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For office use only
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7331
Problem Chosen
A
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2010 Mathematical Contest in Modeling (MCM) Summary Sheet
Abstract
The “sweet spot” has drawn a lot of people’s attention, no matter its position or its effect. In
this paper, based on the laws of conservation of linear and angular momentum and the
concept of the coefficient of restitution, a validated finite element model is developed.
Though this idea is not new, our model has its own properties. Simple and useful are its
greatest characteristics, and it arrives some results that agree with recognized conclusion on
undisputed problems, while on controversial problems, it provides with reasonable
explanations toward the situations.
First, we develop a basic model. We discover that the “sweet spot” is relative to the cg of the
bat, and there isn’t a simple linear relation between them, explaining the empirical finding.
The “sweet spot” of C243 we arrive is 6.49 inch apart from the bat end. By random
simulation, we also get a result that for any bat, the “sweet spot” of it is [12.75,14.75] (cm)
apart from the center of mass with 80% degree of confidence.
Second, we provide a new insight into the effect of corking. We produce 1000 possible
stochastic(I,M) of corking bats based on Monte-Carlo simulation , and then compare the
average outcome with the maximum speed of C243. With statistical analysis, we find that a
corking bat can be better than a wood bat with a probability of 61.4% ,but the speed of it (less
than 45.4m/s) does not make a obvious difference from a wood bat (44.1m/s) according to the
study of Daniel A. Russell, Ph.D.
Third, with general values of wood and aluminum bats, we find that the “sweet spot” of
aluminum bats is wider and the speed is greater(48.8m/s),which will easily clear the wall for a
home run(47.5m/s).Thus a metal bat should be prohibited.
As there are many variables in our model, a sensitivity analysis is given. We discover
robustness of the model is good, except that the changing of the MOI will lead to relatively
bigger variations of the “sweet spot”.
A further model based on the sensitivity analysis is also provided.
Team #7331
Page 1 of 32
Hit a Randy Johnson fastball
1 Introduction
1.1 The definition of the “sweet spot”
There are many definition of the “sweet spot”:
● Center of percussion for a 15 cm pivot point [1]
When the ball hits the bat, it produces a translation that pushes the hands back and a
rotation that pulls the hands forward. When a ball is hit at the center of percussion (CoP)
for the pivot point, these two movements cancel out, and the batter feels no sting.
● Node of fundamental vibration mode[1] [2] [3] [4]
The node of the fundamental vibration mode is the point where the fundamental vibration
mode of the bat has a null point. To find this node, with your fingers and thumb, grip a bat
about 6 in from the knob. Tap the barrel at various points with an impact hammer. The
point where you feel no vibration and hear almost nothing (except the high frequency crack)
is the node. A rubber mallet could be used in place of an impact hammer: The point is that
the hammer itself should not produce any noise.
● Maximum energy transfer area [1] [6]
A collision at the maximum energy transfer point transfers the most energy to the ball.
● Maximum batted-ball speed area[1][7][8]
There is a place on the bat that produces the maximum batted ball speed.
● Maximum coefficient of restitution area
The maximum coefficient of restitution point is the point that produces the maximum CoR
for a bat-ball collision[3][9].
● Minimum sensation area[2][10]
There is an area that minimizes the total (translation plus rotation plus vibration) energy in
the handle. This area depends on the fundamental mode, the second mode, and the center of
percussion.
● Joy spot [11]
Ted Williams said that hitting the ball at the joy spot makes you the happiest.
In this paper ,the definition of the “sweet spot” is where maximum power is transferred to the
ball when hit. In other words, the “sweet spot” is where produces the maximum barrel ball
speed.
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Page 2 of 32
1.2 The controversy of the effect of corking
Does corking can enhances the “sweet spot” effect? There is a hot controversy about it. Some
players believe so, but the other not . The advantages and disadvantages have been shown as
below
1) Advantages
● When the bat is filled with cork or rubber, it will reduce the mass and the MOI.
Therefore ,the weight is concentrated closer to the hands and the speed of swing the bat
vM 1 will get larger. All other things being equal, a higher swing speed gives rise to a higher
hit ball speed and greater distance on a long fly ball.
● The lighter weight and smaller swing weight also lead to better bat control. which has
a beneficial effect for a contact-type hitter. The batter can accelerate the bat with high
speed more quickly with a corked bat, allowing the batter to react to the pitch more
quickly, wait longer before committing on the swing, and more easily change in
mid-swing.
2) Disadvantages
● By reducing the weight at the barrel end of the bat, the energy transferring from the
bat is reduced . As we known that The efficiency of the bat in transferring energy to the
ball in part depends on the weight of the part of the bat near the impact point of the ball. A
heavier bat will produce a higher hit ball speed than a lighter bat.
After all, they just have a qualitative explanation, but not a quantitative one. Here, we provide
a new insight into this problem.
1.3 Wood bats and aluminum bats
There are many different behaviors between wood bats and aluminum bats. But it is no
controversy that aluminum bats perform better than wood bats. All these papers point out that
aluminum bats beat wood bats for the following reason:
1) Aluminum bats can be swung faster
Because the barrel of an aluminum bat is hollow, the distribution of mass along the length of a
metal bat is considerably different than it is for a solid wood bat. Specially, the difference
show up in the location of the center-of-mass (CM) ,otherwise known as the balance point.
The closer the CM to the handle of the bat, the easier it is to swing the bat.
2) Aluminum bats have the "trampoline effect"
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When a ball hits a wood bat, it compresses to nearly half its original diameter, losing up to
75% of its initial energy to internal friction forces during this compression. In a hollow bat,
however, the bat barrel compresses somewhat like a spring, when the ball impacts it. This
means that the ball is not compressed as much and therefore loses less energy to internal
friction forces. Furthermore, most of the energy temporarily stored in the barrel is returned to
the ball, and the energy which is lost in the bat compression is a small fraction of what would
have been lost in the ball if it had impacted a wood bat instead.
