Detection of Baseball Bat Alterations Using
Mass and Stiffness Skyline Optimization Procedures
John O'Callahan, PhD, PE
Modal Analysis and Controls Laboratory
University of Massachusetts Lowell
Lowell, MA 01854
ABSTRACT
A procedure is developed to characterize MLB baseball bats
with an analytical model (Finite Element Model) to determine
geometric properties such as mass center, inertia properties,
mass and stiffness matrices and an experimental model to
obtain dynamic properties such as mode shapes and frequencies. With this data, a signature is developed relative to
the bat’s serial number and is kept as the original reference
data. At any time later, a bat could be recalled for reanalysis
on a random basis or if a bat is questioned for alterations.
During the testing, simple geometric tests are made to determine some of the mass properties; but more importantly,
an experimental modal survey is performed to obtain new
shapes and frequencies. With these new eigenparameters
along with the original reference mass and stiffness matrices, a Mass and Stiffness Skyline Optimization (MSSO) is
performed to identify any element changes in mass and/or
stiffness of the reference model. If a bat were “corked”, the
procedure will trend element changes with a minimum number of modes and frequencies in the region of the barrel of
the bat.
NOMENCLATURE
Matrices:
[A] [B]
[C]
[E]
[G ] [H]
[I]
[M] [K ]
[P] [Q] [R ] [S]
[U]
[W ]
[c]
[m] [k ]
[Ω ] [ω ]
2
2
localization matrices
solution coefficient matrix
normalized updated modal matrix
generalized solution matrices
identity matrix
mass and stiffness matrices
partition matrices in the coefficient matrix
normalized analytical modal matrix
solution weighting matrices
element connection matrix
element mass and stiffness matrices
spectral matrix of eigenvalues
Vectors:
{a} {b} packed column vectors for element mass / stiffness
{d}
{v}
{x}
{y}
{α} {β}
right hand side vector of solution equation
partition of RHS vector of solution equation
solution vector of mass and stiffness coefficient
solution vector of mass and stiffness element
changes
percent change for element mass and stiffness
Subscripts:
I
model of updated or improved system
MK
vectors of mass and stiffness variables
S
model of original FE system
i
element index for assembly
[ ]n
single subscript on matrix symbol implies square
matrix
Superscripts:
T
transpose
e M e K number of mass and stiffness elements
i
element index for assembly
m
dimension of the modal space model
n
dimension of full space model
r, s, t, u order indices of coefficient matrices
t M t K number of mass and stiffness coefficients
-1
matrix inverse
†
pseudo inverse
1.
INTRODUCTION
A pitcher, a bat, a ball and a batter generally control the
main variables of the American pastime game of baseball.
The main objective of the pitcher is to prevent the batter of
hitting the ball. Approximately 15 years ago, the league
questioned that the ball was being modified due to the fact
that more home runs were being hit than previously. It was
proven that the ball was not altered but the increase could
be have been due the some of the new ballparks construction affecting the ball aerodynamics. The conclusion was that
the ball had a “rabbit in it” or it was “juiced”. Knowing what
we know today, it was definitely the batter that was “juiced”
using body-altering drugs.
The MLB sets regulations on the bat as well as the ball. All
wood bats are manufactured for individual players and are
numbered for reference and league consistency. The batter
has the choice of selecting a smaller / lighter or longer /
heavier bat within league regulations. Alterations to the bat
such as adding or removing material from the bat are illegal.
Adding mass could increase the impacted ball to go further if
the batter has the power to handle the bat. Removing mass
could increase the batter ability to contact the ball easier. In
both cases, a core diameter is removed from the barrel of
the bat and an alternate heavier or lighter material is added
and a core plug inserted.
A previous study [1] has been used to estimate mass
changes of a baseball bat using the Analytical Model Improvement (AMI) [2]. The study surveyed the system matrices to estimate the region and approximate change in mass
properties. The AMI can be used as the first step and then a
localization technique [3] could estimate amount and location
of element change. The difficulty with this technique is that
system change matrices will produce changes outside the
FE element assembly skyline (further discussed in the next
section).
2.
THEORETICAL BACKGROUND
FE system and then the updated / improved /modified system of equations.
