Modal Analysis of Rectangular Simply-Supported
Functionally Graded Plates
by
Wesley L Saunders
An Engineering Project Submitted to the Graduate
Faculty of Rensselaer Polytechnic Institute
in Partial Fulfillment of the
Requirements for the degree of
MASTER OF ENGINEERING IN MECHANICAL ENGINEERING
Approved:
______________________________________________________
Professor Ernesto Gutierrez-Miravete, Engineering Project Adviser
Rensselaer Polytechnic Institute
Hartford, CT
December, 2011
i
© Copyright 2011
by
Wesley L Saunders
All Rights Reserved
ii
CONTENTS
LIST OF TABLES.............................................................................................................iv
LIST OF FIGURES............................................................................................................v
NOMENCLATURE..........................................................................................................vi
ACKNOWLEDGMENT..................................................................................................vii
ABSTRACT .................................................................................................................viii
1. Introduction…………………………………………………………………………..1
1.1 Background…………………………………………………………………...1
1.1.1
Functionally Graded Materials…………………………………...1
1.1.2
Modal Analysis…………………………………………………...1
1.2 Problem Description………………………………………………………….2
2. Methodology……………………………………………………………………….…4
2.1 Finite Element Modeling……………………………………………………..4
2.2 Modal Analysis……………………………………………………………….4
2.3 Mori-Tanaka Estimation of FGM Material Properties……………………….5
3. Results and Discussions.....……………………………………………………….…..8
3.1 Isotropic (Case A)…………………………………………………………….8
3.2 Linear (Case B)……………………………………………………………...13
3.3 Power Law n=2 (Case C)……………………………………………………15
3.4 Power Law n=10 (Case D)………………………………………………….18
3.5 Comparison to Efraim [2]…………………………………………………...22
4. Conclusions…………………………………………………………………….…....24
5. References…………………………………………………………………………...26
6. Appendices………………………………………………………………………….27
6.1 Appendix A – Mesh Selection………………………………………………27
6.2 Appendix B – Mesh Extension Procedure…………………………………..29
6.3 Appendix C – Mori-Tanaka FEA Inputs……………………………………33
6.4 Appendix D – MATLAB Files……………………………………………...43
iii
LIST OF TABLES
Table 1: Case Descriptions..………………...................................................................... 2
Table 2: Material Properties………………...................................................................... 3
Table 3: Vc Equations for Different FGM.. ...................................................................... 6
Table 4: Isotropic Frequency Results.................................................................................9
Table 5: Linear FGM Frequency Results.........................................................................13
Table 6: Power Law n=2, FGM Frequency Results…………………………………….15
Table 7: Power Law n=10, FGM Frequency Results…………………………………...18
Table 8: FEA and Efraim [2] Comparison………………...……………………………23
iv
LIST OF FIGURES
Figure 1: Simply-Supported Boundary Conditions…....................................................... 4
Figure 2: Graphical Representation of Vc of Different FGM Through Thickness of
Specimen.............................................................................................................................6
Figure 3: Isotropic Steel and Aluminum Mode Shape Results.........................................10
Figure 4: Isotropic Alumina Mode Shape Results............................................................11
Figure 5: Isotropic Zirconia Mode Shape Results............................................................11
Figure 6: Linear FGM Mode Shape Results, Aluminum-Zirconia h=0.025m and
h=0.05m............................................................................................................................14
Figure 7: Power Law n=2, Mode Shape Results, Steel-Alumina, h=0.05m.....................16
Figure 8: Power Law n=2, Mode Shape Results, Steel-Alumina, h=0.025m and
Aluminum Zirconia, h=0.025m and h=0.05m…………………………………..…...….17
Figure 9: Power Law n=10, Mode Shape Results, Steel-Alumina, h=0.05m...................19
Figure 10: Power Law n=10, Mode Shape Results, Aluminum-Zirconia, h=0.05m……19
Figure 11: Power Law n=10, Mode Shape Results, Steel-Alumina and AluminumZirconia, h=0.025m……………………………………………………………………..20
Figure 12: Physics-Controlled Mesh for Isotropic Cases……………………………….27
Figure 13: User-Generated Mesh for FGM Cases………………………………………28
Figure 14: Isotropic Mesh Extension……………………………………………………30
Figure 15: FGM Mesh Extension…………………………………………………....….31
Figure 16: Plot of MT Young’s Modulus for Steel-Alumina, n=2 Power Law,
h=0.05m…………………………………………………………………………………33
Figure 17: Plot of MT Poisson Ratio for Steel-Alumina, n=2 Power Law,
h=0.05m…………………………………………………………………………………34
v
NOMENCLATURE
ρ
Density (kg/m3)
E
Modulus of Elasticity (Pa)
υ
Poisson’s Ratio (dimensionless)
D
Flexural Rigidity (Pa·m3)
h
Thickness (m)
a
Plate Length (m)
b
Plate Width (m)
f
Frequency (Hz)
VC
Volume Fraction, material 1, ceramic (dimensionless)
VM
Volume Fraction, material 2, metal (dimensionless)
VB
Volume Fraction of material 1 at bottom of plate (dimensionless)
VT
Volume Fraction of material 1 at top of plate (dimensionless)
K
Shear Modulus (Pa)
µ
Bulk Modulus (Pa)
n
Power (dimensionless)
z
thickness direction (m)
λ
Lamé’s first parameter (dimensionless)
vi
ACKNOWLEDGMENT
I would like to thank my family for continued support and inspiration during my
education. I would also like to thank the faculty at RPI Hartford, especially Professor
Ernesto Gutierrez-Miravete, for their guidance.
vii
ABSTRACT
The aim of this project is to perform a modal analysis to determine the natural
frequencies and mode shapes of functionally graded materials (FGM) using Finite
Element Analysis (FEA). FGM are defined as an anisotropic material whose physical
properties vary continuously throughout the volume, either randomly or strategically, to
achieve desired characteristics or functionality [1]. A modal analysis will be performed
on isotropic cases with FEA and compared its known theoretical solution. FGM are then
chosen in increasing complexity and a modal analysis is performed on each case. The
FGM are modeled using the Mori-Tanaka (MT) estimation scheme. The analysis
performed is compared to the natural frequencies and mode shapes of the constituent
materials computed in the isotropic cases. The natural frequencies of select FGM are
compared with known solutions in literature, if they exist or compared to analytic
models for computing FGM [2]. In all cases of modal analysis, the FEA are validated by
mesh extension trials and convergence tests. The frequencies of the FGM materials were
found to be bounded by the frequencies of the constituent materials. Also, it was
observed that certain constituent materials have greater effects on either the frequency or
mode shape.
viii
1. Introduction
1.1 Background
1.1.1 Functional Graded Materials
FGM are composite materials consisting of a matrix and a particulate. The matrix phase
is the continuous portion and the particulate phase is the dispersed portion. FGM are
typically designed for a specific function or application. Many times they are
manufactured to achieve good strength to weight ratios, good thermal or electrical
conductivity, or various other material advantages.
