The Pennsylvania State University The Graduate School ... · helicopter gearbox isolation using periodically layered fluidic isolators ... chapter 3 passive high frequency gearbox

The Pennsylvania State University

The Graduate School

Department of Mechanical and Nuclear Engineering

HELICOPTER GEARBOX ISOLATION USING PERIODICALLY

LAYERED FLUIDIC ISOLATORS

A Thesis in

Mechanical Engineering

by

Joseph Thomas Szefi

2003 Joseph Thomas Szefi

Submitted in Partial Fulfillmentof the Requirements

for the Degree of

Doctor of Philosophy

August 2003

The Pennsylvania State University

We approve the thesis of Joseph Thomas Szefi.

Date of Signature

Edward C. SmithAssociate Professor of Aerospace EngineeringThesis Co-AdvisorCo-chair of Committee

George A. LesieutreProfessor of Aerospace EngineeringThesis Co-AdvisorCo-chair of Committee

Kon-Well WangWilliam E. Diefenderfer Chaired Professor in

Mechanical EngineeringCo-chair of Committee

Gary H. KoopmannDistinguished Professor of Mechanical

Engineering

Richard C. BensonProfessor of Mechanical EngineeringHead of the Department of Mechanical

Engineering

iii

ABSTRACT

In rotorcraft transmissions, vibration generation by meshing gear pairs is a

significant source of vibration and cabin noise. This high-frequency gearbox noise is

primarily transmitted to the fuselage through rigid connections, which do not appreciably

attenuate vibratory energy. The high-frequency vibrations typically include discrete

gear-meshing frequencies in the range of 500 – 2000 Hz, and are often considered

irritating and can reduce pilot effectiveness and passenger comfort.

Periodically-layered isolators were identified as potential passive attenuators of

these high frequency vibrations. Layered isolators exhibit transmissibility “stop bands,”

or frequency ranges in which there is very low transmissibility. An axisymmetric model

was developed to accurately predict the location of these stop bands for isolators in

compression. A Ritz approximation method was used to model the axisymmetric elastic

behavior of layered cylindrical isolators. This model of layered isolators was validated

with experiments.

The physical design constraints of the proposed helicopter gearbox isolators were

then estimated. Namely, constraints associated with isolator mass, axial stiffness,

geometry, and elastomeric fatigue were determined. The passive performance limits of

layered isolators were then determined using a design optimization methodology

employing a simulated annealing algorithm. The results suggest that layered isolators

cannot always meet frequency targets given a certain set of design constraints.

iv

Many passive and active design enhancements were considered to address this

problem, and the use of embedded inertial amplifiers was found to exhibit a combination

of advantageous effects. The first benefit was a lowering of the beginning stop band

frequency, and thus a widening of the original stop band. The second was a tuned

absorber effect, where the elastomer layer stiffness and the amplified tuned mass

combined to act as a vibration absorber within the stop band. The use of embedded fluid

elements was identified as an efficient means of implementing inertial amplification.

When elastomer layers are compressed quasi-statically, the actual measured axial

stiffness is quite higher the than one-dimensional stiffness predicted on the basis of a

Young’s modulus. Because of this effect, layered isolators can be designed to

accommodate the high axial stiffnesses required for helicopter gearbox supports, while

also providing broadband high frequency attenuation.

TABLE OF CONTENTS

LIST OF FIGURES..................................................................................................viii

LIST OF TABLES ...................................................................................................xiii

ACKNOWLEDGMENTS ........................................................................................xv

Chapter 1 INTRODUCTION..................................................................................1

1.1 Background and Motivation.........................................................................11.2 Literature Review ........................................................................................5

1.2.1 Sources of High Frequency Noise Inside Helicopter Cabins...............51.2.2 Rotorcraft Interior Noise Control .......................................................9

1.2.2.1 Passive Treatments...................................................................91.2.2.2 Active Noise Cancellation........................................................111.2.2.3 Active Structural Vibration Control..........................................121.2.2.4 Minimum Noise Transmission from Gearbox...........................13

1.2.3 Periodically Layered Media for High Frequency Isolation..................161.3 Research Objectives.....................................................................................17

Chapter 2 PERIODICALLY LAYERED ISOLATORS FOR HIGHFREQUENCY ISOLATION .............................................................................21

2.1 One Dimensional Finite Elements Analysis .................................................212.2 Three Dimensional Finite Element Analysis ................................................23

2.2.1 One Dimensional Stiffness Correction Factors ...................................262.2.1.1 Shape Factor Influence on Higher Modes.................................30

2.2.2 Table Look-up Methodology..............................................................332.3 Axisymmetric Approximation Method.........................................................34

2.3.1 Natural Frequency of 3-D Cylinders of Finite Length.........................342.3.2 Numerical Results for a Single Cell....................................................382.3.3 Analysis of Layered Isolators Using Component Mode Method.........42

2.4 Experimental Validation ..............................................................................48

Chapter 3 PASSIVE HIGH FREQUENCY GEARBOX ISOLATION....................59

3.1 Design Constraints.......................................................................................59

vi

3.1.1 Mass Constraints................................................................................603.1.2 Axial Stiffness Constraints.................................................................603.1.3 Elastomeric Bearing Stress Constraints ..............................................63

3.2 Design Optimization....................................................................................653.3 Passive Performance Limits.........................................................................68

Chapter 4 DESIGN ENHANCEMENTS FOR IMPROVED ISOLATORPERFORMANCE .............................................................................................74

4.1 Summary of Active, Semi-Active Concepts to Improve IsolatorPerformance ...............................................................................................744.1.1 Emdedded Terfenol-D Actuators in Layered Isolators........................76

4.2 Embedded Vibration Absorbers to Improve Isolator Performance................81

Chapter 5 PERIODICALLY LAYERED ISOLATORS WITH EMBEDDEDINERTIAL AMPLIFIERS.................................................................................84

5.1 Embedded Inertial Amplifiers......................................................................845.2 Effect of Embedded Inertial Amplifiers on Layered Isolator Frequency

Response ....................................................................................................875.3 Vibration Absorber Effect............................................................................895.4 Typical Frequency Response Shapes of Isolator with Embedded

Amplifiers ..................................................................................................925.5 Effect of Shape Factor and Passive Stop Band Location on Response

Behavior.....................................................................................................945.5.1 High Shape Factor Behavior ..............................................................945.5.2 Low Shape Factor Behavior ...............................................................97

5.6 Structural Periodicity of Embedded Amplifier Design..................................1025.7 Passive Stop Band Limitations on Embedded Amplifier Design...................1035.8 Fluid Elements as Efficient Implementation of Inertial Amplification..........105

Chapter 6 ANALYSIS OF LAYERED ISOLATOR EFFECTIVENESS FORGEARBOX ISOLATION..................................................................................108

6.1 The Vibration Isolation Problem..................................................................1086.2 Effectiveness of Layered Isolators with Embedded Fluid Elements..............110

6.2.1 Isolator Effectiveness with Non-Negligible Isolator Mass ..................1106.2.2 Source and Receiver Mobility Approximations ..................................1116.2.3 Layered Isolator Effectiveness Prediction...........................................112

Chapter 7 EXPERIMENTAL VALIDATION OF FLUID-FILLED ISOLATORCONCEPT ........................................................................................................115

7.1 Single-Celled Fluidic Specimen Testing ......................................................1157.2 Three-Celled Fluidic Specimen Testing .......................................................119

vii

Chapter 8 CONCLUSIONS AND RECOMMENDATIONS FOR FUTUREWORK..............................................................................................................133

8.1 Conclusions .................................................................................................1338.2 Recommendations for Future Work .............................................................138

8.2.1 Light, Compact Isolator Design..........................................................1398.2.2 Expansion of Design Optimization Routine........................................1408.2.3 Stiffness and Fatigue Testing. ............................................................1418.2.4 Conceptual Strut/Isolator Configuration.............................................1418.2.5 Construction of a Demonstration Layered Helicopter Gearbox

Isolator ................................................................................................1448.2.6 Semi-active Tuning of Fluid Isolator Inner Diameter .........................144

BIBLIOGRAPHY ....................................................................................................147

Appendix A DERIVATION OF THE EQUATIONS OF MOTION FOR AVIBRATING CYLINDER AS PRESENTED BY HEYLIGER [51]..................160

Appendix B EXPLICIT FORMS OF [M] AND [K] MATRICES ASREPORTED BY HEYLIGER [51] ....................................................................164

Appendix C MATLAB® CODE: DESIGN OPTIMIZATION FOR LAYEREDISOLATORS IN COMPRESSION....................................................................166

Appendix D MATLAB CODE: TRANSMISSIBILITY OF LAYEREDISOLATORS WITH EMBEDDED FLUID ELEMENTS..................................182

Appendix E ANNOTATED DRAWINGS OF LAYERED SPECIMEN WITHEMBEDDED FLUID ELEMENTS...................................................................192

Appendix F CAUSES OF VARIATION IN THE DYNAMIC MATERIALPROPERTIES OF ELASTOMERS ...................................................................208

Dynamic Amplitude Dependence ......................................................................208Temperature Dependence ..................................................................................211Preload Dependence ..........................................................................................213Frequency Dependence......................................................................................214

LIST OF FIGURES

Figure 1.1: Gearbox and mounting for the Cormorant (EH-101).(http://www.dnd.ca/menu/SAR/eng/cormorant/Tour/breakaway/main_gearbox.htm)................................................................................................................2

Figure 1.2: Periodically Layered Isolator in Compression. .......................................3

Figure 1.3: Illustration of Wave Dynamics for frequencies Within and Outside ofStop Band. (http://www.seasalum.ucla.edu/pdf/UEfall02.pdf) ..........................4

Figure 1.4: Lynx Gearbox Noise Identification [12]. ................................................7

Figure 1.5: Growth in Combined Gearbox and Soundproofing Weight at ConstantPower, Constant Design Noise Level [25]. ........................................................10

Figure 1.6: Schematic of ANC Architecture [30]......................................................13

Figure 1.7: Proposed Implementation of Layered Isolators in Gearbox SupportStruts.(http://www.dnd.ca/menu/SAR/eng/cormorant/Tour/breakaway/main_gearbox.htm)................................................................................................................18

Figure 2.1: Comparison of One-Dimensional Transmissibility Predictions forFEM and Floquet Theory for a 3-celled Isolator. ...............................................23

Figure 2.2: First Four Mode Shapes of a Three-celled Isolator in Compression. .......24

Figure 2.3: First Thickness Modes Associated with the Stop Band End FrequencyUsing 1-D and Axisymmetric Models................................................................26

Figure 2.4: Simple Isolator / Mass System in One Dimension. .................................29

Figure 2.5: Simple Isolator / Mass System Modeled with Axisymmetric Elements...29

Figure 2.6: (rS)E vs. Shape Factor. ............................................................................32

Figure 2.7: 1-D /3-D Design Space for (rF)B and (rF)E...............................................33

ix

Figure 2.8: Illustration of Fixed-Free Boundary Condition for a Single Cell. ............37

Figure 2.9: Higher and Lower Approximations for Modes 1 and 2 of a SingleCell. ..................................................................................................................42

Figure 2.10: Experimental Set-up.............................................................................48

Figure 2.11: Experimental and Analytical Transmissibilities for Specimen 1 with4 Cells. ..............................................................................................................50



Figure 2.14: Experimental and Analytical Transmissibilities for Specimen 1 with1 Cell.................................................................................................................51




Figure 2.18: Experimental and Analytical Transmissibilities for Specimen 2 with1 Cell.................................................................................................................54

Figure 2.19: Comparison of Experimental Transmissibilities for Varying Numberof Cells for Specimen 1. ....................................................................................56

Figure 2.20: Comparison of Experimental Transmissibilities for Varying Numberof Cells for Specimen 2. ....................................................................................57

Figure 2.21: Experimental and Analytical 4-Celled Transmissibilities ofSpecimen 1........................................................................................................58

Figure 2.22: Experimental and Analytical 4-Celled Transmissibilities ofSpecimen 3........................................................................................................58

Figure 3.1: Schematic of Helicopter Transmission Mountings. .................................62

Figure 3.2: 20 year-old layered elastomeric bearing supporting bridge deck inEngland [64]......................................................................................................63

x

Figure 3.3: Schematic of Design Variables...............................................................66

Figure 4.1: Model Schematic of Layered Isolator with Embedded Terfenol-DActuator. ...........................................................................................................77

Figure 4.2: Transmissibility of Three-Layered Isolator with Embedded Terfenol-D Actuator.........................................................................................................80

Figure 4.3: Transmissibility of Isolator with and without Embedded Terfenol-DActuator. ...........................................................................................................81

Figure 4.4: Schematic of Layered Isolator with Embedded Vibration Absorbers. .....82

Figure 4.5: Transmissibility of Example Layered Isolator. .......................................83

Figure 5.1: Schematic of Isolator Configuration with Embedded InertialAmplifiers. ........................................................................................................86

Figure 5.2: Schematic of one inertial amplifier. ........................................................87

Figure 5.3: Mode Shapes at Beginning and End Stop Band Frequencies ofLayered Isolator with Embedded Inertial Amplifiers..........................................89

Figure 5.4: Schematic of Subsystem of Layered Isolator with Embedded InertialAmplifier [54]. ..................................................................................................90

Figure 5.5: Transmissibility of Example Isolator with and without InertialAmplifiers. ........................................................................................................92

Figure 5.6: Axisymmetric Frequency Response Shapes of Example Isolator withEmbedded Amplifiers at Tuned Absorber Frequencies. .....................................93

Figure 5.7: Example Isolator Response Shapes (High Shape Factor, S = 2.6). ..........96

Figure 5.8: Transmissibility and Response Shapes for Layered Isolator withShape Factor = 0.2.............................................................................................98

Figure 5.9: Transmissibility of Low Shape Factor Isolator with and withoutEmbedded Inertial Amplifiers............................................................................99

Figure 5.10: Response Shapes of a Single Layer of the Low Shape FactorExample Isolator with Embedded Inertial Amplifiers: (a) m = 10.6 g, (b) m =5.6 g, (c) m = 3.3 g.............................................................................................101

Figure 5.11: Transmissibility of Example Isolator with and without InertialAmplifiers which Maintain Isolator Periodicity. ................................................103

xi

Figure 5.12: Transmissibility of Stiffened Layered Isolator Example with andwithout Inertial Amplifiers. ...............................................................................105

Figure 5.13: Cut-Away View of a Fluidlastic® Mount [54]......................................106

Figure 6.1: Schematic and Mobility Diagram of Source, Isolator, and Receiver[77]. ..................................................................................................................109

Figure 6.2: Experimental Source and Receiver Mobilities: (a) … Experimental, zdir., - - - Approximation, z dir. (b) …. Experimental, z dir., - - -Approximation, z dir. [23]. ................................................................................112

Figure 6.3: Mobilities of Layered Isolator with Embedded Fluid Elements...............113

Figure 6.4: Analytical Effectiveness of Layered Isolator with Embedded FluidElements. ..........................................................................................................114

Figure 7.1: Transmissibility and Schematic of Single-Celled Specimen toCharacterize Elastomer. .....................................................................................116

Figure 7.2: Single-Layered Specimen with Embedded Fluid Element: (a)Illustration, (b) Cross-section.............................................................................117

Figure 7.3: Analytical and Experimental Transmissibilities of Single-LayeredSpecimen with Embedded Fluid Element...........................................................118

Figure 7.4: Cross-sectional View of Three-celled Specimen with Embedded FluidElements. ..........................................................................................................121

Figure 7.5: Analytical and Experimental Transmissibilities of Three-celledSpecimen to Characterize Elastomer..................................................................123

Figure 7.6: Experimental and Analytical Transmissibility Comparison of OriginalThree-celled Configuration................................................................................124

Figure 7.7: Analytical and Experimental Transmissibilities of Single-celledSpecimen with Embedded Fluid Element...........................................................126

Figure 7.8: Analytical and Experimental Transmissibilities of Two-celledSpecimen with Embedded Fluid Elements. ........................................................128

Figure 7.9: Analytical and Experimental Transmissibilities of Three-celledSpecimen with Embedded Fluid Elements. ........................................................129

Figure 7.10: Experimental Transmissibility Comparison of One, Two, and Three-celled Specimens with Embedded Fluid Elements. ............................................130

xii

Figure 7.11: Transmissibility Comparison of Three-celled Specimen with andwithout Embedded Fluid Elements. ...................................................................131

Figure 7.12: Transmissibilities of Three-Celled Specimen in OriginalConfiguration and Rebuilt Configuration. ..........................................................132

Figure 8.1: Conceptual Configuration for Lower Fluidic Isolator Height. .................140

Figure 8.2: Conceptual Strut Configuration to Ensure Compressive Loads onEmbedded Layered Isolator. ..............................................................................143

Figure 8.3: Conceptual Contractable Inner Diameter Using SMA Wires for Semi-actively Tuning Fluidic Isolators. ......................................................................146

Figure A.1: Geometry of Cylinder ............................................................................162

Figure F.1: Example of Payne Effect for Typical Filled Elastomer [79]. ..................209

Figure F.2: Effect of Dynamic Strain at Room Temperature for Gum and CarbonBlack-Filled Rubber for ε = 0.17, 0.35 0.5, 0.8, 0.88 [80]. .................................210

Figure F.3: Predicted Shear Modulus vs. Frequency at Different Temperatures forTypical Unfilled Elastomer [82]. .......................................................................212

Figure F.4: Modeled Driving Point Stiffness Magnitude and Phase versusFrequency of Sample Isolator for 0% (----), 10% (- - - ), and 20% (- . - .)Precompression at (a) –50 oC, (b)-25 oC, (c) 0 oC, (d) 25 oC, and (e) 50 oC[82]. ..................................................................................................................213

Figure F.5: Unfilled Material Shear Modulus vs. Frequency [79]. ............................215

Figure F.6: Young’s Modulus versus Reduced Frequency for Rubber IsolatorMaterial [83]. ....................................................................................................215

Figure F.7: Stiffness vs. Dynamic Amplitude for Single Frequency andSuperimposed Frequencies for Carbon Black-Filled Isolator [79]. .....................216

LIST OF TABLES

Table 2.1: Shape Factor for Rectangular and Circular Isolators.................................28

Table 2.2: (3-D/1-D) Stiffness Ratios for Different Modes of a 3-celled LayeredIsolator. .............................................................................................................31

Table 2.3: (3-D/1-D) Stiffness Ratios for End Frequency of 3-celled Isolators..........32

Table 2.4: Single Cell Natural Frequencies Using Assumed Modes MethodCompared to 3-D FEM Results..........................................................................40

Table 2.5: Rate of Convergence Comparison for a Single Cell..................................40

Table 2.6: Reduced Powers of ‘r’ and ‘z’ for Ur and Uz to ApproximateBeginning and End Stop Band Frequencies to within 5% Error..........................41

Table 2.7: Summary of Specimen Properties ............................................................49

Table 3.1: Combination of Design Variable and Constraint Values for PassivePerformance Limits of Layered Isolators. ..........................................................69

Table 3.2: Passive Performance Design Runs, Isolator Mass = 2 kg..........................72

Table 3.3: Passive Performance Design Runs, Isolator Mass = 4 kg..........................73

Table 4.1: Optimized Properties of Layered Isolator of Example. .............................82

Table 5.1: Summary of Isolator Eigenvalues with and without Embedded InertialAmplifiers. ........................................................................................................88

Table 5.2: Summary of Example Inertial Amplifier Properties..................................88

Table 5.3: Summary of Inertial Amplifier Properties in Low Shape FactorExample. ...........................................................................................................99

Table 5.4: : Summary of Inertial Amplifier Properties to Maintain IsolatorPeriodicity. ........................................................................................................102

Table 5.5: Summary of Inertial Amplifier Properties in Figure 5.12. ........................105

xiv

Table 7.1: Dimensions of Fluid Elements in Three-layered Specimen.......................122

xv

ACKNOWLEDGMENTS

Foremost, I would like to thank my advisors, Dr. Edward C. Smith and Dr.

George A. Lesieutre, for their encouragement and guidance during my graduate studies at

Penn State. They are both fine educators and superb advisors. In addition, I would like

my committee members, Dr. Kon-Well Wang and Dr. Gary H. Koopmann for their

helpful participation. I would also like to thank all the organizations responsible for

funding this research, including the National Rotorcraft Technology Center, United

Technologies Research Center, and the Lord Corporation, as well as the Vertical Flight

Foundation for their generous scholarship. I especially would like to thank the

individuals at Lord for their indispensable efforts and helpful advice, including Donald

Russell, Scott Redinger, Tejbans Kohli, and Oon Hock Yeoh.

xvi

LIST OF SYMBOLS

[A] Total Constraint matrix

[A1] Dependent constraint matrix

[A2] Independent constraint matrix

a Shorter length of amplified mass lever arm

b Longer length of amplified mass lever arm

bk Coefficients of radial displacement functions

[c] Damping matrix of a single cell

ce Wave propagation speed in elastomer

cw Speed of sound in water

C11, C22, C33, C55,C12, C13, C23 Constitutive material constants

CpT Capacitance of material measured under constant stress

d Isolator diameter

dcell Cell diameter

dk Coefficients of axial displacement functions

dmax Maximum isolator diameter

dmin Minimum isolator diameter

E Isolator effectiveness

Ee Elastomer young’s modulus

xvii

Em Metal young’s modulus

f1 First standing wave frequency of column of water

f1-D General one dimensional isolator natural frequency

f3-D General three dimensional isolator natural frequency

fi Frequency of isolation of fluid-filled isolator

fn1 First natural frequency of single cell

fn2 Second natural frequency of single cell

fB Beginning stop band frequency

fE End stop band frequency

fT Elastomer layer thickness mode natural frequency

fTB Target beginning stop band frequency

fTE Target end stop band frequency

F Force on isolator end

Fa Actuator point force

Fb Base force

Fi Applied input force on isolator

Fy Material yield strength

FO Force on isolator from source side

FR Force on isolator from receiver side

G* Complex shear modulus

Gemax Maximum elastomer shear modulus

Gemin Minimum elastomer shear modulus

xviii

Ge Elastomer shear modulus

Gm Metal shear modulus

Gmmax Maximum metal shear modulus

Gmmin Minimum metal shear modulus

Go Shear modulus at reference temperature

GT Shear modulus at Temperature, T

[G] 3-DOF to 2-DOF transformation matrix

h Layered isolator height

J Quadratic objective function

[k] Stiffness matrix of single cell

k Spring stiffness

k* Effective stiffness of piezoelectric element

kE Effective short circuit material stiffness

k1-D One dimensional isolator stiffness

k3-D Three dimensional isolator stiffness

kA Axial stiffness

kAmax Maximum axial stiffness

kAmin Minimum axial stiffness

keff Effective three dimensional isolator stiffness

kL Lateral stiffness

kLmax Maximum lateral stiffness

kLmin Minimum lateral stiffness

xix

kP Material planar electromechanical coupling coefficient

[K] Stiffness matrix for single cell

[K11], [K22],[K12], [K21] Portions of global stiffness matrix

l Length of general, rectangular isolator

L Length of bound column of water

Lr Cylinder radius

Lz Cylinder half-height

[m] Mass matrix of single cell

m Isolator mass

m1 Mass attached to base pivot of lever arm

m2 Mass attached to top pivot of lever arm

mt Tuned mass at end lever arm

[M] Mass matrix for single cell

M Mobility

M1b Mobility of source-side, with receiver side fixed

M2b Mobility of receiver side, with source side fixed

M1f Mobility of source side, with receiver side free

MI Isolator mobility

MR Receiver mobility

MS Source mobility

[M11], [M22],[M12], [M21] Portions of global stiffness matrix

xx

n Number of cells in layered isolator

{p} Generalized coordinates vector

{P} Total generalized forces vector

{q} Independent coordinates vector

{Q} Transformed independent coordinates vector

r Displacement in radial direction

rF 3-D to 1-D frequency ratio

rS 3-D to 1-D stiffness ratio

R Non-dimensional cylinder radius

R Amplification ratio

S Elastomer shape factor

t General isolator thickness

t11, t12, t21, t22 Components of system transfer matrix

te Elastomer layer thickness

temax Maximum elastomer layer thickness

temin Minimum elastomer layer thickness

tm Metal layer thickness

T Isolator transmissibility

[T] System transfer matrix

u Isolator radial direction

u1 Axial displacement of base pivot of lever arm

u2 Axial displacement of top pivot of lever arm

xxi

ru Evaluated displacement constraint in rth cell

su Evaluated displacement constraint in sth cell

Ur Radial displacement component

Uz Axial displacement component

V Velocity of isolator end

V Volume of cylinder

VO Velocity of source side of isolator

VR Velocity of receiver side of isolator

VRO Velocity of source with massless, rigid connection to receiver

w Width of general, rectangular isolator

w Isolator axial direction

[W1] Penalty weighting matrix for displacement

[W2] Penalty weighting matrix for actuator force

x Displacement of the top metal layer in layered isolator

xb Displacement input at isolator base

xo Baseline displacement of metal layer

xt Transmitted displacement amplitude of isolator

z Displacement in axial direction

Z Non-dimensional cylinder height

α Beginning stop band frequency weighting factor

α Intermediate variable in mobility definitions

α(s) Non-dim. ratio of electrical impedance to electrical impedance

xxii

[β] Constraint transformation matrix

β End stop band frequency weighting factor

χT Empirical shift function

ε1, ε2, ε3, ε5 Strain components in cylindrical coordinates

εrr, εθθ, εzz, εrz Strain components in cylindrical coordinates

φ Empirically derived material property for elastomer

φuj Raidal polynominal displacement functions

φwj Axial polynominal displacement functions

η Complex viscosity

νe Elastomer Poisson’s ratio

νm Metal Poisson’s ratio

θ Angular direction in cylindrical coordinates

λ1 Wavelength of first standing wave in column of water

ρe Elastomer density

ρemax Maximum elastomer density

ρemin Minimum elastomer density

ρm Metal density

ρmmax Maximum metal density

ρmmin Minimum metal density

σι ith stress component in cylindrical coordinates

σc Quasi-static compressive stress

xxiii

σ CDYN Dynamic compressive stress

σrr, σzz, σrz Stress components in cylindrical coordinates

ω Frequency

ωn Natural frequency

( )d Contribution from dependent variables

( )f Contribution from independent variables

( )r Contribution from the rth cell

( )s Contribution from the sth cell

( )c Cosine component

( )s Cosine component

Chapter 1

INTRODUCTION

1.1 Background and Motivation

Dynamic excitations generated by meshing gear pairs is a significant source of

vibration and cabin noise in helicopter transmissions. This high-frequency gearbox noise

is primarily transmitted to the fuselage through rigid connections, which do not

appreciably attenuate vibratory energy (Figure 1.1). The close proximity of the

transmission and cabin in rotorcraft causes interior noise levels that are significantly

higher than those in fixed wing aircraft. The high-frequency vibrations typically include

discrete gear-meshing frequencies in the range of 500 – 4000 Hz, and are often

considered irritating. This high frequency noise can reduce pilot effectiveness and

passenger comfort.

Although elastomeric isolators are frequently used for passive isolation of

mechanical components, these typically operate at relatively low frequencies ( < 100 Hz).

Wave effects occur in conventional isolators at high frequencies when the elasticity and

the distributed mass of the mount interact to create sharp transmissibility peaks [1]. Such

2

isolators are not effective at reducing the transmission of higher frequency vibro-

acoustic energy because of this inherent continuous distribution of mass and stiffness.

There is a need to conceptualize and demonstrate methods of achieving greater levels of

isolation, in excess of 10 dB, over the 500 – 2000 Hz range. Fully active approaches,

while offering the potential for high performance, are complex and tend to have poor

power-off behavior.

Multi-layered isolators have potential to substantially reduce noise transmission

over a relatively large frequency range [2, 3]. Such isolators consist of multiple identical

cells, each containing a dense, stiff layer in combination with a softer, light layer (Figure

1.2). From a modal-dynamic perspective, the behavior is similar to that of a multi-stage

isolator [7]. Alternatively. from a one-dimensional wave-dynamic perspective, the

transmission and reflection of stress waves at interfaces between dissimilar materials

leads to “stop bands” in frequency, within which transmitted waves are highly attenuated

[4-6].

Figure 1.1: Gearbox and mounting for the Cormorant (EH-101).(http://www.dnd.ca/menu/SAR/eng/cormorant/Tour/breakaway/main_gearbox.htm)

TypicalRigid Struts

3

In Figure 1.3, the simple schematic illustrates of the one-dimensional wave

dynamics for frequencies within and outside of the stop band. At frequencies within the

stop band, the waves reflected from the denser material are in phase and act to cancel the

incident wave. As a result, the total transmitted wave is significantly attenuated. At

frequencies outside of the stop band, the reflected waves are out of phase. The total

transmitted wave is therefore not appreciably attenuated.

Another performance benefit of layered isolators is their inherent high axial

stiffness. When the elastomer layers of a layered isolator are compressed quasi-statically,

the actual measured axial stiffness is significantly higher the than one-dimensional

stiffness predicted on the basis of a Young’s modulus. A well-documented method to

account for the difference between the effective three-dimensional stiffness and the

predicted one-dimensional stiffness is the use of a one-dimensional stiffness correction

factor [8 ,9]. Essentially, the shape factor accounts for the discrepancy between the

predicted one-dimensional stiffness and the measured effective one-dimensional stiffness

of an elastomer isolator. Because of this effect, layered isolators can be designed with

Figure 1.2: Periodically Layered Isolator in Compression.

High-FrequencyForce Excitation

Attenuated ForceTransmission

MetalElastomer

4

the high axial stiffness required for helicopter gearbox supports, while also providing

broadband high frequency attenuation. Layered isolators may therefore provide an

elegant solution to the high frequency gearbox noise problem in helicopter cabins.

Figure 1.3: Illustration of Wave Dynamics for frequencies Within and Outside of StopBand. (http://www.seasalum.ucla.edu/pdf/UEfall02.pdf)

Frequencies Within Stop Band

Frequencies Outside of Stop Band

AmplitudeAttenuated

AmplitudeUnattenuated

5

1.2 Literature Review

1.2.1 Sources of High Frequency Noise Inside Helicopter Cabins

Through numerous experimental and analytical studies, the gearbox has been

widely established as the primary source of high frequency noise in rotorcraft cabins.

Pollard has reviewed gearbox noise generation and its transmission via the gearbox and

other airframe structures of the Westland Lynx helicopter [10]. The noise is reported to

be a result of force-fluctuations from the elastic deformation of the gear teeth under load

and tooth manufacturing errors. The fluctuations result in non-uniform gear rotation and

dynamic forces are generated at gear meshing frequencies and harmonics. The forces in

turn excite the gear shafts in torsional, axial, and lateral modes, which cause the bearings

to displace and thus the gearbox casing vibrates and radiates noise. As a result, the

harmonic content of noise radiated to interior microphones is markedly similar to that of

vibrations measured via accelerometers placed on the gearbox casing. Gear meshing

vibrations are directly transmitted to the fuselage because the gearbox is essentially

rigidly mounted to the airframe. Vibration levels of up to 31.6 g at gear meshing

frequencies were measured on the Lynx during flight. Shake tests of the airframe, where

the excitations were applied to the gearbox feet locations, show that the vibration levels

on the airframe were of the same order as the input levels at the gearbox feet. The

vibratory energy is reported to be transmitted to the cabin with little reduction causing

6

individual interior panels to vibrate at large amplitudes and hence radiate noise in the

cabin.

A similar study of the of the Army’s OH-58 helicopter was reported by Coy, et al.

[11]. Flight test data suggest that the planetary gear train is the major source of high

frequency cabin noise. The authors note that it is particularly difficult to block the

structure borne path of interior noise, because the gearbox case and its mounts are an

integral part of the lift-load bearing path. The transmission mounts must be strong

enough to support the entire helicopter by transferring the lift-load from the rotor blades

to the airframe and rigid enough to ensure stable control of the helicopter. Because the

stiff mounts transmit noise exceedingly well, the sound is transmitted to the cabin

directly. Experimental data indicate that the most troublesome noise occurs at the

planetary gear mesh frequency, and thus efforts should be made to attenuate the lowest

frequency gear noise, or 400 – 1000 Hz.

A joint research effort between Aerospatiale and Westland was presented by

D’Ambra and his associates [12]. The authors report that helicopter noise in the audio

range is quite high and exposure time for operations without a protective helmet is

extremely limited. Furthermore, in military helicopters, where available weight for

soundproofing is limited, sound levels obtained after treatment remain in excess of those

desired for reasonable communication. The dominant frequencies composing the cabin

noise resulting from the SA330 Puma gearbox and Lynx gearbox were examined. They

were found to range from 500 – 5000 Hz (Figure 1.4). The researchers note, however,

that the important, acoustically subjective frequencies range from 500 – 2000 Hz.

7

A number of research efforts have focused on predicting helicopter interior noise

levels using Statistical Energy Analysis (SEA) [13-20]. The SEA approach evaluates the

power flow between mechanical structures and/or acoustical spaces. These efforts were

motivated by a need for comprehensive analytic models of the entire aircraft to evaluate

potential noise control measures. One important conclusion should be noted in particular.

Yoerkie, et al., and Morgan, et al., report that using the SEA model, the most efficient

means of reducing noise can be achieved with high frequency vibration isolation between

the gearbox and fuselage [13, 18].

Figure 1.4: Lynx Gearbox Noise Identification [12].

8

Many helicopter transmissions are connected to the fuselage via cylindrical

struts, as in Figure 1.1. Consequently, much research has been devoted to studying the

mechanisms of noise transmission through these support struts. Brennan et al., have

conducted a one-dimensional analysis using the mechanical impedance method, which

allows for the strut to be characterized by two parameters at each end: complex force and

velocity [21]. Both the lateral and longitudinal vibrations through the strut were

examined separately. The longitudinal vibrations are reported to be dominant at low

frequencies, but lateral vibrations become increasingly important at higher frequencies.

Throughout the whole frequency range examined (0 – 10 kHz), however, longitudinal

vibrations appear to have a larger influence on transmitted force than lateral vibrations.

The effectiveness of a 1 mm thick layer of elastomer for vibration isolation was also

examined. The shear stiffness of the elastomeric layer is much lower than the axial

stiffness because of the shape factor stiffness effect. Although it may be possible to

attenuate high frequency flexural vibrations, the authors conclude that a 1 mm thick layer

of elastomer is not an effective isolation treatment for longitudinal vibrations.

Brennan et al. continued this research with an experimental study of the noise

propagation through helicopter support struts [22]. Two simple analytical models of an

EH101 helicopter gearbox strut were first developed. Simulations from the analytical

models were then compared to experimental data, and the main features of the dynamic

behavior of the strut were described with these models. The contributions of

longitudinal, lateral, and torsional vibrations through the strut were ranked based on the

amount of kinetic energy transferred to a receiving structure. The strut was excited so

9

that each of the motions was excited with an equal source strength. The dominant

motion was found to be longitudinal, although lateral and torsional motions were found to

be important at certain flexural resonance frequencies of the strut.

Another analytical and experimental study was performed by Ohlrich using a ¾

scale model of a medium sized helicopter, the BK 117 [23]. In this research effort, the

goal was to determine a suitable source descriptor defined by a set of terminal source

powers which described the strength of the vibrations transmitted from the gearbox to the

fuselage. The source descriptor method is based on the concept of equivalent sources,

which assumes that a vibrating source can be adequately represented by the complex

vibratory power produced by a set of uncorrelated, equivalent point forces. An important

conclusion of this work is that the power transmitted to the fuselage is dominated by axial

vibrations through the struts.

1.2.2 Rotorcraft Interior Noise Control

1.2.2.1 Passive Treatments

A general methodology of helicopter soundproofing was presented by Marze, et

al., which can be applied to aircraft in general [24]. The methodology consists of three

rational steps: diagnosis, design of soundproofing treatments to cabin to obtain desired

noise reductions, and validation and optimization.

10

In general, nearly all helicopter manufacturers have employed this

methodology to apply soundproofing treatments to their aircraft. These approaches,

however, do not attempt to eliminate the noise at its source (gear meshing), nor do they

attempt to acoustically isolate the fuselage from the source of the vibration. Rather,

additional researchers describe add-on soundproofing methods which inherently carry a

considerable weight penalty [16, 24-26]. Marze et al., note that VIP versions of the

Aerospatiale Dauphin helicopter include soundproofing treatments which are 2-3% gross

weight [24]. Owen et al., propose the construction of an inner cabin composed of foam

and lead walls which can impose a weight penalty of 1-3% gross weight of the Westland

Lynx helicopter [26].

Levine notes that as gearbox technology has advanced, the noise generated at a

given horsepower has increased [25]. The result has been that increase in soundproofing

weight has more than offset savings in gearbox weight (see Figure 1.5).

Figure 1.5: Growth in Combined Gearbox and Soundproofing Weight at Constant Power,Constant Design Noise Level [25].

0

500

1000

1500

2000

1960 1970 1980 1990Design Calendar Year

Wei

ght (

lbs)

Gearbox PlusSoundproofing

Gearbox Only

11

Directly quoting Levine: “Main gearbox isolation from the airframe at acoustic

frequencies provides the most weight efficient means of source noise control… This

eliminates the need for heavy soundproofing treatment over large radiating areas. The

added complexity of rotor controls and engine mounting has limited the incorporation of

this approach into helicopter designs to date, but the economics of large-scale helicopter

market penetration will soon force the issue.”

As installed horsepower grows and structural materials become lighter, the

problem of noise increases. To avoid a high weight penalty involved with passive

treatments, many active approaches have been considered and some implemented to

reduce acoustic vibrations / noise.

1.2.2.2 Active Noise Cancellation

An active noise control approach which has received a notable amount of

attention is the use of loudspeakers to cancel sound waves in an enclosure, or active noise

control (ANC) [27-29]. Efforts are made to control cabin noise using a number of

microphones and speakers placed throughout the aircraft cabin. ANC requires no

knowledge of the noise transmission path, but does require in some approaches that the

number of speakers be at least equal to the number of acoustic modes. At higher

frequencies (> 200 Hz), several hundred modes exist at a given frequency [30]. This type

12

of noise control approach is therefore best suited for lower frequencies and becomes

impractical at higher frequencies.

1.2.2.3 Active Structural Vibration Control

Considerable research has focused on developing the active structural acoustic

control (ASAC) approach for interior noise control at frequencies below 500 Hz [31-34,

38]. This approach uses structural actuators and sensors optimally placed on the fuselage

to minimize overall interior noise. As with other active noise control techniques, ASAC

requires a large number of control sources to provide sufficient global sound reduction

over a wide frequency range.

O’Connell, et al., developed two ASAC systems using several small piezoelectric

patches bonded directly to the structure to cancel interior noise of a MD 900 Explorer

helicopter [36]. The authors report tonal noise reductions of 3 to 5 dB in the passenger

cabin with both systems at frequencies up to approximately 1 kHz. A total of 16

actuators and 16 microphones were employed. Fuller, et al., and Sun, et al., investigated

the use of piezoelectric patches to control interior noise in uniform cylindrical shells [37,

38]. A global noise reduction of 10 dB is reported by Fuller, and 20 dB by Sun.

13

1.2.2.4 Minimum Noise Transmission from Gearbox

Much research has been devoted to canceling high frequency vibrations before

they enter the cabin. At Sikorsky, Active Noise Control (ANC, distinct from Active

Noise Cancellation) has proven to be an effective method of actively controlling gearbox

noise [30]. ANC involves a choke-point methodology, where actuators are placed at the

gearbox connection points to the fuselage (Figure 1.6).

When flight tested in a S-76 helicopter, the ANC system reduced the primary

mesh tone (~800 Hz) by 10-20 dB in a variety of flight conditions. The authors also

report that passive choke-point isolation techniques were also investigated at Sikorsky,

such as elastomeric isolators and vibration isolators. Though the elastomeric isolators

showed promise, they raised certification issues since they were placed directly in the

primary load path. Additionally, they would have a significant impact on many aircraft

system design considerations. The tuned dynamic absorbers were ineffective because of

their narrow operating frequency range and limited effectiveness.

Figure 1.6: Schematic of ANC Architecture [30].

14

Many research efforts are focused on actively canceling high frequency

vibrations as they are transmitted through rigid support struts, as in Figure 1.1. The wide

array of active control concepts to control noise through support struts reflects the

importance placed on reducing high frequency vibrations.

In Germany, Eurocopter Deutschland and the EADS research and technology

group have developed the ‘smart strut’ [39, 40]. The smart strut consists of a

conventional BK117 transmission support strut with piezoelectric patches axially bonded

to the exterior. By actively applying shear forces to the strut surface, the control

algorithm attempts to cancel transmitted high frequency noise. The authors report that an

11 dB reduction at gear-meshing frequency of 1900 Hz was achieved for forward flight

of 60 kt. Other gear meshing frequencies, however, were unattenuated. The high noise

reduction is not attainable for all flight conditions, however. As forward flight speed

increases, the active system becomes less effective. This performance degradation is

reported to be caused by limited actuator performance with increasing vibration levels.

Current work is focused on improving actuator performance via optimized actuator

design.

In England, researchers at the Institute of Sound and Vibration Research at the

university of Southhampton, and researchers at Westland Helicopters, have characterized

the strut transmission problem and have attempted to develop active control strategies

[41]. An EH101 support strut was set up in the laboratory under realistic loading

conditions and three magnetostrictive actuators were clamped around its circumference at

a certain length along the strut. The purpose of the actuators was to introduce secondary

15

vibration in the frequency range of 250 – 1250 Hz to minimize the kinetic energy of

vibration of the receiving structure. This attenuation was calculated with knowledge of

measured frequency response data of the strut and actuators. A calculated attenuation of

around 40 dB in the kinetic energy was experimentally observed at some discrete

frequencies, which did not necessarily correspond to gear meshing frequencies. The

control system was found to be practical at frequencies up to at least 1250 Hz. The

authors concede, however, that two endplate masses supporting the strut strongly affected

the dynamic response of the complete experimental assembly, and that boundary

conditions experienced by a strut under flight conditions would be completely different.

At the University of Maryland, Balachandran, Pelinescu, et al., have investigated

both longitudinal and flexural wave transmission through support struts and active

control strategies, as well [42-48]. The control configuration consists of either a

magnetostrictive actuator or piezoelectric stack attached to the end of a support strut, or

in some cases, clamped at an angle along the length of the strut. Reaction mass is

included in both actuator configurations. For harmonic longitudinal disturbances, an

experimental reduction of up to 30 dB of the transmitted disturbance through the strut

was achieved using two different piezoelectric configurations, whereas only up to 16 dB

reduction was achieved using one magnetostrictive actuator. Current and future work is

focusing on digital implementation of a closed loop control algorithm. The authors note,

however, that in a practical situation, availability of required electrical power, actuator

and sensor bandwidths, and actuator heating effects are major constraints.

16

1.2.3 Periodically Layered Media for High Frequency Isolation

In 1986, Sackman and his associates reported that periodically-layered metallic

and elastomeric shear mountings are potential attenuators of dynamic stresses at high

frequencies [5, 6]. The impedance difference between layers is the attenuation

mechanism, in which an incident wave is scattered and essentially split into a reflected

and refracted wave. The device becomes increasingly effective with a larger impedance

mismatch between the isolator materials.

A one-dimensional analysis of periodically-layered isolators in compression

(Figure 1.1) was presented by Ghosh in 1985 [4]. Motivation for the research effort was

isolation of reactor components and structures from seismic, impact, or other accident-

induced loads. A time-domain solution was obtained for plane stress excitation through

layered composites, which makes use of continuity of stress and displacement at the layer

interfaces. Plane longitudinal stress waves are attenuated in periodically-layered elastic

mounts, whereas no attenuation is exhibited by an undamped homogeneous elastic

medium.

A one-dimensional analysis of layered isolators, based on the theory of shear

waves in infinite, periodically layered media is presented by Sackman, et al. [4, 5].

Floquet theory was used to solve the equations for the propagation of plane waves

through a laminated system of parallel plates. The direction of propagation is normal to

the plates, which are composed of one of two materials. The theory predicts high

frequency “stop bands” within which vibratory energy is attenuated. The analysis

includes a method for predicting the beginning and end frequencies of stop bands. Thus,

17

the layered isolator behaves in some sense as a mechanical notch filter. The existence

of the predicted stop bands was corroborated by testing of layered specimens in shear.

The test specimens were of finite length, and therefore edge effects and reflections from

the top and bottom layers were observed in the experiment. These effects, however, did

not obscure the basic physical phenomenon of stop bands.

The phenomenon of transmissibility stop bands occurring in periodic structures

has been known for over a century. Around the turn of the century, Lord Kelvin

proposed a “mechanical filter” to filter out vibrations at certain frequencies, which was

later experimentally validated by Vincent [49]. The filter consisted of discrete masses

connected by springs. The research efforts of Sackman, et al., and Ghosh are highlighted

because of their focus on elastomer and metal composites.

1.3 Research Objectives

The overall objective of the subject research effort was to develop new concepts

and design methods for periodically layered metal and elastomer isolators in

compression. To accomplish this objective, the three-dimensional elastic behavior of

layered isolators was investigated using both analytical simulation and experimental

testing. The next goal was to evaluate the feasibility of using layered isolators to reduce

noise and vibration transmitted into helicopter cabins by meshing transmission gears. A

major supposition of the research effort is that a choke-point vibration control

methodology can be employed, wherein all flight loads are transmitted through layered

18

elastomeric and metal isolators before they enter the cabin, as pictured in Figure 1.7.

The approach taken to reach these objectives was as follows:

1) An analytical method was developed to accurately predict isolator stop band

beginning and end frequencies. The method accurately captures the axisymmetric elastic

behavior of vibrating cylinders and accommodates varying geometry, elastomer shape

factor, and number of cells. In addition, the method calculates transmissibility as a

function of frequency.

2) Extensive experimental testing of layered specimens was performed to validate

the analysis method. Experimental and analytical transmissibilities of layered test

specimens having the identical elastomer, but different shape factors and different

numbers of cells were compared. The experimental and analytical transmissibilities of

geometrically-similar test specimens with differing elastomeric damping were also

compared.

Figure 1.7: Proposed Implementation of Layered Isolators in Gearbox Support Struts.(http://www.dnd.ca/menu/SAR/eng/cormorant/Tour/breakaway/main_gearbox.htm)

LayeredIsolators

Embedded inSupport Struts

19

3) The design constraints associated with the proposed gearbox high-frequency

isolator were determined. Important issues investigated included isolator mass, stiffness,

fatigue and geometry.

4) A design optimization methodology was developed to evaluate the passive

performance limits of periodically layered isolators to isolate helicopter gearboxes. The

design optimization methodology was evaluated using different combinations of design

constraints, such as restricted mass, axial stiffnesses and geometries.

5) As a result of passive performance limitations of layered isolators, design

enhancements were considered to improve isolator performance to meet isolation

objectives of the gearbox noise problem. Active, semi-active, and passive design

enhancements were all investigated to improve performance and tunability. A passive

enhancement in the form of embedded inertial amplifers was pursued and experimentally

validated.

6) A final step was the development of guidelines and computational tools for

gearbox isolator design to accommodate a variety of helicopter sizes and transmission

configurations. If layered composites prove feasible and attractive as gearbox isolators,

designers will need tools to begin practical evaluations.

20

The use of layered isolators as high frequency gearbox isolators would be a

new and novel approach. Although layered isolators in shear have been investigated in

the literature, layered elastomeric and metal isolators in compression have not been

modeled in any great detail. An axisymmetric approximation method allows for accurate

prediction of layered isolator stop band frequencies in compression. The design of

layered isolators in compression would allow for relatively high isolator quasi-static

stiffness, while ensuring low isolator stiffness over certain predicted stop band frequency

ranges. To provide tunability and improve performance, active, semi-active, and passive

enhancements to layered isolators were investigated. These efforts will yield new

insights into the state-of-the-art of high frequency isolation.

Chapter 2

PERIODICALLY LAYERED ISOLATORS FOR HIGH FREQUENCY

ISOLATION

Periodically layered elastomer and metal composites in shear are known to exhibit

high frequency stop bands in which transmitted vibrations are significantly attenuated [4,

5]. The utility of employing these isolators in compression was investigated, and an

analysis method was developed to predict their high frequency behavior. Finally, a series

of experiments were conducted to validate the analysis.

2.1 One Dimensional Finite Elements Analysis

A one-dimensional finite element analysis of layered isolators was developed for

comparison with the analytical method presented by Sackman, et al. [4, 5], who used

Floquet theory to predict high frequency stop bands. The beginning and end stop band

frequencies are referred to as fB and fE, respectively. Each layer of the layered isolators

was modeled using two-noded axial finite elements. Using the discretized equations of

22

motion, a frequency domain analysis was performed to determine transmissibility of the

multi-layered isolator.

As an example, consider an isolator with 3 steel and elastomer cells, and a cross-

sectional area of 5 cm2, in which both the steel and elastomer layers had a thickness of 1

cm. The material properties were Ee=5MPa, Em = 200 GPa, ρe = 1,000 kg/m3 and

ρm=7,800 kg/m3. The solution converged when each elastomer layer was modeled with

10 elements, and each steel layer with 3 elements. The transmissibility predictions from

both finite element analysis and Floquet theory are presented in Figure 2.1. Both

transmissibility results are identical within the stop band frequency range. Isolator

transmissibility is defined as the transmitted displacement amplitude, xt, divided by the

base input amplitude, xb, and can be written as

( )

( )t

b

xT

x

ωω

= ( 2.1 )

Three discrete isolator resonance peaks are evident from the finite element analysis,

which are not predicted in the analyses in [4] and [5].

This one-dimensional finite element analysis provided a clearer understanding of

layered isolator behavior in shear, but a more detailed finite element analysis needed to

be performed to better understand the three-dimensional behavior of layered isolators in

compression.

23

2.2 Three Dimensional Finite Element Analysis

A detailed finite element analysis of periodically layered isolators was conducted

to gain an improved understanding of three-dimensional effects on isolator performance

[1]. Mode shapes and isolator transmissibility were examined. Parabolic quadrilateral

axisymmetric elements were used to model each layer of the circular isolator [50]. Each

element had eight nodes and forty-eight degrees of freedom. All deflections and rotations

were constrained at the base of the isolator. Transmissibility was calculated in the

frequency domain by dividing the total axial reaction force (obtained using equilibrium

equations), Fb, by the total applied input force at the top of the unconstrained isolator

surface, Fi, and can be written as

Figure 2.1: Comparison of One-Dimensional Transmissibility Predictions for FEM andFloquet Theory for a 3-celled Isolator.

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

1.E+01

1.E+02

1.E+03

0 1000 2000 3000 4000Frequency (Hz)

Tra

nsm

issi

bili

ty

1D FEM

FloquetTheory

Stop Band

fB fE

IsolatorResonance

24

b

i

FT

F= ( 2.2 )

For a typical three-celled isolator, the first four mode shapes are illustrated in Figure 2.2.

For the first n modes, each elastomer layer associated with these frequencies

undergoes approximately uniform axial strain. In fact, the metal layers behave essentially

like discrete masses supported by n axial springs. Invariably, the first mode shows that in

each elastomer layer the strain is either all compression or all tension. In the next (n-1)

modes, the mode shapes of the individual layers are observed to contain different

combinations of layerwise compression and tension. The (n+1)th mode (mode 4 in

Figure 2.2) is the first mode in which each elastomer layer exhibits a ‘thickness’ mode.

Figure 2.2: First Four Mode Shapes of a Three-celled Isolator in Compression.

Mode 1 Mode 2 Mode 3 Mode 4

nth Mode (n+1)th Mode

Stop Band

25

Physically, this mode involves both tension and compression within the elastomer layer

and minimal net axial motion of the constraining metal layers. This mode is associated

with the end of the stop band frequency range.

The mode associated with the end frequency warrants closer examination.

Referencing Sackman’s analysis, z is the ratio of the input frequency to the natural

frequency of the first thickness mode. The thickness mode can be closely approximated

as the first mode under the condition that both ends of a single elastomer layer are

constrained. In the one-dimensional analysis, the natural frequency associated with the

first thickness mode of an elastomer layer, fT, also marks the end of the stop band, fE, and

is expressed as

2e

T Ee

cf f

t= = ( 2.3 )

where ce is the axial wave speed through the elastomer layer, and te is the elastomer layer

thickness. For a given isolator, the end frequency is then associated with the first

thickness mode of a single elastomer layer.

The one-dimensional and axisymmetric thickness modes for a single elastomer

layer are illustrated in Figure 2.3. Note, the one-dimensional model allows only axial

motion, whereas the axisymmetric model captures pronounced lateral motion, as well as

axial motion. Therefore, a discrepancy is expected between the natural frequency

predictions of the two models.

26

A design analysis for layered isolators requires accurate prediction of the highest

isolator natural frequency in which all layers exhibit uniform axial strain. Additionally,

the natural frequency associated with the first thickness mode of a single elastomer layer

must also be accurately predicted.

2.2.1 One Dimensional Stiffness Correction Factors

The first three modes in Figure 2.2 show the significant lateral motion of the

middle of each elastomer layer. This is due to fact that the upper and lower surfaces of

each layer are constrained and that elastomers are nearly incompressible. Consequently,

the effective three-dimensional stiffness of each layer is expected to differ from the

predicted one-dimensional stiffness, which only addresses axial motion.

