Design and Characterization of Tunable Magneto …...ii Design and Characterization of Tunable Magneto-Rheological Fluid-Elastic Mounts By Brian Mitchell Southern Abstract This study

Design and Characterization of Tunable Magneto-Rheological Fluid-Elastic Mounts

By

Brian Mitchell Southern

Thesis submitted to the Faculty of the

Virginia Polytechnic Institute and State University

in partial fulfillment of the requirements for the degree of

Master of Science

in

Mechanical Engineering

Approved:

Dr. Mehdi Ahmadian, Chairman

Dr. Corina Sandu

Dr. Fernando D. Goncalves

April 28, 2008 Blacksburg, Virginia

Keywords: mount, isolator, elastomer, elastic, MR fluid-elastic mount,

magnetorheological fluid, MR fluid, magneto-rheological fluid, tunable

isolator, characterization, semi-active

Copyright© Brian M Southern 2008

ii

Design and Characterization of Tunable Magneto-Rheological Fluid-Elastic Mounts

By

Brian Mitchell Southern

Abstract

This study of adaptable vibration isolating mounts sets out to capture the uniqueness of

magnetorheological (MR) fluid’s variable viscosity rate, and to physically alter the

damping and stiffness when used inside an elastomeric mount. Apparent variable

viscosity or rheology of the MR fluid has dependency on the application of a magnetic

field. Therefore, this study also intends to look at the design of a compact magnetic field

generator which magnetizes the MR fluid to activate different stiffness and damping

levels within the isolator to create an adaptable and tunable feature.

To achieve this adaptable isolator mount, a mold will be fabricated to construct the

mounts. A process will then be devised to manufacture the mounts and place MR fluid

inside the mount for later compatibility with the magnetic field generator. This process

will then produce an MR fluid-elastic mount. Additionally for comparative purposes,

passive mounts will be manufactured with a soft rubber casing and an assortment of metal

and non-metal inserts. Next, the design of the magnetic field generator will be modeled

using FEA magnetic software and then constructed.

Stiffness or force/displacement measurements will then be analyzed from testing the

isolator mount and magnetic field generator on a state-of-the-art vibration dynamometer.

To vary the magnetic flux through the mount, an electro-magnet is used. To analyze the

results, a frequency method of the stiffness will be used to show the isolators adaptation to

various increments of magnetic flux over the sinusoidal input displacement frequencies.

This frequency response of the stiffness will then be converted into a modeling technique

to capture the essence of the dynamics from activating the MR fluid within the isolator

mount.

iii

With this methodology for studying the adaptability of an MR fluid-elastic mount, the

stiffness increases are dependent on the level of magnetic field intensity provided from the

supplied electro-magnet. When the electro-magnet current supply is increased from 0.0 to

2.0 Amps, the mount stiffness magnitude increase is 78% in one of the MR fluid-elastic

mounts. Through comparison, this MR fluid-elastic mount at off-state with zero magnetic

field is similar to a mount made of solid rubber with a hardness of 30 Shore A. With 2

Amps of current, however, the MR fluid-elastic mount has a higher stiffness magnitude

than a rubber mount and resembles a rubber casing with a steel insert.

Moreover, when the current in the electro-magnet is increased from 0.0 to 2.0 Amps

the equivalent damping coefficient in a MR fluid-elastic mount increases over 500% of

the value at 0 Amps at low frequency. Through damping comparisons, the MR fluid-

elastic mount with no current is similar to that of a mount made of solid rubber with a

hardness of 30 Shore A. At full current in the electromagnet, however, the damping in the

MR fluid-elastic mount is greater than any of the comparative mounts in this study.

Therefore, the results show that the MR fluid-elastic mount can provide a wide range

of stiffness and damping variation for real-time embedded applications. Since many

aerospace and automotive applications use passive isolators as engine mounts in

secondary suspensions to reduce transmitted forces at cruise speed, the MR fluid-elastic

mount could be substituted to reduce transmitted forces over a wider range of speeds.

Additionally, this compact MR fluid-elastic mount system could be easily adapted to

many packaging constraints in those applications.

iv

Acknowledgments

First, I want to thank Dr. Mehdi Ahmadian for presenting me with the opportunity to

further my education with Center for Vehicle Systems and Safety (CVeSS). For his

support and continued involvement, I am greatly indebted. Furthermore, I would like to

thank my committee for their contributions. In addition, I would like to recognize Dr.

Fernando Goncalves for his guidance and wisdom. I would like to thank the PERL

laboratory and Dr. Southward for his continued assistance. At PERL, Shawn Emmons

was of valuable help as he provided superb testing assistance for the mounts in this study.

I would like to thank Dr. Brendan Chan for his help and moral support during my time

with the Advanced Vehicle Dynamics Lab (AVDL). For photography and mount

construction, I would like to thank Zac Charlton for his presence and assistance. Florin

Marcu, Ben Langford, and Mohammad Rastgaar provided great assistance in early design

stages and their help is greatly appreciated. Last, I would like to thank the entire CVeSS

family for the friendships earned and the experiences remembered.

Outside of CVeSS, I would like to thank Dr. Clint Dancey and Dr. Harry Robertshaw

as well as the M.E. Dept. for referring and finding teaching assistantships that provided

funding during my graduate career at Virginia Tech. Special thanks go to Scott Allen and

the Physics Dept. machine shop for fabricating quality parts. The quick turn around on

the work by Joe Linkous with Belmont Machining is vastly appreciated. I appreciate the

parts constructed by the M.E. Dept. machine shop. Last, I would like to thank LORD

Corporation for donating MR fluid and COSMOS for their donation of bobbin spools.

My family has been, in large, a support and driving factor for my achievements.

Therefore, I would like to thank them for their love and support, especially my father

Mike Southern. I would like to thank my grandmother Shirley Southern for her financial

contributions toward my graduate degree. In final, I would also like to remember and

thank my late grandfather Mose Southern for his love of life and resiliency with terminal

cancer and therefore dedicate this thesis to his memory.

v

Content ABSTRACT.................................................................................................................................................... II ACKNOWLEDGMENTS .................................................................................................................................. IV CONTENT...................................................................................................................................................... V LIST OF FIGURES........................................................................................................................................ VIII LIST OF TABLES .........................................................................................................................................XVI

1. INTRODUCTION................................................................................................................................ 1

1.1 OVERVIEW..................................................................................................................................... 1 1.2 MOTIVATION ................................................................................................................................. 2 1.3 OBJECTIVES ................................................................................................................................... 3 1.4 APPROACH..................................................................................................................................... 3 1.5 OUTLINE........................................................................................................................................ 3

2. BACKGROUND................................................................................................................................... 4

2.1 MR FLUID HISTORY AND DEVICES: LITERATURE REVIEW ............................................................ 4 2.1.1 MR Fluid Devices..................................................................................................................... 4 2.1.2 MR Fluid Operation................................................................................................................. 6

2.2 HYDRAULIC MOUNTS: LITERATURE REVIEW ................................................................................ 9 2.3 MR MOUNTS: LITERATURE REVIEW........................................................................................... 10

2.3.1 Magnetorheological Elastomers ............................................................................................ 10 2.3.2 Magnetorheological Fluid-Elastomers .................................................................................. 11 2.3.3 Additional MR Mounts ........................................................................................................... 13

2.4 VIBRATION ANALYSIS TECHNIQUES............................................................................................ 16 2.4.1 Linear Static Spring Stiffness ................................................................................................. 16 2.4.2 Linear Spring Stiffness, Viscous and Hysteretic Damping..................................................... 16 2.4.3 Linear Approximation ............................................................................................................ 17 2.4.4 Frequency Response Modeling .............................................................................................. 18

2.5 SUMMARY OF LITERATURE REVIEW ............................................................................................ 19

3. MR FLUID-ELASTIC MOUNT DESIGN AND FABRICATION................................................ 21

3.1 MAGNETIC CIRCUITRY PRINCIPALS............................................................................................. 21 3.2 MAGNETIC SYSTEM ..................................................................................................................... 23

3.2.1 Magnetic System Design ........................................................................................................ 24 3.2.2 Iteration Stage: Magnetic System Design ............................................................................. 37

3.3 ELASTIC MOUNT DESIGN ............................................................................................................ 42 3.3.1 Elastic Mount Design............................................................................................................. 42 3.3.2 Elastic Mount Fabrication ..................................................................................................... 45

iiivv

viiixvi

vi

3.3.3 Metal-Elastic Mount Fabrication........................................................................................... 50 3.4 DESIGN OF EXPERIMENT.............................................................................................................. 54 3.5 SUMMARY ................................................................................................................................... 59

4. MOUNT STIFFNESS AND DAMPING CHARACTERIZATION .............................................. 60

4.1 ELASTIC PARAMETRIC ANALYSIS ................................................................................................ 60 4.1.1 Static Force-Displacement Analysis and Results ................................................................... 61 4.1.2 Force-Displacement Analysis ................................................................................................ 66 4.1.3 Force-Amplitude Analysis...................................................................................................... 73 4.1.4 Processing Analysis Method Evaluation................................................................................ 81

4.2 MOUNT PARAMETRIC RESULTS ................................................................................................... 83 4.2.1 MR fluid- Elastic Mount Parameters ..................................................................................... 84 4.2.2 Passive Elastic Parameters.................................................................................................... 93 4.2.3 Discrete Comparison of Stiffness Magnitude....................................................................... 102 4.2.4 Mount Comparison .............................................................................................................. 114

4.3 DISCUSSIONS ............................................................................................................................. 121

5. MR FLUID ELASTIC MOUNT MODELING AND CHARACTERIZATION ....................... 123

5.1 NON-PARAMETRIC MODELING APPROACH................................................................................ 123 5.1.1 MR Fluid Metal-Elastic Mount Modeling ............................................................................ 123 5.1.2 Nominal Parameter Results and Comparison...................................................................... 126 5.1.3 Nominal Parameter Relationship......................................................................................... 132

5.2 MODEL SIMULATION AND COMPARISON ................................................................................... 134 5.2.1 MR fluid Metal-Elastic Mount Simulation ........................................................................... 134 5.2.2 Model Error Evaluation....................................................................................................... 137

5.3 DAMPING MODELING APPROACH .............................................................................................. 142 5.3.1 MR Fluid-Elastic Mount Damping Model............................................................................ 143 5.3.2 MR Fluid-Elastic Mount Damping Simulation .................................................................... 144

5.4 SUMMARY ................................................................................................................................. 146 5.4.1 Non-Parametric Simulation and Evaluation Remarks ......................................................... 147 5.4.2 Damping Simulation and Evaluation Remarks .................................................................... 148

6. CONCLUSIONS AND PROSPECTIVE RESEARCH................................................................. 149

6.1 SUMMARY ................................................................................................................................. 149 6.2 RECOMMENDATIONS ................................................................................................................. 153 6.3 FUTURE WORK .......................................................................................................................... 155

REFERENCES ........................................................................................................................................... 157

APPENDIX A: MOUNT AND MAGNETIC DESIGN SCHEMATICS............................................... 162

vii

APPENDIX B: RESULTS......................................................................................................................... 176

APPENDIX C: DATA PROCESSING CODE ........................................................................................ 183

APPENDIX D: EARLY STAGES OF MOUNT DESIGN AND FABRICATION .............................. 204

viii

List of Figures

Figure 2-1: Polarization and alignment of ferrous iron in MR fluid, adapted from Ahn

et al. [8]. ..............................................................................................................5

Figure 2-2: MR fluid in valve mode with applied magnetic field, adapted from [20]. ...6

Figure 2-3: MR fluid in shear mode with applied magnetic field, adapted from [20]. ...7

Figure 2-4: MR fluid in squeeze mode setup prior to axial force with an applied

magnetic field, adapted from [20]. .....................................................................7

Figure 2-5: MR fluid in squeeze mode operation with axial force and applied magnetic

field. ....................................................................................................................8

Figure 2-6: Ferrous particle aggregation in squeeze mode operation after experiencing

a compressive load, adapted from [22]...............................................................8

Figure 2-7: Two chamber passive hydraulic fluid mount with decoupler, adapted from

[24]......................................................................................................................9

Figure 2-8: (a) Zero field curing, and (b) 100 mT field curing of polyurethane MR

elastomer with carbonyl-iron particles, adapted from [30]. .............................11

Figure 2-9: Magneto-rheological fluid-elastomer study by Wang, adapted from [35]. 12

Figure 2-10: Squeeze mode MR fluid mount by Nguyen et al., adapted from [37]........13

Figure 2-11: Single chamber MR fluid mount, adapted from Ahn et al. [8]...................14

Figure 2-12: Single pumper semi-active mount proposed by Vahdati in [42]. ...............15

Figure 2-13: MR fluid mount by Choi et al., adapted from [43]. ....................................15

Figure 3-1: Isometric view of mount and magnetic system design...............................23

Figure 3-2: (a) Elastic Casing sectional view, (b) Elastic Casing with magnetic-pole

plate inserts sectional view, and (c) isometric view of metal-elastic casing. ...24

Figure 3-3: Cross-sectional view of empty metal-elastic casing and magnetic system

with test fixtures. ..............................................................................................25

Figure 3-4: (a) Mount and magnet system cross-section view; (b) cross section

modeled in FEMM with field lines...................................................................28

Figure 3-5: B-H curves for MRF-122, MRF-132, MRF-140, and MRF-145 with field

intensity in fluid gap generated by a 3 Amp current supply.............................29

Figure 3-6: Simulated (a) Flux density for mount system and (b) magnetic flux

magnitude for MRF-122 with 3 Amps of current supplied to the electro coil. 30

ix







Figure 3-10: (a) Magnitude of magnetic field intensity at the center of the fluid gap in

the mount with various MR fluids. ...................................................................34

Figure 3-11: Yield stress in MR fluids marked with the maximum yield stress achieved

in each fluid from a 3 Amp current supply to the mount system. ....................34

Figure 3-12: Simulated flux density magnitude plot using MRF-145 in FEMM for

mount system in the fluid gap at the magnetic-pole plate boundary. ...............35

Figure 3-13: Simulated flux magnitude plot using MRF-140 in FEMM for mount system

at the (a) center of the fluid gap and at the (b) upper-pole plate boundary. .....36

Figure 3-14: Magnetic system iteration-1 (a) model and (b) simulation contour plot of

lower fluid cavity boundary, in FEMM software. ............................................38









Figure 3-19: Three plate mold for manufacturing elastic mounts. ..................................45

Figure 3-20: Vacuum Pump and Bell Jar. .......................................................................46

Figure 3-21: PolyTekTM polyurethane (Parts A and B), scales, and dispensing syringe.46

Figure 3-22: Dispensing Polyurethane components by weight. ......................................47

Figure 3-23: Mixing polyurethane, degassing polyurethane, and degassed polyurethane

processes. ..........................................................................................................47

x

Figure 3-24: Polyurethane being poured into the syringe (left) and then injected into the

mold (right). ......................................................................................................48

Figure 3-25: Halves are demolded and prepped (left) then returned to the mold with a

bead of uncured polyurethane and aluminum insert (right)..............................49

Figure 3-26: Elastic casing mounts with 6061 aluminum, air, 1018 steel, and solid 30 D

polyurethane (rubber). ......................................................................................49

Figure 3-27: Upper-pole plate (top) and magnetic-pole plate (bottom) made of 12L14

Steel with epoxy primer....................................................................................50

Figure 3-28: Pole plates inserted into mold, upper plate first (left) and then magnetic-

pole plate (right), prior to injecting polyurethane.............................................51

Figure 3-29: Prepped-pole plate casing halves returned to the mold (left) and a finished

metal-elastic casing (right). ..............................................................................52

Figure 3-30: Metal-elastic casing and funnel for filling MR fluid-elastic casing. ..........52

Figure 3-31: Degassing MR fluid during the process of filling the metal-elastic case. ..53

Figure 3-32: Weighing the plugged MR fluid-elastic mount with MRF-145 fluid in the

metal-elastic case. .............................................................................................54

Figure 3-33: Roehrig-EMA Shock Dynamometer and Desktop Computer running Shock

6.0 software, adapted from [45]........................................................................55

Figure 3-34: Test Setup of mount and magnetic system in the Roehrig EMA

Dynamometer. ..................................................................................................56

Figure 3-35: Ramp displacement input for quasi-static testing on the shock dyno.........57

Figure 3-36: Sine displacement input for dynamic testing at 1 Hz on shock dyno.........58

Figure 4-1: Force-displacement plotting method example on a MR fluid-elastic mount

with MRF-145 fluid..........................................................................................61

Figure 4-2: Quasi-Static force-displacement analysis for (a) MR fluid-elastic 1 and (b)

MR fluid-elastic 2 both with MRF-145 fluid. ..................................................63

Figure 4-3: Force-displacement analysis for (a) MR Fluid-Elastic 3 with MRF-145

fluid and (b) Metal-Elastic 3B with no fluid displaced with ramp input at 0.00,

0.50, 1.00, 1.50, and 2.00 Amp.........................................................................65

Figure 4-4: Force-displacement plotting method example with hysteretic content. .....67

xi

Figure 4-5: Force-displacement processing for (a) MR fluid-elastic 1 with MRF-145,

(b) MR fluid-elastic 2 with MRF-145, (c) MR fluid-elastic 3 with MRF-145

and (d) MR fluid-elastic 3B with no fluid. .......................................................70

Figure 4-6: Force-displacement processing for passive mount with (a) air, (b) rubber,

(c) steel, and (d) aluminum inserts from a sinusoidal input of 1-Hz. ...............72

Figure 4-7: Force-amplitude method analysis example for processing transmitted force

data....................................................................................................................75

Figure 4-8: Force-amplitude data processing and model for (a) MR fluid-elastic 1 with

MRF-145, (b) MR fluid-elastic 2 with MRF-145, (c) MR fluid-elastic 3, and

(d) MR fluid-elastic 3B with no fluid. .............................................................78

Figure 4-9: Force-amplitude data processing and model for passive mounts with (a) air,

(b) rubber, (c) steel, and (d) aluminum inserts. ...............................................80

Figure 4-10: Processing method evaluation for MR Fluid-Elastic 1 with force-time

method (left) and force-displacement method (right) from a sinusoidal input of

1 Hz. ................................................................................................................83

Figure 4-11: MR fluid-elastic 1 mount (MRF-145) (a) stiffness |F|/X, and (b) damping

Ceq results obtained from analysis. ..................................................................86





Figure 4-14: Blank metal-elastic case MRE 3B (a) stiffness |F|/X, and (b) damping Ceq

results obtained from analysis. .........................................................................90

Figure 4-15: Passive mount with air insert (a) stiffness |F|/X, and (b) damping Ceq

results obtained from analysis. .........................................................................94

Figure 4-16: Passive mount with 30 D rubber insert (a) stiffness |F|/X, and (b) damping


Figure 4-17: Passive mount with 1018 steel insert (a) stiffness |F|/X, and (b) damping


Figure 4-18: Passive mount with 6061 aluminum insert (a) stiffness |F|/X, and (b)

damping Ceq results obtained from analysis. ...................................................99

xii

Figure 4-19: Comparing stiffness magnitude of a metal-elastic case (MRE 3B) to an

elastic case (AIR) mount at (a) 0 Amps and (b) 2 Amps of current. ..............104

Figure 4-20: Comparing stiffness magnitude of a metal-elastic case (MRE 3B) to a MR

fluid-elastic mount at (a) 0 Amps and (b) 2 Amps of current. .......................105

Figure 4-21: Comparing stiffness magnitude of a solid elastic case (RUB) to a MR fluid-

elastic mount at (a) 0 Amps and (b) 2 Amps of current. ................................106

Figure 4-22: Comparing stiffness magnitude of an elastic case with steel insert (STE) to

a MR fluid-elastic mount at (a) 0 Amps and (b) 2 Amps of current. .............107

Figure 4-23: Comparing stiffness magnitude of an elastic case with al. insert (ALU) to a

MR fluid-elastic mount at (a) 0 Amps and (b) 2 Amps of current. ................108

Figure 4-24: Comparing damping of a metal-elastic case (MRE 3B) to an elastic case

(AIR) mount at (a) 0 Amps and (b) 2 Amps of current. .................................109

Figure 4-25: Comparing damping of a metal-elastic case (MRE 3B) to a MR fluid-

elastic mount at (a) 0 Amps and (b) 2 Amps of current. ................................110

Figure 4-26: Comparing damping of a solid elastic case (RUB) to a MR fluid-elastic

mount at (a) 0 Amps and (b) 2 Amps of current.............................................111

Figure 4-27: Comparing damping of an elastic case with steel insert (STE) to a MR

fluid-elastic mount at (a) 0 Amps and (b) 2 Amps of current. .......................112

Figure 4-28: Comparing damping of an elastic case with aluminum insert (ALU) to a

MR fluid-elastic mount at (a) 0 Amps and (b) 2 Amps of current. ................113

Figure 4-29: Comparative (a) stiffness |F|/X, and (b) damping Ceq results obtained at

0.00-Amps from force-amplitude and force-displacement analysis,

respectively. ....................................................................................................115



respectively. ....................................................................................................116



respectively. ....................................................................................................118

Figure 5-1: Selecting a transfer function to model the stiffness magnitude in the

frequency domain. ..........................................................................................125

xiii

Figure 5-2: Nominal gain, K, as a function of current for MR fluid-elastic mount. ...127

Figure 5-3: Nominal (a) zero-damping ratio and (b) pole-damping ratio as a function of

current for each MR fluid-elastic mount ........................................................129

Figure 5-4: Nominal (a) zero-frequency and (b) pole-frequency as a function of current

for each MR fluid-elastic mount model..........................................................131

Figure 5-5: Non-parametric damping ratio relationship, ζ/α, at each current setting for

MR fluid-elastic mount models. .....................................................................132

Figure 5-6: Non-parametric stiffness ratio relationship, ωn2/β2, at each current setting

for MR fluid-elastic mount models.................................................................133

Figure 5-7: Stiffness simulation results for MR fluid-elastic 1 mount at 0.5 Amp

current increments. .........................................................................................135

Figure 5-8: Stiffness simulation results for MR fluid-elastic 2 mount at 0.5 Amp

current increments. .........................................................................................136

Figure 5-9: Stiffness simulation results for MR fluid-elastic 3 mount at all current

settings. ...........................................................................................................137

Figure 5-10: Maximum and mean error for the transfer function when compared to the

stiffness magnitude vales for (a) MR fluid-elastic 1, (b) MR fluid-elastic 2, and

(c) MR fluid-elastic 3......................................................................................140

Figure 5-11: Discrete model error for (a) MR fluid-elastic 1, (b) MR fluid-elastic 2,

and (c) MR fluid-elastic 3 from simulation at all current settings.................142

Figure 5-12: Damping simulation results for MR Fluid-Elastic 1 mount at full range of

current settings................................................................................................144





Figure 6-1: Automotive friendly design for an MR fluid-elastic mount. ....................154

Figure A-1: Top plate schematic of three plate mold...................................................164

Figure A-2: Middle plate schematic of three plate mold..............................................165

Figure A-3: End plate schematic of three plate mold...................................................166

Figure A-4: Lower housing base and core schematic to magnetic system...................167

xiv

Figure A-5: Upper housing schematic to magnetic system..........................................168

Figure A-6: Spacer schematic to lower housing in magnetic system...........................168

Figure A-7: Upper-pole plate schematic for metal-elastic case mount. .......................169

Figure A-8: Magnetic-pole plate schematic for metal-elastic case mount. ..................169

Figure A-9: Lower housing test fixture schematic for Roehrig Dynamometer............170

Figure A-10: Upper Housing Test Fixture for Roehrig Dynamometer. .........................170

Figure A-11: Elastic case mount chronology from initial case half mount to finalized MR

Fluid-Elastic mount in a full elastic case. .......................................................171

Figure A-12: Paraphernalia readied for manufacturing an elastic case mount. .............172

Figure A-13: Polyurethane in a degassing chamber under 28inHg to remove air. ........173

Figure A-14: De-molding the half cases of the mount from the 3-plate mold. ..............173

Figure A-15: Each half of the elastic case after removal of central parting lines from

middle plate of mold. ......................................................................................174

Figure A-16: Degreased and abraded elastic case halves ready to be inserted in top and

bottom mold plates to create the full elastic case with hollow insert cavity. .174

Figure A-17: Prepped halves placed in top and bottom plate with a bead of polyurethane

on the face of the elastic case half. .................................................................175

Figure A-18: Universal jig used to secure elastic case and position fluid syringe for MR

fluid injection into the empty case cavity. ......................................................175

Figure D-1: First generation mold housing and plugs used for molding the lower

section of an elastomeric case.........................................................................204

Figure D-2: First generation mold and three plugs with lower section of elastic case

with an aluminum insert pictured beside a full elastic case mount. ...............205

Figure D-3: Lower section of elastic case with insert placed inside first generation mold

and readied for upper section..........................................................................205

Figure D-4: First generation electromagnet and test fixture with an MR fluid-elastic

mount in an elastic case. .................................................................................206

Figure D-5: First generation magnetic circuitry layout with an MR fluid-elastic mount

positioned above the magnet poles similar to an MR damper configuration. 206

Figure D-6: First generation electromagnet housing schematic...................................207

xv

Figure D-7: Testing first generation electromagnet on MRF-128 fluid-elastic mount in

an elastic case with 28% by volume ferrous particle fluid using quick connect

adapters on the shock dyno.............................................................................208

Figure D-8: Second generation Electromagnet Aluminum Frame also known as

Iteration 1 in Chapter 3. ..................................................................................209

Figure D-9: Second generation electromagnet flanged core also known as Iteration 1 in

Chapter 3.........................................................................................................209

Figure D-10: Second generation electromagnet coils for flanged core with 21 AWG, 23

AWG, and 24 AWG magnet wire at 500, 750, and 1000 turns, respectively.210

Figure D-11: Testing second generation electromagnet on elastic case mount with MRF-

128 which is a 28% by volume ferrous particle fluid. ....................................210

xvi

List of Tables

Table 3-1: Dimensions and material properties for the magnetic system components as

well as packaging and testing dimensions. ..........................................................25

Table 3-2: Durometer rating comparison chart for conceptual understanding of the Shore

A hardness selected for the elastomeric casing material, adapted from [55]. .....43

Table 3-3: Polyurethane metal-elastic and elastic casing dimensions with internal cavity

dimensions for the specified insert. .....................................................................44

Table 3-4: Mount naming nomenclature for abbreviations and legends. .........................44

Table 3-5: Test matrix for dynamic testing of MR fluid-elastic mounts with MRF-145

fluid and passive mounts with air, rubber, steel and aluminum inserts. ..............58

Table 4-1: Static stiffness values for MR fluid-elastic mounts and passive mounts with

air, rubber, steel, and aluminum inserts at an index 0.25 Amp. ..........................66

Table 4-2: Comparative stiffness and RMS-Error obtained from force-time and force-

displacement analysis. .........................................................................................82

Table 4-3: Stiffness magnitude of metal-elastic case mounts at all current settings. .......91

Table 4-4: Equivalent damping in metal-elastic case mounts at all current settings........92

Table 4-5: Stiffness magnitude results for passive elastic case mounts air, rubber, steel

and aluminum at all current settings. .................................................................100

Table 4-6: Equivalent damping results for passive elastic case mount air, rubber, steel

and aluminum at all current settings. .................................................................101

Table 4-7: Stiffness magnitude comparison for MR fluid-elastic and passive mounts at

settings of 0.00, 1.00, 2.00 Amp........................................................................119

Table 4-8: Equivalent damping comparison for MR fluid-elastic and passive mounts at

settings of 0.0, 1.0, and 2.0 Amp.......................................................................120

Table 5-1: Damping model and exponential coefficient values for MR fluid-elastic 1, 2,

and 3 mounts......................................................................................................143

Table A- 1: Bill of Materials without cost estimates for mount and magnet system and

manufacture. ......................................................................................................163

Table B-1: Passive mount damping analysis results for the air, rubber, steel, and

aluminum inserts................................................................................................176

xvii

Table B-2: MR fluid-elastic mount damping analysis results for MRE’s and blank MRE

3. .......................................................................................................................177

Table B-3: MR fluid-elastic mount and passive mount damping analysis comparison

chart. ..................................................................................................................177

Table B-4: MR Fluid-elastic mount Stiffness Analysis Results for MRE’s and blank

MRE 3................................................................................................................178

Table B-5: Passive mount stiffness analysis results for the air, rubber, steel, and

aluminum inserts at 0.50 Amp current indexing. ..............................................179

Table B-6: MR fluid-elastic and passive mount stiffness analysis comparison chart.....179

Table B-7: MR Fluid-elastic mount parameters from force-amplitude and displacement

modeling analysis at 0, 1, and 2-Amp current settings......................................180

Table B-8: MR Fluid-elastic mount error comparison between force-amplitude |F|/X and

force-displacement Kx, sampled at 0, 1, 2-Amp for MR fluid-elastic mounts 1, 2

and 3...................................................................................................................181

Table B-9: Nominal transfer function parameters used to simulate the results in section

5.2. .....................................................................................................................182

1

1. Introduction

This chapter presents an overview of vibration isolation and absorber uses within many

applications. This discussion is then extended to using magneto-rheological fluid to create

a tunable isolator. A motivation section is presented second and discusses the driving

factors that lead to the pursuit of this research. Furthermore, an objectives section

presents a desired list of deliverables from this research. An approach section then

discusses the methods for achieving those objectives. Finally, the last section lays out the

organization for the remainder of the work.

1.1 Overview

In the world today, processing equipment, machinery, and machine operators are just a

few of the entities that come into contact with oscillatory transmitted forces. Over time,

these transmitted forces can degrade machine alignment or cause operator fatigue.

Therefore, many absorption and isolation mounting platforms have been generated to

reduce transmitted force from motor and foundation disturbances. While an absorber may

be an elastomer tuned for one input, an isolator is generally a fluid filled elastomer which

provides damping and reduces transmitted forces over a larger bandwidth. Unfortunately,

passive devices generally are unable to account for startup modes from a motor or engine

since the absorption is designed to occur at a set engine speed or operating point.

Fortunately, since earlier notions of active mount technology [1], tunable isolators are

available and can be manipulated by a control policy to reduce transmitted forces at both

start-up and across the range of engine speeds. Moreover, some tunable devices take

advantage of magnetorheological (MR) fluid which operates by application of a magnetic

field. This magnetic field changes the apparent viscosity of the fluid and alters the

stiffness and damping within the isolator to maximize isolation. Therefore, transmitted

forces to the chassis or operator can be reduced over a larger range of disturbances with

the tunable stiffness and damping feature. These disturbances can be characterized by

revolutions per minute (RPM) in an engine that pass to a chassis or seismic tremors that

pass to a foundation. Additionally, when an isolator is used between an engine and a

chassis, the mount is considered a secondary suspension.

2

With the previous in mind, many of the available mount configurations for MR fluid

isolators are bulky with large masses due to the necessity of an electromagnet activation

device. This can be true particularly when the magnetic field must travel through an

elastomer containing the MR fluid. Thus, the magnetic circuitry is inefficient which also

necessitates a more powerful magnet. This added weight can be counterproductive and

difficult to package. Subsequently, not many magnetorheological (MR) fluid isolators

have been used on wide scale applications. MR fluid mounts, however, can reduce noise-

vibration-harshness (NVH) over a much larger range of disturbances than standard

absorbers and hydraulic mounts. Therefore, the purpose within this research is to create a

slender mount with an efficient and low-profile magnetic activation system with

aspirations of launching more MR fluid mount devices into industrial, automotive, and

aerospace applications. While not limiting the overall use of the MR fluid mounts,

automotive applications include secondary suspensions in vehicles such as engine,

transmission, seat, and sensor mounts.

1.2 Motivation

The motivation for this research is to build on the successes of others within

magnetorheological (MR) fluid-elastomer devices and further create an efficient and

desirable, low-profile packaging. Therefore, creating a design with high magnetic

efficiency supplied to activate the MR fluid is of particular importance. Once more, the

necessity for low-profile packaging provides a semi-active isolator as a shelf readied

substitute for passive absorbers or isolators. The difference in using a semi-active mount

as an engine mount, which has tunable stiffness and damping, is that it can better reduce

transmitted forces from an engine at various engine speeds or RPMs. Most passive

mounts, however, are only designed to reduce transmitted forces at a set operating speed

which is usually referred to as cruise speed. Therefore, it is the intentions of the author to

help bring MR fluid isolators from the laboratory to industry by designing a convenient

package for the MR fluid mount and magnetic activation device.

3

1.3 Objectives

The primary objectives of this research are to:

1. provide further evaluation and analysis of magnetorheological (MR) fluid-

elastic mounts beyond what is currently available in open literature,

2. compare the performance of MR fluid-elastic mounts with various passive

mounts of the same configuration, and

3. provide guidelines for design and fabrication of MR fluid-elastic mounts.

1.4 Approach

The approach that we adapted for reaching the above is one of building, testing, iterating,

and re-testing a number of fluid-elastic mounts with various configurations. Specifically,

we performed the following:

• Design and built molds for fabricating the mounts

• Enacted a number of mold iterations to achieve the most favorable

configuration for the mount

• Fabricated the mounts with different inserts including aluminum, steel, air,

rubber, and MR fluid

• Tested the mounts on a dynamic characterization test rig (also known as a

“shock dyno”)

• Analyzed and evaluated the results

• Simulated the results in the frequency domain

1.5 Outline

Chapter 2 presents a background on magnetorheological fluid (MR fluid) and its

application within vibration isolation devices. With an innovative approach for an elastic

mount, Chapter 3 presents the design of a metal-elastic case isolator and magnetic system.

The results from thorough testing are presented in Chapter 4 and a comparative study is

finalized. Chapter 5 presents a simulation of the results for the MR fluid-elastic mount.

Finally, Chapter 6 presents the conclusions and prospective research for future work.

4

2. Background

The background chapter begins by providing an overview of MR fluid history, MR fluid

devices, and primary modes of operation. Next, conventional hydraulic mounts are

presented. A section is then devoted to magnetorheological mounts which includes

elastomer and fluid incased in elastomers. Furthermore, useful vibration analysis

techniques and theory are presented in the third section. Each topic is then briefly

reviewed in the summary section.

2.1 MR Fluid History and Devices: Literature Review

MR fluid, which simply adds metal filings and particles to a fluid, was discovered by

Jacob Rabinow in 1948 [2]. With this smart material discovery, the rheology of the fluid

in the presence of an applied magnetic field could be altered. To achieve this semi-active

property, ferrous iron particles are dispersed in a carrier fluid similar to damper oil.

Therefore, MR fluid acts like a common damper oil during off-state or with zero magnetic

field. With the application of an applied magnetic field, the fluid is similar to toothpaste

as modeled with Bingham plastic flow [3]. This change in MR fluid is studied by testing

the yield stress at various magnetic field intensities. In addition to yield stress testing, MR

fluid has been studied and tested at high velocity, high shear rates [4]. With the

aforementioned basics of MR fluid, the following discussion presents MR fluid devices

and a more detailed look at the modes of operation when using MR fluid.

2.1.1 MR Fluid Devices

Several common devices have emerged such as fluid mounts, linear dampers, vibration

dampers, and rotary brakes to take advantage of the unique properties of MR fluid [5].

Moreover, this section presents MR fluid devices and the properties of MR fluid.

As with any device, an underlying technology enables certain functionality. The

capability of MR fluid lies in its ability to change the apparent viscosity proportional to an

applied magnetic field due to the polarization of ferrous magnetic particles as seen in

Figure 2-1. This apparent viscosity change is actually due to altering the yield stress of

the MR fluid. The iron particles are usually in a carrier fluid such as hydrocarbon oil,

water, or silicone [6]. The ferrous particles of iron may be from 1-20 microns in size [7].

5

Many variations of the quantity of ferrous iron to fluid ratios exist for MR fluid. To retain

a flowing fluid, the percentage of ferrous particles is typically limited to 20-40% in the

composition of the MR fluid. Through magnetic activation at varied magnetic field

intensities, MR fluid changes its apparent viscosity which is related to the content of

ferrous particles. Therefore, this rheology behavior has enabled many passive devices to

be operated with multifunctional capability to provide semi-active control.

Figure 2-1: Polarization and alignment of ferrous iron in MR fluid, adapted from Ahn et al. [8].

MR fluid has a very fast response time of less than 10 ms. when a magnetic field is

applied [9, 10]. This extremely fast and adaptive behavior allows MR fluid to be

controlled with an applied magnetic field. Moreover, the fast and reversible rheology

helped MR fluid progress into automotive applications like the shock absorber. Since

shock absorbers (dampers) dissipate energy based on the viscosity of the damper fluid, the

viscosity is selected to offer either a comfortable ride or a responsive handling ride in the

primary suspension of a vehicle. Moreover, with MR fluid in a damper, both of these ride

characteristics can be achieved. The Chevrolet Corvette equipped with MR fluid dampers

uses magnetic selective ride control (MSRC) to provide a comfortable ride or improve

handling at the touch of a button [11]. Audi also offers optional magnetic ride equipment

on the TT model [12]. Improved ride comfort and advanced handling are just a few of the

characteristics that MR fluid provides to the automotive community [13-15]. Control

policies such as hybrid control have been studied in detail to understand transient

6

performances in such applications [16]. Furthermore, skyhook and groundhook control

policies are combined in hybrid control.

In addition to consideration as a semi-active suspension device, MR fluid has been

modeled and used for clutches and drum brakes, too [17, 18]. These types of devices

place the fluid in direct shear mode. Many standard friction based clutches have a short

service lifetime, which is especially true if heavy slipping occurs during power

transmission. Using an MR fluid clutch, however, would allow gradual slipping during

power transmission without causing the clutch to fail.