Finite Element Matrices
The FE method is used to develop the constraints needed
for the optimization/localization process in order to allow the
engineer determine system changes or element changes of
an updated / improved / modified system. The constraint
equations come from the definitions of element connectivity
from the local element coordinate space to the finite element
assembly of global model space. The equations governing
mass and stiffness connectivity is given as
[m ] = [c ] [m ] [c ]
(1)
[k ] = [c ] [k ] [c ]
(2)
[m ] and [k ] are the ‘i th’ mass and stiffness ele(i) T
ln
(i)
n
(i)
(i) T
ln
(i)
n
(i )
l
(i )
ln
(i)
l
ln
(i)
(i)
where
l
l
ment matrices expressed in their local element references
coordinates (in ‘l’ space), c ( i ) ln is the ‘i th’ element connec-
[ ]
[ ]
tivity matrix from local to global references and m ( i )
[ ]
n
and
(i )
The procedure proposed to determine if the baseball bat has
been altered, involves three phases of study: one phase of
modeling and two phases of experimental testing. The first
phase requires the newly manufactured bat to be modeled
using finite element methods (see Figure 1 for model representation) producing element / assembled mass and stiffness matrices to define the analytical reference state. With
this model, preliminary and pre-test modal information can
be determined such as inertia properties of center of mass
and mass moment of inertia quantities.
The mass and stiffness matrices are then used to analytically determine eigenparameters of frequencies and mode
shapes. All of the above information can be used to assist in
the initial dynamic testing or pre-test of the actual baseball
bat.
k n are the ‘i th’ mass and stiffness element matrices expressed in their assembled global references coordinates (in
‘n’ space). It is noted that the element connectivity of mass
and stiffness could be different.
The finite assembly of these element matrices is then given
as
[M ] = ∑ [m ]
(3)
[K ] = ∑ [k ]
(4)
eM
S
(i)
n
n
i =1
eK
S
(i)
n
n
i =1
[ ]
where M S
[ ]
and K S
n
n
are the FE assembled mass and
stiffness matrices expressed in ‘n’ dimension space. e M and
The next two phases require an experimental test on the bat
by suspending the bat in an approximate free-free condition.
The second phase is the actual test of the original bat to
determine its signature of frequencies and mode shapes.
This information is kept on file to represent the dynamic state
of the reference system.
e K represent the number of mass and stiffness elements,
respectively used in the assembly process. The superscript
‘S’ denotes that these matrices are the original FE system
matrices to which the optimization / localization constraints
must follow.
The third phase would be a second modal test after there is
a suspicion that the bat has been altered. A re-test of the bat
would generate new eigenparameters of frequency and
mode shapes to be used in the MSSO [3] optimization to
determine any model changes in the bat and to approximately determine “where” and “how much”. The following
sections describe the method of solution.
To enhance the solution of the unknowns of the system matrices or the localization of the element change parameters,
the difference between the original FE model and the modified model is needed and given as
[K ] = [K ] + δ[K]
[M ] = [M ] + δ[M]
I
S
n
I
The development of this approach requires the definitions of
system matrices of mass and stiffness determined from the
FE method since the optimization of the system matrices
and/or the localization of the element change matrices requires the element connectivity to set matrix constraints. We
need to develop system definitions that define the original
n
S
n
2.1 System Definitions
n
n
n
(5)
(6)
where δ[K ]n , δ[M ]n are the system change matrices of stiffness and mass respectively.
Eigensystem Definitions
In order to estimate modified system matrices, the eigenequations of motion must be defined for both the original
FE system as well as the modified / test system. These
equations describe the connection between the mass and
stiffness matrices as they relate to the eigenparameters of
the equation of motion.
The original system FE eigenequation is given as
[K ] [U]
S
n
nm
[ ] [U] [ω ]
= MS
2
n
nm
m
(7)
where [U ]nm is the FE system modal matrix containing ‘m’
columns representing the original modes of the system,
ω 2 m is the corresponding spectral diagonal matrix containing the eigenvalues or square of the natural frequencies.
[ ]
[E ]nm [E ]Tnm [K I ]n − [E ]nm [Ω 2 ]m [E ]Tnm [M I ]n = [0]n
This equation involves the updated stiffness and mass matrices as in a system energy balance equation. The effect of