A well-documented method to account for the difference between the effective

three-dimensional stiffness and the predicted one-dimensional stiffness is the use of a

one-dimensional stiffness correction factor, or shape factor. Essentially, the shape factor

accounts for the discrepancy between the predicted one-dimensional stiffness and the

Figure 2.3: First Thickness Modes Associated with the Stop Band End Frequency Using1-D and Axisymmetric Models.

One Dimension Axisymmetric

d/2 d/2

Tension

Compression

zz

27

measured effective one-dimensional stiffness of an elastomer isolator. Shape factor is

defined as the ratio of one bonded area to the force-free area of an elastomer isolator.

Expressions for shape factor for different geometries have been formulated both

empirically, and analytically [7, 8]. The total displacement of a bonded isolator can be

considered to arise from a superposition of two displacements [7]: (a) those caused by the

same deformation of an unbonded isolator (between frictionless rigid surfaces), and (b)

the shear displacements necessary to restore points in the bonded planes to their original

position. Gent and his co-authors provide a comprehensive discussion [7].

The one-dimensional quasi-static stiffness of a single elastomer layer, without

including bonding shear effects, is given as

1e e

De

E Ak

t− = ( 2.4 )

where Ee is the Young’s modulus, Ae is cross-sectional area, and te is the elastomer

thickness. To simplify the design of isolators, the one-dimensional stiffness is multiplied

by an empirically determined function of the shape factor to approximate the effective

isolator stiffness. This is illustrated in table 1 for rectangular and circular isolators.

28

The effective stiffness is then given as

21 (1 2 )eff Dk k Sφ−= + ( 2.5 )

where S is the shape factor, and φ is an empirically derived factor to account for

experimental deviations in the behavior of different elastomers [8]. To simplify the

discussion, it is assumed φ = 1.

The shape factor is used for the design of elastomeric mounts for low frequency

isolation. For a simple one-dimensional isolator / mass system shown in Figure 2.4,

where d is isolator diameter, the first natural frequency can be predicted with

eff

n

k

Mω = ( 2.6 )

Table 2.1: Shape Factor for Rectangular and Circular Isolators

d

t

w

l

t 2 2 2 ( )

lw lwS

tw tl t l w= =

+ +

2

44

dd

Sdt t

π

π= =

29

For the same system, Figure 2.5 shows the axisymmetric mode shape associated

with this natural frequency. This mode shape was determined using axisymmetric

elements [50].

Recalling Figure 2.2, this mode shape is similar to the shapes exhibited by the

individual elastomer layers for the first n isolator natural frequencies of an n-celled

isolator. The nth natural frequency is the beginning frequency. For the one dimensional

analysis, it would therefore appear that applying the shape factor to each layer of the

isolator is appropriate for predicting the first n natural frequencies in three-dimensions.

Figure 2.4: Simple Isolator / Mass System in One Dimension.

Figure 2.5: Simple Isolator / Mass System Modeled with Axisymmetric Elements.

keff

Mωn

d

M

z

ωn

d/2

30

2.2.1.1 Shape Factor Influence on Higher Modes

To investigate the validity of this assumption, a typical three-celled layered

isolator with a shape factor equal to 1 was modeled using axisymmetric elements [50].

The isolator had the following properties: n=3, d = 10 cm, Ee = 15 MPa, Em = 200 GPa,

ρe=1,000 kg/m3, ρm = 7,800 kg/m3, te=2.5 cm, and tm=1.0 cm.

The elastomer layers were each modeled with five parabolic quadrilateral

elements across te, and five elements across d/2. The steel layers were modeled with 1

element across tm, and five elements across d/2.

The first 5 natural frequencies were then compared to the natural frequencies

resulting from a one-dimensional analysis. A 3-D to 1-D stiffness ratio, rS, was then

calculated using the formula

223 3

2

1 1

D DF

D D

S

k fr r

k f− −

− −

= = = ( 2.7 )

where rF is the 3-D to 1-D frequency ratio. Table 2.2 shows the isolator properties and

results. As expected, the ratio for the first mode is approximately equal to 3. The ratios

for subsequent natural frequencies show a decreasing trend. The third mode, associated

with the beginning frequency, shows a ratio of 1.93. The decreasing ratio values suggest

that the quasi-static shape factor cannot be directly utilized for accurate prediction of the

nth isolator natural frequency, or the beginning frequency.

31

A similar investigation was also performed for the stiffness ratio of the (n+1)th

isolator natural frequency, or the end frequency. This natural frequency was calculated

using both axisymmetric FEM and the one-dimensional prediction method for different

elastomer layer geometries, or different shape factors. The isolator had the same

properties as the previous investigation, except for te, which varied to account for

different shape factors.

Table 2.3 shows the stiffness ratios for the different cases. As expected, the

smaller the shape factor, the closer the stiffness ratio comes to unity. As shape factor

increases, however, the stiffness ratio decreases. These trends are illustrated in Figure

2.6.

Table 2.2: (3-D/1-D) Stiffness Ratios for Different Modes of a 3-celled Layered Isolator.

1 177 306 2.992 506 786 2.41

1 3 3 747 1037 1.934 2493 2409 0.935 2596 2412 0.866 2713 2415 0.79

ShapeFactor

(1+2S2) Mode 3-D fn.1-D fn rS

32

Table 2.3: (3-D/1-D) Stiffness Ratios for End Frequency of 3-celled Isolators.

Figure 2.6: (rS)E vs. Shape Factor.

0.5 1224 1320.7 1.16 0.75 1839 2002 1.19 1 2449 2652 1.17 2 4899 5180 1.12 3 7351 7334 1.00 4 9798 9407 0.92 5 12247 11329 0.86 6 14685 13235 0.81 7 17153 15169 0.78 8 19596 17092 0.76

(rS)EShapeFactor 1-D 3-D(Hz) (Hz)

0.6

0.7

0.8

0.9

1.0

1.1

1.2

1.3

0 1 2 3 4 5 6 7 8 9Shape Factor

rE(rS)E

33

2.2.2 Table Look-up Methodology

A preliminary method was developed to accurately predict beginning and end

frequencies in three dimensions. This was accomplished by mapping out a design space

with the following discrete design variables: shape factor, S, number of layers, n, and

isolator diameter, d. This forms a three-dimensional design space that is illustrated

graphically in Figure 2.7. A look-up method approach was used so that computationally

expensive axisymmetric finite element modeling could be avoided in future design

optimization routines.

For given isolator properties, each ‘node’ in this space contains the 3-D to 1-D

frequency ratios for both the beginning and end frequencies, expressed as

( ) 3

1

D

F B

D B

fr

f

−

−

=� ��

, ( ) 3

1

D

F E

D E

fr

f

−

−

=� ��

( 2.8 )

Figure 2.7: 1-D /3-D Design Space for (rF)B and (rF)E.

12

34 2 4 6 8

5 cm

10 cm

20 cm

34

56

78

Dia

met

erLay

ers

Shape Factor

34

The 1-D frequencies are calculated using the method described in [5] and [6], and

the 3-D frequencies were calculated using axisymmetric elements [50]. Based on 1-D

calculations, the 3-D modal behavior is then linearly interpolated.

To accurately capture axisymmetric behavior, however, this approach requires

numerous axisymmetric FEM solutions for the entire design space of possible isolator

geometries. A more elegant method to predict isolator behavior was therefore pursued.

2.3 Axisymmetric Approximation Method

A more accurate and general method was developed for predicting the

axisymmetric behavior of layered isolators and is summarized here [2]. The method

combines a Ritz approximation method with a component mode synthesis technique. For

given isolator geometry, material properties, and number of cells, the axisymmetric stop

band frequencies and the frequency-dependent transmissibility are calculated.

2.3.1 Natural Frequency of 3-D Cylinders of Finite Length

A closed-form solution for natural modes of a vibrating cylinder is

mathematically very difficult to obtain. Thus, several approximation methods have been

pursued. Heyliger presents a technique for calculating the natural frequencies of the

axisymmetric vibrations of anisotropic and isotropic cylinders of finite length [51]. This

35

method is general and can be applied to cylinders having a variety of boundary conditions

and material properties.

The displacement components that describe the axisymmetric motion of an elastic

cylinder can be expressed as

( , )ru U r z= , ( , )zw U r z= ( 2.9 )

where u and w are independent displacements in the radial and axial directions,

respectively. The strain components for axisymmetric motion can be written in

cylindrical coordinates as

,1 r

urr ∂

∂== εε 2 ,u

rθθε ε= = ,3 z

wzz ∂

∂== εε z

u

r

wrz ∂

∂+∂∂== εε5 ( 2.10 )

The reader is referred to Heyliger’s analysis for a detailed derivation of the

equations of motion and possible boundary conditions [51]. A summary of the derivation

is presented in Appendix A for completeness. Analytical solutions to the equations of

motion are difficult to obtain, and thus approximate solutions for the governing equations

of motion are constructed by using the assumed modes method, in which u and w are

approximated by finite linear combinations of the form

1

( , ) ( , )N

uk k

k

u r z b r zφ=

=� , 1

( , ) ( , )M

wk k

k

w r z d r zφ=

=� ( 2.11 )

Here φuk and φw

k are known functions of position, n represents the number of terms for

the displacement components, and bk and dk are constants.

36

Selection of the approximating functions, φ, is somewhat arbitrary [51]. Several

requirements must be met, however, to guarantee that the approximations will converge

to the exact solution. The functions, φuk and φw

k, must meet the requirement of continuity

as required by the variational statement, they must satisfy the homogeneous form of the

essential, geometric boundary conditions, and they must be linearly independent and

complete [51]. Natural boundary conditions are contained in the variational statement of

the problem, and need not be explicitly satisfied.

In the current work, a single elastomer/metal cell is first modeled in a fixed-free

condition. Multiple cell analyses will subsequently be combined in a complete isolator

analysis. For numerical calculations, it is convenient to non-dimensionalize the cylinder

geometry by mapping the original cylinder to a cylinder with a radius and half-height of 1

using the transformations R = r / Lr, and Z = z / Lz. Here, Lr is the cylinder radius and Lz

is the cylinder half-height (see Figure 2.8). At the bottom of the elastomer layer, the

non-dimensional height is z = -1. The displacement at z = -1 of the elastomer layer is

fixed at u = 0 and w = 0, as in Figure 2.8. The metal layer, attached to the top of the

elastomer layer at the non-dimensional height of z = +1, is modeled as a plane mass and

is rigid and infinitely thin. Therefore, at z = 1, u = 0 and w is free, and the rigid plate

translates vertically while remaining horizontal. This boundary condition effectively

restricts any rocking motion of the top plate. The metal layer is modeled as a plane mass

because all the strain energy is assumed to be in the elastomer layer.

37

Several different sets of functions could be selected to obtain an approximate

solution. In this work, they must satisfy the boundary conditions u = 0, w = 0 at z = -1,

and imposed conditions u = 0, w unconstrained and uniform at z = 1. A series of

approximating functions were developed that consist of power series in the r and z

directions and satisfy the preceding conditions. The functions for the radial direction can

be summarized as

1 0

( , ) ( 1)( 1)n m

u i jk

i j

r z r z z zφ= =

= + −�� , 1..[ ( 1)]k n m= × + ( 2.12 )

where n and m are chosen to vary the maximum power of variables, ‘r’ and ‘z’. All

approximating functions in the u direction must include an ‘r’ term because they describe

an axisymmetric displacement. In the axial direction the functions are formed slightly

differently,

1 ( 1),w zφ = + 0 0

( , ) ( 1)( 1) ( 1)s t

w i jk

i j

r z r z z z zφ= =

� �= + − + +� �� ,

2..[( 1) ( 1)]k s t= + × +

( 2.13 )

Figure 2.8: Illustration of Fixed-Free Boundary Condition for a Single Cell.

r

z

z = -1

z = +1

r = 1r = 0

z = 0

Rigid, Thin MetalLayer

38

where again, s and t are chosen to vary the maximum power of variables ‘r’ and ‘z’ . All

terms for the w direction need not include an ‘r’ term. However, any term multiplied by

‘r’ is forced to zero at z = +1 to satisfy the imposed boundary condition of uniform axial

displacement of the top plate. Alternatively, the displacement in the w direction can only

have a radial dependence from –1 < z < +1. Substitution of approximation functions in

Equation 2.11 into the weak form of the governing equations yields an eigenvalue

problem of the form

[ ] [ ]( )2 { } {0}K M pρω− = ( 2.14 )

The vector {p} contains the coefficients b and d from Equation 2.11 which correspond to

different cylinder mode shapes. The explicit forms of matrices [M] and [K] in terms of φu

and φw are reported by Heyliger [51], and can be found in Appendix B.

2.3.2 Numerical Results for a Single Cell

Numerical simulations were performed for a single cell to compare the natural

frequency predictions using the Ritz method to axisymmetric finite element results. The

first and second modes were examined, because these modes are the component mode

shapes important to the nth and (n+1)th isolator modes, respectively. An n-celled isolator

is illustrated in Figure 2.2. The boundary conditions were fixed-free, as illustrated in

Figure 2.8. The general material properties used were: Gm = 100 Pa, νm = 0.3, ρm = 10

39

kg/m2, Ge = 1 Pa, νe = 0.5, ρe = 1 kg/m3, te = 1 m, and dcell = 1 m. Note that the units for

ρmetal are mass per unit area, as appropriate for a vanishingly-thin metal layer.

The axisymmetric finite element model consisted of parabolic quadrilateral

axisymmetric elements, with 20 elements across the radius, and 20 elements through the

thickness of the elastomer layer. The strain in the metal layer is insignificant, and thus

only 5 elements are used across its thickness. This number of elements was found to be

sufficient for convergence over the frequency range of interest. The frequencies

calculated using the finite element analysis were

fn1 = 0.094 Hz

fn2 = 0.917 Hz

These values are used in subsequent comparisons to the Ritz predictions as baseline

cases. Note that the material properties and isolator dimensions are arbitrary for this

case, so that these frequencies do not have a physical significance related to layered

isolators. This case is only used to validate the mathematics of the linear approximation

method.

The first and second natural frequencies calculated with the Ritz method are

presented in Table 2.4. A percent error is also calculated with reference to the

axisymmetric finite element predictions. Combinations of the powers of the ‘r’ and ‘z’

polynomials are used in the approximating functions for both Ur and Uz. To simplify the

tabular results, the power of ‘r’ in both the Ur and Uz approximating functions is the same

for a given result. Similarly, the power of ‘z’ is the same in both the radial and vertical

approximating functions. As can be observed from Equation 2.12, the lowest possible

40

power of ‘z’ in the radial direction is 2. This is therefore the lowest power used for both

directions.

As seen in Table 2.4, as the powers of ‘r’ and ‘z’ increase, the Ritz predictions

appear to converge to the axisymmetric finite element results. Table 2.5 lists the rates of

convergence for both methods. The first and second natural frequencies of a single cell

are computed for different number degrees of freedom in the elastomer layer. The Ritz

method is observed to be significantly more accurate per degree of freedom.

Table 2.4: Single Cell Natural Frequencies Using Assumed Modes Method Compared to3-D FEM Results.

Table 2.5: Rate of Convergence Comparison for a Single Cell.

zp p 2 3 4 5

(Hz) % Error (Hz) % Error (Hz) % Error (Hz) % Errorrq q

fn1 0.4594 388 0.0983 4.5 0.0982 4.4 0.0959 2.01

fn2 11.2542 1128 5.6200 513 0.9745 6.3 0.9744 6.3

fn1 0.4594 388 0.0982 4.5 0.0981 4.4 0.0951 1.22

fn2 11.2542 1128 5.4220 491 0.9379 2.3 0.9377 2.3

fn1 0.4594 388 0.0982 4.5 0.0981 4.3 0.0950 1.13

fn2 11.2472 1127 5.4211 491 0.9338 1.9 0.9335 1.8

Ritz Method Axisymmetric FEMTotal DOF fn1 (Hz) fn2 (Hz) Total DOF fn1 (Hz) fn2 (Hz)

3 0.4594 11.254 10 0.4576 6.76110 0.0983 5.6200 32 0.1041 4.06821 0.0981 0.9379 66 0.1002 1.00536 0.0950 0.9335 170 0.0967 0.953655 0.0943 0.9179 640 0.0948 0.926978 0.0942 0.9177 2480 0.0940 0.9168

Approximation Method

41

For computational efficiency, the size of the eigenvalue problem should not be

excessively large. Therefore, a frequency prediction with an error of 5% percent or less

is deemed acceptably accurate for the current analysis. The case with the lowest powers

of ‘r’ and ‘z’ where this occurs for both modes 1 and 2, is a power of 2 for ‘r’, and a

power of 4 for ‘z’.

To further reduce the size of [M] and [K] in Equation 2.14, higher-order

approximating functions can be removed from both Ur and Uz. The powers of ‘r’ and ‘z’

in Table 2.6 were found to be sufficient to maintain at most a 5% error for the first and

second natural frequencies.

The approximate value of the first natural frequency was in error by 4.5%.

Likewise, the prediction of the second frequency was in error by 2.33%. In Figure 2.9,

the lower-order mode shapes appear to be reasonably good approximations of the higher-

order mode shapes.

Table 2.6: Reduced Powers of ‘r’ and ‘z’ for Ur and Uz to Approximate Beginning andEnd Stop Band Frequencies to within 5% Error.

Power

Ur r = 2, z = 3 n = 2, m = 1 in equation (4)Uz r = 1, z =4 s = 1, t = 2 in equation (5)

42

2.3.3 Analysis of Layered Isolators Using Component Mode Method

A method was developed by Hurty for analyzing complex structures that can be

divided into interconnected components [52]. For this work, a single component is

considered to be a combination of an elastomer and metal layer, or a cell in a periodically

layered isolator. All strain is considered to be in the elastomer portion of the cell.

Displacements of each cell are expressed in terms of generalized coordinates, {p}, and

are defined by assumed displacement modes [51]. In this work, the assumed modes take

the form of Equations. 2.12 and 2.13 and include a rigid body mode in the z direction

with constant displacement. This mode allows for rigid body motion of cells when

interconnected.

When continuity conditions are imposed at cell boundaries, a set of constraint

equations results which expresses kinematic relationships among the coordinates

Figure 2.9: Higher and Lower Approximations for Modes 1 and 2 of a Single Cell.

Mode 1 Mode 2 Mode 1 Mode 2

Higher Order Lower Order

UrPower of ‘r’: 3Power of ‘z’: 5 Uz

Power of ‘r’: 3Power of ‘z’: 5 Ur

Power of ‘r’: 2Power of ‘z’: 3 Uz

Power of ‘r’: 1Power of ‘z’: 4

43

associated with different cells [52]. These constraint equations are used to determine a

set of overall system (isolator) generalized coordinates equal to the total number of cell

coordinates minus the number of constraint equations. The relationship between the sets

of cell generalized coordinates and the set of isolator generalized coordinates is expressed

in the transformation matrix, [β]. Isolator mass, stiffness, and damping matrices are

obtained through this transformation. Forces on component cells are also transformed

into total system forces in this way. A set of equations of motion for the entire isolator

results.

The procedure used to obtain isolator equations of motion, as presented by Hurty

[52], is summarized here. The equation of motion for the sth cell of the isolator can be

expressed as

[ ] { } [ ] { } [ ] { } { ( )}s s s s s s sm p c p k p P t+ + =�� ( 2.15 )

where

{ } ,{ } ,{ }s s sp p p� �� = column vectors of cell generalized displacements, velocities, and

accelerations

[ ] ,[ ] ,[ ]s s sm c k = square matrices of cell generalized masses, damping, and stiffnesses.

{ ( )}sP t = column vector of generalized forces applied to the sth cell. These

include forces transmitted through constraints as well as externally

applied forces [52].

44

Using Equation 2.15, equations are written for all isolator cells. The sets of cell

equations of motion are grouped together in matrix form to create a total isolator set of

equations:

[ ]{ } [ ]{ } [ ]{ } { ( )}m p c p k p P t+ + =�� ( 2.16 )

When forming [m], [c], and [k], it is desirable to group cell generalized coordinates

together, as in

1

2

{ }{ }

.

.{ }

{ }{ }

.

.

r

s

pp

ppp

� �� = � ��

,

1

2

{ }( ){ }( )

.

.{ ( )}

{ }( ){ }( )

.

.

r

s

P tP t

P tP tP t

� �� = � ��

( 2.17 )

Grouped in this way, the mass matrix takes the form

1

2

[ ] . . . . . . 0. [ ] .. .. . .

[ ]. [ ] .. [ ] .. . .0 . . . . . . .

r

s

mm

mm

m

� �� =� ��

( 2.18 )

45

The damping and stiffness matrices take a similar form [52]. Equation 2.16 can be

considered a group of unconnected sets of cell equations of motion. When displacement

constraints are imposed at cell boundaries, a set of constraint equations results among the

elements of {p}. If there are m elements in vector {p}, and k constraint equations relating

them, then there will be n = m - k independent coordinates in the isolator equations of

motion. This independent set of isolator coordinates is designated {q}, and is directly

related to {p} through a linear transformation. The transformation can be derived such

that

{ } [ ]{ }p qβ= ( 2.19 )

The transformation matrix, [β], has dimensions m × n where m > n. The construction of

matrix [β] can be completed with knowledge of the displacement constraints among the

isolator cells. Suppose that a displacement constraint exists between cell r and s, such that

( , 1) ( , 1)r su r z u r z= − = = ( 2.20 )

If all displacement constraints between cells are written in terms of vector {p}, then the

entire set of constraints can be written in matrix form as

[ ]{ } {0}A p = ( 2.21 )

46

where [A] is a rectangular matrix with dimensions k × m. Because m > k, [A] may be

partitioned as

1 2[ ] [ ]A A A= ( 2.22 )

where [A1] is a square matrix with dimensions k × k. Equation 2.21 can then be

rewritten as

1 2[ ]{ } [ ]{ } {0}d fA p A p+ = ( 2.23 )

Here, {p}d and {p}f are subsets of {p} and are the dependent and independent variables,

respectively. The subsets must be chosen such that matrix [A1] is nonsingular, or

invertible [52]. The dependent variables can then be explicitly expressed in terms of the

independent variables as

11 2{ } [ ] [ ]{ }d fp A A p−= − ( 2.24 )

From this, the relationship between the entire set isolator variables, {p} and the

independent set, {p}f can be derived as

11 2

[ ]{ } { }

[ ] [ ]f

fd

p Ip p

p A A−

� � � �= =� � � �−� �

( 2.25 )

Equation 2.25 can be rewritten, and thus the transformation can be stated as

47

{ } [ ]{ } fp pβ= ( 2.26 )

Substituting Equation 2.26 into Equation 2.16 and premultiplying all terms by [β]T, the

isolator equations of motion can be stated:

[ ] [ ][ ]{ } [ ] [ ][ ]{ }T Tf fm p c pβ β β β+�� [ ] [ ][ ]{ } [ ] { ( )}T T

fk p P tβ β β+ = ( 2.27 )

To perform this substitution, the vector {p} has been arranged so that all dependent

variables are below the independent variables. Therefore, the rows and columns of the

original matrices [m], [c], [k] and rows of vector {P(t)} must be rearranged accordingly.

The following identities can be defined [52]

[ ] [ ] [ ][ ]TM mβ β= ( 2.28 )

[ ] [ ] [ ][ ]TC cβ β= ( 2.29 )

[ ] [ ] [ ][ ]TK kβ β= ( 2.30 )

{ ( )} [ ] { ( )}TQ t P tβ= ( 2.31 )

An eigenvalue analysis can then be performed using the system matrices [M] and

[K] to obtain the nth and (n+1)th natural frequencies. These correspond to the beginning

and end stop band frequencies.

48

2.4 Experimental Validation

Experimental tests were performed for various layered test specimens to verify

the analytical prediction method. The experiments were performed by attaching a given

specimen to a rigid base, which in turn was attached to a mechanical shaker. The shaker

input was a series of chirp signals each spanning 400 Hz. To measure the transmissibility

of a test specimen, one accelerometer was placed on top of the specimen, and another

was placed on the rigid base. The signals were then fed into a Fourier analyzer. In this

way, the specimen transmissibilities were directly measured. The experimental set-up is

pictured in Figure 2.10 .

The first specimen initially consisted of 4 cells. The elastomer was a lightly

damped material and the metal layer was steel. The elastomer material properties were

Figure 2.10: Experimental Set-up.

SignalConditioner

HPAnalyzer

Shaker Amplifier

LayeredSpecimen

49

Ge = 0.6 MPa, and ρe = 1,000 kg/m3. The elastomer was assumed to be incompressible,

which corresponds to a Poisson’s ratio of ν = 0.5. To avoid mathematical singularities,

the Poisson’s ratio was approximated at ν = 0.499. The specimen geometry was te = 1

cm, tm = 1 cm, and d = 4 cm. This geometry corresponds to an elastomer shape factor of

1. The important specimen properties are summarized in Table 2.7.

In Figure 2.11, the experimental and analytical transmissibilities are plotted for

Specimen 1 with four layers. The elastomer had not been characterized at high

frequencies, and thus an initial estimate of the shear modulus for analytical predictions

was obtained from initial low frequency characterization. The shear modulus was then

adjusted so that the analytical plot matched the experimental results near the beginning of

the stop band. Similarly, the loss factor was estimated at 0.05 by matching resonance

peak height. To validate the analytical method, the resulting material property values

were to be used to predict transmissibilities of an additional specimen composed of the

same elastomer, but having a different geometry. Although the model accommodates

frequency-dependent material properties, using constant values for the test frequency

range was adequate to validate the analytical method.

Table 2.7: Summary of Specimen Properties

Specimen 1 Specimen 2 Specimen 3

d (cm) 4.0 2.54 4.0telas (cm) 1.0 1.0 1.0tsteel (cm) 1.0 0.64 1.0

Gelas (MPa) 0.6 0.6 3.06Loss Factor (η) 0.05 0.05 0.15

50

Figure 2.11: Experimental and Analytical Transmissibilities for Specimen 1 with 4 Cells.

.


1.E-111.E-101.E-091.E-081.E-071.E-061.E-051.E-041.E-031.E-021.E-011.E+001.E+011.E+02

10 100 1000 10000Frequency (Hz)

Noise FloorT

rans

mis

sibi

lity

ExperimentalAnalytical

1.E-09

1.E-08

1.E-07

1.E-06

1.E-05

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

1.E+01

1.E+02

10 100 1000 10000Frequency (Hz)

Noise Floor

Tra

nsm

issi

bili

ty


51


Figure 2.14: Experimental and Analytical Transmissibilities for Specimen 1 with 1 Cell.

1.E-06

1.E-05

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

1.E+01

1.E+02

10 100 1000 10000Frequency (Hz)

Noise FloorTra

nsm

issi

bil i

ty


1.E-03

1.E-02

1.E-01

1.E+00

1.E+01

1.E+02

10 100 1000 10000Frequency (Hz)

Tra

nsm

issi

bili

ty


52

In Figures 2.11-2.14 , experimental and analytical results are shown for Specimen

1 with 4, 3, 2, and 1 cells. For all cases, the first n resonance peaks are nearly coincident.

The stop band locations are also accurately predicted. For the 4, 3, and 2 cell cases, a

discrepancy exists between experiment and the analytical predictions for stop band depth.

This discrepancy may be because of the existence of an experimental noise floor, below

which transmissibilities cannot be accurately measured.

The second specimen tested consisted of 4 cells, as well. The elastomer used was

the same as in Specimen 1. The metal layer was again steel. The specimen geometry

was changed to te = 1 cm, tm = 0.64 cm, and d = 2.54 cm. This geometry corresponds to

an elastomer shape factor of 0.64. By changing the geometry of the second specimen, but

using the same elastomer as in Specimen 1, the analytical method was to be validated.


1.E-07

1.E-06

1.E-05

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

1.E+01

1.E+02

10 100 1000 10000Frequency (Hz)

Noise FloorTra

nsm

issi

bil i

ty


53

.


.


1.E-05

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

1.E+01

1.E+02

10 100 1000 10000Frequency (Hz)

Noise Floor

Tra

nsm

issi

bil i

ty


1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

1.E+01

1.E+02

10 100 1000 10000Frequency (Hz)

Noise Floor

Tra

nsm

issi

bili

ty


54

In Figures 2.15-2.18 experimental and analytical transmissibilities are plotted for

Specimen 2, with varying numbers of cells. For the analytical predictions, the same

value of shear modulus was used as determined for Specimen 1. Similar to Specimen 1,

the results for Specimen 2 show a discrepancy between experimental and analytical

results for stop band depth in the 4 cell case. However, the predictions for the 3, 2, and 1

cell cases are nearly colinear with experimental results over the entire frequency range.

In the transmissibility plots, a noise floor of 1.4 x 10-4 is shown. The noise floor

is calculated using the following relation:

min

max max

( )

( ) ( )top res

bot bot

a aNF

a a= = ( 2.32 )

.

Figure 2.18: Experimental and Analytical Transmissibilities for Specimen 2 with 1 Cell.

1.E-03

1.E-02

1.E-01

1.E+00

1.E+01

1.E+02

10 100 1000 10000Frequency (Hz)

Tra

nsm

issi

bili

ty


55

where amin and amax are the accelerations of the specimen top and bottom, respectively,

and ares is the accelerometer resolution, quoted at 0.005 g. The maximum experimental

acceleration of the specimen base was measured to be 36 g at 1.5 kHz. The minimum

measured transmissibility for the 3 and 4 cell cases may have been obscured by the

presence of a noise floor.

The results in Figures 2.11-2.18 show that the analytical model can accurately

predict the locations of the beginning stop band frequencies, assuming that the correct

material properties are known. The experimental end frequencies are not well-defined

and, as a result, the accuracy of the end frequency prediction is difficult to assess. The

model accurately predicts stop band depths for both 1 cell cases, as well as the 3 and 2

cell cases for Specimen 2. The minimum measured stop band depth for both the 3 and 4

celled cases is a transmissibility of around 1 x 10-4, or nearly coincident with the noise

floor. Although the minimum measured transmissibility may have been affected by the

noise floor, this attenuation factor of 10,000 would be sufficient for most vibration

control applications.

In Figures 2.19 and 2.20, experimental transmissibilities are compared for

varying number of layers for both Specimens 1 and 2. Comparing the stop band depths

of both the 1 and 2 cell cases, a full order of magnitude of reduction is gained with an

increase of 1 to 2 cells for both specimens. Similarly, increasing from 2 to 3 cells

reduces the transmissibility by an additional order of magnitude for both specimens. The

effect of increasing the number of cells from 3 to 4, however, is not certain because of the

noise floor location. Although the change in stop band depth cannot be reliably measured

56

when going from 3 to 4 cells, the transmissibility roll-off rate following the beginning of

the stop band is significantly increased. A practical design could therefore limit the

minimum number of cells to three or four to ensure a pronounced stop band attenuation

effect.

The effect of increased elastomer damping was also investigated. An additional

layered specimen (3) was constructed that had the same geometry as Specimen 1. An

elastomer with approximately the same shear modulus as that used in Specimen 1, but

with a significantly higher loss factor, was desired. An elastomer having a modulus

nearly identical to that elastomer 1 at low frequencies was thus selected. Upon tuning the

modulus to align the experimental and analytical results, however, it was found to be

400% higher than the low frequency value. Nevertheless, a transmissibility comparison

between Specimens 1 and 3 is made in Figures 2.21 and 2.22. The loss factor of

Figure 2.19: Comparison of Experimental Transmissibilities for Varying Number ofCells for Specimen 1.

1.E-05

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

1.E+01

1.E+02

10 100 1000 10000

Frequency (Hz)

1 Cell

2 Cells

3 Cells

4 Cells

Noise Floor

Tra

nsm

issi

bil i

ty

57

elastomer 3 was approximately 0.15. The difference between the higher and lower

damping is most noticeable when comparing the first resonance peaks of Specimens 1

and 3. The first peak of Specimen 1 is about a factor of ten higher than that of Specimen

3. Both specimen transmissibilities, however, reach an approximately equal minimum

value between 1 x 10-3 and 1 x 10-4. Although the minimum measured transmissibilities

may coincide with the noise floor, only measurements within a relatively small frequency

range would have been obscured within the stop bands. Also, the transmissibility roll-off

does not change appreciably with higher damping. Therefore, the basic stop band

attenuation characteristics do not appear to be significantly affected by the addition of

modest damping. Modest elastomer damping, however, could help minimize vibration

transmission at frequencies below the stop band.

.

Figure 2.20: Comparison of Experimental Transmissibilities for Varying Number ofCells for Specimen 2.

1.E-05

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

1.E+01

1.E+02

10 100 1000 10000Frequency (Hz)

1 Cell

2 Cells

3 Cells

4 Cells

Noise Floor

Tra

nsm

issi

bili

ty

58

Figure 2.21: Experimental and Analytical 4-Celled Transmissibilities of Specimen 1.

.

Figure 2.22: Experimental and Analytical 4-Celled Transmissibilities of Specimen 3.

1.E-111.E-101.E-091.E-081.E-071.E-061.E-051.E-041.E-031.E-021.E-011.E+001.E+011.E+02

10 100 1000 10000Frequency (Hz)

η = 0.05

Noise FloorT

rans

mis

sibi

l ity


1.E-111.E-101.E-091.E-081.E-071.E-061.E-051.E-041.E-031.E-021.E-011.E+001.E+011.E+02

10 100 1000 10000Frequency (Hz)

η = 0.15

Noise Floor

Tra

nsm

issi

bili

ty


Chapter 3

PASSIVE HIGH FREQUENCY GEARBOX ISOLATION

Layered isolators in compression were experimentally validated to exhibit high

frequency transmissibility stop bands. An analytical investigation was then conducted to

determine the feasibility of using layered isolators as helicopter gearbox isolators. The

gearbox design problem was characterized, and the isolator approximation method was

then used in concert with a design optimization routine to determine passive performance

limits of layered isolators.

3.1 Design Constraints

Initial design considerations for helicopter gearbox isolation were investigated. In

particular, design constraints associated with isolator mass, axial stiffness, and

elastomeric fatigue were estimated.

60

3.1.1 Mass Constraints

Soundproofing materials have long been used in helicopter interiors to reduce

cabin noise. Depending on the helicopter’s use as either military or commercial, a certain

amount of added soundproofing mass is generally allowed.

Owen, et al., report that military standards limit the amount of soundproofing

mass to 40-50 lbs. (18-23 kg) [26]. For a helicopter weighting 12,000 lbs., the added

mass is approximately 0.4% of the gross weight. In contrast, it is reported by Marze et

al., that for the VIP versions of the SA 365N Dauphin with a gross weight of 4,000 lbs.,

soundproofing may comprise 2-3% gross weight [24]. This correlates to 80-120 lbs. (36-

54 kg) of added mass.

In correspondence with Sikorsky Corporation, realistic isolator mass constraints

were determined. For a helicopter with gross weight of 12,000 lbs., 50 lbs. of added

soundproofing mass would be acceptable, with 75 lbs. as a maximum. This correlates to

0.4 – 0.6 % gross weight. To ensure a conservative amount of mass added for isolation

purposes, the isolator mass design constraint was chosen to be less than 0.5 % gross

weight of the helicopter in question.

3.1.2 Axial Stiffness Constraints

Typical modern helicopters require gearbox mountings to be nearly rigid to

minimize the relative deflections of the high speed engine-tranmission shafts [53], and to

61

allow for maintenance-free drive shaft couplings [54]. As helicopter gross weight

increases, so do the transmission mount stiffness requirements. In [54], the Bell Model

427 is reported to have quasi-static mount stiffnesses of 13,000 lb/in at two fuselage

connection points, for a total axial stiffness of 26,000 lb/in (4.6 MN/m). The Model 427

has a gross weight of 6,000 lbs. Welsh and fellow researchers retrofitted a modified S-

76B with a ACSR system [53]. The original transmission mounting stiffness at one of

four connection points is reported to be 80,000 lb/in (14 MN/m). The S-76B has a gross

weight of 12,000 lbs. Both transmission mountings are illustrated in Figure 3.1.

The stiffnesses of the proposed layered isolators should be comparable to existing

gearbox support stiffnesses, because they are being considered as possible replacement

mountings in existing helicopter designs. Four different axial stiffness values were

therefore chosen for a series of design runs in the subsequent design optimization

analysis. They are axial stiffnesses of 1, 5, 10, and 15 MN/m (or 5,700 lb/in, 29,000

lb/in, 57,000 lb/in, and 86,000 lb/in).

62

Figure 3.1: Schematic of Helicopter Transmission Mountings.

GW = 12,000 lbs

Typical Stiffness atFour Feet

kaxial = 14 MN/m

Fluidlastic®

Isolatorskaxial = 1.2 MN/m

Bell Model 427 Pylon Isolation System

S-76B Main Gearbox Supports

GW = 6,000 lbs

63

3.1.3 Elastomeric Bearing Stress Constraints

The design of elastomeric bearings has been investigated in numerous references

as applied to helicopter hub bearings and structural engineering applications, particularly

bridge design [55-64]. Natural rubber bearings have been in documented service

supporting bridges for more than 30 years in the United Kingdom and the United States

with no serious reports of deterioration [64]. A typical bridge bearing designed to

account for axial, shearing, and rotational motions in shown in Figure 17.

In 1987, the National Cooperative Highway Research Program published a report

which was intended to develop better understanding of elastomeric bearing behavior, as

well as formulate more rational design specifications [60]. Experimental tests were

performed to assess bearing compression, rotation, shear, stability, fatigue, and low

temperature behavior.

Figure 3.2: 20 year-old layered elastomeric bearing supporting bridge deck in England[64].

64

For bearings fixed against translation (no rotation or shearing), the researchers

report that a conservative value for the maximum compressive stress to avoid bearing

delamination is

2.00( ) 12.00C eG S MPaσ ≤ ≤ ( 3.1 )

where Ge is elastomer shear modulus, and S is shape factor. Similarly, fatigue tests

suggest a maximum dynamic compressive stress of

1.00( )C eG Sσ ≤ ( 3.2 )

Additionally, the metal layers may experience high tensile loading as result of

high compressive stresses in the bearing. Theoretical predictions of the reinforcement

stress were thus determined for circular bearings. A conservative, minimum thickness of

the metal layer to ensure that maximum steel stress is ½ Fy, the yield strength, is

expressed as

1.5(2 )e Cm

y

tt

F

σ≥ ( 3.3 )

where tm is metal thickness, and te is elastomer thickness.

Gembler, et al., reported that BK 177 helicopter strut loads resulting from

momentary flight loads are quite high, and vary from –10 kN to 30 kN [39]. Loads

experienced by different transmission mountings would depend on gross weight and

mounting configuration of the given helicopter. The BK 117 is a medium sized

65

helicopter and for the purposes of determining passive performance limits, 30 kN is

considered the highest quasi-static loading condition to which the layered isolators would

be subjected. Thus, both the preceding stress and thickness constraints have been

implemented in the subsequent layered isolator design cases.

3.2 Design Optimization

An optimization problem that incorporates various isolator parameters and design

constraints is summarized here. The objective of the optimization is to arrive at a design

which minimizes the deviation of the beginning and end frequencies, fB and fE, from

target frequencies, fTB and fTE, while maintaining certain design criteria, such as limits on

axial stiffness, isolator mass, isolator stress, geometry, and material properties. A

schematic describing the design variables is illustrated in Figure 3.3.

A simulated annealing optimization algorithm was developed to solve this

problem. This particular algorithm was chosen because it could be adapted with relative

ease to accommodate the discrete design variable, n, or number of layered cells. The

algorithm finds a solution by repeatedly perturbing the current design variables.

Improved values of the objective function are always accepted and replace the current

design for the next iteration. A more detailed description of this algorithm can be found

in Ref. 65.

66

The optimization problem for the design of layered isolators can be formally

written as

Minimize

TB B TE E

TB TE

abs absf f f f

f fβα +

� � � �� − −� � � ��

� � � �( 3.4 )

Subject to

min 0A Ak k− + ≤ ( 3.5 )

max 0A Ak k− ≤ ( 3.6 )

min 0L Lk k− + ≤ ( 3.7 )

Figure 3.3: Schematic of Design Variables.

Ge, ρe

Gm, ρm

tm

te

hn = 3

d

67

max 0L Lk k− ≤ ( 3.8 )

( ) /m et h nt n= − ( 3.9 )

min maxe e eG G G≤ ≤ ( 3.10 )

min maxm m mG GG ≤ ≤ ( 3.11 )

min maxe e eρ ρ ρ≤ ≤ ( 3.12 )

min maxm m mρ ρ ρ≤ ≤ ( 3.13 )

min maxe e et t t≤ ≤ ( 3.14 )

min maxd d d≤ ≤ ( 3.15 )

min maxh h h≤ ≤ ( 3.16 )

min maxh h h≤ ≤ ( 3.17 )

max 0mm − ≤ ( 3.18 )

2.00 0C eG Sσ − ≤ ( 3.19 )

12.00 0Cσ − ≤ ( 3.20 )

1.00 0DYNC eG Sσ − ≤ ( 3.21 )

68

where the design variables are

Ge = Shear modulus of elastomerGm = Shear modulus of metalρe = Density of elastomerρm = Density of metal

te = Thickness of elastomer layerd = Isolator diameterh = Isolator heightn = Number of cells

with constants

fTB = Target beginning frequencyfTE = Target end frequencymmax = Max. isolator massα, β = Weighting factorsGemin,Gemax = Min. and Max. Ge Gmmin, Gmmax = Min. and Max. Gm

ρemin, ρemax = Min. and Max. ρe

ρmmin, ρmmax = Min. and Max. ρm

temin, temax = Min. and Max. te

dmin, demax = Min. and Max. dhmin, hmax = Min. and Max. hnmin, nmax = Min. and Max. nkAmin, kAma x= Min. and Max. axial quasi-static stiffnesskLmin, kLmax = Min. and Max. lateral quasi-static stiffness

and with the intermediate variables

m = Isolator Masstm = Layer thickness of the metalkA = Isolator axial quasi-static stiffnesskL = Isolator lateral quasi-static stiffness

σC = Compressive stressσCDYN = Compressive dynamic stressS = Shape Factor

3.3 Passive Performance Limits

With knowledge of the overall design space for various helicopters, a broad

assessment of the passive performance limits of layered gearbox isolators can be

69

determined. These were determined using the Ritz approximation method in concert with

the design optimization routine.

A series of design cases were performed with fB = 400 Hz, and fE = 2000 Hz.

Because the lower frequency vibrations are often the most troublesome (~500 Hz), the

beginning stop band frequency is weighted more heavily in Equation (3.4) , with α = 100

and β = 1, where α and β correspond to the beginning and end frequencies, respectively.

The combinations of design variable and constraint values used in the design cases are

summarized in Table 3.1.

Tables 3.2 and 3.3 contain the results for the isolator mass equal to 2 and 4 kg,

respectively. The metal density was fixed to represent aluminum. This is not a critical

variable, but rather it is the amount of mass in the metal layer that is important. Given a

different density, the metal thickness would simply be adjusted so that the total mass

remained the same. Through correspondence with Lord Corporation, a realistic lower

limit on elastomer shear modulus was set at 0.34 MPa. Also, the upper limit on

elastomer density was set at 1,100 kg/m3, and in each run was reached.

Some general trends can be observed from the results. They include:

Table 3.1: Combination of Design Variable and Constraint Values for PassivePerformance Limits of Layered Isolators.

Isolator Mass

(kg)

Isolator Diameter

(cm)

Isolator Axial Stiffness

(MN/m)

m = 2, 4 d = 6, 8, 10, 12, 14, 16, 18, 20 kAXIAL 1, 5, 10, 15

70

1) For a given mass and stiffness, as diameter increases, the beginning stop band

frequency, fB, is lowered more readily. As diameter is increased, the minimum

axial stiffness constraint can be satisfied with a lower elastomer modulus because

of elastomer shape factor effect. A lower stop band beginning frequency can

therefore be realized more readily at larger diameters.

2) As fB is lowered in the above trend, however, the end frequency, fE, may also be

lowered below the target of 2000 Hz. This is because as the diameter is

increased, the frequency span of the stop band decreases. Because the beginning

frequency was more heavily weighted in the calculations, the stop band is located

closer the beginning target frequency.

3) For a given mass and diameter, as the stiffness constraint increases, the stop band

frequencies increase. This observation might be expected to be caused by the fact

that as the stiffness constraint increases, the elastomer modulus should increase,

and therefore the natural frequencies of the isolator should also increase.

Contrarily, the elastomer modulus actually tends to decrease with increasing axial

stiffness. The results suggest that for higher stiffness isolators, lower stop band

frequencies are possible with thinner elastomer layers having lower modulus

values. For lower stiffness isolators, lower stop band frequencies are possible

with higher modulus elastomers with a higher elastomer thickness.

An important general conclusion which is suggested by Tables 3.2 and 3.3 is that

there often occur cases where the beginning stop band frequencies are above 500 Hz.

71

This frequency is often cited as the beginning of the irritating frequency range emitting

by helicopter gearboxes, and therefore simple layered isolators may not perform

adequately for all design cases. Consequently, an investigation of possible layered

isolator design enhancements was executed to improve isolator performance in the

frequency range immediately below the beginning stop band frequency.

72

Table 3.2: Passive Performance Design Runs, Isolator Mass = 2 kg.

( ) indicates actual optimized value

Axial Stiffness (MN/m)

Stop Band Begin

Frequency (Hz)

Stop Band End

Frequency (Hz)

GELAS

(MPa)

ρ ELAS

(kg/m3)

ρ METAL

(kg/m3)

t ELAS

(cm)t METAL

(cm)

1 (1.17) 739 2369 16.52 1100 2643 4.847 6.6635 1300 5721 5.32 1100 2643 1.213 8.405

10 1753 7396 4.43 1100 2643 0.868 8.54515 (15.87) 2038 7452 1.94 1100 2643 0.548 8.69

1 (1.16) 691 1790 5.65 1100 2643 3.79 3.4335 1127 3465 1.82 1100 2643 1.197 4.517

10 1445 4113 1.21 1100 2643 0.805 4.66715 1895 5619 3.93 1100 2643 1.061 4.576

1 (1.05) 621 1255 3.1 1100 2643 4.06 1.5135 959 2372 1.15 1100 2643 1.373 2.639

10 1191 2916 0.78 1100 2643 0.934 2.81915 1398 3337 0.83 1100 2643 0.827 2.867

1 (1.1) 549 995 1.7 1100 2643 3.846 0.6245 774 1727 0.64 1100 2643 1.425 1.636

10 937 2098 0.44 1100 2643 0.976 1.81415 1052 2341 0.36 1100 2643 0.792 1.901

1 500 969 0.75 1100 2643 2.703 0.5115 628 1327 0.43 1100 2643 1.519 0.993

10 772 1643 0.35 1100 2643 1.113 1.1715 889 1893 0.35 1100 2643 0.97 1.224

1 (3.43) 470 942 0.37 1100 2643 1.975 0.4335 526 1071 0.35 1100 2643 1.688 0.539

10 731 1493 0.62 1100 2643 1.629 0.57415 795 1647 0.46 1100 2643 1.27 0.725

1 (3.66) 418 814 0.34 1100 2643 2.211 0.055 461 914 0.35 1100 2643 1.985 0.164

10 578 1171 0.35 1100 2643 1.558 0.33915 660 1348 0.35 1100 2643 1.353 0.426

1 (9.87) 506 1008 0.34 1100 2643 1.8 0.055 (9.87) 506 1008 0.34 1100 2643 1.8 0.05

10 508 1013 0.34 1100 2643 1.787 0.0515 587 1182 0.37 1100 2643 1.599 0.136

d = 14 cm

d = 16 cm

d = 18 cm

d = 20 cm

d = 6 cm

d = 8 cm

d = 10 cm

d = 12 cm

73

Table 3.3: Passive Performance Design Runs, Isolator Mass = 4 kg.

( ) indicates actual optimized value


Stop Band Begin

Frequency (Hz)

Stop Band End

Frequency (Hz)

GELAS

(MPa)

ρ ELAS

(kg/m3)

ρ METAL

(kg/m3)

t ELAS

(cm)t METAL

(cm)

1 (1.06) 500 2000 25 1015 2643 7.413 14.4245 941 5646 4.51 1100 2643 1.136 17.365

10 1277 7218 3.31 1100 2643 0.78 17.51815 1565 7972 5.02 1100 2643 0.783 17.516

1 472 1608 6.87 1100 2643 4.603 8.1215 886 3570 2.57 1100 2643 1.363 9.465

10 1222 4822 3.41 1100 2643 1.168 9.53515 1481 5843 4.31 1100 2643 1.097 9.578

1 446 1225 3.25 1100 2643 4.255 4.6525 783 2390 1.15 1100 2643 1.371 5.848

10 1050 3020 1.25 1100 2643 1.102 5.95815 1211 3480 0.98 1100 2643 0.877 6.056

1 420 980 1.74 1100 2643 3.931 2.815 672 1701 0.64 1100 2643 1.424 3.865

10 847 2065 0.47 1100 2643 1.002 4.03315 1014 2457 0.59 1100 2643 0.939 4.068

1 (1.16) 400 809 1.26 1100 2643 4.096 1.4995 572 1329 0.44 1100 2643 1.534 2.632

10 738 1677 0.44 1100 2643 1.203 2.75615 1003 2342 1.44 1100 2643 1.582 2.616

1 (1.3) 400 698 1.06 1100 2643 4.363 0.2435 498 1089 0.38 1100 2643 1.74 1.754

10 632 1387 0.37 1100 2643 1.353 1.92915 725 1590 0.36 1100 2643 1.167 2.024

1 (3.31) 400 809 0.43 1100 2643 2.485 0.7235 441 920 0.37 1100 2643 2.023 1.12

10 558 1183 0.37 1100 2643 1.589 1.31115 634 1349 0.34 1100 2643 1.35 1.416

1 (4.75) 400 789 0.39 1100 2643 2.431 0.2495 400 799 0.35 1100 2643 2.291 0.369

10 493 1013 0.35 1100 2643 1.804 0.85215 573 1183 0.39 1100 2643 1.618 0.926

d = 14 cm

d = 16 cm

d = 18 cm

d = 20 cm

d = 6 cm

d = 8 cm

d = 10 cm

d = 12 cm

Chapter 4

DESIGN ENHANCEMENTS FOR IMPROVED ISOLATOR

PERFORMANCE

In the previous chapter, the passive performance of layered isolators was

determined to be inadequate for some design cases. The most apparent performance

limitation was an ability to place the stop band beginning frequency, fB, at low enough

frequencies (~500 Hz), given a set of design constraints. Many active, semi-active, and

passive design enhancements were therefore considered to improve isolator performance

and are briefly summarized here. More specifically, attempts were made to increase the

attenuation at frequencies below fB, and to lower fB to effectively widen the stop band.

4.1 Summary of Active, Semi-Active Concepts to Improve Isolator Performance

The benefits of replacing the conventional elastomer material with

magnetorheological (MR) elastomers were considered. MR elastomers are viscoelastic

solids whose mechanical properties are controllable by applied magnetic fields due to

small iron particles introduced before curing [65-70]. The Young’s modulus has been

75

experimentally observed to increase up to 50% relative to the zero-magnetic-field value.

The change in natural frequency of a given system can be estimated using the familiar

equation /k m . The stop band frequencies could therefore be increased as much as

22% from their zero-magnetic-field values. Because the most critical problem involves

lowering fB, this concept was not pursued.

The performance benefits of replacing the elastomer layers with electrically

shunted layers of either polyvinylidene fluoride (PVDF) or piezoceramic polymer

composites were also investigated. PVDF’s are polymers with modest piezoelectric

coupling effects. Ceramic polymer composites are materials which combine the desirable

properties of piezoelectric sensitivity and mechanical flexibility. This is accomplished by

creating a near-homogeneous solid composed of a piezoceramic and polymer.

Electrically shunting either of these changes their effective stiffness. An expression was

developed for the effective stiffness, k*, of a piezoelectric element undergoing dynamic

strains [71]:

��

�

�

��

�

�

+−+=

)(11

2

2

*

sk

kkk

p

pE

α( 4.1 )

where kE is the effective short circuit material stiffness, kp is the material planar

electromechanical coupling coefficient, α(s) is the nondimensional ratio of the electrical

impedance of the material (i.e. 1/sCpT, where Cp

T is the capacitance of the material

measured under constant stress) to the electrical impedance of the shunt circuit, and s is

the Laplace parameter [72]. Ceramic polymer composites typically have larger

76

electromechanical coupling coefficients than PVDFs, and can range from 0.6 - 0.7.

Using Equation 4.1, this would enable the material stiffness to increase by up to a factor

of 2. This stiffness increase would translate into a 41% increase in stop band frequencies.

Unfortunately, the lowest Young’s modulus possible for these materials is nearly an order

of magnitude higher than that of the elastomer. Therefore, the short circuit stop band

frequencies would be too high, and could only be tuned to higher frequencies. Although

this may be an attractive semi-active approach for other applications, it does not appear

appropriate for helicopter gearbox isolation.

