FinalReport Modal Analysis

8/13/2019 FinalReport Modal Analysis

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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 InstituteHartford, CT

December, 2011


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Copyright 2011

by

Wesley L Saunders

All Rights Reserved


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CONTENTS

LIST OF TABLES.............................................................................................................iv

LIST OF FIGURES............................................................................................................v

NOMENCLATURE..........................................................................................................viACKNOWLEDGMENT..................................................................................................vii

ABSTRACT .................................................................................................................viii

1. Introduction..11.1Background...1

1.1.1 Functionally Graded Materials...11.1.2 Modal Analysis...1

1.2Problem Description.22. Methodology.4

2.1Finite Element Modeling..42.2Modal Analysis.42.3Mori-Tanaka Estimation of FGM Material Properties.5

3. Results and Discussions........83.1Isotropic (Case A).83.2Linear (Case B)...133.3Power Law n=2 (Case C)153.4Power Law n=10 (Case D).183.5Comparison to Efraim [2]...22

4. Conclusions.....245. References...266. Appendices.27

6.1Appendix A Mesh Selection276.2Appendix B Mesh Extension Procedure..296.3Appendix C Mori-Tanaka FEA Inputs336.4Appendix D MATLAB Files...43


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LIST OF TABLES

Table 1: Case Descriptions........................................................................ 2

Table 2: Material Properties...................................................................... 3

Table 3: VcEquations 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


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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=0.05m19

Figure 11: Power Law n=10, Mode Shape Results, Steel-Alumina and Aluminum-

Zirconia, h=0.025m..20

Figure 12: Physics-Controlled Mesh for Isotropic Cases.27

Figure 13: User-Generated Mesh for FGM Cases28

Figure 14: Isotropic Mesh Extension30

Figure 15: FGM Mesh Extension.....31

Figure 16: Plot of MT Youngs Modulus for Steel-Alumina, n=2 Power Law,

h=0.05m33

Figure 17: Plot of MT Poisson Ratio for Steel-Alumina, n=2 Power Law,

h=0.05m34


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NOMENCLATURE

Density (kg/m3)

E Modulus of Elasticity (Pa)

Poissons Ratio (dimensionless)

D Flexural Rigidity (Pam3)

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)

Bulk Modulus (Pa)

n Power (dimensionless)

z thickness direction (m)

Lams first parameter (dimensionless)


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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.


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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.


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1. Introduction

1.1Background1.1.1 Functional Graded MaterialsFGM 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 AnalysisMany 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.


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1.2Problem DescriptionThis 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 simply-supported 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.


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For each material above, Table 1 contains there material properties of interest.

Table 2: Material Properties

Material E [Pa] [kg/m ]

Steel 1e11 .3 7800

Aluminum 7.5e9 .33 2700

Alumina 3e11 .27 3690

Zirconia 2e11 .3 5700


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2. Methodology

2.1Finite Element ModelingCOMSOL 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

2.2Modal AnalysisThe 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:


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(1)

(1a)

2.3Mori-Tanaka Estimation of FGM PropertiesTo 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 theFGM, 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 Youngs modulus of the FGM can be determined

by inserting equations (3) thru (4a) into equations (5), (5a), and (6).

(5)

(5a)


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(6)

For Cases B-D, the VCis a function of thickness of the plate (z). It is assumed that the

VCand the VMcombine to unity and each material is at 100% at its respective plate face.

Thus, VT= 1 and VB= 0. Expressions for VCobtained for the three cases investigated

are shown in Table 2 and graphical representations appear in Figure 2.

Table 3: VCEquations for Different FGM

Case VcEquation

Linear

(7)

Power Law, n=2

(8)

Power Law, n=10

(9)

Average (All Cases)

(10)

Figure 2: Graphical Representation of VCof Different FGM Through

Thickness of Specimen


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With these relations of VC known, the Mori-Tanaka estimation can used to predict

material properties in Cases B-D.


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3. Results and Discussion3.1Isotropic (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 isotropicresults 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.