[E ]nm [E ]Tnm is similar to the PM projector operator and carries
the pertinent information of the updated system. The effecT
tive stiffness / mass relation of [E ]nm Ω 2 m [E ]nm carries the
[ ]
updated frequency information. Equation (9) can then be
written as
[P ] [K ] − [Q ] [M ] = [0]
m
I
n
The unknown modified or test system of equations involving
the mass and stiffness matrices and the known ‘m’ mode
shapes and frequencies, are given as
[K ] [E] = [M ] [E] [Ω ]
(8)
where
[K ] , [M ] are the modified system matrices,
and [E ] , [Ω ] are the measured modal and spectral matriI
I
n
nm
I
n
nm
m
I
n
m
n
I
n
n
n
(10)
where the projector operator for MSSO is given as
[P ] = [E] [E]
T
nm
m
n
nm
(11)
2
n
2
nm
(9)
m
ces. The superscript ‘Ι’ denotes the improved or modified
system.
2.2 Optimization / Localization Methods
The optimization methods require the estimating the system
matrices from FE matrices using the measured modal information of frequencies and mode shapes. If updating process
is not constrained to lie within the skyline of assembly, then
the process is defined as the AMI (Analytical Model Improvement)[2]. The difficulty with this method is that the element changes cannot be determined accurately since information has “leaked out” of the skyline.
To insure that there is “no leakage” from the skyline, constraint equations are applied to the process produces either
the SSO (Stiffness Skyline Optimization) [4-8] or the MSSO
(Mass and Stiffness Skyline Optimization) [3]. The techniques using the predicting of stiffness coefficients changes
inside the skyline of the assembly process is known as the
projector method (PM). From this method, a similar method
is given that includes the mass and stiffness optimizations.
Using this approach along with the system changes identified with the FE definitions of connectivity, the change in
element / model parameters or model localization method is
determined.
Mass and Stiffness Skyline Optimization (MSSO)
and the effective stiffness / mass coefficient matrix is given
as
[Q ]
m
n
[ ]
quantity [E ]nm [E ]nm is still key matrix operator to determine
both the mass and stiffness changes of the system matrices.
T
Therefore, rather than pre-multiplying the transpose of the
eigenequation of motion by M I n [E]nm for the PM method,
[ ]
we will only pre-multiply by [E ]nm producing
m
T
nm
(12)
Unless we are going to track changes in the system mass
and stiffness matrices, we will need additional equations
beyond those in Equation (10) since the equations are not in
standard form to generate a solution. The additional equations that can be used come from the estimation of the modal mass and/or modal stiffness of the updated system as
[E ]Tnm [M I ]n [E]nm = [I]m
(13)
[E ]Tnm [K I ]n [E ]nm = [Ω 2 ]m
(14)
If the mass and stiffness terms of the updated system matrices are to be determined directly as in the PM method, then
only one of the above modal equations can be used since
they will produce identical matrix coefficient contributions to
the solution coefficient matrix.
The above stated equations must be reformed into a standard solution form in order to obtain a robust solution. The
rank of Equations (10, 13, 14) together is equal to ‘3m’ requiring more eigenpairs that could possibly be determined.
Therefore we must re-format the unknown mass and stiffness matrices into vector form using either the upper or
lower triangular portions of the matrices since they are symmetric.
The modified or improved mass matrix requires symmetry as
[M ] = [M ]
I
n
If we are going to determine the mass matrix or mass
changes in the system matrix as well the stiffness matrix as
in the PM method, we must modify the approach. We have
to use a different operator similar to the projector operator
since the mass matrix M I n is now unknown. The matrix
[ ] [E]
= [E ]nm Ω 2
I T
n
(15)
and is repacked into a vector using the upper or lower triangular portion as
[M ] ⇒ {x }
I
M
n
g
(16)
where ‘g’ is the length of the vector given as
g=
n (n + 1)
2
(17)
In a similar fashion, the updated stiffness matrix requirement
of symmetry and vector repacking are given respectively as
[K ] = [K ]
[K ] ⇒ {x }
I T
n
I
n
I
(18)
K
n
(19)
g
Once we have defined the solution unknowns, then the solution coefficient matrices will have to be altered in order to
match the variable reordering. In doing so, the rank of the
solution matrix becomes must more robust.
Applying the above constraints and repacking the equations
into normal form, the energy balance equation and modal
mass and stiffness equations become
[] { }
[] { }
[]
 {0}g
− Q gg  K
  x g   M
R hg   M  =  v h
x
[0]hg   g   v K h
 P gg