3) Aluminum bats don't break
Wood bats, especially the thin handled ash bats used by a majority of today's players, have a
tendency to break upon impact with a ball near the middle of the bat.
2 Problem Analysis
2.1 The restatement of the problem
Our definition of the “sweet spot” is where maximum power is transferred to the ball when hit,
in other words, the “sweet spot” is where produces the maximum barrel ball speed.
In this case, it can be regarded as a optimization problem . and our objective is to find where
produces the maximum barrel ball speed on a bat, which may be different among
wood ,corking
and aluminum baseball bat。
so our tasks are clear as follows :
 Make a suitable Assumption for our models，including the definition of the sweet spot， ;
◆ Based on the Assumption, develop a model to find the sweet spot of baseball bat which is
and explain that why this spot is not at the end of the bat.
◆ Use the model to compare the effect of the “sweet spot” of wood bats ,corking bats and
aluminum bats basing on both theoretical analysis and experimental simulation.
2.2 The Layout of Problem Solving
We solve this problem with Theoretical analysis method and Experimental simulation
method.
◆ Firstly, we develop a basic model. Based on the law of conservation of linear and angular
momentum and the concept of the coefficient of restitution, a validated finite element
model is developed. To get a universal result that where is the “sweet spot”, we may
produce numerous enough possible kinds of baseball bat by stochastic simulation. And
then based on the model, explain that why the spot is not at the end of the bat.
◆ Secondly, As the MOI and mass of corking bats varies in practical situation, the data of
them is hard to get. And if a corking bat is permitted, then people will try corking in all
kinds of ways to get a perfect effect. Thus, we consider all kinds of corking. Here,
different kinds of corking are described as different (I,M).
We will produce numerous enough possible stochastic(I,M) of a corking bat many times
Team #7331
Page 4 of 32
base on Monte-Carlo simulation , and compare the average outcome to the maximum
speed of C243. After statistical analysis, we can see the percentage of corking bats bit
wood bats, and also the advantage corking bats over wood bats. Then a explanation why
Major League Baseball prohibits “corking” can be obtain.
◆ Thirdly, with general values of wood and aluminum bats, we can compare the effect of
the “sweet spot” of wood bats and aluminum bats. The study of Daniel A. Russell, Ph.D.
[12]
can provide a reference standard .
3 Basic Assumptions
1) There is no external force exerted on the bat, and it behaves as a free body on the short time
scale of the collision.
2) For a sufficiently short or a sufficiently stiff bat, it is a rigid body.
3) As the relation between MOI and mass is uncertain, we assume that it is stochastic.
4) As the range and distribution of MOI and mass for existent bats are not available, we
assume that the range is [0.9 I CM /0.9M, 1.1 I CM /1.1M], where the term I CM and M are
MOI and mass of C243.
4 Model Design
4.1 The basic Model
Over the years, papers have addressed both experimental and theoretical issues associated
with the baseball–bat collision.
Brody studied the vibrational spectrum of a hand-held bat during and after the collision and
showed that the bat behaves as a free body on the short time scale of the collision. In other
words, it can assume that there is no external reaction force from the hand. In addition, for a
sufficiently short or a sufficiently stiff bat, the collision between a bat and a ball can be
analyzed using a rigid body approximation, as illustrated in Figure 1 .
=
CM
.
b
.
.
Figure 1. baseball–bat collision
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Hence, the system of both the bat and the ball must follow the law of conservation of
linear and angular momentum. Based on these two world universal fundamental laws, we
develop our validated finite element model as follows:
4.1.1 The model of the bat
As we know, there are rotational motion and transnational motion in batting. Ted Williams
emphasized the importance of rotation, and Charlie Lau emphasized translation. In his book
The Physics of Baseball, Adair reviews the different aspects of a bat-ball collision, and
emphasize the effect of both the rotational and transnational motion. The swing is mostly
translation in the beginning stages and then mostly rotation just before hitting the ball. The
situation can be easily described by Figure 2.
Figure 2. Motion of the swinging bat
So here, we consider both of them.
Note that the swing consists of two kinds of rotations: one centered in the batter’s body and
the other between the batters hands, which can be drawn as shown in Figure 3.
Figure 3. Variables in our model
Then the linear velocity of the bat at the cg ( vI ) and at the point of impact ( vM ) before a
collision with the ball is
vM  ( A  B)wb  BwW
(1)
vI  ( A  B  C ) wb  CwW
(2)
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Where wb —— the rotation of the body
wW —— the rotation of the batter’s wrists during the swing
Combining the two equations above, we can obtain
vI  (wb  wW )C  vM
(3)
The formula can be also transform into
vI  wM  C  vM
(4)
With wM replaces ( wb  wW ) for convenience
4.1.2 The model of the ball
Compared to the bat, the model of the ball is much more simple. Assume that the rotational
kinetic energy of the ball is negligible when compared to the translational kinetic
energy(generally 1800 rpm to 85 mph). And here we only consider the speed of ball just after
collision. Let vm denote the speed of the ball.
4.1.3 The model of the collision of bat and ball
During the collision, as has assumed that the bat behaves as a free body, hence the system of
bat and ball obeys the law of conservation of momentum. It can be described by
MvM 1  mvm1  MvM 2  mvm 2
(5)
Where M —— the mass of bat
m —— the mass of ball
At the same time, suppose that the force exerted on the bat from the impact with the ball is –F,
resulting in a torque on the bat about its cg is –CF.
Equating this torque over time t to the change in angular momentum yields for the bat
 CFt  (wM 1  wM 2 ) I CM
(6)
Ft  (vm1  vm 2 )m
(7)
For the ball,
Combine the two expression, we obtain
( wM 1  wM 2 ) I CM  (vm1  vm 2 )m  0
4.1.4 The definition of COR
(8)
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Page 7 of 32
As the impact is not at the cg location of the bat, COR here is defined as:
e

vI 2  vm 2
vm1  vI 1
vM 2  vm 2  CwM 2
vm1  vM 1  CwM 1
(9)
Thus , we can gain the batted-ball velocity vm 2 that
vm 2
m mC 2
 vm1 (e 

)  (1  e)(vM 1  CwM 1 )
M I CM

m mC 2
1

M
I CM
(10)
4 .1.5 Simulations and result
Instead of doing complex calculation, we choose to draw a curve to make the description.
