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

Transcription

1 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, 28 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 28

2 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. ii

3 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. to 2. 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 3 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. to 2. Amps the equivalent damping coefficient in a MR fluid-elastic mount increases over 5% of the value at Amps at low frequency. Through damping comparisons, the MR fluidelastic mount with no current is similar to that of a mount made of solid rubber with a hardness of 3 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. iii

4 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. iv

5 Content ABSTRACT... ii II ACKNOWLEDGMENTS... iv IV CONTENT... vv LIST OF FIGURES... viii VIII LIST OF TABLES...XVI xvi 1. INTRODUCTION OVERVIEW MOTIVATION OBJECTIVES APPROACH OUTLINE BACKGROUND MR FLUID HISTORY AND DEVICES: LITERATURE REVIEW MR Fluid Devices MR Fluid Operation HYDRAULIC MOUNTS: LITERATURE REVIEW MR MOUNTS: LITERATURE REVIEW Magnetorheological Elastomers Magnetorheological Fluid-Elastomers Additional MR Mounts VIBRATION ANALYSIS TECHNIQUES Linear Static Spring Stiffness Linear Spring Stiffness, Viscous and Hysteretic Damping Linear Approximation Frequency Response Modeling SUMMARY OF LITERATURE REVIEW MR FLUID-ELASTIC MOUNT DESIGN AND FABRICATION MAGNETIC CIRCUITRY PRINCIPALS MAGNETIC SYSTEM Magnetic System Design Iteration Stage: Magnetic System Design ELASTIC MOUNT DESIGN Elastic Mount Design Elastic Mount Fabrication v

6 3.3.3 Metal-Elastic Mount Fabrication DESIGN OF EXPERIMENT SUMMARY MOUNT STIFFNESS AND DAMPING CHARACTERIZATION ELASTIC PARAMETRIC ANALYSIS Static Force-Displacement Analysis and Results Force-Displacement Analysis Force-Amplitude Analysis Processing Analysis Method Evaluation MOUNT PARAMETRIC RESULTS MR fluid- Elastic Mount Parameters Passive Elastic Parameters Discrete Comparison of Stiffness Magnitude Mount Comparison DISCUSSIONS MR FLUID ELASTIC MOUNT MODELING AND CHARACTERIZATION NON-PARAMETRIC MODELING APPROACH MR Fluid Metal-Elastic Mount Modeling Nominal Parameter Results and Comparison Nominal Parameter Relationship MODEL SIMULATION AND COMPARISON MR fluid Metal-Elastic Mount Simulation Model Error Evaluation DAMPING MODELING APPROACH MR Fluid-Elastic Mount Damping Model MR Fluid-Elastic Mount Damping Simulation SUMMARY Non-Parametric Simulation and Evaluation Remarks Damping Simulation and Evaluation Remarks CONCLUSIONS AND PROSPECTIVE RESEARCH SUMMARY RECOMMENDATIONS FUTURE WORK REFERENCES APPENDIX A: MOUNT AND MAGNETIC DESIGN SCHEMATICS vi

7 APPENDIX B: RESULTS APPENDIX C: DATA PROCESSING CODE APPENDIX D: EARLY STAGES OF MOUNT DESIGN AND FABRICATION vii

8 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 [2]....6 Figure 2-3: MR fluid in shear mode with applied magnetic field, adapted from [2]....7 Figure 2-4: MR fluid in squeeze mode setup prior to axial force with an applied magnetic field, adapted from [2]....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) 1 mt field curing of polyurethane MR elastomer with carbonyl-iron particles, adapted from [3] Figure 2-9: Magneto-rheological fluid-elastomer study by Wang, adapted from [35]. 12 Figure 2-1: 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] 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 Figure 3-3: Cross-sectional view of empty metal-elastic casing and magnetic system with test fixtures 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-14, 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. 3 viii

9 Figure 3-7: Simulated (a) Flux density for mount system and (b) magnetic flux magnitude for MRF-132 with 3 Amps of current supplied to the electro coil. 31 Figure 3-8: Simulated (a) Flux density for mount system and (b) magnetic flux magnitude for MRF-14 with 3 Amps of current supplied to the electro coil. 32 Figure 3-9: Simulated (a) Flux density for mount system and (b) magnetic flux magnitude for MRF-14 with 3 Amps of current supplied to the electro coil. 33 Figure 3-1: (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 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 Figure 3-13: Simulated flux magnitude plot using MRF-14 in FEMM for mount system at the (a) center of the fluid gap and at the (b) upper-pole plate boundary Figure 3-14: Magnetic system iteration-1 (a) model and (b) simulation contour plot of lower fluid cavity boundary, in FEMM software Figure 3-15: Magnetic system iteration-2 (a) model and (b) simulation contour plot of lower fluid cavity boundary, in FEMM software Figure 3-16: Magnetic system iteration-3 (a) model and (b) simulation contour plot of lower fluid cavity boundary, in FEMM software....4 Figure 3-17: Magnetic system iteration-4 (a) model and (b) simulation contour plot of lower fluid cavity boundary, in FEMM software Figure 3-18: Magnetic system iteration-5 (a) model and (b) simulation contour plot of lower fluid cavity boundary, in FEMM software Figure 3-19: Three plate mold for manufacturing elastic mounts...45 Figure 3-2: Vacuum Pump and Bell Jar Figure 3-21: PolyTek TM 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 ix