FGM differ from traditional
composites in that their material properties vary continuously, where the composite
changes at each laminate interface [1]. FGM accomplish this by gradually changing the
volume fraction of the materials which make up the FGM [1]. As discussed in Vel et al
[1], FGM can be modeled by using the Mori-Tanaka (MT) method of estimation of
material properties. This estimation scheme is further discussed in section 2.2.
1.1.2 Modal Analysis
Many times when using FGM, or any material, in a structural or mechanical application,
it often required to know the acoustic properties of the material. This project aims to
calculate the natural frequencies and accompanying mode shapes of the FGM, which are
the key parameters when considering acoustic performance. To determine these
parameters, modal analysis will be used. Modal analysis is a method of determining the
natural acoustic characteristics, or dynamic response, of materials. The analysis involves
imposing an excitation into the structure and finding the frequencies at which the
structure resonates, (i.e. when the excitation and the vibration response match). A typical
modal analysis will return multiple frequencies, each with an accompanying
displacement field which the structure experiencing at that frequency. Each frequency is
known as a “mode” and the displacement field is known as the “mode shape”.
1
1.2 Problem Description
This project aims to discover the natural frequencies and accompanying mode shapes of
the FGM, which are the key parameters when considering acoustic performance, then
explore and discuss the difference between the cases. This project performs modal
analysis on simply-supported plates and for the following cases:
Table 1: Case Descriptions
Case
Functionality
Materials
Steel
A
Isotropic
Aluminum
Alumina
Zirconia
B
Linear
Aluminum-Zirconia
C
Power Law n=2
Steel-Alumina
Aluminum-Zirconia
D
Power Law n=10
Steel-Alumina
Aluminum-Zirconia
All cases consider 1m x 1m plates, with thicknesses of 0.025m and 0.05m, and simplysupported boundary conditions imposed. For each case, four frequencies and mode
shapes are extracted from the COMSOL Multiphysics software. The frequencies from
Case A are compared to the exact solution5 of an isotropic plate given by Timoshenko
[3]. For Cases B-D, each material is assumed to be 100% and 0% at each face of the
plate. Each case will be compared its constituent isotropic case to determine the
contribution of each material to the frequency and mode shape. Select results of the
FGM are compared to equation 9 of Efraim [2].
Cases A-D were chosen so a variety of material conditions could be investigated. For
instance, the n=10 power law case represents a material that is made up of one material
for the large majority of the material, then has a thin layer of another material on the top.
This case simulates a metal with a thin ceramic coating.
2
For each material above, Table 1 contains there material properties of interest.
Table 2: Material Properties
Material
E [Pa]
ν
ρ [kg/m3]
Steel
1e11
.3
7800
Aluminum
7.5e9
.33
2700
Alumina
3e11
.27
3690
Zirconia
2e11
.3
5700
3
2. Methodology
2.1 Finite Element Modeling
COMSOL software was used in the modeling of the isotropic and FGM plates.
Appendix A has details on the specific mesh selection that went into each model.
Different elements and mesh densities were used for isotropic and FGM plates.
To simulate the simply-supported boundary conditions, specified displacements were
placed on the bottom edges and at one bottom corner. Edges 1-4 were constrained in the
z-direction, and corner 5 was constrained in the x, y, and z-directions.
Figure 1: Simply-Supported Boundary Conditions
2.2 Modal Analysis
The modal analysis is performed by using the eigenfrequency module, in the Solid
Mechanics physics section of COMSOL. For each case, four frequencies and four mode
shapes are computed. The mode shapes are plotted for each frequency over the
geometry.
The exact solution for the isotropic case is taken from Timoshenko [3] and given as:
4
(1)
(1a)
2.3 Mori-Tanaka Estimation of FGM Properties
To capture the effective material properties of the FGM at a point in the thickness, the
Mori-Tanaka (MT) estimation is used. MT is dependent on the volume fractions of each
constituent materials making up the FGM at a certain point.
For a given FGM, the MT estimation, as derived in Reference [4], gives three material
properties of interest in the modal analysis. The first and simplest is the density of the
FGM, which follows the general mixture rule form [1] (Note M-metal, C-ceramic):
(2)
The last two items are the shear and bulk moduli, given by the following:
(3)
(4)
(4a)
Using elasticity, the Poisson ratio and Young’s modulus of the FGM can be determined
by inserting equations (3) thru (4a) into equations (5), (5a), and (6).
(5)
(5a)
5
(6)
For Cases B-D, the VC is a function of thickness of the plate (z). It is assumed that the
VC and the VM combine to unity and each material is at 100% at its respective plate face.
Thus, VT = 1 and VB = 0. Expressions for VC obtained for the three cases investigated
are shown in Table 2 and graphical representations appear in Figure 2.
Table 3: VC Equations for Different FGM
Case
Vc Equation
Linear
(7)
Power Law, n=2
(8)
Power Law, n=10
(9)
Average (All Cases)
(10)
Figure 2: Graphical Representation of VC of Different FGM Through
Thickness of Specimen
6
With these relations of VC known, the Mori-Tanaka estimation can used to predict
material properties in Cases B-D.
7
3. Results and Discussion
3.1 Isotropic (Case A)
The isotropic case is performed for three reasons: 1) To have a reliable check/validation
of the modal analysis procedure carried out in COMSOL, 2) To have known isotropic
results to use in comparison to the results for when given materials are used as
constituents in FGM, and 3) To find a suitable thickness of a FGM plate so that error
between theoretical values and FEA values is not too large (approx. <5%).
Both the 0.025m and the 0.05m isotropic plates showed good agreement with their
respective theoretical values. This validates the FEA approach used in COMSOL for
further use on the FGM. The 0.05m thick plate was found to be the thickest plate without
the error of the isotropic frequencies exceeding approximately 5% of the theoretical
value. This is significant because the FGM should be as thick as possible so that the
functional dependency of the material properties through the thickness can be adequately
captured. Also, the isotropic values are used for comparison to validate the results for the
modal analysis of the FGM, so error should be kept as low as possible.