From the above model, using parameters of C243 wood bat.
Table 1: The parameters of C243 wood bat
Parameters of C243
Stated length (in)
34
Mass ( kg )
0.905
I CM ( kg / m 2 )
0.048
Measured d knobcm (cm)
57
And data from current literature
Table 2: The other parameters
vM 1 ( m/s)
vm1 ( m/s)
m (kg)
e
30
34
0.145
0.36
we can now determine the sweet spot of the bat
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The batted-ball velocity
45
Velocity(m/s)
44
43
42
41
40
39
0
0.05
0.1
0.15
0.2
0.25
0.3
Distance from cg(m)
Figure 4. The batted-ball velocity vs. Distance from cg
we can obtain that the sweet spot which transfers the greatest kinetic energy is 0.139 m apart
from the cg, and thus 0.164 m apart from the barrel bat end. This result is equal to 6.49 in unit
of inch, which agrees with previous study.
To get a universal result, this is not enough. Hence, we choose to produce many kinds of bats
(with different mass and inertia ) randomly by simulation. As mass and inertia of all kinds of
bats varies in a small range, we assume that mass varies between 0.9 M and 1.1 M , and
similarly, inertia varies between
0.9 I CM and 1.1 I CM . By M changing from 0.9 M to
1.1 M , and I CM changing from 0.9 I CM to 1.1 I CM equably, we obtain 1000 kinds of bats.
Team #7331
Page 9 of 32
Simulate start
produce the Location of
the impact randomly
Calculate the batted
ball speed on the spot
Search the
maximum speed
Returen the
sweet spot
end
Figure 5. The flow chart of the simulations
Count the numbers of C
between[12,12.5],[12.5,13.0],[13.0,13.5],[13.5,14.0],[14.0,14.5],[14.5,15.0],[15.0,15.5]when
it provides a maximum vm 2 in all kinds of stochastic situation. We get result as follows
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The histogram of 1000 sweet spots
200
The number of sweet spots
180
160
140
120
100
80
60
40
20
0
11.5
12
12.5
13
13.5
14
14.5
15
15.5
Distance from cg(cm)
Figure 6. The histogram of 1000 sweet spots
As we can see, the probability of C between [12,12.5] and [15.0,15.5] are relatively small
(0.03 and 0.081 respectively) , and the probability of C between [12.5,15.0] is equal in a
certain degree. For simplification, we neglect the probability between the former two interval,
and assume it is a homogeneous distribution between the latter one. With knowledge of
probability theory, we arrive that when degree of confidence being 80%, sweet spot of any bat
is in a range of [12.75,14.75] apart from the center of mass.
We can easily see that the range of sweet spot does not include the barrel bat end ( the cg is 32
cm apart from the end on average). The sweet spot is more relative to the cg, and from the
model, we know that the distance and the maximum vm 2 is not a simple linear relation, so
the sweet spot can be in the barrel bat end provides that cg is close to the end enough, but for
the current shape of barrel bat, it is impossible.
4.2 The development of the model
4.2.1 The effect of corking
There is a hot controversy about whether corking can enhances the “sweet spot” effect. Some
players believe so, but at the same time, some study point out no. They insist that there is not
only advantages but also disadvantages of it, which cancelled out each other.
After all, they just have a qualitative explanation, but not a quantitative one. Here, we provide
a new insight into this problem.
4.2.2 Corking bat Vs Wood bat model
Base on the current study, we summarize the difference between a corking bat and a wood bat
model:
Team #7331
Page 11 of 32
1) the mass of a corking bat is less than a wooden one
2) the MOI of a corking bat is less than a wooden one
3) the swing speed of a corking bat is bigger than a wooden one
There are three variable, for simplification, the development of the model is necessary.
By a further study into this question, we find that there is a relation between MOI and swing
speed.
Fleisig , et al. [15]investigated the effect of bat mass and inertia on swing velocity at the
American Sports Medicine Institute (ASMI). They found that the bat swing speeds increased
as the bat MOI decreased. This finding was based on the linear velocity data because an
ANOVA analysis revealed significant differences among the linear velocities but not for the
angular velocities.
Based on the regression, the batted-ball bat speed (linear velocity of the sweet spot in mph)
can be predicted by
v  69.6  48.7I
(11)
Where I —— the MOI about the bat handle( lbf  ft  s 2 )
According to the parallel axis theorem, we obtain
I  I CM  MB 2
(12)
So the equation (11) can be changed to
vM 1  31.1267  18.6297( I CM  MB 2 )
(13)
Where
I CM —— the MOI of the bat ( kg / m 2 )
M
B
—— the mass of the bat
—— the distance between the bat handle and the center of quality focus
So our model can be developed as below
 vm1 (e 
vm 2 
m mC 2

)  (1  e){31.1267  18.6297( I CM  MB 2 )  CwM 1}
M I CM
m mC 2
1

M I CM
4.2.3 Simulation and Results
a)The layout of the Simulation
(14)
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Page 12 of 32
As the MOI and mass of corking bats varies in practical situation, the data of them is hard to
get. And if a corking bat is permitted, then people will try corking in all kinds of ways to get a
perfect effect. Thus, we consider all kinds of corking. Here, different kinds of corking are
described as different (I,M).
We will produce numerous enough possible stochastic(I,M) of a corking bat many times base
on Monte-Carlo simulation , and compare the average outcome to the maximum speed of
C243. After statistical analysis, we can see the percentage of corking bats bit wood bats, and
also the advantage corking bats over wood bats.
We use the model simulation to find out the influence on the speed of batted ball bases on our
model with the help of the MATLAB.