[0]hg
S
 hg
[]
{ }
{ }





(20)
where the projectors of Equations (10) given by Equations
(11, 12) are rewritten respectively as
[P ] [K ] ⇒ [P ] {x }
[Q ] [M ] ⇒ [Q ] {x }
m
I
K
n
gg
n
m
I
n
(21)
g
M
n
gg
(22)
g
and modal mass and stiffness matrices given in Equations
(13, 14) are rewritten respectively as
[E] [M ]n [E]nm = [I]m ⇒ [R ]hg {x }g = {v }h
(23)
[E]Tnm [K I ]n [E]nm = [Ω 2 ]m ⇒ [S ]hg {x K }g = {v K }h
(24)
T
nm
I
M
M
with the order of the repacked modal equations ‘h’ given as
m( m + 1)
(25)
h=
2
Defining the total number equations for solution as
s = g + 2h
(26)
and the maximum number of solution variables of stiffness
and mass terms of the updated system matrices as
r = 2g
(27)
(28)
where
[]
 P gg
[C]sr = [0]hg
[]
S
 hg
[]
[]
− Q gg 

R hg 
[0]hg 
{ }
{ }
(29)
= nm −
hg
hg
(32)
m(m − 1) m( m + 1)
+
= m(n + 1)
2
2
This rank can be increased once the system difference
equations defined in Equations (5 and 6) are used producing
an equation rank of
q = q′ +
m(m + 1)
2
2
= m(n + 3 ) + m
2
2
(33)
The coefficient matrix with this rank will be developed later in
the paper when the localization method is used.
In order to obtain a solution, the number of solution unknowns of ‘2g’ will have to be reduced. The way that this is
accomplished is by partitioning out the zero stiffness and
mass terms outside of the skyline or matrix profile of the FE
assembly process. Equation (28) is partitioned such that the
number of unknowns ‘r’ given as
r=t+u
(34)
producing
{x}
[[C]st [C]su ] t  = {d′}s
{
 0}u 
(35)
where ‘t’ is the number of stiffness and mass coefficients
respectively, inside the matrix profile and is given as
t = tK + tM
(36)
and ‘u’ denotes number of zero coefficients of both stiffness
and mass outside the skyline of assembly. The resulting
equation for solution becomes
[C]st {x}t = {d′}s
rank ([C]st ) = min(q, t )
(37)
{ }
{ }
 x
 {0}g 
′
{d }s =  v M h 
 vK 
h

(31)
(38)
The rank of solution will most likely be controlled by ‘t” since
we can supply enough modal information to guarantee that
‘q’ will be greater than ‘t’. The solution vector of unknowns is
then given as
 x KI
(30)
{ }
{ }
([ ] ) + rank([S ] ),2g )
( ([ ] )
= min rank P gg + rank R
{x}t = 
 x K g 

M
 x g 
{x}r = 
q′ = rank ([C]sr )
where the rank of the equations is now given as
then Equation (20) can be written as
[C]sr {x}r = {d′}s
The rank of the solution [C]sr matrix is determined from the
rank of projector matrix and one of the modal matrix coefficient matrices as
MI


tM 

tK
(39)
which only involves the coefficients of the stiffness and mass
matrices under the profile of assembly. The superscript ‘KI’
represents the improved stiffness terms while ‘MI’ denotes
the improved mass matrix terms.
2.3 Localization Methods
The above optimization method is further enhanced by constraining the solution variables in Equation (39) by the con-
nectivity matrices and the change of the FE parameters of
the assembly. To accomplish this, the percent change in the
aggregate element properties are written as constant or proportional factor of the original FE matrices (local coordinates)
and are given as
[ ] [ ]
δ[m ] = α [m ]
δ k ( i ) l = βi k ( i )
(i)
(40)
l
(i)
l
(41)
l
i
where α i , β i are the aggregate element change factors for
stiffness and mass elements respectively.
These change matrices are then expressed in terms of the
system change matrices by using the element connectivity
matrices along with the FE element assembly process producing
eK
[ ] [ ][ ]
δ[K ]n = ∑ c ( i ) ln βi k ( i ) l c ( i )
i =1
eM
T
T
(43)
ln
The upper or lower tridiagonal portions of these matrices are
packed in a similar fashion as in Equations (16 and 19) to
produce a standard matrix form of stiffness and mass respectively as
{ } = [B] {β}
eK
δ[K ]n ⇒ ∑ β i b ( i )
i =1
t KeK
tK
K
− [Q]gt M [A ]t M e M 
[R ]ht M [A]t M e M 
[0]heM 
K K
K
K
 {β}
{y}e = { }e
 α
K
eM
K K
(49)