In summary, many applications exist for using MR fluid in either shear mode, valve

mode or in squeeze mode. Most of these uses have been studied and implemented in the

automotive industry. The rest, however, remain waiting for an initial startup investment

for a currently available market.

2.1.2 MR Fluid Operation

With the aforementioned magnetic particles suspended in a carrier fluid, several modes of

operation can occur. Therefore, this section presents the operational modes of MR fluid.

The primary mode of fluid operation for a damper is valve mode. Valve mode uses

the flow of the fluid passing between magnetic poles, as seen in Figure 2-2, which is also

referred to as pressure driven flow mode as described by Lord Materials Division [19].

During valve mode, the applied magnetic field is varied across the fluid gap to cause an

apparent viscosity change in the fluid. If used in a damper, the applied magnetic field

through the fluid can alter the energy dissipated by the damper. Therefore, the damper

may offer a soft ride or a stiff ride.

Figure 2-2: MR fluid in valve mode with an applied magnetic field, adapted from [20].

7

Another mode of operation in MR fluid is called direct shear mode. Rotary devices

such as brakes place MR fluid into direct shear mode by having a stationary magnetic hub

with fluid around the circumference contained by an outer drum. Without a magnetic

field, the fluid experiences normal shear forces while the drum revolves, but as the fluid is

energized with magnetic field intensity the shear force is increased. In detail, Figure 2-3 is

a representation of MR fluid in shear mode. Other products such as exercise equipment

and clutches can also take advantage of using MR fluid in direct shear mode.

Figure 2-3: MR fluid in shear mode with an applied magnetic field, adapted from [20].

The last operation mode most relevant to this research is squeeze mode. Squeeze

mode is similar to the buckling of a columnar structure of magnetic particles as shown in

Figure 2-4 which has been adapted from [20]. The magnetic field is aligned axially with

the applied force to create chains of the ferrous magnetic particles [20, 21]. The strength

of these chains is dependent on the magnetic field intensity. This operational mode is

typically used in mounts that experience small amplitudes of displacement. Additionally,

the ferrous particles may be embedded in an elastomer as opposed to being in a fluid.

Figure 2-4: MR fluid in squeeze mode setup prior to axial force with an applied magnetic field, adapted from [20].

8

With the columnar structures in place from a magnetic field, the fluid then has to push

through these structures when an external force is applied. Additionally, the columnar

structures are being buckled during this compression. With an applied field, however, the

axial compressive strength of the MR fluid resists this compression [21]. Since the fluid

is assumed incompressible, an elastic deformation at the boundary has to occur to allow

the displaced fluid to move as seen in Figure 2-5. Therefore, an elastic container or

expandable diaphragm is necessary to make use of the MR fluid in squeeze mode

operation.

Figure 2-5: MR fluid in squeeze mode operation with axial force and applied magnetic field.

As the fluid is squeezed, the ferrous iron particles tend to aggregate as discussed by

Goncalves et al. [22]. This is better seen in Figure 2-6 where the aggregation of the

particles has occurred. This aggregation adds to the compressive strengthening effect of

the MR fluid, but is not stated to add the same in extension strengthening when the fluid is

unloaded. Therefore, squeeze mode operation may increase the hysteresis between

loading and unloading the fluid or the dynamic damping element when placed in an

elastomer as seen in the work by York et al. [23].

Figure 2-6: Ferrous particle aggregation in squeeze mode operation after experiencing a compressive load, adapted from [22].

Magnetic field

9

2.2 Hydraulic Mounts: Literature Review

This section presents a general overview of passive hydraulic fluid mounts. A hydraulic

mount is then illustrated and briefly discussed.

The configuration for a hydraulic mount, seen in Figure 2-7a [24], passes fluid

through the inertia track to create damping [25]. Standard hydraulic mounts of this nature

are generally placed between an engine and a chassis. The forces transmitted by the

engine are reduced by the mount with the stiffness of the elastic casing and the damping

created by the fluid being passed through the inertia track. Additionally, a pressure

differential between chambers moves the decoupler as seen in Figure 2-7b with the flow

Q. Within the dynamics of this mount, the force transmitted due to an input displacement

is rationalized in a mathematical model by Christopherson et al. [24]. A model with a

displacement induced decoupler is also presented in the work by Christopherson.

Moreover, Ahn et al. study the desirable transmissibility by developing a genetic

algorithm [26]. Such modeling and prebuild techniques are essential to fabricating a

hydraulic mount for a desired application.

Figure 2-7: Two chamber passive hydraulic fluid mount with decoupler, adapted from [24].

In summary, passive hydraulic mounts are not always set to the desired point of

operation after fabrication [27]. Many hydraulic isolators have to be tuned through costly

iterations, however, methods exist to model the behavior of the mount prior to fabrication

(a) (b)

Decoupler Q

To Engine

To Chassis

10

[28]. Nonetheless, passive hydraulic mounts when used as isolators have given the

automotive community improved transmissibility as compared to the use of rubber

absorbers.

2.3 MR Mounts: Literature Review

This section presents current MR fluid mount devices which have been designed and

tested by either research institutions or industry suppliers. These devices include

magnetorheological elastomers, magnetorheological fluid-elastomers, magneto-

rheological fluid powertrain mounts, and various configurations of fluid mounts just to

name a few. All the while, the main purpose for each mount is to attenuate vibration over

a larger operating range of force disturbances. To take advantage of this characteristic,

Koo et al. as well as other researchers, have investigated control policies for tuned

vibration absorbers which could be used to control MR fluid mounts [29]. This literature

review, however, does not present any further control policies.

2.3.1 Magnetorheological Elastomers

Magnetorheological elastomers, which are composite materials of an elastic element with

embedded magnetic particles, have been investigated and modeled by many researchers.

The magnetic particles are suspended in the elastomer and may be aligned with an applied

magnetic field while the elastomer is cured. This applied field causes the microstructures

of the iron particles to form chains during the curing as described by Boczkowska [30]

and adapted in Figure 2-8b with 100 mT field [30]. Conversely, no chains are noticed in

the absence of applied field while curing in Figure 2-8a. The elastic material used can

range from silicon gels, polyurethane, natural rubber, and foams.

11

Figure 2-8: (a) Zero field curing, and (b) 100 mT field curing of polyurethane magnetorheological elastomer with carbonyl-iron particles, adapted from [30].

Experimentally, Zhou has reported a 55% increase in average shear modulus during

magnetic activation [31]. Shen has indicated through experimental testing that a

polyurethane MR elastomer experiences a 28% increase in modulus [32]. Gong has also

shown that a 60% increase in modulus has been achieved [33]. This is a small sample of

the many available successes that researchers have reported with magnetorheological

elastomers.

In summary, magnetorheological elastomers hold high potential within the tunable

vibration isolation market. The use of these smart material absorbers is likely to grow as

the need for more advanced vibration control is realized.

2.3.2 Magnetorheological Fluid-Elastomers

Magnetorheological fluid-elastomers similar to that presented in this research are

described by an elastomer casing filled with MR fluid. The MR fluid is activated with an

applied magnetic field. So far, limited designs and testing have been published, but the

results have shown great potential as a tunable vibration isolator as seen in the work by

Wang [34].

Of the current designs, Wang et al. has shown that an MR fluid-elastomer undergoing

an oscillatory input can have approximately a 75% increase in output force with the

addition of a magnetic field [35]. This mount is an elastomer casing with MR fluid in the

Field

Direction

No Applied

Field

12

center cavity as seen in Figure 2-9. The system setup places one magnetic pole directly

below the fluid chamber, separated by the elastomer, and a magnetic shield above the

mount. As the mount is compressed, the MR fluid is operated in squeeze mode.

Figure 2-9: Magnetorheological fluid-elastomer study by Wang, adapted from [35].

A second published study by York et al. of similar design to Wang’s has shown the

capacity for tunable damping and dynamic stiffness [23]. The magnetic circuit, however,

has been altered to place the poles of the electromagnet directly above and below the fluid

chamber for improved magnetic efficiency. This design uses a large magnetic field

generator which may be difficult to package. The magnetic field intensity, however, is

able to achieve a sufficient amount of flux density in the fluid. Moreover, Gordaninejad

has patented select configurations of MR fluid-elastomers [36] which are generalized by

the research of Wang and York. This patent details many unique squeeze mode

configurations and arrangements of the fluid-elastomers as well as orientations of the

applied magnetic field. Therefore, these configurations also offer many designs for

further experimental testing and evaluation.

Another style of squeeze mode MR fluid mount by Nguyen et al. [37] is illustrated in

Figure 2-10. Nguyen presents a mathematical model and further presents a numerical

analysis for this mount. To make use of the MR fluid in squeeze mode, a quasi-piston is

placed above a layer of MR fluid in the cavity to interact with the magnetic field

generated by the coil across the fluid gap. This field increases the compressive strength of

the fluid and thereby alters the mounts relative stiffness. The fluid is contained in an

elastomeric shell denoted by the crosshatching in the illustration.

MR Fluid

Elastic Casing

13

Figure 2-10: Squeeze flow mode MR fluid mount by Nguyen et al., adapted from [37].

In summary, MR fluid-elastomers have excellent capability as tunable damping and

dynamic stiffness isolators. Preliminary results by Wang and York et al. have opened the

field for further investigation within these styles of mounts for further casing and

electromagnetic design. Unfortunately, few experimental exploration designs have been

presented by researchers and there is much exploring which can take place for these

devices. One major aspect which should be further investigated is an efficient magnetic

circuit with desirable packaging characteristics.

2.3.3 Additional MR Mounts

Many additional magnetorheological mounts exist which are built on the premise of a

traditional automotive powertrain mount similar to the standard hydraulic fluid mount.

Therefore, this section presents additional MR fluid mounts similar in design to passive

hydraulic fluid mounts.

As stated earlier, passive mounts typically have an upper and lower chamber

separated by an inertia track to create damping where the fluid passes between chambers.

As the fluid is being passed from the upper chamber, the lower chamber expands with a

diaphragm to collect the fluid. Some MR fluid mounts, however, only have a single

chamber as seen in the design shown in Figure 2-11. The MR fluid in this mount is

energized with an applied magnetic field to increase the stiffness of the mount. Here, Ahn

Flux Path

Piston MR Fluid

14

et al. has represented the dynamic stiffness K* with the Laplace function contained within

the illustration [8]. This dynamic stiffness was determined through bond graph modeling.

Figure 2-11: Single chamber MR fluid mount, adapted from Ahn et al. [8].

Moreover, performance analysis within the means of altered variables for MR fluid in

mounts has been numerically simulated and studied by Ahmadian et al. [38]. Furthermore,

semi-active MR fluid mounts have been presented by a number of researchers and have

found their way into limited applications, such as the Delphi’s powertrain motor mount

[39]. Delphi’s mount is a direct replacement for standard automotive engine mounts.

This type of mount can reduce the transmitted vibrations from the engine to the chassis

over a wide range of engine RPMs or during cylinder deactivation. Additionally, several

styles of Delphi’s hydraulic MR powertrain mounts have been patented [40, 41].

A single pumper semi-active fluid mount design has been proposed and simulated by

Vahdati as seen in Figure 2-12 [42]. This research suggests that the dynamic stiffness

which is typically a parameter of frequency can be altered by the MR fluid under magnetic

field activation. Altering the dynamic stiffness allows for a tunable notch frequency

making the mount suited to a wider range of disturbance frequencies.

x(t)

Coil

Flux

Path Core

MR Fluid

15

Figure 2-12: Single pumper semi-active mount proposed by Vahdati, adapted from [42].

Another unique styling for an MR fluid mount has been designed by Choi et al. [43].

This design isolates a piston within a fluid cavity filled with MR fluid as seen in the cross-

sectional view in Figure 2-13. The magnetic flux is directed toward the fluid cavity by the

magnetic poles which encapsulate the coil. Upon activation, the piston motion is damped

by the MR fluid and further damped with increased current in the coil. Therefore, this

style of fluid mount combines damper and mount technology.

Figure 2-13: MR fluid mount by Choi et al., adapted from [43].

In summary, MR mount technology is readily available. Many proposed designs,

simulations, and experimental analyses have shown the merits of using MR fluid in

isolation technology. Fortunately for researchers, however, there are many opportunities

Inertia

Track

Elastomeric

Magnetic

Pole Height Coil

Flux Path

MR Fluid

Piston

Elastic

16

still available for further exploration of MR fluid in isolation technology. Some of these

opportunities include elastomeric casing design, electromagnet design, and the

configuration of both in an efficient package. Furthermore, as more precise vibration

isolation needs arise within the automotive sector, manufacturing industry, and bio-

dynamic applications then MR fluid mount technology will be a readied contender.

2.4 Vibration Analysis Techniques

The purpose of the vibration analysis techniques section is to present common methods

used to parameterize dynamic systems. These techniques highlight linear stiffness and

hysteretic damping, but are not necessarily limited to linear systems. Furthermore, an

oscillatory force output method is employed to increase accuracy of stiffness estimations.

2.4.1 Linear Static Spring Stiffness

Linear spring stiffness is a straight forward measurement. Most mechanical vibration

analysis state that the spring force is

f = k kx (2.1)

where k is the stiffness, and x is the displacement [44]. Plotting force as a function of the

displacement allows many solvers to approximate the slope or the spring stiffness

k = kfx

(2.2)

At static loading, this method is quite useful to recover the actual stiffness.

2.4.2 Linear Spring Stiffness, Viscous and Hysteretic Damping

Many absorption systems have more elements at work than just the spring and must be

measured simultaneously rather than sequentially. With a linear spring and viscous

damper in parallel, the transmitted force becomes

F(t) = k (t) + c (t)eqx x⋅ ⋅ (2.3)

where ceq is the equivalent damping coefficient and (t)x is the velocity. Therefore, a

simple division is no longer possible, but instead the force is plotted against the

displacement and the average spring stiffness is extracted. The area contained within the

17

hysteresis loop can be measured as the energy dissipated, EΔ . Additionally, the damping

coefficient can then be determined from this dissipated energy

2eqEcXπω

Δ= (2.4)

where ω and X are the frequency and magnitude of the oscillatory input displacement,

respectively. York employed this method for calculating the hysteretic damping of an MR

fluid-elastomer [23].

Moreover in regards to hysteretic damping, Inman discusses and presents the stress-

strain relationship. The energy dissipated for the stress-strain lissajou is

2E k Xπ βΔ = (2.5)

where β represents the hysteretic damping constant. If the energy dissipated for a

viscously damped system is compared to that of a hysteretic damped system, the

equivalent damping is

eqkc βω

= (2.6)

where ω is the frequency of the oscillatory input.

In terms of damping, a force-displacement lissajou may be used to visualize the

amount of damping within a system. A damper only exhibits a circular profile within a

force-displacement lissajou and a linear spring exhibits a linear line or slope [45]. The

lissajou is the same as the hysteresis loop, but more commonly used to describe the

physical elements that are transferring the force. Therefore, the combination of both

damper and spring elements results in more of an elliptical pattern within the force-

displacement lissajou. This is useful in interpreting the degree of each element present

within a system such as the MR fluid mount.

2.4.3 Linear Approximation

In the event that a physical system is nonlinear, Dorf et al. has discussed methods for

linear approximation [46]. This method reduces the nonlinear system to an applicable

operating regime in which a linear approximation may be used. Therefore, to approximate

a nonlinear system as a linear system, small changes in the input about the operating point,

18

as described by Dorf, must be linear. Force could then be

( ) ( )kF t k x t b= ⋅ + (2.7)

where x(t) is the displacement and b is the offset. The offset in the case of a spring would

be disregarded because a spring may not produce force at zero displacement. More

commonly, this force model is best suited to ramp inputs for extracting the stiffness,

especially if a preload was on the spring.

Since most mounts are only operated in compression due to a large static preload,

complete unloading is rarely experienced. Therefore when analyzing the results produced

by an input of sinusoidal displacement, it may be necessary to exclude the saturated data

points or the segment of the data that does not produce force during the input cycle. This

approximation regards those data points as being outside the operating range and allows

the characterization of the force data within operating range.

2.4.4 Frequency Response Modeling

Since most vibration isolators are operated across a band of frequencies, it is important to

demonstrate the magnitude of output to input as the frequency is varied. Subsequently,

presenting the frequency response envelope is practical to modeling most physical

systems with either parametric or non-parametric models. Burchett et al. illustrate the

usefulness of the frequency domain plot for obtaining a parametric model of a “spring

mass damper” system [47]. The basis of Burchett’s work is to select a transfer function

applicable to the frequency domain plot. The frequency plot consists of the magnitude of

the output displacement divided by the magnitude of the input force as a function of

frequency. Within this domain, system zeros and poles can be observed more readily.

Because the stiffness magnitude | | /F X is of most importance within material testing

for transmitted force, the input should be a known displacement to generate an output

force [25]. The oscillatory input displacement may have the form

0( ) sin(2 )x t X f Xπ ρ= + + (2.8)

where X is the displacement amplitude, f is the input frequency, ρ is the input phase, and

19

X0 is the displacement offset. A resultant oscillatory force may then take the form

0( ) | | sin(2 )F t F f Fπ ϕ= + + (2.9)

where |F| is the force amplitude, f is the output frequency, φ is the output phase, and F0 is

the force offset. Each frequency input test then produces a stiffness magnitude which may

be plotted to obtain the frequency response. An additional phase difference within the

frequency response may also be viewed, however, this may not provide as much help

when calculating the transmissibility ratio which is defined by the magnitude of the output

divided by the magnitude of the input.

In summary, this section has provided a brief overview of analysis methods. These

analysis methods are useful for parameterizing and characterizing the dynamics of a

vibratory isolator. Therefore, these techniques will be employed during the analysis of the

MR fluid-elastic mount.

2.5 Summary of Literature Review

In the preceding sections of the background on magneto-rheological fluid history and

available MR devices, the specific properties of the fluid were discussed. Different types

of devices such as MR elastomers, MR fluid-elastomers, and MR fluid hydraulic mounts

have been discussed. An analysis section was then presented to plan methods for

measuring the static and dynamic parameters of an MR fluid-elastic mount.

Among the specific properties, the micron sized magnetic particles are activated by a

magnetic field and suspended in a carrier fluid. MR fluid may be operated in valve mode,

direct shear mode, and squeeze mode. Squeeze mode is the most significant of the

operating modes for conducting the design and configuration of an MR fluid-elastic

mount.

Many researchers have experimented with magnetorheological elastomers and were

able to achieve significant increases in the elastomers modulus with an applied magnetic

field. MR mounts and other such hydraulic fluid mounts are no longer experimental as

Delphi anticipates to implement their hydraulic mount on vehicles [39]. Gordaninejad has

patented the magnetorheological fluid-elastomer [36]. York and Wang have both

experimentally tested a MR fluid-elastomer and showed the validity of their

20

configurations [23, 35]. Other methods for mount configuration and designing a magnetic

system, however, have yet to be addressed.

The vibration analysis section provided a basic fundamental approach to determining

static and dynamic parameters. Static stiffness, dynamic stiffness, and damping methods

used by Inman were shown for hysteretic materials [44]. Linear approximation as detailed

by Dorf et al. was presented to recommend a linear analysis about a specific operating

point [46]. Furthermore, frequency domain modeling was discussed as proposed by

Burchett et al. for parametric and non-parametric modeling of physical systems [47].

21

3. MR Fluid-Elastic Mount Design and Fabrication

This chapter is devoted to the design of a magnetorheological fluid-elastic mount and

magnetic system with additional mounts only for later comparisons. First, the magnetic

circuitry principals are presented to facilitate the reader’s understanding of electromagnets

with MR fluid. Next, the design of a unique and compact magnetic system configuration

is presented and validated with a magnetic modeling program called Finite Element

Method Magnetics (FEMM) [48]. Additional designs that were less than desired are also

presented in the magnetic system design section. Third, the elastic mount system is

presented, which includes the design, selection of materials, and fabrication for both the

elastic and metal-elastic case mounts. Lastly, the design of experiment is presented which

discusses the testing procedures and test equipment.

3.1 Magnetic Circuitry Principals

Most MR fluid devices are operated using an electromagnet, permanent magnet or a

combination of the two. Electromagnets involve some predetermined wire gauge wrapped

a specific number of turns around a core of low-carbon, magnetic steel. Unfortunately, a

sufficient amount of coil turns and current will not increase the likelihood of bridging a

poorly designed magnetic circuit gap. Therefore, understanding the principal theory to

magnetic circuitry is the first step to building an appropriate electromagnet, but this

should be validated in lieu of the circuit layout. Fundamentally, this section provides a

brief overview of magnetic circuit theory.

Developing a magnetic circuit begins with the magnetic permeability μ of the

materials that make up the circuit. Selecting materials that readily pass magnetic flux

helps the circuit maintain efficiency. Any air gaps, however, will consequently degrade

magnetic field intensity H and should be avoided or at least used as a passage for MR

fluid. If air gaps are necessary, then using more turns of magnet wire and higher current

maybe necessary to achieve the desired magnetic field intensity. MR fluid permeability is

dependent on the percentage of magnetic particles that make up the fluid. MR fluids have

nonlinear B-H curves and the permeability is not a direct constant.

22

Further design considerations are selecting a magnetic circuit that can saturate the

MR fluid to ensure the most yield stress from the fluid. Consequently, there is limited

research on the compressive yield stress for MR fluid in squeeze mode. The compressive

yield stress is also referred to as the squeeze strengthen effect. Therefore, axial squeeze

strengthening of an MR fluid may require a much higher level of magnetic field intensity

to reach saturation. Until more information is known on the squeeze strengthen effect of

MR fluid, using the yield stress versus field strength data when designing the magnetic

system is the best option.

After the desired field intensity is established for the MR fluid, a corresponding

operating point in the electromagnet material is determined. The operating point B for the

fluid can be found from a B-H curve. The cross-sectional areas of each material are also

taken into account when determining the operating point and the magnetic intensity. Then

utilizing Kirchoff’s law in magnetic circuit form

n nNi H L= ∑ ⋅ (3.1)

the number of turns N and current i are related to the sum of the material magnetic

intensity Hn and material length Ln. Now, the current and number of turns needed for the

electrical circuit are calculated from equation (3.1).

Because space constraints exist in most electromagnetic activated devices, the wire

gauge and number of turns may have to be compromised. Subjectively, passing more

current through a smaller diameter wire is not a practical alternative. The designer should

evaluate the power supply, wire gauge, number of turns, and packaging during the design

stages for an electrical coil.

23

3.2 Magnetic System

This section covers the magnetic system design. In addition to the magnetic system, the

elastic casing design is conceived within this section since the two components are

necessary to make an efficient magnetic circuit. Following the viable magnetic system

design, some earlier design iterations are presented.

Prior to continuing, an overview of the terminology and magnetic components is

presented in Figure 3-1. This isometric view of the mount and system design is for

clarification through the remainder of the document. Some of the major components for

the design are the metal-elastic case mount and the magnetic systems upper and lower

housing. The lower housing contains a magnetic core with a concentric coil bobbin which

is locked in place by a spacer. Additionally, the spacer provides a flush surface for the

mount. Detailed design and modeling of the mount and magnetic system is presented

next.

Figure 3-1: Isometric view of mount and magnetic system design.

Magnet Upper Housing

Metal-Elastic Case Mount

Spacer

Coil-Bobbin

Magnet Lower Housing

24

3.2.1 Magnetic System Design

The magnetic system design proposed in this research first takes aim at removing the

restrictions within the elastic casing that impede magnetic field intensity. The first step in

removing these restrictions is by eliminating the non-magnetic elastic casing as seen in

Figure 3-2a. This part of the elastic comes into contact with the core of the magnet and is

replaced with a magnetic-pole plate, shown in Figure 3-2c, which directs the magnetic

field across the cross-sectional area of the MR fluid cavity gap. Next, a return path is

added to direct the magnetic field from the fluid and complete the loop. The full metal-

elastic case is shown in Figure 3-2c which is followed by a discussion of design

constraints.

Figure 3-2: (a) Elastic Casing sectional view, (b) Elastic Casing with magnetic-pole plate inserts sectional view, and (c) isometric view of metal-elastic casing.

The design parameters for the mount and magnetic system were constrained due to

availability of tooling, materials, and testing equipment. Initial mount manufacturing used

a three-plate mold to cast the elastic casing. This mold provided an elastic sidewall

thickness to the fluid cavity of 0.375 in. in order to provide sufficient rigidity and prevent

rupturing the elastic casing. Additionally, a thinner sidewall thickness could be used

which would reduce the sidewall rigidity as well as the surface area for attaching the

upper-pole plate. The pole plates integrated into the casing design retained a thickness of

0.125 in., however, the use of a thicker pole plate would have required modifications to

the three-plate mold. Moreover, the thickness of the pole plates could be reduced, but the

upper-pole plate requires a plug which needs sufficient thread length. Table 3-1 shows the

dimensions of interest for the complete design and Figure 3-3 is a cross-sectional view of

Inserted Pole Plates

(a) (b)

(c) Magnetic-Pole Plate

Removed Elastic Fluid Cavity

25

complete design. Additional schematics and specifications are covered in Appendix A for

each component.

Table 3-1: Dimensions and material properties for the magnetic system components as well as packaging and testing dimensions.

Figure 3-3: Cross-sectional view of empty metal-elastic casing and magnetic system with test fixtures.

Test Fixture

Extruded Lower

Housing Spacer

Coil Bobbin

Test Fixture

Bulge Space

Air Gap MR Fluid

Upper Housing

Core

26

One of the major parameters incorporated into the mount design is the height of the

fluid cavity gap. This dimension was set to 0.1875 in. to allow sufficient compressive

inputs to be placed across the mount. Since the mount would be operated dynamically

after being loaded statically, the fluid gap was designed to be squeezed up to 25% of the

original fluid gap height which is equivalent to 10% of the overall mount height.

Therefore, when tested a maximum compressive displacement of 1 mm can be applied

comfortably to the mount without crushing the mount. The testing is further explained in

the design of experiment. Shortening the height of the fluid gap, however, may result in

higher yield stresses being achieved in the MR fluid. This increased yield stress would be

noticed as a compressive strengthening effect. Moreover, the sidewall thickness of the

elastic casing could be reduced to allow a larger MR effect to be realized, but design

robustness was considered a top priority.

The remainder of the model design focuses on the electromagnetic activation

components. These components are contained in the top-assembly and the bottom

assembly as listed in Table 3-1. In the top-assembly, an upper housing is used to create an

efficient return path in the magnetic circuit and also constrain the upper-pole plate of the

of the metal-elastic case. The case is then able to sit inside the upper housing which

extends the upper-pole plate toward the extruded lowering housing. This extension

creates a flux return path to the lower housing of the magnetic system. The thickness of

the upper housing provides sufficient thread length for fastening a test fixture at the

perimeter of the housing.

Next, the main focus for the lower housing is a centered magnetic core that mates to

the magnetic-pole plate of the metal-elastic mount. A coil bobbin that would not interfere

with the diameter of the magnetic core was selected from available donated parts.

Therefore, the lower housing model design takes into account the electro coil and metal-

elastic mount elements. A spacer is used to provide a solid base for the mount which also

locks the coil in the lower housing. The extruded lower housing then provides a return

path for the upper housing. Finally, 24 AWG magnet wire at 800 turns was selected to fit

the coil bobbin and to provide a large Ni value with a minimal current supply. Using a

low current supply is necessary to avoid overheating the coil when testing over continuous

27

cycles. Therefore, the design parameters incorporated system integrity with available

tooling, and donated parts and maintain reduced packaging space.

Testing equipment defined the use of a 1/16 in. air gap, as seen in Figure 3-3,

between the upper housing of the mount and the extruded section of the lower housing to

allow for improper axial alignment within the testing equipment. This space prevents any

mode of binding, either axial or torsion, from occurring and possibly adding friction

which might misconstrue the test results. This air gap could be reduced or removed if

placed in a permanent application, but a blow-off route for the air in the bulge volume

would need to be created. The bulge space around the circumference of the elastic casing

is to allow room for expansion of the elastic casing sidewall during compression. This

elastic material thickness adds stiffness to the case design, but may be trimmed if an

insignificant MR effect from the mount is noticed during testing. Test fixtures were also

added to the upper and lower housing to adapt the mount system to a Roehrig shock dyno.

With the aforementioned metal-elastic case, the magnetic model is prototyped in

finite element methods magnetics (FEMM) analysis software. FEMM analyzes the axis-

symmetric vertical cross-section of a magnetic circuit. From this software, contours of the

magnetic flux density |B| as well as the magnetic field intensity |H| in the model can be

extracted. The necessary inputs to create an accurate model are the dimensions, material

properties, coil windings and wire gauge of the system, and the circuit current. The B-H

curves for MR fluid are added to the material library in the FEMM program which is

discussed later.

After inserting the dimensions and material properties of the mount and magnet

system as seen in Figure 3-4a, the magnetic circuit is modeled. Figure 3-4b shows the

FEMM model with magnetic field lines while Figure 3-4a shows the electromagnetic coil,

the upper housing, and the lower magnet housing with core. A couple of other additional

features for the test setup, which are not included in the FEMM model from Figure 3-4b,

are the non-magnetic test fixtures shown in Figure 3-3. These fixtures are required

mounting for the Roehrig shock dyno, but would not be necessary if the mount and

magnetic system were placed in a permanent application. Moreover, the mount could be

rigidly attached to the core of the magnet, but for testing purposes the upper housing is the

only alignment constraint placed on the mount.

28

Figure 3-4: (a) Mount and magnet system cross-section view; (b) cross section modeled in FEMM with field lines.

The last feature of the model was completed with the appropriate B-H curves for the

various MR fluids. For MRF-122, 132, 140, and 145, the B-H curves were determined

using the model

1.1330 01.91 [1 exp( 10.97 )]B H Hμ μ= ⋅Φ ⋅ − − ⋅ ⋅ + ⋅ (3.2)

where B is the magnetic flux, H is the magnetic field intensity, Φ is the percentage of

ferrous iron in the fluid, and μ0 equals 4π10-7 [49]. Therefore, Φ is set to 0.22, 0.32,

0.40, and 0.45 based on the ferrous iron percentage making up the MR fluid. Next, the

magnetic field intensity is increased from 0 to 600 kAmp/m to generate the magnetic flux

in equation (3.2). Additionally, this simulation accurately represents the empirical B-H

curve for each fluid if compared to the product bulletins published by Lord Corp [50-52].

Furthermore, the simulated data was converted into FEMM’s material library as

shown in Figure 3-5. The coil used to generate the magnetic field had 800 turns of 24

AWG magnet wire. Assuming no more than 3 Amps of current would be supplied to the

coil, the magnetic field intensity and magnetic flux are established at the center of the

fluid gap for each MR fluid as marked on each B-H curve. With a 3 Amp current supply,

the magnetic field intensity is 295 kAmp/m in MRF-122, 254 kAmp/m in MRF-132, 218

kAmp/m in MRF-140, and 197 kAmp/m in MRF-145.

Upper Housing

Coil

Lower Housing (a) (b)

Field Lines

MR Fluid

29

0.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

0 100000 200000 300000 400000 500000 600000

H, Amp/m

B, T

esla

MRF-122 MRF-132MRF-140 MRF-145Mid-Gap @ 3 Amp

Figure 3-5: B-H curves for MRF-122, MRF-132, MRF-140, and MRF-145 with field intensity in fluid gap generated by a 3 Amp current supply.

In order to illustrate the effects from the aforementioned fluids used in the fluid

cavity, the following analysis presents a best case scenario for each fluid where the coil is

supplied with 3 Amps of current. Figure 3-9a, using MRF-122, shows the flux density in

the entire system when activated with a 3 Amp current supply. In MRF-122 the fluid

region experiences 0.75 T at the middle section of the gap as seen in Figure 3-6b and

shows the magnitude of the flux against the magnetic-pole plate (bottom), and against the

upper-pole plate (top) of the fluid gap. Switching the simulated fluid to MRF-132, the

flux density increases as depicted in Figure 3-7a in the fluid gap and the middle section

experiences 0.88 T of magnetic flux. Another unique feature in each of these fluid gaps is

that the field direction is normal through the fluid gap; however, at the upper housing the

field is redirected into the return path. This redirection at the top of the fluid gap through

the upper-pole plate causes the magnitude of flux to increase at the top region of the fluid

gap.

30

0.50

0.75

1.00

1.25

1.50

1.75

2.00

-1.00 -0.80 -0.60 -0.40 -0.20 0.00 0.20 0.40 0.60 0.80 1.00

Diameter, in.

|B|,

T

BottomMiddleTop

Figure 3-6: Simulated (a) Flux density for mount system and (b) magnetic flux magnitude for MRF-122 with 3 Amps of current supplied to the electro coil.

22 %

(a)

(b)

31

0.50

0.75

1.00

1.25

1.50

1.75

2.00

-1.00 -0.80 -0.60 -0.40 -0.20 0.00 0.20 0.40 0.60 0.80 1.00

Diameter, in.

|B|,

T

BottomMiddleTop


32 %

(b)

(a)

32

0.50

0.75

1.00

1.25

1.50

1.75

2.00

-1.00 -0.80 -0.60 -0.40 -0.20 0.00 0.20 0.40 0.60 0.80 1.00Diameter, in.

|B|,

T

BottomMiddleTop


Switching to the material properties of MRF-145, the flux density within the MR

fluid gap is approximately 1.0 T and remains consistent through the cross section as seen

in Figure 3-9a when the coil is supplied with 3 Amps of current. To further illustrate the

magnetic flux, the magnitude of magnetic flux at the bottom, middle, and top of the fluid

gap is shown in Figure 3-9b. At the bottom or against the magnetic-pole plate, the flux is

uniform around 1.0 T. In the middle of the fluid gap, however, the magnetic flux

decreases slightly at the circumference.

40 %

(a)

(b)

33

0.50

0.75

1.00

1.25

1.50

1.75

2.00

-1.00 -0.80 -0.60 -0.40 -0.20 0.00 0.20 0.40 0.60 0.80 1.00

Diameter, in.

|B|,

T

BottomMiddleTop


Next, the yield stress of the various MR fluids is generated using Carlson’s yield

stress model

1.5239. . 271700 (0.00633 )Y S C TANH H= ⋅ ⋅Φ ⋅ ⋅ (3.3)

where C equals 1.0 for hydrocarbon oil, Φ is the percentage of ferrous iron in the fluid,

and H is the field intensity in kAmp/m [49]. Before continuing, the yield stress of interest

is determined by the amount of magnetic field intensity from a 3 Amp current supply

produced at the center of the fluid gap as seen in Figure 3-10. The yield stress is then

45 %

(a)

(b)

34

represented in Figure 3-11 for MRF-122, 132, 140, and 145 fluids. The yield stresses

achieved in the fluid are depicted by a marker on each yield curve in Figure 3-11. These

yield stresses are 68 kPa for MRF-145, 59 kPa for MRF-140, 44 kPa for MRF-132, and

26 kPa for MRF-122. Therefore, MRF-145 is used to generate a large MR effect when

activating the fluid in the mount configuration.

100000

150000

200000

250000

300000

350000

400000

-1.00 -0.80 -0.60 -0.40 -0.20 0.00 0.20 0.40 0.60 0.80 1.00

Diameter, in.

|H|,

Am

p/m

MRF122 MRF132MRF140 MRF145

Figure 3-10: Magnitude of magnetic field intensity at the center of the fluid gap in the mount with various MR fluids.

0

10000

20000

30000

40000

50000

60000

70000

80000

90000

100000

0 100000 200000 300000 400000 500000 600000

H, Amp/m

Yiel

d St

ress

, Pa

MRF-122 MRF-132 MRF-140MRF-145 Y.S. @ 3 Amp

Figure 3-11: Yield stress in MR fluids marked with the maximum yield stress achieved in each fluid from a 3 Amp current supply to the mount system.

35

Since MRF-145 fluid is used in the actual construction of the mounts, the more

extensive modeling analysis uses MRF-145. Furthermore, this simulation produced

magnetic field intensities in the center of the fluid gap from 30 kAmp/m to 197 kAmp/m.

The magnitude of magnetic flux |B| entering the fluid cavity is plotted in Figure 3-12.

Most notable is that the magnetic flux remains uniform as it enters the MR fluid region.

With the current to the coil increased in steps of 0.5 Amp up to 3.0 Amp, the flux

magnitude increases from 0.3 T to 1.0 T. This increased magnitude of flux shows the

magnet system is very capable of activating the MR fluid in this configuration. Therefore,

with the uniform magnetic flux profile and increased flux density per current setting, this

design simulation confirms that the magnetic system is a viable solution.

0.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

-1.00 -0.80 -0.60 -0.40 -0.20 0.00 0.20 0.40 0.60 0.80 1.00

Diameter, in.

|B|,

T

0.5 Amp 1.0 Amp1.5 Amp 2.0 Amp2.5 Amp 3.0 Amp

Figure 3-12: Simulated flux density magnitude plot using MRF-145 in FEMM for

mount system in the fluid gap at the magnetic-pole plate boundary.

In addition to the bottom boundary of the fluid cavity, the magnetic flux magnitude is

also collected for the middle and top sections of the fluid gap. Figure 3-13a shows the

level of activation occurring in the middle of the MR fluid cavity at current settings of 0.5

Amp to 3.0 Amp in 0.5 Amp increments. The middle section displays a uniform magnetic

flux value at the center of the fluid gap. The upper boundary of the fluid cavity is shown

in Figure 3-13b and has a less uniform profile. The flux density increases at the outer

radius of the fluid cavity in the upper section as the magnetic field is directed into the

36

upper-pole plate. To further explain this phenomenon, the magnetic field lines are being

redirected into a horizontal flow and condense into the upper-pole plate and housing to

return to the opposite pole of the electro-magnet. This redirection increases the magnetic

flux density at the perimeter of the upper-pole plate, but overall does not have any adverse

implications on activating the MR fluid in squeeze mode. Therefore, the magnetic flux

density through the fluid cavity is acceptable and is of desired uniformity.

