nuclear chemistry gordon conference june 19, 2003 page 1 how does a baseball bat work? the physics...

Nuclear Chemistry Gordon Conference June 19, 2003 Page 2

1927

Solvay Conference:

Greatest physics team

ever assembled

Baseball and Physics

1927 Yankees:

Greatest baseball team

ever assembled

MVP’s


Introduction to the Ball-Bat Collision

forces large (>8000 lbs!) time short (<1/1000 sec!) ball compresses, stops, expands

kinetic energy potential energy

lots of energy dissipated

bat is flexible bat bends, compresses

the goals... large hit ball speed good “contact”

Alan M. Nathan

I will concentrate primarily on large hit ball speed, but both are important


These movies are owned by CE Composites Baseball (combatbaseball.com), designers and manufacturers of composite baseball bats, Ottawa, Ontario, Canada, and are shown here with their permission.

high-speed video of collision


Kinematics of Ball-Bat Collisionvball vbat

vff ball bat

e-r 1+ev = v v

1+r 1+r

r: bat recoil factor = mball/mbat,eff

(momentum and angular momentum conservation)

e: coefficient of restitution (energy dissipation)

typical numbers: vf = 0.2 vball + 1.2 vbat

eA 1+eA


Kinematics: the recoil factor

r1

r-e eA b

• r = mball/mbat,eff mbat,eff = Ip/b2

typically pivot point is ~6” from knob

• r ~ 0.25 for collision ~6” from barrel end• mass in handle doesn’t help

• larger Ip better but ...


Recent ASA Slow-Pitch Softball Field Tests(L. V. Smith, J. Broker, AMN)

Conclusions: • bat speed depends more on I6 than M:• vbat ~ (1/I6)1/4

• rotation point close to knob

0.94

0.96

0.98

1

1.02

1.04

1.06

6000 7000 8000 9000 10000 11000

Bat Speed at 6" Point vs. MOI

MOI (oz-in2)

dashed: n=0.25solid: n=0.23

0.96

0.98

1

1.02

1.04

24 25 26 27 28 29 30 31 32

W (oz)

Bat Speed at 6" Point vs. W

~(1/M)0.25

fixed M fixed MOI

Ideal bat weight/MOI not easy to determine


Aside: Wood-Aluminum Differences Inertial differences

CM closer to hands, further from barrel for aluminum Mbat,eff smaller

* larger recoil factor r, smaller eA

* effectively, less mass near impact location MOIknob smaller swing speed higher ~cancels for many bats …but definite advantage for contact hitter

Dynamic differences Ball-Bat COR significantly larger for aluminum

more on this later

Alan M. Nathan

advantages for contact hitter:more forgiving on inside pitchesquicker bat


Dynamics of Ball-Bat CollisionCOR and Energy Dissipation

e COR vrel,after/vrel,before

in CM frame: (final KE/initial KE) = e2

baseball on hard floor: e2 = hf/hi 0.25

typically e 0.5 ~3/4 CM energy dissipated!

depends (weakly) on v the bat matters too!

vibrations “trampoline” effect


Bat is flexible on short time scale

Collision excites vibrations

Vibrations reduce COR

Energy going to vibrations depends on

Impact location relative to nodes

Collision time (~0.6 ms) relative to 1/fvib

see AMN, Am. J. Phys, 68, 979 (2000)

Accounting for Energy Dissipation:

Dynamic Model for Ball-Bat Colllision


The Details: A Dynamic Model

x

yEI

x - F

t

yA

2

2

2

2

2

2

-2 0

-1 5

-1 0

-5

0

5

10

15

20

0 5 10 15 20 25 30 35

20

y

z

y

Step 1: Solve eigenvalue problem for free vibrations

Step 2: Ball-bat interaction (F) modeled as nonlinear lossy spring

Step 3: Expand in normal modes and solve

yA x

yEI

x n

2n2

n2

2

2

22n n

n n n n2n

d q F(t) y ( )y( ) q ( )y ( ) q

dt A

zx,t t x


Normal Modes of the Bat:Modal Analysis

-1.5

-1

-0.5

0

0.5

1

0 5 10 15 20

R

t (ms)

time domain

0

0.05

0.1

0.15

0 500 1000 1500 2000 2500

FFT(R)

frequency (Hz)

179

582

1181

1830

2400

frequency domain

FFT

frequencies and shapes

0 5 10 15 20 25 30 35

f2 = 582 Hzf1 = 179 Hz f3 = 1181 Hz

demo


Ball-Bat Force

0

1000

2000

3000

4000

5000

6000

0 0.2 0.4 0.6 0.8 1

Time in milliseconds

F vs. time

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 vs. CM displacement

• Details not important --as long as e(v), (v) about right

• Measureable with load cell


0 5 10 15 20 25 30 35

f1 = 179 Hz

f2 = 582 Hz

Effect of Bat on COR: Vibrations

nodes

0

0.1

0.2

0.3

0.4

0.5

0 2 4 6 8 10 12 14distance from tip (inches)

CORCM

COR depends strongly on impact location

the “sweet spot”