(50)
 {0}g  [P]gt
{d}s = {v M }h  − [0]ht
{v K }  [S]
h

 ht
− [Q]gt M  KS
x
[R ]ht M   MS
x
[0]t M  
{ }
{ }
K
K
K


tM 

tK
(51)
The solution of Equation (48) depends on the rank of the
coefficient matrix Equation (49). As previously discussed, by
solving for changes in matrix coefficients or percent change
of element factors, the rank of Equation (49) will be sufficient
and will form a proper solution.
Solution of System Coefficients or Element Changes
[ ] [ ][ ]
δ[M ]n = ∑ c ( i ) ln α i m ( i ) l c ( i )
i =1
(42)
ln
[P]gt [B]t e
[C]se =  [0]he
[S]ht [B]t e

(44)
eK
The actual solution of Equation (48) for the element change
solution or an equation very similar for the matrix coefficient
changes, is governed by the solution of the homogeneous
and particular forms given by
{x } = [G ] {d} + [H] {x }
(52)
where {x } is the solution vector of updated stiffness and
mass coefficients, {x } is the equivalent vector of original
I
S
t
ts
tt
s
t
I
t
S
t
{ }
eM
δ[M ]n ⇒ ∑ α i a ( i )
i =1
{ }
where b ( i )
tK
{ }
and a ( i )
tM
tM
= [A ]t M e M {α}e M
(45)
are the packed stiffness and mass
FE stiffness and mass coefficients, [G ]ts is the generalized
inverse of [C]st involved in the particular solution, [H ]tt is the
homogeneous coefficient matrix of [C]st and is known as a
{ }
onto the null space of [C]st .
element coefficient vectors, [B]t K e K and [A ]t M e M are the stiff-
projector matrix of x S
ness and mass coefficient matrices of the b ( i )
It can be shown that the least squares minimum norm solution requires that the generalized inverse must be a pseudoinverse of the [C]st and a unique solution will exist as long as
{a }
(i)
tM
{ }
tK
and
vectors, and {β}e K and {α}e M are the solution vectors
of stiffness and mass element changes respectively.
Using Equations (5 and 6), the system change matrices can
be formed as
 δ[K ]n 
I
S