Lastly, it is noted that frequency is shown to be directly proportional to thickness. By
comparing the h=0.05m and h=0.025m columns, the frequency doubles from one case to
another. Inspection of equations (1) and (1a) show this relationship to be true. Another
relationship of note is that faluminum/fsteel ≈ 1.5 and falumina/fzirconia ≈ 1.5.
8
Table 4: Isotropic Frequency Results
h=0.025m
Material
Mode
Frequency Frequency
[Hz]
1 (1,1)
85.10
84.55
2a (1,2)
212.75
2b (2,1)
Percent
[Hz]
[Hz]
0.65
170.20
166.14
2.39
211.79
0.45
425.50
413.73
2.77
212.75
211.84
0.43
425.50
413.95
2.71
3 (2,2)
340.40
338.07
0.68
680.80
651.36
4.32
1 (1,1)
126.59
125.82
0.60
80.06
77.67
2.99
Aluminum 2a (1,2)
316.46
315.1
0.43
200.15
192.90
3.62
2b (2,1)
316.46
315.18
0.41
200.15
192.98
3.62
3 (2,2)
506.34
503.04
0.65
320.14
301.98
5.70
1 (1,1)
212.32
210.88
0.68
424.6
412.41
2.87
2a (1,2)
530.79
528.32
0.47
1061.6
1027.44
3.22
2b (2,1)
530.79
528.44
0.44
1061.6
1027.66
3.20
3 (2,2)
849.26
843.23
0.71
1698.5
1612.9
5.04
1 (1,1)
140.78
139.88
0.64
281.60
273.68
2.81
2a (1,2)
351.96
350.37
0.45
703.90
681.45
3.19
2b (2,1)
351.96
350.46
0.43
703.90
681.56
3.17
3 (2,2)
563.14
559.28
0.69
1126.30
1069.87
5.01
Zirconia
[Hz]
Frequency Frequency
(FEA)
Alumina
(FEA)
Percent
Ref [3]
Steel
Ref [3]
h=0.05m
(m,n)
9
Error
Error
Figure 3: Isotropic Steel and Aluminum Mode Shape Results
The mode shapes shown in Figure 3 are obtained from the FEA modal analysis for
isotropic plates of steel and aluminum. For each, the mode shape is the same, but the
scale of deformation varies for each material. For the purpose of this project, only the
shape shall be considered. The results match the typical simply-supported rectangular
isotropic plate shown in Ferreira, et al [5]. This result acts as a baseline for comparison
when FGM mode shapes are computed.
10
Figure 4: Isotropic Alumina Mode Shape Results
Figure 5: Isotropic Zirconia Mode Shape Results
The two ceramic materials shown in Figured 4 and 5 have modes 1 and 3 which match
nicely with the isotropic metals (steel and aluminum) as well as the results in Ferreira, et
11
al [5]. However, in both ceramics the axis of symmetry for both modes 2a and 2b moved
from its typical location on the diagonals. For alumina mode 2a, it has shifted slightly
downward, and for mode 2b the axis of symmetry is the horizontal neutral axis of the
plate. For zirconia, mode 3 expierences some shifting of its diagonal axis of symmetry
away and mode 2a has approximately the vertical neutral axis as its line of symmetry.
Again these mode shapes are used for comparison to the FGM cases.
12
3.2 Linear (Case B)
The comparison of the FGM frequency solutions in Table 5 to the isotropic solutions
found in section 3.1, shows that the general dependency on thickness found in the
isotropic case still exist. Comparing each case to its individual isotropic, the h=0.05m
case is bounded by the isotropic frequencies of aluminum and zirconia. For example,
taking values from Table 4, 170.20 Hz is the upper bound and 80.06 Hz is the lower
bound, while the first mode frequency for the FGM is 116.01 Hz. However, the
h=0.025m case did not have the same relation to its isotropic frequency, in fact it was
well below the lower bound. The ability to accurately calculate the FGM on a the thinner
0.025m plate is low because there is limited amount of elements that are able to be
modeled through the thickness of the plate (See Appendix B). The fewer elements, the
less information is able to be determined between the two faces of the plate.
Table 5: Linear FGM Frequency Results
h=0.025m
h=0.05m
Frequency
Frequency
(FEA)
(FEA)
[Hz]
[Hz]
1 (1,1)
59.67
116.01
Bottom Material: Aluminum
2a (1,2)
150.29
287.45
Top Material: Zirconia
2b (2,1)
150.31
287.48
3 (2,2)
238.52
447.64
FGM
Mode
(m,n)
13
Figure 6: Linear FGM Mode Shape Results, Aluminum-Zirconia, h=0.025m
and h=0.05m
Figure 6 shows the mode shapes for the Aluminum-Zirconia linear FGM plate. The
general mode shape pattern did not change for either 0.025m or 0.05m plate, only the
scale of deflection. The mode shapes correspond very well to the isotropic case of
aluminum in Figure 3, with one exception. The mode shapes for modes 2a and 2b have
swapped from where they were in the isotropic cases. In Efraim [2] this phenomena of
“interchanging vibrational modes” is attributed to variation in the Poisson ratio of the
constituent materials, which in turns affects the FGM.
14
3.3 Power Law n=2 (Case C)
Reviewing the results for the first n=2 power law FGM, all the frequencies are bounded
by their isotropic constituents. The relationship between frequency and plate thickness is
consistent with what was seen in the linear and isotropic cases. Also, since the n=2
power law dictates that on average nearly 67% (see equation 10) of the plate is the
bottom (metal) material, the FGM frequency should be closer to the bottom isotropic
material. This is indeed the case, as all of the n=2 frequencies are close their metal
constituent’s frequency, and are slightly raised due to the presence of the ceramic
material.
The ratio of the two FGM is fS-A/fA-Z ≈ 2.1. This result shows that a Steel-Alumina FGM,
with its steel constituent having a 1:1.5 frequency ratio with aluminum and its alumina
constituent having a 1.5:1 frequency ratio to zirconia, can be combined to have a 2:1
frequency ratio to the other constituent materials, Aluminum-Zirconia.