Comparison with Data: Ball Exit Speed

Louisville Slugger R161, 33/31

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


time evolution

-50

0

50

100

150

200

0 5 10 15 20 25 30

1-10 ms1 ms intervals

impact point


-4

-2

0

2

4

6

8

10

displacement (mm)

0 - 1 ms0.1 ms intervals

impact point

• rigid-body motion develops only after few ms

• far end of bat has no effect on ball

knob moves after 0.6 ms

collision over after 0.6 ms

nothing on knob end matters

• size, shape• boundary conditions• hands


Data courtesy of Keith Koenig

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)Vf independent of end support


Flexible Bat and the “Trampoline Effect”

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0

10

20

30

40

50

60

70

80

0 2 4 6 8 10 12 14

COR % Energy Dissipated

inches from barrel

Ball

Vibrations

Nodes

COR

Losses in ball anti-correlated with vibrations in bat


The “Trampoline” Effect:

Compressional energy shared between ball and bat

PEbat/PEball = kball/kbat

~75% of PEball dissipated

If some energy stored in bat and if PEbat effectively returned to ball, then COR larger

Effect occurs in tennis, golf, aluminum bats, ...demo


0.5

0.6

0.7

0.8

0.9

1

0.01 0.1 1 10 100

eball-bat

kbat

/kball

eball

= 0.5

ebat

= 1.0

Ideal Situation: like person on trampolinekbat kball: most of energy stored in bat: e ebat

ebat 1: energy stored in bat returned

e 1, independent of eball

The “Trampoline” Effect: A Closer Look

2 22 ball bat bat ball

bat ball

e k e ke

k +k

For wood batkbat 50kball: ~2% of energy stored in bat

ebat doesn’t matter

e eball

For aluminum batkbat 7kball: ~15% of energy stored in batebat 1: energy stored in bat returned

e 1.2 eball “BPF” = 1.20


Bending Modes vs. Hoop Modes

kbat R4: large in barrel

little energy stored

f (170 Hz, etc) > 1/ stored energyvibrations

Net effect: e e0 on sweet spot

ee0 off sweet

spot

kbat (t/R)3: small in barrel

more energy stored

f (1-2 kHz) < 1/ energy mostly restored

Net Effect: e > e0

“BPF” e/e0 = 1.20-1.35!

The “Trampoline” Effect:A Closer Look


Modal analysis: Dan Russell and AMN

hoop modes

bending modes

hoop modes


COR vs. Hoop Mode Frequency

0.40

0.45

0.50

0.55

0.60

0.65

0.70

500 1000 1500 2000

COR-modelCOR-expt

COR

fhoop

(Hz)

Energy left in hoop vibrations


Where Does the Energy Go?

0

50

100

150

200

250

300

350

400

0 0.2 0.4 0.6 0.8 1

Wood Bat

Ball KE

Ball PE

Bat Recoil KE

Bat Vibrational E

Energy (J)

t (ms)

0

50

100

150

200

250

300

350

400

0 0.2 0.4 0.6 0.8 1

Aluminum Bat

Ball KE

Ball PE

Bat Recoil KE

Bat Vibrational E

Energy (J)

t (ms)


Some Interesting Consequences(work in progress)

e/e0 increases with … Ball stiffness Impact velocity Decreasing wall thickness Decreasing ball COR

Note: effects larger for “low-s” (high-performance) than for “high-s” (low-performance) bats

“Tuning a bat” Tune by balancing between storing energy (k

small) and returning it (f large) Tuning not simply related to phase of vibration

at time of ball-bat separation

s kbat/kball

e2 (1+se02 )/(s+1)

e 1 for s << 1


Some Interesting Consequences(work in progress)

USGA “pendulum” test---(Wed. NYT) 4 parameters

mball, mclub, kball, kclub

make mball >> mclub and kball >> kclub

heavy, stiff steel ball on clubhead

collision time determined by mball (known) and kclub

measure collision time to determine kclub

kclub determines trampoline effect

implementation expected Jan. 2004


So What’s the Deal with Corked Bats?

~1” diameter hole ~10” deep; fill with whatever

similar to aluminum bat * easier to swing and control * but less effective at transferring energy

Is there a “trampoline” effect from hole or filler?

probably not Net result:

little or no effect for home run hitter possible advantage for “contact” hitter


Not Corked DATA Corked COR: 0.445 0.005 0.444 0.005

Conclusions: • no trampoline effect!

• no advantage to corkedfor home run hitter

• possible advantage for“contact” hitter

Bat Research Center, UML, Sherwood & amn, Aug. 2001

70

80

90

2 3 4 5 6 7 8 9

vf (mph)


uncorked

corked

calculation


Summary

Dynamic model developed for ball-bat collision

flexible nature of bat included

simple model for ball-bat force

Vibrations play major role in COR for collisions off

sweet spot

Far end of bat does not matter in collision

Physics of trampoline effect mostly understood and

interesting consequences predicted

Corking bat has little effect on home run


And in conclusion...

Thanks for inviting me here

I love talking about this stuff, so ask

me lots of questions!

nuclear chemistry gordon conference june 19, 2003 page 1 how does a baseball bat work? the physics...

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