⇒ x t − x t = δ{x}t
δ[M]n 
{ } { }
{ }
{ }
  [B]t K e K {β}e K 

=
tM 
 [A ]t M e M {α}e M 
tK
(46)
(47)
[C]se {y}e = {d ′}s − [C]st {x }t = {d}s
where
Let the [E ]ar matrix contain “r” experimental “a” space vectors by column that have been scaled consistent with the
analytical modal vectors. These vectors are expanded using
SEREP [9] as
[E ]nr = [T(m )]na [E]ar
(53)
where the SEREP transformation matrix is given as
The generalized solution is formed by using Equations (44
through 47)
S
(1): the modal vectors are consistent and (2): the coefficient
matrix is fully column ranked.
2.4 SEREP Expansion
which are packed using Equations (44 and 45) to produce
 δ x K
δ{x}t =  M
δ x
t
(48)
[T(m )]na = [U]nm [U]gam
(54)
It is assumed that the “r” test modes are represented in the
“m” modes that are used to construct the SEREP [T ]na ma-
trix. The undamped modal matrix [U ]nm is obtained from the
[ ]
[ ]
FE system mass MS n and the stiffness K S n matrices eigensolution given in Equation (7). The resulting expanded
modal vectors [E ]nr are effectively scaled in the same way
as the experimental test vectors and should be consistent
with the FE analytical vectors. This will allow us to perform a
pseudo orthogonality correlation of the two vector sets.
3.
MATHEMATICAL MODEL
As noted above, the first phase of the analysis requires a FE
model of the white birch wood bat. It is assumed that the
material is isotropic even though it is highly orthotropic. The
average isotropic Young’s modulus is taken as 1.8e6 pounds
per square inch and a weight density of 0.02167 pound per
cubic inch. The average geometric properties of length and
diameter of the four main regions of the bat are respectively
given in inches as: nub (1, 1.25), handle (10, 0.75), taper
(10, 1.725) and the barrel (12, 2.7). With the above density
and properties, the model bat weight is 2.1117 pounds. A FE
model of the bat (as shown in Figure 1) is accomplished
using MAT_SAP [11] / MATLAB [12] using 34 nodes forming
68 degrees of freedom (DOFs) and 33 taped planar beam
elements.
would be pretty well defined. It can be shown from the theory
above that the minimum number of modes and frequencies
is six eigenpairs. Knowing this fact and that the system mass
and stiffness properties change discontinuously at the beginning and end of the cork region, it should be expected the
percent change parameters of element mass and stiffness
will vary in an oscillatory manner through the affected area of
change. This fact can be seen in both Figures 2 and 3.
In order to estimate the effective change of element mass
and stiffness parameters, the element α and β factors were
twice averaged from elements 20 to 33 (maximum amplitudes over the barrel of bat) and from 24 through 32 (over
elements within core region). The resulting α mass factor is
computed as -0.091755 (from -0.07393 and -0.10958 average respectively) implying the average change in element
mass is approximately 9.2%. Similarly, the β stiffness factor
is -0.0167355 (from -0.012765 and -0.020706 averages)
projecting the average change in element stiffness is approximately 1.7%.
Fractional Change in Stiffness Distribution
Figure 1 - Finite Element Model of Baseball Bat
Using the above model and Equation (7), the frequencies
and mode shapes of the bat are determined assisting in the
selection of the “a” space DOFs to be used in the experimental test.
Fractional Change in Element Stiffness
0.06
0.04
0.02
0
-0.02
-0.04
-0.06
-0.08
-0.1
-0.12
The second and third phases of the study involve the actual
modal testing of the bat at the “a” DOFs and then expanding
the test shapes to the FE model “n” space using Equation
(53). For the purpose of the analytic study, it is assumed the
eigensolution of the FE model is used as the second phase
results.
-0.14
0
5
10
15
20
25
30
35
Bat Element Number
Figure 2 - Fractional Change in Element Stiffness
Fractional Change in Mass Distribution
0.4
0.3
Fractional Change in Element Mass
To simulate the third phase, a centerline core of one inch in
diameter is assumed drilled in the barrel end of the bat at a
length of 10 inches. Nine inches of the drilled hole is filled
with a lighter cork material of approximately one third the
material density of the wood. One inch of the save core is
placed at the end of the barrel to seal the bat. This model
produced a weight of 2.0095 pounds corresponding a
change of 0.1022 pounds or approximately 4.8 percent. This
model is also used to determine the experimental eigenparameters of an altered bat defined in Equation (8). The
resulting shapes and frequencies are then used with the
analytical model information to optimize the system matrices
and the local element changes.
0.2
0.1
0
-0.1
-0.2
-0.3
-0.4
-0.5
0
4.
RESULTS AND DISCUSSION
It is known that the MSSO procedure requires a generalized
inverse as described in the theory section, which is very
sensitive to the model information as well as the number of
modes used in the analysis. From modal testing of the freefree structure, it is also very difficult to obtain very many
modes. If many modes could be obtained then the solution
5
10
15
20
25
30
35
Bat Element Number
Figure 3 - Fractional Change in Element Mass
Realizing that the change in element stiffness is related to
ratio of element stiffness after and before the updating given
as
4
 πD core



4
 Dcore 
64 
loss of stiffness of core

 (55)

β=
=
=
−

D
original barrel stiffness
E w  πD 4barrel 
 barrel 


L3w  64 
−
Ew
L3w
Then diameter of the core can calculated from the β factor
as
Dcore = D barrel (− β )4
1
(56)
producing an estimate of the core of 0.971 inches or approximately 2.9 percent of model drilled core diameter.
The change in element mass is related to ratio of element
mass after and before the updating given as
α=
=
loss of mass
−ρ πD 2 L + ρ πD 2 L
= w core w 2 core core w
original barrel mass
ρ w πD barrelL w
2
(ρcore − ρw )Dcore
(57)
ρ w D 2barrel
The density ratio of the core to the barrel material can estimated from the α factor as
D

ρcore
= 1 + α barrel 
ρbarrel
 Dcore 
2
(58)
producing of ratio of 0.291 as compared to the filled core
material of 0.333 that of the wood. The percent difference is
approximately a negative 12.8.
5.
CONCLUDING REMARKS
A three-phase procedure is presented where the MSSO optimization method is applied to determine if there have been
any alterations applied to a wood bat since its original construction. The procedure is able to sense stiffness and mass
changes especially in the area of the barrel of the bat due to
a drilling and filling a core with a lighter material. If the bat
was not altered all the changes of mass and/or stiffness
would be approximately zero or in the “noise of the solution”.
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