Step 1: get the value of mass and MOI about the corking bats randomly.
Because mass and inertia of all kinds of bats varies in a small range, we assume that mass
varies between 0.9 M and 1.1 M . Similarly, inertia varies between 0.9 I CM and 1.1 I CM .
By M changing from 0.9 M to 1.1 M , and I CM changing from 0.9 I CM to 1.1 I CM equably,
we obtain 1000 kinds of bat datas.
Step 2 : calculate the maximum value of the batted ball speed v
m2
for each corking bat.
The value
Step 3 : compare v
m2
between the corking bats and the wood bats and draw the distribution
picture .
Step 4 : repeat the Simulation 4 times.
The simulation flow diagram is shown as below:
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Page 13 of 32
Simulate start
Produce the corking
bats randomly
Calculate the speed of
on sweet spot of the
corking bats
Compare the speed of
the corking bats with the
wood bats
Returen the
resuls
end
Figure 7: Simulation flow diagram
b) The results of the Simulation
◆the statistical results of the Simulation are shown as below:
Table 3 :the statistical results of model simulation
Times of simulation
Standard
1
2
3
4
On average
v
m2
 44.43 (m/s)
Times (total: 1000)
probability
647
64.70%
594
59.4%
633
582
614
63.30%
58.20%
61.4%
◆the statistical graphs about the maximum batted ball speed of corking bats on sweet spot :
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The maximum speed of 1000 bats
46
45.5
45.5
The maximum speed(m/s)
The maximum speed(m/s)
The maximum speed of 1000 bats
46
45
44.5
44
43.5
43
45
44.5
44
43.5
0
100
200
300
400
500
600
700
800
900
43
1000
0
100
200
300
400
R(I,m)
Figure 8: the results of 1 th simulation
The maximum speed of 1000 bats
700
800
900
1000
The maximum speed of 1000 bats
46
45.5
45.5
The maximum speed(m/s)
The maximum speed(m/s)
600
Figure 9: the results of 2 th simulation
46
45
44.5
44
43.5
43
500
R(I,m)
45
44.5
44
43.5
0
100
200
300
400
500
600
R(I,m)
700
800
900
1000
43
0
100
200
300
400
500
600
700
800
900
1000
R(I,m)
th
th
Figure 10: the results of 3 simulation
Figure 11: the results of 4
●notes : the red line —— the maximum batted ball speed of wood bats at sweet spot
the blue '*' —— the maximum batted ball speed of corking bats at sweet spot
simulation
Form the table 3 and 4 Figures, we can see that, using the corking bats can do better than the
wood bats at a probability of 70%. It indicates that corking is of some meaning. This may
explain the reason that Major League Baseball prohibiting “corking”.
However ,note that though the corking bat has a certain probability to make the baseball go
faster than the wood bat ,it is about 1m/s faster. But does this 1m/s matter?
By some study, higher batted ball speed can translate to better performance . Daniel A.
Russell, Ph.D. [12] found that a ball hit with a speed of 98.6-mph(equal to 44.1m/s) will not
clear the center field wall, but would land about 388-ft from home plate for an easy pop fly
out. A ball hit with a batted ball speed of 101.5-mph(equal to 45.4m/s ), would barely clear
the centerfield wall, but could be caught by a good fielder with a high leap.
Here , the greatest speed of a wood bat is 44.43m/s, while that of a corking bat is almost less
than 45.4m/s, which would barely clear the centerfield wall, but could be caught by a good
fielder with a high leap. Note that we have include all kinds of cooking by stochastic
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simulation, but the result is still not obvious.
To conclude, corking bat indeed can increase the batted ball speed, but this increment does not
make a obvious difference. Though the Major League Baseball prohibiting “corking” , it is
not necessary if we only consider the batted ball speed.
4.3 Wood bats vs Aluminum bats
Because most of metal bats are made of aluminum and hollow. In this paper ,we research the
batted ball speed of solid-wood bats and hollow-aluminum bats on the sweet spot .
A wood bat and an aluminum bat ,which one can do better? There are many thesis about this
question. After giving an insight into these thesis, we find that
●A wood bat is slightly heavier than an aluminum bat.
●Generally speaking, the MOI of wood bats will be larger than aluminum bats.
●The COR of wood bats is different from the aluminum bats.
●Aluminum bats have the "trampoline effect", it has a influence on the COR.
To reflect the effect of these elements, we put the general data of solid-wood and
hollow-aluminum bats[16] (in table 4) into our development model.
Table 4
Bat
Weight ( kg )
CM ( m )
I ( kg / m 2 )
COR
Wood
0.876
0.5842
0.03752
0.36
Metal
0.828
0.5258
0.05669
0.41
then we can get the results as follow
The different batted-ball velocity about aluminum bats and wood bats
wood
aluminum
50
48
Velocity(m/s)
46
44
42
40
38
36
34
32
30
0
0.05
0.1
0.15
0.2
0.25
Distance from cg(m)
Figure 12. Aluminum bats vs. wood bats
0.3
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We can see that locations of them are different. Sweet spot of a aluminum bat has a bigger
distance from cg. In addition, a aluminum bat give out a greater batted-ball speed vm 2 than a
wooden one obviously(48.8m/s and 43.8m/s respectively), which reflects the performance of
them. More over, when the location of impact is removed from the sweet spot, batted-ball
speed of a aluminum bat decreases much slower than a wooden one, which produces a wide
“sweet zone”.
Which element influences the result most? We can arrive the reason in sensitive analysis
below that the difference of COR is the one, and the famous "trampoline effect" plays a very
important role here.
So what is the meaning? As has been discussed above, higher batted ball speed can translate
to better performance . Daniel A. Russell, Ph.D. [12] shows that a ball hit with a speed of
98.6-mph(equal to 44.1m/s) will not clear the center field wall, but would land about 388-ft
from home plate for an easy pop fly out. However, a ball hit with a speed of 106.5-mph(equal
to 47.5m/s), will easily clear the wall for a home run.