The effectiveness of replacing the metal layers in layered isolators with active

piezoceramic layers was also investigated. The concept was modeled using a one

dimensional finite element model. An open loop control strategy was implemented to

minimize isolator transmissibility by actuating the piezoceramic layers. The small strains

induced in the piezoceramic layers, however, had no visible effect on transmissibility.

For this concept to be effective, the Young’s modulus of the compliant layers must be at

least two orders magnitude below that of the piezoceramic layers.

4.1.1 Emdedded Terfenol-D Actuators in Layered Isolators

The effectiveness of embedding reaction mass actuators in metal layers was

investigated. Terfenol-D actuators have a particularly high force to volume ratio, and an

accurate model of such actuators can be found in [73]. The actuator model was

incorporated into the layered isolator one-dimensional finite element model, and an open

77

loop control strategy was implemented to evaluate the control authority of embedded

actuators. The control strategy was adopted by Anusonti-Inthra [75] from an approach

previously used by Johnson for reduction of low frequency rotor vibrations [74].

In the following example, a single Terfenol-D actuator embedded in the top metal

layer of a three-layered isolator is modeled. A schematic of the model can be seen in

Figure 4.1. The actuator is modeled as a point force, Fa, and the disturbance force, Fb,

acts at the base of the isolator.

The control algorithm is based on the minimization of a quadratic objective

function, J, defined as:

1 2[ ] [ ]TTa aJ x W x F W F= + ( 4.2 )

Figure 4.1: Model Schematic of Layered Isolator with Embedded Terfenol-D Actuator.

Elastomer Layer

Embedded Terfenol-DReaction Mass Actuator

Metal Layer ofLayeredIsolator

x

Fb

Fa

78

where [W1] and [W2] represent penalty weightings on the displacement of the top metal

layer, x, and the actuator force, Fa, respectively. It is assumed that the displacement, x, is

related to the frequency-domain forces, Fa, by the relation

o ax x TF= + ( 4.3 )

where xo is the baseline displacement without actuator force, Fa, and T is the system

transfer matrix. By substituting Eq. 4.2 into Eq. 4.3, a gradient-based method can be

used to minimize J and determine the optimal forces, Fa. By setting / aJ F∂ ∂ = 0, the

optimal input is found to be:

[ ]a oF T x= ( 4.4 )

11 2 1[ ] ([ ] [ ][ ] [ ]) [ ] [ ]T TT T W T W T W−= − + ( 4.5 )

The system transfer matrix, [T], can be determined by perturbing the system with

individual components of actuator force, Fa. The components, Fac and Fa

s, are assumed

to be cosine and sine pairs in the frequency domain, where the superscripts ‘c’ and ‘s’

refer to cosine and sine, respectively. The first column of [T], which corresponds to the

actuator force component Fac, is determined by setting Fa

c to a nonzero value, and the

column is found to be:

79

11

12

cco

ca

sso

ca

x x

Ftt x x

F

� �−� �

� � � �=� � � �−� ��

( 4.6 )

Similarly, when Fas has a nonzero value, the second column of [T] is found to be

21

22

cco

sa

sso

sa

x x

Ftt x x

F

� �−� �

� � � �=� � � �−� ��

( 4.7 )

Once the cosine and sine components of the actuator force are determined, their

magnitudes are compared to the maximum attainable force predicted by the Terfenol-D

actuator model at that particular frequency to ensure realistic actuator performance. If

necessary, the force components are scaled down to reflect realizable forces.

In Figure 4.2, the transmissibility of a layered isolator is plotted for varying

values of base force, Fb, as well as the case when there is no actuation in the actuator. In

Figure 4.3, the same layered isolator is plotted with and without the embedded actuator

for a base force of 40 N. The relevant isolator properties are te = 1.1 cm, tm = 6.0 cm, Ge

= 1.25 MPa, ρe = 1,100 kg/m3, and the metal layers are aluminum.

In Figure 4.2, the actuator natural frequency is observed to be around 700 Hz and

as expected, the actuator is most effective around this frequency. Points 1 and 3

correspond to the two side resonance peaks associated with a tuned absorber, or the

actuator in this case. Point 2 is the antiresonance frequency caused when no power is

supplied to the actuator, and point 4 corresponds to the beginning of the passive stop

80

band. With increasing base force, the effective actuator bandwidth becomes narrower

around the resonance. The results indicate, however, that an embedded Terfenol-D

reaction mass actuator indeed has the authority to actively cancel vibrations before the

onset of the passive stop band.

Although this control strategy appeared effective, this approach would necessarily

include many of the complexities involved with active control. The search for a purely

passive vibration control solution was therefore continued.

Figure 4.2: Transmissibility of Three-Layered Isolator with Embedded Terfenol-DActuator.

102

103

104

10-8

10-6

10-4

10-2

100

102

Frequency (Hz)

Tra

nsm

issi

bili

ty

Layered

PassiveStop Band

Fb = 20 NFb = 40 NFb = 60 NFb = 80 N

NoActuation

1

2

34

81

4.2 Embedded Vibration Absorbers to Improve Isolator Performance

A passive solution which appeared effective was embedded vibration absorbers,

as illustrated in Figure 4.4. By embedding vibration absorbers in each metal layer of the

isolator, the vibrations occurring at frequencies below the stop band could be attenuated.

As an example, suppose there was a need for an isolator with a diameter of 6 cm, an axial

stiffness of 2.4 MN/m (2 times the stiffness at one gearbox connection on the Bell Model

Figure 4.3: Transmissibility of Isolator with and without Embedded Terfenol-D Actuator.

102

103

104

10-8

10-6

10-4

10-2

100

102

Frequency (Hz)

Tra

nsm

issi

bili

tyPassive

Stop Band

Fb = 40 N, withActuator

NoActuator

82

427), and with a mass less than or equal to 3 kg. The properties of the optimized isolator

are summarized in Table 4.1.

As can be observed in Table 4.1, the stop band beginning frequency, fB, is 684 Hz,

which is too high for the gearbox application. If three embedded absorbers are added to

the isolator, each having a absorber mass of 1 kg, then the isolator will sufficiently

Figure 4.4: Schematic of Layered Isolator with Embedded Vibration Absorbers.

Table 4.1: Optimized Properties of Layered Isolator of Example.

Elastomer LayerStiffness

Embedded VibrationAbsorber

Metal Layer ofLayeredIsolator

VibrationInput


Stop Band Begin

Frequency (Hz)

Stop Band End

Frequency (Hz)

GELAS

(MPa)

ρ ELAS

(kg/m3)

ρ METAL

(kg/m3)

t ELAS

(cm)t METAL

(cm)

2.4 684 2916 0.36 1100 2643 0.587 13.12

83

attenuate over the frequency range of interest. This is illustrated in Figure 4.5. However,

the isolator mass has now been doubled to a value of 6 kg. In general, if conventional

vibration absorbers are used to attenuate frequencies below the stop band, the weight

penalty is prohibitive. Therefore, different means of effectively amplifying the absorber

mass were investigated.

Figure 4.5: Transmissibility of Example Layered Isolator.

101

102

103

104

10-10

10-8

10-6

10-4

10-2

100

102

104

Frequency (Hz)

AbsorberFrequencies

Minimum DesiredAttenuation Level

( < 0.01 )

Without AbsorbersWith Absorbers


(<0.01)

AbsorberFrequencies

Chapter 5

PERIODICALLY LAYERED ISOLATORS WITH EMBEDDED

INERTIAL AMPLIFIERS

In the previous chapter, a series of design enhancements were considered to

improve isolator performance. The use of embedded vibration absorbers appeared

effective, although they necessitated too much added isolator mass. Several

configurations were therefore considered to effectively amplify absorber mass. An

attractive configuration is illustrated in Figure 5.1. This configuration was found to

exhibit a combination of advantageous benefits, including inertial amplification and

vibration absorber effects. A detailed investigation into the performance effects of these

amplifiers was conducted. The use of embedded fluid elements was determined to be an

efficient method to realize inertial amplification in the isolators.

5.1 Embedded Inertial Amplifiers

The inertial amplification effect can be observed when relative motion occurs

between layers. As the schematic in Figure 5.1 suggests, relative motion necessarily

85

causes the lever arm to rotate, thus causing motion of the tuned mass. Because the tuned

mass is attached to the end of the lever arm, the inertia of the metals layer is effectively

increased when relative motion occurs.

The vibration absorber effect results from interaction between the elastomer layer

stiffness and the tuned mass. At certain tuned frequencies, the isolator will experience

antiresonance conditions independent of layer masses. These tuned frequencies are only

dependent on the amount of tuned mass, length of lever arms, and stiffness of the

elastomer layers.

Consider an inertial amplification element alone, as in Figure 5.2. Equations of

motion can be written for the mechanical system in Figure 5.2, and can be expressed as

2

1

0 0 0 00 0 0 00 0 0t t

uu

m u

� � � � � �� =� � � �

� � � � � � � � � �

��

��

��

( 5.1 )

Assuming that R=b/a, a kinematic constraint equation exists that uses the length ratio, R,

to relate the motion of the tuning mass, ut, to the u2, and u1, [54]

1 2( ) ( 1)tu R u R u= − − ( 5.2 )

This constraint equation can be used to reduce the three degree of freedom system to two

degree of freedom system, by first creating a transformation matrix, G, as in

{ } [ ]{ }3 2DOF DOFu G u= ( 5.3 )

86

{ }22

11

1 00 1

1t

uu

uu

u R R

� � � �� =� �

� �� − � �

( 5.4 )

If the equations of motion in Equation 5.1 are pre and post-multiplied by [G], then the

reduced degree of freedom equations can be written as [54]

{ } { }22

21

( 1) ( 1) 00( 1)

t t

t t

R m R R m uuR R m R m

� �− − − =� �− −� �

��

��

( 5.5 )

In this inertial matrix, the tuned mass is effectively being amplified approximately

by the square of the lever arm ratio, R. An inertial amplifier can be attached to each

metal layer in the three dimensional model by providing appropriate constraint equations

for Equation 5.5.

Figure 5.1: Schematic of Isolator Configuration with Embedded Inertial Amplifiers.

Elastomer LayerStiffness

Embedded Inertial Amplifier

Rigid Link

Metal Layerof Layered

Isolator

Vibration Input

Tuned Mass on LeverArm

ba

87

5.2 Effect of Embedded Inertial Amplifiers on Layered Isolator Frequency Response

The first advantageous effect of these inserted elements is that the relative motion

between layers necessitates relative motion between the lever arm mass and the layers.

This serves to amplify the effective dynamic inertia of the metal layers when relative

motion occurs. This can readily be observed most by examining the first n eigenvalues of

an n-celled isolator with and without embedded inertial amplifiers. In Table 5.1, the first

three eigenvalues of the previous example isolator are listed with and without embedded

inertial amplifiers. The length of the lever arm portions, a and b, and tuned masses are

listed in Table 5.2.

Figure 5.2: Schematic of one inertial amplifier.

a

b

u1

u2

mt R = b / a

88

The beginning stop band frequency, fB, has been lowered to 473 Hz from 684 Hz

with the addition of inertial amplifiers. Because the end stop band mode shape involves

little relative motion between layers, the end frequency is not significantly affected. This

is evidenced by the stop band mode shapes pictured Figure 5.3. The stop band frequency

range has been expanded to include the target frequency range (500 - 2000 Hz). The

differences in tuned mass values can be explained by considering the tuned masses as

part of vibration absorbers. This concept will be explained in the next section.

Table 5.1: Summary of Isolator Eigenvalues with and without Embedded InertialAmplifiers.

Without InertialAmplifiers

Frequency (Hz)

With InertialAmplifiers

Frequency (Hz)

Mode 1 183 180Mode 2 494 383Mode 3 684 473

Table 5.2: Summary of Example Inertial Amplifier Properties.

Tuned Mass(g)

a(cm)

b(cm)

Layer 1 8.3 1 9Layer 2 5.6 1 9Layer 3 1.9 1 9

89

5.3 Vibration Absorber Effect

Consider a subsystem of a layered isolator with masses m1 and m2 and elastomer

spring stiffness, k, and, in addition, an embedded inertial amplifier as in Figure 5.4.

Figure 5.3: Mode Shapes at Beginning and End Stop Band Frequencies of LayeredIsolator with Embedded Inertial Amplifiers.

fb = 473 Hz

m = 8.3 g

m = 5.6 g

m = 1.9 g

fe = 2788 HzStop Band

Little RelativeMotion of MetalLayers at End of

Stop Band

Significant RelativeMotion of Metal

Layers at Beginningof Stop Band

90

The equations of motion for the system shown in Figure 5.4 can written as

{ } { } { }22 2 2

21 11

( 1) ( 1) 00( 1)

t t

t t

m R m R R m u uk ku uk kR R m m R m

� �+ − − − −� �+ =� � � �−− − + � ��

��

��

( 5.6 )

In order to determine the isolation frequency for which m1 is completely isolated

from the motion of m2 and mt, the motion of m1 is set equal to zero, u1 = 0. Then the

lower homogeneous equation of the reduced set of equations of motion is solved to yield

1

2 ( 1)( )it

kf

R R mπ=

−( 5.7 )

Thus, the frequency for which m1 is isolated is independent of both masses m1 and

m2 [54]. This illustrates how the elastomer layer stiffness and the amplified tuned mass

combine to act as a vibration absorber.

In Figure 5.5, the transmissibility of the example isolator is shown with and

without the embedded inertial amplifiers from Table 5.2. The addition of the inertial

Figure 5.4: Schematic of Subsystem of Layered Isolator with Embedded InertialAmplifier [54].

a

b

u1

u2

mt R = b / a

m2

m1

k

91

amplifiers lowers the first three isolator eigenvalues, and thus increases the size of the

stop band. Three tuned absorber frequencies can also be identified at 509, 612, and 928

Hz.

When traditional, single-degree-of-freedom vibration absorbers are added to

vibrating mechanical systems to attenuate a response at a particular frequency, there are

typically two additional resonance peaks created in the frequency response curve. This

effect can be observed in Figure 4.5, where traditional absorbers were embedded in the

metal layers of the example isolator. This detrimental effect is sometimes mitigated with

the addition of absorber damping.

The configuration proposed in Figure 5.1, however, does not increase the number

of degrees of freedom of the isolator. No additional resonance peaks are created on either

side of the absorber frequencies as a result, and no internal damping is needed to control

the isolator response within the stop band.

92

5.4 Typical Frequency Response Shapes of Isolator with Embedded Amplifiers

In Figure 5.6, the axisymmetric response shapes of the example isolator are

shown for the identified tuned frequencies in Figure 5.5. The first response shape at 509

Hz shows the top embedded inertial amplifier and the top elastomer layer as the active

tuned absorber. As noted in the figure, the vibration absorber effect can be visualized as

the compressed elastomer force nearly canceling the inertial amplifier force on the second

Figure 5.5: Transmissibility of Example Isolator with and without Inertial Amplifiers.

101

102

103

104

10-10

10-8

10-6

10-4

10-2

100

102

Minimum DesiredAttenuation LevelAchieved in TargetFrequency Range

( < 0.01)

Tuned AbsorberFrequencies

1 23



Frequency (Hz)

FrequencyRange ofInterest




Achieved in TargetFrequency Range

(<0.01)

Tuned AbsorberFrequencies


93

metal layer. Thus, little axial force is transmitted through the rest of the isolator. This is

also evidenced by the lack of elastomer deformation in the bottom layer.

Similarly, in the second response shape at 612 Hz, the elastomeric force and

amplifier inertial force are nearly equal and opposite in the middle layer, and thus little

force is transmitted to the bottom layer. Finally, the third response shape at 928 Hz

shows the active absorber effect in the bottom layer, where the opposing forces are

cancelled at the isolator base.

In the top layer of the second and third response shapes, the amplifier and

elastomer response shape are nearly identical to those of the case when the top absorber

is active at 509 Hz. Although it may appear like a similar response condition, there are

no antiresonance conditions in the top layer at these higher frequencies, and thus some

Figure 5.6: Axisymmetric Frequency Response Shapes of Example Isolator withEmbedded Amplifiers at Tuned Absorber Frequencies.

1st Tuned FrequencyResponse Shape

f = 509 Hz

2nd Tuned FrequencyResponse Shape

f = 612 Hz

3rd Tuned FrequencyResponse Shape

f = 928 Hz

DrivingForce

Active Absorber

Active Absorber

Active Absorber

m = 8.3 g

m = 5.6 g

m = 1.9 g

Elastomer Force =Inertial Force



94

force is transmitted to the middle layer. The same comment can be applied to the middle

layer response when the third and second response shapes are compared.

5.5 Effect of Shape Factor and Passive Stop Band Location on Response Behavior

Because of the high axial stiffness design constraint for gearbox isolation,

optimized layered isolator designs often require a relatively high elastomer shape factor

(> 1) for a single layer. For example, a single elastomer layer of the example isolator has

a shape factor of 2.6. Subsequently, the implications of employing a higher elastomer

shape factor for the embedded isolator design will be examined.

5.5.1 High Shape Factor Behavior

In Figure 5.7, response shapes of the example isolator without inertial amplifiers

are shown. Three different frequency locations were chosen to represent typical response

shapes at certain frequencies of interest. The first response shape represents the

frequency range starting at the beginning stop band frequency and ending at the isolator

antiresonance frequency, or point number two. In this region, the dominant mode in the

isolator response is the beginning stop band mode, illustrated in Figure 2.2. This mode

shape is characterized by either uniform axial tension or compression in the elastomer

layer.

95

The second point corresponds to the response shape where the isolator

experiences an antiresonance. The elastomer layers at this frequency exhibit shapes

where the imposed axial motion of the metal layers is translated into radial deformations

and overall isolator dynamic stiffness is at a minimum. This point also corresponds to a

transition in modal dominance, or the point at which the end stop band mode becomes

more dominant. This mode shape is also illustrated in Figure 2.2, and consists of

elastomer deformations that consist of both axial tension and compression within a single

layer.

The third response shape corresponds to this region where the end stop band

mode is most dominant. At higher shape factors, this mode shape is characterized by

little axial motion. Little relative motion between the layers would correspond to little

motion of an embedded amplifier. Consequently, it has been observed through numerous

analytical design cases that embedded inertial amplifiers can only function as vibration

absorbers below this antiresonance frequency, which corresponds to point two in Figure

5.7. It is important to note that this condition only applies to higher shape factor

elastomer layers (� 1). For lower shape factors, different response behavior can be

observed.

96

Figure 5.7: Example Isolator Response Shapes (High Shape Factor, S = 2.6).

Response Shapef = 1142 Hz



101

102

103

104

10-8

10-6

10-4

10-2

100

102

Frequency (Hz)

Tra

nsm

issi

bili

ty

13

2

1 2 3

Frequency Rangefor Possible

AntiresonanceConditions

Minimum ForceTransmitted

DrivingForce

97

5.5.2 Low Shape Factor Behavior

In Figure 5.8, response shapes of the layered isolator described in the first line of

Table 3.3 are illustrated. The elastomer layers in this isolator have a shape factor of 0.2,

which is quite low in comparison to those of other optimized cases. Low shape factors

correspond to thicker elastomer layers, and consequently such isolators exhibit slightly

different dynamics which warrant closer inspection.

Similar to Figure 5.7, the response shapes in Figure 5.8 represent typical response

shapes at certain frequencies of interest. The dominant mode in the first response shape

is the beginning stop band mode. At point two, the response shape corresponds to the

lowest transmissibility exhibited by the isolator. As expected, this response shape

consists of a combination of the beginning and end stop band mode shapes. Contrary to

the higher shape factor isolators, there is no defined antiresonance frequency. Instead,

there is a gradual modal shift toward the end stop band mode shape, as illustrated in the

third response shape. Because of the thicker elastomer layers in this lower shape factor

isolator, there will be more relative axial motion between layers in the vicinity of the end

stop band frequency than in higher shape factor isolators.

98

Figure 5.8: Transmissibility and Response Shapes for Layered Isolator with Shape Factor= 0.2.

101

102

103

104

10-4

10-3

10-2

10-1

100

101

102

103

104

Frequency (Hz)

Tra

nsm

issi

bili

ty

1

3

2




1 2 3

99

In Figure 5.9, the transmissibility of the low shape factor isolator is plotted with

and without the inertial amplifiers summarized in Table 5.3. The transmissibility curve

with amplifiers shows that six tuned absorber frequencies result from the addition of one

amplifier per layer.

Table 5.3: Summary of Inertial Amplifier Properties in Low Shape Factor Example.

Tuned Mass(g)

A(cm)

b(cm)


Figure 5.9: Transmissibility of Low Shape Factor Isolator with and without EmbeddedInertial Amplifiers.

101

102

103

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

101

102

103

Frequency (Hz)

Tra

nsm

issi

bili

ty

With AmplifiersWithout Amplifiers

1

23 4

5

6Six Tuned Absorber

Frequencies withThree Amplifiers

100

Presumably, each amplifier is causing two tuned absorber frequencies as it

interacts with the associated elastomer layer. To better understand the dynamics of this

example isolator, each layer should be examined independently. In Figure 5.10, the

transmissibilities of the independent layers are plotted for a fixed-free condition. In

addition, the response shapes at the absorber frequencies are illustrated.

As expected, each amplifier is indeed causing two antiresonance frequencies, all

of which correspond to those in Figure 5.9. In section (a), a 10.6 g mass attached to the

lever arm results in the antiresonance frequencies 554 and 1902 Hz. As the amplifier

mass decreases in sections (b) and (c), the first antiresonance frequency is observed to

increase to 774 and 1093 Hz, respectively. Conversely, the second antiresonance

frequencies in sections (a), (b), and (c) are observed to decrease from 1902 to 1816 and

finally to 1119 Hz, respectively. The response shapes in section (c) appear to be

converging, so that, at a certain amplifier mass, the frequencies will coincide. This

behavior can be explained in terms of the effective dynamic stiffness of the elastomer

layer over the stop band frequency range. The elastomer layer dynamic stiffness has a

certain value at the beginning stop band frequency, and gradually decreases until the

frequency of lowest transmissibility. At that point, the dynamic stiffness then begins to

rise as the frequency approaches the end of the stop band. This change in the dynamic

stiffness explains why the antiresonance frequencies converge with decreasing amplifier

mass.

101

Figure 5.10: Response Shapes of a Single Layer of the Low Shape Factor ExampleIsolator with Embedded Inertial Amplifiers: (a) m = 10.6 g, (b) m = 5.6 g, (c) m = 3.3 g.

101

102

103

104

10-4

10-3

10-2

10-1

100

101

102

103

Frequency (Hz

Tra

nsm

issi

bili

ty


1


2

1 2

101

102

103

104

10-4

10-3

10-2

10-1

100

101

102

103

Frequency (Hz

Tra

nsm

issi

bili

ty


1


21

2

101

102

103

104

10-4

10-3

10-2

10-1

100

101

102

103

Frequency (Hz

Tra

nsm

issi

bili

ty


1


21

2

(a)

(b)

(c)

m = 10.6 g

m = 5.6 g

m = 3.3 g

Top Layer

Middle Layer

Bottom Layer

102

5.6 Structural Periodicity of Embedded Amplifier Design

In the previous discussions about the use of embedded inertial amplifiers, the

tuned masses have different values which correspond to different tuned absorber

frequencies. Different values of tuned masses, however, technically result in a loss of

isolator periodicity. It would be useful, therefore, to examine a case where all the tuned

masses are similar.

In Figure 5.11, the transmissibility of the example isolator is plotted with inertial

amplifiers that maintain isolator periodicity, aperiodic inertial amplifiers as described in

Table 5.2, ands the case without inertial amplifiers. The periodic amplifier properties are

listed in Table 5.4. As expected, the periodic amplifiers resonate at the same resonance

frequency, and the attenuation at that frequency is pronounced. Adding aperiodicity to

the amplifiers has the effect of widening the tuned absorber frequency range, but also the

effect of decreasing the attenuation level at the original, center frequency. This behavior

is typical of the effects of adding slight aperiodicities to periodic systems. This figure

also clearly illustrates how the addition of relatively small amounts of periodic amplified

mass in the metal layers results in a dramatic shift in stop band location.

Table 5.4: : Summary of Inertial Amplifier Properties to Maintain Isolator Periodicity.

Tuned Mass(g)

A(cm)

B(cm)


103

5.7 Passive Stop Band Limitations on Embedded Amplifier Design

Another important observation can be stated regarding Figure 5.9. The isolator

transmissibility curve without embedded amplifiers shows that the passive stop band

frequency range is 500 – 2000 Hz. Although this is the frequency range of interest, the

addition of the amplifiers serves to increase the attenuation of the isolator by nearly two

Figure 5.11: Transmissibility of Example Isolator with and without Inertial Amplifierswhich Maintain Isolator Periodicity.

101

102

103

104

10-8

10-6

10-4

10-2

100

102

Frequency (Hz)

Tra

nsm

issi

bilit

y

Three Co-ResonantAbsorbers


With IdenticalInertial Amplifiers

With Non-identicalInertial Amplifiers

104

orders of magnitude over the entire range. Moreover, numerous analytical case studies

have shown that embedded amplifiers provide maximum attenuation benefits when the

original stop band frequencies span the frequency range of interest.

In Figure 5.5, the layered isolator alone has a stop band ranging from 684 Hz to

2916 Hz. Inertial amplifiers are added to lower the beginning stop band frequency to

below 500 Hz. Although the isolator attenuation is improved at frequencies from 500 to

1000 Hz, the attenuation is diminished from 1000 to 2000 Hz. Absorber performance is

therefore best when the original stop band is as close to the desired range as possible.

The following example will illustrate this point further.

Suppose that the example isolator from Figure 5.5 is now needed to be three

times stiffer, and so an elastomer with a stiffness modulus of three times the initial value

is used. The passive stop band will now range from 1140 Hz to 5011 Hz, as illustrated in

Figure 5.12. Table 5.5 summarizes the amplifier properties which lower the beginning

stop band frequency to 581 Hz.

The length of section b of the lever arm is set to be 6 cm. Because the amount of

amplifier mass should not be prohibitive, the extent to which the beginning stop band can

be lowered is limited. In Figure 5.12, the minimum desired attenuation level is not

achieved over the frequency range of interest. To be most effective, therefore, an isolator

with embedded amplifiers should have an original stop band which spans the desired

frequency range as closely as possible, given physical design constraints.

105

5.8 Fluid Elements as Efficient Implementation of Inertial Amplification

An efficient means of implementing an inertial amplification device, as in Figure

5.4, is in the form of a fluid element. Lord Corporation’s Fluidlastic® mounts, designed

Table 5.5: Summary of Inertial Amplifier Properties in Figure 5.12.

Tuned Mass(g)

a(cm)

b(cm)


Figure 5.12: Transmissibility of Stiffened Layered Isolator Example with and withoutInertial Amplifiers.

101

102

103

104

10-10

10-8

10-6

10-4

10-2

100

102

Tra

nsm

issi

bili

ty

Frequency (Hz)

With AmplifiersWithout Amplifiers


MinimumDesired

AttenuationLevel

106

to isolate primary blade passage frequencies, incorporate a dense fluid acting as a tuned

mass of a vibration absorber. A cut-away view of a fluid-filled mount is shown in Figure

5.13 [54].

These fluid-filled mounts consist of an inner cylinder and an outer cylinder

concentrically bonded with elastomers to form two chambers. The chambers are joined

by a tuning port through the inner cylinder which, when filled with a dense, proprietary

fluid, serves as a tuned mass. As the inner cylinder moves up or down, the dense fluid is

forced through the tuning port. The inertial force of the accelerated fluid cancels the

elastomeric spring force at a discrete frequency, or the isolation frequency [54, 76].

Figure 5.13: Cut-Away View of a Fluidlastic® Mount [54].

PARTITIONLESSVOLUME

COMPENSATOR(PAT> PENDING)

UPPER FLUIDRESERVOIR

INNER CYLINDER

ELASTOMERICELEMENTS

TUNINGPORT

LOWER FLUIDRESERVOIR

107

A mechanical analogy, as in Figure 5.4, is helpful in understanding the dynamics

of these mounts. Here, R is the outer-to-inner chamber area ratio, mt is the mass of the

fluid in the tuning port, and k is the elastomeric spring rate in the mount.

In the current work, the main aspect of the fluid-filled mount of interest is the

inertial amplification effect. The elastomeric spring rate would be provided by the

elastomer layer already existing in layered isolators. Therefore, the fluid element would

serve as an inertial amplifier of the metal layers and also serve as part of a vibration

absorber. The isolation frequency would be determined by the stiffness of the elastomer

layer and the amount of mass in the inner chamber of the fluid element.

The diameter of the inner chamber could perhaps be semi-actively altered to

change the tuning frequency. Because transmission noise is tonal in nature, this would be

ideal to improve gearbox isolation performance. Another issue to consider is that fact

that elastomer material properties will change with changing operating conditions.

Causes for material property variation include temperature, frequency, and dynamic

strain. A brief discussion of this issue is provided in Appendix F. As the elastomer

material properties change, the inner cylinder could be tuned to track particularly

irritating disturbances.

Chapter 6

ANALYSIS OF LAYERED ISOLATOR EFFECTIVENESS FOR

GEARBOX ISOLATION

A number of performance benefits were suggested by transmissibility curves of

layered isolators with embedded fluid elements. These plots, however, did not include

the dynamic influence of the helicopter gearbox and fuselage. A more appropriate

isolator characteristic , isolator effectiveness, was therefore defined and calculated in the

case when a layered isolator is placed between a helicopter gearbox and fuselage.

6.1 The Vibration Isolation Problem

The general vibration isolation problem is often characterized by a source, an

isolator, and a receiver, as in Figure 6.1. The components are represented here in terms

of mobility, which is defined as

VM

F= ( 6.1 )

109

Ungar et al. have thoroughly examined the high-frequency vibration isolation

problem and various parameters to measure isolator performance. The most common

measure of performance is isolator transmissibility, T, and can defined as

R

O

VT

V= ( 6.2 )

where VR is the velocity of the receiver side of the isolator, and VO is the velocity of the

source side in Figure 6.1. The transmissibility is not a property of only the isolator, but

also the source and receiver. The dependence of isolation on the entire system is perhaps

more apparent using another measure of isolator performance, that is, the isolation

effectiveness, E [77]. The effectiveness is defined as the ratio of the receiver amplitude

when the source and receiver are rigidly connected, to the receiver amplitude when an

isolator is inserted, and can be written as

Figure 6.1: Schematic and Mobility Diagram of Source, Isolator, and Receiver [77].

IsolatorSource Receiver

MISource MR

VO VR

FO FR

110

RO

R

VE

V= ( 6.3 )

where the subscript O refers to a rigid and massless connection between source and

receiver, and where the absence of the subscript refers to when an isolator is in place.

The effectiveness is thus a measure of how well an isolator performs in a given situation,

with E > 1 corresponding to improved performance as compared to a rigid connection

[77]. If a massless isolator is considered, the isolator effectiveness can be expressed as

1 I

S R

ME

M M= +

+( 6.4 )

Thus, according to Equation 6.4, the isolator mobility, MI, must exceed the sum MS + MR

considerably if the isolator is to be effective.

6.2 Effectiveness of Layered Isolators with Embedded Fluid Elements

6.2.1 Isolator Effectiveness with Non-Negligible Isolator Mass

At high frequencies, the distribution of mass in layered isolators is critical in

defining stop band locations. An expression must be derived which accounts for isolator

mass to accurately predict layered isolator effectiveness, because Equation 6.4 does not

account for mass effects.

111

The system in Figure 6.1 can be analyzed without the assumption of negligible

isolator mass if the isolator is considered to be a general linear “four-pole” system [78].

As such, the isolator effectiveness can now be expressed as

1 2 1

1 1S SR

S R b b f

M MME

M M M M M

α � �= + + +� �� + � �

( 6.5 )

where

22 1 1

1 1 1 1

b b fM M Mα� �

= −� ��

( 6.6 )

Here M1b is the mobility of the isolator measured on the source side, with the receiver

side fixed (VR=0), M1f is the same, but with the receiver side free (VR≠0), and M2b is the

mobility measured on the receiver side with the source side fixed (VO=0) [77].

6.2.2 Source and Receiver Mobility Approximations

Ohlrich examined the prediction of sound transmission through struts connecting

the gearbox and fuselage using the source descriptor method [23]. To characterize the

transmission problem, the mobilities of both the gearbox and fuselage connection points

were experimentally measured.

112

In Figure 6.2, the gearbox (source) and strut/fuselage (receiver) experimental

mobilities of a realistic ¾ -scale laboratory model of a medium sized BK117 helicopter

are plotted [23]. The experimental values were approximated linearly in log space.

These approximations were used to assess the potential effectiveness of a layered isolator

with embedded fluid elements placed in between the gearbox and fuselage.

6.2.3 Layered Isolator Effectiveness Prediction

The mobilities M1b, M1f , and M2b of the layered isolator with embedded fluid

elements described in the previous chapter can be calculated using the isolator model, and

are plotted in Figure 6.3.

Figure 6.2: Experimental Source and Receiver Mobilities: (a) … Experimental, z dir., - - -Approximation, z dir. (b) …. Experimental, z dir., - - - Approximation, z dir. [23].

Gearbox

ConnectionPoint Strut

ConnectionPoint

Fuselage

MS MR

113

The effectiveness of the isolator can then be calculated using Equations 6.5 and

6.6, and is plotted in Figure 6.4. For much of the frequency range of interest, namely 500

– 2000 Hz, the effectiveness has a value of at least 200. At tuned isolator frequencies, the

effectiveness can exceed 1000, and approach 10,000. Therefore, layered isolators with

embedded fluid elements could provide a passive or perhaps semi-active noise control

solution over a wide frequency band. Additionally, the fluid ports could be tuned to track

tonal gearbox disturbances and would enable high isolator effectiveness at discrete

frequencies.

Figure 6.3: Mobilities of Layered Isolator with Embedded Fluid Elements.

100

101

102

103

104

105

10-6

10-5

10-4

10-3

10-2

10-1

Mo

bilit

y, (

m/N

s)

Frequency (Hz)

M1bM1fM2b

114

Figure 6.4: Analytical Effectiveness of Layered Isolator with Embedded Fluid Elements.

102

103

104

100

101

102

103

104

Frequency (Hz)

Effe

ctiv

enes

s

TargetFrequency

Range

Chapter 7

EXPERIMENTAL VALIDATION OF FLUID-FILLED ISOLATOR

CONCEPT

Analytical predictions suggested that fluid elements could be employed as high

frequency inertial amplifiers. Experimental specimens were therefore designed,

fabricated, and tested to validate the analytical results. The experimental

transmissibilities of a set of fluidic layered specimens were obtained and compared to

their predicted transmissibilities.

7.1 Single-Celled Fluidic Specimen Testing

Before building a multi-layered isolator with embedded fluid elements, a single-

celled specimen was first designed and constructed to experimentally validate the

concept. In Figure 7.2 , a schematic and cross-sectional view of such a single-celled

specimen is shown. The specimen was constructed using aluminum and elastomer parts,

and water was used as the tuning fluid.

116

To produce accurate analytical predictions of the specimen’s behavior, the

elastomer material properties needed to be experimentally estimated. This was

accomplished using a simpler one-celled specimen with the same elastomer material,

illustrated in Figure 7.1. The elastomer density was measured to be 1,029 kg/m3, and the

Poisson’s ratio was assumed to be 0.499, or that of a nearly incompressible solid. The

analytical transmissibility curve, generated using the axisymmetric model, was then

positioned to match the experimental results by adjusting the shear modulus and

hysteretic loss factor. The shear modulus and loss factor which provided a reasonable fit

were 1.05 MPa and 0.12, respectively.

Figure 7.1: Transmissibility and Schematic of Single-Celled Specimen to CharacterizeElastomer.

1.E-03

1.E-02

1.E-01

1.E+00

1.E+01

100 1000 10000Frequency (Hz)

Tra

nsm

issi

bili

ty

Experimental

Analytical

To Shaker

Steel Layer

ElastomerLayer

Accelerometers1“

117

The experimental results for both specimens were obtained by attaching the

specimen to a rigid base, which in turn was attached to a mechanical shaker. The shaker

input was a series of chirp signals each spanning 200 Hz. To measure the transmissibility

of a test specimen, one accelerometer was placed on top of the metal layer, and another

was placed on the rigid base. The signals were then fed into a Fourier analyzer. In this

way, the specimen transmissibilities were directly measured.

Figure 7.2: Single-Layered Specimen with Embedded Fluid Element: (a) Illustration, (b)Cross-section.

InnerCylinder

OuterCylinder

VolumeCompensa

torFlexibleSeals

ElastomerLayer

RigidLinks

TuningPort

PressureValve

To Shaker

(a) (b)

11/4“

3/8“

1 3/4“

1 1/4“

118

Both the experimental and analytical transmissibilities for the specimen pictured

in Figure 7.2 are plotted in Figure 7.3. The curves are nearly coincident for frequencies

greater than 400 Hz, and both indicate a tuned resonance condition near 600 Hz. Two

distinct resonance peaks, however, can be observed in the experimental plot, whereas

only one is predicted analytically. This discrepancy may be due to the fact that the all

parts of the specimen are considered rigid in the model. In reality, the rigid links and

inner cylinder may not be stiff enough at high frequencies to be considered rigid, and

therefore their dynamics may not be captured in the model.

Figure 7.3: Analytical and Experimental Transmissibilities of Single-Layered Specimenwith Embedded Fluid Element.

1.E-02

1.E-01

1.E+00

1.E+01

10 100 1000Frequency (Hz)

Tra

nsm

issi

bili

ty

Experimental

Analytical

AbsorberFrequency

119

Problems with the flexible seals visible in the cross-sectional view in Figure 7.2

rendered more extensive experimental tests of this specimen difficult. Through

correspondence with Lord Corporation, a suggested internal pressure of 10 – 50 psi was

advised to ensure the desired fluid effect. The seals, however, would fail after being

exposed to a pressure of 10 psi for approximately ten minutes. The results in Figure 7.3,

however, indicate that embedded fluid elements are capable of functioning as mass

amplifying elements at high frequencies.

7.2 Three-Celled Fluidic Specimen Testing

As a result of the previous encouraging test, the design of a three-celled specimen

with fluid elements was initiated, the purpose of which would be to validate the predicted

performance benefits of fluid elements in a multi-layered specimen. Preliminary designs

were completed at Penn State in collaboration with Lord Corporation design engineers.

The final technical drawings for the specimen parts are illustrated in Appendix 5, and

were produced using Ironcad design software.

A cross section of the final design is illustrated in Figure 7.4. The design consists

of three cells, with a fluid element embedded in each metal layer. An individual cell

consists of an inner cylinder, which is rigidly connected to the layer below via two rigid

links on opposite sides. The inner diameter is gradually increased at either side of the

cylinder, to minimize fluid viscous effects. The inner cylinder is connected to the outer

120

cylinder by what is ideally a perfectly flexible seal. As a first attempt to satisfy this

design criterion, the original specimen configuration was designed using reinforced fabric

material. Reinforced fabric was presumed to be quite flexible in the configuration shown

in Figure 7.4, but also to resist any bulging that might be caused by fluid pressure

variations. This behavior is important to ensure that fluid is forced anuularly through the

inner cylinder. The design also includes a fluid reservoir which is separated from the

inner cylinder by a orifice plate, and allows for the fluid to be pressurized. The elastomer

was chosen to be similar to that used in previous specimens to help predict specimen

performance. The original design also specified Lord’s proprietary dense fluid to be

used, which has a density of 1,770 kg/m3. Although all the inner diameters in Figure 7.4

are illustrated as equal, Table 7.1 lists actual dimensions.

121

Figure 7.4: Cross-sectional View of Three-celled Specimen with Embedded FluidElements.

InnerCylinder

OuterCylinder

VolumeCompensator

Flexible Seals (k~0)

ElastomerLayer

RigidLinks

TuningPort

To Shaker

122

Similar to the single-celled fluid test, a simpler specimen was also constructed

consisting of elastomer layers of the same material and diameter of that of the fluid

specimen. In this case, a three-celled specimen was built. By matching analytical

predictions with the experimental results, the elastomer could then be characterized. The

elastomer was assumed to be incompressible, and the properties were determined to be G

= 0.73 MPa and η = 0.07. The experimental and analytical results of this specimen are

plotted in Figure 7.5.

The original configuration of the fluid-filled layered specimen was fabricated at

Lord corporation. This configuration included the use of reinforced fabric as the flexible

seals. Upon qualitative inspection, however, the seals appeared to be much stiffer than

anticipated. This observational judgement was experimental verified by measuring the

specimen’s transmissibility. In Figure 7.6, the experimental transmissibility is compared

to the design prediction. The large discrepancy between the curves suggests either that

the performance benefits of fluid-filled layered isolators are not experimentally

verifiable, or that the specimen seals were in fact much too stiff to be considered very

flexible.

Table 7.1: Dimensions of Fluid Elements in Three-layered Specimen.

OuterDiameter

(cm)

InnerDiameter

(cm)

PortLength

(cm)Cell 1 5.4 2 3

Cell 2 5.4 2.5 3

Cell 3 5.4 3 3

123

Therefore, in an attempt to increase seal flexibility, the specimen was dismantled

and the utility of a variety of candidate materials as substitute seals was assessed.

Because the specimen design includes seal clamping mechanisms to ensure a tight seal,

whatever seal material is used will be subject to a substantial compressive stress on both

the inner and outer cylinder sides. As a result of Poisson’s ratio, much of the seal

material would then be forced into the annular gap between the cylinders, and thus

drastically increase seal stiffness. Therefore, a practical seal which both held pressure

and allowed for unobstructed annular motion remained elusive at first.

Figure 7.5: Analytical and Experimental Transmissibilities of Three-celled Specimen toCharacterize Elastomer.

1.E-07

1.E-06

1.E-05

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

1.E+01

1.E+02

10 100 1000 10000Frequency (Hz)

Tra

nsm

issi

bil

ity

Experimental

Analytical

124

The first material tried was a very flexible elastomer with the same thickness as

the reinforced fabric. This material choice resulted in the identification of the Poisson’s

ratio bulging effect, and consequently a variety of other materials were tried. The next

material used was a bicycle inner tube rubber with a thickness of 1/32”, or about half of

the original reinforced fabric thickness. The bulging effect, however, remained a

problem with this material, as well. Next, a gasket material used to seal pipe joints with a

1/32” thickness was chosen. Because this material was more stiff than the inner tube

rubber, it was desired to minimize the bulging effect. Unfortunately, this material also

failed to provide the necessary flexibility. It was concluded that a much thinner seal

material would be needed.

Figure 7.6: Experimental and Analytical Transmissibility Comparison of Original Three-celled Configuration.

1.E-05

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

1.E+01

1.E+02

10 100 1000 10000Frequency (Hz)

Tra

nsm

issi

bil

ity

Experimental

Analytical

PredictedAbsorber

Frequencies

PerformanceDiscrepancy

125

Consequently, a durable nylon with a thickness of about 1/64” was tried. This

material provided much flexibility, although its ability to hold pressure remained in

doubt. In the first attempt with the nylon, only one ply was used per seal and a flexible

silicon adhesive was used to adhere the seals to the metal. The test cell was pressurized

with air up to 50 psi, and the nylon appeared to withstand the higher pressure. When the

inner cylinder was moved annularly, however, the pressure escaped. Upon inspection of

the seal, it was concluded that the pressure escaped from the stretching of one of the

screw holes necessary for the clamping mechanism. Therefore, another seal attempt was

made using a strong, non-flexible epoxy to bond the seal and metal. Also, two nylon

plies were used per seal to ensure their integrity. Initially, this configuration appeared

successful, but after pressurizing the cell to 30 psi, the air began to leak. Because the

strong epoxy ensured that the nylon was not slipping under the metal clamps, the air was

assumed to have escaped from small gaps in the bonds. This problem was solved by

using a permanent, aerosol sealant to close any remaining holes in the seals. The

resulting configuration provided airtight seals at high pressures, while also allowing for

the inner cylinders to readily deflect. The inner cylinders were then filled with water and

pressurized to a working pressure of 20 psi. Next, the metal bases of each outer cylinder

were bonded to elastomer layers. The cells were then bonded together from the base cell

up to allow for transmissibility testing of one, two, and finally the whole three-celled

specimen.

126

In Figure 7.7, the analytical and experimental transmissibilities of the base cell

are plotted. The fluid element dimensions in the analytical model are again those listed in

Table 7.1, with the base cell denoted as Cell 3. The fluid density is now that of water, or

1,000 kg/m3. The mass of the outer cylinder and connected inner parts was measured to

be 1.4 kg. As can observed in the figure, both curves agree quite well for frequencies

below the tuned absorber frequency at around 700 Hz. After the absorber frequency,

however, the experimental plot rises at a steeper slope than the analytical. This

discrepancy may be a result of failing to accurately model the fluid dynamics after the

anitresonance point, such as not accounting for wave dynamics of the water.

Figure 7.7: Analytical and Experimental Transmissibilities of Single-celled Specimenwith Embedded Fluid Element.

1.E-02

1.E-01

1.E+00

1.E+01

1.E+02

10 100 1000 10000Frequency (Hz)

Tra

nsm

issi

bil

ity

Experimental

Analytical AbsorberFrequency

127

For instance, the tuning port of the inner cylinder is bound on both sides by flat

surfaces, and the fluid can be considered as a simple column of water bound on both

ends. In a simple column of fluid, standing waves develop at certain resonance

frequencies when excited. Standing waves occurring in the fluidic isolator could cause

an increase in the overall isolator transmissibility.

The lowest frequency in which a standing wave could occur is given by

11 2w wc c

fLλ

= = ( 7.1 )

where f1 is the first standing wave frequency, λ1 is the associated wavelength, cw is the

speed of sound in water at 20° C (1482 m/s), and L is the total length of the inner column

of water. For inner column length of L = 0.043 m, the first standing wave frequency is f1

= 17,300 Hz. The frequency range of interest for these isolators is 500 – 2000 Hz, and

therefore axial standing waves should not be the cause of the transmissibility

discrepancy. Coupling of the lateral and axial motion of the water, however, may be the

cause. A detailed computational analysis of the fluid dynamics may be necessary to

determine how the fluid motion may be affecting the measured isolator transmissibility.

128

In Figure 7.8, the analytical and experimental transmissibilities of two cells (Cells

2 and 3 in Table 7.1) are plotted. Again, good agreement can be observed between the

two curves before the absorber frequencies, which the analytical model predicts to be

around 600 and 700 Hz. At higher frequencies the transmissibilities once more diverge.

The roll-off rate of the two-celled specimen is significantly higher than the one-celled

case, as can be observed in Figure 7.10. It should be noted that the metal layer mass of

the base cell has now increased to 2.0 kg because of the addition of the rigid links and

also the inner cylinder of the second cell, which are both rigidly connected to the base

outer cylinder.

Figure 7.8: Analytical and Experimental Transmissibilities of Two-celled Specimen withEmbedded Fluid Elements.

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

1.E+01

1.E+02

10 100 1000 10000Frequency (Hz)

Tra

nsm

issi

bil

ity

Experimental

AnalyticalAbsorber

Frequencies

129

Finally, the analytical and experimental transmissibilities of the entire three-celled

specimen are plotted in Figure 7.9. As with the previous two figures, the two curves

indicate a close agreement between predicted and actual performance. Tuned absorber

frequencies are observed in the experimental plot at 500, 600, and 700 Hz, but are not

well defined due to elastomer hysteretic damping. Again, the experimental

transmissibility rises faster than the analytical after the absorber frequencies. A more

detailed model may be required to accurately capture the fluid dynamics at these higher

frequencies. However, the basic predicted performance benefits of embedded fluid

elements in layered isolators are clearly validated. Namely, they are the inertial

amplification effect to lower the first n isolator eigenvalues and the tuned absorber effect.

Figure 7.9: Analytical and Experimental Transmissibilities of Three-celled Specimenwith Embedded Fluid Elements.

1.E-05

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

1.E+01

1.E+02

10 100 1000 10000Frequency (Hz)

Tra

nsm

issi

bil

ity

Experimental

Analytical

AbsorberFrequencies

130

In Figure 7.10, the experimental transmissibilities of the one, two, and three-

celled fluid-filled specimens are compared. Similar to earlier tests of non-fluidic

specimens with differing number of cells, the addition of cells lowers the attenuation in

the stop band and increases the initial roll-off rate of the stop band. Also, the tuned

absorber frequencies do not change with the addition of more cells, but the attenuation at

those frequencies increases with increasing cells.

The effect of the fluid elements on specimen transmissibility can be observed in

Figure 7.11. If the fluid elements are removed in the analytical model, then the

beginning stop band frequency will be increased from 340 to 460 Hz. Additionally, there

will be no absorber effect at discrete frequencies.

Figure 7.10: Experimental Transmissibility Comparison of One, Two, and Three-celledSpecimens with Embedded Fluid Elements.

1.E-05

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

1.E+01

1.E+02

10 100 1000 10000

Frequency (Hz)

Tra

nsm

issi

bil

ity

1 Cell

2 Cells

3 Cells

131

In Figure 7.12, the specimen transmissibilities for the original Lord configuration

and the rebuilt configuration are plotted. Because the fluid in the original configuration

is nearly twice as dense as the water used the rebuilt version, the model predicts that the

original version should have a lower frequency stop band than the latter version. In the

figure, however, the original configuration is clearly much stiffer than the second

configuration, indicating that the reinforced fabric seals are not appropriate for this

application. This comparison serves to emphasize the importance of constructing such

isolators with seals that have very low stiffness relative to the elastomer layers.

Figure 7.11: Transmissibility Comparison of Three-celled Specimen with and withoutEmbedded Fluid Elements.

1.E-08

1.E-07

1.E-06

1.E-05

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

1.E+01

1.E+02

10 100 1000 10000Frequency (Hz)

Tra

nsm

issi

bil

ity

AbsorberFrequencies

Experimental (Fluid)

Analytical (Fluid)

Analytical (No Fluid Elements)

132

Figure 7.12: Transmissibilities of Three-Celled Specimen in Original Configuration andRebuilt Configuration.

1.E-05

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

1.E+01

1.E+02

10 100 1000 10000Frequency (Hz)

Tra

nsm

issi

bil

ity

Original Configuration

Rebuilt Configuration

133

Chapter 8

CONCLUSIONS AND RECOMMENDATIONS FOR FUTURE WORK

8.1 Conclusions

A Ritz approximation method was developed to model the axisymmetric dynamic

behavior of layered isolators. A single cell was modeled in a fixed-free condition, the

first two modes of which are subsequently used to provide estimates for the beginning

and end stop band frequencies. To accurately predict the first two natural frequencies of

a cell to within 5% error, a certain minimum power of the variables, ‘r’ and ‘z’, was

required for the both the radial and axial directions. For the radial direction, the powers

of ‘r’ and ‘z’ were two and three, respectively. For the axial direction, they were one and

four, respectively.

An n-celled isolator model was developed using the Ritz approximation method

combined with a modal synthesis method. The natural frequencies were found to agree

with 2-D axisymmetric finite element predictions. The model enabled the prediction the

134

nth and (n+1)th isolator modes, which correspond to the stop band beginning and end

frequencies.

The isolator model was validated with experiments. Experimental and analytical

transmissibilities were compared for two specimens with the same elastomer, but

different shape factors. The elastomer properties used for analytical predictions were

determined by matching analytical and experimental transmissibilities of the first

specimen. The properties were then used to predict the behavior of the second specimen.

In both cases, the transmissibilities before the start of the stop band show close

agreement. For the four and three cell cases, analytical transmissibilities lie below the

experimental results, although the experimental noise floor may affect these results. A

minimum experimental transmissibility of about 1 x 10-4 was observed for these three and

four cell cases.

Experimental and analytical transmissibilities were also compared for two

specimens fabricated with two different low modulus elastomers, one highly damped, and

one lowly damped. The experimental results show that stop band effectiveness is not

appreciably affected by addition of modest damping.

The physical design constraints of the proposed helicopter gearbox isolators were

estimated. Namely, constraints associated with isolator mass, axial stiffness, geometry,

and elastomeric fatigue were gathered from the literature and correspondence with

industry. The passive performance limits of layered isolators were then determined using

a design optimization methodology employing a simulated annealing algorithm. A series

of design runs were performed with target beginning and end stop band frequencies of

135

400 Hz and 2000 Hz, respectively. The results suggest that layered isolators cannot

always meet frequency targets given a certain set of design constraints. Most commonly,

the beginning stop band frequency cannot be lowered sufficiently to attenuate lower

frequencies effectively (~500 Hz).

Many active control schemes were considered to improve isolator performance at

frequencies below the beginning stop band frequency. Although effective, a purely

passive, or perhaps semi-active solution was desired to reduce complexity.

The use of embedded inertial amplifiers was found to exhibit a combination of

advantageous effects. The first benefit was a lowering of the beginning stop band

frequency, and thus a widening of the original stop band. The second was a tuned

absorber effect, where the elastomer layer stiffness and the amplified tuned mass

combined to act as a vibration absorber within the stop band. The use of embedded fluid

elements was identified as an efficient means of implementing inertial amplification.

