How Does a Baseball Bat Work: Dynamics of the Ball-Bat Collision Page 1 When Ash Meets Cowhide: The Physics of the Ball-Bat Collision Alan M. Nathan University

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


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


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?


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


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”


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….


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


• 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


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:


Putting it all together…..


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


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



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


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


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


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

FFT


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


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


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


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


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


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)


flexible bat

rigid bat

Louisville SluggerR161 (33", 31 oz)

CM nodes

Application to realistic conditions:

(90 mph ball; 70 mph bat at 28”)


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


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"


-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


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


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


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


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


-0.2

-0.1

0

0.1

0.2

20 25 30

vf (mph)


wood versus aluminum

woodaluminum

150 mph ballstationary bat

Performance Comparison

40

60

80

100

20 25 30

vf (mph)


wood versus aluminum

wood

aluminum

70 mph ballpivoted bat


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


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!

Documents

How Does a Baseball Bat Work: Dynamics of the Ball-Bat Collision Page 1 When Ash Meets Cowhide: The Physics of the Ball-Bat Collision Alan M. Nathan University