As a result, a person who with a aluminum bat will easily clear the wall for a home run. The
skill of the sports man will not fully revealed, and this is not fair for those that with a wood
bat. The wider sweet zone also add to this situation.
To conclude, if a aluminum bat is permitted, the skill of the sports man will not fully revealed,
and this is not fair for those that with a wood bat. The aluminum bat should be prohibited.
5 Sensitivity Analysis
By changing the parameters in the model, we can analyze the sensitivity of our model.
First, We vary the value of m , M and v m1 one by one. When we vary a parameter we
obtain three curves about the distance from cg and the bat ball speed. So we can get three
Figures as follows .
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45
m=0.14kg
m=0.145kg
m=0.15kg
Velocity(m/s)
44
43
42
41
40
39
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
Distance from cg(m)
Figure 13. change the mass of ball
45
M=0.85kg
M=0.9kg
M=0.95kg
Velocity(m/s)
44
43
42
41
40
39
0
0.05
0.1
0.15
0.2
0.25
0.3
Distance from cg(m)
Figure 14. change the mass of bat
0.35
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46
vm 1 = 3 0 m / s
vm 1 = 3 5 m / s
vm 1 = 4 0 m / s
Velocity(m/s)
45
44
43
42
41
40
39
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
Distance from cg(m)
Figure 15. change the velocity of ball
From the Figures above, we can see that although we vary the mass of ball, the mass of bat
and the velocity of ball respectively, the distance of the sweet spot from the center of gravity
is around 0.13m all the time. And the change of the maximum speed is less than 2m/s. So the
stability and applicability of our model is high.
Then we vary the Swing Speed vM 1 and the COR respectively, also we obtain two Figures as
follows
VM1=27m/s
VM1=30m/s
VM1=33m/s
Velocity(m/s)
48
46
44
42
40
38
36
0
0.05
0.1
0.15
0.2
0.25
0.3
Distance from cg(m)
Figure 16. change the velocity of bat
0.35
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47
e=0.32
e=0.36
e=0.4
Velocity(m/s)
46
45
44
43
42
41
40
39
38
37
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
Distance from cg(m)
Figure 17. change the value of COR
From the Figures, we can see that with the vM 1 or the COR increased, the maximum speed
change remarkably. But the “sweet spot” is almost invariant. It also illustrates this method is
suitable to find the sweet spot.
Finally, we vary the moment of inertia of the baseball bat about its center of gravity I CM . The
result is shown in Figure 18. Unfortunately, the sweet spot change with the I CM increased. It
is the primary weakness of our model. However, it is somewhat reasonable, because I CM is a
relatively complex term, thus the way it influences the distance is also hard to control.
45
Ic m= 0. 044k g. m2
Ic m= 0. 048k g. m2
Ic m= 0. 052k g. m2
Velocity(m/s)
44
43
42
41
40
39
38
0
0.05
0.1
0.15
0.2
0.25
0.3
Distance from cg(m)
0.35
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Page 20 of 32
Figure 18. change the moment of inertia
6 Strengths and Weaknesses
6.1 Strengths
 Our model bases on the world universal suitable fundamental laws, there is less error
comes from theory inaccuracy
 It is very simple and easy to put into practice
 by the sensitivity analysis, we can see that robustness of the model is good to a certain
extent
 the data used in the model in easy to get and determine
 the result of the model agrees with proverbial conclusion and current study on some
undisputed problems, while on controversial problems, it provides with reasonable
explanations toward the situations
 with the help of simulation, the model gives a universal result
6.2 Weaknesses
 the barrel bat is assumed as a rigid body approximately, and neglects the flexibility,
which may bring some inaccuracy
 the COR is treated as a constant for a certain bat, which is not strictly true in reality.
 When confront uncertain distribution, we assumes that it is homogeneous, which may not
agree with practice
7
Further Works
From the sensitivity analysis above, we can see that COR influences the batted ball speed
most. In order to get a relatively accurate batted ball speed, we must treat COR strictly.
In our model, we treat COR as a constant value for a certain bat. But a further insight into it,
we discover that it is mainly relative to the mass of the ball and the bat, and where the ball
hits the bat. As different location of impact will influence the value of COR, this may also
result in the mistaken the sweet spot.
In this case, we try to give a more precise value of COR. But it seems that COR does not have
a directly relation with cg or oscillation center, let alone relation with the knob or end of the
bat.
Fortunately, By Alan M. Nathan[17], we get a expression of e
e  q  qr  r
Where q is collision efficiency, and r is recoil factor
(15)
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and
Page 21 of 32
r
m
Me f f
(16)
where m due to the mass of a ball, and M eff due to the effective mass of a bat.
According to Chris Ray[18]，we have
M
z2
 (1  2 ) 1
M eff
kT
(17)
Where M is the actual bat mass, z is the axial distance between the impact point and center
of mass, and kT is the centroidal transverse radius of gyration.
Then r can be described as
z 2 1
)
kT2
M
m(1 
r
(18)
For q, base on Alan M. Nathan[4], we get the following picture, which provides us with a
relation between q and cg
Figure 19. Collision efficiency vs. cg
Then q can be described by a function
q  f (C )
(19)
Where C is the distance between cg and the location of impact
The function can be obtained by experiment.
with eq (15),(18),(19), we arrive the expression of e as follows
f (C )  (1 
e  f (C ) 
M
z 2 1
) m
kT2
z 2 1
)
kT2
M
m(1 

(20)
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Page 22 of 32
Substitute the expression for e in our model, we obtain
vm 2
z 2 1
z 2 1
f (C )  (1  2 ) m m(1  2 )
m mC 2
kT
kT
 vm1 ( f (C ) 



)
M
M
M
I CM


m mC 2
1

M
I CM
z2
z2
f (C )  (1  2 ) 1 m m(1  2 ) 1
kT
kT
(1  f (C ) 

)(vM 1  CwM 1 )
M
M
m mC 2
1

M
I CM
(21)
8 References
[1] P. Brancazio, SportScience: Physical Laws and Optimum Performance.New York: Simon and
Schuster, 1984.