0.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

-1.00 -0.80 -0.60 -0.40 -0.20 0.00 0.20 0.40 0.60 0.80 1.00

Diameter, in.

|B|,

T


0.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

-1.00 -0.80 -0.60 -0.40 -0.20 0.00 0.20 0.40 0.60 0.80 1.00

Diameter, in.

|B|,

T


Figure 3-13: Simulated flux magnitude plot using MRF-140 in FEMM for mount system at the (a) center of the fluid gap and at the (b) upper-pole plate boundary.

(b)

(a)

37

The last analysis for this mount system looks at the estimated power usage of the

magnetic coil. The simulated resistance from the FEMM program is 10.3 Ohms for the

800 turn coil with 24 AWG magnet wire. This resistance would require approximately a

30 V power supply at 3 Amp. To reduce overheating the coil, however, a maximum

current of 2 Amp will be invoked during testing. Therefore, the projected yield stress

achieved in the fluid will be 59 kPa with a 2 Amp current supply to the coil.

In summary, an effective magnetic circuit has been simulated, analyzed, and

considered for proper functionality. This system configuration enables advanced

magnetic flux efficiency within the MR fluid cavity using MRF-145 as the simulation has

shown. The magnetic flux density in the fluid has a variable range from 0.3 to 1.0 T with

a current input of 0.5 to 3.0 Amps, respectively. The coil, however, will be operated to a

maximum of 2 Amps which produces a projected yield stress of 59 kPa in the fluid.

Furthermore, packaging of the system has remained compact within the means of the

available components to a total height of 2.62 in. and a diameter of 3.75 in. while using a

low-profile mount.

3.2.2 Iteration Stage: Magnetic System Design

This section illustrates and briefly analyzes the design process which occurred prior to

realization of the final magnetic system design previously presented. As with most

research, a unique and efficient approach is seldom realized at first and design iterations

must occur. This section, however, does not contain all configurations, but instead shows

the basic iterations to present the envelope of the design phase. Other earlier

electromagnet designs and configurations are presented in Appendix D. Furthermore,

these iterations are presented to provide the aspiring mount designer with failed designs

and prevent any recurrence of these designs.

The following designs, unless otherwise noted, use a 24 AWG coil with 1000 turns

supplied with 3.0 Amps of current, and a fluid cavity filled with MRF-145 fluid. Test

frame adapters are not labeled since they are not part of the magnetic circuitry.

Polyurethane is assumed to have the same magnetic permeability as air and therefore is

given the same material property as air by omitting the casings boundary. Additionally,

each design, unless otherwise noted, uses a basic elastic casing of polyurethane with a

38

diameter of 2.375 in. and a height of 0.4375 in. The fluid cavity diameter is 1.625 in. and

the height is 0.1875 in.

The first design iteration in Figure 3-14a uses a flanged magnetic core. A coil bobbin

is placed between the flanges and then inset in an aluminum frame. An upper shield is

used to try and gather the magnetic field across the MR fluid gap. Activating this circuit

with the flanges only loops the magnetic field directly back to the opposite pole.

Therefore, this design passes minimal magnetic flux density into the desired fluid cavity

region as seen in Figure 3-14b and is rejected. Further details for iteration 1 are continued

in Appendix D in the second generation electromagnet section.

Figure 3-14: Magnetic system iteration-1 (a) model and (b) simulation contour plot of lower fluid cavity boundary, in FEMM software.

Similar to iteration-1, the next configuration uses a top flange and removes the

bottom flange as seen in Figure 3-15a. The remaining components are identical. Still,

very little flux density is passed into the fluid cavity as seen in Figure 3-15b. The use of

the single flange still loops the magnetic flux density to the opposite pole and is therefore

discarded.

MRF

Upper

Shield

Flanged

Magnet

Core

Coil

(a) (b)

39


The third attempt was to completely eliminate the flanges from the magnets core.

Unfortunately, a large gap then existed between the fluid cavity and the core of the

magnet as seen in Figure 3-16a. Once again, limited magnetic flux was passed to the fluid

cavity as seen in the flux density plot of Figure 3-16b. This iteration, however, was not a

complete loss, since the direction of the magnetic field was oriented toward the fluid

cavity.

MRF

Upper

Shield

Flanged

Magnet

Core

Coil

(a) (b)

40


With gained understanding of electromagnetic theory, a fourth iteration was then

pursued with the foresight of eliminating the elastic casing that was plaguing the magnetic

efficiency. As seen in Figure 3-17a, the region below the fluid cavity which had been

occupied by the elastomeric casing now contains a magnetic-pole plate. This pole plate

removes the reluctance associated with the magnetic field bridging the air gap and creates

a directional flow to the fluid cavity. Furthermore, a sufficient return path for the field

was adjoined to the top of the fluid cavity by using the upper-pole plate. This upper-pole

plate extended out over the lower housing and placed the air gap on the outer perimeter of

the housing. Sufficient fluid cavity activation was then realized as the lower boundary of

the cavity experienced approximately 0.83 T as seen in Figure 3-17b.

MRF

Upper

Shield

Magnet

Core

Coil

(a) (b)

41


After iteration-4, the first thought was to reduce the perimeter air gap even further,

but doing this might cause the upper and lower housing to bottom out while restricting the

elastic sidewall from having sufficient room to bulge. The second thought was to place

the mount on a spacer to prevent the mount from compressing on the coil windings as

depicted in Figure 3-18a. Parallel research lead to the donation of a coil bobbin, and the

system began to take on constraints. Doubtful that the coil would hold 1000 turns of 24

AWG magnet wire, the windings were reduced to 695 turns in this iteration. Additionally,

the upper-pole plate was placed in an upper housing assembly which would allow for the

system to be attached to the test setup. Upon activation and with the reduced turns of the

coil, the magnetic flux density was fairly uniform at 0.73 T as shown in the flux density

plot of Figure 3-18b.

MRF

Upper -

pole

plate

Magnet

Core

Coil

Magnet

-pole

plate

(a) (b)

42


Each iteration proved to be an excellent resource for the overall system design. The

knowledge obtained here as well as the iterations presented in Appendix D was funneled

to the finalized system. This led to a relatively compact system configuration and a

uniform magnetic flux profile within the MR fluid cavity. Furthermore, these inefficient

designs should be avoided and are mainly added to present the mount designer with

unsuccessful attempts at configuring a mount and magnetic system.

3.3 Elastic Mount Design

This section is devoted to the material selection, design, and fabrication of the elastic

mounts tested in this study. The first section here focuses on materials for the basic elastic

mount casing as well as the metal-elastic mount casing. The next section discusses the

fabrication for the elastic mount which is then followed by the metal-elastic mount

fabrication.

3.3.1 Elastic Mount Design

The elastic mount developed in this study will be in contact with magneto-rheological

fluid which uses a hydrocarbon-based carrier fluid; however, other carrier fluids are

readily available. Prior to selecting an elastomer the following two constraints had to be

MRF

Upper -

Housing

Magnet

Core

Coil

Magnet

-pole

Spacer

(a) (b)

43

met: resistance to degradation from oil or hydrocarbons, and soft available durometer

ratings. Using Lord Corp. compatibility chart for MR fluid, polyurethane has a rating of

good for its compatibility with MR fluid [53]. Therefore, polyurethane with a soft

durometer rating (PolyTek.Corp Poly 74-30) was selected with a hardness of 30 Shore A

(30 Durometer) [54]. The purpose of having a low durometer rating is to avoid

overshadowing the effects of the activated MR fluid. For a better comprehension of this

durometer rating, Table 3-2 provides a comparison of typical products that are everyday

items.

Table 3-2: Durometer rating comparison chart for conceptual understanding of the Shore A hardness selected for the elastomeric casing material, adapted from [55].

Since the premise of the mount is for use as a vibration absorber in a machine or an

engine application, the dimensions are kept relatively compact. These dimensions

coincide with the magnetic system design as discussed in Section 3.2, but are listed again

in Table 3-3 for each type of insert that is used for the comparative study. The MRF-145

MR fluid is injected with a volume of approximately 6.4 cc with an approximate mass of

27 g. An elastic case cavity is left empty which is referred to as the air insert. The solid

rubber elastomeric case is constructed with polyurethane and is referred to as the 30

durometer polyurethane insert. Last, two metal inserts made of 1018-Steel and 6061-

Aluminum are used.

44

Table 3-3: Polyurethane metal-elastic and elastic casing dimensions with internal cavity dimensions for the specified insert.

Mount Mount Insert Insert Height Diameter Height Diameter

P.U. Casing with Type of Insert

inch inch inch inch MRF-145 0.4375 2.375 0.1875 1.625 Air 0.4375 2.375 0.1875 1.625 30 D Polyurethane 0.4375 2.375 0.1875 1.625 1018 Steel 0.4375 2.375 0.1875 1.625 6061 Aluminum 0.4375 2.375 0.1875 1.625

As noticed in Table 3-3, the insert selection contained the following materials: MRF-

145, air, 30 durometer polyurethane, 1018-Steel, and 6061-Aluminun. MRF-145 fluid

was used since it contains 45% by volume of ferrous magnetic particles, which should be

capable of producing significant axial compressive strength changes during magnetic

activation. The magnetic 1018 Steel insert was used for the possibility of networking with

an applied magnetic field and to provide an upper boundary stiffness during the

comparative study. A nonmagnetic 6061 Aluminum insert was used to counter the

previous hypothesis of the 1018 Steel networking with applied magnetic field. Several

passive elements such as AIR and 30 durometer polyurethane were also used to set the

lower boundary stiffness. The AIR filled elastic case is additionally used for stiffness

comparison to an empty metal-elastic case. From this point forward, the aforementioned

polyurethane casing and type of insert will have the nomenclature shown in Table 3-4.

Table 3-4: Mount naming nomenclature for abbreviations and legends. Name Type of Insert MRE MRF-145 AIR Air RUB 30-D Polyurethane STE 1018 Steel ALU 6061 Aluminum

For the metal-elastic case design, the upper-pole plate and magnetic-pole plate

material are made from 12L14 Steel. This steel is machined easily and has superior

magnetic properties. Regardless of the steel selected, an epoxy primer substrate is

required for the polyurethane to bond to the pole plate. Therefore, Omni-MP172 epoxy

primer was selected to create the desired chemical bond between the polyurethane and

45

12L14 Steel. Additionally, an etching primer (SEM#39693) was used prior to the layer of

epoxy primer as added insurance.

3.3.2 Elastic Mount Fabrication

In addition to selecting the materials for the mount, careful consideration of devising the

correct manufacturing process for those materials is of critical importance. Fabricating the

mounts requires an adequate mold to cast the elastic casing, a vacuum degassing chamber

to remove air from the elastomer prior to casting, proper laboratory equipment, and many

techniques that will be explained in this section and further continued in Appendix A.

After having decided the key dimensions of the mount specimens to manufacture, the

pattern can then be copied to a mold. Since the mounts have an inner chamber or cavity, a

three plate mold is needed. The three plates come into contact and require sealing

between each plate which is accomplished with use of axial face o-rings. Depending on

the apparatus or method used to inject the uncured elastomer, the mold will undoubtedly

need to be air tight. The mold used in this study is shown in Figure 3-19 and highlights

the three plates and axial o-ring gland. Further shop schematics and details of the mold

are presented in Appendix A.

Figure 3-19: Three plate mold for manufacturing elastic mounts.

Next, a vacuum pump and degassing chamber are needed as depicted in Figure 3-20.

The pump used in the fabrication is rated at 3.0 cfm and is able to pull a vacuum of 28

inHg when connected to the bell jar. Degassing is generally related to the surface tension

of the fluid or elastomer. Fortunately, this vacuum is sufficient for degassing

Top Plate Mid Plate Bottom Plate

O-Ring

46

polyurethane early in the pot life as well as the MR fluid, but a more viscous elastomer

may require higher vacuum.

Figure 3-20: Vacuum Pump and Bell Jar.

Readying the needed components is the most crucial step and should be done prior to

mixing the resin and catalyst fluid of the polyurethane. This list of components includes

disposable cups, syringes, beakers, and scales as seen in Figure 3-21. The disposable cups

are used to transfer the resin and catalyst to the mixing beaker located on the zeroed

scales. Plastic syringes are used to inject the polyurethane into the mold cavity through

the sprue tunnel of the mold.

Figure 3-21: PolyTekTM polyurethane (Parts A and B), scales, and dispensing

syringe.

Bell Jar

Vacuum Pump

Disposable cup Scales Beaker Syringe

47

Polyurethane is mixed by weight ratio and requires an accurate set of scales. The ill-

flowing resin is the more viscous of the two components and should be dispensed first. A

weight reading of the resin is acquired and the catalyst is added to double the weight

reading as seen in Figure 3-22. Careful consideration must be given to the volume of

polyurethane with respect to the volume of the beaker since the degassing process may

cause the polyurethane to boil out of the beaker. After dispensing, the polyurethane is

mixed as seen in Figure 3-23, placed in the degassing canister where the entrapped air is

removed leaving a degassed elastomer ready for use.

Figure 3-22: Dispensing Polyurethane components by weight.

Figure 3-23: Mixing polyurethane, degassing polyurethane, and degassed polyurethane processes.

Mixed Degassing Degassed

Catalyst Part B

Resin Part A

48

Polyurethane is now ready to be poured into the injection syringe. Pouring above the

syringe several inches helps release any remaining entrapped air as seen in Figure 3-24.

The syringe is held needle up and depressed to evacuate air which also dispenses the

polyurethane. With the mold rigidly attached to workstation and sprue tubes in place, the

polyurethane is injected as illustrated in Figure 3-24. Excess polyurethane is injected to

ensure all air pockets are removed from the cavity. Capping the sprue entrance is

important to prevent the material from flowing back out after the syringe is removed. The

polyurethane is allowed to cure for at least 12 hours prior to being demolded.

Figure 3-24: Polyurethane being poured into the syringe (left) and then injected into the mold (right).

Material cure time may vary, but since the mold makes two halves that need to be

attached it is generally best to demold before the material has completely set. The two

shells have parting lines as well as sprue channels that have to be removed. Upon removal

of the unwanted polyurethane, the halves are degreased and replaced in the mold similar

to the arrangement in Figure 3-25. Notice that extra material is removed from the sprue

entrance or exit as well as around the sides of the mount to allow the next layer of

polyurethane to seep outward.

Sight Window

Excess polymer material

Filling Syringe Injecting

49

Figure 3-25: Halves are demolded and prepped (left) then returned to the mold with a bead of uncured polyurethane and aluminum insert (right).

Uncured polyurethane was applied to one side of the soon to be mount and a 6061

Aluminum insert was placed in the cavity. The uncured polyurethane was spread around

the inner face of the mount on the bottom plate. Similar methods are employed for each

non-liquid insert. The final products are depicted in Figure 3-26 and include the 6061

Aluminum (ALU), 1018 Steel (STE), Air (AIR), and 30 D polyurethane (RUB) insert

mounts.

Figure 3-26: Elastic casing mounts with 6061 aluminum, air, 1018 steel, and solid 30 D polyurethane (rubber).

Aluminum Air Steel Rubber

50

3.3.3 Metal-Elastic Mount Fabrication

In response to producing a more efficient magnetic system design, the elastic casing was

modified. The modifications were made within the limits of the available tooling to

enable a quick turn around. For all intents and purposes, the metal-elastic case may only

be filled with a mobile fluid or gas while solid non-deforming inserts used in the case

would be unfeasible for a vibration absorber.

Returning to the modification, the addition of a surface ground lower magnetic-pole

plate and an upper-pole plate as illustrated in Figure 3-27 was required. The mount

diameter and height remained the same, but the lower magnetic-pole plate replaces the

0.125-in.of polyurethane beneath the internal cavity. Polyurethane above the internal

cavity was replaced by the upper-pole plate which also has a thickness of 0.125-in.

Figure 3-27: Upper-pole plate (top) and magnetic-pole plate (bottom) made of 12L14 Steel with epoxy primer.

The pole plates require a meticulous process before they are mold worthy. This

process includes the following steps: residue removal, sanding, chamfering, etch priming,

epoxy priming, and scuffing. An etching primer is used as an initial substrate to allow

adhesion between the metal and the epoxy primer. The epoxy primer is a necessary

substrate to create a chemical bonding surface so the polyurethane will stick. Before

combining the mold, the epoxy-coated 12L14 metal inserts are placed in the prepped

mold. Embosses on each side of the middle mold ensures that the pole plates are parallel

Sanded

Chamfer

51

during manufacture. Special care has to be administered to keep from contaminating the

epoxy surfaces as they are placed in the mold as seen in Figure 3-28 and when combining

the mold plates.

Figure 3-28: Pole plates inserted into mold, upper plate first (left) and then magnetic-pole plate (right), prior to injecting polyurethane.

Once more, the polyurethane may now be injected into the mold as described in the

previous elastic fabrication process. Similarly, the de-molded polyurethane half must be

prepped and then replaced in the mold as illustrated on the left in Figure 3-29. A small

bead of uncured polyurethane is spread on the inside face of the halves and the mold is

reconnected keeping each half parallel. Once the polyurethane has cured, the metal-

elastic casing is then de-molded. A viable metal-elastic casing, after removing the parting

line material, as depicted in the right quadrant of Figure 3-29, is now ready to be filled

with MR fluid.

Emboss

52

Figure 3-29: Prepped-pole plate casing halves returned to the mold (left) and a finished metal-elastic casing (right).

With the metal-elastic casing ready to be loaded with MR fluid, the degassing process

is once again necessary to remove entrapped gases from the MR fluid. A special funnel

that screws into the upper plate of the casing enables easy transition of fluid to the internal

cavity. The casing with the attached funnel is placed in the bell jar. The funnel,

illustrated in Figure 3-30, is filled with MRF-145 fluid and allows the fluid to drain in the

internal cavity of the casing.

Figure 3-30: Metal-elastic casing and funnel readied for filling MR fluid-elastic casing.

Since the fluid has a high viscosity and does not flow readily, an external pressure

technique is employed to force the degassed fluid into the cavity of the casing. Moreover,

Parting Line

Upper Half Lower Half

Funnel

53

when degassing the fluid, seen in Figure 3-31, a vacuum or negative pressure is created

within the cavity and when releasing the canister vacuum the MR fluid drains into the

cavity with the aid of the atmospheric pressure outside the funnel. This pressure

differential allows MR fluid to be pulled through the funnel by the vacuum pressure inside

the cavity. MR fluid is continually added as the mount is being filled. Repeating this

process ensures the cavity contains only MR fluid and that the MR fluid is thoroughly

degassed.

Figure 3-31: Degassing MR fluid during the process of filling the metal-elastic case.

After removal of the funnel, a socket head cap screw with thread sealant is used to

plug the upper-pole plate. The plug, which is illustrated in Figure 3-32, is then torqued

and the fluid is sealed inside the cavity. The MR fluid-elastic mount is weighed as seen in

Figure 3-32. The dry weight of the metal-elastic casing was 111.0g and loaded with MR

fluid the weight was 137.7g. Therefore, the mass of the MR fluid contained in the mount

is approximately 27 g. This mass could be checked based on the density of the fluid and

volume of the cavity to ensure the cavity is full of MR fluid.

Entrapped

Air

54

Figure 3-32: Weighing the plugged MR fluid-elastic mount with MRF-145 fluid in the metal-elastic case.

As stated earlier, the metal-elastic casing was built within the limits of available

tooling. The tooling consisted of the mold and injection equipment used for the elastic

case. Increasing the structural robustness of the metal-elastic case would require a new

mold. The new mold would have to allow a spacer with magnetic-pole plate to be molded

simultaneously to the elastic case. Further discussion on the spacer with pole plate is

presented in the recommendations section of chapter 6.

3.4 Design of Experiment

This section presents the testing equipment and lists the basic testing setup. After the

basic testing setup, the quasi-static stiffness testing (QST), and the dynamic stiffness

testing (DST) protocols are listed for the MR fluid-elastic mounts and the comparative

passive mounts.

To achieve both the quasi-static and dynamic stiffness testing, an electromagnetic

linear actuator (EMA) dynamometer is employed. The Roehrig-EMA dynamometer

shown in Figure 3-33 is run by a desktop computer via Roehrig-Shock 6.0 software. The

hardware within the EMA measures input displacement and velocity, while the load cell

measures force. The linear actuator has a resolution of 0.25-177.0 mm and able to

produce harmonic inputs up to 100 Hz. Additionally, the Interface brand loadcell is able

to measure forces of up to 2000 lbf. Not pictured is the power supply used to supply the

needed current setting to the coil of the mount magnetic system during testing. This

Plug

55

power supply is a GW Instek GPS-2303 DC power supply and has a current resolution of

0.01 Amp and Voltage resolution of 0.1 V.

Figure 3-33: Roehrig-EMA Shock Dynamometer and Desktop Computer running Shock 6.0 software, adapted from [45].

Eight mounts were tested using the basic experiment setup. These mount consist of

three MRF-145 fluid filled metal-elastic cases (MRE 1-3), one empty metal-elastic case

(MRE 3B), one empty elastic case (AIR), one solid elastic case (RUB), and two metal

inserts in an elastic case (STE, ALU). The experimental setup for testing these mounts is

shown in Figure 3-34 which provides a brief overview of the equipment and important

features. The standard protocols for the test setup are:

• Turn on the EMA and desktop computer

• Raise Crossbar and lock Clamps

• Thread Test Fixture and Lower Housing assembly to the 2000 lb. Load Cell

• Thread Test Fixture and Upper Housing assembly to the Linear Actuator

• Place desired mount for testing in Upper Housing

• Open Shock 6.0 and zero the Load Cell

56

• Unlock Clamps and lower Crossbar

• Allow Crossbar to load approximately 100 N on Mount and lock Clamps

• Fasten Circuit Leads to the coil

• Load test profile in Shock 6.0, run a warm-up session, run test and acquire

results

Figure 3-34: Test Setup of mount and magnetic system in the Roehrig EMA Dynamometer.

As discussed in the protocol list, the crossbar is loaded on the mount at approximately

100 N to establish consistent experimental setup. Due to the variations in stiffness of each

mount, this added load does not generate a standard displacement, but keeps the initial

setup standardized. The mounts are then run through a warm-up period prior to data

collection. Unfortunately, a temperature reading is not possible due to the lower housing

blocking the infrared temperature sensor. Therefore, the warm-up was assumed complete

by running a multi-frequency test at a current setting of 2.0 Amp. This warm-up was used

in both the quasi-static and dynamic testing formats for three metal-elastic case mounts

filled with MRF-145 (MRE 1-3), one empty metal-elastic case mount (MRE 3B), one

Test Fixture

Linear Actuator

Crossbar

Clamps

Test Fixture

Lower Housing

Circuit Leads

Load Cell

57

empty elastic case mount (AIR), one solid elastic case (RUB), one elastic case mount with

steel insert (STE), and one elastic case mount with aluminum insert (ALU).

In the quasi-static experimental testing, a ramp input from 0 to 1 mm over 3 sec. is

used for the soft core mounts and a ramp input from 0 to 0.5 mm over 3 sec. is used for

the solid core mounts as illustrated in Figure 3-35. The current setting was varied from 0

to 2 Amp at 0.25 Amp increments. The acquired force was measured during the

compression for later processing. For further explanation, the soft core mounts consist of

the three different metal-elastic case mounts with MRF-145 fluid (MRE 1-3), the empty

metal-elastic case mount (MRE 3B), the elastomeric case with hollow cavity (AIR), and

the solid polyurethane elastomeric mount (RUB). The solid core mounts refer to the

elastic case mounts with steel insert (STE) and aluminum insert (ALU).

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.5 1 1.5 2 2.5 3

Time, sec.

Dis

plac

men

t, m

m

1mm Ramp0.5mm Ramp

Figure 3-35: Ramp displacement input for quasi-static testing on the shock dyno.

The format for generating dynamic data is shown in the test matrix in Table 3-5. A

maximum amplitude of 0.5 mm is used for each sinusoidal test with the displacement

bounds of 0 to 1.0 mm as shown in Figure 3-36. The solid core mounts denoted with an

asterisk use a reduced amplitude of 0.25 mm and compression range of 0 to 0.5 mm. The

primary reason for reducing the displacement across the solid core mount is to protect the

58

shock dyno and load cell. This amplitude reduction does not have any consequence since

the force amplitude is the most important result measured from the generated data. The

second setting is the current which is incremented at 0.5 Amp for all tested frequencies.

Additional current increments of 0.25 Amp are used on the MRE mounts to provide

deeper characterization and analysis. A frequency band of 1 to 35 Hz is applied to each

mount, but beyond 35 Hz at test amplitude the EMA dynamometer becomes unsteady.

Therefore, higher frequency testing is not pursued.

Table 3-5: Test matrix for dynamic testing of MR fluid-elastic mounts with MRF-145 fluid and passive mounts with air, rubber, steel and aluminum inserts.

* denotes decreased test amplitude on specimen from 0.5 mm to 0.25 mm.

0

0.25

0.5

0.75

1

0 0.2 0.4 0.6 0.8 1

Time, sec.

Dis

plac

emen

t, m

m

0.50mm Sine at 1 Hz

0.25mm Sine at 1 Hz

Figure 3-36: Sine displacement input for dynamic testing at 1 Hz on shock dyno.

59

3.5 Summary

In summary, this chapter has explained the circuit principals which lead to the design of

the mount and magnet system. This mount and magnet system was then validated using

the FEMM analysis software and found to supply approximately 0.8 T of magnetic flux

density to the desired fluid cavity at 2.0 Amp and 1.0 T at 3.0 Amp which produces

approximately 68 kPa of yield stress in the fluid. Several design iterations are also shown

and briefly presented. The elastic mount design section presented the selection materials

and fabrication for the elastic and metal-elastic case mounts. Further mount fabrication

processes are presented in Appendix A. With the mount and magnet system readied, the

previous section covered the Roehrig EMA dynamometer testing equipment and the

design of experiment formulated for the purpose of measuring static and dynamic stiffness

results.

60

4. Mount Stiffness and Damping Characterization

In this chapter, the test data is processed using the vibration analysis techniques from

Chapter 2 to provide a parametric analysis. Therefore, this analysis begins by processing

the quasi-static test data and extracting the parametric stiffness values. Dynamic test data

is then processed using the force-displacement plotting method to obtain both the

parametric stiffness values as well as the equivalent damping coefficient values. In

addition to the force-displacement plotting method, the force-amplitude and displacement-

amplitude values are used to find the magnitude of the dynamic stiffness. An evaluation

section then compares both methods used for obtaining the dynamic stiffness.

After the stiffness and damping analysis, the parametric values are presented in the

results section. This result section characterizes each individual mount by representing

the corresponding stiffness and damping values in the frequency domain. The frequency

domain allows for a more invasive understanding of the mounts parameters, which will

later lead to system identification processing. Moreover, a comparison section inspects

the response of all mounts in an observatory frequency response plot at selected current

settings. This comparison section also concludes on the values of the MR fluid- elastic as

well as the comparative mounts of this study.

4.1 Elastic Parametric Analysis

Herein, this section presents the bounty of this research through a parametric analysis of

each mount, but also states deficiencies to keep the mount and magnet system design in

check. First, the quasi-static processing uses the force-displacement plotting method to

acquire the static stiffness which resulted by applying the ramp displacement input. This

method is then carried to the second section for processing the dynamic stiffness and the

equivalent damping coefficient. Next, the force-amplitude method is used to determine

the dynamic stiffness. A concluding section compares the dynamic stiffness analysis

methods and presents the accuracy through a normalized root-mean-square-error analysis.

61

4.1.1 Static Force-Displacement Analysis and Results

After thorough testing using the ramp displacement inputs, the quasi-static data is

processed with the force-displacement plotting method and presented. Following the

processing, the results from the quasi-static testing are presented.

A ramp displacement input was applied across each mount at different current

settings. The data collected organized in Matlab for processing with a force-displacement

plotting method. The force-displacement plotting method refers to a linear least-squares

curve fitting analysis in Matlab that renders the slope of the force versus displacement plot

as seen in Figure 4-1. More commonly, the slope k is the stiffness of the mount and

related to the force

kF k x= ⋅ (4.1)

where x is the displacement and kx is the linear model shown in the example plot. The

nonlinear section was then discarded from the analysis on the basis of the linear

approximation method since the mount would be operated using a higher static load [46].

The additional offset produced from the linear approximation method is also discarded

since a spring’s force is dependent on displacement.

0 0.2 0.4 0.6 0.8 10

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

Displacement, mm

Forc

e, N

MRE 1,QST,0.0ALinear Model,KxNonlinear Region

Figure 4-1: Force-displacement plotting method example on a MR fluid-elastic mount with MRF-145 fluid.

Discarded

Region

Model,Kx

62

Furthermore, the least-squares method reduces the sum of the square residual of the

data point value and model point value in the loading cycle such that

( )2

1

N

i ii

d mε=

= ∑ − (4.2)

where di is the data point and mi is the model point. The stiffness values determined

using the above method were limited to the loading cycle of the ramp input and not

averaged with the unloading cycle. The unloading cycle receives less force due to the

agglomeration of the ferrous particles in the fluid and was disregarded to process the

quasi-static stiffness. For comparing the quasi-static stiffness k evaluated by the above

model, the stiffness gain s.g. is calculated as

2 0

0

. . 100%k ks gk−

= × (4.3)

where k2 is the stiffness at 2.0 Amp, and k0 is the stiffness at 0.0 Amps or off-state.

Using the aforementioned force-displacement modeling strategy, the force data is

plotted for MR Fluid-Elastic 1 (MRE 1) in Figure 4-2a and for MR Fluid-Elastic 2 (MRE

2) in Figure 4-2b. Each subplot represents current settings incremented at 0.5 Amp.

Additionally, each model is plotted in the lower right subplot to illustrate the change in

stiffness due to the magnetic field. Both MRE 1 and MRE 2 have increased quasi-static

stiffness values ranging from 2587 to 3688 N/mm and 2905 to 3603 N/mm, respectively.

Therefore, from off-state to activated state at 2.0 Amp a stiffness gain of approximately

42% and 24% is recognized in MRE 1 and MRE 2, respectively. The gain in MRE 2 is

lower due to the zero current stiffness which was 300 N/mm stiffer than MRE 1, but the

stiffness at 2.0 Amps of current is comparable.

63

0 0.2 0.4 0.6 0.8 10

1000

2000

3000

4000

5000

Displacement, mm

Forc

e, N

MRE 1,QST,0.0ALinear Model,Kx

0 0.2 0.4 0.6 0.8 10

1000

2000

3000

4000

5000

Forc

e, N

Displacement, mm

0 0.2 0.4 0.6 0.8 10

1000

2000

3000

4000

5000

Displacement, mm

Forc

e, N


0 0.2 0.4 0.6 0.8 10

1000

2000

3000

4000

5000

Displacement, mm

Forc

e, N


0 0.2 0.4 0.6 0.8 10

1000

2000

3000

4000

5000

Displacement, mm

Forc

e, N


0 0.2 0.4 0.6 0.8 10

1000

2000

3000

4000

5000

Displacement, mm

Forc

e, N


Kx,0.0A

Kx,0.5A

Kx,1.0AKx,1.5A

Kx,2.0A

0 0.2 0.4 0.6 0.8 10

1000

2000

3000

4000

5000

Displacement, mm

Forc

e, N


0 0.2 0.4 0.6 0.8 10

1000

2000

3000

4000

5000

Forc

e, N

Displacement, mm

0 0.2 0.4 0.6 0.8 10

1000

2000

3000

4000

5000

Displacement, mm

Forc

e, N


0 0.2 0.4 0.6 0.8 10

1000

2000

3000

4000

5000

Displacement, mm

Forc

e, N


0 0.2 0.4 0.6 0.8 10

1000

2000

3000

4000

5000

Displacement, mm

Forc

e, N


0 0.2 0.4 0.6 0.8 10

1000

2000

3000

4000

5000

Displacement, mm

Forc

e, N


Kx,0.0A

Kx,0.5A

Kx,1.0AKx,1.5A

Kx,2.0A

Figure 4-2: Quasi-Static force-displacement analysis for (a) MR fluid-elastic 1 and (b) MR fluid-elastic 2 both with MRF-145 fluid.

Once more, the force-displacement method is illustrated for the remaining metal-

elastic test specimens. The force-displacement plots for MR fluid-elastic 3 (MRE 3) and

(b)

(a)

64

the empty metal-elastic case mount (MRE 3B) are plotted in Figure 4-3a and b,

respectively. MRE 3 has a lower stiffness than MRE 1 and MRE 2, with the off-state

value of 2292 N/mm. More so for MRE 3, the stiffness gain was lower at approximately

17%. However, the unfilled metal-elastic MRE 3B showed only the effects of the applied

magnetic field pulling down on the load cell. This is evident as the 0 Amp test produced a

stiffness of 575 N/mm and the 2 Amp test lowered the stiffness to 533 N/mm which is an

8% reduction.

To further substantiate the quasi-static stiffness analysis, the remainder of the mounts

are not plotted and instead valued for each test in Table 4-1. Each mount is then listed

with the stiffness value for the current setting from 0.00-2.00 Amps. The solid elastic

mount (RUB) is in the vicinity of the MRF-145 fluid filled metal-elastic mount stiffness

values at the zero current setting. With increased current settings, MRE 1 and 2’s stiffness

approaches the elastic case with metal insert mount (ALU and STE) stiffness. The

aluminum insert mount (ALU) stiffness shows an upward trend in stiffness, but has an s.g.

of only 3% and is considered negligible. The empty elastic case (AIR) has a stiffness

value similar to the empty metal-elastic case (MRE 3B) which suggests the sidewall

thickness of the polyurethane generates the casing stiffness and the pole plates add only

minimal stiffness.

65

0 0.2 0.4 0.6 0.8 10

1000

2000

3000

4000

5000

Displacement, mm

Forc

e, N


0 0.2 0.4 0.6 0.8 10

1000

2000

3000

4000

5000

Forc

e, N

Displacement, mm

0 0.2 0.4 0.6 0.8 10

1000

2000

3000

4000

5000

Displacement, mm

Forc

e, N


0 0.2 0.4 0.6 0.8 10

1000

2000

3000

4000

5000

Displacement, mm

Forc

e, N


0 0.2 0.4 0.6 0.8 10

1000

2000

3000

4000

5000

Displacement, mm

Forc

e, N


0 0.2 0.4 0.6 0.8 10

1000

2000

3000

4000

5000

Displacement, mm

Forc

e, N


Kx,0.0A

Kx,0.5A

Kx,1.0AKx,1.5A

Kx,2.0A

0 0.2 0.4 0.6 0.8 10

1000

2000

3000

4000

5000

Displacement, mm

Forc

e, N

MRE 3B,QST,0.0ALinear Model,Kx

0 0.2 0.4 0.6 0.8 10

1000

2000

3000

4000

5000

Forc

e, N

Displacement, mm

0 0.2 0.4 0.6 0.8 10

1000

2000

3000

4000

5000

Displacement, mm

Forc

e, N


0 0.2 0.4 0.6 0.8 10

1000

2000

3000

4000

5000

Displacement, mm

Forc

e, N


0 0.2 0.4 0.6 0.8 10

1000

2000

3000

4000

5000

Displacement, mm

Forc

e, N


0 0.2 0.4 0.6 0.8 10

1000

2000

3000

4000

5000

Displacement, mm

Forc

e, N


Kx,0.0A

Kx,0.5A

Kx,1.0AKx,1.5A

Kx,2.0A

Figure 4-3: Force-displacement analysis for (a) MR Fluid-Elastic 3 with MRF-145 fluid and (b) Metal-Elastic 3B with no fluid displaced with ramp input at 0.00, 0.50, 1.00, 1.50, and 2.00 Amp.

(a)

(b)

66

Table 4-1: Static stiffness values for MR fluid-elastic mounts and passive mounts with air, rubber, steel, and aluminum inserts at an index 0.25 Amp.

As the aforementioned suggests, the solid elastic mount is representative of a lower

boundary and the aluminum mount is useful for an upper boundary in the comparisons to

follow at the end of this chapter. More importantly though, MRE 1, 2, and 3 stiffness

parameters have shown each specimen to work with increased stiffness gains of 42%,

24%, and 17%, respectively. On the other hand, there are likely differences in each mount

which occurred during the manufacturing process. One difference may be an uncontrolled

depth of epoxy primer substrate which would alter magnetic flux density. In conclusion,

the quasi-static testing and analysis was successful and the analysis is progressed to the

dynamic testing data.

4.1.2 Force-Displacement Analysis

After acquisition of the dynamic test data from applying the test matrix in Table 3-5, the

analysis is now directed toward the dynamic stiffness as well as the equivalent damping

coefficient analysis. Although the previous quasi-static section presented results, this

section is limited to the methods of processing, and does not present the parametric

results. In addition to prevent redundancy, this section shows force-displacement data

plotted for each mount obtained from the 1 Hz oscillatory input at current settings of 0,

0.5, 1.0, 1.5, and 2.0 Amps.

The force-displacement analysis for the dynamic test data uses the same technique as

presented in 4.1.1 for linear approximation and modeling. A hysteretic content, however,

is contained within the dynamic testing. Therefore, an example plot with hysteresis is

67

presented in Figure 4-4. The force-displacement model equation (4.1) is loaded to fit the

region F(x) and the data outside this region is discarded. Prior to determining the stiffness

and damping, the force data is standardized by removing the added force of the crossbar

and load cell. This standardization should also eliminate the offset pull force of the

magnet on the load cell during activated current settings.