The effectiveness of such passively enhanced isolators was analytically

determined in terms of source and receiver mobilities, where an effectiveness greater than

unity is desirable. In a realistic design case, the example isolator effectiveness was found

to have values of at least 200 over the frequency range of interest. At tuned isolator

frequencies, the effectiveness could exceed 1000. Thus, layered isolators with embedded

fluid elements appear to provide a passive noise control solution over a wide frequency

band, and, in addition, enable high isolator effectiveness at particularly troublesome

frequencies. In future isolators, the fluid ports could be semi-actively tuned to track tonal

136

disturbances and account for any changes in elastomeric stiffness properties caused by

varying flight conditions.

A series of experimental tests were performed to validate the analytical

predictions for fluid-filled isolators. A preliminary single-celled fluid specimen was

designed and constructed at Penn State to determine whether or not embedded fluid

elements could be used as inertial amplifers and mass elements of tuned absorbers at high

frequencies. The experimental and analytical transmissibility curves for this specimen

were nearly colinear for the entire frequency test range (10 – 800 Hz), although two

distinct resonance peaks were observed in the experimental plot, whereas only one was

predicted analytically. This behavior discrepancy was assumed to be caused by a lack of

rigidity in the specimen.

As a result of this test, the design of a three-layered fluid specimen was initiated,

the purpose of which was to validate the predicted performance benefits of fluid elements

in a multi-layered specimen. Preliminary designs were completed at Penn State in

collaboration with Lord Corporation design engineers. In the initial design, reinforced

fabric was assumed to be a suitable material for a flexible seal. Upon obtaining

experimental data, however, the fabric proved to be too stiff. The utility of a number of

candidate materials as seals was therefore assessed, and a thin, durable nylon material

was finally chosen. The specimen was then reconstructed and water was used as the

tuning fluid.

Experimental transmissibility of one, two, and three cells of the fluid specimen

was obtained and compared with analytical predictions. The curves for all cases show

137

good agreement, although there was some divergence after the tuned absorber

frequencies, which occur at around 500, 600, and 700 Hz. The results indicate that the

basic predicted performance benefits of embedded fluid elements in layered isolators

were validated.

When the curves for differing number of cells were compared, the curves reveal

that the addition of cells lowered the attenuation in the stop band and increased the initial

roll-off rate of the stop band. Also, the tuned absorber frequencies did not change with

the addition of more cells, but the attenuation at those frequencies increased with

increased number of cells.

A comparison plot between the original isolator configuration with stiffer fabric

seals and the later reconstructed version with more flexible nylon seals reveals that seal

stiffness is a crucial design aspect of high frequency fluid isolators. The onset of the stop

band differed by nearly 200 Hz between the two curves, and the location of the tuned

frequencies differed by nearly 600 Hz.

The development of an axisymmetric approximation method provides a new,

elegant, and efficient means to predict the high frequency, stop band behavior of layered

isolators in compression. Such isolators are ideally suited for a number of broadband

high frequency isolation design problems, particularly those that require high isolator

stiffness as is the case with the helicopter gearbox isolation problem. When partnered

with the simulated annealing design optimization routine, the new design method is

138

capable of determining optimized designs for a given array of design constraints. Thus,

the method could be used to design layered isolators for nearly any application.

The layered isolator design problem examined in this research effort involves a

unique set of constraints. The extremely high axial stiffness requirement for helicopter

gearbox supports results in layered isolators with beginning stop band frequencies that

are often too high (> 500 Hz). A novel solution to lower the stop band was therefore

pursued which combines mass amplification and the layered isolator concept, where the

mass amplification is provided by fluid elements. Fluid elements have been used in low

frequency isolators (< 50 Hz), but their use in higher frequency applications has not been

investigated previously. By utilizing the elastomer layers already present in layered

isolators as stiffness elements, the fluid elements not only provide mass amplification, but

also behave as high frequency vibration absorbers within the stop band. The predicted

performance of such modified layered isolators was validated experimentally. This work

has therefore yielded the development and validation of a design method for passive, high

frequency, broadband isolators suitable for helicopter gearbox isolation, as well as a

novel solution to a variety of other high frequency isolation problems.

8.2 Recommendations for Future Work

Several recommendations for future work of this research effort can be suggested

to expedite the realization of the use of layered isolators in helicopter gearbox mountings.

139

8.2.1 Light, Compact Isolator Design

In some applications, including in helicopters, a possible drawback to the use of

high-frequency fluidic isolators could be their weight and height. Because their design

necessitates rigid links between layers, initial proof-of-concept specimens were designed

with the links as outer concentric parts. This meant that not only was the overall isolator

diameter increased, but the need to attach the links to the middle of the inner cylinders

resulted in a minimum inner cylinder length. In the initial designs, it is this minimum

length which constrains the overall isolator length. The rigid outer links also add much

unwanted weight to the isolator, particularly when they are metal links, as in the case of

the experimental three-layered specimen.

In Figure 8.1, a conceptual alternative design is illustrated. In this design, the

rigid links are attached to the bottom of the inner cylinders and they pass through both the

metal and elastomer layers. The flexible seals would now be located between the bottom

metal layers and the rigid links. An isolator of this sort could have a much smaller

overall height than those with configurations similar to those of previous test specimens.

Also, because the rigid links would be linking the inner cylinders to the layer below in a

straight path, the links could be made significantly smaller and therefore, lighter.

140

8.2.2 Expansion of Design Optimization Routine

The design optimization routine was designed to optimize layered isolators

without considering embedded fluid elements. The routine could be expanded to include

fluid element design variables to account for such performance criteria as depth of stop

band around target antiresonance frequencies and stop band frequency locations. The

fluid element amplification ratio and tuning port length could be also be optimized.

Figure 8.1: Conceptual Configuration for Lower Fluidic Isolator Height.

InnerCylinder

OuterCylinder

Flexible Seals(k~0) Elastomer

Layer

RigidLinks

TuningPort

Flexible Layer(k~0)

141

8.2.3 Stiffness and Fatigue Testing.

Two of the most important performance characteristics of layered helicopter

gearbox isolators would be their axial stiffness and fatigue limits. Stiffness testing of

layered specimens should be conducted to verify the actual quasi-static stiffness is as

predicted using elastomer shape factor. The effect of precompression on isolator

stiffness, briefly discussed in Appendix F, should be examined more thoroughly. Also, a

means to constrain or perhaps stiffen the isolators in the lateral direction should be

considered. A more rigorous investigation into fatigue limits should be pursued, as well.

Both of these tasks should probably be performed in coordination with the Lord

Corporation, because of its extensive knowledge of elastomer behavior.

8.2.4 Conceptual Strut/Isolator Configuration

As the only rigid connection between the fuselage and gearbox in many

helicopters, support struts carry the entire weight of the fuselage in addition to any forces

developed during transitional flight conditions. These forces would primarily result in a

tensile load in the strut. When not in flight, however, these struts support the entire

weight of the gearbox and rotor, and thus are subjected to compressive loads. Therefore,

this change in force must be considered if an isolator is to be placed in series between the

fuselage and gearbox.

142

In Figure 8.2.a, a conceptual gearbox support strut with an embedded layered

isolator is illustrated. The design consists of a set of outer and inner cylindrical sections,

with the isolator embedded in the inner cylinder. The design is such, that if either a

tensile or compressive load is applied to the strut, the entire magnitude of the force is

transferred through the isolator as a compressive force. Because failure of a layered

isolator in tension would rely on the integrity of an elastomer-metal bond alone, aircraft

safety design guidelines and certification would require that such an isolator remain in

compression.

143

In Figure 8.2.b , the load path for a compressive load is highlighted through the

strut. The schematic illustrates how the load would travel through the top portion of the

outer cylinder, then would be transferred through the isolator, and finally would pass

Figure 8.2: Conceptual Strut Configuration to Ensure Compressive Loads on EmbeddedLayered Isolator.

Load Path inCompression

Load Pathin Tension

ConceptualGearbox SupportStrut with Layered

Isolator

DisplacementStops

InnerCylinder

OuterCylinder Isolator in

Compression

(a) (b) (c)

144

through the lower portion of the inner cylinder. Figure 8.2.c illustrates the load path for

when tensile load is applied to the strut. The load will again be passed through the upper

portion of the outer cylinder. Next, the load will be transferred through bolts, which will

serve to compress the isolator, and are connected to the opposite end of the isolator.

Finally, the load will travel through the entire length of the inner cylinder. In this way,

regardless of the sign of the load on the strut, and compressive load will be applied to the

isolator.

8.2.5 Construction of a Demonstration Layered Helicopter Gearbox Isolator

Upon completion of the above recommendations, a proof-of-concept gearbox

isolator/mounting should be designed, fabricated, and tested to demonstrate the efficacy

using layered isolators for helicopter gearbox isolation. The design should satisfy all

mounting performance criteria of a specific helicopter model.

8.2.6 Semi-active Tuning of Fluid Isolator Inner Diameter

A long term research task should be the development of means to semi-actively

tune layered gearbox isolators. As reported in the first chapter of this thesis, irritating

high-frequency gearbox noise is tonal in nature. Because various troublesome tones may

become more or less prominent in changing flight conditions, providing a means to semi-

actively tune a fluidic isolator would allow for better overall interior noise control.

145

In Figure 8.3, a conceptual cell configuration is shown involving a SMA (shape

memory alloy) wire and a contractable inner cylinder. When current is passed through

the SMA wire, it would be heated and consequently would contract axially. By attaching

a coiled wire to the contractable inner cylinder, a given cell could be semi-actively tuned

so that the absorber frequencies would coincide with whichever tones are most prominent

in the gearbox noise signature. For example, if the wire is coiled 10 times at a diameter

of 6 cm, then the total wire length would be about 180 cm. Assuming a maximum strain

of 5%, heating of the wire would cause nearly a 10 cm decrease in the inner area

circumference. This significant change in inner diameter would translate into dramatic

changes in amplification ratio, R, and thus large changes in tuning frequency. Although

SMA wires would require little power and would provide a means to tune a fluidic

isolator, any suitable electroactive material incorporated appropriately could be

envisioned to semi-actively tune the inner cylinders.

146

Figure 8.3: Conceptual Contractable Inner Diameter Using SMA Wires for Semi-activelyTuning Fluidic Isolators.

ContinuousSMA wire

OuterCylinder

ContractableInner

CylinderFlexible

SealsFixed Wire

Ends

FlexibleOverlapping

Sleeve

BIBLIOGRAPHY

1. Snowden, J.C., Vibration Isolation: Use and Characterization, Applied Research

Laboratory, The Pennsylvania State University, Sponsored by the National

Bureau of Standards, pp. 25-37, May 1979.

2. Szefi, J.T., Smith, E.C., Lesieutre, G.A., “Analysis and Design of High Frequency

Periodically Layered Isolators in Compression,” ,” 41st Structures, Structural

Dynamics and Materials Conference, April 3-6, 2000, Atlanta, GA, 2000, pp. 306-

321, 2000.

3. Szefi, J.T., Smith, E.C. and Lesieutre, G.A., “Formulation and Validation of a

Ritz-Based Analytical Model for Design of Periodically-Layered Isolators in

Compression,” 42st Structures, Structural Dynamics and Materials Conference,

Seattle, WA, April 16-19, 2001, pp. 3690-3704, 2001.

4. Ghosh, A.K., “Periodically Layered Composites for Attenuation of Dynamic

Loads,” Nuclear Engineering and Design, Vol. 84, pp. 53-58, 1985.

148

5. Sackman, J.L., Kelly, J.M. and Javid, A.E., “Layered Notch Filter for High

Frequency Dynamic Isolation,” ASME, Pressure Vessels and Piping Conference

and Exhibition, Chicago, IL, Vol. 113, pp. 55-64, 1986.

6. Sackman, J.L., Kelly, J.M. and Javid, A.E., “A Layered Notch Filter for High-

Frequency Dynamic Isolation,” Journal of Pressure Vessel Technology, Vol. 111,

pp. 17-24, 1989.

7. Mead, D.J., Passive Vibration Control, Chichester, N.Y.: Wiley, 1999.

8. Gent, A.N., Meinecke, E.A. “Compression, Bending, and Shear of Bonded

Rubber Blocks,” Poly. Eng. and Science, Vol. 10, no. 1, pp. 48-53, 1970.

9. Sheridan, P.M., James, F.O., Miller, T.S., “Design of Components” in Gent, A.N.,

Ed., Engineering with Rubber: How to Design Rubber Components, Hanser

Publishers, Munich, pp. 214-215, 1992.

10. Pollard, J.S., “Helicopter Gear Noise and its Transmission to the Cabin,” 3rd

European Rotorcraft and Powered Lift Aircraft Forum, Aix-en-Provence, France,

pp. 52.1-52.10, 1977.

11. Coy, J.J., Handschuh, R.F., Lewicki, D.G., Huff, R.G., Krejsa, E.A., Karchmer,

A.M. “Identification and Proposed Control of Helicopter Transmission Noise at

the Source,” NASA/Army Rotorcraft Technology. v.1 Aerodynamics, and

Dynamics and Aeroelasticity, Moffett Field, CA, pp. 1045-1065, 1988.

149

12. D’Ambra, F., Leverton, J.W., “Design Implications of Recent Gearbox Noise and

Vibration Studies,” 4th European Rotorcraft and Powered Lift Aircraft Forum, pp.

56.1-56.22, 1978.

13. Yoerkie, C.A., Moore, J.A., “Statistical Energy Analysis Modeling of Helicopter

Cabin Noise,” 39th Annual Forum of the American Helicopter Society, pp. 458-

471, 1983.

14. Yoerkie, C.A, Moore, J.A., Manning, J.E., “Development of Rotorcraft Interior

Noise Control Concepts, Phase I: Definition Study,” NASA Contractor Report

166101, 1983.

15. Yoerkie, C.A., Gintoli, P.J., Moore, J.A., “Development of Rotorcraft Interior

Noise Control Concepts, Phase II: Full Scale Testing,” NASA Contractor Report

172594, 1986.

16. Yoerkie, C.A., Gintoli, P.J., Ingraham, S.T., Moore, J.A., “Development of

Rotorcraft Interior Noise Control Concepts, Phase III: Development of Noise

Control Concepts,” NASA Contractor Report 178172, 1987.

17. Yoerkie, C.A., Ingraham, S.T., Moore, J.A., “Treated Cabin Acoustic Prediction

using Statistical Energy Analysis,” 43rd Annual Forum and Technology Display of

the American Helicopter Society, St. Louis, Missouri, pp. 461-468, 1987.

150

18. Morgan, H.G., Pao, S.P., Powell, C.A., “Recent Langley Helicopter Acoustics

Contributions,” NASA/Army Rotorcraft Technology v.1 Aerodynamics, and

Dynamics and Aeroelasticity, Moffett Field, CA, pp. 1003-1044, 1988.

19. O’Connell, J., “Predicting Rotorcraft Transmission Noise,” 48th Forum of the

American Helicopter Society, pp. 919-929, 1992.

20. Mathur, G.P., Manning, J.E., “Analytical Prediction and Flight Test Evaluation of

Bell ACAP Helicopter Cabin Noise,” 44th Forum of the American Helicopter

Society, pp. 719-729, 1988.

21. Brennan, M.J., Pinnington, R.J., Elliott, S.J., “Mechanisms of Noise Transmission

through Helicopter Gearbox Support Struts,” Journal of Vibrations and Acoustics,

Vol. 116, pp. 548-554, 1994.

22. Brennan, M.J., Elliott, S.J., Heron, K.H., “Noise Propagation through Helicopter

Support Struts – An Experimental Study,” Journal of Vibration and Acoustics,

Vol. 120, pp. 695-704, 1998.

23. Ohlrich, M., “Terminal Source Power for Predicting Structureborne Sound

Transmission from a Main Gearbox to a Helicopter Fuselage,” Proceedings of

Internoise 1995, pp. 555-558, 1995.

24. Marze, H.J., D’Ambra, F.N., “Helicopter Internal Noise Treatment: Recent

Methodologies and Practical Applications,” 11th European Rotorcraft Forum:

Paper No. 10, London, England, pp. 10.1-10.13, 1985.

151

25. Levine, L.S., “Reducing the Cost Impact of Helicopter Internal Noise Control,”

36th Annual Forum of the American Helicopter Society, pp. 80.59.1-80.59.9,

1980.

26. Owen, S., Woodward, M.C.A., Pollard, J.S. “Demonstration of Potential

Acoustic Gains from Conventional Cabin Soundproofing Treatments,” 4th

European Rotorcraft and Powered Lift Aircraft Forum: Paper No. 60, Stresa,

Italy, 1978.

27. Dorling, C.M., Eatwell, G.P., Hutchins, S.M., Ross, C.F., Sutcliffe, S.G.C.,

“Demonstration of Active Noise Reduction in an Aircraft Cabin,” Journal of

Sound and Vibration, Vol. 128, no. 2, pp. 358-360, 1989.

28. Elliott, S.J., Nelson, P.A., Stothers, I.M., Boucher, C.C., “Preliminary Results of

In-flight Experiments on the Active Control of Propeller-Induced Cabin Noise,”

Journal of Sound and Vibration, Vol. 128, pp. 355-357, 1989.

29. Jolly, M.R., Norris, M.A., Rossetti, D.J., “Adaptive Control of Helicopter Cabin

Noise,” Proceedings of the ASME Dynamic Systems and Control Division, Vol.

57-2, pp. 837-848, 1995.

30. Millott, T.A., Welsh, W.A., Yoerkie, C.A., MacMartin, D.G., Davis, M.W.,

“Flight Test of Active Gear-Mesh Noise Control on the S-76 Aircraft,” 54th

Annual Forum of the American Helicopter Society, pp. 241-249, 1998.

152

31. Grewal, A., Pavel, L., Zimcik, D.G., Leigh, B. Xu, W., “Application of

Feedforward and Feedback Structural Control for Aircraft Cabin Noise

Reduction,” Proceedings of the 1998 39th AIAA/ASME/ASCE/AHS/ASC

Structures, Structural Dynamics, and Materials Conference and Exhibit and

AIAA/ASME/AHS Adaptive Structures Forum, pp. 2265-2275, 1998.

32. Grewal, A., Zimcik, D.G., Leigh, B., “Feedforward Piezoelectric Structural

Control: An Application to Aircraft Cabin Noise Reduction,” Journal of Aircraft,

Vol. 38, no. 1, pp.164-173, 2001.

33. Hirsch, S.M., Jayachandran, V., Sun, J.Q., “Structural-Acoustic Control for

Quieter Aircraft Interior-Smart Trim Technology,” Composite Structures, Vol. 42,

pp. 189-202, 1998.

34. Palumbo, D., Cabell, R., Sullivan, B., “Flight Test of Active Structural Acoustic

Noise Control System,” Journal of Aircraft, Vol. 38, no. 2, pp. 277-284, 2001.

35. Sun, J.Q., Hirsch, S.M., Proceedings of SPIE - The International Society for

Optical Engineering, Vol. 3041, pp. 839-845, 1997.

36. O’Connell, J., Mathur, G., JanakiRam, R. Johnson, M., Rosetti, D.J., “Helicopter

Cabin Noise Reduction Using Active Structural Acoustic Control,” 57th Annual

Forum of the American Helicopter Society, 2001.

153

37. Fuller, C.R., Snyder, S.D., Hansen, C.H., Silcox, R.J., “Active Control of Interior

Noise in Model Aircraft Fuselages Using Piezoceramic Actuators,” AIAA

Journal, Vol. 30, no. 11, pp. 2613-2617, 1992.

38. Sun, J.Q., Norris, M.A., Rossetti, D.J., Highfill, J.H., “Distributed Piezoelectric

Actuators for Shell Interior Noise Control,” Journal of Vibration and Acoustics,

Vol. 118, pp. 676-681, 1996.

39. Gembler, W., Schweitzer, H., “Helicopter Interior Noise Reduction by Active

Gearbox Struts,” 54th Annual Forum of the American Helicopter Society, pp. 216-

229, 1998.

40. Bebesel, M., Maier, R. Hoffmann, F., “Reduction of Interior Noise in Helicopters

by Using Active Gearbox Struts – Results of Flight Tests,” 27th European

Rotorcraft Forum, pp. 1.1-1.8, 2001.

41. Sutton, T.J., Elliott, S.J., Brennan, M.J., “Active Isolation of Multiple Structural

Waves on a Helicopter Gearbox Support Strut,” Journal of Sound and Vibration,

Vol. 205, no. 1, pp. 81-101, 1997.

42. Pelinescu, I., Balachandran, B., “Active Control of Vibration Transmission

through Struts,” Proceedings of SPIE, Vol. 3329, pp. 596-607, 1998.

43. Pelinescu, I., Balachandran, B., “Control of Wave Transmission through Struts,”

Proceedings of the SPIE Conference on Smart Structures and Integrated Systems,

Newport Beach, CA, 80-91, 1999.

154

44. Pelinescu, I., Balachandran, B., “Analytical and Experimental Investigations into

Active Control of Wave Transmission through Gearbox Struts,” Proceedings of

SPIE, Vol. 3985, pp. 76-85, 2000.

45. Pelinescu, I., Balachandran, B., “Analytical Study of Active Control of Wave

Transmission through Cylindrical Struts,” Smart Materials and Structures, 10,

121-136, 2001a.

46. Pelinescu, I., Balachandran, B., ”Experimental Study of Active Control of Wave

Transmission through Hollow Cylindrical Struts,” Proceedings of SPIE, Vol.

4327, pp. 1-12, 2001b.

47. Mahapatra, D.R., Gopalakrishnan, S., Balachandran, B., “Active Feedback

Control of Multiple Waves in Helicopter Gearbox Support Struts,” Smart

Materials and Structures, Vol. 10, pp. 1046-1058, 2001.

48. Ortel, D.J., Balachandran, B., “Control of Flexural Wave Transmission through

Struts,” Proceedings of the SPIE Conference on Smart Structures and Integrated

Systems, Newport Beach, CA, pp. 567-577, 1999.

49. Vincent, Phil. Mag.,Vol. 46, pp. 537, 1898.

50. Lawry, M.H., I-DEAS� Master Series, Student Guide, Structural Dynamics

Research Corporation, 1997.

155

51. Heyliger, P.R. “Axisymmetric Free Vibrations of Finite Anisotropic Cylinders,”

Journal of Sound and Vibration, Vol. 124, pp. 507-520, 1991.

52. Hurty, W.C., “Dynamic Analysis of Structural Systems Using Component

Modes,” AIAA Journal, Vol. 3, pp. 678-685, 1965.

53. Welsh, W.A., Von Hardenberg, P.C.;Von Hardenberg, P.W., Staple, A.E., “Test

and Evaluation of Fuselage Vibration Utilizing Active Control of Structural

Response,” 46th Annual Forum Proceedings of the American Helicopter Society.

Part 1 (of 2), May 21-23 1990, Washington, DC, USA, pp. 21-37, 1990.

54. Smith, M.R., Redinger, W.S., “The Model 427 Pylon Isolation System,” 55th

Annual Forum of the American Helicopter Society, Montreal, Quebec, Canada,

1999.

55. Donguy, P., “Development of a Helicopter Rotor Hub Elastomeric Bearing,”

Journal of Aircraft, Vol. 17, No. 5, pp. 346-350, 1980.

56. Long, J.E. , Bearings in Structural Engineering, NY, New York, Halsted Press,

1974.

57. Mertz, D.R., Kulicki, J.M., “Bridge design by the AASHTO LRFD Bridge Design

Specifications,” Structures Congress - Proceedings, v1, 1996, Building an

International Community of Structural Engineers, Proceedings of the 14th

Structures Congress. Part 1 (of 2), Chicago, IL, USA, pp. 1-8, 1996.

156

58. Roeder, C.W., Stanton, J.F., “Elastomeric Bearings: State-of-the-Art,” Journal of

Structural Engineering, Vol. 109, No. 12, pp. 2853-2871, 1982.

59. Roeder, C.W., Stanton, J.F., Taylor, A.W., “Fatigue of Steel-Reinforced

Elastomeric Bearings,” Journal of Structural Engineering, Vol. 116, No. 2, pp.

407-426, 1987a.

60. Roeder, C.W., Stanton, J.F., Taylor, A.W., “Performance of Elastomeric

Bearings,” National Cooperative Highway Research Program Report, Oct, 1987,

100p, 1987b.

61. Roeder, C.W., Stanton, J.F., Feller, T., “Low-Temperature Performance of

Elastomeric Bearings,” Journal of Cold Regions Engineering, Vol 4, No. 3, pp.

113-132, 1990.

62. Roeder, C.W., Stanton, J.F., “State-of-the-Art Elastomeric Bridge Bearing

Design,” ACI Structural Journal, Vol. 88, No. 1, pp. 31-41, 1991.

63. Stanton, J.F., Roeder, C.W., “Elastomeric Bearings: An Overview,” Concrete

International: Design and Construction, Vol. 14, No. 1, pp. 41-46, 1992.

64. Stevenson, A., Campion, R.P., “Durability” in Gent, A.N., Ed., Engineering with

Rubber: How to Design Rubber Components, Hanser Publishers, Munich, pp.

179-183, 1992.

157

65. Seeley, C.E., Chattopadhyay, A., “Development of a Hybrid Technique for

Continuous/Discrete Optimization,” Proc. 6th Annual AIAA/NASA/ISSMO Symp.

On Multidisciplinary Analysis and Optimization (Bellevue, WA), 1996, pp. 1430-

1440, 1996.

66. Carlson, J.D., Jolly, M.R., “MR Fluid, Foam, and Elastomer Devices,”

Mechatronics, Vol. 10, pp. 555-569, 2000.

67. Ginder, J.M., Nichols, M.E., Elie, L.D., Tardiff, J.L., “Magnetorheological

Elastomers: Properties and Applications,” Proceedings of SPIE, Vol. 3675, pp.

131-138, 1999.

68. Ginder, J.M., Nichols, M.E., Elie, L.D., Clark, S.M., “Controllable-Stiffness

Components Based on Magnetorheological Elastomers,” Proceedings of SPIE,

Vol. 3985, pp. 418-425, 2000.

69. Jolly, M.R., Carlson, J.D., Munoz, B.C., Bullions, T.A., “The

Magnetoviscoelastic Response of Elastomer Composites Consisting of Ferrous

Particles Embedded in a Polymer Matrix,” Journal of Intelligent Material Systems

and Structures, Vol.7, pp. 613-622, 1996a.

70. Jolly, M.R., Carlson, J.D., Munoz, B.C., “A Model of the Behaviour of

Magnetorheological Materials,” Smart Materials and Structures, Vol.5, No.5, pp.

607-614, 1996b.

158

71. Davis, C., "A Tunable Piezoceramic Vibration Absorber," Ph. D. Thesis, Dept. of

Aersp. Eng., The Pennsylvania State University, University Park, PA, 1997.

72. Davis, C. L., G.A. Lesieutre, "Actively tuned Solid-State Vibration Absorber

Using Capacitive Shunting of Piezoelectric Stiffness", Journal of Sound and

Vibration, Vol. 232, No. 3, pp. 601-617, 2000.

73. Dozor, D.M., Jagannathan, S., Gerver, M.J., Fenn, R.C., Logan, D.M., Berry, J.R.,

“High Force to Volume Terfenol-D Reaction Mass Actuator: Modeling and

Design,” Proceeding of SPIE, Vol. 2717, pp. 576-586, 1996.

74. Johnson, W., “Self-tuning Regulators for Multicyclic Control of Helicopter

Vibration,” NASA Technical Paper 1996, March 1982.

75. Anusonti-Inthra, Phuriwat, “Semi-Active Control of Helicopter Vibration Using

Controllable Stiffness and Damping Devices,” Ph. D. Thesis, Dept. of Aersp.

Eng., The Pennsylvania State University, University Park, PA, 2002.

76. McGuire, D.P., “Fluidlastic® Dampers and Isolators for Vibration Control in

Helicopters,” 50th Annual Forum of the American Helicopter Society, pp. 295-

303, 1994.

77. Ungar, E.E., Dietrich, C.W., “High Frequency Vibration Isolation,” Journal of

Sound and Vibration, Vol. 4, pp. 224-241, 1961.

159

78. Molloy, C.T., Mechanical Impedance Methods, “Section 4: Four Pole Parameters

in Vibration Analysis,” ed. by R. Plunkett, New York: ASME, 1958.

79. Sjoberg, Mattias, “On Dynamic Properties of Rubber Isolators,” Ph.D. Thesis,

Dept. of Vehicle Engineering, Royal Institute of Technology, Stockholm,

Sweden, 2002.

80. Scobbo, J.J., “Characterization of Filled Elastomers by Dynamic Strain

Amplification at High Frequencies,” Polymer Testing, Vol 9, No. 6, pp.405-420,

1990.

81. Kari, L., Sjoberg, M., “Temperature Dependent Stiffness of a Precompressed

Rubber Isolator in the Audible Frequency Range,” Manuscript submitted to

International Journal of Solids and Structures, 2002.

82. Kari, L., “Dynamic Transfer Stiffness Measurements of Vibration Isolators In

theAudible Frequency Range,” J. of Noise Control Engineering, Vol. 49, No. 2,

pp. 88-102, 2001.

83. DeJong, R.G., Ermer, G.E., Paydenkar, C.S., Remtema, T.M., “High Frequency

Dynamic Properties of Rubber Isolation Elements, Proceedings – National

Conference on Noise Control Engineering, Part 1 (of 3), Apr 5-9, 1998, Ypsilanti,

MI, USA.

Appendix A

DERIVATION OF THE EQUATIONS OF MOTION FOR A

VIBRATING CYLINDER AS PRESENTED BY HEYLIGER [51]

The following derivation accompanies the discussion in of the development of an

approximation method for layered isolators presented in Chapter 2. It is adapted from

Heyliger’s original derivation presented in Ref. 51.

The non-zero strain components for axisymmetric motion can be written (in

cylindrical coordinates) as

1 ,rr

u

rε ε ∂= =

∂ 2

u

rθθε ε= = , 3 ,zz

w

zε ε ∂= =

∂ 5 rz

w u

r zε ε ∂ ∂= = +

∂ ∂( A.1 )

Hamiltion’s principle is then used to obtain the equations of motion, which can be

expressed as

{ } 2 21 1 2 2 3 3 5 5

0 0

10 ( )

2

t t

V V

dVdt u w dVdtσ δε σ δε σ δε σ δε δ ρ= − + + + + +� � � � � � ( A.2 )

161

where V is the volume of the cylinder, /u u t= ∂ ∂� , t is time, ρ is the density of the

material, σι is the ith stress component, with the subscript notation matching that of the

strains in Eq. (A.1) , and δ is the variational operator.

In this derivation, there are no external forces and the cylinder is assumed to be

stress free. The constitutive relations are written in matrix form as

11 12 131

12 22 232

3 13 23 33

4 55

000

0 0 0

rr rr

zz zz

rz rz

C C CC C CC C C

C

θθ θθ

σ σ εσ σ εσ σ εσ σ γ

� ��

= = � ��

( A.3 )

For numerical calculations, it is convenient to non-dimensionalize the cylinder

geometry by mapping the original cylinder to a cylinder with a radius and half-height of 1

using the transformations

/R r L= , / (1/ ) /rr L R∂ ∂ = ∂ ∂

/ zZ z L= / (1/ ) /zz L Z∂ ∂ = ∂ ∂( A.4 )

Here Lr is the radius of the cylinder, Lz = h/2, and h is the total height of the cylinder, as

shown in Figure A.1.

162

The domain of integration is also transformed and can be rewritten as

2r zdV rdrd dz RL L dRd dZθ θ= = ( A.5 )

For convenience, it has been assumed that this transformation has been completed, but

the upper-case nomenclature is dropped for the transformed coordinates.

The variational form of the governing equations can be derived by substituting

Eqs. (A.1) and (A.3) into Eq. (A.2). The variational form of the equations of motion is

Figure A.1: Geometry of Cylinder

2Lr

2Lz

z

r

163

11 12 122 2 2

1 10

r r rV

C u u C u C uu u

L r r r L r r L r

δ δδ� ∂ ∂ ∂ ∂= + +� ∂ ∂ ∂ ∂�

�

13 13 231 1

r z r z r z

C C Cu w w wu u

r L L r z L L z r L L z

δ δδ∂ ∂ ∂ ∂+ + +∂ ∂ ∂ ∂

3322552 2 2

1 1 1

r z r z

CC w w w uu u C

r L L z z L r L z

δδ� �∂ ∂ ∂ ∂+ + +� �∂ ∂ ∂ ∂� �

1 1( )

r z

w uu u w w dV

L r L z

δ δ ρ δ δ�� ∂ ∂ �× + + + �� ∂ ∂ ��

��

( A.6 )

in V. Additionally, by integrating by parts the boundary conditions can be found to be

specify u or

11 12 13 r

u wC C u C n

r z

∂ ∂� �+ +� �∂ ∂� �55 z rr r rz z

w uC n n n

r zσ σ∂ ∂� �+ + = +� �∂ ∂� �

( A.7 )

specify v or

55 13 23 33

1r z

w u u wC n C C u C n

r z r r z

∂ ∂ ∂ ∂� � � �+ + + +� � � �∂ ∂ ∂ ∂� � � �rz r zz zn nσ σ= + ( A.8 )

on Γ. Here Γ represents the surface of the cylinder , and nr and nz are the components of

the outward unit normal vector to this surface in r and z directions. For this work, u and v

are both specified.

Appendix B

EXPLICIT FORMS OF [M] AND [K] MATRICES AS REPORTED BY

HEYLIGER [51]

The following matrix definitions accompany the discussion in of the development

of an approximation method for layered isolators presented in Chapter 2, and were

originally presented by Heyliger in Ref. 51. The axisymmetric matrices, [M] and [K], are

explicitly defined for a vibrating isotropic cylinder.

The matrices in the eigenvalue problem can be written as

{ }{ }

{ }{ }

{ }{ }

11 12 112

21 22 22

0000

b bK K Md dK K M

ρω� � � � � �� − =� � � � � ��

� � � � �( B.1 )

Here

11 u uij i j

V

M rdrd dzφ φ θ= � ( B.2 )

165

22 w wij i j

V

M rdrd dzφ φ θ= � ( B.3 )

11 11 122 2

1u uu uj ju ui i

ij j ir rV

C CK

L r r r L r r

φ φφ φ φ φ� � �∂ ∂∂ ∂= + +� � �� ∂ ∂ ∂ ∂� � ��

�

55222 2 2

1uuju u i

i jr z

CCrdrd dz

r L L z z

φφφ φ θ�∂∂+ + �∂ ∂ ��

( B.4 )

12 13 23 551w w wu uj j jui i

ij ir z r z r zV

C C CK

L L r z r L L z L L z r

φ φ φφ φφ� �∂ ∂ ∂∂ ∂= + +� �∂ ∂ ∂ ∂ ∂� ��

�

21( ) jirdrd dz Kθ× =

( B.5 )

22 33 552 2

w ww wj ji i

ijz r

C CK rdrd dz

L z z L r r

φ φφ φ θ� �∂ ∂∂ ∂= +� �∂ ∂ ∂ ∂� ��

( B.6 )

Appendix C

MATLAB® CODE: DESIGN OPTIMIZATION FOR LAYERED

ISOLATORS IN COMPRESSION

The following is the MATLAB® code for the design optimization of layered

isolators in compression. The code consists of a main program, and functions ‘initial’,

‘fcomp’, ‘calc3D’, ‘calcaxstiff2’, and ‘calclatstiff2’. Design constraints and objectives

are specified in the function ‘initial’.

Main Program

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% "Design of periodically layered isolators in compression using simulated annealing"%% Rotorcraft Center of Excellence% Penn State University% Joseph T. Szefi% [email protected]% 9/1/2003%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

clear all;close all;

% Grub=x(1);% Gmet=x(2);% rhorub=x(3);% rhomet=x(4);% n=x(5)% trub=x(6);% tmet=(h-n*trub)/n;% d = x(7);% h = x(8);

% Get variable values from 'intial.m'[btarget, etarget, masstarg, alphab, alphae, beta, minaxstiff, maxaxstiff,... minlatstiff, maxlatstiff, maxatten, Grubmin, Grubmax, Gmetmin, Gmetmax, rhorubmin, rhorubmax,... tmetmin, trubmin, rhometmin, rhometmax, nmin, nmax, dmin, dmax, hmin, hmax, elastomer, maxmass,Fcompress] = initial;

% Open output filefid = fopen('compout.m','w');

% Define optimization parameter valuesnvar = 8;eps = 1E-4;epsr = 1E-3;step = 1;

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% The following variables determine the amount of parameter% sampling during optimization.% ntp = overall iterations a given 'temperature'% ncyc = # of variable samplings at a given temperature%

167

% Increasing either will, in general, increase running time, but will also better% ensure that a minimum is found

ntp = 20;ncyc = 10;

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

ntot = 1000000;Irepeat = 10;

% Read intial temperature, reduction parameter, step factorT = 200;rp = 0.6;sf = 2;rs = 0.8;

% Check for number of layer errorsif nmin < 2 fprintf('Error: Check minimum number of layers\n'); fprintf(fid,'Error: Check minimum number of layers\n'); fclose(fid); break;end

if (nmax > 15) fprintf('Error: Check maximum number of layers\n'); fprintf(fid,'Error: Check maximum number of layers\n'); fclose(fid); break;end

if (nmin > nmax) fprintf('Error: Minimum number of layers greater than maximum number of layers\n'); fprintf(fid,'Error: Minimum number of layers greater than maximum number of layers\n'); fclose(fid); break;end

nmin = nmin - .5;nmax = nmax + 0.4999;

% Define min and maxxmin(1,1) = Grubmin;xmin(1,2) = Gmetmin;xmin(1,3) = rhorubmin;xmin(1,4) = rhometmin;xmin(1,5) = nmin;xmin(1,6) = trubmin;xmin(1,7) = dmin;xmin(1,8) = hmin;

xmax(1,1) = Grubmax;xmax(1,2) = Gmetmax;xmax(1,3) = rhorubmax;xmax(1,4) = rhometmax;xmax(1,5) = nmax;xmax(1,7) = dmax;xmax(1,8) = hmax;

for count = 1:nvar if count==6 x(5) = round(x(5)); xmax(1,6)=(hmax-tmetmin*x(5))/x(5); end x(count) = xmin(count) + rand(1)*(xmax(count)-xmin(count));end

xopt = x;xs = x;for c = 1:nvar, st(c) = step*(xmax(c)-xmin(c)); ar(c) = 1;endnfv = 1;

n = xs(5);h = xs(8);

tmet = (h-xs(6)*xs(5))/xs(5);trub = xs(6);

if (tmet < tmetmin)|(tmet < 0) xs(6)=(h-tmetmin*n)/n;end

if (trub < trubmin) xs(6)=trubmin;end

F = fcomp(xs, btarget, etarget, masstarg, alphab, alphae, beta,... minaxstiff, maxaxstiff, minlatstiff, maxlatstiff, maxatten, maxmass,Fcompress);

Fmin = F;ict = 0;Fold = F;cr = 0;firsttry = 1;x=xs;while (1 > 0) d = x(7); h = x(8); axstiff = calcaxstiff2(x); latstiff = calclatstiff2(x); n=x(5); mass = n*d^2*pi/4*((h-n*x(6))/n*x(4) + x(6)*x(3)); atten = 0; SF = d/4/x(6); minSF = Fcompress/(d^2*pi/4)/2/x(1); [threedstart,threedend] = calc3D(x); fprintf('-------------------------------------------------------------------------------------------------------------------------\n'); fprintf(' Iteration'); fprintf(fid,'-------------------------------------------------------------------------------------------------------------------------\n'); fprintf(fid,' Iteration'); if firsttry == 1 fprintf(' 0 | %6.0f | %6.0f |\n', btarget, etarget); fprintf(fid,' 0 | %6.0f | %6.0f |\n', btarget, etarget); firsttry = 0; else if cr < 10 fprintf(' %0.0f | %6.0f | %6.0f |\n', cr, btarget, etarget);

168

fprintf(fid,' %0.0f | %6.0f | %6.0f |\n', cr, btarget, etarget); else fprintf(' %0.0f | %6.0f | %6.0f |\n', cr, btarget, etarget); fprintf(fid,' %0.0f | %6.0f | %6.0f |\n', cr, btarget, etarget); end end fprintf('-------------------------------------------------------------------------------------------------------------------------\n'); fprintf('| # of cells | Elas. thick. | Met. thick. | Beg. Freq. | End Freq. | Elas. rho | Met. rho | Isolator Mass \n'); fprintf('| | (cm) | (cm) | (Hz) | (Hz) | (kg/m^3) | (kg/m^3) | (kg)\n'); fprintf('-------------------------------------------------------------------------------------------------------------------------\n'); fprintf('%8.0f %15.3f %16.3f %14.0f %14.0f %13.0f %13.0f %14.2f\n',n, (x(6)*100),((h-x(5)*x(6))/x(5)*100), threedstart,threedend,x(3),x(4),mass); fprintf('=========================================================================================================================\n'); fprintf('| G Elas. | G Met. | Axial Stiff. | Lat. Stiff. | d | h | S.F. | min. S.F. \n'); fprintf('| (MPa) | (GPa) | (MN/m) | (MN/m) | (cm) | (cm) | | \n'); fprintf('-------------------------------------------------------------------------------------------------------------------------\n'); fprintf('%8.2f %16.2f %16.2f %14.2f %13.2f %14.2f %15.2f %14.2f\n',(x(1)/1e6),(x(2)/1e9),(axstiff/1e6),(latstiff/1e6),d*100,h*100,SF,minSF); fprintf('\n'); fprintf(fid,'-------------------------------------------------------------------------------------------------------------------------\n'); fprintf(fid,'| # of cells | Elas. thick. | Met. thick. | Beg. Freq. | End Freq. | Elas. rho | Met. rho | Isolator Mass \n'); fprintf(fid,'| | (cm) | (cm) | (Hz) | (Hz) | (kg/m^3) | (kg/m^3) | (kg)\n'); fprintf(fid,'-------------------------------------------------------------------------------------------------------------------------\n'); fprintf(fid,'%8.0f %15.3f %16.3f %14.0f %14.0f %13.0f %13.0f %14.2f\n',n, (x(6)*100),((h-x(5)*x(6))/x(5)*100), threedstart,threedend,x(3),x(4),mass); fprintf(fid,'=========================================================================================================================\n'); fprintf(fid,'| G Elas. | G Met. | Axial Stiff. | Lat. Stiff. | d | h | S.F. | min. S.F. \n'); fprintf(fid,'| (MPa) | (GPa) | (MN/m) | (MN/m) | (cm) | (cm) | | \n'); fprintf(fid,'-------------------------------------------------------------------------------------------------------------------------\n'); fprintf(fid,'%8.2f %16.2f %16.2f %14.2f %13.2f %14.2f %15.2f%14.2f\n',(x(1)/1e6),(x(2)/1e9),(axstiff/1e6),(latstiff/1e6),d*100,h*100,SF,minSF); fprintf(fid,'\n'); cr = cr + 1; track(cr,1:8) = x; track(cr,9) = (h-x(5)*x(6))/x(5); track(cr,10) = F; for itp = 1:ntp,

for icyc = 1:ncyc,

for ih = 1:nvar,

xs(ih) = x(ih) + (2*rand(1)-1) * st(ih);

if ih==5 xs(5) = round(xs(5)); end

if ih==6 xmax(1,6)=(h-tmetmin*xs(5))/xs(5); end

if (xs(ih) < xmin(ih)) | (xs(ih) > xmax(ih))

xs(ih) = xmin(ih) + rand(1) * (xmax(ih)-xmin(ih));

% Round n to nearest variable if ih==5 xs(5) = round(xs(5)); xmax(1,6) = (hmax-tmetmin*xs(5))/xs(5); oldnumofcells = xs(5); end end

n = xs(5); h = xs(8);

%xs(6) = .0062;

tmet = (h-xs(6)*xs(5))/xs(5); trub = xs(6);

%Make sure that thicknesses are positive

if (tmet < tmetmin) | (tmet < 0) xs(6)=(h-tmetmin*n)/n; end

if (trub < trubmin) xs(6)=trubmin; end

Fs = fcomp(xs, btarget, etarget, masstarg, alphab, alphae, beta,... minaxstiff, maxaxstiff, minlatstiff, maxlatstiff, maxatten, maxmass,Fcompress);

nfv = nfv + 1; if Fs <= F; %Point is accepted %fprintf('\nLower Point Accepted\n'); x(ih) = xs(ih); F = Fs; if Fs < Fmin %Set to minimum for c2 = 1:nvar, xopt(c2) = xs(c2); end Fmin = Fs; end else %Apply Metropolis criterion for acceptance P = exp((F-Fs)/T); if rand(1) < P x(ih) = xs(ih); F = Fs; else %Rejection of a point xs(ih) = x(ih); ar(ih) = ar(ih) - 1/ncyc; end end end end

%Adjust step size so half of points are accepted for c3 = 1:nvar, if ar(c3) > 0.6 st(c3) = st(c3)*(1+sf*(ar(c3)-0.6)/0.4); else if ar(c3) < 0.4 st(c3) = st(c3)/(1+sf*(0.4-ar(c3))/0.4); end end if st(c3) > (xmin(c3) - xmax(c3)) st(c3) = xmin(c3) - xmax(c3);

169

end ar(c3) = 1; end end

%Acceptance Criterion Ftol = eps + epsr*abs(Fmin); if (Fmin <= Fold) & (Fold-Fmin < Ftol) ict = ict+1; if ict >= Irepeat fprintf('\n\nExit Loop by Consectutive Minima\n\n'); fprintf(fid,'\n\nExit Loop by Consectutive Minima\n\n'); break; end else ict = 0; end if nfv > ntot fprintf('\n\nExit Loop by Too Many Function Evaluations\n\n'); fprintf(fid,'\n\nExit Loop by Too Many Function Evaluations\n\n'); break; end %Reduce the Temperature for the Next Iteration T = rp*T; st = rs*st; x = xopt; F = Fmin; Fold = F;endx = xopt;c = [1:1:size(track,1)];d = x(7);h = x(8);axstiff = calcaxstiff2(x);latstiff = calclatstiff2(x);n=x(5);mass = n*d^2*pi/4*((h-n*x(6))/n*x(4) + x(6)*x(3));atten = 0;SF = d/4/x(6);minSF = Fcompress/(d^2*pi/4)/2/x(1);[threedstart, threedend] = calc3D(x);fprintf('-------------------------------------------------------------------------------------------------------------------------\n');fprintf(' Optimizied Design');fprintf(fid,'-------------------------------------------------------------------------------------------------------------------------\n');fprintf(fid,' Optimized Design');fprintf(' | %6.0f | %6.0f |\n', btarget, etarget);fprintf(fid,' | %6.0f | %6.0f |\n', btarget, etarget);fprintf('-------------------------------------------------------------------------------------------------------------------------\n');fprintf('| # of cells | Elas. thick. | Met. thick. | Beg. Freq. | End Freq. | Elas. rho | Met. rho | Isolator Mass \n');fprintf('| | (cm) | (cm) | (Hz) | (Hz) | (kg/m^3) | (kg/m^3) | (kg)\n');fprintf('-------------------------------------------------------------------------------------------------------------------------\n');fprintf('%8.0f %15.3f %16.3f %14.0f %14.0f %13.0f %13.0f %14.2f\n',n, (x(6)*100),((h-x(5)*x(6))/x(5)*100), threedstart,threedend,x(3),x(4),mass);fprintf('=========================================================================================================================\n');fprintf('| G Elas. | G Met. | Axial Stiff. | Lat. Stiff. | d | h | S.F. | min. S.F. \n');fprintf('| (MPa) | (GPa) | (MN/m) | (MN/m) | (cm) | (cm) | | \n');fprintf('-------------------------------------------------------------------------------------------------------------------------\n');fprintf('%8.2f %16.2f %16.2f %14.2f %13.2f %14.2f %15.2f %14.2f\n',(x(1)/1e6),(x(2)/1e9),(axstiff/1e6),(latstiff/1e6),d*100,h*100,SF,minSF);fprintf('\n');fprintf(fid,'-------------------------------------------------------------------------------------------------------------------------\n');fprintf(fid,'| # of cells | Elas. thick. | Met. thick. | Beg. Freq. | End Freq. | Elas. rho | Met. rho | Isolator Mass \n');fprintf(fid,'| | (cm) | (cm) | (Hz) | (Hz) | (kg/m^3) | (kg/m^3) | (kg)\n');fprintf(fid,'-------------------------------------------------------------------------------------------------------------------------\n');fprintf(fid,'%8.0f %15.3f %16.3f %14.0f %14.0f %13.0f %13.0f %14.2f\n',n, (x(6)*100),((h-x(5)*x(6))/x(5)*100), threedstart,threedend,x(3),x(4),mass);fprintf(fid,'=========================================================================================================================\n');fprintf(fid,'| G Elas. | G Met. | Axial Stiff. | Lat. Stiff. | d | h | S.F. | min. S.F. \n');fprintf(fid,'| (MPa) | (GPa) | (MN/m) | (MN/m) | (cm) | (cm) | | \n');fprintf(fid,'-------------------------------------------------------------------------------------------------------------------------\n');fprintf(fid,'%8.2f %16.2f %16.2f %14.2f %13.2f %14.2f %15.2f %14.2f\n',(x(1)/1e6),(x(2)/1e9),(axstiff/1e6),(latstiff/1e6),d*100,h*100,SF,minSF);fprintf(fid,'\n');

st = fclose(fid);

170

Function - Initial

function [btarget, etarget, masstarg, alphab, alphae, beta, minaxstiff, maxaxstiff,... minlatstiff, maxlatstiff, maxatten, Grubmin, Grubmax, Gmetmin, Gmetmax, rhorubmin, rhorubmax,... tmetmin, trubmin, rhometmin, rhometmax, nmin, nmax, dmin, dmax, hmin, hmax,elastomer, maxmass, Fcompress] = initial()

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% "Design of periodically layered isolators in shear using simulated annealing"%% Rotorcraft Center of Excellence% Penn State University% Joseph T. Szefi% [email protected]% 9/1/2003%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% Lower limits on the thicknesses of the metal and elastomer layers.

tmetmin = .0005; % meterstrubmin = .01; % meters

% Isolator diameter limits

dmin = .2; % metersdmax = .2001; % meters

%----------------------------------------------------------------------

% Isolator height limits

hmin = (tmetmin+trubmin)*3; % metershmax = .5; % meters

%----------------------------------------------------------------------

% Possible number of cells in isolator.% The main program currently allows a minimum nmin = 2 and a maximum nmax = 15.

nmin = 3;nmax = 3;

%----------------------------------------------------------------------

% Target beginning frequency for the stop band

btarget = 400; % hertz

% Target end frequency for the stop band

etarget = 2000; % hertz

%----------------------------------------------------------------------

% A nominal target isolator mass

masstarg = 1; % kilograms

%----------------------------------------------------------------------

% Weighting factors for objective function terms

% Weighting factor for beginning frequency term in objective function.% Typically equal to 1.alphab = 500;

% Weighting factor for end frequency term in objective function.% Typically equal to 1.alphae = 1;

% Weighting factor for isolator mass term.% Should range from 0 for no mass dependence to% 500 for mass term dominancebeta = 1;

%----------------------------------------------------------------------

% Isolator axial stiffness limits.% Limits of 0 and 1e20 relax the constraint

minaxstiff = 1000000; % Newton / metermaxaxstiff = 1e30; % Newton / meter

%----------------------------------------------------------------------

% Isolator lateral stiffness limits.% Limits of 0 and 1e20 relax the constraint

minlatstiff = 0; % Newton / metermaxlatstiff = 1e20; % Newton / meter

%----------------------------------------------------------------------

% Maximum attenuation allowed in isolator design.

maxatten = .01; % Transmissibility

%----------------------------------------------------------------------%----------------------------------------------------------------------% Limits on elastomer shear modulus.% (Erubmin = 1378951/3 Pa is 200/3 psi, or lowest value physically possible for elastomers)

% Enable these values for non-specific elastomer

%Grubmin = 50*6894.757; % PascalsGrubmax = 10000*6894.757; % PascalsGrubmin = .34e6; % Pascals

elastomer = 0; %Disregard for compression optimization

%----------------------------------------------------------------------%----------------------------------------------------------------------

% Limits on metal Young's modulus.

171

% The stop band frequencies are typically insensitive to the metal's Young's modulus% The metal's density is the more important parameter

Gmetmin = 2.0e11/3; % PascalsGmetmax = 2.01e11/3; % Pascals

%----------------------------------------------------------------------

% Limits on the density of the elastomer.% Elastomer densities are typically around 1000 kg/m^3

rhorubmin = 1000; % kg / m^3rhorubmax = 1100; % kg / m^3

% 30% vol. ferrite MR ELASTOMER%rhorubmin = 3330; % kg / m^3%rhorubmax = 3333; % kg / m^3

% range of elastomer density / MR ELASTOMER%rhorubmin = 1000; % kg / m^3%rhorubmax = 3333; % kg / m^3

%----------------------------------------------------------------------

% Limits on the density of the metal.

% For this design parameter, the user may want to first treat the parameter like% a continuous variable, ranging from the low density of aluminum to a high density of% tungsten.% Then for subsequent fine tuning, the user can limit the density to specific ranges,% corresponding to specific metals. Some density limits of different metals have been included.