Table 6: Power Law n=2, FGM Frequency Results
h=0.025m
h=0.05m
Frequency
Frequency
(FEA)
(FEA)
[Hz]
[Hz]
1 (1,1)
113.71
222.23
Bottom Material: Steel
2a (2,1)
284.47
551.71
Top Material: Alumina
2b (1,2)
284.50
551.71
3 (2,2)
451.85
861.17
1 (1,1)
54.88
107.11
Bottom Material: Aluminum
2a (2,1)
137.2
265.16
Top Material: Zirconia
2b (1,2)
137.21
265.2
3 (2,2)
217.72
412.47
FGM
Mode
(m,n)
15
Figure 7: Power Law n=2, Mode Shape Results, Steel-Alumina, h=0.05m
It is worth noting that even though the frequency favors the metal material for power
law, n=2; the mode shape is greatly influenced by the prescence of ceramic in the
material. Modes 2a and 2b almost resemble indentically what modes 2a and 2b of the
istropic zirconia in Figure 5 look like. This interesting because there is no zirconia
present in this FGM. The istropic steel mode 2a, once symetrical on the diagonal has
now become symetric on the vertical neutral axis line of the plate. Mode 2b has
experienced the shifting of the symmetry line away from the corners, as has been shown
for all ceramic plates, similar to the alumina isotropic plate in Figure 4.
16
Figure 8: Power Law n=2, Mode Shape Results, Steel Alumina, h=0.025m, and
Aluminum Zirconia, h=0.025m and h=0.05m
Figure 8 shows each of the three remaining cases computed for the power law, n=2 case.
Each case (h=0.025m Aluminum-Zirconia, h=0.05m Aluminum-Zirconia, and 0.025
Steel-Aluminum), are similar to the linear FGM in Figure 6 and had the same changes
from the isotropic mode shapes.
17
3.4 Power Law n=10 (Case D)
The frequencies for the power law n=10 case were as expected in all cases. Both
h=0.05m cases and the h=0.025m Steel-Alumina case have frequencies just above what
would be an all metal plate (Equation 10 shows VC=9.1%). Since the ceramics inherently
have higher natural frequencies, it was expected that the FGM frequencies would be
altered slightly due to the presence of the ceramic. The standard checks of the
frequencies performed for other cases still hold for the power law n=10 case. The
frequency-thickness relationship holds and each frequency is bounded by its isotropic
constituents.
The frequency ratio of this case drops slightly to fS-A/fA-Z ≈ 1.95. This result again
shows that a Steel-Alumina FGM, with its steel constituent having a 1:1.5 frequency
ratio with aluminum and its alumina constituent having a 1.5:1 frequency ratio to
zirconia, can be combined to have a 2:1 frequency ratio to the other constituent
materials, Aluminum-Zirconia. This result is apparent even when the FGM is all metal
with a thin ceramic coating.
Table 7: Power Law n=10, FGM Frequency Results
h=0.025m
h=0.05m
Frequency
Frequency
(FEA)
(FEA)
[Hz]
[Hz]
1 (1,1)
96.44
188.41
Bottom Material: Steel
2a (2,1)
241.25
467.67
Top Material: Alumina
2b (1,2)
241.27
467.67
3 (2,2)
383.07
729.74
1 (1,1)
49.51
96.34
Bottom Material: Aluminum
2a (2,1)
123.76
238.65
Top Material: Zirconia
2b (1,2)
123.76
238.65
3 (2,2)
196.17
370.62
FGM
Mode
(m,n)
18
Figure 9: Power Law n=10, Mode Shape Results, Steel-Alumina, h=0.05m
Figure 10: Power Law n=10, Mode Shape Results, Aluminum-Zirconia, h=0.05m
19
The case showed in Figure 9 follows closely on what was discussed for the same two
materials in the n=2 power law in Figure 7. The presence of some ceramic, even though
it is n=10 power and can be considered a thin coating on the top of the plate, has a
significant effect on the mode shape. Mode 2a has been converted almost entirely into an
alumina isotropic mode 2b, further illustrating the dominate presence the ceramic has on
the mode shape and another example of the “interchanging vibrational modes” pointed
out by Efraim [2]. Figure 10 likewise shows the ceramic coating having a greater effect
on the mode shape and starting to resemble the isotropic mode shape of the zirconia
plate.
Figure 11: Power Law n=10, Mode Shape Results, Steel-Alumina and AluminumZirconia, h=0.025m
This is another case where the thinner plate did not behave the same as the thicker plate.
There are two possible explanations for this, first being one that has been discussed that
exploring thinner plates was difficult because of the how many elements could be
distributed through the thickness of the plate. There does not seem to be enough to
capture the effect of the ceramic on the metal. The second explanation is that even if
20
there were enough elements, the ceramic thickness is dependent on the overall thickness
of the plate, so it may be extraordinarily difficult to get a accurate view of what is
happening on this plate.
21
3.5 Comparison to Efraim [2]
Efraim [2] introduced a method where a frequency could be determined for a FGM if the
natural frequencies of isotropic case for the constituents in a corresponding isotropic
case, (i.e. same boundary conditions, size, and thickness) were known. This method was
very relevant to the project at hand. The main result is given by equation (11) (Equation
[9] in [2]).
(11)
where:
(11a)
(11b)
(11c)
(11d)
[Note: Some subscripts have been changed to match nomenclature of this paper. Also,
11c and 11d were dimensionless in Efraim [2], however to match COMSOL results they
were made to have dimensions]
Using MATLAB, select cases of the Mori-Tanaka based FEA results were compared to
the Efraim [2] formula results.
The results are not as close as one would like (typically around <5%). The Efraim [2]
formula consistently produced results that were between 6-11% lower that what was
produced with COMSOL. The reason for this discrepancy could be contributed to a few
factors. First, the Efraim values were calculated using the isotropic frequencies from
Timoshenko [3] and equation (1). There was already said to be a measure of error
between the FEA and the Timoshenko [3] values in section 3.1, Table 4. More likely
22
though is the comparison between two fundamentally different approaches of dissecting
FGM material properties. The only facet that MT (used in the FEA) and Efraim have in
common is that they both consider density to be a rule of mixtures, which is logical
because density is an intrinsic property of a material. The remaining two key material
properties are handled very differently. First, the Poisson ratio is neglected in the Efraim
discussion. For Mori-Tanaka, as seen in Vel et al [1], the Poisson ratio is an extremely
detailed and probabilistic approach, using “random distribution of isotropic particles in
an isotropic matrix” [1] to determine the physical properties of the material. The same
probabilistic method developed by Mori-Tanaka [4] and implemented by Vel et al [1] is
used to determine the value of Young’s modulus. Efraim uses two “dimensionless
frequency parameters” which are singularly dependent on one of the materials Young’s
modulus and a rule of mixtures Young’s modulus (see equation 11a).