0 0.2 0.4 0.6 0.8 10

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

Displacement, mm

Forc

e, N

MRE 1,2.0A,1hzRegion,F(x)Model,Kx

Figure 4-4: Force-displacement plotting method example with hysteretic content.

Next, the equivalent damping coefficient is calculated as

2eqEcXπω

Δ= (4.4)

where EΔ is the energy dissipated per cycle, ω is the input frequency in rad/s, and X is the

displacement amplitude [44]. Matlab software is used to calculate the dissipated energy

of the same region used to model the stiffness. The energy dissipated, seen within the

loading and un-loading region in Figure 4-2, was found using the polyarea.m function in

Matlab. The altered region may decrease the damping coefficient, but this was deemed

non-contributory in the overall study.

Discarded

Region

Loading

Un-Loading

68

In the metal-elastic case testing, additional current settings at 0.25 Amp increments

were included. The purpose for adding these extra current tests was to provide a more

thorough investigation of the dynamics associated with the MR fluid-elastic mounts. All

processing was completed for each frequency from 1-35 Hz with the amplitude of 0.50

mm for the oscillatory input, and current increments of 0.25 Amp.

The dynamic stiffness model Kx, found through a reduction of the sum of least

squares, was fit to the test data taken from the 1 Hz input for the metal-elastic case

mounts. MR fluid-elastic 1, 2, 3, and empty 3B are presented in Figure 4-5a, b, c, and d,

respectively. These dynamic figures are in the same layout as the quasi-static figures.

Each model is compiled in the southeast subplot for visual comparisons. The stiffness

model seen as the blue-dashed line typically centers the loading and unloading sections of

the force data. As noticed, a large hysteresis exists between the loading and unloading

cycle. Therefore, the stiffness becomes an average of the loading and unloading.

Unfortunately, the unloading cycle is nonlinear which may be caused by the

agglomeration of the ferrous particles in the MR fluid. This nonlinearity causes more

error to propogate into the results. A nonlinear model for the stiffness, however, is not

pursued.

0 0.2 0.4 0.6 0.8 10

1000

2000

3000

4000

5000

Displacement, mm

Forc

e, N


0 0.2 0.4 0.6 0.8 10

1000

2000

3000

4000

5000

Forc

e, N

Displacement, mm

0 0.2 0.4 0.6 0.8 10

1000

2000

3000

4000

5000

Displacement, mm

Forc

e, N


0 0.2 0.4 0.6 0.8 10

1000

2000

3000

4000

5000

Displacement, mm

Forc

e, N


0 0.2 0.4 0.6 0.8 10

1000

2000

3000

4000

5000

Displacement, mm

Forc

e, N


0 0.2 0.4 0.6 0.8 10

1000

2000

3000

4000

5000

Displacement, mm

Forc

e, N


0.0A0.5A1.0A1.5A2.0A

Figure 4-5: (continue)

(a)

69

0 0.2 0.4 0.6 0.8 10

1000

2000

3000

4000

5000

Displacement, mm

Forc

e, N


0 0.2 0.4 0.6 0.8 10

1000

2000

3000

4000

5000

Forc

e, N

Displacement, mm

0 0.2 0.4 0.6 0.8 10

1000

2000

3000

4000

5000

Displacement, mm

Forc

e, N


0 0.2 0.4 0.6 0.8 10

1000

2000

3000

4000

5000

Displacement, mm

Forc

e, N


0 0.2 0.4 0.6 0.8 10

1000

2000

3000

4000

5000

Displacement, mm

Forc

e, N


0 0.2 0.4 0.6 0.8 10

1000

2000

3000

4000

5000

Displacement, mm

Forc

e, N


0.0A0.5A1.0A1.5A2.0A

0 0.2 0.4 0.6 0.8 10

1000

2000

3000

4000

5000

Displacement, mm

Forc

e, N


0 0.2 0.4 0.6 0.8 10

1000

2000

3000

4000

5000

Forc

e, N

Displacement, mm

0 0.2 0.4 0.6 0.8 10

1000

2000

3000

4000

5000

Displacement, mm

Forc

e, N


0 0.2 0.4 0.6 0.8 10

1000

2000

3000

4000

5000

Displacement, mm

Forc

e, N


0 0.2 0.4 0.6 0.8 10

1000

2000

3000

4000

5000

Displacement, mm

Forc

e, N


0 0.2 0.4 0.6 0.8 10

1000

2000

3000

4000

5000

Displacement, mm

Forc

e, N


0.0A0.5A1.0A1.5A2.0A


(b)

(c)

70

0 0.2 0.4 0.6 0.8 10

1000

2000

3000

4000

5000

Displacement, mm

Forc

e, N

MRE 3B,0.0A,1hzRegion,F(x)Model,Kx

0 0.2 0.4 0.6 0.8 10

1000

2000

3000

4000

5000

Forc

e, N

Displacement, mm

0 0.2 0.4 0.6 0.8 10

1000

2000

3000

4000

5000

Displacement, mm

Forc

e, N


0 0.2 0.4 0.6 0.8 10

1000

2000

3000

4000

5000

Displacement, mm

Forc

e, N


0 0.2 0.4 0.6 0.8 10

1000

2000

3000

4000

5000

Displacement, mm

Forc

e, N


0 0.2 0.4 0.6 0.8 10

1000

2000

3000

4000

5000

Displacement, mm

Forc

e, N


0.0A0.5A1.0A1.5A2.0A

Figure 4-5: Force-displacement processing for (a) MR fluid-elastic 1 with MRF-145, (b) MR fluid-elastic 2 with MRF-145, (c) MR fluid-elastic 3 with MRF-145 and (d) MR fluid-elastic 3B with no fluid.

The force-displacement processing section is now turned to the elastic case mounts.

The displacement is reduced to an amplitude input of 0.25 mm for STE and ALU. This

reduced amplitude ensured that the solid core mount would not run out of the elastic

region and cause subsequent damage to the testing equipment. Current settings during the

passive mount testing are incremented from 0.00-2.00 Amps at 0.50 Amp steps.

Processing of the elastic case mount demonstrated that these mounts are not a

function of the applied magnetic field. This is seen in Figure 4-6a-d for the air, rubber,

steel, and aluminum mounts by viewing the modeled force Kx plotted in the southeast

subplots. The damping also appears to be relatively low for these mounts as the force-

displacement lissajous have thin profiles. The coefficient of equivalent damping for these

mounts is shown later in the results section.

(d)

71

0 0.2 0.4 0.6 0.8 10

1000

2000

3000

4000

5000

Displacement, mm

Forc

e, N

AIR,0.0A,1hzRegion,F(x)Model,Kx

0 0.2 0.4 0.6 0.8 10

1000

2000

3000

4000

5000

Forc

e, N

Displacement, mm

0 0.2 0.4 0.6 0.8 10

1000

2000

3000

4000

5000

Displacement, mm

Forc

e, N


0 0.2 0.4 0.6 0.8 10

1000

2000

3000

4000

5000

Displacement, mm

Forc

e, N


0 0.2 0.4 0.6 0.8 10

1000

2000

3000

4000

5000

Displacement, mm

Forc

e, N


0 0.2 0.4 0.6 0.8 10

1000

2000

3000

4000

5000

Displacement, mm

Forc

e, N


0.0A0.5A1.0A1.5A2.0A

0 0.2 0.4 0.6 0.8 10

1000

2000

3000

4000

5000

Displacement, mm

Forc

e, N

RUB,0.0A,1hzRegion,F(x)Model,Kx

0 0.2 0.4 0.6 0.8 10

1000

2000

3000

4000

5000

Forc

e, N

Displacement, mm

0 0.2 0.4 0.6 0.8 10

1000

2000

3000

4000

5000

Displacement, mm

Forc

e, N


0 0.2 0.4 0.6 0.8 10

1000

2000

3000

4000

5000

Displacement, mm

Forc

e, N


0 0.2 0.4 0.6 0.8 10

1000

2000

3000

4000

5000

Displacement, mm

Forc

e, N


0 0.2 0.4 0.6 0.8 10

1000

2000

3000

4000

5000

Displacement, mm

Forc

e, N


0.0A0.5A1.0A1.5A2.0A


(b)

(a)

72

0 0.2 0.4 0.6 0.8 10

1000

2000

3000

4000

5000

Displacement, mm

Forc

e, N

STE,0.0A,1hzRegion,F(x)Model,Kx

0 0.2 0.4 0.6 0.8 10

1000

2000

3000

4000

5000

Forc

e, N

Displacement, mm

0 0.2 0.4 0.6 0.8 10

1000

2000

3000

4000

5000

Displacement, mm

Forc

e, N


0 0.2 0.4 0.6 0.8 10

1000

2000

3000

4000

5000

Displacement, mm

Forc

e, N


0 0.2 0.4 0.6 0.8 10

1000

2000

3000

4000

5000

Displacement, mm

Forc

e, N


0 0.2 0.4 0.6 0.8 10

1000

2000

3000

4000

5000

Displacement, mm

Forc

e, N


0.0A0.5A1.0A1.5A2.0A

0 0.2 0.4 0.6 0.8 10

1000

2000

3000

4000

5000

Displacement, mm

Forc

e, N

ALU,0.0A,1hzRegion,F(x)Model,Kx

0 0.2 0.4 0.6 0.8 10

1000

2000

3000

4000

5000

Forc

e, N

Displacement, mm

0 0.2 0.4 0.6 0.8 10

1000

2000

3000

4000

5000

Displacement, mm

Forc

e, N


0 0.2 0.4 0.6 0.8 10

1000

2000

3000

4000

5000

Displacement, mm

Forc

e, N


0 0.2 0.4 0.6 0.8 10

1000

2000

3000

4000

5000

Displacement, mm

Forc

e, N


0 0.2 0.4 0.6 0.8 10

1000

2000

3000

4000

5000

Displacement, mm

Forc

e, N


0.0A0.5A1.0A1.5A2.0A

Figure 4-6: Force-displacement processing for passive mount with (a) air, (b) rubber, (c) steel, and (d) aluminum inserts from a sinusoidal input of 1-Hz.

(d)

(c)

73

In summary, the force-displacement processing was completed for the gathered data

as designed in section 3.4. The techniques for processing equivalent damping and

dynamic stiffness using the force-displacement plotting methods were presented for each

mount at the 1 Hz frequency. In the MR fluid-elastic mounts, the dynamic stiffness was

found by averaging the loading and unloading cycles. The linear stiffness model,

however, does not accurately represent the unloading cycle. Therefore, a method for

determining the stiffness magnitude of the MR fluid-elastic mounts as well as the passive

mounts is presented next.

4.1.3 Force-Amplitude Analysis

Within this section, the force-amplitude processing techniques are discussed for extracting

the stiffness magnitude of the mounts. A brief analysis is then presented for each mount

and shown for testing at a frequency of 1 Hz.

As discussed in section 2.3.4, the force-amplitude processing can be used for later

frequency response modeling. Therefore, the force-amplitude method is used to model

both the time response of the force data F(t) and the input x(t). The input displacement

model is

0( ) sin( )x t X t Xω ρ= ⋅ ⋅ + + (4.5)

where X is the displacement amplitude, X0 is the static displacement offset and ρ is the

phase. The displacement model is required due to the resolution of the electromagnetic

actuator at increased frequency shortening the requested amplitude. The model for the

time response of the transmitted force is

0( ) | | sin( )F t F t Fω ϕ= ⋅ ⋅ + + (4.6)

where |F| is the force amplitude, F0 is the offset force, and φ is the phase.

Similar to the force-displacement method, the force data is standardized to remove

the force added by preload during initial setup. In the force model, an additional

saturation removal function

0[1 ( sin( ) )]2

SATSAT

sign X X t XF ω ρ+ + ⋅ ⋅ + −⎡ ⎤= ⎢ ⎥⎣ ⎦ (4.7)

74

where XSAT is the displacement value at the start of saturation, is used to remove the

saturation as the mount is completely unloaded. Therefore, the force-amplitude model for

the data is represented as

[ ] 00

[1 ( sin( ) )]( ) | | sin( )2

SATsign X X t XF t F t F ω ρω ϕ + + ⋅ ⋅ + −⎡ ⎤= ⋅ ⋅ + + ⋅ ⎢ ⎥⎣ ⎦ (4.8)

where |F| is the amplitude of the force, and X is the amplitude of the displacement. With

these amplitudes, a magnitude relationship which relates to all physical elements of the

mounts is used to characterize the mount. Therefore, the relationship for the stiffness

magnitude is

| |. . FS MX

= (4.9)

with the units of N/mm and is thus called the stiffness magnitude.

The force-amplitude method is then applied to the data as seen in the solid line in the

example plot of Figure 4-7. The force, which does not account for the saturation, extends

well beyond the empirical data. The saturation removal function, however, is able to

represent the empirical data. At high current settings the data was more difficult to model

as this example shows, but at lower currents the model was able to converge more readily

to the data.

75

0 0.5 1 1.5 2-5000

-4000

-3000

-2000

-1000

0

1000

2000

3000

4000

5000

Time, s

Forc

e, N

MRE 1,2.0A,1hzModel,F(t)No Saturation

Figure 4-7: Force-amplitude method analysis example for processing transmitted force data.

With the above mentioned analysis, the force model is then applied to the data using

the fit.m function in Matlab. All data acquired through the dynamic testing is processed

with this method for each mount. To remain consistent, the 1 Hz frequency testing is used

to illustrate this method for the following current settings of 0, 0.5, 1.0, 1.5, and 2.0 Amp.

Additionally, the displacement is not plotted since the fitting was trivial. The extracted

amplitude of the force and the amplitude of the displacement are later used to calculate the

stiffness magnitude as shown in equation (4.9). This stiffness magnitude, however, does

not represent the physical stiffness element from any of the tested mounts and instead

encompasses all of the physical elements in the mount.

After testing with the 1.0 mm amplitude displacement inputs, Figure 4-8a, b, c and d

show the force-amplitude processing for MR fluid-elastic mounts 1, 2, 3, and the empty

case 3B, respectively. The data is then represented by the thick line and the model is

shown as a thin, red line. The force-amplitude model is able to approximate the force data

XSAT

Loading

UnLoading

No Saturation

76

at the low current settings more readily than at the higher current settings. This is

apparent as the peak of the output declines with a non-sinusoidal slope with increased

current in MRE 1-3. As stated earlier, this unloading difference is likely due to the

ferrous particles in the MR fluid aggregating after being compressed from the loading

cycle. Nonetheless, the force-amplitude model shows a higher degree of coherence to the

data than the force-displacement model.

0 0.5 1 1.5 20

1000

2000

3000

4000

5000

Time, s

Forc

e, N

MRE 1,0.0A,1hzModel,F(t)

0 0.5 1 1.5 20

1000

2000

3000

4000

5000

Forc

e, N

Time, s

0 0.5 1 1.5 20

1000

2000

3000

4000

5000

Time, s

Forc

e, N


0 0.5 1 1.5 20

1000

2000

3000

4000

5000

Time, s

Forc

e, N


0 0.5 1 1.5 20

1000

2000

3000

4000

5000

Time, s

Forc

e, N


0 0.5 1 1.5 20

1000

2000

3000

4000

5000

Time, s

Forc

e, N


0.0A0.5A1.0A1.5A2.0A


(a)

77

0 0.5 1 1.5 20

1000

2000

3000

4000

5000

Time, s

Forc

e, N


0 0.5 1 1.5 20

1000

2000

3000

4000

5000

Forc

e, N

Time, s

0 0.5 1 1.5 20

1000

2000

3000

4000

5000

Time, s

Forc

e, N


0 0.5 1 1.5 20

1000

2000

3000

4000

5000

Time, s

Forc

e, N


0 0.5 1 1.5 20

1000

2000

3000

4000

5000

Time, s

Forc

e, N


0 0.5 1 1.5 20

1000

2000

3000

4000

5000

Time, s

Forc

e, N


0.0A0.5A1.0A1.5A2.0A

0 0.5 1 1.5 20

1000

2000

3000

4000

5000

Time, s

Forc

e, N


0 0.5 1 1.5 20

1000

2000

3000

4000

5000

Forc

e, N

Time, s

0 0.5 1 1.5 20

1000

2000

3000

4000

5000

Time, s

Forc

e, N


0 0.5 1 1.5 20

1000

2000

3000

4000

5000

Time, s

Forc

e, N


0 0.5 1 1.5 20

1000

2000

3000

4000

5000

Time, s

Forc

e, N


0 0.5 1 1.5 20

1000

2000

3000

4000

5000

Time, s

Forc

e, N


0.0A0.5A1.0A1.5A2.0A


(b)

(c)

78

0 0.5 1 1.5 20

1000

2000

3000

4000

5000

Time, s

Forc

e, N

MRE 3B,0.0A,1hzModel,F(t)

0 0.5 1 1.5 20

1000

2000

3000

4000

5000

Forc

e, N

Time, s

0 0.5 1 1.5 20

1000

2000

3000

4000

5000

Time, s

Forc

e, N


0 0.5 1 1.5 20

1000

2000

3000

4000

5000

Time, s

Forc

e, N


0 0.5 1 1.5 20

1000

2000

3000

4000

5000

Time, s

Forc

e, N


0 0.5 1 1.5 20

1000

2000

3000

4000

5000

Time, s

Forc

e, N


0.0A0.5A1.0A1.5A2.0A

Figure 4-8: Force-amplitude data processing and model for (a) MR fluid-elastic 1 with MRF-145, (b) MR fluid-elastic 2 with MRF-145, (c) MR fluid-elastic 3, and (d) MR fluid-elastic 3B with no fluid.

The elastic case mounts are processed using the force-amplitude method as shown in

Figure 4-9a, b, c, and d for the air, rubber, steel, and aluminum mounts. The force results

were generated with the sinusoidal input displacement of 0.5 mm for the soft core elastic

case mounts and 0.25 mm for the metal core elastic mounts. The force model is applied

and approximates the force results for the elastic case mounts without a problem. Upon

inspection, however, the force results from the metal insert mounts are similar to the solid

elastic case mount because the displacement amplitude had been reduced. The models are

collected in the southeast subplot for each current, but without any difference in

incremented current settings they are difficult to discern.

(d)

79

0 0.5 1 1.5 20

1000

2000

3000

4000

5000

Time, s

Forc

e, N

AIR,0.0A,1hzModel,F(t)

0 0.5 1 1.5 20

1000

2000

3000

4000

5000

Forc

e, N

Time, s

0 0.5 1 1.5 20

1000

2000

3000

4000

5000

Time, s

Forc

e, N


0 0.5 1 1.5 20

1000

2000

3000

4000

5000

Time, s

Forc

e, N


0 0.5 1 1.5 20

1000

2000

3000

4000

5000

Time, s

Forc

e, N


0 0.5 1 1.5 20

1000

2000

3000

4000

5000

Time, s

Forc

e, N


0.0A0.5A1.0A1.5A2.0A

0 0.5 1 1.5 20

1000

2000

3000

4000

5000

Time, s

Forc

e, N

RUB,0.0A,1hzModel,F(t)

0 0.5 1 1.5 20

1000

2000

3000

4000

5000

Forc

e, N

Time, s

0 0.5 1 1.5 20

1000

2000

3000

4000

5000

Time, s

Forc

e, N


0 0.5 1 1.5 20

1000

2000

3000

4000

5000

Time, s

Forc

e, N


0 0.5 1 1.5 20

1000

2000

3000

4000

5000

Time, s

Forc

e, N


0 0.5 1 1.5 20

1000

2000

3000

4000

5000

Time, s

Forc

e, N


0.0A0.5A1.0A1.5A2.0A


(a)

(b)

80

0 0.5 1 1.5 20

1000

2000

3000

4000

5000

Time, s

Forc

e, N

STE,0.0A,1hzModel,F(t)

0 0.5 1 1.5 20

1000

2000

3000

4000

5000

Forc

e, N

Time, s

0 0.5 1 1.5 20

1000

2000

3000

4000

5000

Time, s

Forc

e, N


0 0.5 1 1.5 20

1000

2000

3000

4000

5000

Time, s

Forc

e, N


0 0.5 1 1.5 20

1000

2000

3000

4000

5000

Time, s

Forc

e, N


0 0.5 1 1.5 20

1000

2000

3000

4000

5000

Time, s

Forc

e, N


0.0A0.5A1.0A1.5A2.0A

0 0.5 1 1.5 20

1000

2000

3000

4000

5000

Time, s

Forc

e, N

ALU,0.0A,1hzModel,F(t)

0 0.5 1 1.5 20

1000

2000

3000

4000

5000

Forc

e, N

Time, s

0 0.5 1 1.5 20

1000

2000

3000

4000

5000

Time, s

Forc

e, N


0 0.5 1 1.5 20

1000

2000

3000

4000

5000

Time, s

Forc

e, N


0 0.5 1 1.5 20

1000

2000

3000

4000

5000

Time, s

Forc

e, N


0 0.5 1 1.5 20

1000

2000

3000

4000

5000

Time, s

Forc

e, N


0.0A0.5A1.0A1.5A2.0A

Figure 4-9: Force-amplitude data processing and model for passive mounts with (a) air, (b) rubber, (c) steel, and (d) aluminum inserts.

In summary, all of the acquired dynamic data was processed using the force-

amplitude method. The MR fluid-elastic mounts at increased current settings showed less

(c)

(d)

81

convergence with the method, but for the most part adhered to the data. The elastic case

force model responded in accord to the data. Therefore, this method has proved to be a

useful processing tool in extracting the envelope of the mount dynamics.

4.1.4 Processing Analysis Method Evaluation

To determine whether or not to present the dynamic stiffness resuts or the stiffness

magnitude results the RMS-error for each method is calculated. Therefore, this section

compares the RMS-error for the force-displacement processing method and the force-

amplitude processing method.

As the processing progressed, the force-displacement method looked at the stiffness

of the force-displacement data in N/mm and the RMS-error is therefore in N/mm. To

remove the units for comparison, the error is normalized to a percentage. The force-

amplitude model, however, only extracts the magnitude of the force in N, and then the

amplitude of the displacement separately in mm. Therefore, the separated errors for the

force-amplitude method are combined in the error calculation

2 2| |/ ( ) ( )F X F t x tE e e= + (4.10)

where eF(t) is the normalized RMS-error in the force F(t), and ex(t) is the normalized RMS-

error in the displacement x(t).

After normalizing the RMS-error, a sample of the error for both methods is tabulated

in Table 4-2 under the RMS-error header. The remainder of the RMS-error comparison is

shown in Appendix B. The sample, however, consists of each mount within this study at

the 1 Hz test case for 0.0 Amps. Upon inspection, the error is typically lower in the force-

amplitude method. The stiffness magnitude from the force-amplitude method and the

stiffness from the force-displacement method are presented beneath the method header.

The unit values for both methods are relatively close which suggests that extracting the

stiffness magnitude approximates the dynamic stiffness results. The stiffness magnitude

results, however, are dependent on more than just the input displacement and therefore

cannot be considered equal to the dynamic stiffness. Additionally, the stiffness magnitude

can be fitted with a transfer function and further presented in a transmissibility ratio which

is the main reason for continuing with results obtained from the force-amplitude method.

82

Table 4-2: Comparative stiffness and RMS-Error obtained from force-time and force-displacement analysis.

To further illustrate the selection of the force-amplitude method for the stiffness

magnitude, the processing methods are compared graphically. This comparison uses

results from the 1 Hz case at an applied current of 0.0, 1.0, and 2.0 Amps. Results from

the metal-elastic case are shown in Figure 4-10 for MRE 1. Additional annotation is

added in each subplot which states both the normalized RMS-error and the stiffness

magnitude. For MRE 1, the error increases from 5% at 0 Amp to 13% at 2 Amp in the

force-amplitude method. Moreover, in the force-displacement method the error increases

from 5% to 18% from 0 to 2 Amp. Similar processing error was found in the remainder of

the fluid filled metal-elastic case mounts.

83

0 0.5 1 1.5 20

1000

2000

3000

4000

5000

Time, s

Forc

e, N

|F| /X=2806-N/mm,

rmse = 5%MRE 1,0.0A,1hzModel,F(t)

0 0.2 0.4 0.6 0.8 10

1000

2000

3000

4000

5000

Displacement, mm

Forc

e, N

K=2695-N/mm,

rmse = 5%


0 0.5 1 1.5 20

1000

2000

3000

4000

5000

Time, s

Forc

e, N

|F| /X=4192-N/mm,


0 0.2 0.4 0.6 0.8 10

1000

2000

3000

4000

5000

Displacement, mm

Forc

e, N

K=3268-N/mm,

rmse = 16%


0 0.5 1 1.5 20

1000

2000

3000

4000

5000

Time, s

Forc

e, N

|F| /X=5852-N/mm,


0 0.2 0.4 0.6 0.8 10

1000

2000

3000

4000

5000

Displacement, mm

Forc

e, N

K=3883-N/mm,

rmse = 18%


Figure 4-10: Processing method evaluation for MR Fluid-Elastic 1 with force-time method (left) and force-displacement method (right) from a sinusoidal input of 1 Hz.

After evaluating both methods, it was determined more accurate to use the processing

results from the force-amplitude analysis. This extracted stiffness magnitude represents

all the elements of the system and not just the actual stiffness. The actual mount stiffness,

however, would only be an average between the loading and unloading cycle from the

force-displacement analysis. Therefore, the force-displacement results are not pursued.

Without a counter comparison, however, the equivalent damping coefficient results are

used from the force-displacement method.

4.2 Mount Parametric Results

Using the frequency domain plots, as alluded to earlier, is the basis for the presentation of

the results. The stiffness magnitude, which encompasses all the elements in the mount,

and the equivalent damping are plotted in the frequency domain [47]. This section first

focuses on each metal-elastic case mount results and is then followed by the elastic case

84

mount results. The final section makes a comparison at current settings of 0, 1.0, and 2.0

Amps for all the mounts of this study.

4.2.1 MR fluid- Elastic Mount Parameters

This section is limited to the presentation of the stiffness magnitude results and the

equivalent damping results for the individual mounts tested. The main objective here is to

characterize the stiffness magnitude gains caused by increasing the current supplied to the

coil. Additionally, the equivalent damping is also presented for each mount as the current

supply to the coil is increased.

With the force-amplitude processing completed, a consistent analysis of the results is

undertaken and the stiffness magnitude is plotted in the frequency domain from 0-35 Hz

for each mount. The quasi-static values from earlier processing are also included in the

stiffness magnitude and used to represent the stiffness magnitude at 0 Hz. Additionally,

the range for the stiffness magnitude is 0-10,000 N/mm, while the range for the equivalent

damping coefficient is 0-165 Ns/mm. Remembering that the metal-elastic case was tested

at 0.25 Amp increments, there are nine current settings with unique marker and line

combinations as depicted in the legend. Five current settings then exist for the elastic case

results.

For determining the increase in stiffness magnitude due to an applied magnetic field,

a stiffness magnitude evaluation quotient is used. The stiffness evaluation quotient is

calculated as

35 35

00 0

. . 35

00

1 1| | / | | /

1 | | /

f fAmp Ampf f

S M

f Ampf

F X F XN N

UF X

N

= =

=

⎡ ⎤ ⎡ ⎤−⎣ ⎦ ⎣ ⎦=

⎡ ⎤⎣ ⎦

∑ ∑

∑ (4.11)

where |F|/X is the stiffness magnitude, f is the frequency, and Amp is the current setting.

This evaluation is similar to an output gain produced by increased magnetic field, but the

stiffness magnitude results are summed and averaged across the range of frequencies.

Therefore, equation (4.11) takes into account the envelope of the stiffness magnitude over

all frequencies when altered by an applied current. Additionally, the equivalent damping

evaluation quotient is calculated in the same manner as equation (4.11) by replacing the

85

stiffness magnitude with the equivalent damping values. Using the evaluation quotients, a

nominal gain in the measured result can be associated to the applied current.

The stiffness magnitude for MR fluid-elastic 1 is presented in Figure 4-11a. The

obvious result is a drastic increase in stiffness magnitude or |F|/X with applied current.

More notable is that the increase in stiffness is steadily increasing per current setting

which suggests the fluid could tolerate a higher level of magnetic flux density. The

stiffness evaluation quotient increased 78% at a 2 Amp current setting in MRE 1 above

the 0 Amp current setting. Concurrently, the equivalent damping for MRE 1 is plotted in

Figure 4-11b. The damping also increased with added current, but on a larger scale than

the stiffness magnitude. The result from 2 Amp current is more than 500% in the

equivalent damping evaluation quotient which shows that the mount has a high capacity

for damping. The damping, however, occurs mainly at low frequency inputs.

Furthermore, the trend of the damping is exponentially decaying with increased

frequency.

0 5 10 15 20 25 30 350

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

Frequency, Hz

Stif

fnes

s M

agni

tude

, N/m

m

MRE 1,|F|/X,0.00-AMRE 1,|F|/X,0.25-AMRE 1,|F|/X,0.50-AMRE 1,|F|/X,0.75-AMRE 1,|F|/X,1.00-AMRE 1,|F|/X,1.25-AMRE 1,|F|/X,1.50-AMRE 1,|F|/X,1.75-AMRE 1,|F|/X,2.00-A

Figure 4-11: (continue) (a)

86

0 5 10 15 20 25 30 350

20

40

60

80

100

120

140

160

Frequency, Hz

Dam

ping

, Ns/

mm

MRE 1,Ceq,0.00-AMRE 1,Ceq,0.25-AMRE 1,Ceq,0.50-AMRE 1,Ceq,0.75-AMRE 1,Ceq,1.00-AMRE 1,Ceq,1.25-AMRE 1,Ceq,1.50-AMRE 1,Ceq,1.75-AMRE 1,Ceq,2.00-A

Figure 4-11: MR fluid-elastic 1 mount (MRF-145) (a) stiffness |F|/X, and (b) damping Ceq results obtained from analysis.

The stiffness magnitude for MR fluid-elastic 2 is presented in Figure 4-12a. The

stiffness evaluation quotient for MRE 2 increases by 57% when the current is set to 2

Amps. The equivalent damping for MRE 2 is plotted in Figure 4-12b. The quotient

increase in the damping for MRE 2 is also significant at 430% when the current is set to 2

Amps. Therefore, the same statements can be made about this mount as were made for

MRE 1.

(b)

87

0 5 10 15 20 25 30 350

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

Frequency, Hz

Stif

fnes

s M

agni

tude

, N/m

m


0 5 10 15 20 25 30 350

20

40

60

80

100

120

140

160

Frequency, Hz

Dam

ping

, Ns/

mm



Moving to the last of the fluid filled metal-elastics, the stiffness magnitude and

damping for MR fluid-elastic 3 is presented in Figure 4-13a and b, respectively. The

(a)

(b)

88

stiffness magnitude for MRE 3, as expected, was less than the other two fluid filled

mounts and had a stiffness evaluation quotient of 46%. As disappointing as this may be in

regards to the other MR fluid-elastic mounts, the stiffness is still very appreciable. The

damping evaluation quotient in this mount increased to 170% as the current was set to 2.0

Amp. As mentioned earlier in the quasi-static results, MRE 3 may have been less efficient

due to a higher profile of primer and polyurethane on the face of the pole plate.

0 5 10 15 20 25 30 350

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

Frequency, Hz

Stif

fnes

s M

agni

tude

, N/m

m



(a)

89

0 5 10 15 20 25 30 350

20

40

60

80

100

120

140

160

Frequency, Hz

Dam

ping

, Ns/

mm



The last metal-elastic case mount up for discussion is MRE 3B. With no fluid, this

mount preformed passively as seen in Figure 4-14. Therefore, the only conclusions to be

made are that the metal-elastic case has no significant impacts on the mount during

magnetic activation. The case design itself, however, contributes 10 Ns/mm of damping

at 1 Hz and approximately 700 N/mm of stiffness magnitude at all frequencies to the

results of MR fluid-elastic 3 filled with fluid.

(b)

90

0 5 10 15 20 25 30 350

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

Frequency, Hz

Stif

fnes

s M

agni

tude

, N/m

m

MRE 3B,|F|/X,0.00-AMRE 3B,|F|/X,0.25-AMRE 3B,|F|/X,0.50-AMRE 3B,|F|/X,0.75-AMRE 3B,|F|/X,1.00-AMRE 3B,|F|/X,1.25-AMRE 3B,|F|/X,1.50-AMRE 3B,|F|/X,1.75-AMRE 3B,|F|/X,2.00-A

0 5 10 15 20 25 30 350

20

40

60

80

100

120

140

160

Frequency, Hz

Dam

ping

, Ns/

mm

MRE 3B,Ceq,0.00-AMRE 3B,Ceq,0.25-AMRE 3B,Ceq,0.50-AMRE 3B,Ceq,0.75-AMRE 3B,Ceq,1.00-AMRE 3B,Ceq,1.25-AMRE 3B,Ceq,1.50-AMRE 3B,Ceq,1.75-AMRE 3B,Ceq,2.00-A

Figure 4-14: Blank metal-elastic case MRE 3B (a) stiffness |F|/X, and (b) damping Ceq results obtained from analysis.

(b)

(a)

91

For further presentation, the stiffness magnitude results from the force-amplitude

analysis are tabulated in Table 4-3. These results are limited to the metal-elastic case and

show MR fluid-elastic 1, MR fluid-elastic 2, MR fluid-elastic 3, and the empty metal-

elastic case (MRE 3B). All nine current settings are represented in rows and the stiffness

magnitude value is beneath the associated frequency. MRE 1 stiffness magnitude values

are marginally lower than MRE 2. The stiffness magnitude values do decrease at higher

frequency and this primarily the result of the damping element decreasing in the MR fluid

as discussed next.

Table 4-3: Stiffness magnitude of metal-elastic case mounts at all current settings.

92

In addition to the stiffness magnitude, the equivalent damping coefficient is tabulated

in Table 4-4. Upon inspection of the table, the subsequent rows represent the applied

current and the columns represent the harmonic input frequency. The most damping

occurs at 1 Hz for each mount and the values are very similar for MR fluid-elastic 1 and

MR fluid-elastic 2. The large drop in damping at high frequency suggests that the loading

and unloading cycles are converging. This convergence may indicate that the ferrous

particles in the MR fluid are no longer being repositioned into columnar structures and

that the displacement input is being transferred primarily through elastic casing of the

mount. Therefore, the reduction in damping impacts the overall stiffness magnitude of the

MR fluid-elastic mounts.

Table 4-4: Equivalent damping in metal-elastic case mounts at all currents.

93

In summary, MR fluid-elastic 1 and MR fluid-elastic 2 are very similar. The

stiffness evaluation quotient for MRE 1 was 78% and the damping evaluation quotient

was 500%. The stiffness evaluation quotient for MRE 2 was 57% and the damping

evaluation quotient was 430%. The stiffness evaluation quotient for MR fluid-elastic 3

was 46% and the damping evaluation quotient was 170%. Therefore, each fluid filled MR

fluid-elastic mount showed large scale increases in stiffness magnitude values. The

equivalent damping values, however, decayed as the input frequency increased. This

decay was seen in the stiffness magnitude values. Nonetheless, the results are conclusive

that an applied magnetic field to the MR fluid is able to change the stiffness magnitude in

this mount configuration.

4.2.2 Passive Elastic Parameters

This section is directed to comparing the elastic case mount. The results are utilized from

the force-displacement method for the equivalent damping and from the force-amplitude

method for the stiffness magnitude.

The stiffness magnitude for the elastic mount with air insert is plotted in Figure

4-15a. The frequency for the air mount was only tested at 1-10, 20, and 30 Hz.

Nonetheless, this empty elastic cavity shows that no impending force is added with the

magnetic field. Additionally, the equivalent damping plotted in Figure 4-15b does not

show any change with the applied field. Of course, this style mount has a high reluctance

for the magnetic flux density to pass and is relatively compliant with a stiffness of 500

N/mm and the damping maximum of 4 Ns/mm. Moreover, the elastic air insert mount’s

average stiffness is only 200 N/mm softer than the metal-elastic case mount.

94

0 5 10 15 20 25 30 350

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

Frequency, Hz

Stif

fnes

s M

agni

tude

, N/m

m

AIR,|F|/X,0.00-AAIR,|F|/X,0.50-AAIR,|F|/X,1.00-AAIR,|F|/X,1.50-AAIR,|F|/X,2.00-A

0 5 10 15 20 25 30 350

20

40

60

80

100

120

140

160

Frequency, Hz

Dam

ping

, Ns/

mm

AIR,Ceq,0.00-AAIR,Ceq,0.50-AAIR,Ceq,1.00-AAIR,Ceq,1.50-AAIR,Ceq,2.00-A

Figure 4-15: Passive mount with air insert (a) stiffness |F|/X, and (b) damping Ceq results obtained from analysis.

(b)

(a)

95

The solid rubber mount results are plotted in Figure 4-16. The stiffness for RUB, as

in Figure 4-16a, remained consistent around 2300 N/mm. The damping showed a slight

decay from 20 Ns/mm as seen in Figure 4-16b. As mentioned earlier, this mount is

representative of a bottom boundary for the comparative study with the metal-elastic case

mounts. A slightly higher durometer elastomer, however, would make for a better

comparison to the off-state MR fluid-elastic mounts.

0 5 10 15 20 25 30 350

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

Frequency, Hz

Stif

fnes

s M

agni

tude

, N/m

m

RUB,|F|/X,0.00-ARUB,|F|/X,0.50-ARUB,|F|/X,1.00-ARUB,|F|/X,1.50-ARUB,|F|/X,2.00-A


96

0 5 10 15 20 25 30 350

20

40

60

80

100

120

140

160

Frequency, Hz

Dam

ping

, Ns/

mm

RUB,Ceq,0.00-ARUB,Ceq,0.50-ARUB,Ceq,1.00-ARUB,Ceq,1.50-ARUB,Ceq,2.00-A

Figure 4-16: Passive mount with 30 D rubber insert (a) stiffness |F|/X, and (b) damping Ceq results obtained from analysis.