% Low range%rhometmin = 700; % kg / m^3%rhometmax = 7820; % kg / m^3

% Wide range%rhometmin = 500; % kg / m^3%rhometmax = 19000; % kg / m^3

% Aluminum rangerhometmin = 2642.999; % kg / m^3rhometmax = 2643; % kg / m^3

% Steel%rhometmin = 7800; % kg / m^3%rhometmax = 7820; % kg / m^3

% Lead%rhometmin = 11370; % kg / m^3%rhometmax = 11371; % kg / m^3

% Tungsten%rhometmin = 19000; % kg / m^3%rhometmax = 19001; % kg / m^3

% iron%rhometmin = 7873;%rhometmax = 7874;

%----------------------------------------------------------------------

%Define maximum mass for mount

maxmass =2;

%----------------------------------------------------------------------

% Define maximum compressive force

Fcompress = 30000;

172

Function - fcomp

function F = fcomp(x, btarget, etarget, masstarg, alphab, alphae, beta,... minaxstiff, maxaxstiff, minlatstiff, maxlatstiff, maxatten, maxmass, Fcompress)


Grub=x(1);Gmet=x(2);rhorub=x(3);rhomet=x(4);trub=x(6);n=x(5);d = x(7);h = x(8);tmet=(h-n*trub)/n;

% Calculate isolator massmass = n*d^2*pi/4*(tmet*rhomet + trub*rhorub);

%Calculate start and end of stop band, and maximum attenuation acheived[threedstart,threedend] = calc3d(x);

%Calculate axial and lateral stiffnessesaxialstiff = calcaxstiff2(x);lateralstiff = calclatstiff2(x);

%Calculate violations of stiffness constraintsconaxstiffmin = -axialstiff + minaxstiff;conaxstiffmax = axialstiff - maxaxstiff;conlatstiffmin = -lateralstiff + minlatstiff;conlatstiffmax = lateralstiff - maxlatstiff;conmassmax = mass - maxmass;

%Choose penalty for objective functionconaxstiffmin = max(0,conaxstiffmin);conaxstiffmax = max(0,conaxstiffmax);conlatstiffmin = max(0,conlatstiffmin);conlatstiffmax = max(0,conlatstiffmax);conmassmax = max(0,conmassmax);

%conatten = max(0,(atten-maxatten));conatten = 0;

%Ensure start frequency is lower than end frequencyB = threedstart;E = threedend;gap = E-B;mid = (E+B)/2;targmiddle = (btarget+etarget)/2;

if gap < 0 gap = 1e10; mid = 1e10;elseif gap >0 gap = 0;end

%Objective Function

% WEIGHTING FACTORS - defined in function 'initial.m'% alphab = weighting factor for beginning frequency (default = 1)% alphae = weighting factor for end frequency (default = 1)% beta = weighting factor for mass term (default = 0)

% Calculate compressive stress requirements stress <= 2GS, or dynamic stress <= GS

compstress = Fcompress / (pi*d^2/4);

dyncompstress = 0.1*compstress;

shapefactor = d/4/trub;

if compstress > 2*Grub*shapefactor | dyncompstress > Grub*shapefactor constress = 10000;else constress = 0;end

F = alphab*abs((btarget-B)/btarget) + alphae*abs((etarget-E)/etarget) + beta*(mass-masstarg)/(masstarg*100) ... + gap + conaxstiffmin + conaxstiffmax + conlatstiffmin + conlatstiffmax + conatten*10000 + conmassmax*10000 + constress;

% if F is negative, rejectif real(F) < 0 F = -F*1000;end

173

Function – calc3D

function [threedstart,threedend] = calc3D(x)

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% "Design of periodically layered isolators in compression using simulated annealing"%% Rotorcraft Center of Excellence% Penn State University% Joseph T. Szefi% [email protected]% 9/1/2003%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Grub = x(1);Gmet = x(2);rhorub = x(3);rhomet = x(4);N = x(5);trub = x(6);d = x(7);height = x(8);tmetactual = (x(8)-N*trub)/N;

h = .00000000000001;

tmet = .00000000000001;n = 1;

h = h + trub;

metthick = tmet/h*2;Lz= h/2;Lr= d/2;

%Elastomer properties

rho1 = rhorub;nu1 = .499;G1 = Grub;

E1 = G1*2*(1+nu1);

Q1 = E1*(1-nu1)/(1+nu1)/(1-2*nu1);S1 = nu1/(1-nu1);

C111= Q1;C122= Q1;C133= Q1;C112= Q1*S1;C123= Q1*S1;C113= Q1*S1;C155= Q1*(1-2*nu1)/2/(1-nu1);

%Metal Properties%Metal layer is approximated with very thin layer with appropriate density

rho2 = rhomet;nu2 = .499;G2 = Gmet;

E2 = G2*2*(1+nu2);

Q2 = E2*(1-nu2)/(1+nu2)/(1-2*nu2);S2 = nu2/(1-nu2);


umax = 4;wmax = 8;

% Calculate components of global mass matrix

% The global matrices are in the following form

% |K11||K12|{a} - omega^2*|M11| |0|{a} = {0}% |K21||K22|{b} |0| |M22|{b} {0}%

clen = 2/n;

tbegin = -1;tend = -1 + clen;rho2 = rho2*tmetactual/tmet;

m = zeros((umax+wmax));k = zeros((umax+wmax));

%Loop through number of cells and calculate M and K matrices.for cell=1:n

% Calculation of M11 and M22 for elastomer and metalthickness = tend - metthick;

m(1,1)=2*pi*(1/30*(thickness)^5-1/30*tbegin^5-1/9*(thickness)^3+1/9*tbegin^3+1/6*tend-1/6*metthick-1/6*tbegin)*Lr^2*Lz*rho1... +2*pi*(1/30*tend^5-1/30*(thickness)^5-1/9*tend^3+1/9*(thickness)^3+1/6*metthick)*Lr^2*Lz*rho2;m(1,2)=2*pi*(1/25*(thickness)^5-1/25*tbegin^5-2/15*(thickness)^3+2/15*tbegin^3+1/5*tend-1/5*metthick-1/5*tbegin)*Lr^2*Lz*rho1... +2*pi*(1/25*tend^5-1/25*(thickness)^5-2/15*tend^3+2/15*(thickness)^3+1/5*metthick)*Lr^2*Lz*rho2;m(1,3)=2*pi*(1/36*(thickness)^6-1/36*tbegin^6-1/12*(thickness)^4+1/12*tbegin^4+1/12*(thickness)^2-1/12*tbegin^2)*Lr^2*Lz*rho1... +2*pi*(1/36*tend^6-1/36*(thickness)^6-1/12*tend^4+1/12*(thickness)^4+1/12*tend^2-1/12*(thickness)^2)*Lr^2*Lz*rho2;m(1,4)=2*pi*(1/30*(thickness)^6-1/30*tbegin^6-1/10*(thickness)^4+1/10*tbegin^4+1/10*(thickness)^2-1/10*tbegin^2)*Lr^2*Lz*rho1... +2*pi*(1/30*tend^6-1/30*(thickness)^6-1/10*tend^4+1/10*(thickness)^4+1/10*tend^2-1/10*(thickness)^2)*Lr^2*Lz*rho2;m(2,2)=2*pi*(1/20*(thickness)^5-1/20*tbegin^5-1/6*(thickness)^3+1/6*tbegin^3+1/4*tend-1/4*metthick-1/4*tbegin)*Lr^2*Lz*rho1... +2*pi*(1/20*tend^5-1/20*(thickness)^5-1/6*tend^3+1/6*(thickness)^3+1/4*metthick)*Lr^2*Lz*rho2;m(2,3)=2*pi*(1/30*(thickness)^6-1/30*tbegin^6-1/10*(thickness)^4+1/10*tbegin^4+1/10*(thickness)^2-1/10*tbegin^2)*Lr^2*Lz*rho1... +2*pi*(1/30*tend^6-1/30*(thickness)^6-1/10*tend^4+1/10*(thickness)^4+1/10*tend^2-1/10*(thickness)^2)*Lr^2*Lz*rho2;m(2,4)=2*pi*(1/24*(thickness)^6-1/24*tbegin^6-1/8*(thickness)^4+1/8*tbegin^4+1/8*(thickness)^2-1/8*tbegin^2)*Lr^2*Lz*rho1... +2*pi*(1/24*tend^6-1/24*(thickness)^6-1/8*tend^4+1/8*(thickness)^4+1/8*tend^2-1/8*(thickness)^2)*Lr^2*Lz*rho2;m(3,3)=2*pi*(1/42*(thickness)^7-1/42*tbegin^7-1/15*(thickness)^5+1/15*tbegin^5+1/18*(thickness)^3-1/18*tbegin^3)*Lr^2*Lz*rho1... +2*pi*(1/42*tend^7-1/42*(thickness)^7-1/15*tend^5+1/15*(thickness)^5+1/18*tend^3-1/18*(thickness)^3)*Lr^2*Lz*rho2;m(3,4)=2*pi*(1/35*(thickness)^7-1/35*tbegin^7-2/25*(thickness)^5+2/25*tbegin^5+1/15*(thickness)^3-1/15*tbegin^3)*Lr^2*Lz*rho1...

174

+2*pi*(1/35*tend^7-1/35*(thickness)^7-2/25*tend^5+2/25*(thickness)^5+1/15*tend^3-1/15*(thickness)^3)*Lr^2*Lz*rho2;m(4,4)=2*pi*(1/28*(thickness)^7-1/28*tbegin^7-1/10*(thickness)^5+1/10*tbegin^5+1/12*(thickness)^3-1/12*tbegin^3)*Lr^2*Lz*rho1... +2*pi*(1/28*tend^7-1/28*(thickness)^7-1/10*tend^5+1/10*(thickness)^5+1/12*tend^3-1/12*(thickness)^3)*Lr^2*Lz*rho2;m(5,5)=rho1*Lr^2*Lz*2*pi*(1/2*tend-1/2*metthick-1/2*tbegin) + rho2*Lr^2*Lz*pi*metthick;m(5,6)=rho1*Lr^2*Lz*2*pi*(1/4*(thickness)^2-1/4*tbegin^2+1/2*tend-1/2*metthick-1/2*tbegin)... + rho2*Lr^2*Lz*2*pi*(1/4*tend^2-1/4*(thickness)^2+1/2*metthick);m(5,7)=rho1*Lr^2*Lz*2*pi*(1/9*(thickness)^3-1/9*tbegin^3+1/6*tend-1/6*metthick-1/6*tbegin+1/4*(thickness)^2-1/4*tbegin^2)... + rho2*Lr^2*Lz*2*pi*(1/9*tend^3-1/9*(thickness)^3+1/6*metthick+1/4*tend^2-1/4*(thickness)^2);m(5,8)=rho1*Lr^2*Lz*2*pi*(1/6*(thickness)^3-1/6*tbegin^3+1/4*(thickness)^2-1/4*tbegin^2)... + rho2*Lr^2*Lz*2*pi*(1/6*tend^3-1/6*(thickness)^3+1/4*tend^2-1/4*(thickness)^2);m(5,9)=rho1*Lr^2*Lz*2*pi*(1/12*(thickness)^4-1/12*tbegin^4+1/12*(thickness)^2-1/12*tbegin^2+1/2*tend-1/2*metthick-1/2*tbegin)... + rho2*Lr^2*Lz*2*pi*(1/12*tend^4-1/12*(thickness)^4+1/12*tend^2-1/12*(thickness)^2+1/2*metthick);m(5,10)=rho1*Lr^2*Lz*2*pi*(1/8*(thickness)^4-1/8*tbegin^4+1/2*tend-1/2*metthick-1/2*tbegin)... + rho2*Lr^2*Lz*2*pi*(1/8*tend^4-1/8*(thickness)^4+1/2*metthick);m(5,11)=rho1*Lr^2*Lz*2*pi*(1/15*(thickness)^5-1/15*tbegin^5-1/9*(thickness)^3+1/9*tbegin^3+1/4*(thickness)^2-1/4*tbegin^2+... 1/2*tend-1/2*metthick-1/2*tbegin) + rho2*Lr^2*Lz*2*pi*(1/15*tend^5-1/15*(thickness)^5-1/9*tend^3+1/9*(thickness)^3+... 1/4*tend^2-1/4*(thickness)^2+1/2*metthick);m(5,12)=rho1*Lr^2*Lz*2*pi*(1/10*(thickness)^5-1/10*tbegin^5-1/6*(thickness)^3+1/6*tbegin^3+1/4*(thickness)^2-1/4*tbegin^2+... 1/2*tend-1/2*metthick-1/2*tbegin) + rho2*Lr^2*Lz*2*pi*(1/10*tend^5-1/10*(thickness)^5-1/6*tend^3+1/6*(thickness)^3+... 1/4*tend^2-1/4*(thickness)^2+1/2*metthick);m(6,6)=rho1*Lr^2*Lz*2*pi*(1/6*(thickness)^3-1/6*tbegin^3+1/2*(thickness)^2-1/2*tbegin^2+1/2*tend-1/2*metthick-1/2*tbegin)... + rho2*Lr^2*Lz*2*pi*(1/6*tend^3-1/6*(thickness)^3+1/2*tend^2-1/2*(thickness)^2+1/2*metthick);m(6,7)=rho1*Lr^2*Lz*2*pi*(1/12*(thickness)^4-1/12*tbegin^4+5/18*(thickness)^3-5/18*tbegin^3+1/3*(thickness)^2-1/3*tbegin^2+... 1/6*tend-1/6*metthick-1/6*tbegin) + rho2*Lr^2*Lz*2*pi*(1/12*tend^4-1/12*(thickness)^4+5/18*tend^3-5/18*(thickness)^3+... 1/3*tend^2-1/3*(thickness)^2+1/6*metthick);m(6,8)=rho1*Lr^2*Lz*2*pi*(1/8*(thickness)^4-1/8*tbegin^4+1/3*(thickness)^3-1/3*tbegin^3+1/4*(thickness)^2-1/4*tbegin^2)... + rho2*Lr^2*Lz*2*pi*(1/8*tend^4-1/8*(thickness)^4+1/3*tend^3-1/3*(thickness)^3+1/4*tend^2-1/4*(thickness)^2);m(6,9)=rho1*Lr^2*Lz*2*pi*(1/12*(thickness)^4-1/12*tbegin^4+1/18*(thickness)^3-1/18*tbegin^3+1/15*(thickness)^5-1/15*tbegin^5+... 1/3*(thickness)^2-1/3*tbegin^2+1/2*tend-1/2*metthick-1/2*tbegin) + rho2*Lr^2*Lz*2*pi*(1/12*tend^4-1/12*(thickness)^4+... 1/18*tend^3-1/18*(thickness)^3+1/15*tend^5-1/15*(thickness)^5+1/3*tend^2-1/3*(thickness)^2+1/2*metthick);m(6,10)=rho1*Lr^2*Lz*2*pi*(1/10*(thickness)^5-1/10*tbegin^5+1/8*(thickness)^4-1/8*tbegin^4+1/4*(thickness)^2-1/4*tbegin^2+... 1/2*tend-1/2*metthick-1/2*tbegin) + rho2*Lr^2*Lz*2*pi*(1/10*tend^5-1/10*(thickness)^5+1/8*tend^4-1/8*(thickness)^4+... 1/4*tend^2-1/4*(thickness)^2+1/2*metthick);m(6,11)=rho1*Lr^2*Lz*2*pi*(1/18*(thickness)^6-1/18*tbegin^6+1/18*(thickness)^3-1/18*tbegin^3+1/15*(thickness)^5-1/15*tbegin^5-... 1/12*(thickness)^4+1/12*tbegin^4+1/2*(thickness)^2-1/2*tbegin^2+1/2*tend-1/2*metthick-1/2*tbegin)... + rho2*Lr^2*Lz*2*pi*(1/18*tend^6-1/18*(thickness)^6+1/18*tend^3-1/18*(thickness)^3+1/15*tend^5-1/15*(thickness)^5-... 1/12*tend^4+1/12*(thickness)^4+1/2*tend^2-1/2*(thickness)^2+1/2*metthick);m(6,12)=rho1*Lr^2*Lz*2*pi*(1/12*(thickness)^6-1/12*tbegin^6+1/10*(thickness)^5-1/10*tbegin^5-1/8*(thickness)^4+1/8*tbegin^4+... 1/2*(thickness)^2-1/2*tbegin^2+1/2*tend-1/2*metthick-1/2*tbegin) + rho2*Lr^2*Lz*2*pi*(1/12*tend^6-1/12*(thickness)^6+... 1/10*tend^5-1/10*(thickness)^5-1/8*tend^4+1/8*(thickness)^4+1/2*tend^2-1/2*(thickness)^2+1/2*metthick);m(7,7)=rho1*Lr^2*Lz*2*pi*(1/20*(thickness)^5-1/20*tbegin^5+2/9*(thickness)^3-2/9*tbegin^3+1/12*tend-1/12*metthick-... 1/12*tbegin+1/6*(thickness)^4-1/6*tbegin^4+1/6*(thickness)^2-1/6*tbegin^2) + rho2*Lr^2*Lz*2*pi*(1/20*tend^5-... 1/20*(thickness)^5+2/9*tend^3-2/9*(thickness)^3+1/12*metthick+1/6*tend^4-1/6*(thickness)^4+1/6*tend^2-... 1/6*(thickness)^2);m(7,8)=rho1*Lr^2*Lz*2*pi*(5/24*(thickness)^4-5/24*tbegin^4+2/9*(thickness)^3-2/9*tbegin^3+1/15*(thickness)^5-... 1/15*tbegin^5+1/12*(thickness)^2-1/12*tbegin^2) + rho2*Lr^2*Lz*2*pi*(5/24*tend^4-5/24*(thickness)^4+2/9*tend^3-... 2/9*(thickness)^3+1/15*tend^5-1/15*(thickness)^5+1/12*tend^2-1/12*(thickness)^2);m(7,9)=rho1*Lr^2*Lz*2*pi*(1/24*(thickness)^4-1/24*tbegin^4+7/24*(thickness)^2-7/24*tbegin^2+1/24*(thickness)^6-... 1/24*tbegin^6+1/15*(thickness)^5-1/15*tbegin^5+1/6*tend-1/6*metthick-1/6*tbegin+1/6*(thickness)^3-1/6*tbegin^3)... + rho2*Lr^2*Lz*2*pi*(1/24*tend^4-1/24*(thickness)^4+7/24*tend^2-7/24*(thickness)^2+1/24*tend^6-1/24*... (thickness)^6+1/15*tend^5-1/15*(thickness)^5+1/6*metthick+1/6*tend^3-1/6*(thickness)^3);m(7,10)=rho1*Lr^2*Lz*2*pi*(1/18*(thickness)^6-1/18*tbegin^6+1/24*(thickness)^4-1/24*tbegin^4+1/9*(thickness)^3-1/9*tbegin^3+... 1/6*tend-1/6*metthick-1/6*tbegin+1/10*(thickness)^5-1/10*tbegin^5+1/4*(thickness)^2-1/4*tbegin^2) +... rho2*Lr^2*Lz*2*pi*(1/18*tend^6-1/18*(thickness)^6+1/24*tend^4-1/24*(thickness)^4+1/9*tend^3-1/9*(thickness)^3+... 1/6*metthick+1/10*tend^5-1/10*(thickness)^5+1/4*tend^2-1/4*(thickness)^2);m(7,11)=rho1*Lr^2*Lz*2*pi*(-1/30*(thickness)^5+1/30*tbegin^5+1/28*(thickness)^7-1/28*tbegin^7+1/4*(thickness)^3-... 1/4*tbegin^3+1/18*(thickness)^6-1/18*tbegin^6+1/3*(thickness)^2-1/3*tbegin^2+1/6*tend-1/6*metthick-1/6*tbegin)... + rho2*Lr^2*Lz*2*pi*(-1/30*tend^5+1/30*(thickness)^5+1/28*tend^7-1/28*(thickness)^7+1/4*tend^3-1/4*(thickness)^3+... 1/18*tend^6-1/18*(thickness)^6+1/3*tend^2-1/3*(thickness)^2+1/6*metthick);m(7,12)=rho1*Lr^2*Lz*2*pi*(1/21*(thickness)^7-1/21*tbegin^7+2/9*(thickness)^3-2/9*tbegin^3-1/30*(thickness)^5+... 1/30*tbegin^5+1/3*(thickness)^2-1/3*tbegin^2-1/24*(thickness)^4+1/24*tbegin^4+1/6*tend-1/6*metthick-1/6*tbegin+... 1/12*(thickness)^6-1/12*tbegin^6) + rho2*Lr^2*Lz*2*pi*(1/21*tend^7-1/21*(thickness)^7+2/9*tend^3-2/9*(thickness)^3-... 1/30*tend^5+1/30*(thickness)^5+1/3*tend^2-1/3*(thickness)^2-1/24*tend^4+1/24*(thickness)^4+1/6*metthick+1/12*tend^6-... 1/12*(thickness)^6);m(8,8)=rho1*Lr^2*Lz*2*pi*(1/10*(thickness)^5-1/10*tbegin^5+1/4*(thickness)^4-1/4*tbegin^4+1/6*(thickness)^3-1/6*tbegin^3)... + rho2*Lr^2*Lz*2*pi*(1/10*tend^5-1/10*(thickness)^5+1/4*tend^4-1/4*(thickness)^4+1/6*tend^3-1/6*(thickness)^3);m(8,9)=rho1*Lr^2*Lz*2*pi*(1/18*(thickness)^6-1/18*tbegin^6+2/9*(thickness)^3-2/9*tbegin^3+1/15*(thickness)^5-1/15*tbegin^5+... 1/24*(thickness)^4-1/24*tbegin^4+1/4*(thickness)^2-1/4*tbegin^2) + rho2*Lr^2*Lz*2*pi*(1/18*tend^6-1/18*(thickness)^6+... 2/9*tend^3-2/9*(thickness)^3+1/15*tend^5-1/15*(thickness)^5+1/24*tend^4-1/24*(thickness)^4+1/4*tend^2-1/4*(thickness)^2);m(8,10)=rho1*Lr^2*Lz*2*pi*(1/12*(thickness)^6-1/12*tbegin^6+1/10*(thickness)^5-1/10*tbegin^5+1/6*(thickness)^3-1/6*tbegin^3+... 1/4*(thickness)^2-1/4*tbegin^2) + rho2*Lr^2*Lz*2*pi*(1/12*tend^6-1/12*(thickness)^6+1/10*tend^5-1/10*(thickness)^5+... 1/6*tend^3-1/6*(thickness)^3+1/4*tend^2-1/4*(thickness)^2);m(8,11)=rho1*Lr^2*Lz*2*pi*(1/21*(thickness)^7-1/21*tbegin^7+1/24*(thickness)^4-1/24*tbegin^4+1/18*(thickness)^6-... 1/18*tbegin^6-1/15*(thickness)^5+1/15*tbegin^5+1/3*(thickness)^3-1/3*tbegin^3+1/4*(thickness)^2-1/4*tbegin^2)... + rho2*Lr^2*Lz*2*pi*(1/21*tend^7-1/21*(thickness)^7+1/24*tend^4-1/24*(thickness)^4+1/18*tend^6-1/18*... (thickness)^6-1/15*tend^5+1/15*(thickness)^5+1/3*tend^3-1/3*(thickness)^3+1/4*tend^2-1/4*(thickness)^2);m(8,12)=rho1*Lr^2*Lz*2*pi*(1/14*(thickness)^7-1/14*tbegin^7+1/12*(thickness)^6-1/12*tbegin^6-1/10*(thickness)^5+... 1/10*tbegin^5+1/3*(thickness)^3-1/3*tbegin^3+1/4*(thickness)^2-1/4*tbegin^2) + rho2*Lr^2*Lz*2*pi*(1/14*tend^7-... 1/14*(thickness)^7+1/12*tend^6-1/12*(thickness)^6-1/10*tend^5+1/10*(thickness)^5+1/3*tend^3-1/3*(thickness)^3+... 1/4*tend^2-1/4*(thickness)^2);m(9,9)=rho1*Lr^2*Lz*2*pi*(1/30*(thickness)^5-1/30*tbegin^5+1/28*(thickness)^7-1/28*tbegin^7+1/36*(thickness)^3-... 1/36*tbegin^3+1/6*(thickness)^2-1/6*tbegin^2+1/6*(thickness)^4-1/6*tbegin^4+1/2*tend-1/2*metthick-1/2*tbegin)... + rho2*Lr^2*Lz*2*pi*(1/30*tend^5-1/30*(thickness)^5+1/28*tend^7-1/28*(thickness)^7+1/36*tend^3-1/36*(thickness)^3+... 1/6*tend^2-1/6*(thickness)^2+1/6*tend^4-1/6*(thickness)^4+1/2*metthick);m(9,10)=rho1*Lr^2*Lz*2*pi*(1/21*(thickness)^7-1/21*tbegin^7+1/30*(thickness)^5-1/30*tbegin^5+5/24*(thickness)^4-5/24*tbegin^4+... 1/12*(thickness)^2-1/12*tbegin^2+1/2*tend-1/2*metthick-1/2*tbegin) + rho2*Lr^2*Lz*2*pi*(1/21*tend^7-1/21*(thickness)^7+... 1/30*tend^5-1/30*(thickness)^5+5/24*tend^4-5/24*(thickness)^4+1/12*tend^2-1/12*(thickness)^2+1/2*metthick);m(9,11)=rho1*Lr^2*Lz*2*pi*(1/32*(thickness)^8-1/32*tbegin^8-1/36*(thickness)^6+1/36*tbegin^6+1/16*(thickness)^4-... 1/16*tbegin^4+1/3*(thickness)^2-1/3*tbegin^2-1/18*(thickness)^3+1/18*tbegin^3+2/15*(thickness)^5-2/15*tbegin^5+... 1/2*tend-1/2*metthick-1/2*tbegin) + rho2*Lr^2*Lz*2*pi*(1/32*tend^8-1/32*(thickness)^8-1/36*tend^6+1/36*(thickness)^6+... 1/16*tend^4-1/16*(thickness)^4+1/3*tend^2-1/3*(thickness)^2-1/18*tend^3+1/18*(thickness)^3+2/15*tend^5-... 2/15*(thickness)^5+1/2*metthick);m(9,12)=rho1*Lr^2*Lz*2*pi*(1/24*(thickness)^8-1/24*tbegin^8-1/36*(thickness)^6+1/36*tbegin^6-1/9*(thickness)^3+... 1/9*tbegin^3+1/6*(thickness)^5-1/6*tbegin^5+1/3*(thickness)^2-1/3*tbegin^2+1/24*(thickness)^4-1/24*tbegin^4+... 1/2*tend-1/2*metthick-1/2*tbegin) + rho2*Lr^2*Lz*2*pi*(1/24*tend^8-1/24*(thickness)^8-1/36*tend^6+1/36*(thickness)^6-... 1/9*tend^3+1/9*(thickness)^3+1/6*tend^5-1/6*(thickness)^5+1/3*tend^2-1/3*(thickness)^2+1/24*tend^4-1/24*(thickness)^4+... 1/2*metthick);m(10,10)=rho1*Lr^2*Lz*2*pi*(1/14*(thickness)^7-1/14*tbegin^7+1/4*(thickness)^4-1/4*tbegin^4+1/2*tend-1/2*metthick-1/2*tbegin)... + rho2*Lr^2*Lz*2*pi*(1/14*tend^7-1/14*(thickness)^7+1/4*tend^4-1/4*(thickness)^4+1/2*metthick);m(10,11)=rho1*Lr^2*Lz*2*pi*(-1/18*(thickness)^6+1/18*tbegin^6+1/24*(thickness)^8-1/24*tbegin^8-1/9*(thickness)^3+... 1/9*tbegin^3+1/6*(thickness)^5-1/6*tbegin^5+1/8*(thickness)^4-1/8*tbegin^4+1/4*(thickness)^2-1/4*tbegin^2+... 1/2*tend-1/2*metthick-1/2*tbegin) + rho2*Lr^2*Lz*2*pi*(-1/18*tend^6+1/18*(thickness)^6+1/24*tend^8-... 1/24*(thickness)^8-1/9*tend^3+1/9*(thickness)^3+1/6*tend^5-1/6*(thickness)^5+1/8*tend^4-1/8*(thickness)^4+... 1/4*tend^2-1/4*(thickness)^2+1/2*metthick);m(10,12)=rho1*Lr^2*Lz*2*pi*(1/16*(thickness)^8-1/16*tbegin^8-1/12*(thickness)^6+1/12*tbegin^6+1/5*(thickness)^5-... 1/5*tbegin^5+1/8*(thickness)^4-1/8*tbegin^4-1/6*(thickness)^3+1/6*tbegin^3+1/4*(thickness)^2-1/4*tbegin^2+1/2*tend... -1/2*metthick-1/2*tbegin) + rho2*Lr^2*Lz*2*pi*(1/16*tend^8-1/16*(thickness)^8-1/12*tend^6+1/12*(thickness)^6+... 1/5*tend^5-1/5*(thickness)^5+1/8*tend^4-1/8*(thickness)^4-1/6*tend^3+1/6*(thickness)^3+1/4*tend^2-1/4*(thickness)^2+... 1/2*metthick);m(11,11)=rho1*Lr^2*Lz*2*pi*(1/36*(thickness)^9-1/36*tbegin^9+11/60*(thickness)^5-11/60*tbegin^5-1/14*(thickness)^7+... 1/14*tbegin^7-1/6*(thickness)^4+1/6*tbegin^4+1/9*(thickness)^6-1/9*tbegin^6-1/18*(thickness)^3+1/18*tbegin^3+... 1/2*(thickness)^2-1/2*tbegin^2+1/2*tend-1/2*metthick-1/2*tbegin) + rho2*Lr^2*Lz*2*pi*(1/36*tend^9-1/36*(thickness)^9+... 11/60*tend^5-11/60*(thickness)^5-1/14*tend^7+1/14*(thickness)^7-1/6*tend^4+1/6*(thickness)^4+1/9*tend^6-1/9*(thickness)^6-... 1/18*tend^3+1/18*(thickness)^3+1/2*tend^2-1/2*(thickness)^2+1/2*metthick);m(11,12)=rho1*Lr^2*Lz*2*pi*(5/36*(thickness)^6-5/36*tbegin^6+1/27*(thickness)^9-1/27*tbegin^9-2/21*(thickness)^7+... 2/21*tbegin^7-5/24*(thickness)^4+5/24*tbegin^4-1/9*(thickness)^3+1/9*tbegin^3+7/30*(thickness)^5-7/30*tbegin^5+... 1/2*(thickness)^2-1/2*tbegin^2+1/2*tend-1/2*metthick-1/2*tbegin) + rho2*Lr^2*Lz*2*pi*(5/36*tend^6-5/36*(thickness)^6+... 1/27*tend^9-1/27*(thickness)^9-2/21*tend^7+2/21*(thickness)^7-5/24*tend^4+5/24*(thickness)^4-1/9*tend^3+1/9*(thickness)^3+... 7/30*tend^5-7/30*(thickness)^5+1/2*tend^2-1/2*(thickness)^2+1/2*metthick);m(12,12)=rho1*Lr^2*Lz*2*pi*(1/18*(thickness)^9-1/18*tbegin^9-1/7*(thickness)^7+1/7*tbegin^7+1/6*(thickness)^6-1/6*tbegin^6+... 3/10*(thickness)^5-3/10*tbegin^5-1/4*(thickness)^4+1/4*tbegin^4-1/6*(thickness)^3+1/6*tbegin^3+1/2*(thickness)^2-... 1/2*tbegin^2+1/2*tend-1/2*metthick-1/2*tbegin) + rho2*Lr^2*Lz*2*pi*(1/18*tend^9-1/18*(thickness)^9-1/7*tend^7+... 1/7*(thickness)^7+1/6*tend^6-1/6*(thickness)^6+3/10*tend^5-3/10*(thickness)^5-1/4*tend^4+1/4*(thickness)^4-... 1/6*tend^3+1/6*(thickness)^3+1/2*tend^2-1/2*(thickness)^2+1/2*metthick);

175

temp = triu(m(1:umax,1:umax),1);temp = m(1:umax,1:umax) + temp';m(1:umax, 1:umax) = temp;temp = triu(m((umax+1):(umax+wmax),(umax+1):(umax+wmax)),1);temp = m((umax+1):(umax+wmax),(umax+1):(umax+wmax)) + temp';m((umax+1):(umax+wmax),(umax+1):(umax+wmax)) = temp;

%Calculation of K11 for elastomer and metal

k(1,1)=2*pi*(1/20*(4*C111/Lr^2+C122/Lr^2+4*C112/Lr^2)*Lr^2*Lz*((thickness)^5-tbegin^5)+1/3*(2/3*C155/Lz*Lr^2+... 1/4*(-8*C111/Lr^2-8*C112/Lr^2-2*C122/Lr^2)*Lr^2*Lz)*((thickness)^3-tbegin^3)+1/4*(4*C111/Lr^2+C122/Lr^2+... 4*C112/Lr^2)*Lr^2*Lz*(tend-metthick-tbegin))+2*pi*(1/20*(4*C211/Lr^2+C222/Lr^2+4*C212/Lr^2)*Lr^2*Lz*(tend^5-... (thickness)^5)+1/3*(2/3*C255/Lz*Lr^2+1/4*(-8*C211/Lr^2-8*C212/Lr^2-2*C222/Lr^2)*Lr^2*Lz)*(tend^3-(thickness)^3)+... 1/4*(4*C211/Lr^2+C222/Lr^2+4*C212/Lr^2)*Lr^2*Lz*metthick);k(1,2)=2*pi*(1/15*(2*C111/Lr^2+C122/Lr^2+3*C112/Lr^2)*Lr^2*Lz*((thickness)^5-tbegin^5)+1/3*(4/5*C155/Lz*Lr^2+... 1/3*(-4*C111/Lr^2-6*C112/Lr^2-2*C122/Lr^2)*Lr^2*Lz)*((thickness)^3-tbegin^3)+1/3*(2*C111/Lr^2+C122/Lr^2+... 3*C112/Lr^2)*Lr^2*Lz*(tend-metthick-tbegin))+2*pi*(1/15*(2*C211/Lr^2+C222/Lr^2+3*C212/Lr^2)*Lr^2*Lz*(tend^5-... (thickness)^5)+1/3*(4/5*C255/Lz*Lr^2+1/3*(-4*C211/Lr^2-6*C212/Lr^2-2*C222/Lr^2)*Lr^2*Lz)*(tend^3-(thickness)^3)+... 1/3*(2*C211/Lr^2+C222/Lr^2+3*C212/Lr^2)*Lr^2*Lz*metthick);k(1,3)=2*pi*(1/24*(4*C111/Lr^2+C122/Lr^2+4*C112/Lr^2)*Lr^2*Lz*((thickness)^6-tbegin^6)+1/4*(C155/Lz*Lr^2+... 1/4*(-8*C111/Lr^2-8*C112/Lr^2-2*C122/Lr^2)*Lr^2*Lz)*((thickness)^4-tbegin^4)+1/2*(-1/3*C155/Lz*Lr^2+... 1/4*(4*C111/Lr^2+C122/Lr^2+4*C112/Lr^2)*Lr^2*Lz)*((thickness)^2-tbegin^2))+2*pi*(1/24*(4*C211/Lr^2+C222/Lr^2+... 4*C212/Lr^2)*Lr^2*Lz*(tend^6-(thickness)^6)+1/4*(C255/Lz*Lr^2+1/4*(-8*C211/Lr^2-8*C212/Lr^2-... 2*C222/Lr^2)*Lr^2*Lz)*(tend^4-(thickness)^4)+1/2*(-1/3*C255/Lz*Lr^2+1/4*(4*C211/Lr^2+C222/Lr^2+... 4*C212/Lr^2)*Lr^2*Lz)*(tend^2-(thickness)^2));k(1,4)=2*pi*(1/18*(2*C111/Lr^2+C122/Lr^2+3*C112/Lr^2)*Lr^2*Lz*((thickness)^6-tbegin^6)+1/4*(6/5*C155/Lz*Lr^2+... 1/3*(-4*C111/Lr^2-6*C112/Lr^2-2*C122/Lr^2)*Lr^2*Lz)*((thickness)^4-tbegin^4)+1/2*(-2/5*C155/Lz*Lr^2+... 1/3*(2*C111/Lr^2+C122/Lr^2+3*C112/Lr^2)*Lr^2*Lz)*((thickness)^2-tbegin^2))+2*pi*(1/18*(2*C211/Lr^2+... C222/Lr^2+3*C212/Lr^2)*Lr^2*Lz*(tend^6-(thickness)^6)+1/4*(6/5*C255/Lz*Lr^2+1/3*(-4*C211/Lr^2-... 6*C212/Lr^2-2*C222/Lr^2)*Lr^2*Lz)*(tend^4-(thickness)^4)+1/2*(-2/5*C255/Lz*Lr^2+1/3*(2*C211/Lr^2+... C222/Lr^2+3*C212/Lr^2)*Lr^2*Lz)*(tend^2-(thickness)^2));k(2,2)=2*pi*(1/10*(C111/Lr^2+C122/Lr^2+2*C112/Lr^2)*Lr^2*Lz*((thickness)^5-tbegin^5)+1/3*(C155/Lz*Lr^2+... 1/2*(-2*C111/Lr^2-4*C112/Lr^2-2*C122/Lr^2)*Lr^2*Lz)*((thickness)^3-tbegin^3)+1/2*(C111/Lr^2+C122/Lr^2+... 2*C112/Lr^2)*Lr^2*Lz*(tend-metthick-tbegin))+2*pi*(1/10*(C211/Lr^2+C222/Lr^2+2*C212/Lr^2)*Lr^2*Lz*(tend^5-... (thickness)^5)+1/3*(C255/Lz*Lr^2+1/2*(-2*C211/Lr^2-4*C212/Lr^2-2*C222/Lr^2)*Lr^2*Lz)*(tend^3-(thickness)^3)+... 1/2*(C211/Lr^2+C222/Lr^2+2*C212/Lr^2)*Lr^2*Lz*metthick);k(2,3)=2*pi*(1/18*(2*C111/Lr^2+C122/Lr^2+3*C112/Lr^2)*Lr^2*Lz*((thickness)^6-tbegin^6)+1/4*(6/5*C155/Lz*Lr^2+... 1/3*(-4*C111/Lr^2-6*C112/Lr^2-2*C122/Lr^2)*Lr^2*Lz)*((thickness)^4-tbegin^4)+1/2*(-2/5*C155/Lz*Lr^2+... 1/3*(2*C111/Lr^2+C122/Lr^2+3*C112/Lr^2)*Lr^2*Lz)*((thickness)^2-tbegin^2))+2*pi*(1/18*(2*C211/Lr^2+... C222/Lr^2+3*C212/Lr^2)*Lr^2*Lz*(tend^6-(thickness)^6)+1/4*(6/5*C255/Lz*Lr^2+1/3*(-4*C211/Lr^2-6*C212/Lr^2-... 2*C222/Lr^2)*Lr^2*Lz)*(tend^4-(thickness)^4)+1/2*(-2/5*C255/Lz*Lr^2+1/3*(2*C211/Lr^2+C222/Lr^2+... 3*C212/Lr^2)*Lr^2*Lz)*(tend^2-(thickness)^2));k(2,4)=2*pi*(1/12*(C111/Lr^2+C122/Lr^2+2*C112/Lr^2)*Lr^2*Lz*((thickness)^6-tbegin^6)+1/4*(3/2*C155/Lz*Lr^2+... 1/2*(-2*C111/Lr^2-4*C112/Lr^2-2*C122/Lr^2)*Lr^2*Lz)*((thickness)^4-tbegin^4)+1/2*(-1/2*C155/Lz*Lr^2+... 1/2*(C111/Lr^2+C122/Lr^2+2*C112/Lr^2)*Lr^2*Lz)*((thickness)^2-tbegin^2))+2*pi*(1/12*(C211/Lr^2+C222/Lr^2+... 2*C212/Lr^2)*Lr^2*Lz*(tend^6-(thickness)^6)+1/4*(3/2*C255/Lz*Lr^2+1/2*(-2*C211/Lr^2-4*C212/Lr^2-... 2*C222/Lr^2)*Lr^2*Lz)*(tend^4-(thickness)^4)+1/2*(-1/2*C255/Lz*Lr^2+1/2*(C211/Lr^2+C222/Lr^2+... 2*C212/Lr^2)*Lr^2*Lz)*(tend^2-(thickness)^2));k(3,3)=2*pi*(1/28*(4*C111/Lr^2+C122/Lr^2+4*C112/Lr^2)*Lr^2*Lz*((thickness)^7-tbegin^7)+1/5*(3/2*C155/Lz*Lr^2+... 1/4*(-8*C111/Lr^2-8*C112/Lr^2-2*C122/Lr^2)*Lr^2*Lz)*((thickness)^5-tbegin^5)+1/3*(-C155/Lz*Lr^2+... 1/4*(4*C111/Lr^2+C122/Lr^2+4*C112/Lr^2)*Lr^2*Lz)*((thickness)^3-tbegin^3)+1/6*C155/Lz*Lr^2*(tend-metthick-tbegin))+... 2*pi*(1/28*(4*C211/Lr^2+C222/Lr^2+4*C212/Lr^2)*Lr^2*Lz*(tend^7-(thickness)^7)+1/5*(3/2*C255/Lz*Lr^2+... 1/4*(-8*C211/Lr^2-8*C212/Lr^2-2*C222/Lr^2)*Lr^2*Lz)*(tend^5-(thickness)^5)+... 1/3*(-C255/Lz*Lr^2+1/4*(4*C211/Lr^2+C222/Lr^2+4*C212/Lr^2)*Lr^2*Lz)*(tend^3-(thickness)^3)+1/6*C255/Lz*Lr^2*metthick);k(3,4)=2*pi*(1/21*(2*C111/Lr^2+C122/Lr^2+3*C112/Lr^2)*Lr^2*Lz*((thickness)^7-tbegin^7)+1/5*(9/5*C155/Lz*Lr^2+... 1/3*(-4*C111/Lr^2-6*C112/Lr^2-2*C122/Lr^2)*Lr^2*Lz)*((thickness)^5-tbegin^5)+1/3*(-6/5*C155/Lz*Lr^2+... 1/3*(2*C111/Lr^2+C122/Lr^2+3*C112/Lr^2)*Lr^2*Lz)*((thickness)^3-tbegin^3)+1/5*C155/Lz*Lr^2*(tend-metthick-tbegin))+... 2*pi*(1/21*(2*C211/Lr^2+C222/Lr^2+3*C212/Lr^2)*Lr^2*Lz*(tend^7-(thickness)^7)+1/5*(9/5*C255/Lz*Lr^2+... 1/3*(-4*C211/Lr^2-6*C212/Lr^2-2*C222/Lr^2)*Lr^2*Lz)*(tend^5-(thickness)^5)+1/3*(-6/5*C255/Lz*Lr^2+... 1/3*(2*C211/Lr^2+C222/Lr^2+3*C212/Lr^2)*Lr^2*Lz)*(tend^3-(thickness)^3)+1/5*C255/Lz*Lr^2*metthick);k(4,4)=2*pi*(1/14*(C111/Lr^2+C122/Lr^2+2*C112/Lr^2)*Lr^2*Lz*((thickness)^7-tbegin^7)+1/5*(9/4*C155/Lz*Lr^2+... 1/2*(-2*C111/Lr^2-4*C112/Lr^2-2*C122/Lr^2)*Lr^2*Lz)*((thickness)^5-tbegin^5)+1/3*(-3/2*C155/Lz*Lr^2+1/2*(C111/Lr^2+... C122/Lr^2+2*C112/Lr^2)*Lr^2*Lz)*((thickness)^3-tbegin^3)+1/4*C155/Lz*Lr^2*(tend-metthick-tbegin))+... 2*pi*(1/14*(C211/Lr^2+C222/Lr^2+2*C212/Lr^2)*Lr^2*Lz*(tend^7-(thickness)^7)+1/5*(9/4*C255/Lz*Lr^2+... 1/2*(-2*C211/Lr^2-4*C212/Lr^2-2*C222/Lr^2)*Lr^2*Lz)*(tend^5-(thickness)^5)+1/3*(-3/2*C255/Lz*Lr^2+1/2*(C211/Lr^2+... C222/Lr^2+2*C212/Lr^2)*Lr^2*Lz)*(tend^3-(thickness)^3)+1/4*C255/Lz*Lr^2*metthick);

temp = triu(k(1:umax,1:umax),1);temp = k(1:umax,1:umax) + temp';k(1:umax, 1:umax) = temp;


k(1,5)=0;k(1,6)=2*pi*(1/9*(2*C113/Lr/Lz+C123/Lr/Lz)*Lr^2*Lz*((thickness)^3-tbegin^3)+1/3*(-2*C113/Lr/Lz-C123/Lr/Lz)*Lr^2*Lz*(tend-... metthick-tbegin))+2*pi*(1/9*(2*C213/Lr/Lz+C223/Lr/Lz)*Lr^2*Lz*(tend^3-(thickness)^3)+1/3*(-2*C213/Lr/Lz-... C223/Lr/Lz)*Lr^2*Lz*metthick);k(1,7)=2*pi*(1/16*(4*C113/Lr/Lz+2*C123/Lr/Lz+2*C155/Lr/Lz)*Lr^2*Lz*((thickness)^4-tbegin^4)+... 1/9*(2*C113/Lr/Lz+C123/Lr/Lz)*Lr^2*Lz*((thickness)^3-tbegin^3)+1/8*(-4*C113/Lr/Lz-2*C123/Lr/Lz-... 2*C155/Lr/Lz)*Lr^2*Lz*((thickness)^2-tbegin^2)+1/3*(-2*C113/Lr/Lz-C123/Lr/Lz)*Lr^2*Lz*(tend-metthick-tbegin))+... 2*pi*(1/16*(4*C213/Lr/Lz+2*C223/Lr/Lz+2*C255/Lr/Lz)*Lr^2*Lz*(tend^4-(thickness)^4)+... 1/9*(2*C213/Lr/Lz+C223/Lr/Lz)*Lr^2*Lz*(tend^3-(thickness)^3)+1/8*(-4*C213/Lr/Lz-2*C223/Lr/Lz-... 2*C255/Lr/Lz)*Lr^2*Lz*(tend^2-(thickness)^2)+1/3*(-2*C213/Lr/Lz-C223/Lr/Lz)*Lr^2*Lz*metthick);k(1,8)=2*pi*(1/12*(4*C113/Lr/Lz+2*C123/Lr/Lz)*Lr^2*Lz*((thickness)^4-tbegin^4)+1/9*(2*C113/Lr/Lz+... C123/Lr/Lz)*Lr^2*Lz*((thickness)^3-tbegin^3)+1/6*(-4*C113/Lr/Lz-2*C123/Lr/Lz)*Lr^2*Lz*((thickness)^2-... tbegin^2)+1/3*(-2*C113/Lr/Lz-C123/Lr/Lz)*Lr^2*Lz*(tend-metthick-tbegin))+2*pi*(1/12*(4*C213/Lr/Lz+... 2*C223/Lr/Lz)*Lr^2*Lz*(tend^4-(thickness)^4)+1/9*(2*C213/Lr/Lz+C223/Lr/Lz)*Lr^2*Lz*(tend^3-(thickness)^3)+... 1/6*(-4*C213/Lr/Lz-2*C223/Lr/Lz)*Lr^2*Lz*(tend^2-(thickness)^2)+1/3*(-2*C213/Lr/Lz-C223/Lr/Lz)*Lr^2*Lz*metthick);k(1,9)=2*pi*(1/20*(3*C123/Lr/Lz+6*C113/Lr/Lz+2*C155/Lr/Lz)*Lr^2*Lz*((thickness)^5-tbegin^5)+... 1/3*(1/4*(-8*C113/Lr/Lz-2*C155/Lr/Lz-4*C123/Lr/Lz)*Lr^2*Lz+1/3*(2*C113/Lr/Lz+C123/Lr/Lz)*Lr^2*Lz)*((thickness)^3-... tbegin^3)+1/4*(2*C113/Lr/Lz+C123/Lr/Lz)*Lr^2*Lz*(tend-metthick-tbegin)+1/3*(-2*C113/Lr/Lz-... C123/Lr/Lz)*Lr^2*Lz*(tend-metthick-tbegin))+2*pi*(1/20*(3*C223/Lr/Lz+6*C213/Lr/Lz+2*C255/Lr/Lz)*Lr^2*Lz*(tend^5-... (thickness)^5)+1/3*(1/4*(-8*C213/Lr/Lz-2*C255/Lr/Lz-4*C223/Lr/Lz)*Lr^2*Lz+1/3*(2*C213/Lr/Lz+... C223/Lr/Lz)*Lr^2*Lz)*(tend^3-(thickness)^3)+1/4*(2*C213/Lr/Lz+C223/Lr/Lz)*Lr^2*Lz*metthick+... 1/3*(-2*C213/Lr/Lz-C223/Lr/Lz)*Lr^2*Lz*metthick);k(1,10)=2*pi*(1/15*(6*C113/Lr/Lz+3*C123/Lr/Lz)*Lr^2*Lz*((thickness)^5-tbegin^5)+1/9*(-6*C113/Lr/Lz-... 3*C123/Lr/Lz)*Lr^2*Lz*((thickness)^3-tbegin^3))+2*pi*(1/15*(6*C213/Lr/Lz+3*C223/Lr/Lz)*Lr^2*Lz*(tend^5-... (thickness)^5)+1/9*(-6*C213/Lr/Lz-3*C223/Lr/Lz)*Lr^2*Lz*(tend^3-(thickness)^3));k(1,11)=2*pi*(1/24*(4*C123/Lr/Lz+8*C113/Lr/Lz+2*C155/Lr/Lz)*Lr^2*Lz*((thickness)^6-tbegin^6)+... 1/16*(-12*C113/Lr/Lz-2*C155/Lr/Lz-6*C123/Lr/Lz)*Lr^2*Lz*((thickness)^4-tbegin^4)+... 1/9*(2*C113/Lr/Lz+C123/Lr/Lz)*Lr^2*Lz*((thickness)^3-tbegin^3)+1/8*(4*C113/Lr/Lz+2*C123/Lr/Lz)*Lr^2*Lz*((thickness)^2-... tbegin^2)+1/3*(-2*C113/Lr/Lz-C123/Lr/Lz)*Lr^2*Lz*(tend-metthick-tbegin))+2*pi*(1/24*(4*C223/Lr/Lz+8*C213/Lr/Lz+... 2*C255/Lr/Lz)*Lr^2*Lz*(tend^6-(thickness)^6)+1/16*(-12*C213/Lr/Lz-2*C255/Lr/Lz-6*C223/Lr/Lz)*Lr^2*Lz*(tend^4-... (thickness)^4)+1/9*(2*C213/Lr/Lz+C223/Lr/Lz)*Lr^2*Lz*(tend^3-(thickness)^3)+1/8*(4*C213/Lr/Lz+2*C223/Lr/Lz)*Lr^2*Lz*(tend^2-... (thickness)^2)+1/3*(-2*C213/Lr/Lz-C223/Lr/Lz)*Lr^2*Lz*metthick);k(1,12)=2*pi*(1/18*(8*C113/Lr/Lz+4*C123/Lr/Lz)*Lr^2*Lz*((thickness)^6-tbegin^6)+1/12*(-6*C123/Lr/Lz-... 12*C113/Lr/Lz)*Lr^2*Lz*((thickness)^4-tbegin^4)+1/9*(2*C113/Lr/Lz+C123/Lr/Lz)*Lr^2*Lz*((thickness)^3-... tbegin^3)+1/6*(4*C113/Lr/Lz+2*C123/Lr/Lz)*Lr^2*Lz*((thickness)^2-tbegin^2)+1/3*(-2*C113/Lr/Lz-... C123/Lr/Lz)*Lr^2*Lz*(tend-metthick-tbegin))+2*pi*(1/18*(8*C213/Lr/Lz+4*C223/Lr/Lz)*Lr^2*Lz*(tend^6-... (thickness)^6)+1/12*(-6*C223/Lr/Lz-12*C213/Lr/Lz)*Lr^2*Lz*(tend^4-(thickness)^4)+... 1/9*(2*C213/Lr/Lz+C223/Lr/Lz)*Lr^2*Lz*(tend^3-(thickness)^3)+1/6*(4*C213/Lr/Lz+2*C223/Lr/Lz)*Lr^2*Lz*(tend^2-... (thickness)^2)+1/3*(-2*C213/Lr/Lz-C223/Lr/Lz)*Lr^2*Lz*metthick);k(2,5)=0;k(2,6)=2*pi*(1/6*(C113/Lr/Lz+C123/Lr/Lz)*Lr^2*Lz*((thickness)^3-tbegin^3)+1/2*(-C113/Lr/Lz-C123/Lr/Lz)*Lr^2*Lz*(tend-... metthick-tbegin))+2*pi*(1/6*(C213/Lr/Lz+C223/Lr/Lz)*Lr^2*Lz*(tend^3-(thickness)^3)+1/2*(-C213/Lr/Lz-... C223/Lr/Lz)*Lr^2*Lz*metthick);k(2,7)=2*pi*(1/12*(2*C113/Lr/Lz+2*C123/Lr/Lz+2*C155/Lr/Lz)*Lr^2*Lz*((thickness)^4-tbegin^4)+1/6*(C113/Lr/Lz+... C123/Lr/Lz)*Lr^2*Lz*((thickness)^3-tbegin^3)+1/6*(-2*C113/Lr/Lz-2*C123/Lr/Lz-2*C155/Lr/Lz)*Lr^2*Lz*((thickness)^2-... tbegin^2)+1/2*(-C113/Lr/Lz-C123/Lr/Lz)*Lr^2*Lz*(tend-metthick-tbegin))+2*pi*(1/12*(2*C213/Lr/Lz+2*C223/Lr/Lz+... 2*C255/Lr/Lz)*Lr^2*Lz*(tend^4-(thickness)^4)+1/6*(C213/Lr/Lz+C223/Lr/Lz)*Lr^2*Lz*(tend^3-(thickness)^3)+... 1/6*(-2*C213/Lr/Lz-2*C223/Lr/Lz-2*C255/Lr/Lz)*Lr^2*Lz*(tend^2-(thickness)^2)+1/2*(-C213/Lr/Lz-...

