introduction: description of ball-bat collision

The Physics of the Ball-Bat Collision UML Seminar March 12, 2003 Page 2

Introduction: Description of 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 goal... large hit ball speed


Kinematics of the Ball-Bat Collision

f ball bat

e - r 1 + ev v v

1 r 1 r

r bat recoil factor = mball/Mbat,effective

e Coefficient of Restitution (COR)

typical numbers:r 0.25 e 0.50 vf = 0.2 vball + 1.2 vbat

vball vbat

vf

Note: this talk focuses entirely on COR

Alan M. Nathan

ideally, want r small (heavy bat), but...


COR and Energy Dissipation

e COR vrel,after/vrel,before

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

e.g., drop ball on hard floor: COR2 = hf/hi 0.25

typically COR 0.5 ~3/4 CM energy dissipated!

depends on impact speed mostly a property of ball but…

the bat matters too! vibrations , “trampoline” effect


Collision excites bending vibrations

Ouch!! Thud!! Sometimes broken bat

Energy lost lower COR, vf

Find lowest mode by tapping

Reduced considerably if

Impact is at a node

Collision time (~0.6 ms) >> Tvib

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

Accounting for Energy Dissipation:

Dynamic Model for Ball-Bat Colllision


ball bat

Mass= 1 2

The Essential Physics: A Toy Model

rigid limit 1

1 on

flexible limit 1

1 on 2 0

0.2

0.4

0.6

0.8

1

0 2 4 6 8 10 12

COR

fvib

fball


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: Nonlinear lossy spring for ball-bat interaction

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

Louisville Slugger R161 (34”, 31 oz)

0 5 10 15 20 25 30 35

f1 = 177 Hz

f2 = 583 Hz

f3 = 1179 Hz

f4 = 1821 Hz

Can easily be measured: Modal Analysis


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


Vibrations and the COR

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

COR maximum near 2nd node

the “sweet spot”


-20

0

20

0 2 4 6 8 10

v (m/s)

t (ms)

Motion of Handle

24”

27”

30” -3

-2

-1

0

1

2

3

0 0.5 1 1.5 2

y (mm)

t (ms)

impact at 27"

13 cm

• Center of Percussion close to lowest node @ 27”• Coincides neither with max COR @ 29”

…nor with max. vf

• Far end of bat doesn’t matter mass, grip, …

Some interesting insights:


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

Ballleaves

bat

Conclusions:

• Knob end doesn’t matter

• Batter’s grip doesn’t matter

• vibrations and rigid motion indistinguishable on short time scale


Bounce superballs from beam (Rod Cross)

Conclusion:Nothing on far end of beam matters


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:A Closer Look

Compressional energy shared between ball and bat PEbat/PEball = kball/kbat (= s) PEball mostly dissipated (75%) BPF = Bat Proficiency Factor e/e0

Ideal Situation: like person on trampoline kball >>kbat: most of energy stored in bat f >>1: stored energy returned e2 (s+e0

2)/(s+1) 1 for s >>1

eo2 for s <<1


Trampoline Effect: Toy Model, revisited

ball bat

Mass= 1 2

0.5

0.52

0.54

0.56

0.58

0.6

0.62

0.5 1 1.5 2 2.5 3

COR

f

COR

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.5 1 1.5 2 2.5 3

energy fraction

f

dissipated

ball

vibrations


Bending Modes vs. Shell Modes

k R4: large in barrel little energy stored

f (170 Hz, etc) > 1/ energy lost to vibrations

Net effect: BPF 1

k (t/R)3: small in barrel

more energy stored

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

Net Effect: BPF > 1

The “Trampoline” Effect:A Closer Look


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)

BPF increases with … Ball stiffness Impact velocity Decreasing wall thickness Decreasing ball COR

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

“Tuning a bat” Tuning due to balance between storing energy

(k small) and returning it (f large) Tuning not related to phase of vibration at time

of ball-bat separation

s kball/kbat

e2 (s+e02 )/(s+1)

BPF = e/e0


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

should be tested experimentally

introduction: description of ball-bat collision

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