Transcript Physics of the Trampoline Effect in Baseball and Softball Bats

Physics of the Trampoline Effect
baseball, golf, tennis, ...
Alan M. Nathana, Daniel Russellb, Lloyd Smithc
aUniversity
of Illinois at Urbana-Champaign
bKettering University
cWashington State University
The “Trampoline” Effect:
A Simple Physical Picture
• Two springs mutually compress each other
– KE  PE  KE
• PE shared between “ball spring” and “bat spring”
– PE stored in ball mostly dissipated
– PE stored in bat mostly restored
• Net effect: less overall energy dissipated
– e  e0: the trampoline effect
• e0  COR for ball on rigid surface
– 1-e02 = fraction of ball PE dissipated
• e  COR for ball on flexible surface
– 1-e2 = fraction of initial ball KE lost to ball
The Essential Physics: Toy Model
• Cross (tennis, M=0)
• Cochran (golf)
• Naruo & Sato (baseball)
M
m
ball
• Numerically solve ODE to get e = vf/vi
– Energy lost (e<1) due to...
• Dissipation in ball
• Vibrations in bat
• Essentially a 3-parameter problem:
– e0
• Controls dissipation of energy stored in ball
– rk  kbat/kball = PEball/PEbat
• Controls energy fraction stored in bat
– rm  m/M
kbat
kball
• f  (rk/rm) ( depends mainly on ball)
• Controls energy transferred to bat (vibrations)
bat
Energy Flow
Energy Fraction
Energy Fraction
1
1
PE-Ball
0.8
0.8
PE-Ball
0.6
0.6
r =25 r =1
k
r =10 r =1
k
m
KE-Ball
m
0.4
0.4
KE-Ball
0.2
0.2
E-Bat
0
0
0.2
0.4
0.6
t (ms)
0.8
wood-like: rk=75
(very stiff bat)
E-Bat
1
0
0
0.2
0.4
0.6
t (ms)
0.8
1
aluminum-like: rk=10
(less stiff bat)
e
f
2
0.60
kball
kbat
e
e = 0.50
1.5
0
M
f
0.40
m
ball
1
bat
0.30
1. Strong coupling limit:
rk>>1, f>1
Ebat/Eball<<1
e = e0
2. Weak coupling limit:
rk<<1, f<<1
m on M
e=(e0-m/M)/(1+m/M)
3. Intermediate coupling
rk>1, f>1
e > e0
0.5
rm= m/M=0.25
0.20
0
energy fraction
0.80
25
50
r
75
0
100
75
100
k
dissipation in ball
0.60
0.40
ball
0.20
vibrations in bat
0.0
0
25
50
r
k
Dependence
on
r
=
m/M
m
e
f
5
solid: r =1/4
0.70
m
4
dashed: r =1
m
0.60
e
f
3
e = 0.50
0
2
0.40
1 f=1.1
0.30
0
25
• M  f max @ smaller rk
50
r
75
0
100
k
• Conclude: e depends on both rk and rM
Not unique function of f
• Limiting case: rk<<1 and f>>1 (rm0) (thin flexible membrane)
e1, independent of e0
Important Results
(all confirmed experimentally)
• Harder ball or softer bat decreases rk,
increases e
• Nonlinear baseball: ekball increases with vi
 e/e0 increases with vi
0.60
e
e = 0.50 as e increases
1.5
• e/e0 (“BPF”)
decreases
0
1.4
f
0.40
• Collision time increases
as
r
decreases
k
e =0.45
0
e/e 1.3
0
f
2
1.5
1
0
USGA pendulum test
1.2
0.30
0.5
e =0.50
0
1.1
1.0
0.20
5
10
r
0
15
k
25
20
50
r
k
75
0
100
Realizing the Trampoline Effect
in Baseball/Softball Bats
Bending Modes
kbat  R4: large in barrel
 little energy stored
vs.
Hoop Modes
kbat  (t/R)3: small in barrel
 more energy stored
f (170 Hz, etc)  < 1
f (1-2 kHz)  > 1
 stored energyvibrations
 energy mostly restored
Net effect:
e  e0 on sweet spot
Net Effect:
e<<e0 off sweet spot
e/e0 = 1.20-1.35
no trampoline effect
trampoline effect
Trampoline Effect:
Softball vs. Baseball
COR
0.70
0.65
baseball
0.60
0.55
Net result:
softball
•ordering reversed
•should be tested experimentally
0.50
0.45
500
bb< sb  curve
“stretches” to higher f
1000
1500
f
(Hz)
hoop
2000
2500
Summary
• Simple physical model developed for
trampoline effect
• Model qualitatively accounts for observed
phenomena with baseball/softball bats
– Both rk and rM are important
– e/e0 not a bat property independent of e0
• Relative performance of bats depends on
the ball!
– But this needs to be tested