As you may recall, the steel mount with a 1018-steel insert was built to see if an

effect from the material property would be present in the magnetic field. Figure 4-17a

plots the stiffness magnitude and does not show any appreciable change due to the applied

magnetic field. The stiffness does vary slightly, but stays near 4900 N/mm. The damping

as seen in Figure 4-17b is not altered by the magnetic field and exhibits an exponential

decay which starts around 60 N/mm. Therefore, the use of a steel insert can only be used

as an upper bound in comparison to the MR fluid-elastic mounts and that magnetic flux

does not alter stiffness.

(b)

97

0 5 10 15 20 25 30 350

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

Frequency, Hz

Stif

fnes

s M

agni

tude

, N/m

m

STE,|F|/X,0.00-ASTE,|F|/X,0.50-ASTE,|F|/X,1.00-ASTE,|F|/X,1.50-ASTE,|F|/X,2.00-A

0 5 10 15 20 25 30 350

20

40

60

80

100

120

140

160

Frequency, Hz

Dam

ping

, Ns/

mm

STE,Ceq,0.00-ASTE,Ceq,0.50-ASTE,Ceq,1.00-ASTE,Ceq,1.50-ASTE,Ceq,2.00-A

Figure 4-17: Passive mount with 1018 steel insert (a) stiffness |F|/X, and (b) damping Ceq results obtained from analysis.

(b)

(a)

98

Moreover, the aluminum insert mount, which is non-magnetic, was built to counter

the assertion that a magnetic insert material property would have a presence in the

magnetic field. The stiffness magnitude plotted in Figure 4-18a for the aluminum insert

mount, however, shows that the aluminum mount is very similar to the STE mount with a

stiffness of approximately 5000 N/mm. Additionally, the damping is plotted in Figure

4-18b and has an exponential decay trend that starts around 60 Ns/mm. Therefore, the

behavior of either the ALU or STE mount can be used as an upper bound for the MRE

mounts.

0 5 10 15 20 25 30 350

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

Frequency, Hz

Stif

fnes

s M

agni

tude

, N/m

m

ALU,|F|/X,0.00-AALU,|F|/X,0.50-AALU,|F|/X,1.00-AALU,|F|/X,1.50-AALU,|F|/X,2.00-A


99

0 5 10 15 20 25 30 350

20

40

60

80

100

120

140

160

Frequency, Hz

Dam

ping

, Ns/

mm

ALU,Ceq,0.00-AALU,Ceq,0.50-AALU,Ceq,1.00-AALU,Ceq,1.50-AALU,Ceq,2.00-A

Figure 4-18: Passive mount with 6061 aluminum insert (a) stiffness |F|/X, and (b) damping Ceq results obtained from analysis.

With the passive isolators plotted, the stiffness results are then tabulated for all

currents in Table 4-5. At most, the elastic material can be shown to have a stiffness of

approximately 2000 N/mm for the rubber mount. With a solid insert, the elastic

compressive strength is greatly affected and more than doubles to approximately 5000

N/mm. Therefore, the aluminum or steel mount can be set as the upper bound for the

overall stiffness comparisons. On the other hand, the rubber mount can be used as a lower

bound for the overall stiffness comparisons.

(b)

100

Table 4-5: Stiffness magnitude results for passive elastic case mounts air, rubber, steel and aluminum at all current settings.

The damping is presented in Table 4-6 for the passive mounts for each current setting.

The rubber mount has a damping value of approximately 20 Ns/mm at the 1 Hz frequency

which is similar in value to the off-state MR fluid metal-elastic case. The damping in the

empty elastic case was lower than the empty metal-elastic case at approximately 4 Ns/mm

at 1 Hz. With the steel metal insert, the damping achieved a high of 63.6 Ns/mm at 1 Hz,

but remained comparable to the aluminum mount.

101

Table 4-6: Equivalent damping results for passive elastic case mount air, rubber, steel and aluminum at all current settings.

In summary, the passive isolator results have been presented for both stiffness

magnitude and equivalent damping. The stiffness magnitude of the rubber mount was

shown to make a good candidate for the lower bound in the stiffness comparisons to

follow. Additionally, the steel and aluminum insert mount had a stiffness of

approximately 5000 N/mm and can be used as an upper bound in the comparative study.

102

4.2.3 Discrete Comparison of Stiffness Magnitude

This section compares the stiffness magnitude of the MR fluid-elastic mount to each

passive mount. After the stiffness magnitude comparisons, the extracted damping for the

MR fluid-elastic mount is compared to the damping of the passive mounts. Since the

stiffness magnitude takes into account all the dynamic elements of the mount, these

comparisons are not representing the actual stiffness element of the mount.

The first comparison looks at the empty metal-elastic case and the empty elastic case

as seen in Figure 4-19a and b with the current at 0 and 2 Amps. The metal-elastic case

has a similar stiffness magnitude to the elastic case which suggests the sidewall of both

cases have similar attributes. Moreover, a MR fluid-elastic mount almost triples the

stiffness magnitude of the metal-elastic case as seen in Figure 4-20a at 0 Amps. After the

current is increased to 2 Amps, the MR fluid-elastic mount stiffness magnitude has

increased substantially compared to the empty metal-elastic case as seen in Figure 4-20b.

Next, the MR fluid-elastic mount stiffness magnitude is compared to the solid elastic

case mount in Figure 4-21a and b. This comparison shows the MR fluid-elastic mount to

have a similar result to the passive solid elastic case mount when the current supply is

zero. Activating the coil with 2 Amps of current, however, dramatically increases the

stiffness magnitude of the MR fluid-elastic mount and is no longer comparable to the

passive solid elastic mount. Keeping the coil energized with 2 Amps of current, the MR

fluid-elastic mount does become comparable to an elastic casing with a metal insert as

seen in Figure 4-22b or Figure 4-23b. Therefore, the stiffness magnitude of the metal-

elastic case shows a large MR effect when filled with MRF-145 fluid and activated over 2

Amps. This activation allows the MR fluid-elastic mount to have a broad range of

stiffness magnitudes as seen in the stiffness magnitude figures.

With the comparison of the stiffness magnitude completed for an MR fluid-elastic

mount, the damping is compared. In Figure 4-24, the damping for the empty metal-elastic

case is compared to the damping in the empty elastic case. The damping in the empty

cases is not altered by an applied magnetic field. The damping in both casing styles is

similar. The damping is slightly increased by adding the pole plates to the mount.

Moreover, adding MRF-145 fluid to the metal-elastic case increases the damping at 1 Hz

from 9 Ns/mm to 26 Ns/mm as seen in Figure 4-25a.

103

Furthermore, the equivalent damping values at high frequency suddenly drop which is

most likely due to the fast displacement input bypassing the fluid cavity and going

through the sidewalls of the elastic region of the case. For clarity, as the MR fluid is

placed in squeeze mode, the agglomeration of the ferrous iron particles at low frequency

causes the large hysteresis in the unloading cycle. This unloading may be thought of as

pulling on a loose column of the ferrous iron particles. With increased frequency, the iron

particles are not restored into a respective column in the applied magnetic field.

Therefore, the loading cycle does not compress on the iron particles in the fluid, which

becomes similar to the unloading cycle and less energy is dissipated by the MR fluid.

Next, the damping of the MR fluid-elastic mount is compared to the damping of the

solid elastic case as seen in Figure 4-26a and b. At zero current the MR fluid-elastic

mount has a damping value similar to the solid elastic case, but when the current is

increased to 2 Amps there is no similarity. The MR fluid-elastic mount has more than

twice the damping of the elastic mounts with metal inserts when the coil is supplied with 2

Amps of current as seen in Figure 4-27b and Figure 4-28b. Therefore, the MR fluid-

elastic mount has a large capacity for damping at low frequency which ranges from 30-

160 Ns/mm with an applied magnetic field. An additional benefit is that damping decays

at high frequency which would make the MR fluid-elastic mount suitable as an absorber

where low damping is desired.

104

0 5 10 15 20 25 30 35

1000

2000

3000

4000

5000

6000

Frequency, Hz

Stif

fnes

s M

agni

tude

, N/m

m

MRE 3B,|F|/X,0.0AmpAIR,|F|/X,0.0Amp

0 5 10 15 20 25 30 35

1000

2000

3000

4000

5000

6000

Frequency, Hz

Stif

fnes

s M

agni

tude

, N/m

m

MRE 3B,|F|/X,2.0AmpAIR,|F|/X,2.0Amp

Figure 4-19: Comparing stiffness magnitude of a metal-elastic case (MRE 3B) to an elastic case (AIR) mount at (a) 0 Amps and (b) 2 Amps of current.

(a)

(b)

105

0 5 10 15 20 25 30 35

1000

2000

3000

4000

5000

6000

Frequency, Hz

Stif

fnes

s M

agni

tude

, N/m

m

MRE 1,|F|/X,0.0AmpMRE 3B,|F|/X,0.0Amp

0 5 10 15 20 25 30 35

1000

2000

3000

4000

5000

6000

Frequency, Hz

Stif

fnes

s M

agni

tude

, N/m

m

MRE 1,|F|/X,2.0AmpMRE 3B,|F|/X,2.0Amp

Figure 4-20: Comparing stiffness magnitude of a metal-elastic case (MRE 3B) to a MR fluid-elastic mount at (a) 0 Amps and (b) 2 Amps of current.

(a)

(b)

106

0 5 10 15 20 25 30 35

1000

2000

3000

4000

5000

6000

Frequency, Hz

Stif

fnes

s M

agni

tude

, N/m

m

MRE 1,|F|/X,0.0AmpRUB,|F|/X,0.0Amp

0 5 10 15 20 25 30 35

1000

2000

3000

4000

5000

6000

Frequency, Hz

Stif

fnes

s M

agni

tude

, N/m

m

MRE 1,|F|/X,2.0AmpRUB,|F|/X,2.0Amp

Figure 4-21: Comparing stiffness magnitude of a solid elastic case (RUB) to a MR fluid-elastic mount at (a) 0 Amps and (b) 2 Amps of current.

(a)

(b)

107

0 5 10 15 20 25 30 35

1000

2000

3000

4000

5000

6000

Frequency, Hz

Stif

fnes

s M

agni

tude

, N/m

m

MRE 1,|F|/X,0.0AmpSTE,|F|/X,0.0Amp

0 5 10 15 20 25 30 35

1000

2000

3000

4000

5000

6000

Frequency, Hz

Stif

fnes

s M

agni

tude

, N/m

m

MRE 1,|F|/X,2.0AmpSTE,|F|/X,2.0Amp

Figure 4-22: Comparing stiffness magnitude of an elastic case with steel insert (STE) to a MR fluid-elastic mount at (a) 0 Amps and (b) 2 Amps of current.

(a)

(b)

108

0 5 10 15 20 25 30 35

1000

2000

3000

4000

5000

6000

Frequency, Hz

Stif

fnes

s M

agni

tude

, N/m

m

MRE 1,|F|/X,0.0AmpALU,|F|/X,0.0Amp

0 5 10 15 20 25 30 35

1000

2000

3000

4000

5000

6000

Frequency, Hz

Stif

fnes

s M

agni

tude

, N/m

m

MRE 1,|F|/X,2.0AmpALU,|F|/X,2.0Amp

Figure 4-23: Comparing stiffness magnitude of an elastic case with aluminum insert (ALU) to a MR fluid-elastic mount at (a) 0 Amps and (b) 2 Amps of current.

(a)

(b)

109

0 5 10 15 20 25 30 350

20

40

60

80

100

120

140

160

Frequency, Hz

Dam

ping

, Ns/

mm

MRE 3B,Ceq,0.00-AAIR,Ceq,0.00-A

0 5 10 15 20 25 30 350

20

40

60

80

100

120

140

160

Frequency, Hz

Dam

ping

, Ns/

mm

MRE 3B,Ceq,2.00-AAIR,Ceq,2.00-A

Figure 4-24: Comparing damping of a metal-elastic case (MRE 3B) to an elastic case (AIR) mount at (a) 0 Amps and (b) 2 Amps of current.

(a)

(b)

110

0 5 10 15 20 25 30 350

20

40

60

80

100

120

140

160

Frequency, Hz

Dam

ping

, Ns/

mm

MRE 1,Ceq,0.00-AMRE 3B,Ceq,0.00-A

0 5 10 15 20 25 30 350

20

40

60

80

100

120

140

160

Frequency, Hz

Dam

ping

, Ns/

mm

MRE 1,Ceq,2.00-AMRE 3B,Ceq,2.00-A

Figure 4-25: Comparing damping of a metal-elastic case (MRE 3B) to a MR fluid-elastic mount at (a) 0 Amps and (b) 2 Amps of current.

(a)

(b)

111

0 5 10 15 20 25 30 350

20

40

60

80

100

120

140

160

Frequency, Hz

Dam

ping

, Ns/

mm

MRE 1,Ceq,0.00-ARUB,Ceq,0.00-A

0 5 10 15 20 25 30 350

20

40

60

80

100

120

140

160

Frequency, Hz

Dam

ping

, Ns/

mm

MRE 1,Ceq,2.00-ARUB,Ceq,2.00-A

Figure 4-26: Comparing damping of a solid elastic case (RUB) to a MR fluid-elastic mount at (a) 0 Amps and (b) 2 Amps of current.

(a)

(b)

112

0 5 10 15 20 25 30 350

20

40

60

80

100

120

140

160

Frequency, Hz

Dam

ping

, Ns/

mm

MRE 1,Ceq,0.00-ASTE,Ceq,0.00-A

0 5 10 15 20 25 30 350

20

40

60

80

100

120

140

160

Frequency, Hz

Dam

ping

, Ns/

mm

MRE 1,Ceq,2.00-ASTE,Ceq,2.00-A

Figure 4-27: Comparing damping of an elastic case with steel insert (STE) to a MR fluid-elastic mount at (a) 0 Amps and (b) 2 Amps of current.

(a)

(b)

113

0 5 10 15 20 25 30 350

20

40

60

80

100

120

140

160

Frequency, Hz

Dam

ping

, Ns/

mm

MRE 1,Ceq,0.00-AALU,Ceq,0.00-A

0 5 10 15 20 25 30 350

20

40

60

80

100

120

140

160

Frequency, Hz

Dam

ping

, Ns/

mm

MRE 1,Ceq,2.00-AALU,Ceq,2.00-A

Figure 4-28: Comparing damping of an elastic case with aluminum insert (ALU) to a MR fluid-elastic mount at (a) 0 Amps and (b) 2 Amps of current.

(a)

(b)

114

4.2.4 Mount Comparison

As this may be questioned, comparing the two styles of mount casing is inconsistent but

does shed light on the overall impact of activating the MR fluid within the metal-elastic

case. The main goal of this comparative study, however, is to organize the MR fluid-

elastic mounts for likeliness with passive mounts. Therefore, this section presents a

comparison of the elastic and metal-elastic case mounts and provides a further qualitative

analysis of the significance of activating MR fluid within the metal-elastic casing.

Continuing from the discrete comparisons, the metal-elastic case mount results are

plotted with the elastic case mounts at 0.0 Amp in Figure 4-29a and b for stiffness and

damping, respectively. At 0.0 Amp the MRE mounts all have a similar profile with MRE

2 showing the most stiffness (left). The damping as seen in Figure 4-29b is quite the

opposite since MRE 3 has higher damping than MRE 1 and 2 at 0.0 Amp.

0 5 10 15 20 25 30 350

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

Frequency, Hz

Stif

fnes

s M

agni

tude

, N/m

m

MRE 1,|F|/X,0.0AmpMRE 2,|F|/X,0.0AmpMRE 3,|F|/X,0.0AmpMRE 3B,|F|/X,0.0AmpAIR,|F|/X,0.0AmpRUB,|F|/X,0.0AmpSTE,|F|/X,0.0AmpALU,|F|/X,0.0Amp


(a)

115

0 5 10 15 20 25 30 350

20

40

60

80

100

120

140

160

Frequency, Hz

Dam

ping

, Ns/

mm

MRE 1,Ceq,0.00-AMRE 2,Ceq,0.00-AMRE 3,Ceq,0.00-AMRE 3B,Ceq,0.00-AAIR,Ceq,0.00-ARUB,Ceq,0.00-ASTE,Ceq,0.00-AALU,Ceq,0.00-A

Figure 4-29: Comparative (a) stiffness |F|/X, and (b) damping Ceq results obtained at 0.00-Amps from force-amplitude and force-displacement analysis, respectively.

Previously, the stiffness magnitude of MRE 3B in Figure 4-29a, an empty metal-

elastic case is relatively similar to the empty elastic case of AIR and provides some

consistency for the cross-comparative study of the casings. More still, the MR fluid-

elastic mounts at the 0.0 Amp setting are close to a solid rubber mount in both stiffness

magnitude and damping. So, with no magnetic field intensity, the behavior of the MR

fluid-elastic mount is comparable with that of the rubber elastic mount. The significance

here is that in application the MR fluid-elastic mount would by very similar to a solid

rubber mount given that the cofigurations were consistent. A higher durometer rating for

the rubber, however, would have more similarity.

As the current is increased to 1.00 Amp, the mount stiffness and damping results are

then configured in Figure 4-30a and b, respectively. By this current, the damping of the

MR fluid-elastic mounts has increased significantly, but almost at the same rate. Within

the stiffness magnitude range, MRE 1 and MRE 2 have started to leave MRE 3 behind

(b)

116

showing that the three mounts have slight variation in stiffness build. As a whole, MRE

1 and 2 have started to approach the stiffness magnitude of the solid metal insert mounts.

0 5 10 15 20 25 30 350

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

Frequency, Hz

Stif

fnes

s M

agni

tude

, N/m

m


0 5 10 15 20 25 30 350

20

40

60

80

100

120

140

160

Frequency, Hz

Dam

ping

, Ns/

mm



(b)

(a)

117

Presenting the final current of 2.0 Amp, the mount stiffness magnitude and damping

are plotted in Figure 4-31a and b, respectively. At lower frequency, MR fluid-elastic 1

and MR fluid-elastic 2 have increased in stiffness magnitude at nearly the same rate. The

damping increase for MRE 1 and MRE 2 was also similar. As for the boundary, MRE 1

and MRE 2 produced more stiffness magnitude than the solid metal insert in the elastic

case mount at low frequency. The significance of this increase, qualitatively, is that fluid

region has been energized to practically a solid. With increased frequency the MR fluid-

elastic mount has become compliant as noted by the continued drop in stiffness

magnitude. Thus with the dial of current supply, an MR fluid-elastic mount is as rigid as

an elastic mount with a steel insert at low frequency. This adjustability signifies that an

MR fluid-elastic mount has a large range of both stiffness and damping characteristics

which would be desirable in many oscillatory devices that operate at various speeds.

0 5 10 15 20 25 30 350

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

Frequency, Hz

Stif

fnes

s M

agni

tude

, N/m

m



(a)

118

0 5 10 15 20 25 30 350

20

40

60

80

100

120

140

160

Frequency, Hz

Dam

ping

, Ns/

mm



Further mount comparison is implemented by tabulating the stiffness magnitude in

Table 4-7. Herein, this table shows the stiffness magnitude results from 0.00, 1.00, and

2.00 Amp current supply. MR fluid-elastic 1 and MR fluid-elastic 3 have similar

stiffness magnitude values at 0.00 Amp with MR fluid-elastic 2 having the higher stiffness

magnitude. At 1.00 and 2.00 Amp, however, MRE 1 and MRE 2 have a comparable

stiffness magnitude. The variation between MRE 3 and the other MR fluid-elastic mounts

shows that a fabrication error resulted and caused the discrepancy. Nonetheless, the

mount and magnetic system design has accomplished a priority objective in light that a

tunable dynamic stiffness characteristic is present as a function of the magnetic field

intensity. Additionally, the MRE’s show controllable behavior and symmetry to the

applied magnetic field in stiffness magnitude with incremental current.

(b)

119

Table 4-7: Stiffness magnitude comparison for MR fluid-elastic and passive mounts at settings of 0.00, 1.00, 2.00 Amp

For a final comparison, the damping from the force-displacement analysis is

presented in Table 4-8. The aluminum and steel insert mounts have approximately the

same damping across the board. Initially, MR fluid-elastic 1 and MR fluid-elastic 2 have

comparable damping which is slightly higher than a solid rubber mount. As the current is

increased, MRE 1 and MRE 2 show very high damping at just 1 Amp in comparison to

the solid metal insert mounts. This increased damping suggests that an MRE mount has a

tunable damping element at low frequency. At high frequency, however, the damping

decays. These damping features would allow the MR fluid-elastic mount to be a useful

isolator across a wide frequency band.

120

Table 4-8: Equivalent damping comparison for MR fluid-elastic and passive mounts at settings of 0.0, 1.0, and 2.0 Amp

In summary, the fabrication, and testing of MR fluid-elastic 1 and MR fluid-elastic 2

has shown that repeatability within results can be achieved. Additionally, all the MR

fluid-elastic mounts have shown stiffness magnitude and damping controllability with the

applied magnetic field. A comparison between the stiffness magnitudes showed that at

0.0 Amp an MRE behaves like a solid rubber mount and at 2.0 Amps an MRE behaves

like a solid metal insert mount. This validated the mounts adaptability across a lower and

upper boundary with the turn of a current dial. Furthermore, the damping or the stiffness

magnitude of the MRE’s has consistent controllability making this mount and magnetic

system design useful to a broad range of disturbance inputs.

121

4.3 Discussions

In this chapter, a parametric analysis and comparison was undertaken. The methodology

used to process the quasi-static results was derived from a linear force-displacement

plotting method. This method was then applied to the dynamic testing data to extract the

stiffness magnitude. Furthermore, the force-displacement method provided the energy

dissipated which was used to determine an equivalent damping coefficient.

After using the force-displacement method, an amplitude method was employed to

extract the magnitude of the force and displacement. This force and displacement was

then converted into a stiffness magnitude. Upon evaluation of both methods for collecting

the stiffness and stiffness magnitude, the force-amplitude method was found to have less

error than the force-displacement method. Therefore, the stiffness magnitude from the

force-amplitude method was plotted in the frequency domain for presenting the results.

Within the results section, each mount was then analyzed in an independent

presentation. MR fluid-elastic 1, 2, and 3 mounts from 0 to 2.0 Amp showed significant

stiffness magnitude increases in the evaluation quotient at 78%, 57% and 46%,

respectively. Additionally, the equivalent damping of the MR fluid-elastic case mounts

showed even greater increases in the damping evaluation quotient at 500%, 430%, and

170%. Therefore, the results proved the validity of the design and the magnetic circuitry.

During the discrete comparisons, the MR fluid-elastic mount damping values were

explained in detail. This explanation of damping suggested that at higher frequencies the

MR fluid was being bypassed due to the agglomeration of the ferrous iron particles in the

fluid and that the loading cycle and unloading cycle had more similarity. Moreover at

high frequency, the energy being dissipated was less dependent on the ferrous iron

particles in the MR fluid and more dependent on the elastic sidewall of the mount.

For understanding the range of the stiffness magnitude in an MR fluid-elastic mount,

four passive isolator results were compared to the MR fluid-elastic mounts. This

comparison shed light on how the stiffness being altered in the MRE can be related to

different passive mounts. The results processed from the passive isolators weighted the

aluminum and steel insert mount as an upper boundary, and the rubber mount was used as

a lower boundary. At 0.0 Amp, the MR fluid-elastic mounts had a stiffness magnitude

comparable to the rubber mount which has a solid construction of polyurethane. At 2.0

122

Amp, however, MRE 1 and MRE 2 exceeded the stiffness magnitude for the steel insert

mount at low frequency. Therefore, this comparison provided a qualitative tool for

understanding the range available for an MR fluid-elastic mount to be altered across.

123

5. MR Fluid Elastic Mount Modeling and Characterization

This chapter presents a preliminary model for the isolator system. The first section

derives a basic isolator transmissibility ratio and shows a non-parametric stiffness

magnitude model for the metal-elastic case. A proposed transmissibility relationship is

then shown for using the MRE as an isolator. After validation of the non-parametric

model, a comparative study is undertaken for the nominal transfer function parameters.

The basis for pursing a model is to extract the isolator’s dependency on current. This

current dependency for the model parameters could be evaluated by a control policy to

select a current setting which would give the desired attenuation if used as a semi-active

isolator. More importantly, the transfer function is devised to have a plug-in capability

within a system specific derivation of force transmissibility as demonstrated [56].

Furthermore, an exponential model is used to represent the dynamic damping of the

MR fluid-elastic mounts as a function of applied current and frequency. This approach

uses the trend of the results to form an exponential damping model.

5.1 Non-Parametric Modeling Approach

This section uses the techniques as mentioned in Chapter 2 for devising a transfer function

for the force-amplitude stiffness results. Additionally, this section discusses the

transmissibility ratio and later proposes a model for the transmissibility ratio of MR fluid-

elastic mounts in this study. Nominal parameters for the transfer function are found using

a nonlinear optimizer in Matlab.

5.1.1 MR Fluid Metal-Elastic Mount Modeling

Herein, a transfer function for the stiffness magnitude of the MR fluid-elastic mount is

presented and converted into a transmissibility ratio. A model for a basic oscillatory

imbalanced mass on an isolator with a spring and damper element is used to generate a

generic transmissibility ratio.

Since machinery can typically generate oscillatory forces F0 at the speed of operation

ω, the use of an isolation device can reduce the force transmitted to the platform FT. The

force generated by the machinery and the force transmitted to the foundation define the

ratio of transmissibility TR as

124

0

TFTRF

= (5.1)

With the ratio of transmissibility defined as a target reduction, the force generated by

an oscillatory input is

0 ( ) ( ) ( ) ( )eqf t Mx t c x t kx t= + + (5.2)

where M is the mass of the machine, ceq is the equivalent damping, and k is the stiffness.

Similarly, the force transmitted across most linear elastic isolators [44,56] has the

equation of motion as

( ) ( ) ( )T eqf t c x t kx t= + (5.3)

Upon conversion to the Laplace domain, the force transmitted F(s) and input X(s) transfer

function is

( )( ) eq

F s c s kX s

= + (5.4)

Furthermore, the equation of motion for the complete system has a relationship of output

X(s) to input F0(s) which is combined with equation (5.4) to define the transmissibility as

0 0

( ) ( ) ( )( ) ( ) ( )

F s X s F sTRX s F s F s

= ⋅ = (5.5)

Building on the aforementioned approach, the transfer function for modeling the

metal-elastic case mount is

( ) | |( )MRE

Num s FTF KDen s X

= = (5.6)

where |F|/X is the stiffness magnitude determined in Chapter 4. The transfer function

selected for this model is determined by the stiffness magnitude results of the MR fluid-

elastic mount. Each mount showed characteristics of two poles and two zeros as

illustrated in Figure 5-1 where the initial ramp is dominated by a zero and stopped by a

pole while another pole declines the response until the final zero levels out the response.

Therefore, the proposed transfer function is

2 2

2 2

22

n nMRE

s sTF Ks s

ζω ωαβ β

+ ⋅ +=

+ ⋅ + (5.7)

where K is the gain, ζ is the nominal zero damping ratio, ωn is the nominal zero frequency,

α is the nominal pole damping ratio, and β is the nominal pole frequency. This

125

nomenclature was selected since the variables are located in the quadratic numerator and

denominator like a standard second-order characteristic equation. Therefore, when

discussing the nominal parameters it becomes easier to associate with the non-parametric

transfer function.

0 5 10 15 20 25 30 353500

4000

4500

5000

5500

6000

6500

Frequency, Hz

Stif

fnes

s M

agni

tude

, N/m

m

MRE 1,|F|/X,2.0Amp

Figure 5-1: Selecting a transfer function to model the stiffness magnitude in the frequency domain.

If the machine being isolated was configured with the MR fluid-elastic mount, then a

proposed transmissibility relationship would be

2 2

2 2

2 22

2 2

22( )

22

n n

n n

s sKs sTR s

s sMs Ks s

ζω ωαβ β

ζω ωαβ β

+ ⋅ ++ ⋅ +=

+ ⋅ ++

+ ⋅ +

(5.8)

where M is the mass of the machine. The damping and stiffness terms from equation (5.4)

have been equated to the proposed transfer function TFMRE in equation (5.7) in a black

box approach. Therefore, the transmitted force prior to taking the magnitude would be

Zero

Pole

126

2 2

2 2

0 2 22

2 2

22

22

n n

Tn n

s sKs sF F

s sMs Ks s

ζω ωαβ β

ζω ωαβ β

+ ⋅ ++ ⋅ +=

+ ⋅ ++

+ ⋅ +

(5.9)

where F0 is the imbalance force magnitude. In theory, after determining the nominal

parameters in terms of current for the transfer function and with an appropriate machine

specification, a control strategy could be applied to minimize the transmitted force by

selecting a current.

5.1.2 Nominal Parameter Results and Comparison

This section discusses the methods used to process the nominal parameters for the

proposed transfer function model TFMRE of the stiffness magnitude. After those

techniques have been introduced, the nominal parameter results are compared for MRE 1,

MRE 2, and MRE 3.

Finding the nominal parameters of the model in equation (5.7) requires the cost

function

| |FJ norm TFX

⎡ ⎤= −⎢ ⎥⎣ ⎦ (5.10)

where the difference between the model and the stiffness magnitude are normalized. The

choice function for minimizing the cost function at each current is fminsearch.m, a

nonlinear optimization technique in Matlab, see Appendix C. The initial guesses for the

nominal parameters require iterations before the minimization function provides sufficient

convergence. The resulting nominal parameters are shown in Table B-9 in Appendix B.

First presented is the parameter for the gain K of the transfer function as seen in

Figure 5-2. As the current is increased from 0-2.0 Amp, the gain has an upward trend for

MRE 1, MRE 2, and MRE 3. For all intents and purposes, the gain is quasi-linear and a

gain model as a function of current I can be used to eliminate the gain variable in equation

(5.7). Thus, a linear model for the gain with dependency on current is fitted to the

nominal parameters as follows:

MRE 1, 824.7 2570;MRE 2, 586.4 3060;MRE 3, 453.3 2514;

I

I

I

K IK IK I

= ⋅ += ⋅ += ⋅ +

(5.11)

127

Although, the purpose here is to generalize the nominal parameter with an acceptable

model, a more complicated model may be used to yield a better degree of accuracy for the

gain KI.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

Current, A

Gai

n, K

MRE 1MRE 2MRE 3

Figure 5-2: Nominal gain, K, as a function of current for each MR fluid-elastic mount.

Next, the nominal damping ratios of the zero ζ and pole α are plotted in Figure 5-3a,

and b, respectively. The damping ratios increase with applied current and other than a

couple of outliers, are symmetric. Therefore, a simple linear model is fit to the damping

ratio parameters for ζ and α as a function of current as follows:

MRE 1, 0.8315 1.257; 0.3394 1.480;MRE 2, 0.8153 1.609; 0.2593 1.559;MRE 3, 0.6154 1.812; 0.2397 1.648;

I I

I I

I I

I II II I

ζ αζ αζ α

= ⋅ + = ⋅ += ⋅ + = ⋅ += ⋅ + = ⋅ +

(5.12)

As with the model for the gain, the damping ratio models can also be used in place of the

variables ζ and α. Moreover, the significance of the non-parametric nominal damping

128

parameters increasing with current is suggestive of increasing the parametric damping

within the isolator.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Current, A

Zero

-Dam

ping

Rat

io, ζ

MRE 1MRE 2MRE 3


(a)

129

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Current, A

Pol

e-D

ampi

ng R

atio

, α

MRE 1MRE 2MRE 3

Figure 5-3: Nominal (a) zero-damping ratio and (b) pole-damping ratio as a

function of current for each MR fluid-elastic mount

The nominal frequency is illustrated in Figure 5-4a and b for the zero ωn and pole β,

respectively. Unlike the damping ratios which increased with current, the non-parametric

core frequencies of the transfer function decline with current. This decline in frequency

for ωn and β is quasi-linear for all three MREs and is also modeled as a function of current

as follows:

,

,

,

MRE 1, 3.712 16.5; 3.518 16.7;MRE 2, 2.275 14.5; 2.107 15.0;MRE 3, 0.503 11.7; 0.169 12.6;

n I I

n I I

n I I

I II II I

ω β

ω β

ω β

= − ⋅ + = − ⋅ +

= − ⋅ + = − ⋅ +

= − ⋅ + = − ⋅ +

(5.13)

Since the squared frequency in a parametric view point is the ratio of stiffness to mass, a

putative claim can be made that the stiffness component of the transfer function decreases

with applied current.

(b)

130

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

2

4

6

8

10

12

14

16

18

20

Current, A

Zero

-Fre

quen

cy, ω

n

MRE 1MRE 2MRE 3


(a)

131

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

2

4

6

8

10

12

14

16

18

20

Current, A

Pol

e-Fr

eque

ncy,

β

MRE 1MRE 2MRE 3

Figure 5-4: Nominal (a) zero-frequency and (b) pole-frequency as a function of current for each MR fluid-elastic mount model

With subsequent models for the nominal parameter variables of the transfer function

devised as a function of current, the transfer function can be reduced to a dependency of

current. This eliminates the need to keep track of the nominal parameters granted an

accurate nominal value model was found. For MR fluid-elastic 1, equations (5.11), (5.12),

and (5.13) are inserted in the transfer function of equation (5.7) to produce an

approximation transfer function for MRE 1 as

2 2

1 2 2

( 3.1) ( 6.2 ( 4.4) ( 1.5) 14 ( 4.4) )8252.4 ( 4.7) ( 4.4) 12 ( 4.7)MRE

I s I I s ITFs I I s I

+ ⋅ − ⋅ − ⋅ + ⋅ + ⋅ −= ⋅

− ⋅ − ⋅ + ⋅ + ⋅ − (5.14)

where I is the applied current. This approximation approach could also be used for MRE

2 and MRE 3. Although, this methodology for the transfer function is not as accurate as

using the nominal parameters found from the cost function, this method does reduce the

complexity and computation required to achieve a desirable transmissibility.

(b)

132

Furthermore, it is important to note that initial guesses played a substantial role in the

solution for the nominal parameters. Each parameter could be selected or made constant

and still give a desirable TFMRE. Therefore, investigation of the relationship between the

zero and pole for both nominal damping relationship and nominal frequency relationship

is used to substantiate this methodology.

5.1.3 Nominal Parameter Relationship

As aforementioned, the relationship of the zero and pole, dislodging the gain K, has more

significance than the value of the nominal parameters by themselves. Figure 5-5

illustrates the relationship for the damping ratio of ζ/α and is consistent with the earlier

discussion to show the relationship increasing linearly. This relationship was also used to

manually solve for the nominal damping ratios within an acceptable range as previously

shown in Figure 5-3. For protocol, this relationship is not used to generate any of the

simulation results.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Current, A

Zero

/Pol

e D

ampi

ng R

atio

ζ/ α

MRE 1MRE 2MRE 3

Figure 5-5: Non-parametric damping ratio relationship, ζ/α, at each current setting for MR fluid-elastic mount models.

133

An additional relationship exists between the nominal frequency of the zero and pole,

but represented in a normalized stiffness form. The stiffness relationship is based on the

squared frequency

2 Z P

Z P

km

ω −

−

= (5.15)

where kZ-P is the nominal stiffness of either the zero or pole, and mZ-P is a normalized

mass. By alteration, the Zero/Pole stiffness relationship kR is defined as

2,2

n ZR

P

kωβ

= (5.16)

where nominal frequency parameters for the zero and pole are squared prior to division.

This Zero/Pole stiffness relationship is then illustrated in Figure 5-6 for each MRE. As

alluded to earlier, the stiffness relationship is linear and can also be used to manually

solve the nominal frequency parameters for the transfer function.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Current, A

Zero

/Pol

e S

tiffn

ess

Rat

io, ω

2 n/ β2

MRE 1MRE 2MRE 3

Figure 5-6: Non-parametric stiffness ratio relationship, ωn2/β2, at each current

setting for MR fluid-elastic mount models.

134

5.2 Model Simulation and Comparison

This section presents the MR fluid-elastic mount simulation in conjunction with the

empirical stiffness results. Additionally, the error associated with the model is presented

last.

5.2.1 MR fluid Metal-Elastic Mount Simulation

Herein, the simulation for the metal-elastic case is produced from the nominal parameters

for the transfer function. Each plot contains both the empirical stiffness |F|/X and the

modeled stiffness TF. The axis is held constant for the domain with 0 to 35 Hz and for the

range with 2000 to 7,000 N/mm. To reduce congestion, current increments at 0.5 Amp

are illustrated. Additionally, the stiffness magnitude |F|/X is depicted with a colored

marker for each current while the model TF has a solid red line.

The nominal parameters from the system identification method found for MRE 1 are

now used to generate a simulation of the stiffness magnitude |F|/X. The frequencies are

loaded and the model is simulated for MRE 1 as seen in Figure 5-7. This comparison

shows that the model is valid for MRE 1 and sufficiently replicates the stiffness

magnitude results. At 0 Hz, the model is able to achieve the quasi-static stiffness and

from there increase to the stiffness magnitude at 1 Hz. Typically, 1 Hz is the largest

stiffness magnitude that the model achieves and from there the model decreases with the

slope of the stiffness data before following the plateau to 35 Hz.