176

C223/Lr/Lz)*Lr^2*Lz*metthick);k(2,8)=2*pi*(1/8*(2*C113/Lr/Lz+2*C123/Lr/Lz)*Lr^2*Lz*((thickness)^4-tbegin^4)+1/6*(C113/Lr/Lz+C123/Lr/Lz)*... Lr^2*Lz*((thickness)^3-tbegin^3)+1/4*(-2*C113/Lr/Lz-2*C123/Lr/Lz)*Lr^2*Lz*((thickness)^2-tbegin^2)+1/2*(-C113/Lr/... Lz-C123/Lr/Lz)*Lr^2*Lz*(tend-metthick-tbegin))+2*pi*(1/8*(2*C213/Lr/Lz+2*C223/Lr/Lz)*Lr^2*Lz*(tend^4-(thickness)^4)+... 1/6*(C213/Lr/Lz+C223/Lr/Lz)*Lr^2*Lz*(tend^3-(thickness)^3)+1/4*(-2*C213/Lr/Lz-2*C223/Lr/Lz)*Lr^2*Lz*(tend^2-... (thickness)^2)+1/2*(-C213/Lr/Lz-C223/Lr/Lz)*Lr^2*Lz*metthick);k(2,9)=2*pi*(1/15*(3*C123/Lr/Lz+3*C113/Lr/Lz+2*C155/Lr/Lz)*Lr^2*Lz*((thickness)^5-tbegin^5)+1/3*(1/3*(-4*C113/Lr/Lz-2*... C155/Lr/Lz-4*C123/Lr/Lz)*Lr^2*Lz+1/2*(C113/Lr/Lz+C123/Lr/Lz)*Lr^2*Lz)*((thickness)^3-tbegin^3)+1/3*(C113/Lr/Lz+C123/Lr/Lz)... *Lr^2*Lz*(tend-metthick-tbegin)+1/2*(-C113/Lr/Lz-C123/Lr/Lz)*Lr^2*Lz*(tend-metthick-tbegin))+2*pi*(1/15*(3*... C223/Lr/Lz+3*C213/Lr/Lz+2*C255/Lr/Lz)*Lr^2*Lz*(tend^5-(thickness)^5)+1/3*(1/3*(-4*C213/Lr/Lz-2*C255/Lr/Lz-... 4*C223/Lr/Lz)*Lr^2*Lz+1/2*(C213/Lr/Lz+C223/Lr/Lz)*Lr^2*Lz)*(tend^3-(thickness)^3)+1/3*(C213/Lr/Lz+C223/Lr/Lz)*... Lr^2*Lz*metthick+1/2*(-C213/Lr/Lz-C223/Lr/Lz)*Lr^2*Lz*metthick);k(2,10)=2*pi*(1/10*(3*C113/Lr/Lz+3*C123/Lr/Lz)*Lr^2*Lz*((thickness)^5-tbegin^5)+1/6*(-3*C113/Lr/Lz-3*C123/Lr/Lz)... *Lr^2*Lz*((thickness)^3-tbegin^3))+2*pi*(1/10*(3*C213/Lr/Lz+3*C223/Lr/Lz)*Lr^2*Lz*(tend^5-(thickness)^5)+1/6*... (-3*C213/Lr/Lz-3*C223/Lr/Lz)*Lr^2*Lz*(tend^3-(thickness)^3));k(2,11)=2*pi*(1/18*(4*C123/Lr/Lz+4*C113/Lr/Lz+2*C155/Lr/Lz)*Lr^2*Lz*((thickness)^6-tbegin^6)+1/12*(-6*C113/Lr/Lz-... 2*C155/Lr/Lz-6*C123/Lr/Lz)*Lr^2*Lz*((thickness)^4-tbegin^4)+1/6*(C113/Lr/Lz+C123/Lr/Lz)*Lr^2*Lz*((thickness)^3-... tbegin^3)+1/6*(2*C113/Lr/Lz+2*C123/Lr/Lz)*Lr^2*Lz*((thickness)^2-tbegin^2)+1/2*(-C113/Lr/Lz-C123/Lr/Lz)*Lr^2*Lz*... (tend-metthick-tbegin))+2*pi*(1/18*(4*C223/Lr/Lz+4*C213/Lr/Lz+2*C255/Lr/Lz)*Lr^2*Lz*(tend^6-(thickness)^6)+1/12*... (-6*C213/Lr/Lz-2*C255/Lr/Lz-6*C223/Lr/Lz)*Lr^2*Lz*(tend^4-(thickness)^4)+1/6*(C213/Lr/Lz+C223/Lr/Lz)*Lr^2*Lz*... (tend^3-(thickness)^3)+1/6*(2*C213/Lr/Lz+2*C223/Lr/Lz)*Lr^2*Lz*(tend^2-(thickness)^2)+1/2*(-C213/Lr/Lz-C223/Lr/Lz)... *Lr^2*Lz*metthick);k(2,12)=2*pi*(1/12*(4*C113/Lr/Lz+4*C123/Lr/Lz)*Lr^2*Lz*((thickness)^6-tbegin^6)+1/8*(-6*C123/Lr/Lz-6*C113/Lr/Lz)*... Lr^2*Lz*((thickness)^4-tbegin^4)+1/6*(C113/Lr/Lz+C123/Lr/Lz)*Lr^2*Lz*((thickness)^3-tbegin^3)+1/4*(2*C113/Lr/Lz... +2*C123/Lr/Lz)*Lr^2*Lz*((thickness)^2-tbegin^2)+1/2*(-C113/Lr/Lz-C123/Lr/Lz)*Lr^2*Lz*(tend-metthick-tbegin))+... 2*pi*(1/12*(4*C213/Lr/Lz+4*C223/Lr/Lz)*Lr^2*Lz*(tend^6-(thickness)^6)+1/8*(-6*C223/Lr/Lz-6*C213/Lr/Lz)*Lr^2*Lz*... (tend^4-(thickness)^4)+1/6*(C213/Lr/Lz+C223/Lr/Lz)*Lr^2*Lz*(tend^3-(thickness)^3)+1/4*(2*C213/Lr/Lz+2*C223/Lr/Lz)*... Lr^2*Lz*(tend^2-(thickness)^2)+1/2*(-C213/Lr/Lz-C223/Lr/Lz)*Lr^2*Lz*metthick);k(3,5)=0;k(3,6)=2*pi*(1/12*(2*C113/Lr/Lz+C123/Lr/Lz)*Lr^2*Lz*((thickness)^4-tbegin^4)+1/6*(-2*C113/Lr/Lz-C123/Lr/Lz)*Lr^2*Lz*... ((thickness)^2-tbegin^2))+2*pi*(1/12*(2*C213/Lr/Lz+C223/Lr/Lz)*Lr^2*Lz*(tend^4-(thickness)^4)+1/6*(-2*C213/Lr/... Lz-C223/Lr/Lz)*Lr^2*Lz*(tend^2-(thickness)^2));k(3,7)=2*pi*(1/20*(2*C123/Lr/Lz+4*C113/Lr/Lz+3*C155/Lr/Lz)*Lr^2*Lz*((thickness)^5-tbegin^5)+1/12*(2*C113/Lr/Lz+... C123/Lr/Lz)*Lr^2*Lz*((thickness)^4-tbegin^4)+1/12*(-4*C155/Lr/Lz-2*C123/Lr/Lz-4*C113/Lr/Lz)*Lr^2*Lz*... ((thickness)^3-tbegin^3)+1/6*(-2*C113/Lr/Lz-C123/Lr/Lz)*Lr^2*Lz*((thickness)^2-tbegin^2)+1/4*C155*Lr*... (tend-metthick-tbegin))+2*pi*(1/20*(2*C223/Lr/Lz+4*C213/Lr/Lz+3*C255/Lr/Lz)*Lr^2*Lz*(tend^5-(thickness)^5)... +1/12*(2*C213/Lr/Lz+C223/Lr/Lz)*Lr^2*Lz*(tend^4-(thickness)^4)+1/12*(-4*C255/Lr/Lz-2*C223/Lr/Lz-4*C213/Lr/Lz)*... Lr^2*Lz*(tend^3-(thickness)^3)+1/6*(-2*C213/Lr/Lz-C223/Lr/Lz)*Lr^2*Lz*(tend^2-(thickness)^2)+1/4*C255*Lr*metthick);k(3,8)=2*pi*(1/15*(4*C113/Lr/Lz+2*C123/Lr/Lz)*Lr^2*Lz*((thickness)^5-tbegin^5)+1/12*(2*C113/Lr/Lz+C123/Lr/Lz)*... Lr^2*Lz*((thickness)^4-tbegin^4)+1/9*(-4*C113/Lr/Lz-2*C123/Lr/Lz)*Lr^2*Lz*((thickness)^3-tbegin^3)+1/6*... (-2*C113/Lr/Lz-C123/Lr/Lz)*Lr^2*Lz*((thickness)^2-tbegin^2))+2*pi*(1/15*(4*C213/Lr/Lz+2*C223/Lr/Lz)*Lr^2*Lz*... (tend^5-(thickness)^5)+1/12*(2*C213/Lr/Lz+C223/Lr/Lz)*Lr^2*Lz*(tend^4-(thickness)^4)+1/9*(-4*C213/Lr/Lz-2*C223/Lr/Lz)... *Lr^2*Lz*(tend^3-(thickness)^3)+1/6*(-2*C213/Lr/Lz-C223/Lr/Lz)*Lr^2*Lz*(tend^2-(thickness)^2));k(3,9)=2*pi*(1/24*(3*C123/Lr/Lz+6*C113/Lr/Lz+3*C155/Lr/Lz)*Lr^2*Lz*((thickness)^6-tbegin^6)+1/4*... (1/4*(-8*C113/Lr/Lz-4*C123/Lr/Lz-4*C155/Lr/Lz)*Lr^2*Lz+1/3*(2*C113/Lr/Lz+C123/Lr/Lz)*Lr^2*Lz)*((thickness)^4-tbegin^4)... +1/2*(1/4*(2*C113/Lr/Lz+C155/Lr/Lz+C123/Lr/Lz)*Lr^2*Lz+1/3*(-2*C113/Lr/Lz-C123/Lr/Lz)*Lr^2*Lz)*... ((thickness)^2-tbegin^2))+2*pi*(1/24*(3*C223/Lr/Lz+6*C213/Lr/Lz+3*C255/Lr/Lz)*Lr^2*Lz*(tend^6-(thickness)^6)... +1/4*(1/4*(-8*C213/Lr/Lz-4*C223/Lr/Lz-4*C255/Lr/Lz)*Lr^2*Lz+1/3*(2*C213/Lr/Lz+C223/Lr/Lz)*Lr^2*Lz)... *(tend^4-(thickness)^4)+1/2*(1/4*(2*C213/Lr/Lz+C255/Lr/Lz+C223/Lr/Lz)*Lr^2*Lz+1/3*(-2*C213/Lr/Lz-C223/Lr/Lz)... *Lr^2*Lz)*(tend^2-(thickness)^2));k(3,10)=2*pi*(1/18*(6*C113/Lr/Lz+3*C123/Lr/Lz)*Lr^2*Lz*((thickness)^6-tbegin^6)+1/12*(-6*C113/Lr/Lz-3*C123/Lr/Lz)... *Lr^2*Lz*((thickness)^4-tbegin^4))+2*pi*(1/18*(6*C213/Lr/Lz+3*C223/Lr/Lz)*Lr^2*Lz*(tend^6-(thickness)^6)+1/12*... (-6*C213/Lr/Lz-3*C223/Lr/Lz)*Lr^2*Lz*(tend^4-(thickness)^4));k(3,11)=2*pi*(1/28*(4*C123/Lr/Lz+8*C113/Lr/Lz+3*C155/Lr/Lz)*Lr^2*Lz*((thickness)^7-tbegin^7)+1/20*(-12*C113/Lr/Lz-6*... C123/Lr/Lz-4*C155/Lr/Lz)*Lr^2*Lz*((thickness)^5-tbegin^5)+1/12*(2*C113/Lr/Lz+C123/Lr/Lz)*Lr^2*Lz*... ((thickness)^4-tbegin^4)+1/12*(4*C113/Lr/Lz+C155/Lr/Lz+2*C123/Lr/Lz)*Lr^2*Lz*((thickness)^3-tbegin^3)+1/6*... (-2*C113/Lr/Lz-C123/Lr/Lz)*Lr^2*Lz*((thickness)^2-tbegin^2))+2*pi*(1/28*(4*C223/Lr/Lz+8*C213/Lr/Lz+3*C255/Lr/Lz)*... Lr^2*Lz*(tend^7-(thickness)^7)+1/20*(-12*C213/Lr/Lz-6*C223/Lr/Lz-4*C255/Lr/Lz)*Lr^2*Lz*(tend^5-... (thickness)^5)+1/12*(2*C213/Lr/Lz+C223/Lr/Lz)*Lr^2*Lz*(tend^4-(thickness)^4)+1/12*(4*C213/Lr/Lz+... C255/Lr/Lz+2*C223/Lr/Lz)*Lr^2*Lz*(tend^3-(thickness)^3)+1/6*(-2*C213/Lr/Lz-C223/Lr/Lz)*Lr^2*Lz*(tend^2-(thickness)^2));k(3,12)=2*pi*(1/21*(8*C113/Lr/Lz+4*C123/Lr/Lz)*Lr^2*Lz*((thickness)^7-tbegin^7)+1/15*(-6*C123/Lr/Lz-12*C113/Lr/Lz)*... Lr^2*Lz*((thickness)^5-tbegin^5)+1/12*(2*C113/Lr/Lz+C123/Lr/Lz)*Lr^2*Lz*((thickness)^4-tbegin^4)+1/9*(4*C113/Lr/Lz... +2*C123/Lr/Lz)*Lr^2*Lz*((thickness)^3-tbegin^3)+1/6*(-2*C113/Lr/Lz-C123/Lr/Lz)*Lr^2*Lz*((thickness)^2-tbegin^2))+... 2*pi*(1/21*(8*C213/Lr/Lz+4*C223/Lr/Lz)*Lr^2*Lz*(tend^7-(thickness)^7)+1/15*(-6*C223/Lr/Lz-12*C213/Lr/Lz)*Lr^2*Lz*... (tend^5-(thickness)^5)+1/12*(2*C213/Lr/Lz+C223/Lr/Lz)*Lr^2*Lz*(tend^4-(thickness)^4)+1/9*(4*C213/Lr/Lz+2*C223/Lr/Lz)... *Lr^2*Lz*(tend^3-(thickness)^3)+1/6*(-2*C213/Lr/Lz-C223/Lr/Lz)*Lr^2*Lz*(tend^2-(thickness)^2));k(4,5)=0;k(4,6)=2*pi*(1/8*(C113/Lr/Lz+C123/Lr/Lz)*Lr^2*Lz*((thickness)^4-tbegin^4)+1/4*(-C113/Lr/Lz-C123/Lr/Lz)*Lr^2*Lz... *((thickness)^2-tbegin^2))+2*pi*(1/8*(C213/Lr/Lz+C223/Lr/Lz)*Lr^2*Lz*(tend^4-(thickness)^4)+1/4*... (-C213/Lr/Lz-C223/Lr/Lz)*Lr^2*Lz*(tend^2-(thickness)^2));k(4,7)=2*pi*(1/15*(2*C123/Lr/Lz+2*C113/Lr/Lz+3*C155/Lr/Lz)*Lr^2*Lz*((thickness)^5-tbegin^5)+1/8*(C113/Lr/Lz+C123/Lr/Lz)... *Lr^2*Lz*((thickness)^4-tbegin^4)+1/9*(-4*C155/Lr/Lz-2*C123/Lr/Lz-2*C113/Lr/Lz)*Lr^2*Lz*((thickness)^3-tbegin^3)+... 1/4*(-C113/Lr/Lz-C123/Lr/Lz)*Lr^2*Lz*((thickness)^2-tbegin^2)+1/3*C155*Lr*(tend-metthick-tbegin))+... 2*pi*(1/15*(2*C223/Lr/Lz+2*C213/Lr/Lz+3*C255/Lr/Lz)*Lr^2*Lz*(tend^5-(thickness)^5)+1/8*(C213/Lr/Lz+C223/Lr/Lz)... *Lr^2*Lz*(tend^4-(thickness)^4)+1/9*(-4*C255/Lr/Lz-2*C223/Lr/Lz-2*C213/Lr/Lz)*Lr^2*Lz*(tend^3-(thickness)^3)+... 1/4*(-C213/Lr/Lz-C223/Lr/Lz)*Lr^2*Lz*(tend^2-(thickness)^2)+1/3*C255*Lr*metthick);k(4,8)=2*pi*(1/10*(2*C113/Lr/Lz+2*C123/Lr/Lz)*Lr^2*Lz*((thickness)^5-tbegin^5)+1/8*(C113/Lr/Lz+C123/Lr/Lz)*Lr^2*Lz*... ((thickness)^4-tbegin^4)+1/6*(-2*C113/Lr/Lz-2*C123/Lr/Lz)*Lr^2*Lz*((thickness)^3-tbegin^3)+1/4*... (-C113/Lr/Lz-C123/Lr/Lz)*Lr^2*Lz*((thickness)^2-tbegin^2))+2*pi*(1/10*(2*C213/Lr/Lz+2*C223/Lr/Lz)*Lr^2*Lz*... (tend^5-(thickness)^5)+1/8*(C213/Lr/Lz+C223/Lr/Lz)*Lr^2*Lz*(tend^4-(thickness)^4)+1/6*(-2*C213/Lr/Lz-2*C223/Lr/Lz)... *Lr^2*Lz*(tend^3-(thickness)^3)+1/4*(-C213/Lr/Lz-C223/Lr/Lz)*Lr^2*Lz*(tend^2-(thickness)^2));k(4,9)=2*pi*(1/18*(3*C123/Lr/Lz+3*C113/Lr/Lz+3*C155/Lr/Lz)*Lr^2*Lz*((thickness)^6-tbegin^6)+1/4*(1/3*... (-4*C113/Lr/Lz-4*C123/Lr/Lz-4*C155/Lr/Lz)*Lr^2*Lz+1/2*(C113/Lr/Lz+C123/Lr/Lz)*Lr^2*Lz)*((thickness)^4-tbegin^4)... +1/2*(1/3*(C113/Lr/Lz+C155/Lr/Lz+C123/Lr/Lz)*Lr^2*Lz+1/2*(-C113/Lr/Lz-C123/Lr/Lz)*Lr^2*Lz)*((thickness)^2-tbegin^2))... +2*pi*(1/18*(3*C223/Lr/Lz+3*C213/Lr/Lz+3*C255/Lr/Lz)*Lr^2*Lz*(tend^6-(thickness)^6)+1/4*(1/3*(-4*C213/Lr/Lz-4*C223... /Lr/Lz-4*C255/Lr/Lz)*Lr^2*Lz+1/2*(C213/Lr/Lz+C223/Lr/Lz)*Lr^2*Lz)*(tend^4-(thickness)^4)+1/2*(1/3*(C213/Lr/Lz+C255... /Lr/Lz+C223/Lr/Lz)*Lr^2*Lz+1/2*(-C213/Lr/Lz-C223/Lr/Lz)*Lr^2*Lz)*(tend^2-(thickness)^2));k(4,10)=2*pi*(1/12*(3*C113/Lr/Lz+3*C123/Lr/Lz)*Lr^2*Lz*((thickness)^6-tbegin^6)+1/8*(-3*C113/Lr/Lz-3*C123/Lr/Lz)*... Lr^2*Lz*((thickness)^4-tbegin^4))+2*pi*(1/12*(3*C213/Lr/Lz+3*C223/Lr/Lz)*Lr^2*Lz*(tend^6-(thickness)^6)+1/8*(-3*... C213/Lr/Lz-3*C223/Lr/Lz)*Lr^2*Lz*(tend^4-(thickness)^4));k(4,11)=2*pi*(1/21*(4*C123/Lr/Lz+4*C113/Lr/Lz+3*C155/Lr/Lz)*Lr^2*Lz*((thickness)^7-tbegin^7)+1/15*(-6*C113/Lr/Lz-6*... C123/Lr/Lz-4*C155/Lr/Lz)*Lr^2*Lz*((thickness)^5-tbegin^5)+1/8*(C113/Lr/Lz+C123/Lr/Lz)*Lr^2*Lz*((thickness)^4-... tbegin^4)+1/9*(2*C113/Lr/Lz+C155/Lr/Lz+2*C123/Lr/Lz)*Lr^2*Lz*((thickness)^3-tbegin^3)+1/4*(-C113/Lr/Lz-C123/Lr/Lz)*... Lr^2*Lz*((thickness)^2-tbegin^2))+2*pi*(1/21*(4*C223/Lr/Lz+4*C213/Lr/Lz+3*C255/Lr/Lz)*Lr^2*Lz*(tend^7-(thickness)^7)... +1/15*(-6*C213/Lr/Lz-6*C223/Lr/Lz-4*C255/Lr/Lz)*Lr^2*Lz*(tend^5-(thickness)^5)+1/8*(C213/Lr/Lz+C223/Lr/Lz)*Lr^2*Lz*... (tend^4-(thickness)^4)+1/9*(2*C213/Lr/Lz+C255/Lr/Lz+2*C223/Lr/Lz)*Lr^2*Lz*(tend^3-(thickness)^3)+1/4*(-C213/Lr/Lz-... C223/Lr/Lz)*Lr^2*Lz*(tend^2-(thickness)^2));k(4,12)=2*pi*(1/14*(4*C113/Lr/Lz+4*C123/Lr/Lz)*Lr^2*Lz*((thickness)^7-tbegin^7)+1/10*(-6*C123/Lr/Lz-6*C113/Lr/Lz)*... Lr^2*Lz*((thickness)^5-tbegin^5)+1/8*(C113/Lr/Lz+C123/Lr/Lz)*Lr^2*Lz*((thickness)^4-tbegin^4)+1/6*(2*C113/Lr/Lz+... 2*C123/Lr/Lz)*Lr^2*Lz*((thickness)^3-tbegin^3)+1/4*(-C113/Lr/Lz-C123/Lr/Lz)*Lr^2*Lz*((thickness)^2-tbegin^2))+... 2*pi*(1/14*(4*C213/Lr/Lz+4*C223/Lr/Lz)*Lr^2*Lz*(tend^7-(thickness)^7)+1/10*(-6*C223/Lr/Lz-6*C213/Lr/Lz)*Lr^2*Lz*... (tend^5-(thickness)^5)+1/8*(C213/Lr/Lz+C223/Lr/Lz)*Lr^2*Lz*(tend^4-(thickness)^4)+1/6*(2*C213/Lr/Lz+2*C223/Lr/Lz)*... Lr^2*Lz*(tend^3-(thickness)^3)+1/4*(-C213/Lr/Lz-C223/Lr/Lz)*Lr^2*Lz*(tend^2-(thickness)^2));

%Calculation of K22 for elastomer and metalk(5,5)=0;k(5,6)=0;k(5,7)=0;k(5,8)=0;k(5,9)=0;k(5,10)=0;k(5,11)=0;k(5,12)=0;k(6,6)=pi*C133/Lz*Lr^2*(tend-metthick-tbegin)+pi*C233/Lz*Lr^2*metthick;k(6,7)=2*pi*(1/3*C133/Lz*Lr^2*((thickness)^2-tbegin^2)+1/2*C133/Lz*Lr^2*(tend-metthick-tbegin))+2*pi*(1/3*C233/Lz*... Lr^2*(tend^2-(thickness)^2)+1/2*C233/Lz*Lr^2*metthick);k(6,8)=2*pi*(1/2*C133/Lz*Lr^2*((thickness)^2-tbegin^2)+1/2*C133/Lz*Lr^2*(tend-metthick-tbegin))+2*pi*(1/2*C233/Lz*... Lr^2*(tend^2-(thickness)^2)+1/2*C233/Lz*Lr^2*metthick);k(6,9)=2*pi*(1/3*C133/Lz*Lr^2*((thickness)^3-tbegin^3)+1/6*C133/Lz*Lr^2*(tend-metthick-tbegin))+2*pi*(1/3*C233/Lz*Lr^2*... (tend^3-(thickness)^3)+1/6*C233/Lz*Lr^2*metthick);k(6,10)=pi*C133/Lz*Lr^2*((thickness)^3-tbegin^3)+pi*C233/Lz*Lr^2*(tend^3-(thickness)^3);k(6,11)=2*pi*(1/3*C133/Lz*Lr^2*((thickness)^4-tbegin^4)-1/3*C133/Lz*Lr^2*((thickness)^2-tbegin^2)+1/2*C133/Lz*Lr^2*... (tend-metthick-tbegin))+2*pi*(1/3*C233/Lz*Lr^2*(tend^4-(thickness)^4)-1/3*C233/Lz*Lr^2*(tend^2-(thickness)^2)+... 1/2*C233/Lz*Lr^2*metthick);

177

k(6,12)=2*pi*(1/2*C133/Lz*Lr^2*((thickness)^4-tbegin^4)-1/2*C133/Lz*Lr^2*((thickness)^2-tbegin^2)+1/2*C133/Lz*Lr^2*... (tend-metthick-tbegin))+2*pi*(1/2*C233/Lz*Lr^2*(tend^4-(thickness)^4)-1/2*C233/Lz*Lr^2*(tend^2-(thickness)^2)+1/2*... C233/Lz*Lr^2*metthick);k(7,7)=2*pi*(1/10*C155*Lz*((thickness)^5-tbegin^5)+1/3*(C133/Lz*Lr^2-C155*Lz)*((thickness)^3-tbegin^3)+2/3*C133/Lz*Lr^2*... ((thickness)^2-tbegin^2)+1/2*(C133/Lz^2+C155/Lr^2)*Lr^2*Lz*(tend-metthick-tbegin))+2*pi*(1/10*C255*Lz*(tend^5-... (thickness)^5)+1/3*(C233/Lz*Lr^2-C255*Lz)*(tend^3-(thickness)^3)+2/3*C233/Lz*Lr^2*(tend^2-(thickness)^2)+1/2*... (C233/Lz^2+C255/Lr^2)*Lr^2*Lz*metthick);k(7,8)=2*pi*(4/9*C133/Lz*Lr^2*((thickness)^3-tbegin^3)+5/6*C133/Lz*Lr^2*((thickness)^2-tbegin^2)+1/2*C133/Lz*Lr^2*... (tend-metthick-tbegin))+2*pi*(4/9*C233/Lz*Lr^2*(tend^3-(thickness)^3)+5/6*C233/Lz*Lr^2*(tend^2-(thickness)^2)+... 1/2*C233/Lz*Lr^2*metthick);k(7,9)=2*pi*(1/12*C155*Lz*((thickness)^6-tbegin^6)+1/4*(3/2*C133/Lz*Lr^2-C155*Lz)*((thickness)^4-tbegin^4)+1/3*C133/... Lz*Lr^2*((thickness)^3-tbegin^3)+1/2*(1/6*C133/Lz*Lr^2+1/2*C155*Lz)*((thickness)^2-tbegin^2)+1/6*C133/Lz*Lr^2*... (tend-metthick-tbegin))+2*pi*(1/12*C255*Lz*(tend^6-(thickness)^6)+1/4*(3/2*C233/Lz*Lr^2-C255*Lz)*(tend^4-(thickness)^4)... +1/3*C233/Lz*Lr^2*(tend^3-(thickness)^3)+1/2*(1/6*C233/Lz*Lr^2+1/2*C255*Lz)*(tend^2-(thickness)^2)+1/6*C233/Lz*... Lr^2*metthick);k(7,10)=2*pi*(1/2*C133/Lz*Lr^2*((thickness)^4-tbegin^4)+1/2*C133/Lz*Lr^2*((thickness)^3-tbegin^3))+2*pi*(1/2*C233/Lz... *Lr^2*(tend^4-(thickness)^4)+1/2*C233/Lz*Lr^2*(tend^3-(thickness)^3));k(7,11)=2*pi*(1/14*C155*Lz*((thickness)^7-tbegin^7)+1/5*(2*C133/Lz*Lr^2-C155*Lz)*((thickness)^5-tbegin^5)+1/3*C133/Lz... *Lr^2*((thickness)^4-tbegin^4)+1/3*(-C133/Lz*Lr^2+1/2*C155*Lz)*((thickness)^3-tbegin^3)+1/2*C133/Lz*Lr^2*... (tend-metthick-tbegin))+2*pi*(1/14*C255*Lz*(tend^7-(thickness)^7)+1/5*(2*C233/Lz*Lr^2-C255*Lz)*(tend^5-(thickness)^5)... +1/3*C233/Lz*Lr^2*(tend^4-(thickness)^4)+1/3*(-C233/Lz*Lr^2+1/2*C255*Lz)*(tend^3-(thickness)^3)+1/2*C233/Lz*Lr^2*metthick);k(7,12)=2*pi*(8/15*C133/Lz*Lr^2*((thickness)^5-tbegin^5)+1/2*C133/Lz*Lr^2*((thickness)^4-tbegin^4)-4/9*C133/Lz*Lr^2*... ((thickness)^3-tbegin^3)-1/6*C133/Lz*Lr^2*((thickness)^2-tbegin^2)+1/2*C133/Lz*Lr^2*(tend-metthick-tbegin))+2*pi*... (8/15*C233/Lz*Lr^2*(tend^5-(thickness)^5)+1/2*C233/Lz*Lr^2*(tend^4-(thickness)^4)-4/9*C233/Lz*Lr^2*(tend^3-... (thickness)^3)-1/6*C233/Lz*Lr^2*(tend^2-(thickness)^2)+1/2*C233/Lz*Lr^2*metthick);k(8,8)=2*pi*(2/3*C133/Lz*Lr^2*((thickness)^3-tbegin^3)+C133/Lz*Lr^2*((thickness)^2-tbegin^2)+1/2*C133/Lz*Lr^2*... (tend-metthick-tbegin))+2*pi*(2/3*C233/Lz*Lr^2*(tend^3-(thickness)^3)+C233/Lz*Lr^2*(tend^2-(thickness)^2)+... 1/2*C233/Lz*Lr^2*metthick);k(8,9)=2*pi*(1/2*C133/Lz*Lr^2*((thickness)^4-tbegin^4)+1/3*C133/Lz*Lr^2*((thickness)^3-tbegin^3)+1/6*C133/Lz*... Lr^2*((thickness)^2-tbegin^2)+1/6*C133/Lz*Lr^2*(tend-metthick-tbegin))+2*pi*(1/2*C233/Lz*Lr^2*(tend^4-... (thickness)^4)+1/3*C233/Lz*Lr^2*(tend^3-(thickness)^3)+1/6*C233/Lz*Lr^2*(tend^2-(thickness)^2)+1/6*C233/Lz*Lr^2*metthick);k(8,10)=2*pi*(3/4*C133/Lz*Lr^2*((thickness)^4-tbegin^4)+1/2*C133/Lz*Lr^2*((thickness)^3-tbegin^3))+2*pi*... (3/4*C233/Lz*Lr^2*(tend^4-(thickness)^4)+1/2*C233/Lz*Lr^2*(tend^3-(thickness)^3));k(8,11)=2*pi*(8/15*C133/Lz*Lr^2*((thickness)^5-tbegin^5)+1/3*C133/Lz*Lr^2*((thickness)^4-tbegin^4)-4/9*... C133/Lz*Lr^2*((thickness)^3-tbegin^3)+1/6*C133/Lz*Lr^2*((thickness)^2-tbegin^2)+1/2*C133/Lz*Lr^2*... (tend-metthick-tbegin))+2*pi*(8/15*C233/Lz*Lr^2*(tend^5-(thickness)^5)+1/3*C233/Lz*Lr^2*(tend^4-(thickness)^4)... -4/9*C233/Lz*Lr^2*(tend^3-(thickness)^3)+1/6*C233/Lz*Lr^2*(tend^2-(thickness)^2)+1/2*C233/Lz*Lr^2*metthick);k(8,12)=2*pi*(4/5*C133/Lz*Lr^2*((thickness)^5-tbegin^5)+1/2*C133/Lz*Lr^2*((thickness)^4-tbegin^4)-2/3*C133/Lz*Lr^2*... ((thickness)^3-tbegin^3)+1/2*C133/Lz*Lr^2*(tend-metthick-tbegin))+2*pi*(4/5*C233/Lz*Lr^2*(tend^5-(thickness)^5)... +1/2*C233/Lz*Lr^2*(tend^4-(thickness)^4)-2/3*C233/Lz*Lr^2*(tend^3-(thickness)^3)+1/2*C233/Lz*Lr^2*metthick);k(9,9)=2*pi*(1/14*C155*Lz*((thickness)^7-tbegin^7)+1/5*(9/4*C133/Lz*Lr^2-C155*Lz)*((thickness)^5-tbegin^5)+1/3*(1/2*... C155*Lz+1/2*C133/Lz*Lr^2)*((thickness)^3-tbegin^3)+1/12*C133/Lz*Lr^2*(tend-metthick-tbegin))+2*pi*(1/14*C255*Lz*... (tend^7-(thickness)^7)+1/5*(9/4*C233/Lz*Lr^2-C255*Lz)*(tend^5-(thickness)^5)+1/3*(1/2*C255*Lz+1/2*C233/Lz*Lr^2)*... (tend^3-(thickness)^3)+1/12*C233/Lz*Lr^2*metthick);k(9,10)=2*pi*(3/5*C133/Lz*Lr^2*((thickness)^5-tbegin^5)+1/6*C133/Lz*Lr^2*((thickness)^3-tbegin^3))+2*pi*... (3/5*C233/Lz*Lr^2*(tend^5-(thickness)^5)+1/6*C233/Lz*Lr^2*(tend^3-(thickness)^3));k(9,11)=2*pi*(1/16*C155*Lz*((thickness)^8-tbegin^8)+1/6*(3*C133/Lz*Lr^2-C155*Lz)*((thickness)^6-tbegin^6)+1/4*... (-7/6*C133/Lz*Lr^2+1/2*C155*Lz)*((thickness)^4-tbegin^4)+1/3*C133/Lz*Lr^2*((thickness)^3-tbegin^3)-1/12*C133... /Lz*Lr^2*((thickness)^2-tbegin^2)+1/6*C133/Lz*Lr^2*(tend-metthick-tbegin))+2*pi*(1/16*C255*Lz*(tend^8-... (thickness)^8)+1/6*(3*C233/Lz*Lr^2-C255*Lz)*(tend^6-(thickness)^6)+1/4*(-7/6*C233/Lz*Lr^2+1/2*C255*Lz)*... (tend^4-(thickness)^4)+1/3*C233/Lz*Lr^2*(tend^3-(thickness)^3)-1/12*C233/Lz*Lr^2*(tend^2-(thickness)^2)+... 1/6*C233/Lz*Lr^2*metthick);k(9,12)=2*pi*(2/3*C133/Lz*Lr^2*((thickness)^6-tbegin^6)-1/3*C133/Lz*Lr^2*((thickness)^4-tbegin^4)+1/3*C133/Lz*... Lr^2*((thickness)^3-tbegin^3)-1/6*C133/Lz*Lr^2*((thickness)^2-tbegin^2)+1/6*C133/Lz*Lr^2*(tend-metthick-tbegin))... +2*pi*(2/3*C233/Lz*Lr^2*(tend^6-(thickness)^6)-1/3*C233/Lz*Lr^2*(tend^4-(thickness)^4)+1/3*C233/Lz*Lr^2*(tend^3-... (thickness)^3)-1/6*C233/Lz*Lr^2*(tend^2-(thickness)^2)+1/6*C233/Lz*Lr^2*metthick);k(10,10)=9/5*pi*C133/Lz*Lr^2*((thickness)^5-tbegin^5)+9/5*pi*C233/Lz*Lr^2*(tend^5-(thickness)^5);k(10,11)=2*pi*(2/3*C133/Lz*Lr^2*((thickness)^6-tbegin^6)-1/2*C133/Lz*Lr^2*((thickness)^4-tbegin^4)+1/2*C133/Lz*Lr^2*... ((thickness)^3-tbegin^3))+2*pi*(2/3*C233/Lz*Lr^2*(tend^6-(thickness)^6)-1/2*C233/Lz*Lr^2*(tend^4-(thickness)^4)+... 1/2*C233/Lz*Lr^2*(tend^3-(thickness)^3));k(10,12)=2*pi*(C133/Lz*Lr^2*((thickness)^6-tbegin^6)-3/4*C133/Lz*Lr^2*((thickness)^4-tbegin^4)+1/2*C133/Lz*Lr^2*... ((thickness)^3-tbegin^3))+2*pi*(C233/Lz*Lr^2*(tend^6-(thickness)^6)-3/4*C233/Lz*Lr^2*(tend^4-(thickness)^4)+... 1/2*C233/Lz*Lr^2*(tend^3-(thickness)^3));k(11,11)=2*pi*(1/18*C155*Lz*((thickness)^9-tbegin^9)+1/7*(4*C133/Lz*Lr^2-C155*Lz)*((thickness)^7-tbegin^7)+... 1/5*(1/2*C155*Lz-4*C133/Lz*Lr^2)*((thickness)^5-tbegin^5)+2/3*C133/Lz*Lr^2*((thickness)^4-tbegin^4)+1/3*... C133/Lz*Lr^2*((thickness)^3-tbegin^3)-2/3*C133/Lz*Lr^2*((thickness)^2-tbegin^2)+1/2*C133/Lz*Lr^2*(tend-... metthick-tbegin))+2*pi*(1/18*C255*Lz*(tend^9-(thickness)^9)+1/7*(4*C233/Lz*Lr^2-C255*Lz)*(tend^7-... (thickness)^7)+1/5*(1/2*C255*Lz-4*C233/Lz*Lr^2)*(tend^5-(thickness)^5)+2/3*C233/Lz*Lr^2*(tend^4-... (thickness)^4)+1/3*C233/Lz*Lr^2*(tend^3-(thickness)^3)-2/3*C233/Lz*Lr^2*(tend^2-(thickness)^2)+1/2*C233/Lz*Lr^2*metthick);k(11,12)=2*pi*(16/21*C133/Lz*Lr^2*((thickness)^7-tbegin^7)-16/15*C133/Lz*Lr^2*((thickness)^5-tbegin^5)+5/6*C133/Lz... *Lr^2*((thickness)^4-tbegin^4)+4/9*C133/Lz*Lr^2*((thickness)^3-tbegin^3)-5/6*C133/Lz*Lr^2*((thickness)^2-... tbegin^2)+1/2*C133/Lz*Lr^2*(tend-metthick-tbegin))+2*pi*(16/21*C233/Lz*Lr^2*(tend^7-(thickness)^7)-16/15*C233/Lz*... Lr^2*(tend^5-(thickness)^5)+5/6*C233/Lz*Lr^2*(tend^4-(thickness)^4)+4/9*C233/Lz*Lr^2*(tend^3-(thickness)^3)-... 5/6*C233/Lz*Lr^2*(tend^2-(thickness)^2)+1/2*C233/Lz*Lr^2*metthick);k(12,12)=2*pi*(8/7*C133/Lz*Lr^2*((thickness)^7-tbegin^7)-8/5*C133/Lz*Lr^2*((thickness)^5-tbegin^5)+C133/Lz*Lr^2*... ((thickness)^4-tbegin^4)+2/3*C133/Lz*Lr^2*((thickness)^3-tbegin^3)-C133/Lz*Lr^2*((thickness)^2-tbegin^2)+1/2*... C133/Lz*Lr^2*(tend-metthick-tbegin))+2*pi*(8/7*C233/Lz*Lr^2*(tend^7-(thickness)^7)-8/5*C233/Lz*Lr^2*... (tend^5-(thickness)^5)+C233/Lz*Lr^2*(tend^4-(thickness)^4)+2/3*C233/Lz*Lr^2*(tend^3-(thickness)^3)-... C233/Lz*Lr^2*(tend^2-(thickness)^2)+1/2*C233/Lz*Lr^2*metthick);

temp = triu(k((umax+1):(umax+wmax),(umax+1):(umax+wmax)),1);temp = k((umax+1):(umax+wmax),(umax+1):(umax+wmax)) + temp';k((umax+1):(umax+wmax),(umax+1):(umax+wmax)) = temp;

tbegin = tend;tend = tend + clen;

%Calculation of K21k12 = k(1:umax,(umax+1):(umax+wmax));k21 = k12';k((umax+1):(umax+wmax),1:umax)=k21;

end %Loop on number of cells

%Begin system eigenvalue solutionr=0;n = N;sizeof = umax+wmax;beg = 1;ending = sizeof;for i = 1:n M(beg:ending,beg:ending) = m; K(beg:ending,beg:ending) = k; beg = beg + sizeof; ending = ending + sizeof;end

phis = [0 0 0 0 1 0 0 0 0 0 0 0];

beg = 1;ending = sizeof;for i = 1:(n-1) constraints(1,beg:ending)=0; beg = beg + sizeof; ending = ending + sizeof;endconstraints(n,beg:ending) = phis;

for i = 1:(n*sizeof) if constraints(n,i)==0 reaction(i)=1; else reaction(i)=0; end

178

end

%Put together constraints for remaining layer interfacesphib=phis;phit=[0 0 0 0 1 2 2 2 2 2 2 2];if n > 1 beg = 1;

ending = sizeof;for layer = 1:(n-1)

constraints((layer),beg:ending)= phib; constraints((layer),(beg+sizeof):(ending+sizeof)) = (phit.*(-1)); beg = beg + sizeof; ending = ending + sizeof; end %end layer loopend

oldcon = constraints;

%Determine the dependend and independent variables

for layer = 1:n depend(layer) = 5 + 12*(layer-1);end

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%Rearrange constraint equations so dependent variables are in front%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

numcon = length(constraints(:,1));

for i = 1:numcon place = depend(i); convec(:,i) = constraints(:,place);end

A1 = convec;

conlen = length(constraints(1,:));

concount = 1;a2count = 1;for i = 1:conlen if i~=depend(concount) A2(:,a2count)=constraints(:,i); a2count = a2count + 1; elseif concount < numcon concount = concount + 1; endend

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

G = -1.*inv(A1)*A2;

newsize = length(G);

beta(1:newsize,1:newsize) = eye(newsize);beta((newsize+1):(newsize+n),1:newsize)=G;

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Rearrange original M and K so that pf and pd are in right place%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

numcon = length(constraints(:,1));colsize = length(M(1,:));rowsize = length(M(:,1));oldM=M;oldK=K;

%Move columns

for i = 1:numcon

place=depend(i);

Mcolvec(:,i) = M(:,place); Kcolvec(:,i) = K(:,place);

end %numconcountcon = 1;tempcount = 0;for j = 1:colsize if j ~= depend(countcon) tempcount = tempcount+1; tempM(:,tempcount)=M(:,j); tempK(:,tempcount)=K(:,j); else if countcon < numcon countcon = countcon + 1; end endend

tempM(:,(tempcount+1):(tempcount+numcon))=Mcolvec;tempK(:,(tempcount+1):(tempcount+numcon))=Kcolvec;M = tempM;K = tempK;

%Move rows

for i = 1:numcon

place=depend(i);

Mrowvec(i,:) = M(place,:); Krowvec(i,:) = K(place,:);

end %numcon

countcon = 1;tempcount = 0;for j = 1:rowsize if j ~= depend(countcon) tempcount = tempcount+1; tempM(tempcount,:)=M(j,:); tempK(tempcount,:)=K(j,:); else if countcon < numcon

179

countcon = countcon + 1; end endend

tempM((tempcount+1):(tempcount+numcon),:)=Mrowvec;tempK((tempcount+1):(tempcount+numcon),:)=Krowvec;

M = tempM;K = tempK;

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Mp = beta.'*M*beta;Kp = beta.'*K*beta;

%% EIGENSOLUTION%

[FEvector, FEvalues] = eig(Kp,Mp);

%% SORT EIGENSOLUTION FROM LOW TO HIGH IN FREQUENCY%

[EIGval, EIGord] = sort(diag(FEvalues));FEvalues = diag(EIGval);

freqs=diag((sqrt(FEvalues))/2/pi);

threedstart = freqs(n);threedend = freqs(n+1);

180

Function – calcaxstiff2

function [axialstiff]=calcaxstiff2(x)


%Calculate axial stiffness with current design parameters

trub=x(6);n=x(5);d = x(7);h = x(8);

shapefactor = d/4/trub;E = x(1)*2*(1+.49);Erubeff = E*(1+2*shapefactor^2);area = d^2*pi/4;

axialstiff = Erubeff*area/(trub*n);

181

Function – calclatstiff2

function [latstiff]=calclatstiff2(x)


%Calculate lateral stiffness with current design parameters

trub = x(6);n=x(5);d = x(7);h = x(8);

nu = 0.49;Grub = x(1);area = d^2*pi/4;

latstiff = Grub*area/(n*trub);

Appendix D

MATLAB CODE: TRANSMISSIBILITY OF LAYERED ISOLATORS

WITH EMBEDDED FLUID ELEMENTS

The following is the MATLAB® code calculates the transmissibility of layered

isolators with embedded fluid elements. The isolators parameters are defined in the first

portion of the code.

