Transcript When Ash/Metal Meets Cowhide: The Physics of the Ball

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
Comparative Study of Wood and Aluminum Baseball Bats UC/Davis Seminar April 11, 2003 Page 2
Kinematics of the Ball-Bat Collision
vf  e A v ball  1+e A  v bat
e-r
kinematics: e A =
1 r
eA  “collision efficiency”
r  bat recoil factor = mball/Mbat,effective
e  Coefficient of Restitution (COR)
vball vbat
vf
• superball on massive surface:
r=1
e = 1  eA = 1 and vf = vball + 2 vbat
• baseball on bat:
r  0.25 e  0.50  eA=0.2 and vf = 0.2 vball + 1.2 vbat
Comparative Study of Wood and Aluminum Baseball Bats UC/Davis Seminar April 11, 2003 Page 3
The Bat Recoil Factor r
e-r
eA =
1 r
=
.
.
CM
+
b
Small r is best
r  0.25 typical…depends on….
• mass of bat
• mass distribution of bat
m ball
m ball m ballb 2
r 


m bat,eff m bat I bat,CM
• impact location
Heavier bat is better but….
Comparative Study of Wood and Aluminum Baseball Bats UC/Davis Seminar April 11, 2003 Page 4
Recent ASA Slow-Pitch Softball Field Tests
(L. V. Smith, J. Broker, AMN)
Bat Speed at 6" Point vs. W
Bat Speed at 6" Point vs. MOI
1.06
fixed M
1.04
1.02
1.02
1
1
dashed: n=0.25
solid: n=0.23
0.98
fixed MOIknob
1.04
~(1/M)
0.25
0.98
0.96
0.96
0.94
6000
7000
8000
9000
2
MOI (oz-in )
10000
11000
24
25
26
27
28
29
30
31
32
W (oz)
Conclusion: bat speed more a function of
mass distribution than mass
Comparative Study of Wood and Aluminum Baseball Bats UC/Davis Seminar April 11, 2003 Page 5
COR and Energy Dissipation
(primary focus of this talk)
e  COR  vrel,after/vrel,before
 in CM frame: (final KE/initial KE) = e2

 e.g., drop ball on hard floor: e2 = hf/hi  0.25

typically e  0.5
 ~3/4 CM energy dissipated!


depends on impact speed
the bat matters too!
 vibrations 
“trampoline” effect 
Comparative Study of Wood and Aluminum Baseball Bats UC/Davis Seminar April 11, 2003 Page 6
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

Dynamic differences
 Ball-Bat COR significantly larger for aluminum
Comparative Study of Wood and Aluminum Baseball Bats UC/Davis Seminar April 11, 2003 Page 7
Accounting for Energy Dissipation:
Dynamic Model for Ball-Bat Colllision

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)
Comparative Study of Wood and Aluminum Baseball Bats UC/Davis Seminar April 11, 2003 Page 8
The Essential Physics: A Toy Model
bat
ball
Mass=
e 5
A
10
Fraction of Initial Energy
0.6
0.7
0.5
0.6
0.4
20
bat vibration
vf/vi
bat recoil
0.3
0.5
0.2
0.4
0.1
ball rebound
0.30
00
22
44


66
88
10
10
 1: flexible limit
ball “sees” Ma
 1: rigid limit
ball “sees” Ma+Mb
(5 on 10)
(5 on 30)
Comparative Study of Wood and Aluminum Baseball Bats UC/Davis Seminar April 11, 2003 Page 9
The Details: A Dynamic Model
2y
2  2y 
A 2  F - 2  EI 2 
t
x  x 
02

Step 1: Solve eigenvalue problem for
free vibrations
 2   2 yn 
2


EI


A

n yn
2 
2 
x  x 


Step 2: Nonlinear lossy spring for ballbat interaction
Step 3: Expand in normal modes and
solve
y(x,t )   q n (t )yn ( x)
n
2
51
y
20
01
5
0
5-
y
-10
z
-15
-20
d qn
F(t) yn ( z )
2
 n q n 
2
dt
A
0
5
01
51
02
52
03
53
Comparative Study of Wood and Aluminum Baseball Bats UC/Davis Seminar April 11, 2003 Page 10
Normal Modes of the Bat:
Modal Analysis
frequency domain
time domain
FFT(R)
0.15
1
582
0.5
R
FFT
0
1181
0.1
-0.5
1830
179
0.05
-1
-1.5
2400
0
5
10
t (ms)
f1 = 179 Hz
15
20
0
0
500
f2 = 582 Hz
1000
1500
frequency (Hz)
2000
2500
f3 = 1181 Hz
frequencies and shapes
Comparative Study of Wood and Aluminum Baseball Bats UC/Davis Seminar April 11, 2003 Page 11
Ball-Bat Force
• Details not important
--as long as e(v), (v) about right
• Measureable with load cell
F vs. CM displacement
force (pounds)
F vs. time
1 10
4
8000