[2] R. Cross, “The sweet spot of the baseball bat,” Amer. J. Phys., vol. 66, pp. 772–779, 1998.
[3] A. M. Nathan, “Dynamics of the baseball-bat collision,” Amer. J. Phys., vol. 68, pp. 979–990, 2000.
[4] L. L. Van Zandt, “The dynamical theory of the baseball bat,” Amer. J. Physics, vol. 60, no. 2, pp.
172–181, 1992.
[5] Smith L.V. (2001) Evaluating baseball bat performance, Sports Eng., 4, 205-214.
[6] Vedula, G., and Sherwood, J.A., (2004), An Experimental and Finite Element Study of the
Relationship amongst the ‘Sweet spot’, COP, and Vibration Nodes in BaseballBats, Proceedings of the
5th International Conference on the Engineering of Sport,September, UC Davis, CA
[7] J. J. Crisco, R. M. Greenwald, and L. H. Penna. (1999) Baseball bat Performance: A Batting Cage
Study. [Online]. Available:
http://www.nisss.org/BBSPEED6a.html
[8] http://paws.kettering.edu/~drussell/bats-new/sweetspot.html
[9] G.S. Sawicki, M. Hubbard andW.J. Stronge, “How to hit home runs: Optimum baseball swing
parameters for maximum range trajectories,” Am. J. Phys. 71,1152–1162 (2003).
[10] R. K. Adair, ‘‘The physics of baseball,’’ Phys. Today 48, 26–31 ~May 1995
T. Williams and J. Underwood, The Science of Hitting. New York: Simon and Schuster, 1982.
[11] T. Williams and J. Underwood, The Science of Hitting. New York: Simon and Schuster, 1982.
[12] Daniel A. Russell, Ph.D.”Physics and Acoustics of Baseball & Softball Bats”, Applied Physics,
Kettering University, Flint, MI 48504-4898
[13]Howard Brody, “Models of baseball bats”, Physics department, Universtiy of pennsylvania,
Fhiladelphia, Pennsylvania 19104,1989
[14] J. J. Crisco & R. M. Greenwald, “Medicine & Science in Sports & Exercise 34(10): 1675-1684”, Oct
2002
[15] Fleisig, Glenn S., Zheng, Nigel, Stodden, David, Andrews, James R., "The Relationship Among
Baseball Bat Weight, Moment of Inertia, and Velocity," American Sports Medicine Institute, August 1997
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[16] Daniel A. Russell, Ph.D., “Physics and Acoustics of Baseball & Softball Bats”
[17] Alan M. Nathan 403 Loomis , “Physics 199BBThe Physics of Baseball Fall 2007 Freshman
Discovery Course “,333-0965
[18] Ann Chase1, Mont Hubbard1, Chris Ray2, “How Optimal Baseball Swings Change for Three Levels
of Play “(P103)
9 Appendix
1. The code of Figure 4
clear all
m1=0.145;
m2=0.905;
v1=34.03;
v2=30;
e=0.36;
om2=45;
i=0.048;
b(1)=0;
v1a(1)=(v1.*(e-m1./m2-(m1.*b(1).*b(1))/i)+(1+e).*(v2+b(1).*om2))./(1+m1/m2+(m1.*b(1).*b(1))/i);
for n=2:2936
b(n)=b(n-1)+0.0001;
v1a(n)=(v1.*(e-m1./m2-(m1.*b(n).*b(n))/i)+(1+e).*(v2+b(n).*om2))./(1+m1/m2+(m1.*b(n).*b(n))/i);
end
axis([0,0.3,39,45])
hold on
for n=1:2936
plot(b(n),v1a(n),'r')
end
title(' The batted-ball velocity ','fontsize',12,'fontweight','bold','fontname',' Times New Roman')
xlabel(' Distance from cg(m)','fontsize',12,'fontweight','bold','fontname',' Times New Roman'),
ylabel(' Velocity(m/s)','fontsize',12,'fontweight','bold','fontname',' Times New Roman')
grid
hold off
[Dv1a,bb]=max(v1a);
ss=bb.*0.0001
Dv1a
2. The code of Figure 6
clear all
m1=0.145;
v1=34.03;
d=0.57;
e=0.55;
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om2=45;
m2(1)=0.8+0.2.*rand(1);
i(1)=0.043+0.01*rand(1);
v1a(1)=0;
for k=1:2936
b(k)=0.0001.*k;
if
v1a(1)<(v1.*(e-m1./m2(1)-(m1.*b(k).*b(k))/i(1))+(1+e).*(31.1267-18.6297.*(i(1)+m2(1).*d.*d)+b(k).*om
2))./(1+m1/m2(1)+(m1.*b(k).*b(k))/i(1))
v1a(1)=(v1.*(e-m1./m2(1)-(m1.*b(k).*b(k))/i(1))+(1+e).*(31.1267-18.6297.*(i(1)+m2(1).*d.*d)+b(k).*om
2))./(1+m1/m2(1)+(m1.*b(k).*b(k))/i(1));
bb(1)=b(k);
end
end
for n=2:1000
m2(n)=0.8+0.2.*rand(1);
i(n)=0.043+0.01*rand(1);
v1a(n)=0;
for k=1:2936
b(k)=0.0001.*k;
if
v1a(n)<(v1.*(e-m1./m2(n)-(m1.*b(k).*b(k))/i(n))+(1+e).*(31.1267-18.6297.*(i(n)+m2(n).*d.*d)+b(k).*om
