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Baseball and Physics

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1927 Yankees:

Greatest baseball team

ever assembled

1927

Solvay Conference:

Greatest physicsteam

ever assembled

Baseball and Physics

MVP’s

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 vfdepend on?

“Lab” Frame

Bat Rest Frame

vrel

vball

vbat

vf

eAvrel

Kinematics: Reference Frames

vf = eA vball + (1+eA) vbat

- eA “Collision Efficiency” = “Ball Exit Speed Ratio” - 0.5
- property of ball & bat
- weakly dependent on vrel
- near “sweet spot” eA 0.2 vf 0.2 vball + 1.2 vbat

BESR

Conclusion:vbatmuch more important than vball

vball

Vball,f

Vbat,f

- Momentum Conservation
vball,f = eAvball vbat,f = r(1+eA)vball r = mball/mbat

- Coefficient of Restitution
e (vball,f+vbat,f)/vball

- Energy Conservation
fball = frecoil = fdis =

.

.

.

CM

b

Kinematics:

recoil factorrand coefficient of restitutione

- 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

=

+

0.16 + 0.08

0.24

- All things equal, want rsmall
- But….

.

.

.

CM

b

Kinematics:

Where Does the Energy Go?

- Hit ball
- Recoil of Bat (r)
- Dissipation in ball and bat (e)

=

+

0.16 + 0.08

0.24

- All things equal, want rsmall
- But….

(½ mv2)

What is the Ideal Bat Weight?

Conclusion from Players: Lighter seems to be better!

vbat I-0.3

vbat I-0.5

- vBAT(6”) = 1.2 mph/(1000 oz-in2)(vf=1.5 0.3 mph)

“bounciness” of ball

Kinematics: Coefficient of Restitution (e):

(Energy Dissipation)

- 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!

COR: Is the Ball “Juiced”?

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

CM

12

10

8

6

4

2

0

-2

0

5

10

15

20

25

30

Putting it all together (rigid bat)...

vf = eA vball + (1+eA) vbat

CM

Putting it all together (realistic bat)...

vf = eA vball + (1+eA) vbat

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)

The Essential Physics: A Toy Model

bat

ball

Mass= 1 2 4

rigid

<< 1

m on Ma

(1 on 2)

>> 1

m on Ma+Mb

(1 on 6)

flexible

A Dynamic Model of the Bat-Ball Collision

y

20

Euler-Bernoulli Beam Theory‡

y

z

- 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

f1 = 177 Hz

f3 = 1179 Hz

f2 = 583 Hz

f4 = 1821 Hz

nodes

Normal Modesof the Bat

Louisville Slugger R161 (33”, 31 oz)

Can easily be measured (modal analysis)

Measurements via Modal Analysis

Louisville Slugger R161 (33”, 31 oz)

FFT

frequencybarrel node

ExptCalcExptCalc

17917726.526.6

58258327.828.2

1181117929.029.2

1830182130.029.9

Conclusion: free vibrations

of bat can be well characterized

F=kxn

F=kxm

Model for the Ball

3-parameter problem:

k

nv-dependence of

m COR of ball with rigid surface

Putting it all together….

ball compression

- Procedure:
- specify initial conditions
- numerically integrate coupled equations
- find vf = ball speed after ball and bat separate

General Result

energy in nth mode

Fourier transform

Conclusion:only modes with fn < 1 strongly excited

Results: Ball Exit Speed

Louisville Slugger R161

33-inch/31-oz. wood bat

only lowest mode excited

lowest 4 modes excited

Conclusion:essential physics under control

CM

nodes

Application to realistic conditions:

(90 mph ball; 70 mph bat at 28”)

The “sweet spot”

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”

3

Displacement at 5”

2

1

y (mm)

0

-1

impact at 27"

-2

-3

0

0.5

1

1.5

2

t (ms)

Boundary conditions

- Conclusions:
- size, shape, boundary conditions at far end don’t matter
- hands don’ t matter!

T= 0-1 ms

Time evolution

of the bat

T= 1-10 ms

Time evolution

of the bat

1.0 ms

0.8 ms

0.6 ms

0.4 ms

0.2 ms

impact location

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
- Stiffer for bending
- Less energy lost due to vibrations

- More compressible
- COReff larger

tennis ball/racket

- CM energyshared 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

Ebat/Eball kball/kbat xbat/ xball

>10%

larger!

Wood versus Aluminum:

Dynamics of “Trampoline” Effect

“bell” modes:

“ping” of bat

- Want k small to maximize stored energy
- Want >>1 to minimize retained energy
- Conclusion: there is an optimum

Where Does the Energy Go?

eA

vf

COR

Which Performance Metric?

eA or COR or vf

(my most important slide)

- 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

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!