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Introduction: Description of Ball-Bat Collision. forces large (>8000 lbs!) time short (
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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
The Physics of the Ball-Bat Collision UML Seminar March 12, 2003 Page 3
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
The Physics of the Ball-Bat Collision UML Seminar March 12, 2003 Page 4
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
The Physics of the Ball-Bat Collision UML Seminar March 12, 2003 Page 5
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
The Physics of the Ball-Bat Collision UML Seminar March 12, 2003 Page 6
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 Physics of the Ball-Bat Collision UML Seminar March 12, 2003 Page 7
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
The Physics of the Ball-Bat Collision UML Seminar March 12, 2003 Page 8
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
The Physics of the Ball-Bat Collision UML Seminar March 12, 2003 Page 9
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
The Physics of the Ball-Bat Collision UML Seminar March 12, 2003 Page 10
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”
The Physics of the Ball-Bat Collision UML Seminar March 12, 2003 Page 11
-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:
The Physics of the Ball-Bat Collision UML Seminar March 12, 2003 Page 12
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
The Physics of the Ball-Bat Collision UML Seminar March 12, 2003 Page 13
Bounce superballs from beam (Rod Cross)
Conclusion:Nothing on far end of beam matters
The Physics of the Ball-Bat Collision UML Seminar March 12, 2003 Page 14
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 Physics of the Ball-Bat Collision UML Seminar March 12, 2003 Page 15
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
The Physics of the Ball-Bat Collision UML Seminar March 12, 2003 Page 16
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
The Physics of the Ball-Bat Collision UML Seminar March 12, 2003 Page 17
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
The Physics of the Ball-Bat Collision UML Seminar March 12, 2003 Page 18
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)
The Physics of the Ball-Bat Collision UML Seminar March 12, 2003 Page 19
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
The Physics of the Ball-Bat Collision UML Seminar March 12, 2003 Page 20
The Physics of the Ball-Bat Collision UML Seminar March 12, 2003 Page 21
The Physics of the Ball-Bat Collision UML Seminar March 12, 2003 Page 22
The Physics of the Ball-Bat Collision UML Seminar March 12, 2003 Page 23
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