135

0 5 10 15 20 25 30 352000

2500

3000

3500

4000

4500

5000

5500

6000

6500

7000

Frequency, Hz

Stif

fnes

s, N

/mm

MRE 1,|F|/X,0.00-AMRE 1,TF,0.00-A

MRE 1,|F|/X,0.50-AMRE 1,TF,0.50-A

MRE 1,|F|/X,1.00-AMRE 1,TF,1.00-AMRE 1,|F|/X,1.50-A

MRE 1,TF,1.50-AMRE 1,|F|/X,2.00-AMRE 1,TF,2.00-A

Figure 5-7: Stiffness simulation results for MR fluid-elastic 1 mount at 0.5 Amp current increments.

Using the nominal parameters solved in the transfer function for MR fluid-elastic 2,

the model TF stiffness magnitude is found for each frequency in the specified range. As

shown in Figure 5-8, the simulation of the TF is able to reproduce the |F|/X values for

MRE 2. Therefore, the model is valid for representing the stiffness magnitude results

from MR fluid-elastic 2.

136

0 5 10 15 20 25 30 352000

2500

3000

3500

4000

4500

5000

5500

6000

6500

7000

Frequency, Hz

Stif

fnes

s, N

/mm

MRE 2,|F|/X,0.00-AMRE 2,TF,0.00-A

MRE 2,|F|/X,0.50-AMRE 2,TF,0.50-A



Figure 5-8: Stiffness simulation results for MR fluid-elastic 2 mount at 0.5 Amp current increments.

A final look at the usability of the TF model is shown for MR fluid-elastic 3 in Figure

5-9. The stiffness magnitude simulation was generated with the nominal parameters for

MRE 3 as listed in Table B-9. This mount, however, had lower achieved stiffness

magnitudes |F|/X, but the proposed model accurately predicted the stiffness magnitude for

each current at all input frequencies.

137

0 5 10 15 20 25 30 352000

2500

3000

3500

4000

4500

5000

5500

6000

6500

7000

Frequency, Hz

Stif

fnes

s, N

/mm

MRE 3,|F|/X,0.00-AMRE 3,TF,0.00-A

MRE 3,|F|/X,0.50-AMRE 3,TF,0.50-A



Figure 5-9: Stiffness simulation results for MR fluid-elastic 3 mount at all current settings.

In summary, this section presented the simulation of the proposed transfer function

for each MR fluid-elastic mount. The nominal parameters determined by the

fminsearch.m function in Matlab were used in each respective transfer function model.

The nominal parameters for the zeros and poles were iterated until a sufficient

convergence was achieved. The TF model with the appropriate nominal parameters

replicated the stiffness magnitude results without any visible problems. Therefore, the

usage of this transfer function for all MR fluid metal-elastic cases has been illustrated to

work sufficiently. Next, the error between the model and stiffness magnitude results is

compared to better support the use of the two zero and two pole transfer function model.

5.2.2 Model Error Evaluation

This section presents the error for the simulated model TF values and stiffness magnitude

|F|/X values. First, an equation for estimating the discrete error is devised and presented.

138

Next, the maximum discrete error that occurred as well as the average error is graphically

illustrated. Lastly, the discrete error residuals are plotted and discussed.

The discrete error is determined at each frequency within the model and then

standardized. The equation for this error is

100%f fError

f

y xTF abs

x⎛ ⎞−

= ⋅⎜ ⎟⎜ ⎟⎝ ⎠

(5.17)

where yf is the simulated stiffness magnitude value, xf is the empirical stiffness magnitude

result, and f is the frequency. Additionally, the absolute value of all error points is

averaged which determines the mean error across all the frequencies at a specified current

setting. For comparison, the absolute error range is from 0 to 5% for each 0.25 Amp

current increment.

The maximum discrete error that occurred at each current is shown in Figure 5-10a, b,

and c when modeling the stiffness magnitude of the MR fluid-elastic mounts. The mean

error is below 2% when modeling the stiffness magnitude for MR fluid-elastic 1 and the

maximum error at a single frequency is 4.9%. The maximum error when modeling the

stiffness magnitude for MR fluid-elastic 2 is reduced to 4.6%. The maximum error when

modeling the stiffness magnitude for MR fluid-elastic 3 is further reduced to 3.3%.

Furthermore, the mean error is less than 2% when modeling the stiffness magnitude

results for the MR fluid-elastic mounts. Therefore, the transfer function accurately

models the stiffness magnitude results.

139

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00

Current, Amps

Erro

r, %

MRE 1 TF Max ErrorMRE 1 TF Mean Error

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00

Current, Amps

Erro

r, %



(a)

(b)

140

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00

Current, Amps

Erro

r, %


Figure 5-10: Maximum and mean error for the transfer function when compared to the stiffness magnitude vales for (a) MR fluid-elastic 1, (b) MR fluid-elastic 2, and (c) MR fluid-elastic 3.

For thoroughness, the remainder of the model evaluation is now turned to plotting the

discrete error. The discrete error residuals for MR fluid-elastic 1, 2, and 3 are presented in

Figure 5-11a, b, and c, respectively. The error appears chaotic and therefore the TF model

can be deemed a suitable choice. If the error residuals were biased or showed a uniform

nonconvergence then it would be necessary to choose another model to represent the data.

Each error plot, however, has a similar trend and indicates that the stiffness magnitude

value is being missed consistently by the model for each mount. Nonetheless, the chaotic

nature of the error indicates that the two zero and two pole transfer function model is a

suitable choice for modeling the stiffness magnitude results of the MR fluid-elastic

mounts.

(c)

141

0 5 10 15 20 25 30 35-15

-10

-5

0

5

10

15

Frequency, Hz

Erro

r, %

MRE 1,TF-Error,0.00-AMRE 1,TF-Error,0.25-AMRE 1,TF-Error,0.50-AMRE 1,TF-Error,0.75-AMRE 1,TF-Error,1.00-AMRE 1,TF-Error,1.25-AMRE 1,TF-Error,1.50-AMRE 1,TF-Error,1.75-AMRE 1,TF-Error,2.00-A

0 5 10 15 20 25 30 35-15

-10

-5

0

5

10

15

Frequency, Hz

Erro

r, %



(a)

(b)

142

0 5 10 15 20 25 30 35-15

-10

-5

0

5

10

15

Frequency, Hz

Erro

r, %


Figure 5-11: Discrete model error for (a) MR fluid-elastic 1, (b) MR fluid-elastic 2, and (c) MR fluid-elastic 3 from simulation at all current settings.

In summary, the error showed no visible trends other than consistency between

mounts. Low error was found for each data point and at most reached 4.9% when

modeling the stiffness magnitude for MR fluid-elastic 1. The average error, however,

remained below 2% for modeling the stiffness magnitude from MR fluid-elastic 1. The

average error for modeling the stiffness magnitude of MR fluid-elastic 2 and 3 was

generally less than 1.5%. This has further shown the validity of using the proposed

transfer function model.

5.3 Damping Modeling Approach

In this section, a damping model is proposed and fit to the damping results. By modeling

the damping alone, the dependencies can be extracted. After the model is presented, a

section is devoted to the simulation of the model and compared to the empirical damping

Ceq results.

(c)

143

5.3.1 MR Fluid-Elastic Mount Damping Model

The trend of the damping looks similar to an exponential decay. Therefore, the data is fit

with an exponential model of the form

( )bC fMount aC C e ⋅= (5.18)

where Ca is damping model coefficient, Cb is the exponential coefficient, f is the

frequency. This fit was accomplished in Matlab where the damping model and

exponential coefficients were solved. These coefficients are shown in Table 5-1 for each

current. Additionally, MR fluid-elastic 1 and 2 have comparable coefficients.

Table 5-1: Damping model and exponential coefficient values for MR fluid-elastic 1, 2, and 3 mounts.

Next, the model damping coefficients in Table 5-1 were fitted as a function of current

I and then solved to represent the damping model coefficient Ca’s current dependence for

each mount as follows:

,

,

,

MRE 1, 133 51,MRE 2, 116 68,MRE 3, 83.6 81.5

a I

a I

a I

C IC IC I

= ⋅ +

= ⋅ +

= ⋅ +

(5.19)

With little variation in the exponential coefficient, Cb = -0.6 and the damping model for

MR fluid-elastic 1 is

( 0.6 )1 (133 51) f

MREC I e − ⋅= ⋅ + (5.20)

where I is the input current, and f is the oscillatory input frequency. Therefore, equation

(5.20) for the damping in MRE 1 has been constructed with dependency on current I and

oscillatory frequency f. This model for damping is not used to generate any simulation

results, but only to show the dependency of current within the damping.

144

5.3.2 MR Fluid-Elastic Mount Damping Simulation

The following simulation presented in this section uses equation (5.18) with the

coefficients listed in Table 5-1. This simulation is for the MR fluid-elastic mounts and

shows the model fitted to the damping values for the mounts.

The simulation for the damping values in MR fluid-elastic 1 is shown in Figure 5-12

for all applied currents. The empirical damping results from previous processing have

unique marker and line styles while the model uses a solid red line. For MRE 1, the

model follows the empirical results consistently to 0.5 Amp at all currents. Beyond 4 Hz

and above 0.75 Amps the model is unable to account for the actual damping. Therefore,

the model has low accuracy beyond 4 Hz and doesn’t provide a usable fit.

0 5 10 15 20 25 30 350

20

40

60

80

100

120

140

160

180

200

Frequency, Hz

Est

. Dam

ping

, Ns/

mm

MRE 1,Ceq,0.00-AMRE 1,Model,0.00-AMRE 1,Ceq,0.25-AMRE 1,Model,0.25-AMRE 1,Ceq,0.50-AMRE 1,Model,0.50-AMRE 1,Ceq,0.75-AMRE 1,Model,0.75-AMRE 1,Ceq,1.00-AMRE 1,Model,1.00-AMRE 1,Ceq,1.25-AMRE 1,Model,1.25-AMRE 1,Ceq,1.50-AMRE 1,Model,1.50-AMRE 1,Ceq,1.75-AMRE 1,Model,1.75-AMRE 1,Ceq,2.00-AMRE 1,Model,2.00-A

Figure 5-12: Damping simulation results for MR Fluid-Elastic 1 mount at full range of current settings.

In the same regard as MR fluid-elastic 1, the damping values for MR fluid-elastic 2

are plotted for both the empirical results and the model in Figure 5-13. The exponential

145

model is able to replicate the empirical results as long as the current is lower than 0.75

Amps. As the current is increased with a frequency higher than 4 Hz, the model for MRE

2 is unable to replicate the empirical results.

0 5 10 15 20 25 30 350

20

40

60

80

100

120

140

160

180

200

Frequency, Hz

Est

. Dam

ping

, Ns/

mm



Lastly, the model is simulated for the damping in MR fluid-elastic 3 as shown in

Figure 5-14. The same case exists as with MRE 1 and MRE 2; the model is unable to

represent the higher current damping values above 4 Hz. Therefore, at best the

exponential model may be used for estimation purposes at low frequency. This is

considered acceptable since the majority of the damping produced in the mount occurs at

low frequency.

146

0 5 10 15 20 25 30 350

20

40

60

80

100

120

140

160

180

200

Frequency, Hz

Est

. Dam

ping

, Ns/

mm



Upon inspection of the proposed damping model for MR fluid-elastic 1, 2, and 3,

there is very little accuracy above 4 Hz if the current is increased to more than 0.75 Hz. In

contrast, very little damping is present in the empirical results as the frequency is

increased above 14 Hz. Thus, for estimation purposes at lower frequency, the model from

equation (5.18) would suffice. A better model may exist, but was not pursued since the

damping values quickly approach 0 Ns/mm at high frequency.

5.4 Summary

Herein, the first section summarizes the simulation of the frequency domain stiffness

magnitude results for the MR fluid-elastic mounts. Additionally, an evaluation of the

transfer function nominal parameters is recapped. Finally, the damping model and

simulation for the equivalent damping of the MR fluid-elastic mounts is summarized.

147

5.4.1 Non-Parametric Simulation and Evaluation Remarks

A relationship between the transmitted force and oscillatory input displacement was

presented as the stiffness magnitude. This stiffness magnitude in the frequency domain

for a MR fluid-elastic mount showed characteristics of two zeros and two poles.

Therefore, the transfer function in equation (5.7) was used to model the frequency

response in the Laplace domain.

Prior to simulating the data, a nonlinear optimizer was used to solve for the nominal

parameters in equation (5.7). The nominal parameters were solved for each current setting

within the frequency data, plotted as a function of current, and then evaluated further.

Nominal gain for each of the MR fluid-elastic mounts was plotted and evaluated with a

quasi-linear increasing slope as a function of current. This increase signified that the gain

was representative of the increased stiffness magnitude due to higher levels of current.

Moreover, the nominal zero-pole damping ratio had a positive quasi-linear slope

while the nominal zero-pole frequency had a shallow, negative slope when plotted for

each MR fluid-elastic mount as a function of current. The ensuing result, in a non-

parametric standpoint, is that the magnitude of the MR fluid-elastic mount has a larger

damping element with the increasing magnetic field intensity. The stiffness relationship

obtained from the zero-pole frequency, however, is suggestive that the MR fluid-elastic

mount has a steady stiffness element which is intensified by the gain.

Using the nominal zero and pole values determined by the solver, the transfer

function was simulated for the empirical stiffness magnitude results. Graphically, the

simulations stayed close to the stiffness magnitude for each mount. At worse case using

the discrete error calculation, however, the simulation for MR fluid-elastic mount 1 had an

error of 4.9%. The error trend was plotted and had a non-uniform distribution which

signified the transfer function in equation (5.7) was an appropriate model. Additionally,

MR fluid-elastic mount 2 and 3 had a maximum discrete error of 3.6% and 3.3%,

respectively. Final inspection of the error trend for all of the MR fluid-elastic mounts

showed consistency per mount, but had no visible trend.

Concurrently, the nominal parameters were modeled as a function of current. The

purpose of modeling the parameters as a function of current was to reduce the complexity

of maintaining a possible control policy for the transfer function in equation (5.7). After

148

each parameter was constructed with a basic linear model, a final current dependent model

for the transfer function was presented in equation (5.14). For all intents and purpose this

model is not stated to be accurate nor was it used to simulate the data, however, it may be

useful for future modeling.

5.4.2 Damping Simulation and Evaluation Remarks

The equivalent damping results from the earlier force-displacement analysis were plotted

in the frequency domain. After this representation, the damping had an exponential

decaying trend with frequency. Therefore, the exponential damping model from equation

(5.18) was proposed.

The damping model coefficients and exponential coefficients where discovered by

fitting the model to the equivalent damping results. Typically, the exponential coefficient

was uniform between all current inputs. The damping model coefficient, however,

increased linearly with current. This finding led to an estimation of the damping model

coefficient as a function of current as seen in equation (5.20) for each MR fluid-elastic

mount.

An illustrative analysis was then undertaken for presenting the validity of the

exponential model from equation (5.18). The simulation was shown for the damping

results of the MR fluid-elastic mounts. Inspection indicated that the model was a good fit

for the low frequencies, but tended to diverge from the data toward 0 Ns/mm when

simulated above 4 Hz. This model would be useful at low frequencies for all input

currents; however, the damping at higher frequencies above 0.75 Amp of input current

may not be estimated with confidence using this model.

149

6. Conclusions and Prospective Research

This chapter broadly summarizes the design, testing, and results from this study on a MR

fluid-elastic mount. After the reiteration of results, a recommendation section presents

improvements from hands-on experience. Furthermore, the future work section discusses

where research with an MR fluid-elastic mount may expound.

6.1 Summary

This section is devoted to presenting the major objectives, the objectives delivered, and

conclusions for this research. Of course, many of the objectives were realized and

pursued after an extensive literature review of MR fluid mount technology.

As discussed in Chapter 1, the motivation for this research was to rethink the design

of an MR fluid-elastic mount and the associated magnetic circuit. The major objective of

further evaluating and analyzing MR fluid-elastic mounts laid the foundation for this

research. First and foremost, we wanted to further evaluate the magnetic circuit presented

by Wang et al. [35]. The underlying reason for this evaluation was to establish a more

efficient magnetic circuitry. Moreover, since many designs of other researchers utilize

large magnetic circuits, we decided to take it a step further and configure a smaller

magnetic circuit design. The purpose of an efficient and low-profile mount configuration

was to create a more market friendly isolator with desirable packaging characteristics.

With the redesigning goal in mind, a design process using finite element magnetic

software (FEMM) was undertaken to simulate a circuit of less size with improved

magnetic efficiency. The simulated design received dimensional constraints from donated

parts and the requirement of testing fixtures. The metal-elastic case mount design

coincided with the specifications of a three-plate mold that had been previously designed.

With these constraints in place, the mount and magnetic system design was confirmed in

the simulated model. This analysis of the system, generated with MRF-145 fluid,

rendered a usable magnetic flux density of approximately 1.0 T entering the fluid cavity

with an applied current of 3 Amp. Thus, the magnetic system and mount design was

deemed efficient and space conscious which complimented the first objective and lead to

the selection of materials.

150

Since the mounts were intended to be compact, a 1 mm displacement input was to be

used during testing. Therefore, the fluid cavity gap height was designed to allow the 1

mm input displacement to compress the height of the fluid gap by approximately 25%.

Additionally, to prevent rupturing the MR fluid-elastic mounts, the total height of the

mount allowed the 1 mm compression to squeeze the mount approximately 10% of its

static height. Therefore, the height of the fluid cavity was set to 0.1875 in. and the height

of the mount was set to 0.4375 in. The sidewall of the casing was designed to a thickness

of 0.375 in. to prevent rupturing the mount as well as providing a large surface area to

attach to the upper-pole plate. The fluid gap, however, plays an important role in the MR

effect of the mount. Reducing the fluid gap may increase the MR effect, but this

configuration in the FEMM simulation indicated large magnetic field intensities in the

fluid gap over the range of current increments.

The material selected for the elastomeric case was a 30 durometer rated polyurethane

rubber from PolyTek (PolyTek 74-30). This durometer rating provided a compliant

elastomer with low stiffness that would not overshadow the activation of the MR fluid.

Furthermore, 12L14 steel was selected for the upper and magnetic-pole plate in the metal-

elastic case. The use of the pole plates required an adhesive substrate for the polyurethane

to bond against. So, an etching primer (SEM#39693-Green) was used as an initial

substrate followed by an epoxy primer (Omni-MP172 & MP175) to create a bondable

surface for the polyurethane as recommended by PolyTek.

After material selection, the molding and fabrication process for both the elastic and

metal-elastic case mount was presented. In this section, the reader was familiarized with

the process required to mold the halves, and then mold the final casing. The elastic casing

inserts consisted of air, rubber, aluminum, and steel. In addition, Chapter 3 discussed the

metal-elastic case fabrication. Preparation of the 12L14 steel into useable pole plates was

also discussed. After the metal-elastic case was fabricated, additional processes were used

to fill the cavity with MRF-145 fluid which involved degassing and plugging the fluid

cavity. Therefore, the guidelines for manufacturing the MR fluid-elastic mount (MRE)

were presented which delivered the third objective in this research. Additional guidelines

for manufacturing an elastic case with MR fluid were included in Appendix A.

151

The first objective combined the testing, characterization, and subsequent modeling

of the MR fluid-elastic mount. The elastomer case mounts, however, were presented

through the testing and characterization stages with the MR fluid-elastic mounts as

necessary to complete the second objective. The testing was accomplished using an

electromagnetic linear actuated shock dynamometer (Roehrig EMA). This shock dyno

had a displacement resolution from 0.25-177 mm and therefore, testing was limited to

0.50 mm amplitude for the MR fluid-elastic mounts. The lower resolution of 0.25 mm

was used for the elastomer mounts with metal inserts to prevent damaging the actuator

and the 2000 lb loadcell. Both static and dynamic testing was completed on this shock

dynamometer.

Characterization of the MR fluid-elastic mounts and other passive mounts was

presented in Chapter 4. The first section processed the quasi-static testing data and

presented the results. The static stiffness increased in MRE 1-3, respectively, at 42%,

24%, and 17% from varying the current from 0 Amp to 2 Amps. In the subsequent

sections, the dynamic data was processed and analyzed using a force-displacement

method and a force-amplitude method. An error comparison was then presented for the

two methods on the bases of accuracy and usability. The force-amplitude method was

evaluated to have sufficient accuracy, mostly at higher current inputs, and was selected for

presenting and simulating the results. The equivalent damping parameters, however, were

extracted from the force-displacement method.

From the results, an initial characterization evaluation quotient for both stiffness

magnitude and damping was presented. The increase in stiffness magnitude quotient at 2

Amps of current for MR fluid-elastic 1, 2 and 3 was 78%, 57%, and 46% above the zero

current stiffness magnitude. In addition, the damping quotient at 2 Amps of current for

MRE 1-3 was 500%, 430%, and 170%, respectively, above the zero current equivalent

damping. Furthermore, the stiffness magnitude results were presented in the frequency

domain for each mount. This information was also processed on the elastic case mounts

to determine a suitable boundary for comparison with the MR fluid-elastic mounts. An

upper boundary with the steel mount and a lower boundary with the rubber mount were

used to show the implications that added magnetic field had on the MR fluid-elastic

mount. With the respective boundary, an MRE at zero current had slightly more stiffness

152

than the rubber mount, but activated with 2.0 Amps of current the MRE achieved more

stiffness at low frequency than the steel insert mount. Therefore, an off-state MRE mount

was characterized similar to a solid rubber mount with a 30 Durometer rating and at full

current activation of 2 Amps the mount was stiffer than an elastic mount with a metal

insert at low frequency.

Further discussion indicated that the damping element of the mount decreased at high

frequency. At high frequency inputs, the loading and unloading cycles converged. The

suspected reason for this convergence is that the ferrous particles in the fluid are

aggregating and are not being restored in columnar structures. Therefore, the hysteresis is

decreased and the transmission of the displacement input is bypassing the ferrous particles

in the MR fluid at high frequency inputs.

After the MR fluid-elastic mount and system characterization, a non-parametric

transfer function model was used to represent the system dynamics of the transmitted

force. A zero-pole identification was used to determine an appropriate numerator and

denominator for the Laplace transfer function. This identification approach used the

stiffness magnitude which was determined from the force-amplitude processing method.

Nominal parameters for the model were estimated using a nonlinear optimization

technique and then used to simulate the stiffness magnitude results. This model was

found to be precise and represent the dynamics of the MR fluid-elastic mounts with an

average error below 2%. Additionally, the discrete error calculation was used to view the

maximum error of the transfer function and at most was found to be 4.9%. Furthermore,

the transfer function was converted into a dependency of current for MR fluid-elastic 1,

and also used to represent a basic transmissibility ratio that would arise from oscillatory

input forces. Thus, the primary objectives for this research had been achieved.

In conclusion, a unique magnetic system design configuration has been presented for

an MR fluid-elastic mount beyond currently available and open literature. The guidelines

for the design and fabrication of this MR fluid-elastic mount were also presented. The

configuration was then tested and validated. Stiffness magnitude and damping results

were explored and used to characterize the MR fluid-elastic mount and magnetic circuitry.

Further prudence lead to modeling the system dynamics of the stiffness magnitude results

in the frequency domain with a non-parametric transfer function model. The nominal

153

parameters for the model were calculated and used to replicate the stiffness magnitude

results. Therefore, an MR fluid-elastic mount with tunable stiffness magnitude

characteristics has been covered.

6.2 Recommendations

Along with the success in this research, the author noticed details and design implications

that could easily be resolved. The first major flaw within the metal-elastic case is the

reduced surface area around the magnetic-pole plate. The design was unable to be altered

with the three-plate mold. Therefore, a modified mold that would allow the spacer

(positioned between the magnetic coil and the elastomer case) and the magnetic-pole plate

to be molded to the face of the mount would increase the design robustness. This

modification would create a larger surface area for bonding the polyurethane.

Additionally, this design would eliminate the need for a protective epoxy on the external

face of the mount and allow for improved magnetic efficiency.

Additional modeling should be done with various MR-fluids within a magnetic finite

element software program. More so, the design can be compacted with the reduction of

the lower magnet housing and modeled to achieve a very thin design. The housing used

for the design in this research was made to hold a specified coil and therefore took on

larger than desired dimensions. With a spacer-pole plate combination molded to the face

of the MR-fluid cavity, the pole plate could extend axially and allow a magnetic bobbin to

be placed around the core with a matched housing. This would allow for quick and easy

removal of the coil as well as the mount and magnetic system to be rigidly attached.

As for testing recommendations, a mount of this scale should be tested with an

actuator capable of producing a displacement resolution suitable to the specifications of a

static load rating for the mount. Higher resolution of input displacement with 0.1 mm

would be advisable for the collection of dynamic transmitted force data. The reason for

this recommendation is that during testing with the EMA at oscillatory amplitude

displacements below 0.50 mm, two of the three MRE mounts ruptured at the bonding area

of the magnetic-pole plate and elastomer hull. With confidence, the design robustness can

be greatly increased with a full face spacer-pole plate combination.

154

If additional consultation is needed you may contact the author at {[email protected]}.

In any event, the recommendations are:

• Press fit or adapt a spacer to the magnetic-pole plate to improve surface area

to bond ratio with elastic case half as seen in Figure 6-1 and use a suitable

substrate to bond to the pole plate and spacer prior to molding

• Model a redesign of this mount within a finite element magnetics program

such as FEMM to achieve desired dimensions and yield stress in the MR fluid

cavity based on the MR fluid B-H curve

• Compact the design and include a method for rigid attachment to the lower

housing between the pole plate and core of the magnetic circuit

• Design the upper-pole plate and upper magnetic housing as one rigid piece as

seen in Figure 6-1

• Use an input actuator or shaker equipment with high resolution for

displacement inputs for dynamic testing

Figure 6-1: Automotive friendly design for an MR fluid-elastic mount.

To better clarify the idea for reducing the system size, the MR fluid-elastic mount and

magnetic circuit height are reduced to 0.88 in. as seen in Figure 6-1. This design fits the

spacer to the magnetic-pole plate and combines to form the magnetic-pole core.

Additionally, the fluid gap height and diameter are the same, but the spacer removes the

elastic casing around the pole plate. This design uses a 300 turn coil with 24 AWG

magnet wire for compliance with a 12 V power supply. The magnitude of magnetic flux

155

at the center of the fluid gap is 0.65 T with a 3 Amp current supply if MRF-145 fluid is

used. Furthermore, this design shows the author’s recommendation of creating a more

robust and compact MR fluid-elastic mount system. Now, the discussion is turned to the

future work to follow the current research.

6.3 Future Work

This section presents the future work that could follow from this research, but is not

limited to the topics and suggestions presented.

Granted that a transmissibility model was presented, but not simulated or used within

a specified system, several items that can stem out are:

• Run a modeling simulation for the force transmitted by an oscillatory force

input for a desired system

• Develop a control policy for use with the MR fluid-elastic mount in a

specified system possibly based on the control policies of Koo et al

• Develop a testing protocol and dynamically test the MR fluid-elastic mount in

an isolation scenario between a foundation or a suspended mass subjected to

oscillatory forces

Further testing analysis ideas are:

• Test the MR fluid-elastic mount in a controlled temperature environment to

determine a relationship between operating temperature and performance

• The phase difference between input and output in the frequency domain was

noted, but not presented due to the scope of this work. Therefore, it is

suggested that a study of the phase differences be completed more thoroughly

• Develop an MR fluid-elastic mount design by performing/employing a design

parameter optimization technique to minimize the need for actual testing to

determine the MR effect

• Use a permanent magnet instead of an electro coil for activating the MR fluid.

This would allow a designer to use a MR fluid-elastic mount as a passive

mount with desired stiffness magnitude

156

Mount design alterations for future work are:

• Alter fluid cavity dimensions to see different MR effects in the mount design

and verify if dynamic damping or stiffness is changed

• Develop a parametric model to account for altering fluid cavity to elastic

sidewall thickness

• Use different durometer rated elastomer and determine a suitable combination

for a set percentage by weight MR fluid

• Use an advanced and controllable manufacturing process to limit variability

during fabrication

• Test different MR fluids in the MR fluid-elastic mount

• Embed ferrous iron particles in a polyurethane cavity instead of using MR

fluid while using the current magnetic circuit configuration

Cost and performance analysis for future work:

• Build a desirable MR fluid-elastic mount and determine the number of cycles

before failure of the mount and components

• Determine the cost-performance ratio of an MR fluid-elastic mount with an

applicable controller and compare the results to a cost-performance ratio of

passive mounts

• Devise and build a mold for easier manufacture of the metal-elastic case to

reduce time and cost during fabrication process

157

References

1. Miller, L.R., Ahmadian, M., “Active Mounts - A Discussion of Future

Technological Trends,” Proceedings of Internoise-92. Toronto, Canada, 1992.

2. Rabinow, J., “The magnetic fluid clutch.” AIEE Trans. 67, p. 1308, 1948.

3. Lange, U., Richter, L., Zipser, L., “Flow of Magnetorheological Fluids,” Journal

of Intelligent Material Systems and Structures, Vol. 12, p. 161-164, 2001.

4. Goncalves, F.D., “Characterizing the Behavior of Magnetorheological Fluids at

High Velocities and High Shear Rates,” Ph.D. Dissertation, Virginia Polytechnic

Institute and State University: Blacksburg, VA, 2005.

5. Carlson, J.D., Catanzarite, D.M., Clair, K.A.S., “Commercial Magneto-

Rheological Fluid Devices,” Lord Corporation, Cary, NC.

6. Jolly, M.R., Bender, J.W., Carlson, J.D., “Properties and Applications of

Commercial Magnetorheological Fluids,” SPIE 5th Annual Symposium on Smart

Structures and Materials, San Diego, CA, March 1998.

7. Lord Corporation, “LORD Product Selector Guide,” 2008.

8. Ahn, Y.K., Yang, B., Ahmadian, M., Morishita, S., “A Small-sized Variable-

damping Mount Using Magnetorheological Fluid,” International Journal of

Vehicle System Dynamics, Vol. 16, p. 127-133, 2005.

9. Carlson, J.D., Weiss, K.D., “A growing attraction to magnetic fluids,” Machine

Design, Vol. 66, No. 15, p. 61-64, 1994.

10. Weiss, K.D., Duclos, T.G., Carlson, J.D., Chrzan, M.J., and Margida, A.J., “High

strength magneto - and electro - rheological fluids,” Society of Automotive

Engineers, Vol. 932451, 1993.

11. General Motors Corporation, “Magnetic Selective Ride Control,” 2008.

12. Audi, “Audi Magnetic Ride Suspension,” 2008.

13. Lee, H.S., Choi, S.B., “Control and response characteristics of a

magnetorheological fluid damper for passenger vehicles,” Journal of Intelligent

Material Systems and Structures, Vol. 11 (No.1), p. 80-87, 2000.

158

14. Simon, D.E., “An investigation of the effectiveness of skyhook suspensions for

controlling roll dynamics of sport utility vehicles using magnetorheological

dampers,” Ph.D. Dissertation, Virginia Polytechnic Institute and State University,

Blacksburg, VA. 2001

15. Song, X., Ahmadian, M., Southward, S.C., “An Adaptive Semiactive Control

Algorithm for Vehicle Suspension Systems,” Proceedings of IMECE. Washington,

D.C., 2003.

16. Ahmadian, M., Vahdati, N., “Transient Dynamics of Semiactive Suspensions with

Hybrid Control,” Journal of Intelligent Material Systems and Structures, Vol. 17

(No.2), p. 145-153, 2006.

17. Farjoud, A., Vahdati, N., Fah, Y.F., “Mathematical Model of Drum-type MR

Brakes using Herschel-Bulkley Shear Model,” Journal of Intelligent Material

Systems and Structures, Vol. 00, 2007.

18. Lee, U., Kim, D., Hur, N., Jeon, D., “Design Analysis and Experimental

Evaluation of an MR Fluid Clutch,” Journal of Intelligent Material Systems and

Structures, Vol. 10 (No. 9), p. 701-707, 1999.

19. Lord Materials Division, “Designing with MR Fluids,” Engineering Note, p. 1-5,

1999.

20. Carlson, J.D., Jolly, M.R., “MR fluid, foam and elastomer devices,” Mechatronics,

Vol. 10, p. 555-569, 2000.

21. Mazlan, S.A., Ekreem, N.B., Olabi, A.G., “The performance of

magnetorheological fluid in squeeze mode,” Smart Materials and Structures, Vol.

16, p. 1678-1682, 2007.

22. Goncalves, F.D., Koo, J.H., Ahmadian, M., “A Review of the State of the Art in

Magnetorheological Fluid Technologies – Part 1: MR fluid and MR fluid

models.,” The Shock and Vibration Digest, Vol. 38, 2006.

23. York, D., Wang, X., Gordaninejad, F., “A New MRF-E Vibration Isolator,”

Journal of Intelligent Material Systems and Structures, Vol. 18, p. 1221-1225,

2007.

159

24. Christopherson, J., Jazar, G.N., “Dynamic behavior comparison of passive

hydraulic engine mounts. Part 1: Mathematical analysis,” Journal of Sound and

Vibration, Vol. 290, p. 1040-1070, 2005.

25. Adiguna, H., Tiwari, M., Singh, R., Tseng, H.E., Hrovat, D., “Transient response

of a hydraulic engine mount,” Journal of Sound and Vibration, Vol. 268, p. 217-

248, 2003.

26. Ahn, Y.K., Kim, Y., Yang, B., Ahmadian, M., Baek, W.K., “Optimal Design of

Engine Mount Using an Enhanced Genetic Algorithm and Simplex Method,” The

8th International Congress on Sound and Vibration, Hong Kong, China, 2001.

27. Vahdati, N., Ahmadian, M., “Variable Volumetric Stiffness Fluid Mount Design,”

Shock and Vibration, Vol. 11, p.21-32, 2004.

28. Wang, L.R., Wang, J.C., Hagiwara, I., “An integrated characteristic simulation

method for hydraulically damped rubber mount of vehicle engine,” Journal of

Sound and Vibration, Vol. 286, p. 673-696, 2005.

29. Koo, J.H., Ahmadian, M., Setareh, M., Murray, T.M., “In Search of Suitable

Control Methods for Semi-Active Tuned Vibration Absorbers,” Journal of

Vibration and Control, Vol. 10, p. 163-174, 2004.

30. Boczkowska, A., Awietjan, S.F., Wroblewski, R., “Microstructure-property

relationships of urethane magnetorheological elastomers,” Smart Materials and

Structures, Vol. 16, p. 1924-1930, 2007.

31. Zhou, G.Y., “Complex shear modulus of a magnetorheological elastomer,” Smart

Materials and Structures, Vol. 13, p. 1203-1210, 2004.

32. Shen, Y., Golnaraghi, M.F., Heppler, G.R., “Experimental Research and Modeling

of Magnetorheological Elastomers,” Journal of Intelligent Material Systems and

Structures, Vol. 15, p. 27-35, 2004.

33. Gong, X.L., Zhang, X.Z., and Zhang, P.Q., “Fabrication and characterization of

isotropic magnetorheological elastomers,” Polymer Testing, Vol. 24, p. 669-676,

2005.

160

34. Wang, X., Gordaninejad, F., Hitchcock, G., “A Dynamic Behaviors of Magneto-

Rheological Fluid - Elastomer Composites Under Oscillatory Compression,”

Damping and Isolation, Conference on Smart Materials and Structures, Proc. of

SPIE Vol. 5386, p. 250-258, 2004.

35. Wang, X., Gordaninejad, F., Hitchcock, G., “A magneto-rheological fluid

elastomer vibration isolator,” Smart Structures and Materials, Proc. of SPIE Vol.

5760, p. 217-225, 2005.

36. Gordaninejad, F., Fuchs, A., Wang, X., Hitchcock, G.H., and Xin, M., “Magneto-

rheological fluid encased in flexible materials for vibration control,” U.S. Patent

6,971,491 B1, 2005.

37. Nguyen, T., Schroeder, C., Ciocanel, C., and Elahinia, M., “On the Design and

Control of a Squeeze-Flow Mode Magnetorheological Fluid Mount,” Proc. of the

ASME 2007 IDETC/CIE, Las Vegas, Nevada, 2007.

38. Ahmadian, M., Ahn, Y.A., “Performance Analysis of Magneto-Rheological

Mounts,” Journal of Intelligent Material Systems and Structures, Vol. 10, p. 248-

256, 1999.

39. Delphi Chassis and Steering Systems, “Delphi Magneto-Rheological (MR)

Powertrain Mount,” http://delphi.com/manufacturers/auto/other/mount/magneto, p.

1-4, 2005.

40. Baudendistel, T.A., Tewani, S., Long, M., and Dingle, J., “Hybrid Hydraulic

Mount with Magnetorheological Fluid Chamber,” Delphi Technologies, Inc., U.S.

Patent 6,412,761 B1, 2002.

41. Tewani, S.G., Bodie, M., Kiefer, J., Long, M., Walterbusch, J.,

“Magnetorheological-Fluid Hydraulic Mount,” Delphi Technologies, Inc., U.S.

Patent 7,118,100 B2, 2006.

42. Vahdati, N., Ahmadian, M., “Single Pumper Semi-active Fluid Mount,” Proc. of

ASME 2003 IMECE 2003, International Mechanical Engineers Conference,

Washington, D.C, 2003.

43. Choi, S.B., Hong, S., Choi, Y., Wereley, N., “Vibration control of flexible

structures using MR and piezoceramic mounts,” Smart Structures and Materials,

Proc. of SPIE Vol. 5390, p. 127-134, 2004.

161

44. Inman, D.J., Engineering Vibration, Third Edition, Pearson Education, Inc., Upper

Saddle River, New Jersey, 2008.

45. Emmons, S.G., “Characterizing a Racing Damper's Frequency Dependent

Behavior with an Emphasis on High Frequency Inputs,” M.S. Thesis, Virginia

Polytechnic Institute and State University, Blacksburg, VA, 2007.