2))./(1+m1/m2(n)+(m1.*b(k).*b(k))/i(n))
v1a(n)=(v1.*(e-m1./m2(n)-(m1.*b(k).*b(k))/i(n))+(1+e).*(31.1267-18.6297.*(i(n)+m2(n).*d.*d)+b(k).*om
2))./(1+m1/m2(n)+(m1.*b(k).*b(k))/i(n));
bb(n)=b(k);
end
end
end
for m=1:63
s(m)=0;
for n=1:1000
if bb(n)<0.01*m/2
if bb(n)>=0.01*(m-1)/2
s(m)=s(m)+1;
end
end
end
end
s
t=[12,12.5,13,13.5,14,14.5,15];
ss=[30 165 176 170 190 188
81];
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bar(t,ss)
title('The histogram of 1000 sweet spots','fontsize',12,'fontweight','bold','fontname',' Times New Roman')
xlabel('Distance from cg(cm)','fontsize',12,'fontweight','bold','fontname',' Times New Roman'),
ylabel('The number of sweet spots','fontsize',12,'fontweight','bold','fontname',' Times New Roman')
3. The code of Figure 8,9,10,11
clear all
m1=0.145;
v1=34.03;
d=0.5355;
e=0.36;
om2=45;
maxv=44.43;
s=0;
m2(1)=0.8+0.15.*rand(1);
i(1)=0.043+0.007*rand(1);
v1a(1)=0;
for k=1:3145
b(k)=0.0001.*k;
if
v1a(1)<(v1.*(e-m1./m2(1)-(m1.*b(k).*b(k))/i(1))+(1+e).*(31.1267-1.7797.*(i(1)+m2(1).*d.*d)+b(k).*om2
))./(1+m1/m2(1)+(m1.*b(k).*b(k))/i(1))
v1a(1)=(v1.*(e-m1./m2(1)-(m1.*b(k).*b(k))/i(1))+(1+e).*(31.1267-1.7797.*(i(1)+m2(1).*d.*d)+b(k).*om2
))./(1+m1/m2(1)+(m1.*b(k).*b(k))/i(1));
end
end
if v1a(1)>maxv
s=s+1;
end
for n=2:1000
m2(n)=0.8+0.15.*rand(1);
i(n)=0.043+0.007*rand(1);
v1a(n)=0;
for k=1:3145
b(k)=0.0001.*k;
if
v1a(n)<(v1.*(e-m1./m2(n)-(m1.*b(k).*b(k))/i(n))+(1+e).*(31.1267-1.7797.*(i(n)+m2(n).*d.*d)+b(k).*om2
))./(1+m1/m2(n)+(m1.*b(k).*b(k))/i(n))
v1a(n)=(v1.*(e-m1./m2(n)-(m1.*b(k).*b(k))/i(n))+(1+e).*(31.1267-1.7797.*(i(n)+m2(n).*d.*d)+b(k).*om2
))./(1+m1/m2(n)+(m1.*b(k).*b(k))/i(n));
end
end
if v1a(n)>maxv
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s=s+1;
end
end
hold on
for n=1:1000
plot(n,v1a(n),'*')
end
x=0:1000;
for n=1:1001
y(n)=44.43;
end
plot(x,y,'r','LineWidth',2)
title('The maximum speed of 1000 bats', 'fontsize',12,'fontweight','bold','fontname',' Times New
Roman'),xlabel('R(I,m)', 'fontsize',12,'fontweight','bold','fontname',' Times New Roman'),ylabel('The
maximum speed(m/s)', 'fontsize',12,'fontweight','bold','fontname',' Times New Roman')
s
hold off
fprintf('The percentage of the maximum speed larger than the base one is %03.2f%%',s/10)
4. The code of Figure 12
clear all
mb=0.145;
mw=0.876;
mm=0.828;
vb=34.03;
vw=30.354;
vm=31.6951;
ew=0.36;
em=0.41;
om2=45;
iw=0.03752;
im=0.05669;
bw(1)=0;
bm(1)=0;
maxvw(1)=(vb.*(ew-mb./mw-(mb.*bw(1).*bw(1))/iw)+(1+ew).*(vw+bw(1).*om2))./(1+mb/mw+(mb.*bw
(1).*bw(1))/iw);
maxvm(1)=(vb.*(em-mb./mm-(mb.*bm(1).*bm(1))/im)+(1+em).*(vm+bm(1).*om2))./(1+mb/mm+(mb.*b
m(1).*bm(1))/im);
for n=2:2936
bw(n)=bw(n-1)+0.0001;
bm(n)=bm(n-1)+0.0001;
maxvw(n)=(vb.*(ew-mb./mw-(mb.*bw(n).*bw(n))/iw)+(1+ew).*(vw+bw(n).*om2))./(1+mb/mw+(mb.*bw
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(n).*bw(n))/iw);
maxvm(n)=(vb.*(em-mb./mm-(mb.*bm(n).*bm(n))/im)+(1+em).*(vm+bm(n).*om2))./(1+mb/mm+(mb.*b
m(n).*bm(n))/im);
end
axis([0,0.3,30,51])
hold on
plot(bw,maxvw,'r','LineWidth',2)
plot(bm,maxvm,'b','LineWidth',2)
title('
The
different
batted-ball
velocity
about
aluminum
bats
and
wood
bats
','fontsize',12,'fontweight','bold','fontname',' Times New Roman')
xlabel(' Distance from cg(m)','fontsize',12,'fontweight','bold','fontname',' Times New Roman'),
ylabel(' Velocity(m/s)','fontsize',12,'fontweight','bold','fontname',' Times New Roman')
legend('wood','aluminum')
grid
hold off
5. The code of Figure 13
clear all
m1(1)=0.14;
m1(2)=0.145;
m1(3)=0.15;
m2=0.905;
v1=34.03;
v2=30;
e=0.36;
om2=45;
i=0.048;
for k=1:3
for n=1:2936
b(n)=0.0001*n;
v1a(k,n)=(v1.*(e-m1(k)./m2-(m1(k).*b(n).*b(n))/i)+(1+e).*(v2+b(n).*om2))./(1+m1(k)/m2+(m1(k).*b(n).