Table 8: FEA and Efraim [2] Comparison
h=0.05m
FGM
Mode
(m,n)
Frequency
Efraim [2]
(FEA)
Frequency
[Hz]
[Hz]
Percent
Error
1 (1,1)
188.41
171.17
9.2%
Bottom Material: Steel
2a (2,1)
467.67
427.93
8.5%
Top Material: Alumina
2b (1,2)
467.67
427.93
8.5%
Power Law n=10
3 (2,2)
729.74
684.68
6.2%
1 (1,1)
107.11
94.77
11.5%
Bottom Material: Aluminum
2a (2,1)
265.16
236.93
10.6%
Top Material: Zirconia
2b (1,2)
265.2
236.93
10.7%
Power Law n=2
3 (2,2)
412.47
379.09
8.1%
23
4. Conclusions
This project explored the modes of vibration of simply-supported FGM plates. The end
results were found to match the expectations at the onset of the project. Much of the
work in Case A and Appendix C was to set a level of confidence in the values obtained
going forward. Case A solidified that the plate modeling and the COMSOL FEA
software would give acceptable answers going forward. With confidence in the models
and software, all that would need to be done is to insert the correct physical parameters
for the applicable FGM. Appendix C, specifically the graphing portion, confirmed that
the correct physical parameters were being entered into the COMSOL FEA software.
With those two checks in place, a great level of confidence was achieved in the results of
the project. Any lingering doubts or discrepancies can be attributed inherent error in the
calculations, technological limitations by either the software or operating system, or a
difference in approach as discussed in section 3.5.
The FGM were arbitrarily selected for study, but useful conclusion can be drawn from
the results. As shown and discussed in section 3.3, Table 5 and section 3.4, Table 6 the
Steel Alumina frequencies are approximately double what Aluminum-Zirconia are.
Structurally, this would result is Steel-Alumina members being more rigid and
resonating at double the frequency as Aluminum-Zirconia members.
The FEA results were found to be within 6-11% of the computed values from Efraim
[2]. While the discrepancy between the FEA and Efraim [2] is a bit high, the Efraim [2]
formula is a direct method of calculating FGM frequencies and can be used as a starting
point reference or sanity check on more complex FGM FEA models.
The takeaways from this project are as follows:

When considering a FGM constructed of a metal and a ceramic, the frequency
seems to follow the metal while the mode shape seems to the follow the ceramic

A FGM should be thick enough so that enough data is able to be extracted from
the cross-section of the plate. Many times in this project, the 0.025m plate was
24
simply too thin to fully capture was occur as the frequencies and modes were
calculated through the plate’s thickness.

The Mori-Tanaka estimate is heavy computationally, as demonstrated in
Appendix C. For future use, the use of µ and K should stand to simplify the
calculations.
25
5. References
[1] Senthil S. Vel, R.C. Batra, Three-dimensional exact solution for the vibration of
functionally graded rectangular plates, Journal of Sound and Vibration 272 (2004), pages
703-730
[2] Elia Efraim, Accurate formula for determination of natural frequencies of FGM
plates basing on frequencies of isotropic plates, Engineering Procedia 10 (2011), pages
242-247
[3] S. Timoshenko, D.H. Young, W. Weaver Jr., Vibration Problems in Engineering.
Forth Edition. John Wiley & Sons, 1974, pages 481-502
[4] T. Mori, K. Tanaka, Average stress in matrix and average elastic energy of materials
with misfitting inclusions, Acta Metallurgica 21 (1973), pages 571-574
[5]A.J.M. Ferreira, C.M.C. Roque, E. Carrera, M. Cinefra, Analysis of thick isotropic
and cross-ply laminated plates by radial basis functions and a Unified formulation,
Journal of Sound and Vibration 330 (2011), pages 771-787
26
6. Appendices
6.1 Appendix A – Mesh Selection
Isotropic Cases
The isotropic cases used the default, or “physics-controlled” mesh given by COMSOL,
which are the 3-D tetrahedral elements. The mesh was sized in accordance with
Appendix B. The final mesh size settled on for the isotropic case was “Normal”. The
general mesh depiction is shown below:
Figure 12: Physics-Controlled Mesh for Isotropic Cases
27
FGM Cases
All other cases, for the linear, n=2 and n=10 power law, a different mesh was required to
adequately capture the changing material properties through the thickness of the plate.
The mesh selected was 3-D quad elements using the “Mapped” function, which were
distributed on the face of the plate then swept downward through the thickness of the
plate using the “Swept” function. Again, the mesh size was determined in accordance
with Appendix B for each case. A depiction of the mesh is shown below:
Figure 13: User-Generated Mesh for FGM Cases
28
6.2 Appendix B – Mesh Extension Procedure
Isotropic Cases
For each isotropic case, using the physics-controlled mesh as introduced in Appendix A,
a mesh extension was run on each model. The physic-controlled mesh function has mesh
sizes ranging from “extremely coarse” to “extremely fine”, with default mesh size
parameters built into to each setting. The extension is accomplished by running the
model at each setting until either the model has suitable convergence, or a setting creates
too many elements and the computer cannot process it.
Since the models for the isotropic cases were all the same, with the only the material
properties changing, the mesh extensions all had identical results. Thus, all the isotropic
cases ended up being assigned the same mesh size. Figure 14 shows the extension done
for the isotropic aluminum plate, with thickness of 0.05m, considering first mode
frequencies only.
29
Extremely Coarse, f=107.05 Hz
Coarse, f=84.88 Hz
Normal, f=84.55 Hz
Extra Fine, f=84.22 Hz
Figure 14: Isotropic Mesh Extension
The change from “Normal” to “Extra Fine” was only 0.4% in frequency value but it
required a significant amount of additional elements. Thus, the “Normal” setting was
used.
30
FGM Cases
The FGM cases were more difficult because there are two mesh parameters to vary.
However, if both parameters are varied simultaneously a result can be found. Below is
the procedure, where m is the mapped size in meters and s is the swept size in meters.
The extension is shown for the linear 0.05m thick steel aluminum plate.
m=0.5 s=0.05, f=124.34 Hz
m=0.1 s=0.025, f=116.62 Hz
m=0.07 s=0.01, f=116.26 Hz
m=0.07 s=0.009, f=116.26 Hz
Figure 15: FGM Mesh Extension
31
As shown, the transition from m=0.07 s=0.01 to m=0.07 s=0.009 gives no appreciable
change to the frequency. The m=0.07 s=0.01 setting is right at the limit for what
computer can handle without running out of memory and crashing. For example, a run
with m=0.06 s=0.009 will crash. Thus, m=0.07 and s=0.01 setting was used. As was the
case with the isotropic cases, the linear, n=2 power law, and n=10 power law did not
change other than the material property inputs, so the mesh was the same for all cases for
thickness of 0.05m. For thickness of 0.025m, an analogous procedure was done and it
was found that m=0.125 s=0.008 was the limit for those cases.