46. Dorf, R.C., Bishop, R.H., Modern Control Systems, Ninth Edition, p. 38-40,

Prentice-Hall, Inc, Upper Saddle River, New Jersey, 2001.

47. Burchett, B.T., Layton, R.A., “An Undergraduate System Identification

Experiment,” American Control Conference, Portland, OR, USA, 2005.

48. Meeker, D., Finite Element Method Magnetics (FEMM), Foster-Miller, Inc.,

http://www.foster-miller.com/magnetic_modeling.htm, 2006.

49. Carlson, J. D., “Mr Fluids and Devices in the Real World,” International Journal

of Modern Physics B, Vol. 19, p. 1463-1470, 2005.

50. Lord Corporation, “MRF-122 Product Bulletin,” Lord Technical Data, 2008.



53. Lord Corporation, “Lord Magneto-Rheological Fluids,” Lord Product Selector

Guide, 2008.

54. PolyTek Development Corp., http://www.polytek.com

55. Redwood Plastics Corp., “Hardness Comparison Chart,” Redco Polyurethane

Brochure, 2008.

56. Ogata, K., System Dynamics, Third Edition, p. 302-305, Prenctice Hall, Saddle

River, New Jersey, 1998.

57. Lord Corporation, www.lord.com

58. Cosmo Corporation, http://www.cosmocorp.com/en

59. A.P.W Company Inc., http://www.apwcoinc.com

60. Belmont Machining, Inc., (540) 639-7102

162

Appendix A: Mount and Magnetic Design Schematics First, this appendix presents the bill of materials for the characterized system in this

research. Design schematics for the magnetic system and mount are presented second and

followed by the design of the 3-plate-mold. In final, a section is devoted to the process of

manufacturing an elastic case fluid mount. This process does not detail the fabrication of

the metal-elastic case mount which was covered in chapter 3.

A-1 Bill of materials for the system During the manufacture of the system and components for the MR fluid-elastic mount

design, many process materials were needed as seen Table A- 1. The category is made up

of the components and each component is specified to have a certain material or

specification. The materials used in the manufacturing process have been discussed in

chapter 3 and will be covered more in the last section of Appendix A. Furthermore, cost

estimates within this bill of materials are not presented.

163

Table A- 1: Bill of Materials without cost estimates for mount and magnet system and manufacture.

164

A-2 Dimensional schematics for 3-plate Mold This section presents the shop drawings for the 3-plate-mold. The top plate as seen in

Figure A-1 contains thru holes and an inset cavity. The inset cavity is used to mold half of

the elastic or metal-elastic case. Figure A-2 shows the middle plate which has extrusions

or embosses to create the insert cavity. This middle plate also has an axial face o-ring

location for matting to the top plate and sealing the upper half of the mold. In final a

bottom plate similar to the top plate is shown in Figure A-3 where the only difference is

an o-ring gland for matting to the middle plate. Moreover, when the middle plate is

removed from the mold there is only one axial o-ring gland to secure containment of the

mold cavity.

Figure A-1: Top plate schematic of three plate mold.

165

Figure A-2: Middle plate schematic of three plate mold.

166

Figure A-3: End plate schematic of three plate mold.

167

A-3 Dimensional schematics for the Mount and Magnet Design This section presents the dimensional schematics for the mount and magnet system

design. The lower housing to the magnet is shown in Figure A-4. The core for the

magnet is also depicted and has a small shelf for holding a non-magnetic spacer. Figure

A-5 illustrates the dimensions for the upper housing of the magnetic system, but does not

show the inset plug cavity that had been used to plug the metal-elastic case with fluid.

The aforementioned spacer is shown in Figure A-6 and needs to be fabricated from a non-

magnetic metal. Figure A-7 and Figure A-8 are the upper and magnetic-pole plates for the

metal-elastic case, respectively. The upper plate has a threaded hole in the center which is

used to for filling the case with fluid and for final sealing with a plug. The remaining

fixtures in Figure A-9 and Figure A-10 are developed to adapt to the Roehrig EMA shock

dynamometer and either the upper or lower housing of the magnetic system.

Figure A-4: Lower housing base and core schematic to magnetic system.

168

Figure A-5: Upper housing schematic to magnetic system

Figure A-6: Spacer schematic to lower housing in magnetic system.

169

Figure A-7: Upper-pole plate schematic for metal-elastic case mount.

Figure A-8: Magnetic-pole plate schematic for metal-elastic case mount.

170

Figure A-9: Lower housing test fixture schematic for Roehrig Dynamometer.

Figure A-10: Upper Housing Test Fixture for Roehrig Dynamometer.

171

A-4 Manufacturing a Fluid-Elastic Mount in an Elastic Case This section presents the procedures for manufacturing a fluid-elastic mount in a generic

elastic case as illustrated by the mount chronology in Figure A-11. Many of the processes

contained herein require specialized fabrication equipment. This equipment includes a jig

to hold the mount and inject the fluid, as well as a jig to hold the mold. The equipment is

made universal and is not detailed by a schematic.

Figure A-11: Elastic case mount chronology from initial case half mount to finalized MR Fluid-Elastic mount in a full elastic case.

Procedure for manufacturing a fluid-elastic mount in an elastic case:

1. Gather all supplies needed to make product, refer to Figure A-12

a. Clean and assembled mold

b. Clean tubing (1/4in OD)

c. Dispensing Syringes

d. Mixing cups and stirring sticks

2. Spray the 3-plate mold with poly-release

3. Mix the polyurethane at a 1:1 weight ratio in mixing cup shown in Figure A-12

4. Put polyurethane mixture in degas chamber as seen in Figure A-13

5. Remove polyurethane and pour into syringe

6. Plug syringe into the mold and dispense liquid rubber

Case Half Un-Prepped

Elastic Case

Prepped

Case

Filled

Elastic Case

172

a. Rotate material around in mold to coat the inside walls of the mold

b. Finish dispensing liquid in mold.

7. Allow urethane to cure for 16 hours.

8. Remove case halves from mold as seen in Figure A-14 and Figure A-15 to

clean up parting material on the edges and to wash off release agent

9. Remove some of the outer material as seen in Figure A-16 and put back in top

and bottom plate mold as seen in Figure A-17

a. Repeat steps 2-7 but de-mold within 4 hours

10. With closed circular disk, remove material from one edge and drill small hole

to allow for puncture of needle and wash off release agent

11. Stir and degas MR fluid

12. Place circular polyurethane disk in the jig as seen in Figure A-18

13. Put MR fluid in a syringe and inject fluid in to circular polyurethane disk.

14. Repeat steps 2-7

Figure A-12: Paraphernalia readied for manufacturing an elastic case mount.

173

Figure A-13: Polyurethane in a degassing chamber under 28inHg to remove entrapped air.

Figure A-14: De-molding the half cases of the mount from the 3-plate mold.

174

Figure A-15: Each half of the elastic case after removal of central parting lines from middle plate of mold.

Figure A-16: Degreased and abraded elastic case halves ready to be inserted in top and bottom mold plates to create the full elastic case with hollow insert cavity.

175

Figure A-17: Prepped halves placed in top and bottom plate with a bead of polyurethane on the face of the elastic case half.

Figure A-18: Universal jig used to secure elastic case and position fluid syringe for MR fluid injection into the empty case cavity.

176

Appendix B: Results This appendix contains the results from analyzing the mount parameters. In addition,

comparisons are also shown as well as the error comparison between processing methods.

B-1 Damping Analysis for Passive and MR Fluid-Elastic Mounts This section presents the damping results for the MR fluid-elastic and passive mounts.

Table B-1 lists the damping values for the passive mounts and Table B-2 lists the damping

values for the MR fluid-elastic case mounts with the non-filled metal-elastic case mount

MRE 3B. In final, Table B-3 presents a comparison of the results for the passive mounts

and the MR fluid-elastic mounts.

Table B-1: Passive mount damping analysis results for the air, rubber, steel, and aluminum inserts

177

Table B-2: MR fluid-elastic mount damping analysis results for MRE’s and blank MRE 3.

Table B-3: MR fluid-elastic mount and passive mount damping analysis comparison chart.

178

B-2 Stiffness Results for MR Fluid-Elastic Mounts This section presents the stiffness magnitude |F|/X results in Table B-4 for the MR fluid-

elastic case mounts.

Table B-4: MR Fluid-elastic mount Stiffness Analysis Results for MRE’s and blank MRE 3.

179

B-3 Stiffness Results for Passive and MR fluid-Elastic Mounts This section presents the stiffness results for the passive mounts in Table B-5 which is

followed by a stiffness results comparison for both the MR fluid-elastic mounts and the

passive mounts in Table B-6.

Table B-5: Passive mount stiffness analysis results for the air, rubber, steel, and aluminum inserts at 0.50 Amp current indexing.

Table B-6: MR fluid-elastic and passive mount stiffness analysis comparison chart.

180

B-4 Parameters for f(t) and x(t) Processing This section contains the parameters for f(t) and x(t) as listed in Table B-7. These

parameters were used or found by Program_4 in Appendix C. The major parameters

found include the force magnitude |F|, the offset force F-Offset, the displacement input

magnitude |X|, the displacement offset X-Offset, and the displacement removed for the

saturated force content Xt_SAT.

Table B-7: MR Fluid-elastic mount parameters from force-amplitude and displacement modeling analysis at 0, 1, and 2-Amp current settings.

181

B-5 Processing Evaluation Method Error for MR Fluid-Elastic Mounts This section presents a more thorough error chart as seen in Table B-8 for the error

obtained using either the force-amplitude |F| or the force-displacement Kx processing

methods. As noticed, the error in the force time trace and the error in the displacement

input need to be combined to represent the total error for the force-amplitude method.

The input trace showed slight error at higher frequencies because the shock dyno’s

electromagnetic actuator was being operated at the lower resolution of the available

displacement range.

Table B-8: MR Fluid-elastic mount error comparison between force-amplitude |F|/X and force-displacement Kx, sampled at 0, 1, 2-Amp for MR fluid-elastic mounts 1, 2 and 3.

182

B-6 Transfer Function Nominal Parameters This section contains the converged transfer function parameters in Table B-9 that were

used to construct the transfer function model for each current setting.

Table B-9: Nominal transfer function parameters used to simulate the results in section 5.2.

183

Appendix C: Data Processing Code This appendix presents the MatLab code used for processing the MR fluid-elastic mounts

and comparison mounts from this study. The layout for this appendix is:

1. results processing program, and

2. transfer function modeling program

Section C-1 presents the results processing code for both the force-displacement and

force-amplitude analysis methods. Then section C-2 presents the code for modeling the

stiffness magnitude in the frequency domain for the MR fluid-elastic mounts.

C-1 Results Processing As previously mentioned, this section contains the MatLab code used to analyze the force-

displacement and force-amplitude stiffness. The code is then shown for the:

• force-displacement stiffness,

• force-displacement damping, and

• force-amplitude stiffness.

Results Processing Code:

clc, clear all,close all

% Code loads Shock data from CSV, plots it, and saves it to a Matlab Structure

% The Matlab structure can be loaded in the workspace later for custom data analysis

% F(t) Model = (ampF*sin(2piF*t+phaseF)+off)*(1-sign(xT-X(t)))/2

% x(t) Model = X*sin(2piF*t+phaseX)+offX

% K(x) Model = K*x, or K = F(t)/x(t)

first_row = 28; %first row of data in CSV file

fsamp = 2000; %sample rate, Hz

nfiles = 18; %12 for AIR, 18 for all others %number of files to load #16 [1-25hz]

nfileshold = 18; %set to 18 to print to individual amp excel folder

ampfull = 5;

ampfullset = 5; %Turn on plotters using #5 or 9;

ampfullsetPrint = 5; % Turn on with 5 or 9, if amps = ampfullsetPrint export will happen

a = 1; z = 18; %for i = frequency a to frequency z, 12 or 18

%% %Select the following to make me work

mount = 8; %(1-8), pick mount number to describe elastomer

fileholes = {'MRE 1 DST','MRE 2 DST','MRE 3 DST',...

'MRE 3 Blank DST',...

'AIR DST','RUB DST','STE DST','ALU DST'};

184

innerholes = {'\MRE 1 ','\MRE 2 ','\MRE 3 ','\By Current\MRE 3b ','\AIR DST ','\RUB

','\','\'};

filehole = fileholes{mount}; %input the fileset you want to review

innerhole = innerholes{mount};

loadermounts = {'MRE_1_DST_01mm_','MRE_2_DST_01mm_','MRE_3_DST_01mm_',...

'MRE_3b_DST_01mm_','AIR_DST_01mm_','RUB_DST_01mm_',...

'STE_DST_005mm_','ALU_1_DST_005mm_'};

loadermount = loadermounts{mount};

elastomer = {'MRE_','MRE_','MRE_','MRE_','AIR','RUB_','STE_','ALU_'}; %Model

elastomers = fileholes;

if mount <=4

current = {'000','025','050','075','100','125','150','175','200'};

ampleg = {'0.00-A','0.25-A','0.50-A','0.75-A','1.00-A','1.25-A',...

'1.50-A','1.75-A','2.00-A'};

L40 =

{['|F|/X:',ampleg{1}],['K(x):',ampleg{1}],['|F|/X:',ampleg{3}],['K(x):',ampleg{3}],...

['|F|/X:',ampleg{5}],['K(x):',ampleg{5}],['|F|/X:',ampleg{7}],['K(x):',ampleg{7}],...

['|F|/X:',ampleg{9}],['K(x):',ampleg{9}]};

elseif mount >4

current = {'000','050','100','150','200'};

ampleg = {'0.00-A','0.50-A','1.00-A','1.50-A','2.00-A'};

L40 =

{['|F|/X:',ampleg{1}],['K(x):',ampleg{1}],['|F|/X:',ampleg{2}],['K(x):',ampleg{2}],...

['|F|/X:',ampleg{3}],['K(x):',ampleg{3}],['|F|/X:',ampleg{4}],['K(x):',ampleg{4}],...

['|F|/X:',ampleg{5}],['K(x):',ampleg{5}]};

end

if mount == 5

hzs = {'1hz','2hz','3hz','4hz','5hz','6hz','7hz','8hz','9hz','10hz',...

'20hz','30hz'};

freqfull = [1,2,3,4,5,6,7,8,9,10,20,30];

freqleg = {'1-Hz','2-Hz','3-Hz','4-Hz','5-Hz','6-Hz','7-Hz','8-Hz',...

'9-Hz','10-Hz','20-Hz','30-Hz'};

nfiles = 12;

else

hzs = {'1hz','2hz','3hz','4hz','5hz','6hz','7hz','8hz','9hz','10hz',...

'12hz','14hz','16hz','18hz','20hz','25hz','30hz','35hz'};

freqfull = [1,2,3,4,5,6,7,8,9,10,12,14,16,18,20,25,30,35];

freqleg = {'1-Hz','2-Hz','3-Hz','4-Hz','5-Hz','6-Hz','7-Hz','8-Hz',...

'9-Hz','10-Hz','12-Hz','14-Hz','16-Hz','18-Hz','20-Hz','25-Hz',...

'30-Hz','35-Hz'};

end

marker = {'v','d','^','x','o','p','*','.','+'};

colors1 = {'k','r','g','b','k','','g','b','k','r','g','b'};

lines = {'-',':','-','--','-',':','-','--','-',':','-','--'};

fsize = 8; %font size

tsize = 7; %title font size

xfsize = 10; %x axis font size

yfsize = 10; %y axis font size

185

msize = 5; %markersize

lsize = 8;

for amps = 1:ampfull

for i=a:z

pathname = ['C:\Documents and Settings\Administrator\My Documents\CSV Mount Files\A

CSV Mount Files\',filehole,innerhole,current{amps},'a'];

%path for file window

cd(pathname);

colors = 'bgrcmkbgrcmkbgrcmkbgrcmk';

freq = freqfull(i); %index freq. collected to calculate desired file frequency

%% Manual Load

% [filename, pathname] = uigetfile('*.csv',...

% ['Select CSV Shock Data File #',freqleg{i}]);

% [filename, pathname] = csvread(mountfile,

%%Auto Load

mountfile = [loadermount,current{amps},'a_',hzs{i},'.csv'];

filename = mountfile;

cd(pathname);

dd = csvread(filename,first_row-1,0);

disa = dd(:,1);

Forcea = dd(:,2);

N = length(disa);

disp = disa(1:(N-2));

Forced = Forcea(1:(N-2));

ForceFix = min(Forced);

ForcedNorm = Forced - ForceFix;

Force = smooth(ForcedNorm,10);

dispFix = min(disp);

dispNorm = disp - dispFix;

dis = smooth(dispNorm,10);

N = length(dis);

t = (0:(N-1)) * 1/fsamp;

xTime = 1/freq;

pN = floor(xTime*fsamp);

pTime = (0:(pN-1))*1/fsamp;

pdisA = dis(1:pN);

pdisB = dis(pN+1:2*pN);

pdis = (pdisA + pdisB)./2;

pForceA = Force(1:pN);

pForceB = Force(pN+1:2*pN);

pForce = (pForceA + pForceB)./2;

%_____________________________________________________________

%estimating velocity by differentiating the displacement input

%curve fitting from displacement

xdata = t';

ydata = dispNorm;

if mount == 5

186

xitterfreq =

{'a*sin(2*pi*1*x+b)+c','a*sin(2*pi*2*x+b)+c','a*sin(2*pi*3*x+b)+c',...

'a*sin(2*pi*4*x+b)+c','a*sin(2*pi*5*x+b)+c','a*sin(2*pi*6*x+b)+c',...


'a*sin(2*pi*10*x+b)+c','a*sin(2*pi*20*x+b)+c','a*sin(2*pi*30*x+b)+c'};

else

xitterfreq =

{'a*sin(2*pi*1*x+b)+c','a*sin(2*pi*2*x+b)+c','a*sin(2*pi*3*x+b)+c',...



'a*sin(2*pi*10*x+b)+c','a*sin(2*pi*12*x+b)+c','a*sin(2*pi*14*x+b)+c',..

'a*sin(2*pi*16*x+b)+c','a*sin(2*pi*18*x+b)+c','a*sin(2*pi*20*x+b)+c',..

'a*sin(2*pi*25*x+b)+c','a*sin(2*pi*30*x+b)+c','a*sin(2*pi*35*x+b)+c'};

end

g = fittype(xitterfreq{i});

[Xvec,Xerrors] = fit(xdata,ydata,g,'Start',[max(ydata)/2 0.3 max(ydata)/2]);

DisRMSE = Xerrors.rmse;

NormDisRMSE = (Xerrors.rmse)/(max(ydata)-min(ydata));

d1 = differentiate(Xvec,xdata);

phaseX = Xvec.b;

ampX = Xvec.a;

wX = 2*pi*freqfull(i);

offX =Xvec.c;

freqfindX = freqfull(i);

v = d1;

vdot = v;

vdot(N)= vdot(N-1);

%F(t) Model Section:

%Fit Force Data to solve for phase and frequency

%remove saturated force data by using a sign(x(t)) function

fitterfreq =

strcat('(a*sin(',num2str(wX),'*x+b)+c)*(1+sign(',num2str(offX),'+',num2str(ampX),'*sin(',n

um2str(wX),'*x+(',num2str(phaseX),'))-d))./2');

fitforce = fittype(fitterfreq);

[xTL,xT,pT] = program4fun([mount,amps,i]);

[Fvec, ErrorAmp] = fit(xdata,Force,fitforce,'Start',[max(Force)./1.3 pT

max(Force)./3 xT]);

ampF = Fvec.a;

phaseF = Fvec.b;

offF = Fvec.c;

xT = Fvec.d;

wF = 2*pi*freqfull(i);

freqfindF = freqfull(i);

phasediff = rad2deg(phaseF-phaseX);

LowForce = min(Force);

ErrorF = abs(max(Force)-(abs(ampF) + offF))./max(Force);

AmpRMSE = ErrorAmp.rmse; %square root of the deviation. Syx = root(sum(Xi-

Xci)^2/N) %Xi-data, Xci-model, N-number of data points

187

NormFtRMSE = ErrorAmp.rmse/(max(Force)-min(Force));

Fvecs= ((ampF.*sin(wX*xdata + phaseF)+offF).*(1+sign(offX +

ampX.*sin(wX*xdata+phaseX)-xT))./2);

%Plot bad Force F(t) Models.

if min(ErrorF) >= 0.1

figure(100+i),plot(Fvec,xdata,Force), title(['Failed Inspection

',elastomer{mount},',',current{amps}]),...

grid on, axis tight, xlabel('Time, s'),ylabel('Force,

N'),legend(freqleg{i})

end

phasediffold = phasediff;

[phasediff] = program4phase([phasediffold]);

%% Kx Model Section

xTl = xTL;%0.25; %0.35; %xlow

xTh = 0.01; %xhigh kicks off data at the upper end of domain

linear = excludedata(pdis,pForce,'domain',[xTl,max(pdis)-xTh]);

%look only at data within the domain specified [xlow to xhigh]

dE = polyarea(pdis(~linear),pForce(~linear));

%calculate the area within the domain saturate

ceq = dE/(pi*freq*2*pi*((ampX)^2));

fitslope = fittype('a*x + b');

[Flin,Goodness] = fit(pdis(~linear),pForce(~linear),fitslope,'Start',[2300 -300]);

LinRMSE = Goodness.rmse;

NormKxRMSE = Goodness.rmse/(max(pForce(~linear))-min(pForce(~linear)));

slope = Flin.a;

b = Flin.b;

newdis = pdis(~linear);

linMod = slope*pdis(~linear) + b;

%% %Collect Parameters and Load to Table

row = i+nfiles*(amps-1);

XFHolds(row,1) = freq;

XFHolds(row,2) = abs(ampF)./abs(ampX);

XFHolds(row,3) = slope;

XFHolds(row,4) = phasediff;

XFHolds(row,5) = ceq;

XFHolds(row,6) = ErrorF; %error in amplitude fit

XFHolds(row,7) = NormFtRMSE*100;

XFHolds(row,8) = NormDisRMSE*100;

XFHolds(row,9) = NormKxRMSE*100;

sysparam(row,1) = freq; %Hz

sysparam(row,2) = slope; %F/x

sysparam(row,3) = abs(ampF)./abs(ampX); %|F|/X

sysparam(row,4) = ampF; %|F|

sysparam(row,5) = ampX; %X

sysparam(row,6) = offF; %DC Force N

sysparam(row,7) = phaseF; %rad F

sysparam(row,8) = phaseX; %rad X

sysparam(row,9) = xT; %saturate X value

188

sysparam(row,10) = AmpRMSE; %|F| vs Force(t)

sysparam(row,11) = DisRMSE; %X vs x(t)

sysparam(row,12) = LinRMSE; %F/x vs F/x(T)

sysparam(row,13) = NormFtRMSE*100; %RMSE/(maxF-minF)

sysparam(row,14) = NormDisRMSE*100; %RMSE/(maxX-minX)

sysparam(row,15) = NormKxRMSE*100; %RMSE/(maxpF-minpF)

sysparam(row,16) = ForceFix; rowa = i;

XFHold(rowa,1) = freq;

XFHold(rowa,2) = abs(ampF)./abs(ampX);

XFHold(rowa,3) = slope;

XFHold(rowa,4) = phasediff;

XFHold(rowa,5) = ceq;

XFHold(rowa,6) = ErrorF; %error in amplitude fit

XFHold(rowa,7) = NormFtRMSE*100;

XFHold(rowa,8) = NormDisRMSE*100;

XFHold(rowa,9) = NormKxRMSE*100;

if ampfullset == 5

%% %Figure1s F(t) Model

dks = {'k',':c'};

dk = dks{1};

L1s = {['Data:',freqleg{i}],'Model,F(t)'};

figure(i+0),subplot(3,2,amps),plot(xdata,Force,dk,'LineWidth',2),hold on,

plot(xdata,Fvecs,'m','LineWidth',1),xlabel(''),

ylabel('Force,N','fontsize',yfsize),xlabel('Time, s','fontsize',xfsize), axis([0 max(t) 0

5000]),

legend(L1s,'Location','NorthEast','fontsize',lsize),

%% %Figure20s K(x) Model

L20s = {['Data:',freqleg{i}],'RegionModel','Model-K(x)'};

figure(i+20),subplot(3,2,amps),plot(pdis,pForce,dk,'LineWidth',2),hold on,

plot(pdis(~linear),pForce(~linear),'m','LineWidth',1),hold on,

plot(pdis(~linear),linMod,'--b','LineWidth',1),

legend(L20s,'Location','NorthWest','fontsize',lsize),

ylabel('Force, N','fontsize',yfsize),

xlabel('disp, mm','fontsize',xfsize),

axis([0 1 0 5000])

%% %Figure60s F(t)_Model vs F/x_Model


if (amps==1)||(amps==3)||(amps==5)

spot = 1+(amps-1); %1+(amps-1)/2;%(1+(1-1)/2 = 1,1+(5-

1)/2=3,1+(9-1)/2 =5

figure(i+60),subplot(3,2,spot),plot(xdata,Force,dk,'LineWidth',2), hold on,


ylabel('Force,

N','fontsize',yfsize),legend(L1s,'Location','NorthEast','fontsize',lsize),

xlabel('Time, s','fontsize',xfsize),axis([0 max(t) 0

5000]),

189

text(xTime*.1,4000,['NRMSE =

',num2str(round(NormFtRMSE*100)),'%'],'fontsize',lsize),

figure(i+60),subplot(3,2,spot+1),

plot(pdis,pForce,dk,'LineWidth',2),hold on,

plot(pdis(~linear),pForce(~linear),'m','LineWidth',1),hold

on

plot(pdis(~linear),linMod,'--b','LineWidth',1),hold on


axis([0 1 0 5000]),ylabel('Force,

N','fontsize',yfsize),xlabel('disp, mm','fontsize',xfsize),

text(0.05,2000,['NRMSE =

',num2str(round(NormKxRMSE*100)),'%'],'fontsize',lsize)

end

%% %Figure80s 0,1,2-Amp F(t)_Model vs K(x)_Model

L80s = {['Data:',freqleg{i}],'Model-F(t)'};



spot = 1+(amps-1);%(1+(1-1)/2 = 1, 1+(5-1)/2=3, 1+(9-1)/2 =

5


plot(xdata,Fvecs,'m','LineWidth',1),xlabel('Time, s','fontsize',xfsize),

ylabel('Force,

N','fontsize',yfsize),legend(L80s,'fontsize',lsize), axis([0 max(t) 0 5000]),


',num2str(round(NormFtRMSE*100)),'%'],'fontsize',lsize)




plot(pdis(~linear),linMod,'--b','LineWidth',1),hold on,

text(0.05,2000,['NRMSE =


xlabel('disp, mm','fontsize',xfsize), axis([0 1 0 5000]),

legend(L81s,'Location','NorthWest','fontsize',lsize),xlabel('disp, mm','fontsize',xfsize)

end

elseif ampfullset == 9

%% %Figure1s F(t) Model

dks = {'k',':c'};

dk = dks{1};

L1s = {['Data:',freqleg{i}],'Model,F(t)'};

figure(i+0),subplot(3,3,amps),plot(xdata,Force,dk,'LineWidth',2),hold on,


ylabel('Force,N','fontsize',yfsize),xlabel('Time, s','fontsize',xfsize), axis([0 max(t) 0

5000]),

legend(L1s,'Location','NorthEast','fontsize',lsize),

%% %Figure20s K(x) Model


figure(i+20),subplot(3,3,amps),plot(pdis,pForce,dk,'LineWidth',2),hold on,


190

plot(pdis(~linear),linMod,'--b','LineWidth',1),


ylabel('Force, N','fontsize',yfsize),

xlabel('disp, mm','fontsize',xfsize),

axis([0 1 0 5000])



spot = 1+(amps-1)/2;%(1+(1-1)/2 = 1,



ylabel('Force,

N','fontsize',yfsize),legend(L1s,'Location','NorthEast','fontsize',lsize),

xlabel('Time, s','fontsize',xfsize),axis([0 max(t) 0

5000]),


',num2str(round(NormFtRMSE*100)),'%'],'fontsize',lsize),



plot(pdis(~linear),pForce(~linear),'m','LineWidth',1),hold on

plot(pdis(~linear),linMod,'--b','LineWidth',1),hold on


axis([0 1 0 5000]),ylabel('Force,

N','fontsize',yfsize),xlabel('disp, mm','fontsize',xfsize),

text(0.05,2000,['NRMSE =


end

%% %Figure80s 0,1,2-Amp F(t)_Model vs K(x)_Model

L80s = {['Data:',freqleg{i}],'Model-F(t)'};



spot = 1+(amps-1)/2;%(1+(1-1)/2 = 1, 1+(5-1)/2=3, 1+(9-1)/2


plot(xdata,Fvecs,'m','LineWidth',1),xlabel('Time, s','fontsize',xfsize),

ylabel('Force,

N','fontsize',yfsize),legend(L80s,'fontsize',lsize), axis([0 max(t) 0 5000]),


',num2str(round(NormFtRMSE*100)),'%'],'fontsize',lsize)




plot(pdis(~linear),linMod,'--b','LineWidth',1),hold on,

text(0.05,2000,['NRMSE =


xlabel('disp, mm','fontsize',xfsize), axis([0 1 0 5000]),

legend(L81s,'Location','NorthWest','fontsize',lsize),xlabel('disp, mm','fontsize',xfsize)

end

elseif (ampfullset ~= 5) && (ampfullset ~= 9)

['Figure Printing to Screen is Off']

191

end

%% %I save XFHold at amp(1-9) or 0-2amp

if i == nfileshold

pathname4 = ['C:\Documents and Settings\Administrator\My Documents\CSV

MOUNT FILES\A CSV Mount Files\',filehole,'\Model\Each'];

cd(pathname4)

xlswrite([current{amps},'A_','XFHold_',elastomer{mount},num2str(mount)],XFHold)

end

end %for i=nfiles loop

%% %I make the styles

Stiff = XFHold(:,2);

style1 = [lines{amps},marker{amps},colors1{amps}];

style2 = [':',marker{amps},'m'];

%% %Figure40 Compare Stiffness Vs Frequency

% if amps==1||amps==3||amp==5||amp==7||amp==9

figure(40),plot(XFHold(:,1),XFHold(:,2),style1,'MarkerSize',6),hold on,

xlabel('Frequency, Hz','fontsize',14),ylabel('Stiffness, N/mm','fontsize',14),hold

on,

plot(XFHold(:,1),XFHold(:,3),style2,'MarkerSize',5,'MarkerFaceColor','m'),

legend(L40,'location','SouthEast','fontsize',lsize), axis([0 35 0 7000])

% end

%% %Figure41-43

figure(41),plot(XFHold(:,1),XFHold(:,4),style1),hold on,

xlabel('Frequency, Hz','fontsize',14),ylabel(['Phase,

deg','\circ',],'fontsize',14),

legend(ampleg,'location','NorthEast'), axis([0 35 -30 30])


xlabel('Frequency, Hz','fontsize',14),ylabel('Stiffness, N/mm','fontsize',14),

legend(ampleg,'location','SouthEast'),hold on, axis([0 35 0 7000])


xlabel('Frequency, Hz','fontsize',14),ylabel('Stiffness, N/mm','fontsize',14),

legend(ampleg,'location','SouthEast'),hold on, axis([0 35 0 7000])

%% %Write Data to Files when current = 2.0 or amp == 9

if amps == ampfullsetPrint

pause on

%Store constants, damping, stiffness terms in mat file

pathname3 = ['C:\Documents and Settings\Administrator\My Documents\CSV MOUNT FILES\A

CSV Mount Files\',filehole,'\Model'];

cd(pathname3)

syslabel = {'Frequency','Kx_Model','Ft/X_Model',...

'Ft_amp','Xt_amp','Offset Ft','Phase Ft','Phase Xt',...

'Sat xT','FtRMSE','XtRMSE','KxRMSE','NormFtRMSE',...

'NormDisRMSE','NormKxRMSE','Force Normalized'};

xlswrite('Sysparam_Data',syslabel,[elastomer{mount},num2str(mount)],'A1')

xlswrite('Sysparam_Data',sysparam,[elastomer{mount},num2str(mount)],'A4') %load to

All_Mount.mat

pause(6)

192

xlswrite(['XFHold_Data_',elastomer{mount},num2str(mount)],XFHolds) %load to

comparison spreadsheet

pause(6)

matdata1 = struct('XFHolds',XFHolds); %shockdata1 for 2 column csv

eval(['QuickXF' filename(1:length(filename)-8) ' = matdata1']);

eval(['save ', 'QuickXF', filename(1:length(filename)-8)]) %saves structure file

%Full contains all 1-9amps

pause(6)

matdata2 = struct('sysparam',sysparam);

eval(['Model' filename(1:length(filename)-8) ' = matdata2']);

eval(['save ', 'Model', filename(1:length(filename)-8)])

pause(6)

%%

pathname5 = 'C:\Documents and Settings\Administrator\My Documents\CSV MOUNT FILES\A

CSV Mount Files\Program Folder';

cd(pathname5)

pause(5)

xlswrite('Program_4_Stiffness',syslabel,filehole,'A1')

xlswrite('Program_4_Stiffness',sysparam,filehole,'A4')

%% %Export figures(40-43) when current = 2.0 or amp == 9

pathname2 = ['C:\Documents and Settings\Administrator\My Documents\My Pictures\Chapter

5 Mount\',filehole];

cd(pathname2)

f40 = ['-f',num2str(40)];

print(f40,'-r900','-dtiff','Stiffness_Amplitude_Linear_vs_Freq'),

pause(8)

% figure(40),set(gcf),close gcf;

f41 = ['-f',num2str(41)];

print(f41,'-r900','-dtiff','Phase_Amplitude_Freq')

% figure(41),set(gcf),close gcf;

pause(8)

f42 = ['-f',num2str(42)];

print(f42,'-r900','-dtiff','Stiffness_Amplitude_vs_Freq')

pause(8)

f43 = ['-f',num2str(43)];

print(f43,'-r900','-dtiff','Stiffness_Linear_vs_Freq')

%% %Export figures(1-98) when current = 2.0, or amp == 9

for i = 1:nfiles

% %Figure1s

pause(8)

famp3x3 = ['-f',num2str(i)];

print(famp3x3,'-r300','-dtiff',['Amp_Ft_Fit_',current{amps},'A_',freqleg{i}])

%% %Figure20s

pause(8)

fi20 =['-f',num2str(i+20)];

print(fi20,'-r300','-dtiff',['Line_Kx_Fit_',current{amps},'A_',freqleg{i}])

pause(10)

fi60 = ['-f',num2str(i+60)];

193

print(fi60,'-r300','-dtiff',['Ft_Kx_',current{amps},'A_',freqleg{i}])

pause(6)

fi80 = ['-f',num2str(i+80)];

print(fi80,'-r300','-dtiff',['Ft_Kx_Text',current{amps},'A_',freqleg{i}])

['Printed Figures for Frequency:',freqleg{i},'-Hz']

end

'Exporting Complete'

end

end

FUNCTION FOR PROGRAM 4

function [xTLout,xTsout,pTsout] = program4fun(needing)

mount = needing(1);

amp = needing(2);

freq = needing(3);

if mount == 1

pTs =

[1.5,1.5,1.5,1.5,1.5,1.5,1.5,1.5,1.5,1.5,1.5,1.5,0.2,1.9,1.8,1.1,1.3,0.7];

xTs = (1+(amp-

1)/16)*[0.38,0.37,0.36,0.35,0.34,0.33,0.32,0.3,0.28,0.26,0.25,0.24,0.23,0.22,0.21,0.2,0.2,

0.2];

xTL = 0.3;

elseif mount == 2 %MRE 2

if amp == 2, pTs =

[1.5,1.5,1.5,1.5,1.5,1.5,1.5,1.5,1.5,1.5,1.5,1.5,0.7,1.9,1.1,1.5,1,1.5]

else pTs =

[1.5,1.5,1.5,1.5,1.5,1.5,1.5,1.5,1.5,1.5,1.5,1.5,0,1.5,1.5,1.5,1,1.5];

end

xTs = (1+(amp-

1)/16)*[0.38,0.37,0.36,0.35,0.34,0.33,0.32,0.3,0.28,0.26,0.25,0.24,0.23,0.22,0.21,0.2,0.2,

0.2];

xTL = 0.3;

elseif mount == 3 %MRE 3

if amp == 2, pTs =

[1.5,1.5,1.5,1.5,1.5,1.5,1.5,1.5,1.5,1.5,1.5,1.5,1.5,1.9,1.1,1.5,1,1.5]

else pTs =

[1.5,1.5,1.5,1.5,1.5,1.5,1.5,1.5,1.5,1.5,1.5,1.5,1.2,1.5,1.5,1.5,0,2];

end

xTs = (1+(amp-

1)/16)*[0.38,0.37,0.36,0.35,0.34,0.33,0.32,0.3,0.28,0.26,0.25,0.24,0.23,0.22,0.21,0.2,0.2,

0.2];

xTL = 0.3;

elseif mount == 4 %MRE 3b

xTs =

[0.3,0.3,0.3,0.3,0.3,0.3,0.3,0.3,0.3,0.3,0.3,0.3,0.3,0.3,0.3,0.3,0.3,0.3]; %all amps

pTs =

[0.3,0.3,0.3,0.3,0.3,0.3,0.3,0.3,0.3,0.8,2,0.3,0.3,0.3,0.25,0.3,0.3,0.3];

xTL = 0.3;

elseif mount == 5 %AIR

194

xTs

=[0.125,0.125,0.125,0.125,0.125,0.125,0.125,0.125,0.125,0.125,0.125,0.125]; %all amps

pTs = [0.3,0.3,0.3,0.3,0.3,0.3,0.3,0.3,0.3,2,2,0.3,0.3,0.3,0.25,0.3,0.3,0.3];

xTL = 0.125;

elseif mount == 6 %RUB

pTs = [0.3,0.3,0.3,0.3,0.3,0.3,0.3,0.3,0.3,1,1.15,1.3,2,1.3,1.25,0.3,0.3,0.3]

xTs =

[0.1,0.1,0.1,0.1,0.1,0.1,0.1,0.1,0.1,0.1,0.1,0.1,0.1,0.1,0.1,0.1,0.1,0.1];

xTL = 0.1;

elseif mount == 7 %STE

xTs =

[0.11,0.11,0.11,0.11,0.11,0.11,0.11,0.11,0.11,0.11,0.11,0.11,0.11,0.11,0.11,0.11,0.11,0.11

]; %0A=0.3,0.25A=0.33;

pTs = [1.5,1.5,1.5,1.5,1.5,1.5,1.5,1.5,1.5,1.5,1.5,1.5,3,2.3,3,1.5,3,1.5]; %at

1.5amp

xTL = 0.11;

elseif mount == 8 %ALU

xTs =

[0.11,0.11,0.11,0.11,0.11,0.11,0.11,0.11,0.11,0.11,0.11,0.11,0.11,0.11,0.11,0.11,0.11,0.11

]; %0A=0.3,0.25A=0.33

pTs =

[1.5,1.5,1.5,1.5,1.5,1.5,1.5,1.5,1.5,1.5,1.5,1.5,1.5,1.9,1.8,1.1,1.3,1.5];

xTL = 0.11;

end

xTLout = xTL;

xTsout = xTs(freq);

pTsout = pTs(freq);

%%END OF FUNCTION program4fun

195

C-2 Transfer Function Analysis Code This section presents the code used during the transfer function analysis and simulation

which was shown in chapter 5. Herein, the transfer function modeling and function code

is for:

1. retrieving |F|, and X values from program 4,

2. using fminsearch to find nominal parameters, and

3. simulating TF model and ploting comparison of empirical values.