*b(n))/i);
end
end
%axis([0,0.3,39,45])
hold on
plot(b,v1a(1,:),'r','LineWidth',2)
plot(b,v1a(2,:),'k','LineWidth',2)
plot(b,v1a(3,:),'b','LineWidth',2)
xlabel(' Distance from cg(m)','fontsize',12,'fontweight','bold','fontname',' Times New Roman'),
ylabel(' Velocity(m/s)','fontsize',12,'fontweight','bold','fontname',' Times New Roman')
legend('m=0.14kg','m=0.145kg','m=0.15kg')
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grid
hold off
6. The code of Figure 14
clear all
m1=0.145;
m2(1)=0.85;
m2(2)=0.9;
m2(3)=0.95;
v1=34.03;
v2=30;
e=0.36;
om2=45;
i=0.048;
for k=1:3
for n=1:2936
b(n)=0.0001*n;
v1a(k,n)=(v1.*(e-m1./m2(k)-(m1.*b(n).*b(n))/i)+(1+e).*(v2+b(n).*om2))./(1+m1/m2(k)+(m1.*b(n).*b(n))/
i);
end
end
%axis([0,0.3,39,45])
hold on
plot(b,v1a(1,:),'r','LineWidth',2)
plot(b,v1a(2,:),'k','LineWidth',2)
plot(b,v1a(3,:),'b','LineWidth',2)
xlabel(' Distance from cg(m)','fontsize',12,'fontweight','bold','fontname',' Times New Roman'),
ylabel(' Velocity(m/s)','fontsize',12,'fontweight','bold','fontname',' Times New Roman')
legend('M=0.85kg','M=0.9kg','M=0.95kg')
grid
hold off
7. The code of Figure 15
clear all
m1=0.145;
m2=0.905;
v1(1)=30;
v1(2)=35;
v1(3)=40;
v2=30;
e=0.36;
om2=45;
i=0.048;
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for k=1:3
for n=1:2936
b(n)=0.0001*n;
v1a(k,n)=(v1(k).*(e-m1./m2-(m1.*b(n).*b(n))/i)+(1+e).*(v2+b(n).*om2))./(1+m1/m2+(m1.*b(n).*b(n))/i);
end
end
%axis([0,0.3,39,45])
hold on
plot(b,v1a(1,:),'r','LineWidth',2)
plot(b,v1a(2,:),'k','LineWidth',2)
plot(b,v1a(3,:),'b','LineWidth',2)
xlabel(' Distance from cg(m)','fontsize',12,'fontweight','bold','fontname',' Times New Roman'),
ylabel(' Velocity(m/s)','fontsize',12,'fontweight','bold','fontname',' Times New Roman')
legend('vm1=30m/s','vm1=35m/s','vm1=40m/s')
grid
hold off
8. The code of Figure 16
clear all
m1=0.145;
m2=0.905;
v1=34.03;
v2(1)=27;
v2(2)=30;
v2(3)=33;
e=0.36;
om2=45;
i=0.048;
for k=1:3
for n=1:2936
b(n)=0.0001*n;
v1a(k,n)=(v1.*(e-m1./m2-(m1.*b(n).*b(n))/i)+(1+e).*(v2(k)+b(n).*om2))./(1+m1/m2+(m1.*b(n).*b(n))/i);
end
end
axis([0,0.35,36,49])
hold on
plot(b,v1a(1,:),'r','LineWidth',2)
plot(b,v1a(2,:),'k','LineWidth',2)
plot(b,v1a(3,:),'b','LineWidth',2)
xlabel(' Distance from cg(m)','fontsize',12,'fontweight','bold','fontname',' Times New Roman'),
ylabel(' Velocity(m/s)','fontsize',12,'fontweight','bold','fontname',' Times New Roman')
legend('VM1=27m/s','VM1=30m/s','VM1=33m/s')
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grid
hold off
9. The code of Figure 17
clear all
m1=0.145;
m2=0.905;
v1=34.03;
v2=30;
e(1)=0.32;
e(2)=0.36;
e(3)=0.4;
om2=45;
i=0.048;
for k=1:3
for n=1:2936
b(n)=0.0001*n;
v1a(k,n)=(v1.*(e(k)-m1./m2-(m1.*b(n).*b(n))/i)+(1+e(k)).*(v2+b(n).*om2))./(1+m1/m2+(m1.*b(n).*b(n))/
i);
end
end
%axis([0,0.3,36,49])
hold on
plot(b,v1a(1,:),'r','LineWidth',2)
plot(b,v1a(2,:),'k','LineWidth',2)
plot(b,v1a(3,:),'b','LineWidth',2)
xlabel(' Distance from cg(m)','fontsize',12,'fontweight','bold','fontname',' Times New Roman'),
ylabel(' Velocity(m/s)','fontsize',12,'fontweight','bold','fontname',' Times New Roman')
legend('e=0.32','e=0.36','e=0.4')
grid
hold off
10. The code of Figure 18
clear all
m1=0.145;
m2=0.905;
v1=34.03;
v2=30;
e=0.36;
om2=45;
i(1)=0.044;
i(2)=0.048;
i(3)=0.052;
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for k=1:3
for n=1:2936
b(n)=0.0001*n;
v1a(k,n)=(v1.*(e-m1./m2-(m1.*b(n).*b(n))/i(k))+(1+e).*(v2+b(n).*om2))./(1+m1/m2+(m1.*b(n).*b(n))/i(k
));
end
end
%axis([0,0.3,36,49])
hold on
plot(b,v1a(1,:),'r','LineWidth',2)
plot(b,v1a(2,:),'k','LineWidth',2)
plot(b,v1a(3,:),'b','LineWidth',2)
xlabel(' Distance from cg(m)','fontsize',12,'fontweight','bold','fontname',' Times New Roman'),
ylabel(' Velocity(m/s)','fontsize',12,'fontweight','bold','fontname',' Times New Roman')
legend('Icm=0.044kg.m2','Icm=0.048kg.m2','Icm=0.052kg.m2')
grid
hold off