32
6.3 Appendix C – Mori-Tanaka FEA Inputs
For each FGM case, a MT estimation of density, Young’s Modulus, and Poisson ratio
was completed. The equations for Young’s Modulus and Poisson ratio were checked for
accuracy by graphing them in Maple. (As shown in equation (2), the density equation is
not as involved and can be verified by inspection). The assumption that at each face of
the plate, the material was 100% of one and 0% of the other must hold true in the
computation of the MT estimates. Each graph (or equation) verified that assumption. A
typical graph would look similar to the ones provided for Steel-Alumina plate, with
thickness 0.05m, and power law n=2.
Figure 16: Plot of MT Young’s Modulus for Steel-Alumina, n=2 Power Law,
h=0.05m
33
Figure 17: Plot of MT Poisson Ratio for Steel-Alumina, n=2 Power Law, h=0.05m
With each equation verified, the following inputs from Maple, in “string” format, can be
directly entered into the COMSOL Global Definitions module and the FGM can be
constructed.
h=0.025, Linear, Aluminum-Zirconia
ρ
4200+120000.0000*z
E
-2.880000000*(973375111.0+.4148473450e11*z+.1885180924e13*z^2)*(.3492951908e19+.53897180
73e20*z)/(-288525779.+.1498103751e11*z+.3403896768e12*z^2)/(1633665009.0+.1146932007e12*z)
υ
.1000000000e-1*(-.4293019142e27+.2500717343e29*z+.2854199148e30*z^2)/(.1442628895e26+.7490518756e27*z+.1701948384e29*z^2)
34
h= 0.05, Linear, Aluminum-Zirconia
ρ
4200+60000.00000*z
E
-.9600000000*(2920125333.0+.6222710170e11*z+.1413885693e13*z^2)*(.3492951908e19+.2694859
036e20*z)/(-288525779.+7490518756.0*z+.8509741918e11*z^2)/(1633665009.+.5734660036e11*z)
υ
.1000000000e-1*(-.4293019142e27+.1250358672e29*z+.7135497869e29*z^2)/(.1442628895e26+.3745259378e27*z+.4254870959e28*z^2)
h=0.025, Power Law n=2, Aluminum-Zirconia
ρ
2700+3000*((1/2)+40.00000000*z)^2
E
-48.0*(695408893.0+.1075198378e11*z+.1561187906e13*z^2+.9048868431e14*z^3+.180977
3685e16*z^4)*(.6312189057e19+.1077943615e21*z+.4311774458e22*z^2)/(3688607917.0+.1072616656e12*z+.7694363390e13*z^2+.2723117414e15*z^3+.54462
34825e16*z^4)/(-4700995027.+.2293864014e12*z+.9175456058e13*z^2)
υ
.1000000000e-1*(.2297790131e28+.8575769800e29*z+.4571987579e31*z^2+.9133437273e32*z^3+.182
6687455e34*z^4)/(.7377215834e26+.2145233311e28*z+.1538872678e30*z^2+.5446234827e31*z^3+.108
9246965e33*z^4)
35
h=0.05, Power Law n=2, Aluminum-Zirconia
ρ
2700+3000*((1/2)+20.00000000*z)^2
E
-48.0*(695408893.0+5375991875.0*z+.3902969760e12*z^2+.1131108554e14*z^3+.11311085
54e15*z^4)*(.6312189057e19+.5389718073e20*z+.1077943615e22*z^2)/(3688607917.+.5363083275e11*z+.1923590847e13*z^2+.3403896767e14*z^3+.340389
6767e15*z^4)/(-4700995027.+.1146932007e12*z+.2293864014e13*z^2)
υ
.1000000000e-1*(.2297790131e28+.4287884900e29*z+.1142996895e31*z^2+.1141679659e32*z^3+.114
1679659e33*z^4)/(.7377215834e26+.1072616655e28*z+.3847181694e29*z^2+.6807793534e30*z^3+.680
7793534e31*z^4)
h=0.025, Power Law n=10, Aluminum-Zirconia
ρ
2700+3000*((1/2)+40.00000000*z)^10
E
-4.800000000*(-852900672.0*z-.3013886990e12*z^2-.6143078050e14*z^3.7568738280e16*z^4.4460115620e18*z^5+.3012452220e20*z^6+.1163859987e23*z^7+.1785765035e25*z^
8+.1977548586e27*z^9+.1751879962e29*z^109417613335.+.1274940860e31*z^11+.7649645163e32*z^12+.3765979158e34*z^13+.1
506391663e36*z^14+.4820453323e37*z^15+.1205113330e39*z^16+.2268448622e40*
z^17+.3024598162e41*z^18+.2547030031e42*z^19+.1018812012e43*z^20)*(.144412
3135e22+.5389718073e21*z+.1940298506e24*z^2+.4139303480e26*z^3+.579502487
2e28*z^4+.5563223877e30*z^5+.3708815918e32*z^6+.1695458705e34*z^7+.5086376
36
116e35*z^8+.9042446429e36*z^9+.7233957143e37*z^10)/(1660906543.*z+.59962830
40e12*z^2+.1287830254e15*z^3+.1834005894e17*z^4+.1845084083e19*z^5+.141019
1361e21*z^6+.9534622900e22*z^7+.7183634218e24*z^8+.6272843019e26*z^9+.5297
749833e28*z^10+.3836737913e30*z^11+.2302042748e32*z^12+.1133313353e34*z^1
3+.4533253411e35*z^14+.1450641092e37*z^15+.3626602729e38*z^16+.6826546313
e39*z^17+.9102061751e40*z^18+.7664894106e41*z^19+.3065957642e42*z^205537066745.)/(.1569039304e13+.1146932007e13*z+.4128955226e15*z^2+.8808437815e17*z^3+.123
3181294e20*z^4+.1183854042e22*z^5+.7892360283e23*z^6+.3607936129e25*z^7+.1
082380839e27*z^8+.1924232602e28*z^9+.1539386082e29*z^10)
υ
.1000000000e1*(.9157433387e32*z+.3299530219e35*z^2+.7053459075e37*z^3+.9926903298e39*z
^4+.9671431977e41*z^5+.6749711583e43*z^6+.3603451463e45*z^7+.1806052074e4
7*z^8+.1158872924e49*z^9+.8969949541e50*z^10+.6434280961e52*z^11+.3860568
577e54*z^12+.1900587607e56*z^13+.7602350428e57*z^14+.2432752137e59*z^15+.6
081880343e60*z^16+.1144824535e62*z^17+.1526432713e63*z^18+.1285417022e64*
z^19+.5141668087e64*z^20.1826772039e33)/(.1660906543e31*z+.5996283040e33*z^2+.1287830254e36*z^3+.18
34005894e38*z^4+.1845084083e40*z^5+.1410191361e42*z^6+.9534622900e43*z^7+.