approx quadratic
6000
4000
2000
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
compression (inches)
Comparative Study of Wood and Aluminum Baseball Bats UC/Davis Seminar April 11, 2003 Page 12
Vibrations and the COR
COR
% Energy Dissipated
Nodes
0.55
80
0.5
70
COR
0.45
60
Ball
0.4
50
0.35
40
0.3
30
0.25
20
0.2
10
Vibrations
0.15
0
2
4
6
8
10
inches from barrel
the “sweet
spot”
12
0
14
COR maximum near 2nd node
Comparative Study of Wood and Aluminum Baseball Bats UC/Davis Seminar April 11, 2003 Page 13
Results: Ball Exit Speed
v
final
Louisville Slugger R161
33-inch/31-oz.
wood bat
v /v
/v
CM
initial
final
node
initial
nodes
0.35
0.4 rigid bat
0.3
rigid bat
0.25
0.3
0.2
flexible bat
0.2
0.15
data from Rod Cross
freely suspended bat
v = 2.2 mph
0.1
flexible bat
0.1
0.05
data from Lansmont BBVC
bat pivoted about 5-3/4"
=100 mph
v
initial
i
0
16
20
24
28
distance from knob (inches)
only lowest mode excited
32
0
23
24
25 26 27 28 29 30
distance from knob (inches)
31
lowest 4 modes excited
Conclusion: essential physics under control
Comparative Study of Wood and Aluminum Baseball Bats UC/Davis Seminar April 11, 2003 Page 14
Some interesting insights:
v (m/s)
Motion of Handle
3
2
24”
20
impact at 27"
1
y (mm)
0
27”
0
-1
13 cm
-2
-20
30”
0
2
4
6
t (ms)
8
10
-3
0
0.5
1
t (ms)
1.5
2
• 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, …
Comparative Study of Wood and Aluminum Baseball Bats UC/Davis Seminar April 11, 2003 Page 15
Flexible Bat and the “Trampoline Effect”
COR
% Energy Dissipated
Nodes
0.55
80
0.5
70
COR
0.45
60
Ball
0.4
50
0.35
40
0.3
30
0.25
20
0.2
10
Vibrations
0.15
0
2
4
6
8
10
12
Losses in ball
anti-correlated
with vibrations
in bat
0
14
inches from barrel
Comparative Study of Wood and Aluminum Baseball Bats UC/Davis Seminar April 11, 2003 Page 16
The “Trampoline” Effect:
A Closer Look

Compressional energy shared between ball and
bat
 PEbat/PEball = kball/kbat ( s)
 PEball mostly dissipated (75%)

Ideal Situation: like person on trampoline
 kball >>kbat: most of energy stored in bat
 f >>1: stored energy returned
 e2  (s+e02)/(s+1)
 1 for s >>1
 eo2 for s <<1
Comparative Study of Wood and Aluminum Baseball Bats UC/Davis Seminar April 11, 2003 Page 17
Trampoline Effect:
toy model with dissipation in ball
bat
ball
Mass= 1
COR

2
energy fraction
0.62
0.8
0.6
0.7
dissipated
0.6
0.58
0.5
COR
0.56
0.4
0.54
0.3
kbat>>kball
0.52
0.5
0.5
1
1.5
2
f
2.5
3
ball
0.2
vibrations
0.1
0
0.5
1
1.5
2
2.5
3
f
Comparative Study of Wood and Aluminum Baseball Bats UC/Davis Seminar April 11, 2003 Page 18
The “Trampoline” Effect:
A Closer Look
Bending Modes
vs.
Shell Modes
k  R4: large in barrel
 little energy stored
k  (t/R)3: small in barrel
 more energy stored
f (170 Hz, etc) > 1/
 energy lost to vibrations
f (1-2 kHz) < 1/ 
 energy mostly restored
Net effect: e  e0
Net Effect: e > e0
Comparative Study of Wood and Aluminum Baseball Bats UC/Davis Seminar April 11, 2003 Page 19
Comparative Study of Wood and Aluminum Baseball Bats UC/Davis Seminar April 11, 2003 Page 20
Where Does the Energy Go?
Energy (J)
Energy (J)
400
400
Ball KE
350
Wood Bat
300
Ball KE
350
Aluminum Bat
300
250
250
Ball PE
200
200
150
Ball PE
150
Bat Recoil KE
100
Bat Recoil KE
100
50
Bat Vibrational E
0
0
0.2
0.4
0.6
t (ms)
0.8
1
Bat Vibrational E
50
0
0
0.2
0.4
0.6
t (ms)
0.8
1
Comparative Study of Wood and Aluminum Baseball Bats UC/Davis Seminar April 11, 2003 Page 21
Some Interesting Consequences
(work in progress)
 e/e0 increases with …
s  k /k
 Ball stiffness
 Impact velocity
 Decreasing wall thickness
 Decreasing ball COR
ball
bat
(s+e02 )/(s+1)
e2 
e  1 for s  1
 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
Comparative Study of Wood and Aluminum Baseball Bats UC/Davis Seminar April 11, 2003 Page 22
Some Interesting Consequences
(work in progress)

Simple measurements to predict BPF
 Measure static compression of bat
 Measure frequency of shell modes
 Measure collision time with massive steel ball
mball >> mbat
kball >> kbat
 Collision time = (mball/kbat)
* Similar to USGA method for metal drivers
Comparative Study of Wood and Aluminum Baseball Bats UC/Davis Seminar April 11, 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
Comparative Study of Wood and Aluminum Baseball Bats UC/Davis Seminar April 11, 2003 Page 24