Main Program

clear allclose all

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Transmissibility of layered isolators in compression with embedded% fluid elements%% Rotorcraft Center of Excellence% Penn State University% Joseph T. Szefi% [email protected]% 12/1/2002%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%Define frequencies in log spacestart=1;stop=15000;counter = 400;

start = start*2*pi;stop = stop*2*pi;

w = logspace(log10(start),log10(stop),counter);

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Define weighting factors for different isolator mass, stiffness, damping components.% Indices 1 - 7 refer to primary metal and elastomer layers in isolator.% Index 1 is a stiff, massless layer to distribute force.% Small index values for zero mass or stiffness, or '1' for no change.%% Indices 8 - 13 refer to the embedded fluidlastic elements.% Fluid element 8 is embedded in layer 2, 9 in 3, etc.% Fluidlastic stiffness and damping are essentially zero for prototype isolator

% Mass weighting factorsmfact(1)=.000001;mfact(2)=1;mfact(3)=1;mfact(4)=1;mfact(5)=.0000001;mfact(6)=.0000001;mfact(7)=.0000001;

mfact(8)=1;mfact(9)=1;mfact(10)=1;mfact(11)=.0000001;mfact(12)=.0000001;

183

mfact(13)=.0000001;

%Stiffness weighting factorskfact(1)=1000000;kfact(2)=1;kfact(3)=1;kfact(4)=1;kfact(5)=1000000;kfact(6)=1000000;kfact(7)=1000000;

kfact(8)=.001;kfact(9)=.001;kfact(10)=.001;kfact(11)=.001;kfact(12)=.001;kfact(13)=.001;

%Hysteretic damping weighting factorsbetas(1)=0.0;betas(2)=0.005;betas(3)=0.005;betas(4)=0.005;betas(5)=0.001;betas(6)=0.001;betas(7)=0.001;

betas(8)=.01;betas(9)=.01;betas(10)=.01;betas(11)=.01;betas(12)=.01;betas(13)=.01;

% Define fluid viscous damping factorvisdamp = 0;

%Define elastomer properties and thickness per layerGrub = 1.05e6;rhorub = 1100;nu1=.499;trub = .5*.0254;

%Define metal properties and thickness per layerGmet = 2e11;rhomet = 7800;tmetactual = .0254;

%Define Isolator diameterd = 3*.0254;

%Approximate axial stiffnessSS = d/4/trub;Emod =Grub*3;axialstiff = (1 + 2*(SS)^2)*Emod*d^2*pi/4/trub

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% Fluid element properties% Allows for 6 layers with with embedded fluid elements

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

innerdiam(1) = .02;outerdiam(1) = .052;R(1) = (outerdiam(1)/innerdiam(1))^2;rho(1) =1770;portlength(1) = .02;

innerdiam(2) =.025;outerdiam(2) = .052;R(2) = (outerdiam(2)/innerdiam(2))^2;rho(2) =1770;portlength(2) = .02;

innerdiam(3) = .03;outerdiam(3) = .052;R(3) = (outerdiam(3)/innerdiam(3))^2;rho(3) =1770;portlength(3) = .02;

innerdiam(4) = 3/8*.0254;outerdiam(4) = 1.6*.0254;R(4) = (outerdiam(4)/innerdiam(4))^2;rho(4) =5000;portlength(4) = .1*.0254;



N = 7; % 7 layers, although some layers may be massless, and very stiff, depending on weighting factorsnumofabsorbers = 6;

for mm = 1:N %loop through layers and create each layer independently

h = .00000000000001;tmet = .00000000000001;n = 1;

h = h + trub;

metthick = tmet/h*2;Lz= h/2;Lr= d/2;%metthick%Elastomer properties

rho1 = rhorub;

G1 = kfact(mm)*Grub;

184

E1 = G1*2*(1+nu1);

Q1 = E1*(1-nu1)/(1+nu1)/(1-2*nu1);S1 = nu1/(1-nu1);


%Metal Properties%Metal layer is approximated with very thin layer with appropriate density

rho2 = mfact(mm)*rhomet*tmetactual/tmet;nu2 = .499;G2 = Gmet;

E2 = G2*2*(1+nu2);

Q2 = E2*(1-nu2)/(1+nu2)/(1-2*nu2);S2 = nu2/(1-nu2);


umax = 4;wmax = 8;

% Calculate components of global mass matrix

% The global matrices are in the following form

% |K11||K12|{a} - omega^2*|M11| |0|{a} = {0}% |K21||K22|{b} |0| |M22|{b} {0}%

clen = 2/n;

tbegin = -1;tend = -1 + clen;

m = zeros((umax+wmax));k = zeros((umax+wmax));

% Calculation of M11 and M22 for elastomer and metalthickness = tend - metthick;

m(1,1)=2*pi*(1/30*(thickness)^5-1/30*tbegin^5-1/9*(thickness)^3+1/9*tbegin^3+1/6*tend-1/6*metthick-1/6*tbegin)*Lr^2*Lz*rho1... +2*pi*(1/30*tend^5-1/30*(thickness)^5-1/9*tend^3+1/9*(thickness)^3+1/6*metthick)*Lr^2*Lz*rho2;m(1,2)=2*pi*(1/25*(thickness)^5-1/25*tbegin^5-2/15*(thickness)^3+2/15*tbegin^3+1/5*tend-1/5*metthick-1/5*tbegin)*Lr^2*Lz*rho1... +2*pi*(1/25*tend^5-1/25*(thickness)^5-2/15*tend^3+2/15*(thickness)^3+1/5*metthick)*Lr^2*Lz*rho2;m(1,3)=2*pi*(1/36*(thickness)^6-1/36*tbegin^6-1/12*(thickness)^4+1/12*tbegin^4+1/12*(thickness)^2-1/12*tbegin^2)*Lr^2*Lz*rho1... +2*pi*(1/36*tend^6-1/36*(thickness)^6-1/12*tend^4+1/12*(thickness)^4+1/12*tend^2-1/12*(thickness)^2)*Lr^2*Lz*rho2;m(1,4)=2*pi*(1/30*(thickness)^6-1/30*tbegin^6-1/10*(thickness)^4+1/10*tbegin^4+1/10*(thickness)^2-1/10*tbegin^2)*Lr^2*Lz*rho1... +2*pi*(1/30*tend^6-1/30*(thickness)^6-1/10*tend^4+1/10*(thickness)^4+1/10*tend^2-1/10*(thickness)^2)*Lr^2*Lz*rho2;m(2,2)=2*pi*(1/20*(thickness)^5-1/20*tbegin^5-1/6*(thickness)^3+1/6*tbegin^3+1/4*tend-1/4*metthick-1/4*tbegin)*Lr^2*Lz*rho1... +2*pi*(1/20*tend^5-1/20*(thickness)^5-1/6*tend^3+1/6*(thickness)^3+1/4*metthick)*Lr^2*Lz*rho2;m(2,3)=2*pi*(1/30*(thickness)^6-1/30*tbegin^6-1/10*(thickness)^4+1/10*tbegin^4+1/10*(thickness)^2-1/10*tbegin^2)*Lr^2*Lz*rho1... +2*pi*(1/30*tend^6-1/30*(thickness)^6-1/10*tend^4+1/10*(thickness)^4+1/10*tend^2-1/10*(thickness)^2)*Lr^2*Lz*rho2;m(2,4)=2*pi*(1/24*(thickness)^6-1/24*tbegin^6-1/8*(thickness)^4+1/8*tbegin^4+1/8*(thickness)^2-1/8*tbegin^2)*Lr^2*Lz*rho1... +2*pi*(1/24*tend^6-1/24*(thickness)^6-1/8*tend^4+1/8*(thickness)^4+1/8*tend^2-1/8*(thickness)^2)*Lr^2*Lz*rho2;m(3,3)=2*pi*(1/42*(thickness)^7-1/42*tbegin^7-1/15*(thickness)^5+1/15*tbegin^5+1/18*(thickness)^3-1/18*tbegin^3)*Lr^2*Lz*rho1... +2*pi*(1/42*tend^7-1/42*(thickness)^7-1/15*tend^5+1/15*(thickness)^5+1/18*tend^3-1/18*(thickness)^3)*Lr^2*Lz*rho2;m(3,4)=2*pi*(1/35*(thickness)^7-1/35*tbegin^7-2/25*(thickness)^5+2/25*tbegin^5+1/15*(thickness)^3-1/15*tbegin^3)*Lr^2*Lz*rho1... +2*pi*(1/35*tend^7-1/35*(thickness)^7-2/25*tend^5+2/25*(thickness)^5+1/15*tend^3-1/15*(thickness)^3)*Lr^2*Lz*rho2;m(4,4)=2*pi*(1/28*(thickness)^7-1/28*tbegin^7-1/10*(thickness)^5+1/10*tbegin^5+1/12*(thickness)^3-1/12*tbegin^3)*Lr^2*Lz*rho1... +2*pi*(1/28*tend^7-1/28*(thickness)^7-1/10*tend^5+1/10*(thickness)^5+1/12*tend^3-1/12*(thickness)^3)*Lr^2*Lz*rho2;m(5,5)=rho1*Lr^2*Lz*2*pi*(1/2*tend-1/2*metthick-1/2*tbegin) + rho2*Lr^2*Lz*pi*metthick;m(5,6)=rho1*Lr^2*Lz*2*pi*(1/4*(thickness)^2-1/4*tbegin^2+1/2*tend-1/2*metthick-1/2*tbegin)... + rho2*Lr^2*Lz*2*pi*(1/4*tend^2-1/4*(thickness)^2+1/2*metthick);m(5,7)=rho1*Lr^2*Lz*2*pi*(1/9*(thickness)^3-1/9*tbegin^3+1/6*tend-1/6*metthick-1/6*tbegin+1/4*(thickness)^2-1/4*tbegin^2)... + rho2*Lr^2*Lz*2*pi*(1/9*tend^3-1/9*(thickness)^3+1/6*metthick+1/4*tend^2-1/4*(thickness)^2);m(5,8)=rho1*Lr^2*Lz*2*pi*(1/6*(thickness)^3-1/6*tbegin^3+1/4*(thickness)^2-1/4*tbegin^2)... + rho2*Lr^2*Lz*2*pi*(1/6*tend^3-1/6*(thickness)^3+1/4*tend^2-1/4*(thickness)^2);m(5,9)=rho1*Lr^2*Lz*2*pi*(1/12*(thickness)^4-1/12*tbegin^4+1/12*(thickness)^2-1/12*tbegin^2+1/2*tend-1/2*metthick-1/2*tbegin)... + rho2*Lr^2*Lz*2*pi*(1/12*tend^4-1/12*(thickness)^4+1/12*tend^2-1/12*(thickness)^2+1/2*metthick);m(5,10)=rho1*Lr^2*Lz*2*pi*(1/8*(thickness)^4-1/8*tbegin^4+1/2*tend-1/2*metthick-1/2*tbegin)... + rho2*Lr^2*Lz*2*pi*(1/8*tend^4-1/8*(thickness)^4+1/2*metthick);m(5,11)=rho1*Lr^2*Lz*2*pi*(1/15*(thickness)^5-1/15*tbegin^5-1/9*(thickness)^3+1/9*tbegin^3+1/4*(thickness)^2-1/4*tbegin^2+... 1/2*tend-1/2*metthick-1/2*tbegin) + rho2*Lr^2*Lz*2*pi*(1/15*tend^5-1/15*(thickness)^5-1/9*tend^3+1/9*(thickness)^3+... 1/4*tend^2-1/4*(thickness)^2+1/2*metthick);m(5,12)=rho1*Lr^2*Lz*2*pi*(1/10*(thickness)^5-1/10*tbegin^5-1/6*(thickness)^3+1/6*tbegin^3+1/4*(thickness)^2-1/4*tbegin^2+... 1/2*tend-1/2*metthick-1/2*tbegin) + rho2*Lr^2*Lz*2*pi*(1/10*tend^5-1/10*(thickness)^5-1/6*tend^3+1/6*(thickness)^3+... 1/4*tend^2-1/4*(thickness)^2+1/2*metthick);m(6,6)=rho1*Lr^2*Lz*2*pi*(1/6*(thickness)^3-1/6*tbegin^3+1/2*(thickness)^2-1/2*tbegin^2+1/2*tend-1/2*metthick-1/2*tbegin)... + rho2*Lr^2*Lz*2*pi*(1/6*tend^3-1/6*(thickness)^3+1/2*tend^2-1/2*(thickness)^2+1/2*metthick);m(6,7)=rho1*Lr^2*Lz*2*pi*(1/12*(thickness)^4-1/12*tbegin^4+5/18*(thickness)^3-5/18*tbegin^3+1/3*(thickness)^2-1/3*tbegin^2+... 1/6*tend-1/6*metthick-1/6*tbegin) + rho2*Lr^2*Lz*2*pi*(1/12*tend^4-1/12*(thickness)^4+5/18*tend^3-5/18*(thickness)^3+... 1/3*tend^2-1/3*(thickness)^2+1/6*metthick);m(6,8)=rho1*Lr^2*Lz*2*pi*(1/8*(thickness)^4-1/8*tbegin^4+1/3*(thickness)^3-1/3*tbegin^3+1/4*(thickness)^2-1/4*tbegin^2)... + rho2*Lr^2*Lz*2*pi*(1/8*tend^4-1/8*(thickness)^4+1/3*tend^3-1/3*(thickness)^3+1/4*tend^2-1/4*(thickness)^2);m(6,9)=rho1*Lr^2*Lz*2*pi*(1/12*(thickness)^4-1/12*tbegin^4+1/18*(thickness)^3-1/18*tbegin^3+1/15*(thickness)^5-1/15*tbegin^5+... 1/3*(thickness)^2-1/3*tbegin^2+1/2*tend-1/2*metthick-1/2*tbegin) + rho2*Lr^2*Lz*2*pi*(1/12*tend^4-1/12*(thickness)^4+... 1/18*tend^3-1/18*(thickness)^3+1/15*tend^5-1/15*(thickness)^5+1/3*tend^2-1/3*(thickness)^2+1/2*metthick);m(6,10)=rho1*Lr^2*Lz*2*pi*(1/10*(thickness)^5-1/10*tbegin^5+1/8*(thickness)^4-1/8*tbegin^4+1/4*(thickness)^2-1/4*tbegin^2+... 1/2*tend-1/2*metthick-1/2*tbegin) + rho2*Lr^2*Lz*2*pi*(1/10*tend^5-1/10*(thickness)^5+1/8*tend^4-1/8*(thickness)^4+... 1/4*tend^2-1/4*(thickness)^2+1/2*metthick);m(6,11)=rho1*Lr^2*Lz*2*pi*(1/18*(thickness)^6-1/18*tbegin^6+1/18*(thickness)^3-1/18*tbegin^3+1/15*(thickness)^5-1/15*tbegin^5-... 1/12*(thickness)^4+1/12*tbegin^4+1/2*(thickness)^2-1/2*tbegin^2+1/2*tend-1/2*metthick-1/2*tbegin)... + rho2*Lr^2*Lz*2*pi*(1/18*tend^6-1/18*(thickness)^6+1/18*tend^3-1/18*(thickness)^3+1/15*tend^5-1/15*(thickness)^5-... 1/12*tend^4+1/12*(thickness)^4+1/2*tend^2-1/2*(thickness)^2+1/2*metthick);m(6,12)=rho1*Lr^2*Lz*2*pi*(1/12*(thickness)^6-1/12*tbegin^6+1/10*(thickness)^5-1/10*tbegin^5-1/8*(thickness)^4+1/8*tbegin^4+... 1/2*(thickness)^2-1/2*tbegin^2+1/2*tend-1/2*metthick-1/2*tbegin) + rho2*Lr^2*Lz*2*pi*(1/12*tend^6-1/12*(thickness)^6+... 1/10*tend^5-1/10*(thickness)^5-1/8*tend^4+1/8*(thickness)^4+1/2*tend^2-1/2*(thickness)^2+1/2*metthick);m(7,7)=rho1*Lr^2*Lz*2*pi*(1/20*(thickness)^5-1/20*tbegin^5+2/9*(thickness)^3-2/9*tbegin^3+1/12*tend-1/12*metthick-... 1/12*tbegin+1/6*(thickness)^4-1/6*tbegin^4+1/6*(thickness)^2-1/6*tbegin^2) + rho2*Lr^2*Lz*2*pi*(1/20*tend^5-... 1/20*(thickness)^5+2/9*tend^3-2/9*(thickness)^3+1/12*metthick+1/6*tend^4-1/6*(thickness)^4+1/6*tend^2-... 1/6*(thickness)^2);m(7,8)=rho1*Lr^2*Lz*2*pi*(5/24*(thickness)^4-5/24*tbegin^4+2/9*(thickness)^3-2/9*tbegin^3+1/15*(thickness)^5-... 1/15*tbegin^5+1/12*(thickness)^2-1/12*tbegin^2) + rho2*Lr^2*Lz*2*pi*(5/24*tend^4-5/24*(thickness)^4+2/9*tend^3-... 2/9*(thickness)^3+1/15*tend^5-1/15*(thickness)^5+1/12*tend^2-1/12*(thickness)^2);m(7,9)=rho1*Lr^2*Lz*2*pi*(1/24*(thickness)^4-1/24*tbegin^4+7/24*(thickness)^2-7/24*tbegin^2+1/24*(thickness)^6-... 1/24*tbegin^6+1/15*(thickness)^5-1/15*tbegin^5+1/6*tend-1/6*metthick-1/6*tbegin+1/6*(thickness)^3-1/6*tbegin^3)... + rho2*Lr^2*Lz*2*pi*(1/24*tend^4-1/24*(thickness)^4+7/24*tend^2-7/24*(thickness)^2+1/24*tend^6-1/24*... (thickness)^6+1/15*tend^5-1/15*(thickness)^5+1/6*metthick+1/6*tend^3-1/6*(thickness)^3);m(7,10)=rho1*Lr^2*Lz*2*pi*(1/18*(thickness)^6-1/18*tbegin^6+1/24*(thickness)^4-1/24*tbegin^4+1/9*(thickness)^3-1/9*tbegin^3+...

185

1/6*tend-1/6*metthick-1/6*tbegin+1/10*(thickness)^5-1/10*tbegin^5+1/4*(thickness)^2-1/4*tbegin^2) +... rho2*Lr^2*Lz*2*pi*(1/18*tend^6-1/18*(thickness)^6+1/24*tend^4-1/24*(thickness)^4+1/9*tend^3-1/9*(thickness)^3+... 1/6*metthick+1/10*tend^5-1/10*(thickness)^5+1/4*tend^2-1/4*(thickness)^2);m(7,11)=rho1*Lr^2*Lz*2*pi*(-1/30*(thickness)^5+1/30*tbegin^5+1/28*(thickness)^7-1/28*tbegin^7+1/4*(thickness)^3-... 1/4*tbegin^3+1/18*(thickness)^6-1/18*tbegin^6+1/3*(thickness)^2-1/3*tbegin^2+1/6*tend-1/6*metthick-1/6*tbegin)... + rho2*Lr^2*Lz*2*pi*(-1/30*tend^5+1/30*(thickness)^5+1/28*tend^7-1/28*(thickness)^7+1/4*tend^3-1/4*(thickness)^3+... 1/18*tend^6-1/18*(thickness)^6+1/3*tend^2-1/3*(thickness)^2+1/6*metthick);m(7,12)=rho1*Lr^2*Lz*2*pi*(1/21*(thickness)^7-1/21*tbegin^7+2/9*(thickness)^3-2/9*tbegin^3-1/30*(thickness)^5+... 1/30*tbegin^5+1/3*(thickness)^2-1/3*tbegin^2-1/24*(thickness)^4+1/24*tbegin^4+1/6*tend-1/6*metthick-1/6*tbegin+... 1/12*(thickness)^6-1/12*tbegin^6) + rho2*Lr^2*Lz*2*pi*(1/21*tend^7-1/21*(thickness)^7+2/9*tend^3-2/9*(thickness)^3-... 1/30*tend^5+1/30*(thickness)^5+1/3*tend^2-1/3*(thickness)^2-1/24*tend^4+1/24*(thickness)^4+1/6*metthick+1/12*tend^6-... 1/12*(thickness)^6);m(8,8)=rho1*Lr^2*Lz*2*pi*(1/10*(thickness)^5-1/10*tbegin^5+1/4*(thickness)^4-1/4*tbegin^4+1/6*(thickness)^3-1/6*tbegin^3)... + rho2*Lr^2*Lz*2*pi*(1/10*tend^5-1/10*(thickness)^5+1/4*tend^4-1/4*(thickness)^4+1/6*tend^3-1/6*(thickness)^3);m(8,9)=rho1*Lr^2*Lz*2*pi*(1/18*(thickness)^6-1/18*tbegin^6+2/9*(thickness)^3-2/9*tbegin^3+1/15*(thickness)^5-1/15*tbegin^5+... 1/24*(thickness)^4-1/24*tbegin^4+1/4*(thickness)^2-1/4*tbegin^2) + rho2*Lr^2*Lz*2*pi*(1/18*tend^6-1/18*(thickness)^6+... 2/9*tend^3-2/9*(thickness)^3+1/15*tend^5-1/15*(thickness)^5+1/24*tend^4-1/24*(thickness)^4+1/4*tend^2-1/4*(thickness)^2);m(8,10)=rho1*Lr^2*Lz*2*pi*(1/12*(thickness)^6-1/12*tbegin^6+1/10*(thickness)^5-1/10*tbegin^5+1/6*(thickness)^3-1/6*tbegin^3+... 1/4*(thickness)^2-1/4*tbegin^2) + rho2*Lr^2*Lz*2*pi*(1/12*tend^6-1/12*(thickness)^6+1/10*tend^5-1/10*(thickness)^5+... 1/6*tend^3-1/6*(thickness)^3+1/4*tend^2-1/4*(thickness)^2);m(8,11)=rho1*Lr^2*Lz*2*pi*(1/21*(thickness)^7-1/21*tbegin^7+1/24*(thickness)^4-1/24*tbegin^4+1/18*(thickness)^6-... 1/18*tbegin^6-1/15*(thickness)^5+1/15*tbegin^5+1/3*(thickness)^3-1/3*tbegin^3+1/4*(thickness)^2-1/4*tbegin^2)... + rho2*Lr^2*Lz*2*pi*(1/21*tend^7-1/21*(thickness)^7+1/24*tend^4-1/24*(thickness)^4+1/18*tend^6-1/18*... (thickness)^6-1/15*tend^5+1/15*(thickness)^5+1/3*tend^3-1/3*(thickness)^3+1/4*tend^2-1/4*(thickness)^2);m(8,12)=rho1*Lr^2*Lz*2*pi*(1/14*(thickness)^7-1/14*tbegin^7+1/12*(thickness)^6-1/12*tbegin^6-1/10*(thickness)^5+... 1/10*tbegin^5+1/3*(thickness)^3-1/3*tbegin^3+1/4*(thickness)^2-1/4*tbegin^2) + rho2*Lr^2*Lz*2*pi*(1/14*tend^7-... 1/14*(thickness)^7+1/12*tend^6-1/12*(thickness)^6-1/10*tend^5+1/10*(thickness)^5+1/3*tend^3-1/3*(thickness)^3+... 1/4*tend^2-1/4*(thickness)^2);m(9,9)=rho1*Lr^2*Lz*2*pi*(1/30*(thickness)^5-1/30*tbegin^5+1/28*(thickness)^7-1/28*tbegin^7+1/36*(thickness)^3-... 1/36*tbegin^3+1/6*(thickness)^2-1/6*tbegin^2+1/6*(thickness)^4-1/6*tbegin^4+1/2*tend-1/2*metthick-1/2*tbegin)... + rho2*Lr^2*Lz*2*pi*(1/30*tend^5-1/30*(thickness)^5+1/28*tend^7-1/28*(thickness)^7+1/36*tend^3-1/36*(thickness)^3+... 1/6*tend^2-1/6*(thickness)^2+1/6*tend^4-1/6*(thickness)^4+1/2*metthick);m(9,10)=rho1*Lr^2*Lz*2*pi*(1/21*(thickness)^7-1/21*tbegin^7+1/30*(thickness)^5-1/30*tbegin^5+5/24*(thickness)^4-5/24*tbegin^4+... 1/12*(thickness)^2-1/12*tbegin^2+1/2*tend-1/2*metthick-1/2*tbegin) + rho2*Lr^2*Lz*2*pi*(1/21*tend^7-1/21*(thickness)^7+... 1/30*tend^5-1/30*(thickness)^5+5/24*tend^4-5/24*(thickness)^4+1/12*tend^2-1/12*(thickness)^2+1/2*metthick);m(9,11)=rho1*Lr^2*Lz*2*pi*(1/32*(thickness)^8-1/32*tbegin^8-1/36*(thickness)^6+1/36*tbegin^6+1/16*(thickness)^4-... 1/16*tbegin^4+1/3*(thickness)^2-1/3*tbegin^2-1/18*(thickness)^3+1/18*tbegin^3+2/15*(thickness)^5-2/15*tbegin^5+... 1/2*tend-1/2*metthick-1/2*tbegin) + rho2*Lr^2*Lz*2*pi*(1/32*tend^8-1/32*(thickness)^8-1/36*tend^6+1/36*(thickness)^6+... 1/16*tend^4-1/16*(thickness)^4+1/3*tend^2-1/3*(thickness)^2-1/18*tend^3+1/18*(thickness)^3+2/15*tend^5-... 2/15*(thickness)^5+1/2*metthick);m(9,12)=rho1*Lr^2*Lz*2*pi*(1/24*(thickness)^8-1/24*tbegin^8-1/36*(thickness)^6+1/36*tbegin^6-1/9*(thickness)^3+... 1/9*tbegin^3+1/6*(thickness)^5-1/6*tbegin^5+1/3*(thickness)^2-1/3*tbegin^2+1/24*(thickness)^4-1/24*tbegin^4+... 1/2*tend-1/2*metthick-1/2*tbegin) + rho2*Lr^2*Lz*2*pi*(1/24*tend^8-1/24*(thickness)^8-1/36*tend^6+1/36*(thickness)^6-... 1/9*tend^3+1/9*(thickness)^3+1/6*tend^5-1/6*(thickness)^5+1/3*tend^2-1/3*(thickness)^2+1/24*tend^4-1/24*(thickness)^4+... 1/2*metthick);m(10,10)=rho1*Lr^2*Lz*2*pi*(1/14*(thickness)^7-1/14*tbegin^7+1/4*(thickness)^4-1/4*tbegin^4+1/2*tend-1/2*metthick-1/2*tbegin)... + rho2*Lr^2*Lz*2*pi*(1/14*tend^7-1/14*(thickness)^7+1/4*tend^4-1/4*(thickness)^4+1/2*metthick);m(10,11)=rho1*Lr^2*Lz*2*pi*(-1/18*(thickness)^6+1/18*tbegin^6+1/24*(thickness)^8-1/24*tbegin^8-1/9*(thickness)^3+... 1/9*tbegin^3+1/6*(thickness)^5-1/6*tbegin^5+1/8*(thickness)^4-1/8*tbegin^4+1/4*(thickness)^2-1/4*tbegin^2+... 1/2*tend-1/2*metthick-1/2*tbegin) + rho2*Lr^2*Lz*2*pi*(-1/18*tend^6+1/18*(thickness)^6+1/24*tend^8-... 1/24*(thickness)^8-1/9*tend^3+1/9*(thickness)^3+1/6*tend^5-1/6*(thickness)^5+1/8*tend^4-1/8*(thickness)^4+... 1/4*tend^2-1/4*(thickness)^2+1/2*metthick);m(10,12)=rho1*Lr^2*Lz*2*pi*(1/16*(thickness)^8-1/16*tbegin^8-1/12*(thickness)^6+1/12*tbegin^6+1/5*(thickness)^5-... 1/5*tbegin^5+1/8*(thickness)^4-1/8*tbegin^4-1/6*(thickness)^3+1/6*tbegin^3+1/4*(thickness)^2-1/4*tbegin^2+1/2*tend... -1/2*metthick-1/2*tbegin) + rho2*Lr^2*Lz*2*pi*(1/16*tend^8-1/16*(thickness)^8-1/12*tend^6+1/12*(thickness)^6+... 1/5*tend^5-1/5*(thickness)^5+1/8*tend^4-1/8*(thickness)^4-1/6*tend^3+1/6*(thickness)^3+1/4*tend^2-1/4*(thickness)^2+... 1/2*metthick);m(11,11)=rho1*Lr^2*Lz*2*pi*(1/36*(thickness)^9-1/36*tbegin^9+11/60*(thickness)^5-11/60*tbegin^5-1/14*(thickness)^7+... 1/14*tbegin^7-1/6*(thickness)^4+1/6*tbegin^4+1/9*(thickness)^6-1/9*tbegin^6-1/18*(thickness)^3+1/18*tbegin^3+... 1/2*(thickness)^2-1/2*tbegin^2+1/2*tend-1/2*metthick-1/2*tbegin) + rho2*Lr^2*Lz*2*pi*(1/36*tend^9-1/36*(thickness)^9+... 11/60*tend^5-11/60*(thickness)^5-1/14*tend^7+1/14*(thickness)^7-1/6*tend^4+1/6*(thickness)^4+1/9*tend^6-1/9*(thickness)^6-... 1/18*tend^3+1/18*(thickness)^3+1/2*tend^2-1/2*(thickness)^2+1/2*metthick);m(11,12)=rho1*Lr^2*Lz*2*pi*(5/36*(thickness)^6-5/36*tbegin^6+1/27*(thickness)^9-1/27*tbegin^9-2/21*(thickness)^7+... 2/21*tbegin^7-5/24*(thickness)^4+5/24*tbegin^4-1/9*(thickness)^3+1/9*tbegin^3+7/30*(thickness)^5-7/30*tbegin^5+... 1/2*(thickness)^2-1/2*tbegin^2+1/2*tend-1/2*metthick-1/2*tbegin) + rho2*Lr^2*Lz*2*pi*(5/36*tend^6-5/36*(thickness)^6+... 1/27*tend^9-1/27*(thickness)^9-2/21*tend^7+2/21*(thickness)^7-5/24*tend^4+5/24*(thickness)^4-1/9*tend^3+1/9*(thickness)^3+... 7/30*tend^5-7/30*(thickness)^5+1/2*tend^2-1/2*(thickness)^2+1/2*metthick);m(12,12)=rho1*Lr^2*Lz*2*pi*(1/18*(thickness)^9-1/18*tbegin^9-1/7*(thickness)^7+1/7*tbegin^7+1/6*(thickness)^6-1/6*tbegin^6+... 3/10*(thickness)^5-3/10*tbegin^5-1/4*(thickness)^4+1/4*tbegin^4-1/6*(thickness)^3+1/6*tbegin^3+1/2*(thickness)^2-... 1/2*tbegin^2+1/2*tend-1/2*metthick-1/2*tbegin) + rho2*Lr^2*Lz*2*pi*(1/18*tend^9-1/18*(thickness)^9-1/7*tend^7+... 1/7*(thickness)^7+1/6*tend^6-1/6*(thickness)^6+3/10*tend^5-3/10*(thickness)^5-1/4*tend^4+1/4*(thickness)^4-... 1/6*tend^3+1/6*(thickness)^3+1/2*tend^2-1/2*(thickness)^2+1/2*metthick);

temp = triu(m(1:umax,1:umax),1);temp = m(1:umax,1:umax) + temp';m(1:umax, 1:umax) = temp;temp = triu(m((umax+1):(umax+wmax),(umax+1):(umax+wmax)),1);temp = m((umax+1):(umax+wmax),(umax+1):(umax+wmax)) + temp';m((umax+1):(umax+wmax),(umax+1):(umax+wmax)) = temp;


k(1,1)=2*pi*(1/20*(4*C111/Lr^2+C122/Lr^2+4*C112/Lr^2)*Lr^2*Lz*((thickness)^5-tbegin^5)+1/3*(2/3*C155/Lz*Lr^2+... 1/4*(-8*C111/Lr^2-8*C112/Lr^2-2*C122/Lr^2)*Lr^2*Lz)*((thickness)^3-tbegin^3)+1/4*(4*C111/Lr^2+C122/Lr^2+... 4*C112/Lr^2)*Lr^2*Lz*(tend-metthick-tbegin))+2*pi*(1/20*(4*C211/Lr^2+C222/Lr^2+4*C212/Lr^2)*Lr^2*Lz*(tend^5-... (thickness)^5)+1/3*(2/3*C255/Lz*Lr^2+1/4*(-8*C211/Lr^2-8*C212/Lr^2-2*C222/Lr^2)*Lr^2*Lz)*(tend^3-(thickness)^3)+... 1/4*(4*C211/Lr^2+C222/Lr^2+4*C212/Lr^2)*Lr^2*Lz*metthick);k(1,2)=2*pi*(1/15*(2*C111/Lr^2+C122/Lr^2+3*C112/Lr^2)*Lr^2*Lz*((thickness)^5-tbegin^5)+1/3*(4/5*C155/Lz*Lr^2+... 1/3*(-4*C111/Lr^2-6*C112/Lr^2-2*C122/Lr^2)*Lr^2*Lz)*((thickness)^3-tbegin^3)+1/3*(2*C111/Lr^2+C122/Lr^2+... 3*C112/Lr^2)*Lr^2*Lz*(tend-metthick-tbegin))+2*pi*(1/15*(2*C211/Lr^2+C222/Lr^2+3*C212/Lr^2)*Lr^2*Lz*(tend^5-... (thickness)^5)+1/3*(4/5*C255/Lz*Lr^2+1/3*(-4*C211/Lr^2-6*C212/Lr^2-2*C222/Lr^2)*Lr^2*Lz)*(tend^3-(thickness)^3)+... 1/3*(2*C211/Lr^2+C222/Lr^2+3*C212/Lr^2)*Lr^2*Lz*metthick);k(1,3)=2*pi*(1/24*(4*C111/Lr^2+C122/Lr^2+4*C112/Lr^2)*Lr^2*Lz*((thickness)^6-tbegin^6)+1/4*(C155/Lz*Lr^2+... 1/4*(-8*C111/Lr^2-8*C112/Lr^2-2*C122/Lr^2)*Lr^2*Lz)*((thickness)^4-tbegin^4)+1/2*(-1/3*C155/Lz*Lr^2+... 1/4*(4*C111/Lr^2+C122/Lr^2+4*C112/Lr^2)*Lr^2*Lz)*((thickness)^2-tbegin^2))+2*pi*(1/24*(4*C211/Lr^2+C222/Lr^2+... 4*C212/Lr^2)*Lr^2*Lz*(tend^6-(thickness)^6)+1/4*(C255/Lz*Lr^2+1/4*(-8*C211/Lr^2-8*C212/Lr^2-... 2*C222/Lr^2)*Lr^2*Lz)*(tend^4-(thickness)^4)+1/2*(-1/3*C255/Lz*Lr^2+1/4*(4*C211/Lr^2+C222/Lr^2+... 4*C212/Lr^2)*Lr^2*Lz)*(tend^2-(thickness)^2));k(1,4)=2*pi*(1/18*(2*C111/Lr^2+C122/Lr^2+3*C112/Lr^2)*Lr^2*Lz*((thickness)^6-tbegin^6)+1/4*(6/5*C155/Lz*Lr^2+... 1/3*(-4*C111/Lr^2-6*C112/Lr^2-2*C122/Lr^2)*Lr^2*Lz)*((thickness)^4-tbegin^4)+1/2*(-2/5*C155/Lz*Lr^2+... 1/3*(2*C111/Lr^2+C122/Lr^2+3*C112/Lr^2)*Lr^2*Lz)*((thickness)^2-tbegin^2))+2*pi*(1/18*(2*C211/Lr^2+... C222/Lr^2+3*C212/Lr^2)*Lr^2*Lz*(tend^6-(thickness)^6)+1/4*(6/5*C255/Lz*Lr^2+1/3*(-4*C211/Lr^2-... 6*C212/Lr^2-2*C222/Lr^2)*Lr^2*Lz)*(tend^4-(thickness)^4)+1/2*(-2/5*C255/Lz*Lr^2+1/3*(2*C211/Lr^2+... C222/Lr^2+3*C212/Lr^2)*Lr^2*Lz)*(tend^2-(thickness)^2));k(2,2)=2*pi*(1/10*(C111/Lr^2+C122/Lr^2+2*C112/Lr^2)*Lr^2*Lz*((thickness)^5-tbegin^5)+1/3*(C155/Lz*Lr^2+... 1/2*(-2*C111/Lr^2-4*C112/Lr^2-2*C122/Lr^2)*Lr^2*Lz)*((thickness)^3-tbegin^3)+1/2*(C111/Lr^2+C122/Lr^2+... 2*C112/Lr^2)*Lr^2*Lz*(tend-metthick-tbegin))+2*pi*(1/10*(C211/Lr^2+C222/Lr^2+2*C212/Lr^2)*Lr^2*Lz*(tend^5-... (thickness)^5)+1/3*(C255/Lz*Lr^2+1/2*(-2*C211/Lr^2-4*C212/Lr^2-2*C222/Lr^2)*Lr^2*Lz)*(tend^3-(thickness)^3)+... 1/2*(C211/Lr^2+C222/Lr^2+2*C212/Lr^2)*Lr^2*Lz*metthick);k(2,3)=2*pi*(1/18*(2*C111/Lr^2+C122/Lr^2+3*C112/Lr^2)*Lr^2*Lz*((thickness)^6-tbegin^6)+1/4*(6/5*C155/Lz*Lr^2+... 1/3*(-4*C111/Lr^2-6*C112/Lr^2-2*C122/Lr^2)*Lr^2*Lz)*((thickness)^4-tbegin^4)+1/2*(-2/5*C155/Lz*Lr^2+... 1/3*(2*C111/Lr^2+C122/Lr^2+3*C112/Lr^2)*Lr^2*Lz)*((thickness)^2-tbegin^2))+2*pi*(1/18*(2*C211/Lr^2+... C222/Lr^2+3*C212/Lr^2)*Lr^2*Lz*(tend^6-(thickness)^6)+1/4*(6/5*C255/Lz*Lr^2+1/3*(-4*C211/Lr^2-6*C212/Lr^2-... 2*C222/Lr^2)*Lr^2*Lz)*(tend^4-(thickness)^4)+1/2*(-2/5*C255/Lz*Lr^2+1/3*(2*C211/Lr^2+C222/Lr^2+... 3*C212/Lr^2)*Lr^2*Lz)*(tend^2-(thickness)^2));k(2,4)=2*pi*(1/12*(C111/Lr^2+C122/Lr^2+2*C112/Lr^2)*Lr^2*Lz*((thickness)^6-tbegin^6)+1/4*(3/2*C155/Lz*Lr^2+... 1/2*(-2*C111/Lr^2-4*C112/Lr^2-2*C122/Lr^2)*Lr^2*Lz)*((thickness)^4-tbegin^4)+1/2*(-1/2*C155/Lz*Lr^2+... 1/2*(C111/Lr^2+C122/Lr^2+2*C112/Lr^2)*Lr^2*Lz)*((thickness)^2-tbegin^2))+2*pi*(1/12*(C211/Lr^2+C222/Lr^2+... 2*C212/Lr^2)*Lr^2*Lz*(tend^6-(thickness)^6)+1/4*(3/2*C255/Lz*Lr^2+1/2*(-2*C211/Lr^2-4*C212/Lr^2-... 2*C222/Lr^2)*Lr^2*Lz)*(tend^4-(thickness)^4)+1/2*(-1/2*C255/Lz*Lr^2+1/2*(C211/Lr^2+C222/Lr^2+... 2*C212/Lr^2)*Lr^2*Lz)*(tend^2-(thickness)^2));k(3,3)=2*pi*(1/28*(4*C111/Lr^2+C122/Lr^2+4*C112/Lr^2)*Lr^2*Lz*((thickness)^7-tbegin^7)+1/5*(3/2*C155/Lz*Lr^2+...

186

1/4*(-8*C111/Lr^2-8*C112/Lr^2-2*C122/Lr^2)*Lr^2*Lz)*((thickness)^5-tbegin^5)+1/3*(-C155/Lz*Lr^2+... 1/4*(4*C111/Lr^2+C122/Lr^2+4*C112/Lr^2)*Lr^2*Lz)*((thickness)^3-tbegin^3)+1/6*C155/Lz*Lr^2*(tend-metthick-tbegin))+... 2*pi*(1/28*(4*C211/Lr^2+C222/Lr^2+4*C212/Lr^2)*Lr^2*Lz*(tend^7-(thickness)^7)+1/5*(3/2*C255/Lz*Lr^2+... 1/4*(-8*C211/Lr^2-8*C212/Lr^2-2*C222/Lr^2)*Lr^2*Lz)*(tend^5-(thickness)^5)+... 1/3*(-C255/Lz*Lr^2+1/4*(4*C211/Lr^2+C222/Lr^2+4*C212/Lr^2)*Lr^2*Lz)*(tend^3-(thickness)^3)+1/6*C255/Lz*Lr^2*metthick);k(3,4)=2*pi*(1/21*(2*C111/Lr^2+C122/Lr^2+3*C112/Lr^2)*Lr^2*Lz*((thickness)^7-tbegin^7)+1/5*(9/5*C155/Lz*Lr^2+... 1/3*(-4*C111/Lr^2-6*C112/Lr^2-2*C122/Lr^2)*Lr^2*Lz)*((thickness)^5-tbegin^5)+1/3*(-6/5*C155/Lz*Lr^2+... 1/3*(2*C111/Lr^2+C122/Lr^2+3*C112/Lr^2)*Lr^2*Lz)*((thickness)^3-tbegin^3)+1/5*C155/Lz*Lr^2*(tend-metthick-tbegin))+... 2*pi*(1/21*(2*C211/Lr^2+C222/Lr^2+3*C212/Lr^2)*Lr^2*Lz*(tend^7-(thickness)^7)+1/5*(9/5*C255/Lz*Lr^2+... 1/3*(-4*C211/Lr^2-6*C212/Lr^2-2*C222/Lr^2)*Lr^2*Lz)*(tend^5-(thickness)^5)+1/3*(-6/5*C255/Lz*Lr^2+... 1/3*(2*C211/Lr^2+C222/Lr^2+3*C212/Lr^2)*Lr^2*Lz)*(tend^3-(thickness)^3)+1/5*C255/Lz*Lr^2*metthick);k(4,4)=2*pi*(1/14*(C111/Lr^2+C122/Lr^2+2*C112/Lr^2)*Lr^2*Lz*((thickness)^7-tbegin^7)+1/5*(9/4*C155/Lz*Lr^2+... 1/2*(-2*C111/Lr^2-4*C112/Lr^2-2*C122/Lr^2)*Lr^2*Lz)*((thickness)^5-tbegin^5)+1/3*(-3/2*C155/Lz*Lr^2+1/2*(C111/Lr^2+... C122/Lr^2+2*C112/Lr^2)*Lr^2*Lz)*((thickness)^3-tbegin^3)+1/4*C155/Lz*Lr^2*(tend-metthick-tbegin))+... 2*pi*(1/14*(C211/Lr^2+C222/Lr^2+2*C212/Lr^2)*Lr^2*Lz*(tend^7-(thickness)^7)+1/5*(9/4*C255/Lz*Lr^2+... 1/2*(-2*C211/Lr^2-4*C212/Lr^2-2*C222/Lr^2)*Lr^2*Lz)*(tend^5-(thickness)^5)+1/3*(-3/2*C255/Lz*Lr^2+1/2*(C211/Lr^2+... C222/Lr^2+2*C212/Lr^2)*Lr^2*Lz)*(tend^3-(thickness)^3)+1/4*C255/Lz*Lr^2*metthick);

temp = triu(k(1:umax,1:umax),1);temp = k(1:umax,1:umax) + temp';k(1:umax, 1:umax) = temp;


k(1,5)=0;k(1,6)=2*pi*(1/9*(2*C113/Lr/Lz+C123/Lr/Lz)*Lr^2*Lz*((thickness)^3-tbegin^3)+1/3*(-2*C113/Lr/Lz-C123/Lr/Lz)*Lr^2*Lz*(tend-... metthick-tbegin))+2*pi*(1/9*(2*C213/Lr/Lz+C223/Lr/Lz)*Lr^2*Lz*(tend^3-(thickness)^3)+1/3*(-2*C213/Lr/Lz-... C223/Lr/Lz)*Lr^2*Lz*metthick);k(1,7)=2*pi*(1/16*(4*C113/Lr/Lz+2*C123/Lr/Lz+2*C155/Lr/Lz)*Lr^2*Lz*((thickness)^4-tbegin^4)+... 1/9*(2*C113/Lr/Lz+C123/Lr/Lz)*Lr^2*Lz*((thickness)^3-tbegin^3)+1/8*(-4*C113/Lr/Lz-2*C123/Lr/Lz-... 2*C155/Lr/Lz)*Lr^2*Lz*((thickness)^2-tbegin^2)+1/3*(-2*C113/Lr/Lz-C123/Lr/Lz)*Lr^2*Lz*(tend-metthick-tbegin))+... 2*pi*(1/16*(4*C213/Lr/Lz+2*C223/Lr/Lz+2*C255/Lr/Lz)*Lr^2*Lz*(tend^4-(thickness)^4)+... 1/9*(2*C213/Lr/Lz+C223/Lr/Lz)*Lr^2*Lz*(tend^3-(thickness)^3)+1/8*(-4*C213/Lr/Lz-2*C223/Lr/Lz-... 2*C255/Lr/Lz)*Lr^2*Lz*(tend^2-(thickness)^2)+1/3*(-2*C213/Lr/Lz-C223/Lr/Lz)*Lr^2*Lz*metthick);k(1,8)=2*pi*(1/12*(4*C113/Lr/Lz+2*C123/Lr/Lz)*Lr^2*Lz*((thickness)^4-tbegin^4)+1/9*(2*C113/Lr/Lz+... C123/Lr/Lz)*Lr^2*Lz*((thickness)^3-tbegin^3)+1/6*(-4*C113/Lr/Lz-2*C123/Lr/Lz)*Lr^2*Lz*((thickness)^2-... tbegin^2)+1/3*(-2*C113/Lr/Lz-C123/Lr/Lz)*Lr^2*Lz*(tend-metthick-tbegin))+2*pi*(1/12*(4*C213/Lr/Lz+... 2*C223/Lr/Lz)*Lr^2*Lz*(tend^4-(thickness)^4)+1/9*(2*C213/Lr/Lz+C223/Lr/Lz)*Lr^2*Lz*(tend^3-(thickness)^3)+... 1/6*(-4*C213/Lr/Lz-2*C223/Lr/Lz)*Lr^2*Lz*(tend^2-(thickness)^2)+1/3*(-2*C213/Lr/Lz-C223/Lr/Lz)*Lr^2*Lz*metthick);k(1,9)=2*pi*(1/20*(3*C123/Lr/Lz+6*C113/Lr/Lz+2*C155/Lr/Lz)*Lr^2*Lz*((thickness)^5-tbegin^5)+... 1/3*(1/4*(-8*C113/Lr/Lz-2*C155/Lr/Lz-4*C123/Lr/Lz)*Lr^2*Lz+1/3*(2*C113/Lr/Lz+C123/Lr/Lz)*Lr^2*Lz)*((thickness)^3-... tbegin^3)+1/4*(2*C113/Lr/Lz+C123/Lr/Lz)*Lr^2*Lz*(tend-metthick-tbegin)+1/3*(-2*C113/Lr/Lz-... C123/Lr/Lz)*Lr^2*Lz*(tend-metthick-tbegin))+2*pi*(1/20*(3*C223/Lr/Lz+6*C213/Lr/Lz+2*C255/Lr/Lz)*Lr^2*Lz*(tend^5-... (thickness)^5)+1/3*(1/4*(-8*C213/Lr/Lz-2*C255/Lr/Lz-4*C223/Lr/Lz)*Lr^2*Lz+1/3*(2*C213/Lr/Lz+... C223/Lr/Lz)*Lr^2*Lz)*(tend^3-(thickness)^3)+1/4*(2*C213/Lr/Lz+C223/Lr/Lz)*Lr^2*Lz*metthick+... 1/3*(-2*C213/Lr/Lz-C223/Lr/Lz)*Lr^2*Lz*metthick);k(1,10)=2*pi*(1/15*(6*C113/Lr/Lz+3*C123/Lr/Lz)*Lr^2*Lz*((thickness)^5-tbegin^5)+1/9*(-6*C113/Lr/Lz-... 3*C123/Lr/Lz)*Lr^2*Lz*((thickness)^3-tbegin^3))+2*pi*(1/15*(6*C213/Lr/Lz+3*C223/Lr/Lz)*Lr^2*Lz*(tend^5-... (thickness)^5)+1/9*(-6*C213/Lr/Lz-3*C223/Lr/Lz)*Lr^2*Lz*(tend^3-(thickness)^3));k(1,11)=2*pi*(1/24*(4*C123/Lr/Lz+8*C113/Lr/Lz+2*C155/Lr/Lz)*Lr^2*Lz*((thickness)^6-tbegin^6)+... 1/16*(-12*C113/Lr/Lz-2*C155/Lr/Lz-6*C123/Lr/Lz)*Lr^2*Lz*((thickness)^4-tbegin^4)+... 1/9*(2*C113/Lr/Lz+C123/Lr/Lz)*Lr^2*Lz*((thickness)^3-tbegin^3)+1/8*(4*C113/Lr/Lz+2*C123/Lr/Lz)*Lr^2*Lz*((thickness)^2-... tbegin^2)+1/3*(-2*C113/Lr/Lz-C123/Lr/Lz)*Lr^2*Lz*(tend-metthick-tbegin))+2*pi*(1/24*(4*C223/Lr/Lz+8*C213/Lr/Lz+... 2*C255/Lr/Lz)*Lr^2*Lz*(tend^6-(thickness)^6)+1/16*(-12*C213/Lr/Lz-2*C255/Lr/Lz-6*C223/Lr/Lz)*Lr^2*Lz*(tend^4-... (thickness)^4)+1/9*(2*C213/Lr/Lz+C223/Lr/Lz)*Lr^2*Lz*(tend^3-(thickness)^3)+1/8*(4*C213/Lr/Lz+2*C223/Lr/Lz)*Lr^2*Lz*(tend^2-... (thickness)^2)+1/3*(-2*C213/Lr/Lz-C223/Lr/Lz)*Lr^2*Lz*metthick);k(1,12)=2*pi*(1/18*(8*C113/Lr/Lz+4*C123/Lr/Lz)*Lr^2*Lz*((thickness)^6-tbegin^6)+1/12*(-6*C123/Lr/Lz-... 12*C113/Lr/Lz)*Lr^2*Lz*((thickness)^4-tbegin^4)+1/9*(2*C113/Lr/Lz+C123/Lr/Lz)*Lr^2*Lz*((thickness)^3-... tbegin^3)+1/6*(4*C113/Lr/Lz+2*C123/Lr/Lz)*Lr^2*Lz*((thickness)^2-tbegin^2)+1/3*(-2*C113/Lr/Lz-... C123/Lr/Lz)*Lr^2*Lz*(tend-metthick-tbegin))+2*pi*(1/18*(8*C213/Lr/Lz+4*C223/Lr/Lz)*Lr^2*Lz*(tend^6-... (thickness)^6)+1/12*(-6*C223/Lr/Lz-12*C213/Lr/Lz)*Lr^2*Lz*(tend^4-(thickness)^4)+... 1/9*(2*C213/Lr/Lz+C223/Lr/Lz)*Lr^2*Lz*(tend^3-(thickness)^3)+1/6*(4*C213/Lr/Lz+2*C223/Lr/Lz)*Lr^2*Lz*(tend^2-... (thickness)^2)+1/3*(-2*C213/Lr/Lz-C223/Lr/Lz)*Lr^2*Lz*metthick);k(2,5)=0;k(2,6)=2*pi*(1/6*(C113/Lr/Lz+C123/Lr/Lz)*Lr^2*Lz*((thickness)^3-tbegin^3)+1/2*(-C113/Lr/Lz-C123/Lr/Lz)*Lr^2*Lz*(tend-... metthick-tbegin))+2*pi*(1/6*(C213/Lr/Lz+C223/Lr/Lz)*Lr^2*Lz*(tend^3-(thickness)^3)+1/2*(-C213/Lr/Lz-... C223/Lr/Lz)*Lr^2*Lz*metthick);k(2,7)=2*pi*(1/12*(2*C113/Lr/Lz+2*C123/Lr/Lz+2*C155/Lr/Lz)*Lr^2*Lz*((thickness)^4-tbegin^4)+1/6*(C113/Lr/Lz+... C123/Lr/Lz)*Lr^2*Lz*((thickness)^3-tbegin^3)+1/6*(-2*C113/Lr/Lz-2*C123/Lr/Lz-2*C155/Lr/Lz)*Lr^2*Lz*((thickness)^2-... tbegin^2)+1/2*(-C113/Lr/Lz-C123/Lr/Lz)*Lr^2*Lz*(tend-metthick-tbegin))+2*pi*(1/12*(2*C213/Lr/Lz+2*C223/Lr/Lz+... 2*C255/Lr/Lz)*Lr^2*Lz*(tend^4-(thickness)^4)+1/6*(C213/Lr/Lz+C223/Lr/Lz)*Lr^2*Lz*(tend^3-(thickness)^3)+... 1/6*(-2*C213/Lr/Lz-2*C223/Lr/Lz-2*C255/Lr/Lz)*Lr^2*Lz*(tend^2-(thickness)^2)+1/2*(-C213/Lr/Lz-... C223/Lr/Lz)*Lr^2*Lz*metthick);k(2,8)=2*pi*(1/8*(2*C113/Lr/Lz+2*C123/Lr/Lz)*Lr^2*Lz*((thickness)^4-tbegin^4)+1/6*(C113/Lr/Lz+C123/Lr/Lz)*... Lr^2*Lz*((thickness)^3-tbegin^3)+1/4*(-2*C113/Lr/Lz-2*C123/Lr/Lz)*Lr^2*Lz*((thickness)^2-tbegin^2)+1/2*(-C113/Lr/... Lz-C123/Lr/Lz)*Lr^2*Lz*(tend-metthick-tbegin))+2*pi*(1/8*(2*C213/Lr/Lz+2*C223/Lr/Lz)*Lr^2*Lz*(tend^4-(thickness)^4)+... 1/6*(C213/Lr/Lz+C223/Lr/Lz)*Lr^2*Lz*(tend^3-(thickness)^3)+1/4*(-2*C213/Lr/Lz-2*C223/Lr/Lz)*Lr^2*Lz*(tend^2-... (thickness)^2)+1/2*(-C213/Lr/Lz-C223/Lr/Lz)*Lr^2*Lz*metthick);k(2,9)=2*pi*(1/15*(3*C123/Lr/Lz+3*C113/Lr/Lz+2*C155/Lr/Lz)*Lr^2*Lz*((thickness)^5-tbegin^5)+1/3*(1/3*(-4*C113/Lr/Lz-2*... C155/Lr/Lz-4*C123/Lr/Lz)*Lr^2*Lz+1/2*(C113/Lr/Lz+C123/Lr/Lz)*Lr^2*Lz)*((thickness)^3-tbegin^3)+1/3*(C113/Lr/Lz+C123/Lr/Lz)... *Lr^2*Lz*(tend-metthick-tbegin)+1/2*(-C113/Lr/Lz-C123/Lr/Lz)*Lr^2*Lz*(tend-metthick-tbegin))+2*pi*(1/15*(3*... C223/Lr/Lz+3*C213/Lr/Lz+2*C255/Lr/Lz)*Lr^2*Lz*(tend^5-(thickness)^5)+1/3*(1/3*(-4*C213/Lr/Lz-2*C255/Lr/Lz-... 4*C223/Lr/Lz)*Lr^2*Lz+1/2*(C213/Lr/Lz+C223/Lr/Lz)*Lr^2*Lz)*(tend^3-(thickness)^3)+1/3*(C213/Lr/Lz+C223/Lr/Lz)*... Lr^2*Lz*metthick+1/2*(-C213/Lr/Lz-C223/Lr/Lz)*Lr^2*Lz*metthick);k(2,10)=2*pi*(1/10*(3*C113/Lr/Lz+3*C123/Lr/Lz)*Lr^2*Lz*((thickness)^5-tbegin^5)+1/6*(-3*C113/Lr/Lz-3*C123/Lr/Lz)... *Lr^2*Lz*((thickness)^3-tbegin^3))+2*pi*(1/10*(3*C213/Lr/Lz+3*C223/Lr/Lz)*Lr^2*Lz*(tend^5-(thickness)^5)+1/6*... (-3*C213/Lr/Lz-3*C223/Lr/Lz)*Lr^2*Lz*(tend^3-(thickness)^3));k(2,11)=2*pi*(1/18*(4*C123/Lr/Lz+4*C113/Lr/Lz+2*C155/Lr/Lz)*Lr^2*Lz*((thickness)^6-tbegin^6)+1/12*(-6*C113/Lr/Lz-... 2*C155/Lr/Lz-6*C123/Lr/Lz)*Lr^2*Lz*((thickness)^4-tbegin^4)+1/6*(C113/Lr/Lz+C123/Lr/Lz)*Lr^2*Lz*((thickness)^3-... tbegin^3)+1/6*(2*C113/Lr/Lz+2*C123/Lr/Lz)*Lr^2*Lz*((thickness)^2-tbegin^2)+1/2*(-C113/Lr/Lz-C123/Lr/Lz)*Lr^2*Lz*... (tend-metthick-tbegin))+2*pi*(1/18*(4*C223/Lr/Lz+4*C213/Lr/Lz+2*C255/Lr/Lz)*Lr^2*Lz*(tend^6-(thickness)^6)+1/12*... (-6*C213/Lr/Lz-2*C255/Lr/Lz-6*C223/Lr/Lz)*Lr^2*Lz*(tend^4-(thickness)^4)+1/6*(C213/Lr/Lz+C223/Lr/Lz)*Lr^2*Lz*... (tend^3-(thickness)^3)+1/6*(2*C213/Lr/Lz+2*C223/Lr/Lz)*Lr^2*Lz*(tend^2-(thickness)^2)+1/2*(-C213/Lr/Lz-C223/Lr/Lz)... *Lr^2*Lz*metthick);k(2,12)=2*pi*(1/12*(4*C113/Lr/Lz+4*C123/Lr/Lz)*Lr^2*Lz*((thickness)^6-tbegin^6)+1/8*(-6*C123/Lr/Lz-6*C113/Lr/Lz)*... Lr^2*Lz*((thickness)^4-tbegin^4)+1/6*(C113/Lr/Lz+C123/Lr/Lz)*Lr^2*Lz*((thickness)^3-tbegin^3)+1/4*(2*C113/Lr/Lz... +2*C123/Lr/Lz)*Lr^2*Lz*((thickness)^2-tbegin^2)+1/2*(-C113/Lr/Lz-C123/Lr/Lz)*Lr^2*Lz*(tend-metthick-tbegin))+... 2*pi*(1/12*(4*C213/Lr/Lz+4*C223/Lr/Lz)*Lr^2*Lz*(tend^6-(thickness)^6)+1/8*(-6*C223/Lr/Lz-6*C213/Lr/Lz)*Lr^2*Lz*... (tend^4-(thickness)^4)+1/6*(C213/Lr/Lz+C223/Lr/Lz)*Lr^2*Lz*(tend^3-(thickness)^3)+1/4*(2*C213/Lr/Lz+2*C223/Lr/Lz)*... Lr^2*Lz*(tend^2-(thickness)^2)+1/2*(-C213/Lr/Lz-C223/Lr/Lz)*Lr^2*Lz*metthick);k(3,5)=0;k(3,6)=2*pi*(1/12*(2*C113/Lr/Lz+C123/Lr/Lz)*Lr^2*Lz*((thickness)^4-tbegin^4)+1/6*(-2*C113/Lr/Lz-C123/Lr/Lz)*Lr^2*Lz*... ((thickness)^2-tbegin^2))+2*pi*(1/12*(2*C213/Lr/Lz+C223/Lr/Lz)*Lr^2*Lz*(tend^4-(thickness)^4)+1/6*(-2*C213/Lr/... Lz-C223/Lr/Lz)*Lr^2*Lz*(tend^2-(thickness)^2));k(3,7)=2*pi*(1/20*(2*C123/Lr/Lz+4*C113/Lr/Lz+3*C155/Lr/Lz)*Lr^2*Lz*((thickness)^5-tbegin^5)+1/12*(2*C113/Lr/Lz+... C123/Lr/Lz)*Lr^2*Lz*((thickness)^4-tbegin^4)+1/12*(-4*C155/Lr/Lz-2*C123/Lr/Lz-4*C113/Lr/Lz)*Lr^2*Lz*... ((thickness)^3-tbegin^3)+1/6*(-2*C113/Lr/Lz-C123/Lr/Lz)*Lr^2*Lz*((thickness)^2-tbegin^2)+1/4*C155*Lr*... (tend-metthick-tbegin))+2*pi*(1/20*(2*C223/Lr/Lz+4*C213/Lr/Lz+3*C255/Lr/Lz)*Lr^2*Lz*(tend^5-(thickness)^5)... +1/12*(2*C213/Lr/Lz+C223/Lr/Lz)*Lr^2*Lz*(tend^4-(thickness)^4)+1/12*(-4*C255/Lr/Lz-2*C223/Lr/Lz-4*C213/Lr/Lz)*... Lr^2*Lz*(tend^3-(thickness)^3)+1/6*(-2*C213/Lr/Lz-C223/Lr/Lz)*Lr^2*Lz*(tend^2-(thickness)^2)+1/4*C255*Lr*metthick);k(3,8)=2*pi*(1/15*(4*C113/Lr/Lz+2*C123/Lr/Lz)*Lr^2*Lz*((thickness)^5-tbegin^5)+1/12*(2*C113/Lr/Lz+C123/Lr/Lz)*... Lr^2*Lz*((thickness)^4-tbegin^4)+1/9*(-4*C113/Lr/Lz-2*C123/Lr/Lz)*Lr^2*Lz*((thickness)^3-tbegin^3)+1/6*... (-2*C113/Lr/Lz-C123/Lr/Lz)*Lr^2*Lz*((thickness)^2-tbegin^2))+2*pi*(1/15*(4*C213/Lr/Lz+2*C223/Lr/Lz)*Lr^2*Lz*... (tend^5-(thickness)^5)+1/12*(2*C213/Lr/Lz+C223/Lr/Lz)*Lr^2*Lz*(tend^4-(thickness)^4)+1/9*(-4*C213/Lr/Lz-2*C223/Lr/Lz)... *Lr^2*Lz*(tend^3-(thickness)^3)+1/6*(-2*C213/Lr/Lz-C223/Lr/Lz)*Lr^2*Lz*(tend^2-(thickness)^2));k(3,9)=2*pi*(1/24*(3*C123/Lr/Lz+6*C113/Lr/Lz+3*C155/Lr/Lz)*Lr^2*Lz*((thickness)^6-tbegin^6)+1/4*... (1/4*(-8*C113/Lr/Lz-4*C123/Lr/Lz-4*C155/Lr/Lz)*Lr^2*Lz+1/3*(2*C113/Lr/Lz+C123/Lr/Lz)*Lr^2*Lz)*((thickness)^4-tbegin^4)... +1/2*(1/4*(2*C113/Lr/Lz+C155/Lr/Lz+C123/Lr/Lz)*Lr^2*Lz+1/3*(-2*C113/Lr/Lz-C123/Lr/Lz)*Lr^2*Lz)*... ((thickness)^2-tbegin^2))+2*pi*(1/24*(3*C223/Lr/Lz+6*C213/Lr/Lz+3*C255/Lr/Lz)*Lr^2*Lz*(tend^6-(thickness)^6)... +1/4*(1/4*(-8*C213/Lr/Lz-4*C223/Lr/Lz-4*C255/Lr/Lz)*Lr^2*Lz+1/3*(2*C213/Lr/Lz+C223/Lr/Lz)*Lr^2*Lz)... *(tend^4-(thickness)^4)+1/2*(1/4*(2*C213/Lr/Lz+C255/Lr/Lz+C223/Lr/Lz)*Lr^2*Lz+1/3*(-2*C213/Lr/Lz-C223/Lr/Lz)... *Lr^2*Lz)*(tend^2-(thickness)^2));k(3,10)=2*pi*(1/18*(6*C113/Lr/Lz+3*C123/Lr/Lz)*Lr^2*Lz*((thickness)^6-tbegin^6)+1/12*(-6*C113/Lr/Lz-3*C123/Lr/Lz)... *Lr^2*Lz*((thickness)^4-tbegin^4))+2*pi*(1/18*(6*C213/Lr/Lz+3*C223/Lr/Lz)*Lr^2*Lz*(tend^6-(thickness)^6)+1/12*...

