 9.5 10 10.5 11 11.5

I6"

(103 oz-in2)

knob

(rad/s)

y = m1*(9/m0)^m2

ErrorValue

0.392146.218m1

0.0574220.28747m2

NA3.8574Chisq

NA0.93138R

Crisco/Greenwald Batting Cage Study

woodaluminum

Batting cage study show how bat speed depends on I for college baseball players

~ [1/I6]n

0<n<0.5n 0.3

Other studies show bat speed independent of M for fixed I

Crisco, Greenwald, AMN

15

0.65

0.67

0.69

0.71

0.73

0.75

9000 9500 10000 10500 11000 11500 12000

BESR

MOI (oz-in2)

MOI limit

BESR limittypical wood

max vf

102 mph

97 mph

All bats below horizontal line and to right of vertical line are allowed

Example: 34” Bats=q+1/2

16

What About Corked Bats?or..What was Sammy thinking?

Conclusion:• No increase in BBS

increase in swing speed decrease in collision efficiency

~ [1/I6]n

0<n<0.5

no trampoline effect!

Wood Bat RW01

unmodified

drilled

corked

0.475

0.480

0.485

0.490

0.495

ball-bat COR

17

What About Juiced Baseballs?

Conclusion:No evidence for juiced ball

18

-80

-60

-40

-20

0

20

40

60

80

-2 -1 0 1 2

(inches)

1 = 79 rad/s

1 = 0 rad/s

1 = -72 rad/s

Etopspin

backspin

no spin

Putting spin on the ball: Low speeds

Conclusions: • slide-then-roll model approximately works• curveball is hit with more backspin than fastball

Cross & AMN

19

High-Speed Version:Work in Progress

20

Flight of the Baseball• Gravity• Drag (“air resistance”)• Lift (or “Magnus”)

v

ω

mg

Fd

FM

Courtesy, Popular Mechanics

Fd=½ CDAv2

-v direction

FM = ½ CLAv2 CL = CMR/v

(ω v) direction direction leading edge is turning

21

Measuring Magnus Force Using High-Speed Motion

Analysis

ATEC 2-wheel pitching machine

Motion Analysis System

Baseball with reflecting dot

22

Joe Hopkins

~15 ft

Motion Analysis Geometry

23

Motion Capture System:

• 10 Eagle-4 cameras

• 700 frames/sec

• 1/2000 shutter

• EVaRT 4.0 software

www.motionanalysis.com

Pitching Machine:• project horizontally• 50-110 mph• 1500-4500 rpm

24

Typical Data and Fit

-3000

-2000

-1000

0

1000

2000

3000

1400

1420

1440

1460

1480

1500

1520

0.00 0.04 0.08 0.12 0.16t (s)

z

y

<v>=72 mph =4900 rpm

ay=1.58g

y

z

25

0.0

0.1

0.2

0.3

0.4

0.5

0.6

present

Alaways 2-Seam

Alaways 4-Seam

Watts & Ferrer

Briggs

SHS

RKA-100

0.0 0.2 0.4 0.6 0.8 1.0

CL

S

Results for Lift Coefficient

FL= 1/2ACLv2

S=r/v100 mph, 2000 rpm

S=0.17

Conclusions: --data qualitatively consistent (~20%)

--RKA model inconsistent with data

26

The PITCHf/x Tracking SystemA Quantitative Tool to Study Pitched Baseball Trajectories

27

How Does PITCHf/x Work?

• Two video cameras track baseball in 1/60-sec intervals – usually “high home” and “high first”– third CF camera used establishes ht. of strike zone

• Pattern-recognition software to identify “blobs”• Camera calibration to convert pixels to (x,y,z)

– 9-parameter fit to trajectory– constant acceleration for x(t),y(t),z(t)

• Use fit to calculate lots of stuff– The full trajectory– The “break”– Drag and Magnus forces

28

Example: Bonds’ 756

29

Example: Drag and Drag Coefficients

20k pitches from Anaheim, 2007

drag/W=1@~90 mph

30

Using PITCHf/x to Classify PitchesJon Lester, Aug 3, 2007 @ Seattle

I: 4-seam fastballII: Slider (?) III: 2-seam fastballIV: Curveball

LHPCatcher’s View

spin axis

break direction = -90o

31

How Far Did That Home Run Travel?

• Ball leaves bat• Hits stands D from home plate, H above

ground• How far would it have gone if no obstruction?

32

0

50

100

150

200

0 100 200 300 400 500horizontal distance (ft)

0

5

10

15

20

25

30

35

40

390 400 410 420 430 440 450 460horizontal distance (ft)

400 ft/30 ftRange=415-455Time can resolve

4 s

5 s7 s

See www.hittrackeronline.com

Calculations

33

From PITCHf/x to HITf/xBarry Bond’s 756th Home Run

• PITCHf/x data tracked hit ball over first 20 ft• Precision measurement of endpoint and time-of-flight

• Inferred: v0=112 mph; =270 up; =160 to right of dead center; =1186 rpm (backspin) and 189 rpm (sidespin, breaking to center)

34

Baseball Aerodynamics:Things I would like to know better

• Better data on drag– “drag crisis”?– spin-dependent drag?– drag for v>100 mph

• Dependence of drag & Magnus on seam orientation, surface roughness, …

• Is the spin constant?

35

Trackman: The Wave of the Futuresee www.trackmangolf.com

• Doppler radar to measure radial velocity• 3-detector array to measure phase

– two angles

• Sidebands gives spin magnitude• Result:

– in principle, full trajectory can be reconstructed, including spin and spin axis

– already in use for golf, currently being adapted for baseball

36

Trackman Radar

Monopulse Principle (Phase)

TRANSMITTER

RECEIVER 1

FTX

FTX

FRX1fd1

(fd1 ) - (fd2) = 2**sin()*D*FTX/cRECEIVER 2FRX2

fd2

FTX

D

37thanks to Fredrik Tuxen, CTO of Trackman

38

• Steroids increases muscle mass

• Increased muscle mass increases swing speed

• Increased swing speed increase BBS

• Increased BBS means longer fly balls

• Longer fly balls means more home runs

Steroids and Home Run Productonsee Roger Tobin, AJP, Jan. 2008

39

0

0.0025

0.005

0 100 200 300 400 500

Range (feet)

Pro

bab

ility

(p

er f

t)

Home Run Threshold

10%

To have 10% HR’s,there must be a lotof near-HR’s

Elite hitters:HR/BBIP = ~10%

Thanks to Roger Tobin

40

0

0.0025

0.005

0 100 200 300 400 500

Range (feet)

Pro

bab

ility

(p

er f

t)

Home Run Threshold

10%

14.9%

Change in range distribution when batted ball speed increased by 3%:3% change in BBS gives 50% increase in HR rate!

Baseline

3% speedincrease

Thanks to Roger Tobin

Home Run Distances, 2007www.hittrackeronline.com

Delta = distance beyond fence (ft)

~4% per foot

Tobin’s Conclusion: increase of BBS by few mph can increase HR rate by 30-50%!

42

Work in Progress

• Collision experiments & calculations to elucidate trampoline effect

• New studies of aerodynamics using Trackman and PITCHf/x

• Experiments on high-speed oblique collisions

• A book, with Aussi Rod Cross

Thanks for the invitation and your attention