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Table 4: Isotropic Frequency Results

h=0.025m h=0.05m

MaterialMode

(m,n)

Frequency

Ref [3]

[Hz]

Frequency

(FEA)

[Hz]

Percent

Error

Frequency

Ref [3]

[Hz]

Frequency

(FEA)

[Hz]

Percent

Error

1 (1,1) 85.10 84.55 0.65 170.20 166.14 2.39

Steel 2a (1,2) 212.75 211.79 0.45 425.50 413.73 2.77

2b (2,1) 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

Alumina 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

Zirconia 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


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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.


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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


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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.


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3.2Linear (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

FGMMode

(m,n)

Frequency

(FEA)

[Hz]

Frequency

(FEA)

[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


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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.


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3.3Power 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

constituents 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

FGM

Mode

(m,n)

Frequency

(FEA)

[Hz]

Frequency

(FEA)

[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) 137.2 265.16Top Material: Zirconia 2b (1,2) 137.21 265.2

3 (2,2) 217.72 412.47


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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.


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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.


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3.4Power 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

FGMMode

(m,n)

Frequency

(FEA)

[Hz]

Frequency

(FEA)

[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


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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


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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 Aluminum-

Zirconia, 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


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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.


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3.5Comparison 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


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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 Youngs modulus. Efraim uses two dimensionless

frequency parameters which are singularly dependent on one of the materials Youngs

modulus and a rule of mixtures Youngs modulus (see equation 11a).

Table 8: FEA and Efraim [2] Comparison

h=0.05m

FGMMode

(m,n)

Frequency(FEA)

[Hz]

Efraim [2]Frequency

[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%


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4. ConclusionsThis 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 thework 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 frequencyseems 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 fromthe cross-section of the plate. Many times in this project, the 0.025m plate was


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simply too thin to fully capture was occur as the frequencies and modes were

calculated through the plates thickness.

The Mori-Tanaka estimate is heavy computationally, as demonstrated inAppendix C. For future use, the use of and K should stand to simplify the

calculations.


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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), pages703-730

[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


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6. Appendices6.1Appendix A Mesh SelectionIsotropic 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


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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


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6.2Appendix B Mesh Extension ProcedureIsotropic 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.


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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.


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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


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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.


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6.3Appendix C Mori-Tanaka FEA InputsFor each FGM case, a MT estimation of density, Youngs Modulus, and Poisson ratio

was completed. The equations for Youngs 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 Youngs Modulus for Steel-Alumina, n=2 Power Law,

h=0.05m


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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

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)

.1000000000e-1*(-.4293019142e27+.2500717343e29*z+.2854199148e30*z^2)/(-

.1442628895e26+.7490518756e27*z+.1701948384e29*z^2)


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h= 0.05, Linear, Aluminum-Zirconia

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)

.1000000000e-1*(-.4293019142e27+.1250358672e29*z+.7135497869e29*z^2)/(-

.1442628895e26+.3745259378e27*z+.4254870959e28*z^2)

h=0.025, Power Law n=2, Aluminum-Zirconia

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)

.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)


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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)

.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)


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^10-

9417613335.+.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


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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^20-

5537066745.)/(-

.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)

.1000000000e-

1*(.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)


2700+3000*((1/2)+20.00000000*z)^10

E

-24.0*(-852900671.0*z-.1506943492e12*z^2-.1535769518e14*z^3-

.9460922920e15*z^4-


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.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)

.1000000000e-

1*(.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)


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h=0.025, Power Law n=2, Steel-Alumina

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)

.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)


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)

.1000000000*(-

.1683460106e32+.1259385337e33*z+.3243971949e34*z^2+.2900805097e35*z^3+.290

0805097e36*z^4)/(-


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.5793557129e31+.2953280623e32*z+.1001603514e34*z^2+.1643789556e35*z^3+.164

3789556e36*z^4)

h=0.025, Power Law n=10,Steel-Alumina

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^10-

3996900505.0+.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^20-

3996100402.)/(-

.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)

.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


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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)


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^20-

7992200804.)/(-

.2042763975e14+.3983436853e13*z+.7170186336e15*z^2+.7648198758e17*z^3+.535


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3739131e19*z^4+.2569794783e21*z^5+.8565982609e22*z^6+.1957938882e24*z^7+.2

936908323e25*z^8+.2610585176e26*z^9+.1044234070e27*z^10)

.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)


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6.4Appendix D MATLAB Files

FinalReport Modal Analysis

Documents

Transcript of FinalReport Modal Analysis