Transfer Function Analysis Code: clc, clear all, close all fsamp = 2000; %sample rate, Hz nfiles = 18; %12 for AIR, 18 for all others %number of files to load #16 [1-25hz] nfileshold = 18; %set to 18 to print to individual amp excel folder ampfull = 9; ampfullset = 9; %Turn on plotters using #5 or 9; ampfullsetPrint = 5; % Turn on with 5 or 9, if amps = ampfullsetPrint export will happen a = 1; z = 18; %for i = frequency a to frequency z, 12 or 18 %% %Select the following to make me work mount = 8; %(1-8), pick mount number to describe elastomer mountset = 8; %use 0 for no print and 8 for full print wantprint = 0; %0-off, 1-on for all figures wantsave = 0; %0-off, 1-on for all excel data loadstorage = 1; %%DON"T PRINT OR SAVE IF SET to ONE,1,ONE,1 %% % fileholes = {'MRE 1 DST','MRE 2 DST','MRE 3 DST',... 'MRE 3 Blank DST',... 'AIR DST','RUB DST','STE DST','ALU DST'}; innerholes = {'\MRE 1 ','\MRE 2 ','\MRE 3 ','\By Current\MRE 3b ','\AIR DST ','\RUB ','\','\'}; loadermounts = {'MRE_1_DST_01mm_','MRE_2_DST_01mm_','MRE_3_DST_01mm_',... 'MRE_3b_DST_01mm_','AIR_DST_01mm_','RUB_DST_01mm_',... 'STE_DST_005mm_','ALU_1_DST_005mm_'}; elastomer = {'MRE_','MRE_','MRE_','MRE_','AIR','RUB_','STE_','ALU_'}; %Model elastomers = fileholes; Mount = {'MRE 1','MRE 2','MRE 3','MRE 3B','AIR','RUB','STE','ALU'}; %% Enter Values 1-8, or Run me in a For Loop for i = 1:mount filehole = fileholes{i}; %input the fileset you want to review innerhole = innerholes{i}; loadermount = loadermounts{i}; % currentinc = {'000','050','100','150','200'}; currentinc = {'0.0Amp','0.5Amp','1.0Amp','1.5Amp','2.0Amp'}; if i <=4 current = {'000','025','050','075','100','125','150','175','200'}; ampleg = {'0.00-A','0.25-A','0.50-A','0.75-A','1.00-A','1.25-A',... '1.50-A','1.75-A','2.00-A'}; L40 = {['|F|/X,',ampleg{1}],['K(x),',ampleg{1}],['|F|/X,',ampleg{3}],['K(x),',ampleg{3}],... ['|F|/X,',ampleg{5}],['K(x),',ampleg{5}],['|F|/X,',ampleg{7}],['K(x),',ampleg{7}],... ['|F|/X,',ampleg{9}],['K(x),',ampleg{9}]}; Li0 = {['Data,|F|/X,',ampleg{1}],['Model,TF,',ampleg{1}],['Data,|F|/X,',ampleg{3}],['Model,TF,',ampleg{3}],... ['Data,|F|/X,',ampleg{5}],['Model,TF,',ampleg{5}],['Data,|F|/X,',ampleg{7}],['Model,TF,',ampleg{7}],... ['Data,|F|/X,',ampleg{9}],['Model,TF,',ampleg{9}]}; Li1 = {[Mount{i},',|F|/X,',ampleg{1}],[Mount{i},',TF,',ampleg{1}],[Mount{i},',|F|/X,',ampleg{3}],[Mount{i},',TF,',ampleg{3}],...

196

[Mount{i},',|F|/X,',ampleg{5}],[Mount{i},',TF,',ampleg{5}],[Mount{i},',|F|/X,',ampleg{7}],[Mount{i},',TF,',ampleg{7}],... [Mount{i},',|F|/X,',ampleg{9}],[Mount{i},',TF,',ampleg{9}]}; ampfull = 9; ampin = [0,0.25,0.5,0.75,1.0,1.25,1.5,1.75,2.0]; elseif i >4 current = {'000','050','100','150','200'}; ampleg = {'0.00-A','0.50-A','1.00-A','1.50-A','2.00-A'}; L40 = {['|F|/X,',ampleg{1}],['K(x),',ampleg{1}],['|F|/X,',ampleg{2}],['K(x),',ampleg{2}],... ['|F|/X,',ampleg{3}],['K(x),',ampleg{3}],['|F|/X,',ampleg{4}],['K(x),',ampleg{4}],... ['|F|/X,',ampleg{5}],['K(x),',ampleg{5}]}; Li0 = {['Data,|F|/X,',ampleg{1}],['Model,TF,',ampleg{1}],['Data,|F|/X,',ampleg{2}],['Model,TF,',ampleg{2}],... ['Data,|F|/X,',ampleg{3}],['Model,TF,',ampleg{3}],['Data,|F|/X,',ampleg{4}],['Model,TF,',ampleg{4}],... ['Data,|F|/X,',ampleg{5}],['Model,TF,',ampleg{5}]}; Li1 = {[Mount{i},',|F|/X,',ampleg{1}],[Mount{i},',TF,',ampleg{1}],[Mount{i},',|F|/X,',ampleg{2}],[Mount{i},',TF,',ampleg{2}],... [Mount{i},',|F|/X,',ampleg{3}],[Mount{i},',TF,',ampleg{3}],[Mount{i},',|F|/X,',ampleg{4}],[Mount{i},',TF,',ampleg{4}],... [Mount{i},',|F|/X,',ampleg{5}],[Mount{i},',TF,',ampleg{5}]}; ampfull = 5; ampin = [0,0.5,1,1.5,2]; end if i == 5 hzs = {'1hz','2hz','3hz','4hz','5hz','6hz','7hz','8hz','9hz','10hz',... '20hz','30hz'}; freqfull = [1,2,3,4,5,6,7,8,9,10,20,30]; freqleg = {'1-Hz','2-Hz','3-Hz','4-Hz','5-Hz','6-Hz','7-Hz','8-Hz',... '9-Hz','10-Hz','20-Hz','30-Hz'}; nfiles = 12; else hzs = {'1hz','2hz','3hz','4hz','5hz','6hz','7hz','8hz','9hz','10hz',... '12hz','14hz','16hz','18hz','20hz','25hz','30hz','35hz'}; freqfull = [1,2,3,4,5,6,7,8,9,10,12,14,16,18,20,25,30,35]; freqleg = {'1-Hz','2-Hz','3-Hz','4-Hz','5-Hz','6-Hz','7-Hz','8-Hz',... '9-Hz','10-Hz','12-Hz','14-Hz','16-Hz','18-Hz','20-Hz','25-Hz',... '30-Hz','35-Hz'}; nfiles = 18; end if loadstorage == 0 pathname5 = 'C:\Documents and Settings\Administrator\My Documents\CSV MOUNT FILES\A CSV Mount Files\Program Folder'; cd(pathname5) ndata = xlsread('Program_4_Stiffness.xls',filehole); Freqdata = ndata(:,1); FtStiff = ndata(:,3); KxStiff = ndata(:,2); FtPhase = rad2deg(ndata(:,7)-ndata(:,8)); clear ndata; pathname3 = ['C:\Documents and Settings\Administrator\My Documents\CSV MOUNT FILES\A CSV Mount Files\',filehole,'\Model']; cd(pathname3) filecall = ['XFHold_Data_',elastomer{i},num2str(i),'.xls']; pdata = xlsread(filecall,'Sheet1'); %load to comparison spreadsheet Phasedata = pdata(:,4); clear pdata; end for amps = 1:ampfull %% Non-Parameteric Modeler mountnum = i; if loadstorage == 0 sa = 1+nfiles*(amps-1); %(1,19,37,etc ea = nfiles + nfiles*(amps-1);

197

[base] = program5base(mountnum,amps); XFHold(1:nfiles+1,1) = [0;Freqdata(sa:ea)]; XFHold(1:nfiles+1,2) = [base;FtStiff(sa:ea)]; XFHold(1:nfiles+1,3) = [base;KxStiff(sa:ea)]; XFHold(1:nfiles+1,4) = [0;Phasedata(sa:ea)]; Freqvec = [0;Freqdata(sa:ea)]; StiffVec = XFHold(1:nfiles+1,2); magvecnormdbe=20.*log10(abs(StiffVec)); [ka,za,wa,aa,ba] = program5guess(mountnum,amps); x0 = [ka,za,wa,aa,ba]; [xx,costval] = fminsearch(@(x)program5fun(x,magvecnormdbe),x0); k = xx(1); zeta = xx(2); wn = xx(3); a = xx(4); b = xx(5); HF = tf({[1 2.*zeta.*wn wn.^2]},{[1 2.*a.*b b.^2]}); win = 2*pi*Freqvec; Fin = Freqvec; long = length(Fin); [magmod,phasemod] = bode(k*HF,win); magmodnorm = reshape(magmod,1,long); phasemodnorm = reshape(phasemod,1,long); magmoddb = 20.*log10(magmodnorm); row = amps; ModelHold(row,1) = ampin(amps); ModelHold(row,2) = k; ModelHold(row,3) = zeta; ModelHold(row,4) = wn; ModelHold(row,5) = a; ModelHold(row,6) = b; ModelHold(row,7) = zeta./a; ModelHold(row,8) = wn./b; XFHold(1:nfiles+1,5) = magmodnorm; XFHold(1:nfiles+1,6) = phasemodnorm; XFLabel = {'Frequency','Ft_Stiffness','Kx_Stiffness','Ft_Phase','TF_Stiffness','TF_Phase'}; ModelValue(1:long,1) = Fin; ModelValue(1:long,amps+1) = magmodnorm; DataValue(1:long,1) = Fin; DataValue(1:long,amps+1) = StiffVec; PhaseValue(1:long,1) = Fin; PhaseValue(1:long,amps+1) = XFHold(1:nfiles+1,4); ModelLabel = {'Current','Gain,K','Zeta,Z','Wn','Alpha,A','Beta,B','Damping Ratio, z/a','Stiffness Ratio, w/b'}; ValueLabel = ['Frequency',current]; if amps == ampfull Current = ModelHold(:,1); Gain = ModelHold(:,2); ZetaL = ModelHold(:,3); WnL = ModelHold(:,4); AlphaL = ModelHold(:,5); BetaL = ModelHold(:,6); DampingRatio = ModelHold(:,7); StiffRatio = ModelHold(:,8); end elseif loadstorage == 1 pathstored = 'C:\Documents and Settings\Administrator\My Documents\CSV MOUNT FILES\A CSV Mount Files\Program Folder'; cd(pathstored) ModelHold = xlsread('Program_5_TF_Parameters.xls',filehole); ModelValue = xlsread('Program_5_TF_Values.xls',filehole); DataValue = xlsread('Program_5_FX_Values.xls',filehole); XFHold = xlsread('Program_5_XFHold.xls',filehole); Freqvec = DataValue(:,1); StiffVec = DataValue(:,1+amps); Fin= ModelValue(:,1); magmodnorm = ModelValue(:,1+amps); if amps == ampfull Current = ModelHold(:,1); Gain = ModelHold(:,2); ZetaL = ModelHold(:,3); WnL = ModelHold(:,4); AlphaL = ModelHold(:,5); BetaL = ModelHold(:,6);

198

DampingRatio = ModelHold(:,7); StiffRatio = ModelHold(:,8); end end %% Plot Style Selection marker = {'v','d','^','x','o','p','*','.','+','v','d','^','x'}; colors1 = {'k','r','g','b','k','r','g','b','k','r','g','b'}; colors2 = {'k','m','c','b','k','m','c','b','k','m','c','b'}; colors3 = {'k','r','g','b','k','r','g','b','k','r','g','b'}; lines = {'-',':','-','--','-',':','-','--','-',':','-','--'}; lines1 = {'-',':','-','--','-',':','-','--','-',':'}; fsize = 8; %font size tsize = 7; %title font size xfsize = 10; %x axis font size yfsize = 10; %y axis font size msize = 5; %markersize lsize = 8; fonts = 10; lwide = 1; style2 = [':',marker{amps},'r']; style1c = [':',marker{amps},colors2{amps}]; style2c = ['-','+','r']; style1i = [lines1{i},marker{i},colors2{i}]; style1iq = [lines1{i},marker{i},colors2{i}]; stylem = ['r']; style70 = [':',marker{amps},colors2{amps}]; %figures 70-77 xmin = 0; xmax = 35; ymin = 0; ymax = 10000; %% Figure For Model With Data for each current = 36+20 = 56figures if ampfull == 9 nums = ampfull*(i-1)+amps; elseif ampfull == 5 nums = 36+ampfull*(i-5)+amps; end Lsolo = {[Mount{i},',|F|/X,',ampleg{amps}],[Mount{i},',TF,',ampleg{amps}]}; figure(nums),plot(Freqvec,StiffVec,style1c,'MarkerSize',msize,'LineWidth',lwide),hold on, xlabel('Frequency, Hz','fontsize',fonts),ylabel('Stiffness, N/mm','fontsize',fonts),hold on, plot(Fin,magmodnorm,style2c,'MarkerSize',msize,'LineWidth',1), legend(Lsolo,'location','NorthEast','fontsize',lsize), axis([xmin xmax ymin ymax]) %% Figure For Model Parameter versus Current To show dynamics of the model if amps == ampfull figure(57),plot(Current,DampingRatio,style1iq,'MarkerSize',msize,'LineWidth',lwide),hold on, xlabel('Current, A','fontsize',fonts),ylabel(['Zeta,','\zeta',' / ','Alpha,','\alpha'],'fontsize',fonts), axis([0 2 0 2]),legend(Mount,'fontsize',lsize),hold on figure(58),plot(Current,StiffRatio,style1iq,'MarkerSize',msize,'LineWidth',lwide),hold on, xlabel('Current, A','fontsize',fonts),ylabel(['\omega','/','\beta'],'fontsize',fonts), axis([0 2 0 2]),legend(Mount,'fontsize',lsize),hold on figure(59),plot(Current,Gain,style1iq,'MarkerSize',msize,'LineWidth',lwide),hold on, xlabel('Current, A','fontsize',fonts),ylabel('Gain, K','fontsize',fonts), axis([0 2 0 8000]),legend(Mount,'fontsize',lsize),hold on figure(60),plot(Current,ZetaL,style1iq,'MarkerSize',msize,'LineWidth',lwide),hold on, xlabel('Current, A','fontsize',fonts),ylabel(['Zeta,','\zeta'],'fontsize',fonts), axis([0 2 0 70]),legend(Mount,'fontsize',lsize),hold on figure(61),plot(Current,WnL,style1iq,'MarkerSize',msize,'LineWidth',lwide), hold on, xlabel('Current, A','fontsize',fonts),ylabel('\omega','fontsize',fonts), axis([0 2 0 50]),legend(Mount,'fontsize',lsize),hold on figure(62),plot(Current,AlphaL,style1iq,'MarkerSize',msize,'LineWidth',lwide), hold on, xlabel('Current, A','fontsize',fonts),ylabel(['Alpha,','\alpha'],'fontsize',fonts), axis([0 2 0 70]),legend(Mount,'fontsize',lsize),hold on figure(63),plot(Current,BetaL,style1iq,'MarkerSize',msize,'LineWidth',lwide),hold on, xlabel('Current, A','fontsize',fonts),ylabel('\beta','fontsize',fonts), axis([0 2 0 50]),legend(Mount,'fontsize',lsize),hold on end %% Figure For Model + Data 0-2 Amp Comparison, each mount = 9 or 5 plots on 8 graphs figure(69+i),plot(Freqvec,StiffVec,style70,'MarkerSize',msize),hold on,

199

xlabel('Frequency, Hz','fontsize',fonts),ylabel('Stiffness, N/mm','fontsize',fonts),hold on, plot(Fin,magmodnorm,stylem),%,'MarkerSize',5,'MarkerFaceColor','m'), legend(Li1,'location','NorthEast','fontsize',lsize), axis([xmin xmax ymin ymax]) %% 0.5 Amp increment, Figure Data Comparison, 1-8 mounts Fig(Mount,Current) = 8 plots on 5 graphs if ampfull == 9 if (amps == 1)||(amps ==3)||(amps == 5)||(amps ==7)||(amps == 9) ampinc = 1+(amps-1)/2; ampd = ampinc; L7set = {[Mount{1},',|F|/X',',',currentinc{ampd}],[Mount{2},',|F|/X',',',currentinc{ampd}],... [Mount{3},',|F|/X',',',currentinc{ampd}],[Mount{4},',|F|/X',',',currentinc{ampd}],... [Mount{5},',|F|/X',',',currentinc{ampd}],[Mount{6},',|F|/X',',',currentinc{ampd}],... [Mount{7},',|F|/X',',',currentinc{ampd}],[Mount{8},',|F|/X',',',currentinc{ampd}]}; L7 = L7set; figure(77+ampinc),plot(Freqvec,StiffVec,style1i,'MarkerSize',msize),hold on, xlabel('Frequency, Hz','fontsize',fonts),ylabel('Stiffness, N/mm','fontsize',fonts),hold on, legend(L7,'location','NorthEast','fontsize',lsize), axis([xmin xmax ymin ymax]) end elseif ampfull == 5 ampd = amps; L7set = {[Mount{1},',|F|/X',',',currentinc{ampd}],[Mount{2},',|F|/X',',',currentinc{ampd}],... [Mount{3},',|F|/X',',',currentinc{ampd}],[Mount{4},',|F|/X',',',currentinc{ampd}],... [Mount{5},',|F|/X',',',currentinc{ampd}],[Mount{6},',|F|/X',',',currentinc{ampd}],... [Mount{7},',|F|/X',',',currentinc{ampd}],[Mount{8},',|F|/X',',',currentinc{ampd}]}; L7 = L7set; figure(77+amps),plot(Freqvec,StiffVec,style1i,'MarkerSize',msize),hold on, xlabel('Frequency, Hz','fontsize',fonts),ylabel('Stiffness, N/mm','fontsize',fonts),hold on, legend(L7,'location','NorthEast','fontsize',lsize),axis([xmin xmax ymin ymax]) end %% 0.5 Amp increment, Figure Model Comparison, 1-8 mounts Fig(Mount,Amp) = 8plots on 5graphs if ampfull == 9 if (amps == 1)||(amps ==3)||(amps == 5)||(amps ==7)||(amps == 9) ampinc = 1+(amps-1)/2; ampd = ampinc; L8set = {[Mount{1},',TF',',',currentinc{ampd}],[Mount{2},',TF',',',currentinc{ampd}],... [Mount{3},',TF',',',currentinc{ampd}],[Mount{4},',TF',',',currentinc{ampd}],... [Mount{5},',TF',',',currentinc{ampd}],[Mount{6},',TF',',',currentinc{ampd}],... [Mount{7},',TF',',',currentinc{ampd}],[Mount{8},',TF',',',currentinc{ampd}]}; L8 = L8set; figure(83+ampinc),plot(Fin,magmodnorm,style1i,'MarkerSize',msize),hold on, xlabel('Frequency, Hz','fontsize',fonts),ylabel('Stiffness, N/mm','fontsize',fonts),hold on, legend(L8,'location','NorthEast','fontsize',lsize), axis([xmin xmax ymin ymax]) end elseif ampfull == 5 ampd = amps; L8set = {[Mount{1},',TF',',',currentinc{ampd}],[Mount{2},',TF',',',currentinc{ampd}],... [Mount{3},',TF',',',currentinc{ampd}],[Mount{4},',TF',',',currentinc{ampd}],... [Mount{5},',TF',',',currentinc{ampd}],[Mount{6},',TF',',',currentinc{ampd}],... [Mount{7},',TF',',',currentinc{ampd}],[Mount{8},',TF',',',currentinc{ampd}]}; L8 = L8set; figure(83+amps),plot(Fin,magmodnorm,style1i,'MarkerSize',msize),hold on, xlabel('Frequency, Hz','fontsize',fonts),ylabel('Stiffness, N/mm','fontsize',fonts),hold on, legend(L8,'location','NorthEast','fontsize',lsize), axis([xmin xmax ymin ymax]) end %% Save DATA for Program 5

200

% if wantsave == 1 % if amps == ampfull % pathname6 = 'C:\Documents and Settings\Administrator\My Documents\CSV MOUNT FILES\A CSV Mount Files\Program Folder'; % cd(pathname6) % xlswrite('Program_5_TF_Parameters',ModelLabel,filehole,'A1') % xlswrite('Program_5_TF_Parameters',ModelHold,filehole,'A2') % %ModelHold = xlsread('Program_5_TF_Parameters.xls',filehole); xlswrite('Program_5_TF_Values',ValueLabel,filehole,'A1') xlswrite('Program_5_TF_Values',ModelValue,filehole,'A2') % %ModelValue = % %xlsread('Program_5_TF_Values.xls',filehole); xlswrite('Program_5_FX_Values',ValueLabel,filehole,'A1') xlswrite('Program_5_FX_Values',DataValue,filehole,'A2') % %DataValue = % %xlsread('Program_5_FX_Values.xls',filehole) xlswrite('Program_5_Phase_Values',ValueLabel,filehole,'A1') xlswrite('Program_5_Phase_Values',PhaseValue,filehole,'A2') % %PhaseValue = % %xlsread('Program_5_Phase_Values.xls',filehole) % xlswrite('Program_5_XFHold',XFLabel,filehole,'A1') % xlswrite('Program_5_XFHold',XFHold,filehole,'A2') % clear ModelHold ModelValue DataValue Current Gain ZetaL WnL AlphaL BetaL DampingRatio StiffRatio % clear Freqvec Freqdata Phasedata magmodnorm magvecnormdbe FtPhase FtStiff % clear XFHold XFLabel phasemodnorm % end % end %% Export figures(40-43) when current = 2.0 or amp == 9 if wantprint == 1 if amps == ampfull %print once per ampfull pathname2 = ['C:\Documents and Settings\Administrator\My Documents\My Pictures\Chapter 6 Mount\',filehole]; cd(pathname2) fi69 = ['-f',num2str(69+i)]; callmeMount = [Mount{i},'_Mount_Current_Range']; print(fi69,'-r900','-dtiff',callmeMount),pause(3) end %% Export figures(1:56) if amps == ampfull %numbers the figure 1-56 and prints after everything is complete (low resolution) for cani =1:ampfull if ampfull == 9 nums = ampfull*(i-1)+cani; elseif ampfull == 5 nums = 36+ampfull*(i-5)+cani; end pathname2 = ['C:\Documents and Settings\Administrator\My Documents\My Pictures\Chapter 6 Mount\',filehole]; cd(pathname2) fis = ['-f',num2str(nums)]; callmemountamp = [Mount{i},'_',current{cani},'_single']; print(fis,'-r150','-dtiff',callmemountamp),pause(1) figure(nums),close pause(1) end end %% Export Parameter Vs Current Figures if i == 28%8 pathname9 = 'C:\Documents and Settings\Administrator\My Documents\My Pictures\Chapter 6 Mount'; cd(pathname9),pause(1) print('-f57','-r900', '-dtiff','Damping_Ratio_Plot'),pause(3) print('-f58','-r900', '-dtiff','Stiffness_Ratio_Plot'),pause(3) print('-f59','-r900', '-dtiff','Gain_Plot'),pause(3) print('-f60','-r900', '-dtiff','Zeta_Plot'),pause(3) print('-f61','-r900', '-dtiff','Omega_Plot'),pause(3) print('-f62','-r900', '-dtiff','Alpha_Plot'),pause(3) print('-f63','-r900', '-dtiff','Beta_Plot'),pause(3) end

201

%% Export Figure Data and Model Comparison if amps == 10%ampfull pathname9 = 'C:\Documents and Settings\Administrator\My Documents\My Pictures\Chapter 6 Mount\Comparison'; cd(pathname9),pause(1) print('-f78','-r900', '-dtiff','Mount_Data_000Amp'),pause(3) print('-f79','-r900', '-dtiff','Mount_Data_050Amp'),pause(3) print('-f80','-r900', '-dtiff','Mount_Data_100Amp'),pause(3) print('-f81','-r900', '-dtiff','Mount_Data_150Amp'),pause(3) print('-f82','-r900', '-dtiff','Mount_Data_200Amp'),pause(3) print('-f84','-r900', '-dtiff','Mount_Model_000Amp'),pause(3) print('-f85','-r900', '-dtiff','Mount_Model_050Amp'),pause(3) print('-f86','-r900', '-dtiff','Mount_Model_100Amp'),pause(3) print('-f87','-r900', '-dtiff','Mount_Model_150Amp'),pause(3) print('-f88','-r900', '-dtiff','Mount_Model_200Amp'),pause(3) end end end end

Transfer Function Analysis Code Functions:

The functions used within the transfer function analysis code are: 1. program5base used for predetermined quasi-stiffness values,

2. program5guess used for initial starting points for nominal

parameters,

3. determines the transfer functions nominal values and passes them

back to Program 5

Program5base finds QST Stiffness Results:

function [base] = program5base(mnum,ampere) mount = mnum; amps = ampere; %BaseStiffness acquired through quasi-static testing if mount == 1 BSTF = [2586.8,2586.8,2845.7,3036.9,3235.8,3396.2,3542.8,... 3619.4,3687.6]; elseif mount == 2 BSTF = [2904.549446, 2958.931891,3040.689349,3169.978628,3288.319535,... 3398.493404,3481.212444,3546.520701,3603.394853]; elseif mount == 3 BSTF = [2292.049221,2287.422842,2256.878684,2301.962294,2370.026145,... 2455.585702,2546.980828,2626.811386,2674.065159]; elseif mount == 4 BSTF = [575.0394724,584.9265954,579.2995214,572.0821985,566.420404,... 558.2595196,549.625788,541.0624381,532.6941146]; elseif mount == 5 BSTF = [460,460,460,460,460]; elseif mount == 6 BSTF = [2037.049252,2033.976119,2036.238984,2039.322628,2042.520739]; elseif mount == 7 BSTF = [4212.46,4204.19,4203.09,4205.76,4218.23]; elseif mount == 8 BSTF = [4279.979,4352.173,4374.144,4396.704,4420.925]; end base = BSTF(amps); %%END of program5base Function

202

Program5guess provides an initial starting point for

fminsearch: function [k,z,w,a,b] = program5guess(mountnum,ampere) mount = mountnum; amps = ampere; if mount == 1 k0 = [2753.0097;2858.5164;3057.6273;3278.8856;3509.2417;3859.1168;3948.5543;4082.4133;4104.2792]; zeta0 = [1.8,1.8,1.92,2.07,1.6,1.6,1.85,1.8,1.6]; a0 = [1.3,1.3,1.6,1.13,1.11,1.1,1.26,1.2,1.15]; wn0 = [10,10,10,10,9.08,9.08,9.29,8.8,8.89]; b0 = [11,11,11.7,10.97,10.48,10.48,10.79,10.33,10.48]; % x0 = [zeta0(i),wn0(i),a0(i),b0(i),k0(i)]; elseif mount == 2 k0 = [3307.9,3374.5,3703.1,3992,4450,4747,4840,5008,5156]; zeta0 = [1.560,1.687,1.659,1.954,1.894,2.012,2.295,2.235,2.015]; a0 = [1.393,1.470,1.394,1.528,1.370,1.376,1.501,1.432,1.277]; wn0 = [11.561,11.977,10.228,10.940,9.969,9.475,9.673,9.132,8.868]; b0 = [12.34,12.79,11.29,12.28,11.6,11.2,11.4,10.85,10.61]; elseif mount == 3 k0 = [2672.76,2765.38,2883.11,3107.57,3310.21,3495.82,3577.40,... 3797.99,3909.32]; zeta0 = [1.5352;1.55;1.7471;2.25;2.45;2.9;3.25;3.65;4]; wn0 = [7.8835;9.5;9.9452;11.8196;13.5;14.5;15.5;15.6765;17]; a0 = [1.415;1.4358;1.5645;1.9229;2.0611;2.32;2.5811;2.8693;3.1233]; b0 = [8.4209;10.2944;11.0972;13.46;15.4969;16.4;17.4521;17.4574;19.1828]; elseif mount == 4 k0 = [2672.76,2765.38,2883.11,3107.57,3310.21,3495.82,3577.40,... 3797.99,3909.32]; zeta0 = [1.45;1.52;1.5729;1.59;1.62;1.6531;1.6936;1.7;1.73]; wn0 = [16.5;15.873;15;14.5;14;13;12.4;12.25;11.7]; a0 = [1.43;1.5416;1.6278;1.6219;1.6481;1.717;1.7512;1.7543;1.7647]; b0 = [19.0051;18.0818;17.1068;16.6631;16.2016;15.0767;14.5065;14.4103;13.8684]; elseif mount == 5 k0 = [400,400,400,400,400]; zeta0 = [1.67;1.67;1.67;1.67;1.67]; wn0 = [4.56;4.56;4.56;4.56;4.56]; a0 = [1.6189;1.5847;1.6283;1.638;1.581]; b0 = [5.1064;5.1057;5.1034;5.1024;5.1452]; elseif mount == 6 k0 = [2485;2485;2490;2515;2535]; zeta0 = [3.935;3.935;3.935;3.935;3.935]; wn0 = [9.622;9.622;9.622;9.622;9.622]; a0 = [3.7466;3.7466;3.7466;3.7466;3.7466]; b0 = [10.6453;10.6453;10.6453;10.6453;10.6453]; elseif mount == 7 k0 = [5600,5600,5600,5600,5600]; zeta0 = [7.5092;7.5092;7.5092;7.5092;7.5092]; wn0 = [10.817;10.817;10.817;10.817;10.817]; a0 = [7.4289;7.4289;7.4289;7.4289;7.4289]; b0 = [12.403;12.403;12.403;12.403;12.403]; elseif mount == 8 k0 = [5600,5600,5600,5600,5600]; zeta0 = [7.5092;7.5092;7.5092;7.5092;7.5092]; wn0 = [10.817;10.817;10.817;10.817;10.817]; a0 = [7.4289;7.4289;7.4289;7.4289;7.4289]; b0 = [12.403;12.403;12.403;12.403;12.403]; end k = k0(amps); z = zeta0(amps); w = wn0(amps); a = a0(amps); b = b0(amps); %%END of program5guess Function

203

program5fun determines the transfer functions nominal values

and passes them back to Program 5: function [costout] = program5fun(x,magdata) %H = tf({[1 2.*p(1).*p(2) p(2).^2]},{[1 2.*p(2).*p(4) p(4).^2]}); long = length(magdata); if long == 19 N = 19; Fin = [0,1,2,3,4,5,6,7,8,9,10,12,14,16,18,20,25,30,35]; elseif long == 13 N = 13; Fin = [0,1,2,3,4,5,6,7,8,9,10,20,30]; end win = 2*pi*Fin'; k = x(1); zeta = x(2); wn = x(3); a = x(4); b = x(5); HF = tf({[1 2.*zeta.*wn wn.^2]},{[1 2.*a.*b b.^2]}); % Calculates the magnitude of the system in dB magmod = bode(k*HF,win); magmodnorm = reshape(magmod,1,N); magmod = 20.*log10(magmodnorm); % Experimental data expdata = magdata'; % Calculates the cost of the minimizer costout = norm(magmod - expdata); %%END of program5fun Function

204

Appendix D: Early Stages of Mount Design and Fabrication Due to the nature of this research, early design and fabrication stages that would clutter

the body of the document are presented in this appendix. Moreover, this appendix

presents the earlier stages of mold designs and electromagnet designs. The fabrication

process, however, is presented in a general overview. Following the first generation mold,

a first generation electromagnet is presented. A second generation electromagnet is

presented in the final section of this appendix.

D-1 First Generation Mold and Magnetic Circuit The first generation mold in Figure D-1 shows the mold housing and two plugs. The

housing is made of delrin and the three plugs are made of aluminum. The plugs depicted

in Figure D-1 create the lower section of the elastic case as well as the insert cavity. An

insert may be added after this procedure as seen in Figure D-2 or a top section is placed on

the lower section for later injecting MR fluid. As seen in Figure D-3, the lower section of

the elastic case with insert is ready for a final layer of elastic material that will create the

top section of the mount. This first generation mold was successful at building precise

mounts. Do to the screw in design; however, the de-molding process was quite difficult

and often required the mold housing to be heated to expand away from the plugs.

Figure D-1: First generation mold housing and plugs used for molding the lower section of an elastomeric case.

205

Figure D-2: First generation mold and three plugs with lower section of elastic case with an aluminum insert pictured beside a full elastic case mount.

Figure D-3: Lower section of elastic case with insert placed inside first generation mold and readied for upper section.

Moreover, the first generation mold required extra steps in the manufacture of an

elastic case mount which is demonstrated by the fact that the lower and upper section of

the elastic case are molded separately and not parallel. Therefore, a more expensive three-

plate mold was pursued to expedite the manufacturing process.

The original electromagnet in Figure D-4 was built using available electro-coils and

with a flux design similar to that of an MR damper as seen in Figure D-5. The fixtures for

the shock dyno are also shown, but the magnetic shield opposite of the electromagnet is

not depicted. Upon testing with the elastic case filled with MRF-128 fluid, which is a

206

28% by volume ferrous iron fluid, the magnet was unable to cause any change in

transmitted forced. This magnetic design, however, was inefficient with the elastic case

and was not tested with the metal-elastic case. Therefore, this magnet may have been

useful in activating the MR fluid within a metal-elastic case, but no substantiation is

available to prove or disprove this magnetic circuitry design.

Figure D-4: First generation electromagnet and test fixture with an MR fluid-elastic mount in an elastic case.

Figure D-5: First generation magnetic circuitry layout with an MR fluid-elastic mount positioned above the magnet poles similar to an MR damper configuration.

N S

207

The aforementioned first generation design does not present an axis symmetric profile

as seen in the shop schematic for the magnet housing in Figure D-6. Therefore, this

design is not modeled with finite element magnetic software (FEMM). As shown in

Figure D-7, initial testing on an elastic case MR fluid-elastic mount did not produce any

variation in transmitted force when tested with the shock dyno. During this initial testing

the quick connect adapters are used, but not illustrated in the schematics and instead a

revised housing and fixture are illustrated. Further investigation with the metal-elastic

case mount, however, may prove or disprove this to be a useful magnetic circuitry. The

lack of testing and modeling for this design is due to the fact that the fluid is not activated

in a complete squeeze mode which is the basis for increasing the axial compressive

strength of the MR fluid. Therefore, this electromagnet was not tested anymore and a

more efficient magnet circuit that activates the fluid in squeeze mode was pursued.

Figure D-6: First generation electromagnet housing schematic.

208

Figure D-7: Testing first generation electromagnet on MRF-128 fluid-elastic mount in an elastic case with 28% by volume ferrous particle fluid using quick connect adapters on the shock dyno.

D-2 Second Generation Electromagnet As promised earlier, this section presents the second generation electromagnet. This

electromagnet was also presented as iteration 1 in section 3.2.2.

The aluminum frame shown in Figure D-8 holds the electromagnet flanged core

shown in Figure D-9. Additionally, this frame was integrated to attach to the shock dyno

and avoid adding a test fixture. This magnet was only tested with the earlier elastic case

MR fluid mounts which contained MRF-128 fluid, but was unable to activate the MR

fluid. MRF-128 fluid only contains 28% ferrous particles by volume. The coils used to

activate the flanged electromagnet core are presented in Figure D-10. A test setup is

shown in

209

Figure D-8: Second generation Electromagnet Aluminum Frame also known as Iteration 1 in Chapter 3.

Figure D-9: Second generation electromagnet flanged core also known as Iteration 1 in Chapter 3.

210

Figure D-10: Second generation electromagnet coils for flanged core with 21 AWG, 23 AWG, and 24 AWG magnet wire at 500, 750, and 1000 turns, respectively.

Figure D-11: Testing second generation electromagnet on elastic case mount with MRF-128 which is a 28% by volume ferrous particle fluid.