7183634218e45*z^8+.6272843019e47*z^9+.5297749833e49*z^10+.3836737913e51*z
^11+.2302042748e53*z^12+.1133313353e55*z^13+.4533253411e56*z^14+.14506410
92e58*z^15+.3626602729e59*z^16+.6826546313e60*z^17+.9102061751e61*z^18+.76
64894106e62*z^19+.3065957642e63*z^20-.5537066745e31)
h=0.05, Power Law n=10, Aluminum-Zirconia
ρ
2700+3000*((1/2)+20.00000000*z)^10
E
-24.0*(-852900671.0*z-.1506943492e12*z^2-.1535769518e14*z^3.9460922920e15*z^4-
37
.2787572280e17*z^5+.9413913200e18*z^6+.1818531231e21*z^7+.1395128934e23*z^
8+.7724799160e24*z^9+.3421640551e26*z^10+.1245059434e28*z^11+.3735178302e
29*z^12+.9194285048e30*z^13+.1838857010e32*z^14+.2942171216e33*z^15+.3677
714020e34*z^16+.3461377902e35*z^17+.2307585267e36*z^18+.9716148492e36*z^1
9+.1943229699e37*z^20.1883522667e11)*(.1444123135e22+.2694859036e21*z+.4850746265e23*z^2+.51741
29350e25*z^3+.3621890545e27*z^4+.1738507462e29*z^5+.5795024872e30*z^6+.132
4577114e32*z^7+.1986865670e33*z^8+.1766102818e34*z^9+.7064411272e34*z^10)/
(8304532717.0*z+.1499070760e13*z^2+.1609787817e15*z^3+.1146253683e17*z^4+.
5765887758e18*z^5+.2203424002e20*z^6+.7448924141e21*z^7+.2806107116e23*z^
8+.1225164652e25*z^9+.5173583821e26*z^10+.1873407184e28*z^11+.5620221552e
29*z^12+.1383439151e31*z^13+.2766878303e32*z^14+.4427005284e33*z^15+.5533
756605e34*z^16+.5208241511e35*z^17+.3472161007e36*z^18+.1461962529e37*z^1
9+.2923925059e37*z^20-.5537066745e11)/(.1569039304e13+.5734660036e12*z+.1032238806e15*z^2+.1101054727e17*z^3+.770
7383088e18*z^4+.3699543882e20*z^5+.1233181294e22*z^6+.2818700101e23*z^7+.4
228050151e24*z^8+.3758266801e25*z^9+.1503306720e26*z^10)
υ
.1000000000e1*(.4578716694e32*z+.8248825546e34*z^2+.8816823844e36*z^3+.6204314561e38*z
^4+.3022322493e40*z^5+.1054642435e42*z^6+.2815196455e43*z^7+.7054890912e4
4*z^8+.2263423680e46*z^9+.8759716349e47*z^10+.3141738751e49*z^11+.9425216
252e50*z^12+.2320053231e52*z^13+.4640106463e53*z^14+.7424170340e54*z^15+.9
280212925e55*z^16+.8734318047e56*z^17+.5822878698e57*z^18+.2451738399e58*
z^19+.4903476798e58*z^20.1826772039e33)/(.8304532717e30*z+.1499070760e33*z^2+.1609787817e35*z^3+.11
46253683e37*z^4+.5765887758e38*z^5+.2203424002e40*z^6+.7448924141e41*z^7+.
2806107116e43*z^8+.1225164652e45*z^9+.5173583821e46*z^10+.1873407184e48*z
^11+.5620221552e49*z^12+.1383439151e51*z^13+.2766878303e52*z^14+.44270052
84e53*z^15+.5533756605e54*z^16+.5208241511e55*z^17+.3472161007e56*z^18+.14
61962529e57*z^19+.2923925059e57*z^20-.5537066745e31)
38
h=0.025, Power Law n=2, Steel-Alumina
ρ
7800-4110*((1/2)+40.00000000*z)^2
E
-30.0*(2426636917.0+8690198960.0*z+.1411236495e13*z^2+.8509028296e14*z^3+.1701805
659e16*z^4)*(.7163561075e21+.8171152517e22*z+.3268461007e24*z^2)/(5793557129.+.5906561246e11*z+.4006414055e13*z^2+.1315031645e15*z^3+.263006
3290e16*z^4)/(-.6987577639e11+.1593374741e13*z+.6373498965e14*z^2)
υ
.1000000000*(.1683460106e32+.2518770675e33*z+.1297588780e35*z^2+.2320644077e36*z^3+.464
1288155e37*z^4)/(.5793557129e31+.5906561246e32*z+.4006414055e34*z^2+.1315031645e36*z^3+.263
0063290e37*z^4)
h=0.05, Power Law n=2, Steel-Alumina
ρ
7800-4110*(1/2+20.00000000*z)^2
E
-30.0*(2426636917.0+4345099490.*z+.3528091242e12*z^2+.1063628537e14*z^3+.10636285
37e15*z^4)*(.7163561075e21+.4085576258e22*z+.8171152517e23*z^2)/(5793557129.+.2953280623e11*z+.1001603514e13*z^2+.1643789556e14*z^3+.164378
9556e15*z^4)/(-.6987577639e11+.7966873706e12*z+.1593374741e14*z^2)
υ
.1000000000*(.1683460106e32+.1259385337e33*z+.3243971949e34*z^2+.2900805097e35*z^3+.290
0805097e36*z^4)/(-
39
.5793557129e31+.2953280623e32*z+.1001603514e34*z^2+.1643789556e35*z^3+.164
3789556e36*z^4)
h=0.025, Power Law n=10, Steel-Alumina
ρ
7800-4110*((1/2)+40.00000000*z)^10
E
-12.0*(-145703128.0*z-.5178835835e11*z^2-.1071136740e14*z^3.1378337782e16*z^4.9933943325e17*z^5+.4133164250e18*z^6+.1225056518e22*z^7+.2056143831e24*z^
8+.2316838850e26*z^9+.2058597188e28*z^103996900505.0+.1498600065e30*z^11+.8991600391e31*z^12+.4426634039e33*z^13+.