187

(-6*C213/Lr/Lz-3*C223/Lr/Lz)*Lr^2*Lz*(tend^4-(thickness)^4));k(3,11)=2*pi*(1/28*(4*C123/Lr/Lz+8*C113/Lr/Lz+3*C155/Lr/Lz)*Lr^2*Lz*((thickness)^7-tbegin^7)+1/20*(-12*C113/Lr/Lz-6*... C123/Lr/Lz-4*C155/Lr/Lz)*Lr^2*Lz*((thickness)^5-tbegin^5)+1/12*(2*C113/Lr/Lz+C123/Lr/Lz)*Lr^2*Lz*... ((thickness)^4-tbegin^4)+1/12*(4*C113/Lr/Lz+C155/Lr/Lz+2*C123/Lr/Lz)*Lr^2*Lz*((thickness)^3-tbegin^3)+1/6*... (-2*C113/Lr/Lz-C123/Lr/Lz)*Lr^2*Lz*((thickness)^2-tbegin^2))+2*pi*(1/28*(4*C223/Lr/Lz+8*C213/Lr/Lz+3*C255/Lr/Lz)*... Lr^2*Lz*(tend^7-(thickness)^7)+1/20*(-12*C213/Lr/Lz-6*C223/Lr/Lz-4*C255/Lr/Lz)*Lr^2*Lz*(tend^5-... (thickness)^5)+1/12*(2*C213/Lr/Lz+C223/Lr/Lz)*Lr^2*Lz*(tend^4-(thickness)^4)+1/12*(4*C213/Lr/Lz+... C255/Lr/Lz+2*C223/Lr/Lz)*Lr^2*Lz*(tend^3-(thickness)^3)+1/6*(-2*C213/Lr/Lz-C223/Lr/Lz)*Lr^2*Lz*(tend^2-(thickness)^2));k(3,12)=2*pi*(1/21*(8*C113/Lr/Lz+4*C123/Lr/Lz)*Lr^2*Lz*((thickness)^7-tbegin^7)+1/15*(-6*C123/Lr/Lz-12*C113/Lr/Lz)*... Lr^2*Lz*((thickness)^5-tbegin^5)+1/12*(2*C113/Lr/Lz+C123/Lr/Lz)*Lr^2*Lz*((thickness)^4-tbegin^4)+1/9*(4*C113/Lr/Lz... +2*C123/Lr/Lz)*Lr^2*Lz*((thickness)^3-tbegin^3)+1/6*(-2*C113/Lr/Lz-C123/Lr/Lz)*Lr^2*Lz*((thickness)^2-tbegin^2))+... 2*pi*(1/21*(8*C213/Lr/Lz+4*C223/Lr/Lz)*Lr^2*Lz*(tend^7-(thickness)^7)+1/15*(-6*C223/Lr/Lz-12*C213/Lr/Lz)*Lr^2*Lz*... (tend^5-(thickness)^5)+1/12*(2*C213/Lr/Lz+C223/Lr/Lz)*Lr^2*Lz*(tend^4-(thickness)^4)+1/9*(4*C213/Lr/Lz+2*C223/Lr/Lz)... *Lr^2*Lz*(tend^3-(thickness)^3)+1/6*(-2*C213/Lr/Lz-C223/Lr/Lz)*Lr^2*Lz*(tend^2-(thickness)^2));k(4,5)=0;k(4,6)=2*pi*(1/8*(C113/Lr/Lz+C123/Lr/Lz)*Lr^2*Lz*((thickness)^4-tbegin^4)+1/4*(-C113/Lr/Lz-C123/Lr/Lz)*Lr^2*Lz... *((thickness)^2-tbegin^2))+2*pi*(1/8*(C213/Lr/Lz+C223/Lr/Lz)*Lr^2*Lz*(tend^4-(thickness)^4)+1/4*... (-C213/Lr/Lz-C223/Lr/Lz)*Lr^2*Lz*(tend^2-(thickness)^2));k(4,7)=2*pi*(1/15*(2*C123/Lr/Lz+2*C113/Lr/Lz+3*C155/Lr/Lz)*Lr^2*Lz*((thickness)^5-tbegin^5)+1/8*(C113/Lr/Lz+C123/Lr/Lz)... *Lr^2*Lz*((thickness)^4-tbegin^4)+1/9*(-4*C155/Lr/Lz-2*C123/Lr/Lz-2*C113/Lr/Lz)*Lr^2*Lz*((thickness)^3-tbegin^3)+... 1/4*(-C113/Lr/Lz-C123/Lr/Lz)*Lr^2*Lz*((thickness)^2-tbegin^2)+1/3*C155*Lr*(tend-metthick-tbegin))+... 2*pi*(1/15*(2*C223/Lr/Lz+2*C213/Lr/Lz+3*C255/Lr/Lz)*Lr^2*Lz*(tend^5-(thickness)^5)+1/8*(C213/Lr/Lz+C223/Lr/Lz)... *Lr^2*Lz*(tend^4-(thickness)^4)+1/9*(-4*C255/Lr/Lz-2*C223/Lr/Lz-2*C213/Lr/Lz)*Lr^2*Lz*(tend^3-(thickness)^3)+... 1/4*(-C213/Lr/Lz-C223/Lr/Lz)*Lr^2*Lz*(tend^2-(thickness)^2)+1/3*C255*Lr*metthick);k(4,8)=2*pi*(1/10*(2*C113/Lr/Lz+2*C123/Lr/Lz)*Lr^2*Lz*((thickness)^5-tbegin^5)+1/8*(C113/Lr/Lz+C123/Lr/Lz)*Lr^2*Lz*... ((thickness)^4-tbegin^4)+1/6*(-2*C113/Lr/Lz-2*C123/Lr/Lz)*Lr^2*Lz*((thickness)^3-tbegin^3)+1/4*... (-C113/Lr/Lz-C123/Lr/Lz)*Lr^2*Lz*((thickness)^2-tbegin^2))+2*pi*(1/10*(2*C213/Lr/Lz+2*C223/Lr/Lz)*Lr^2*Lz*... (tend^5-(thickness)^5)+1/8*(C213/Lr/Lz+C223/Lr/Lz)*Lr^2*Lz*(tend^4-(thickness)^4)+1/6*(-2*C213/Lr/Lz-2*C223/Lr/Lz)... *Lr^2*Lz*(tend^3-(thickness)^3)+1/4*(-C213/Lr/Lz-C223/Lr/Lz)*Lr^2*Lz*(tend^2-(thickness)^2));k(4,9)=2*pi*(1/18*(3*C123/Lr/Lz+3*C113/Lr/Lz+3*C155/Lr/Lz)*Lr^2*Lz*((thickness)^6-tbegin^6)+1/4*(1/3*... (-4*C113/Lr/Lz-4*C123/Lr/Lz-4*C155/Lr/Lz)*Lr^2*Lz+1/2*(C113/Lr/Lz+C123/Lr/Lz)*Lr^2*Lz)*((thickness)^4-tbegin^4)... +1/2*(1/3*(C113/Lr/Lz+C155/Lr/Lz+C123/Lr/Lz)*Lr^2*Lz+1/2*(-C113/Lr/Lz-C123/Lr/Lz)*Lr^2*Lz)*((thickness)^2-tbegin^2))... +2*pi*(1/18*(3*C223/Lr/Lz+3*C213/Lr/Lz+3*C255/Lr/Lz)*Lr^2*Lz*(tend^6-(thickness)^6)+1/4*(1/3*(-4*C213/Lr/Lz-4*C223... /Lr/Lz-4*C255/Lr/Lz)*Lr^2*Lz+1/2*(C213/Lr/Lz+C223/Lr/Lz)*Lr^2*Lz)*(tend^4-(thickness)^4)+1/2*(1/3*(C213/Lr/Lz+C255... /Lr/Lz+C223/Lr/Lz)*Lr^2*Lz+1/2*(-C213/Lr/Lz-C223/Lr/Lz)*Lr^2*Lz)*(tend^2-(thickness)^2));k(4,10)=2*pi*(1/12*(3*C113/Lr/Lz+3*C123/Lr/Lz)*Lr^2*Lz*((thickness)^6-tbegin^6)+1/8*(-3*C113/Lr/Lz-3*C123/Lr/Lz)*... Lr^2*Lz*((thickness)^4-tbegin^4))+2*pi*(1/12*(3*C213/Lr/Lz+3*C223/Lr/Lz)*Lr^2*Lz*(tend^6-(thickness)^6)+1/8*(-3*... C213/Lr/Lz-3*C223/Lr/Lz)*Lr^2*Lz*(tend^4-(thickness)^4));k(4,11)=2*pi*(1/21*(4*C123/Lr/Lz+4*C113/Lr/Lz+3*C155/Lr/Lz)*Lr^2*Lz*((thickness)^7-tbegin^7)+1/15*(-6*C113/Lr/Lz-6*... C123/Lr/Lz-4*C155/Lr/Lz)*Lr^2*Lz*((thickness)^5-tbegin^5)+1/8*(C113/Lr/Lz+C123/Lr/Lz)*Lr^2*Lz*((thickness)^4-... tbegin^4)+1/9*(2*C113/Lr/Lz+C155/Lr/Lz+2*C123/Lr/Lz)*Lr^2*Lz*((thickness)^3-tbegin^3)+1/4*(-C113/Lr/Lz-C123/Lr/Lz)*... Lr^2*Lz*((thickness)^2-tbegin^2))+2*pi*(1/21*(4*C223/Lr/Lz+4*C213/Lr/Lz+3*C255/Lr/Lz)*Lr^2*Lz*(tend^7-(thickness)^7)... +1/15*(-6*C213/Lr/Lz-6*C223/Lr/Lz-4*C255/Lr/Lz)*Lr^2*Lz*(tend^5-(thickness)^5)+1/8*(C213/Lr/Lz+C223/Lr/Lz)*Lr^2*Lz*... (tend^4-(thickness)^4)+1/9*(2*C213/Lr/Lz+C255/Lr/Lz+2*C223/Lr/Lz)*Lr^2*Lz*(tend^3-(thickness)^3)+1/4*(-C213/Lr/Lz-... C223/Lr/Lz)*Lr^2*Lz*(tend^2-(thickness)^2));k(4,12)=2*pi*(1/14*(4*C113/Lr/Lz+4*C123/Lr/Lz)*Lr^2*Lz*((thickness)^7-tbegin^7)+1/10*(-6*C123/Lr/Lz-6*C113/Lr/Lz)*... Lr^2*Lz*((thickness)^5-tbegin^5)+1/8*(C113/Lr/Lz+C123/Lr/Lz)*Lr^2*Lz*((thickness)^4-tbegin^4)+1/6*(2*C113/Lr/Lz+... 2*C123/Lr/Lz)*Lr^2*Lz*((thickness)^3-tbegin^3)+1/4*(-C113/Lr/Lz-C123/Lr/Lz)*Lr^2*Lz*((thickness)^2-tbegin^2))+... 2*pi*(1/14*(4*C213/Lr/Lz+4*C223/Lr/Lz)*Lr^2*Lz*(tend^7-(thickness)^7)+1/10*(-6*C223/Lr/Lz-6*C213/Lr/Lz)*Lr^2*Lz*... (tend^5-(thickness)^5)+1/8*(C213/Lr/Lz+C223/Lr/Lz)*Lr^2*Lz*(tend^4-(thickness)^4)+1/6*(2*C213/Lr/Lz+2*C223/Lr/Lz)*... Lr^2*Lz*(tend^3-(thickness)^3)+1/4*(-C213/Lr/Lz-C223/Lr/Lz)*Lr^2*Lz*(tend^2-(thickness)^2));

%Calculation of K22 for elastomer and metalk(5,5)=0;k(5,6)=0;k(5,7)=0;k(5,8)=0;k(5,9)=0;k(5,10)=0;k(5,11)=0;k(5,12)=0;k(6,6)=pi*C133/Lz*Lr^2*(tend-metthick-tbegin)+pi*C233/Lz*Lr^2*metthick;k(6,7)=2*pi*(1/3*C133/Lz*Lr^2*((thickness)^2-tbegin^2)+1/2*C133/Lz*Lr^2*(tend-metthick-tbegin))+2*pi*(1/3*C233/Lz*... Lr^2*(tend^2-(thickness)^2)+1/2*C233/Lz*Lr^2*metthick);k(6,8)=2*pi*(1/2*C133/Lz*Lr^2*((thickness)^2-tbegin^2)+1/2*C133/Lz*Lr^2*(tend-metthick-tbegin))+2*pi*(1/2*C233/Lz*... Lr^2*(tend^2-(thickness)^2)+1/2*C233/Lz*Lr^2*metthick);k(6,9)=2*pi*(1/3*C133/Lz*Lr^2*((thickness)^3-tbegin^3)+1/6*C133/Lz*Lr^2*(tend-metthick-tbegin))+2*pi*(1/3*C233/Lz*Lr^2*... (tend^3-(thickness)^3)+1/6*C233/Lz*Lr^2*metthick);k(6,10)=pi*C133/Lz*Lr^2*((thickness)^3-tbegin^3)+pi*C233/Lz*Lr^2*(tend^3-(thickness)^3);k(6,11)=2*pi*(1/3*C133/Lz*Lr^2*((thickness)^4-tbegin^4)-1/3*C133/Lz*Lr^2*((thickness)^2-tbegin^2)+1/2*C133/Lz*Lr^2*... (tend-metthick-tbegin))+2*pi*(1/3*C233/Lz*Lr^2*(tend^4-(thickness)^4)-1/3*C233/Lz*Lr^2*(tend^2-(thickness)^2)+... 1/2*C233/Lz*Lr^2*metthick);k(6,12)=2*pi*(1/2*C133/Lz*Lr^2*((thickness)^4-tbegin^4)-1/2*C133/Lz*Lr^2*((thickness)^2-tbegin^2)+1/2*C133/Lz*Lr^2*... (tend-metthick-tbegin))+2*pi*(1/2*C233/Lz*Lr^2*(tend^4-(thickness)^4)-1/2*C233/Lz*Lr^2*(tend^2-(thickness)^2)+1/2*... C233/Lz*Lr^2*metthick);k(7,7)=2*pi*(1/10*C155*Lz*((thickness)^5-tbegin^5)+1/3*(C133/Lz*Lr^2-C155*Lz)*((thickness)^3-tbegin^3)+2/3*C133/Lz*Lr^2*... ((thickness)^2-tbegin^2)+1/2*(C133/Lz^2+C155/Lr^2)*Lr^2*Lz*(tend-metthick-tbegin))+2*pi*(1/10*C255*Lz*(tend^5-... (thickness)^5)+1/3*(C233/Lz*Lr^2-C255*Lz)*(tend^3-(thickness)^3)+2/3*C233/Lz*Lr^2*(tend^2-(thickness)^2)+1/2*... (C233/Lz^2+C255/Lr^2)*Lr^2*Lz*metthick);k(7,8)=2*pi*(4/9*C133/Lz*Lr^2*((thickness)^3-tbegin^3)+5/6*C133/Lz*Lr^2*((thickness)^2-tbegin^2)+1/2*C133/Lz*Lr^2*... (tend-metthick-tbegin))+2*pi*(4/9*C233/Lz*Lr^2*(tend^3-(thickness)^3)+5/6*C233/Lz*Lr^2*(tend^2-(thickness)^2)+... 1/2*C233/Lz*Lr^2*metthick);k(7,9)=2*pi*(1/12*C155*Lz*((thickness)^6-tbegin^6)+1/4*(3/2*C133/Lz*Lr^2-C155*Lz)*((thickness)^4-tbegin^4)+1/3*C133/... Lz*Lr^2*((thickness)^3-tbegin^3)+1/2*(1/6*C133/Lz*Lr^2+1/2*C155*Lz)*((thickness)^2-tbegin^2)+1/6*C133/Lz*Lr^2*... (tend-metthick-tbegin))+2*pi*(1/12*C255*Lz*(tend^6-(thickness)^6)+1/4*(3/2*C233/Lz*Lr^2-C255*Lz)*(tend^4-(thickness)^4)... +1/3*C233/Lz*Lr^2*(tend^3-(thickness)^3)+1/2*(1/6*C233/Lz*Lr^2+1/2*C255*Lz)*(tend^2-(thickness)^2)+1/6*C233/Lz*... Lr^2*metthick);k(7,10)=2*pi*(1/2*C133/Lz*Lr^2*((thickness)^4-tbegin^4)+1/2*C133/Lz*Lr^2*((thickness)^3-tbegin^3))+2*pi*(1/2*C233/Lz... *Lr^2*(tend^4-(thickness)^4)+1/2*C233/Lz*Lr^2*(tend^3-(thickness)^3));k(7,11)=2*pi*(1/14*C155*Lz*((thickness)^7-tbegin^7)+1/5*(2*C133/Lz*Lr^2-C155*Lz)*((thickness)^5-tbegin^5)+1/3*C133/Lz... *Lr^2*((thickness)^4-tbegin^4)+1/3*(-C133/Lz*Lr^2+1/2*C155*Lz)*((thickness)^3-tbegin^3)+1/2*C133/Lz*Lr^2*... (tend-metthick-tbegin))+2*pi*(1/14*C255*Lz*(tend^7-(thickness)^7)+1/5*(2*C233/Lz*Lr^2-C255*Lz)*(tend^5-(thickness)^5)... +1/3*C233/Lz*Lr^2*(tend^4-(thickness)^4)+1/3*(-C233/Lz*Lr^2+1/2*C255*Lz)*(tend^3-(thickness)^3)+1/2*C233/Lz*Lr^2*metthick);k(7,12)=2*pi*(8/15*C133/Lz*Lr^2*((thickness)^5-tbegin^5)+1/2*C133/Lz*Lr^2*((thickness)^4-tbegin^4)-4/9*C133/Lz*Lr^2*... ((thickness)^3-tbegin^3)-1/6*C133/Lz*Lr^2*((thickness)^2-tbegin^2)+1/2*C133/Lz*Lr^2*(tend-metthick-tbegin))+2*pi*... (8/15*C233/Lz*Lr^2*(tend^5-(thickness)^5)+1/2*C233/Lz*Lr^2*(tend^4-(thickness)^4)-4/9*C233/Lz*Lr^2*(tend^3-... (thickness)^3)-1/6*C233/Lz*Lr^2*(tend^2-(thickness)^2)+1/2*C233/Lz*Lr^2*metthick);k(8,8)=2*pi*(2/3*C133/Lz*Lr^2*((thickness)^3-tbegin^3)+C133/Lz*Lr^2*((thickness)^2-tbegin^2)+1/2*C133/Lz*Lr^2*... (tend-metthick-tbegin))+2*pi*(2/3*C233/Lz*Lr^2*(tend^3-(thickness)^3)+C233/Lz*Lr^2*(tend^2-(thickness)^2)+... 1/2*C233/Lz*Lr^2*metthick);k(8,9)=2*pi*(1/2*C133/Lz*Lr^2*((thickness)^4-tbegin^4)+1/3*C133/Lz*Lr^2*((thickness)^3-tbegin^3)+1/6*C133/Lz*... Lr^2*((thickness)^2-tbegin^2)+1/6*C133/Lz*Lr^2*(tend-metthick-tbegin))+2*pi*(1/2*C233/Lz*Lr^2*(tend^4-... (thickness)^4)+1/3*C233/Lz*Lr^2*(tend^3-(thickness)^3)+1/6*C233/Lz*Lr^2*(tend^2-(thickness)^2)+1/6*C233/Lz*Lr^2*metthick);k(8,10)=2*pi*(3/4*C133/Lz*Lr^2*((thickness)^4-tbegin^4)+1/2*C133/Lz*Lr^2*((thickness)^3-tbegin^3))+2*pi*... (3/4*C233/Lz*Lr^2*(tend^4-(thickness)^4)+1/2*C233/Lz*Lr^2*(tend^3-(thickness)^3));k(8,11)=2*pi*(8/15*C133/Lz*Lr^2*((thickness)^5-tbegin^5)+1/3*C133/Lz*Lr^2*((thickness)^4-tbegin^4)-4/9*... C133/Lz*Lr^2*((thickness)^3-tbegin^3)+1/6*C133/Lz*Lr^2*((thickness)^2-tbegin^2)+1/2*C133/Lz*Lr^2*... (tend-metthick-tbegin))+2*pi*(8/15*C233/Lz*Lr^2*(tend^5-(thickness)^5)+1/3*C233/Lz*Lr^2*(tend^4-(thickness)^4)... -4/9*C233/Lz*Lr^2*(tend^3-(thickness)^3)+1/6*C233/Lz*Lr^2*(tend^2-(thickness)^2)+1/2*C233/Lz*Lr^2*metthick);k(8,12)=2*pi*(4/5*C133/Lz*Lr^2*((thickness)^5-tbegin^5)+1/2*C133/Lz*Lr^2*((thickness)^4-tbegin^4)-2/3*C133/Lz*Lr^2*... ((thickness)^3-tbegin^3)+1/2*C133/Lz*Lr^2*(tend-metthick-tbegin))+2*pi*(4/5*C233/Lz*Lr^2*(tend^5-(thickness)^5)... +1/2*C233/Lz*Lr^2*(tend^4-(thickness)^4)-2/3*C233/Lz*Lr^2*(tend^3-(thickness)^3)+1/2*C233/Lz*Lr^2*metthick);k(9,9)=2*pi*(1/14*C155*Lz*((thickness)^7-tbegin^7)+1/5*(9/4*C133/Lz*Lr^2-C155*Lz)*((thickness)^5-tbegin^5)+1/3*(1/2*... C155*Lz+1/2*C133/Lz*Lr^2)*((thickness)^3-tbegin^3)+1/12*C133/Lz*Lr^2*(tend-metthick-tbegin))+2*pi*(1/14*C255*Lz*... (tend^7-(thickness)^7)+1/5*(9/4*C233/Lz*Lr^2-C255*Lz)*(tend^5-(thickness)^5)+1/3*(1/2*C255*Lz+1/2*C233/Lz*Lr^2)*... (tend^3-(thickness)^3)+1/12*C233/Lz*Lr^2*metthick);k(9,10)=2*pi*(3/5*C133/Lz*Lr^2*((thickness)^5-tbegin^5)+1/6*C133/Lz*Lr^2*((thickness)^3-tbegin^3))+2*pi*... (3/5*C233/Lz*Lr^2*(tend^5-(thickness)^5)+1/6*C233/Lz*Lr^2*(tend^3-(thickness)^3));k(9,11)=2*pi*(1/16*C155*Lz*((thickness)^8-tbegin^8)+1/6*(3*C133/Lz*Lr^2-C155*Lz)*((thickness)^6-tbegin^6)+1/4*... (-7/6*C133/Lz*Lr^2+1/2*C155*Lz)*((thickness)^4-tbegin^4)+1/3*C133/Lz*Lr^2*((thickness)^3-tbegin^3)-1/12*C133... /Lz*Lr^2*((thickness)^2-tbegin^2)+1/6*C133/Lz*Lr^2*(tend-metthick-tbegin))+2*pi*(1/16*C255*Lz*(tend^8-... (thickness)^8)+1/6*(3*C233/Lz*Lr^2-C255*Lz)*(tend^6-(thickness)^6)+1/4*(-7/6*C233/Lz*Lr^2+1/2*C255*Lz)*... (tend^4-(thickness)^4)+1/3*C233/Lz*Lr^2*(tend^3-(thickness)^3)-1/12*C233/Lz*Lr^2*(tend^2-(thickness)^2)+... 1/6*C233/Lz*Lr^2*metthick);

188

k(9,12)=2*pi*(2/3*C133/Lz*Lr^2*((thickness)^6-tbegin^6)-1/3*C133/Lz*Lr^2*((thickness)^4-tbegin^4)+1/3*C133/Lz*... Lr^2*((thickness)^3-tbegin^3)-1/6*C133/Lz*Lr^2*((thickness)^2-tbegin^2)+1/6*C133/Lz*Lr^2*(tend-metthick-tbegin))... +2*pi*(2/3*C233/Lz*Lr^2*(tend^6-(thickness)^6)-1/3*C233/Lz*Lr^2*(tend^4-(thickness)^4)+1/3*C233/Lz*Lr^2*(tend^3-... (thickness)^3)-1/6*C233/Lz*Lr^2*(tend^2-(thickness)^2)+1/6*C233/Lz*Lr^2*metthick);k(10,10)=9/5*pi*C133/Lz*Lr^2*((thickness)^5-tbegin^5)+9/5*pi*C233/Lz*Lr^2*(tend^5-(thickness)^5);k(10,11)=2*pi*(2/3*C133/Lz*Lr^2*((thickness)^6-tbegin^6)-1/2*C133/Lz*Lr^2*((thickness)^4-tbegin^4)+1/2*C133/Lz*Lr^2*... ((thickness)^3-tbegin^3))+2*pi*(2/3*C233/Lz*Lr^2*(tend^6-(thickness)^6)-1/2*C233/Lz*Lr^2*(tend^4-(thickness)^4)+... 1/2*C233/Lz*Lr^2*(tend^3-(thickness)^3));k(10,12)=2*pi*(C133/Lz*Lr^2*((thickness)^6-tbegin^6)-3/4*C133/Lz*Lr^2*((thickness)^4-tbegin^4)+1/2*C133/Lz*Lr^2*... ((thickness)^3-tbegin^3))+2*pi*(C233/Lz*Lr^2*(tend^6-(thickness)^6)-3/4*C233/Lz*Lr^2*(tend^4-(thickness)^4)+... 1/2*C233/Lz*Lr^2*(tend^3-(thickness)^3));k(11,11)=2*pi*(1/18*C155*Lz*((thickness)^9-tbegin^9)+1/7*(4*C133/Lz*Lr^2-C155*Lz)*((thickness)^7-tbegin^7)+... 1/5*(1/2*C155*Lz-4*C133/Lz*Lr^2)*((thickness)^5-tbegin^5)+2/3*C133/Lz*Lr^2*((thickness)^4-tbegin^4)+1/3*... C133/Lz*Lr^2*((thickness)^3-tbegin^3)-2/3*C133/Lz*Lr^2*((thickness)^2-tbegin^2)+1/2*C133/Lz*Lr^2*(tend-... metthick-tbegin))+2*pi*(1/18*C255*Lz*(tend^9-(thickness)^9)+1/7*(4*C233/Lz*Lr^2-C255*Lz)*(tend^7-... (thickness)^7)+1/5*(1/2*C255*Lz-4*C233/Lz*Lr^2)*(tend^5-(thickness)^5)+2/3*C233/Lz*Lr^2*(tend^4-... (thickness)^4)+1/3*C233/Lz*Lr^2*(tend^3-(thickness)^3)-2/3*C233/Lz*Lr^2*(tend^2-(thickness)^2)+1/2*C233/Lz*Lr^2*metthick);k(11,12)=2*pi*(16/21*C133/Lz*Lr^2*((thickness)^7-tbegin^7)-16/15*C133/Lz*Lr^2*((thickness)^5-tbegin^5)+5/6*C133/Lz... *Lr^2*((thickness)^4-tbegin^4)+4/9*C133/Lz*Lr^2*((thickness)^3-tbegin^3)-5/6*C133/Lz*Lr^2*((thickness)^2-... tbegin^2)+1/2*C133/Lz*Lr^2*(tend-metthick-tbegin))+2*pi*(16/21*C233/Lz*Lr^2*(tend^7-(thickness)^7)-16/15*C233/Lz*... Lr^2*(tend^5-(thickness)^5)+5/6*C233/Lz*Lr^2*(tend^4-(thickness)^4)+4/9*C233/Lz*Lr^2*(tend^3-(thickness)^3)-... 5/6*C233/Lz*Lr^2*(tend^2-(thickness)^2)+1/2*C233/Lz*Lr^2*metthick);k(12,12)=2*pi*(8/7*C133/Lz*Lr^2*((thickness)^7-tbegin^7)-8/5*C133/Lz*Lr^2*((thickness)^5-tbegin^5)+C133/Lz*Lr^2*... ((thickness)^4-tbegin^4)+2/3*C133/Lz*Lr^2*((thickness)^3-tbegin^3)-C133/Lz*Lr^2*((thickness)^2-tbegin^2)+1/2*... C133/Lz*Lr^2*(tend-metthick-tbegin))+2*pi*(8/7*C233/Lz*Lr^2*(tend^7-(thickness)^7)-8/5*C233/Lz*Lr^2*... (tend^5-(thickness)^5)+C233/Lz*Lr^2*(tend^4-(thickness)^4)+2/3*C233/Lz*Lr^2*(tend^3-(thickness)^3)-... C233/Lz*Lr^2*(tend^2-(thickness)^2)+1/2*C233/Lz*Lr^2*metthick);

temp = triu(k((umax+1):(umax+wmax),(umax+1):(umax+wmax)),1);temp = k((umax+1):(umax+wmax),(umax+1):(umax+wmax)) + temp';k((umax+1):(umax+wmax),(umax+1):(umax+wmax)) = temp;

tbegin = tend;tend = tend + clen;

%Calculation of K21k12 = k(1:umax,(umax+1):(umax+wmax));k21 = k12';k((umax+1):(umax+wmax),1:umax)=k21;

if mm == 1 kiso1 = k; miso1 = m; diso1 = k*betas(1)*1i;elseif mm==2 kiso2 = k; miso2 = m; diso2 = k*betas(2)*1i;elseif mm==3 kiso3 = k; miso3 = m; diso3 = k*betas(3)*1i;elseif mm==4 kiso4 = k; miso4 = m; diso4 = k*betas(4)*1i;elseif mm==5 kiso5 = k; miso5 = m; diso5 = k*betas(5)*1i;elseif mm==6 kiso6 = k; miso6 = m; diso6 = k*betas(6)*1i;elseif mm==7 kiso7 = k; miso7 = m; diso7 = k*betas(7)*1i;end

end %loop on matrices

%Determine the individual mass and stiffness properties of each fluid element

for ii=1:6 mabs(ii)= rho(ii)*portlength(ii)*pi*innerdiam(ii)^2/4;end

kr = 1; % Stiffness within fluid element.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%Begin system eigenvalue solutionr=0;n = N;sizeof = umax+wmax;

beg = 1;ending = sizeof;for i = 1:n if i==1 M(beg:ending,beg:ending) = miso1; K(beg:ending,beg:ending) = kiso1 + diso1; elseif i==2 M(beg:ending,beg:ending) = miso2; K(beg:ending,beg:ending) = kiso2 + diso2; elseif i==3 M(beg:ending,beg:ending) = miso3; K(beg:ending,beg:ending) = kiso3 + diso3; elseif i==4 M(beg:ending,beg:ending) = miso4; K(beg:ending,beg:ending) = kiso4 + diso4; elseif i==5 M(beg:ending,beg:ending) = miso5; K(beg:ending,beg:ending) = kiso5 + diso5; elseif i==6 M(beg:ending,beg:ending) = miso6; K(beg:ending,beg:ending) = kiso6 + diso6; elseif i==7 M(beg:ending,beg:ending) = miso7; K(beg:ending,beg:ending) = kiso7 + diso7; end beg = beg + sizeof; ending = ending + sizeof;end

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%Add mass, stiffness, and damping matrices of fluid elements to global matrices.

189

beg = beg-sizeof;ending = ending - sizeof;

for ii = 1:(n-1)

%Fluid element mass matrix M(((ending+(2*ii-1)):(ending+(2*ii))),((ending+(2*ii-1)):(ending+(2*ii)))) = mfact(ii+n)*mabs(ii)*[(R(ii)-1)^2 -R(ii)*(R(ii)-1); -R(ii)*(R(ii)-1) R(ii)^2 ]; %Fluid element stiffness matrix (ideally zero) K(((ending+(2*ii-1)):(ending+(2*ii))),((ending+(2*ii-1)):(ending+(2*ii)))) = kfact(ii+n)*[kr -kr; -kr kr]; %Fluid viscous damping matrix C(((ending+(2*ii-1)):(ending+(2*ii))),((ending+(2*ii-1)):(ending+(2*ii)))) = betas(ii+n)*[visdamp -visdamp; -visdamp visdamp]; %Fluid element hysteretic damping matrix K(((ending+(2*ii-1)):(ending+(2*ii))),((ending+(2*ii-1)):(ending+(2*ii)))) = ... K(((ending+(2*ii-1)):(ending+(2*ii))),((ending+(2*ii-1)):(ending+(2*ii)))) + ... betas(ii+n)*1i*K(((ending+(2*ii-1)):(ending+(2*ii))),((ending+(2*ii-1)):(ending+(2*ii))));

end

oldM=M;%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% Begin constraining layers together

%value of phis when a z = -1phis = [0 0 0 0 1 0 0 0 0 0 0 0];phib=phis;

%Values of phis when z = 1phit=[0 0 0 0 1 2 2 2 2 2 2 2];

if n > 1 beg = 1;

ending = sizeof;for layer = 1:(n-1)

constraints((layer),beg:ending)= phib; constraints((layer),(beg+sizeof):(ending+sizeof)) = (phit.*(-1)); beg = beg + sizeof; ending = ending + sizeof; end %end layer loopend

oldcon = constraints;

%Constrain top and bottom of fluid elements appropriately within isolator

beg = 1;ybeg1= 1;xbeg1 = 2;ybeg2 = 2;

ending = sizeof;for count = 1:(n-1)

beg = beg + sizeof; ending = ending + sizeof;

if count < (n-1)

constraints(2*count-1+n,(beg+sizeof):(ending+sizeof))=phit; constraints(2*count-1+n,(xbeg1+n*sizeof))=-1;

constraints(2*count-2+n,beg:ending) = phit; constraints(2*count-2+n,(ybeg1+n*sizeof))=-1;

elseif count == (n-1)

constraints(2*count-1+n,beg:ending)=phib; constraints(2*count-1+n,(xbeg1+n*sizeof))=-1;

constraints(2*count-2+n,beg:ending) = phit; constraints(2*count-2+n,(ybeg1+n*sizeof))=-1;

end

xbeg1 = xbeg1 + 2; ybeg1 = ybeg1 + 2;

end

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%Predetermined dependent variable locations (Arranged so that A1 is invertible)depend = [5 17 29 41 53 65 85 86 87 88 89 90 91 92 93 94 95 96];

sizedepend = length(depend);origsize = length(oldM(:,1)) - sizedepend;

%Calculate dependent variable location matrix

rowcount = 1;count = 1;for i = 1:origsize if count > sizedepend; count = sizedepend; end if rowcount~=depend(count) T(rowcount,i) = 1; rowcount = rowcount + 1; else T(rowcount+1,i)=1; rowcount = rowcount + 2; count = count + 1; end

end

T(5,79)=1;T(17,80)=1;T(29,81)=1;T(41,82)=1;T(53,83)=1;T(65,84)=1;T(85,85)=1;T(86,86)=1;T(87,87)=1;T(88,88)=1;T(89,89)=1;T(90,90)=1;T(91,91)=1;

190

T(92,92)=1;T(93,93)=1;T(94,94)=1;T(95,95)=1;T(96,96)=1;

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%Rearrange constraint equations so dependent variables are in the beginning%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

numcon = length(constraints(:,1));

for i = 1:numcon place = depend(i); convec(:,i) = constraints(:,place);end

A1 = convec;

conlen = length(constraints(1,:));

concount = 1;a2count = 1;for i = 1:conlen if i~=depend(concount) A2(:,a2count)=constraints(:,i); a2count = a2count + 1; elseif concount < numcon concount = concount + 1; endend

G = -1.*inv(A1)*A2;

newsize = length(G);

beta(1:newsize,1:newsize) = eye(newsize);beta((newsize+1):(newsize+numcon),1:newsize)=G;

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Rearrange original M and K so that pf and pd are in right place%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

M = T'*M*T;K = T'*K*T;C = T'*C*T;

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Mp = beta.'*M*beta;Kp = beta.'*K*beta;Cp = beta.'*C*beta;

%% EIGENSOLUTION%

[FEvector, FEvalues] = eig(Kp,Mp);

%% SORT EIGENSOLUTION FROM LOW TO HIGH IN FREQUENCY%

[EIGval, EIGord] = sort(diag(FEvalues));FEvalues = diag(EIGval);

tempvec = zeros(max(size(EIGord)));for j = 1:max(size(EIGord)) tempvec(:,j) = FEvector(:,EIGord(j));end

%% MASS NORMALIZE MODE SHAPES%

FEvector = real(tempvec*sqrt(inv(tempvec'*Mp*tempvec)));a=FEvector;

freqs=diag((sqrt(FEvalues))/2/pi);

a = beta*a;

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Calculate Transmissibility%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%Calculate generalized forces (at top of mount)

%For 1 to umax (in radial dir.) generalized forces are zero

Force = 1;

for i = 1:(umax+wmax) f(i)=phib(i)*Force;end

F(((n-1)*(umax+wmax)+1):n*(umax+wmax))=f;F((n*(umax+wmax)+1):((umax+wmax)*n+2*numofabsorbers))=0;

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Move rows so dependent forces to end of vector%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

F=T'*F';

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Multiply F by beta to get generalized forces for independent coordinates

191

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Q = beta'*F;

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%Begin Frequency Loop%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

clear irowsize = size(oldM(:,1));

for tt = 1:counter

ww = w(tt);

s = ww*1i; b = Mp.*s^2 + Cp*s + Kp;

%Calculate all the independent variables at this point clear p p = (b\Q)'; oldp = p;

%Multiply by beta to get independent coordinates back p = beta*p';

%Move independent coordinates back to proper position p = T*p;

%Calculate Transmissibility

rowsize = rowsize - 2*numofabsorbers;

%Obtain top and bottom generalized displacement coordinates topps = p(1:sizeof); botps = p(((n-1)*sizeof+1):n*sizeof);

%Top, z = 1 z=1; topdisp = topps(5)*1+ topps(6)*(z+1)+topps(7)*(r*((z-1)*(z+1))+(z+1))+topps(8)*((z-1)*(z+1)+(z+1))+... topps(9)*(r*(z*(z-1)*(z+1))+(z+1))+topps(10)*(z*(z-1)*(z+1)+(z+1))+topps(11)*(r*(z^2*(z-1)*(z+1))+... (z+1))+topps(12)*(z^2*(z-1)*(z+1)+(z+1));

%Bottom, z = -1 z=-1; botdisp = botps(5)*1+ botps(6)*(z+1)+botps(7)*(r*((z-1)*(z+1))+(z+1))+botps(8)*((z-1)*(z+1)+(z+1))+... botps(9)*(r*(z*(z-1)*(z+1))+(z+1))+botps(10)*(z*(z-1)*(z+1)+(z+1))+botps(11)*(r*(z^2*(z-1)*(z+1))... +(z+1))+botps(12)*(z^2*(z-1)*(z+1)+(z+1));

trans(tt) = abs(topdisp/botdisp);

end %freq loop

hertz = w./2/pi;

figure(1);loglog(hertz,trans);xlabel('Frequency (Hz)')ylabel('Transmissibility')grid

Appendix E

ANNOTATED DRAWINGS OF LAYERED SPECIMEN WITH

EMBEDDED FLUID ELEMENTS

The following pages contain the technical drawings for parts of the three-layered

fluidic specimen. The drawings were created using Ironcad® solid modeling design

software.

193

194

195

196

197

198

199

200

201

202

203

204

205

206

207

Appendix F

CAUSES OF VARIATION IN THE DYNAMIC MATERIAL

PROPERTIES OF ELASTOMERS

The analytical prediction method for layered isolators presented in this thesis

assumes constant material properties over the entire frequency range and also does not

account for any strain amplitude dependence in elastomers. In reality, a helicopter with

installed layered isolators will undoubtedly experience a variety of operating conditions

which will affect elastomer dynamic properties. The primary factors influencing these

properties would be amplitude, temperature, preload, and frequency. Therefore, a brief

examination of these issues is appropriate.

Dynamic Amplitude Dependence

The Mullin’s effect and the Payne effect are two well-known dynamic amplitude

phenomena. Mullin’s effect can be observed when a previously unstrained elastomer

specimen is subjected to strain cycles at a constant peak value. During the first few

oscillations, the specimen will experience reduced peak stress [79].

209

The Payne effect is most often observed with an increased amount of elastomer

filler, such as carbon black. It is characterized by a decreasing shear modulus with

increasing dynamic amplitude. In Figure F.1, specimen stiffness versus dynamic strain is

plotted for a typical filled elastomer and the Payne effect can be observed.

In an installed layered isolator, the high frequency dynamic strains that an

elastomer layer would experience would be on the order of 0.1 %. The Payne effect

would therefore not likely be observed at the high frequencies. Thus, variable dynamic

strain should not be an important factor in determining elastomer shear modulus in the

frequency range of interest.

Figure F.1: Example of Payne Effect for Typical Filled Elastomer [79].

210

Scobbo investigated the effects of varying dynamic strain at higher frequencies

(10 - 57 Hz) [80]. A high-frequency dynamic tester was used to characterize gum

elastomers and their carbon black-filled counterparts. An objective was to validate a

strain amplification model at high frequencies. In Figure F.2, the magnitude of the

complex viscosity is plotted versus frequency for different dynamic strains. Complex

viscosity is defined as

*Gηω

= ( F.1 )

where G* is the magnitude of the complex shear modulus. The different dynamic strain

cases are nearly co-linear for both the gum and compound curves. The curves therefore

indicate that, for the material tested, varying dynamic strains at higher frequencies has

Figure F.2: Effect of Dynamic Strain at Room Temperature for Gum and Carbon Black-Filled Rubber for ε = 0.17, 0.35 0.5, 0.8, 0.88 [80].

211

little effect on the elastomer complex viscosity, and therefore has little effect on the

complex modulus.

Unfortunately, no experimental studies which investigated the effect of varying

dynamic strain in the frequency range of 500 – 2000 Hz were found in the literature.

Temperature Dependence

Materials are said to be thermo-rheologically simple when the relation between

the dependence on temperature and frequency can be described by an equation of the

form

0( ) ( , )T TG Gω χ ω= ( F.2 )

where GT and G0 are the shear modulus at temperature T, and at reference temperature T0,

respectively, and χT is an empirically chosen shift function [79]. Elastomers have long

been known to exhibit this sort of behavior. With such a relation, material properties at

variety of temperatures and frequencies can be estimated from limited experimental data.

In [82], the authors first present a comprehensive fractional derivative based

model for elastomer dynamic stiffness which includes temperature, frequency, and

preload dependencies. The frequency and temperature relations for an unfilled, sulfur

cured natural rubber were obtained over a low-frequency band at varying temperatures.

The shear modulus magnitude was measured by a dynamic mechanical thermal analyzer

using a sinusoidal force applied to the sample to produce a bending mode. The shear

212

modulus was then estimated over the frequency range of 20 – 20,000 Hz, a temperature

range of –50 oC to +50 oC, and 0 to 20 % precompression.

In Figure F.3, the sample shear modulus is plotted over the frequency and

temperature range at 0 % precompression. Between 0 oC and +50 oC in the frequency

range of interest (500 – 2000 Hz), the shear modulus varies by 10 % at most. As the

temperature is further decreased to -25 oC, however, the modulus is increased by an order

of magnitude. This sharp increase is most probably a result of encountering the rubber-

to-glassy transition temperature, where material properties are quite sensitive to

temperature changes. For layered isolators, it would be desirable to employ elastomers

that are relatively insensitive to changing operating temperatures. Thus, elastomers

composing layered isolators should have transition temperatures which are as low as

possible, or at least lower than the lowest operating temperature of the helicopter. In this

region, the elastomer loss factor, η, typically will be in its lower value range. A lower

value of loss factor is desirable for a pronounced tuned absorber effect.

Figure F.3: Predicted Shear Modulus vs. Frequency at Different Temperatures forTypical Unfilled Elastomer [82].

Frequency Rangeof Interest

213

Preload Dependence

As reported in the previous section, Kari et al developed a dynamic stiffness

model which includes the effects of precompression [82]. In Figure F.4, the effect of

10% and 20% precompression on the driving point stiffness of a modeled cylindrical

isolator is illustrated at different temperatures.

Figure F.4: Modeled Driving Point Stiffness Magnitude and Phase versus Frequency ofSample Isolator for 0% (----), 10% (- - - ), and 20% (- . - .) Precompression at (a) –50 oC,

(b)-25 oC, (c) 0 oC, (d) 25 oC, and (e) 50 oC [82].

0 %10%20%

214

Driving point stiffness is defined as the driving force divided by the displacement

at one end of the isolator with the other end fixed. As Figure F.4 indicates,

precompression appears to have only a mild effect on isolator stiffness with a maximum

increases of about 20% at 20% precompression. The direct effect of precompression on

shear modulus, however, was not reported. An increase in driving point stiffness,

however, implies an increase in shear modulus at high frequencies.

Frequency Dependence

Figure F.5 indicates the typical relationship between elastomer shear modulus

and frequency. Like temperature dependence plots, rubber, transition, and glassy regions

can be identified. Actual measurements at such high frequencies are difficult to obtain,

and are most often estimated using temperature-frequency shift functions.

In the frequency range of interest (500-2000 Hz), the shear modulus shows little

dependence on frequency. This insensitivity to frequency would be desirable for

elastomers employed in layered isolators.

215

Figure F.5: Unfilled Material Shear Modulus vs. Frequency [79].

Figure F.6: Young’s Modulus versus Reduced Frequency for Rubber Isolator Material[83].

216

The reduced frequency measurements in Figure F.6 of a rubber isolation element

correspond to actual frequencies ranging from 200 to 4000 Hz at the various temperatures

[83]. This experimental data shows that the rubber’s Young’s modulus increases by a

factor of as much as three with increasing frequency.

Like an insensitive temperature dependence, an insensitive frequency dependence

would significantly ease design considerations. Therefore, as Figure F.5 indicates,

elastomers with a relatively high rubber-to-glassy transition frequency should be

employed in layered isolators.

Because isolators are seldom exposed to a single frequency, it is useful to

examine the effects of dual frequency excitation on isolator stiffness properties. In

Figure F.7, isolator stiffness is plotted versus dynamic amplitude for an example isolator.

The solid line indicates the stiffness at 10 Hz with varying amplitude. The dashed line

represents the same conditions with an imposed 1 Hz frequency amplitude. The results

Figure F.7: Stiffness vs. Dynamic Amplitude for Single Frequency and SuperimposedFrequencies for Carbon Black-Filled Isolator [79].

217

suggest that the existence of multiple frequencies can influence the dynamic properties at

a single frequency. Because layered isolators in helicopters would experience a range of

frequency excitations, this effect may need to be considered in layered isolator design.

VITA

Contact InformationOffice Address Home Address229 Hammond Building 1238 Inverary PlaceUniversity Park, PA 16802 State College, PA 16801Phone: 814-865-1986 Email: [email protected]

Phone: 814-234-6654Education

Doctor of Philosophy in Mechanical Engineering, The Pennsylvania State UniversityMasters of Science in Mechanical Engineering, December 2002, The Pennsylvania State UniversityBachelor of Science with High Distinction, University Scholars Program, Mechanical Engineering, May

1997, The Pennsylvania State University

Honors and AwardsAHS Vertical Flight Foundation Scholarship, 2002Rotorcraft Center Fellowship, 1999 and 2000Penn State Honors Program, 1993-1997Phi Beta Kappa Honors Society, 1993-1997National Merit Scholar, 1997

Work ExperienceSummer Internship, Sikorsky Aircraft, 1999Summer Internship, The Ford Motor Company, 1996Mathematics Tutor, Penn State University, 1994-1997

PublicationsSzefi, J.T., Smith, E.C. and Lesieutre, G.A., “Formulation and Validation of a Ritz-based Analytical Model

of High Frequency Periodically Layered Isolators in Compression,” Accepted for Publication in theJournal of Sound and Vibration.

Szefi, J.T., Smith, E.C. and Lesieutre, G.A., “Design, Analysis, and Testing of High-FrequencyPeriodically Layered Isolators with Passive Design Enhancements for Helicopter Gearbox Isolation,”AHS 59th Annual Forum, Phoenix, AZ, 2003.

Szefi, J.T., Smith, E.C. and Lesieutre, G.A., “Design and Analysis of High Frequency Periodically LayeredIsolators for Helicopter Gearbox Isolation,” 44th Structures, Structural Dynamics and MaterialsConference, Norfolk, VA, 2003.

Szefi, J.T., Smith, E.C. and Lesieutre, G.A., “Formulation and Validation of a Ritz-Based Analytical Modelfor Design of Periodically-Layered Isolators in Compression,” 42nd Structures, Structural Dynamicsand Materials Conference, Seattle, WA, 2001.

Szefi, J.T., Smith, E.C. and Lesieutre, G.A., “Analysis and Design of High Frequency Periodically LayeredIsolators in Compression,” 41st Structures, Structural Dynamics and Materials Conference, Atlanta,GA., 2000.

Documents

The Pennsylvania State University The Graduate School ... · helicopter gearbox isolation using periodically layered fluidic isolators ... chapter 3 passive high frequency gearbox