1770653616e35*z^14+.5666091568e36*z^15+.1416522892e38*z^16+.2666396034e39
*z^17+.3555194711e40*z^18+.2993848176e41*z^19+.1197539271e42*z^20)*(.17036
43892e24+.4085576258e23*z+.1470807453e26*z^2+.3137722566e28*z^3+.43928115
93e30*z^4+.4217099129e32*z^5+.2811399419e34*z^6+.1285211163e36*z^7+.385563
3490e37*z^8+.6854459537e38*z^9+.5483567630e39*z^10)/(494060879.*z+.17827286
38e12*z^2+.3823975762e14*z^3+.5428522871e16*z^4+.5415264463e18*z^5+.404512
5655e20*z^6+.2594827751e22*z^7+.1822326757e24*z^8+.1530228723e26*z^9+.1280
427438e28*z^10+.9264073125e29*z^11+.5558443875e31*z^12+.2736464677e33*z^1
3+.1094585871e35*z^14+.3502674786e36*z^15+.8756686966e37*z^16+.1648317547
e39*z^17+.2197756729e40*z^18+.1850742508e41*z^19+.7402970034e41*z^203996100402.)/(.2042763975e14+.7966873706e13*z+.2868074534e16*z^2+.6118559007e18*z^3+.856
5982609e20*z^4+.8223343305e22*z^5+.5482228870e24*z^6+.2506161769e26*z^7+.7
518485307e27*z^8+.1336619610e29*z^9+.1069295688e30*z^10)
υ
.1000000000*(.2761710651e36*z+.9949410357e38*z^2+.2126215229e41*z^3+.29899
28992e43*z^4+.2906311098e45*z^5+.2014296499e47*z^6+.1052402572e49*z^7+.499
9346123e50*z^8+.3017465915e52*z^9+.2284942813e54*z^10+.1634836432e56*z^11
+.9809018593e57*z^12+.4829055307e59*z^13+.1931622123e61*z^14+.6181190793e
40
62*z^15+.1545297698e64*z^16+.2908795668e65*z^17+.3878394223e66*z^18+.3266
016188e67*z^19+.1306406475e68*z^20.1198670100e37)/(.4940608790e35*z+.1782728638e38*z^2+.3823975762e40*z^3+.54
28522871e42*z^4+.5415264463e44*z^5+.4045125655e46*z^6+.2594827751e48*z^7+.
1822326757e50*z^8+.1530228723e52*z^9+.1280427438e54*z^10+.9264073125e55*z
^11+.5558443875e57*z^12+.2736464677e59*z^13+.1094585871e61*z^14+.35026747
86e62*z^15+.8756686966e63*z^16+.1648317547e65*z^17+.2197756729e66*z^18+.18
50742508e67*z^19+.7402970034e67*z^20-.3996100402e36)
h=0.05, Power Law n=10, Steel-Alumina
ρ
7800-4110*(1/2+20.00000000*z)^10
E
-6.0*(-291406257.0*z-.5178835830e11*z^2+.8041395268e25*z^10.5355683710e13*z^3-.3445844450e15*z^4.1241742916e17*z^5+.2583227600e17*z^6+.3828301618e20*z^7+.3212724738e22*z^
8+.1810030351e24*z^9.1598760202e11+.2926953253e27*z^11+.8780859760e28*z^12+.2161442402e30*z^1
3+.4322884803e31*z^14+.6916615682e32*z^15+.8645769606e33*z^16+.8137194928
e34*z^17+.5424796616e35*z^18+.2284124891e36*z^19+.4568249782e36*z^20)*(.17
03643892e24+.2042788129e23*z+.3677018632e25*z^2+.3922153208e27*z^3+.27455
07246e29*z^4+.1317843478e31*z^5+.4392811593e32*z^6+.1004071221e34*z^7+.150
6106832e35*z^8+.1338761628e36*z^9+.5355046513e36*z^10)/(494060879.*z+.89136
43190e11*z^2+.2500834840e25*z^10+.9559939406e13*z^3+.6785653590e15*z^4+.33
84540290e17*z^5+.1264101767e19*z^6+.4054418360e20*z^7+.1423692779e22*z^8+.
5977455948e23*z^9+.9046946412e26*z^11+.2714083924e28*z^12+.6680821966e29*
z^13+.1336164393e31*z^14+.2137863028e32*z^15+.2672328786e33*z^16+.2515132
976e34*z^17+.1676755317e35*z^18+.7060022386e35*z^19+.1412004477e36*z^207992200804.)/(.2042763975e14+.3983436853e13*z+.7170186336e15*z^2+.7648198758e17*z^3+.535
41
3739131e19*z^4+.2569794783e21*z^5+.8565982609e22*z^6+.1957938882e24*z^7+.2
936908323e25*z^8+.2610585176e26*z^9+.1044234070e27*z^10)
υ
.1000000000*(.1380855326e36*z+.2487352589e38*z^2+.2231389466e51*z^10+.2657
769037e40*z^3+.1868705620e42*z^4+.9082222183e43*z^5+.3147338280e45*z^6+.82
21895091e46*z^7+.1952869579e48*z^8+.5893488115e49*z^9+.7982599767e52*z^11
+.2394779930e54*z^12+.5894842905e55*z^13+.1178968581e57*z^14+.1886349729e
58*z^15+.2357937162e59*z^16+.2219234976e60*z^17+.1479489984e61*z^18+.6229
431511e61*z^19+.1245886302e62*z^20.1198670100e37)/(.2470304395e35*z+.4456821595e37*z^2+.1250417420e51*z^10+.4
779969703e39*z^3+.3392826795e41*z^4+.1692270145e43*z^5+.6320508836e44*z^6
+.2027209180e46*z^7+.7118463896e47*z^8+.2988727974e49*z^9+.4523473206e52*
z^11+.1357041962e54*z^12+.3340410983e55*z^13+.6680821965e56*z^14+.1068931
514e58*z^15+.1336164393e59*z^16+.1257566488e60*z^17+.8383776584e60*z^18+.3
530011193e61*z^19+.7060022386e61*z^20-.3996100402e36)
42
6.4 Appendix D – MATLAB Files
43