10 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 661 aluminum, air, 118 steel, and solid 3 D polyurethane (rubber) Figure 3-27: Upper-pole plate (top) and magnetic-pole plate (bottom) made of 12L14 Steel with epoxy primer...5 Figure 3-28: Pole plates inserted into mold, upper plate first (left) and then magneticpole 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) Figure 3-3: Metal-elastic casing and funnel for filling MR fluid-elastic casing 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. software, adapted from [45]...55 Figure 3-34: Test Setup of mount and magnetic system in the Roehrig EMA Dynamometer 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 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.,.5, 1., 1.5, and 2. Amp...65 Figure 4-4: Force-displacement plotting method example with hysteretic content x

11 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....7 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 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....8 Figure 4-1: 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 Figure 4-11: MR fluid-elastic 1 mount (MRF-145) (a) stiffness F /X, and (b) damping Ceq results obtained from analysis...86 Figure 4-12: MR fluid-elastic 2 mount (MRF-145) (a) stiffness F /X, and (b) damping Ceq results obtained from analysis...87 Figure 4-13: MR fluid-elastic 3 mount (MRF-145) (a) stiffness F /X, and (b) damping Ceq results obtained from analysis...89 Figure 4-14: Blank metal-elastic case MRE 3B (a) stiffness F /X, and (b) damping Ceq results obtained from analysis....9 Figure 4-15: Passive mount with air insert (a) stiffness F /X, and (b) damping Ceq results obtained from analysis Figure 4-16: Passive mount with 3 D rubber insert (a) stiffness F /X, and (b) damping Ceq results obtained from analysis...96 Figure 4-17: Passive mount with 118 steel insert (a) stiffness F /X, and (b) damping Ceq results obtained from analysis...97 Figure 4-18: Passive mount with 661 aluminum insert (a) stiffness F /X, and (b) damping Ceq results obtained from analysis...99 xi

12 Figure 4-19: Comparing stiffness magnitude of a metal-elastic case (MRE 3B) to an elastic case (AIR) mount at (a) Amps and (b) 2 Amps of current...14 Figure 4-2: Comparing stiffness magnitude of a metal-elastic case (MRE 3B) to a MR fluid-elastic mount at (a) Amps and (b) 2 Amps of current Figure 4-21: Comparing stiffness magnitude of a solid elastic case (RUB) to a MR fluidelastic mount at (a) Amps and (b) 2 Amps of current Figure 4-22: Comparing stiffness magnitude of an elastic case with steel insert (STE) to a MR fluid-elastic mount at (a) Amps and (b) 2 Amps of current Figure 4-23: Comparing stiffness magnitude of an elastic case with al. insert (ALU) to a MR fluid-elastic mount at (a) Amps and (b) 2 Amps of current Figure 4-24: Comparing damping of a metal-elastic case (MRE 3B) to an elastic case (AIR) mount at (a) Amps and (b) 2 Amps of current...19 Figure 4-25: Comparing damping of a metal-elastic case (MRE 3B) to a MR fluidelastic mount at (a) Amps and (b) 2 Amps of current Figure 4-26: Comparing damping of a solid elastic case (RUB) to a MR fluid-elastic mount at (a) Amps and (b) 2 Amps of current Figure 4-27: Comparing damping of an elastic case with steel insert (STE) to a MR fluid-elastic mount at (a) Amps and (b) 2 Amps of current Figure 4-28: Comparing damping of an elastic case with aluminum insert (ALU) to a MR fluid-elastic mount at (a) Amps and (b) 2 Amps of current Figure 4-29: Comparative (a) stiffness F /X, and (b) damping Ceq results obtained at.-amps from force-amplitude and force-displacement analysis, respectively Figure 4-3: Comparative (a) stiffness F /X, and (b) damping Ceq results obtained at 1.-Amps from force-amplitude and force-displacement analysis, respectively Figure 4-31: Comparative (a) stiffness F /X, and (b) damping Ceq results obtained at 2.-Amps from force-amplitude and force-displacement analysis, respectively Figure 5-1: Selecting a transfer function to model the stiffness magnitude in the frequency domain xii

13 Figure 5-2: Nominal gain, K, as a function of current for MR fluid-elastic mount Figure 5-3: Nominal (a) zero-damping ratio and (b) pole-damping ratio as a function of current for each MR fluid-elastic mount Figure 5-4: Nominal (a) zero-frequency and (b) pole-frequency as a function of current for each MR fluid-elastic mount model Figure 5-5: Non-parametric damping ratio relationship, ζ/α, at each current setting for MR fluid-elastic mount models Figure 5-6: Non-parametric stiffness ratio relationship, ω 2 n /β 2, at each current setting for MR fluid-elastic mount models Figure 5-7: Stiffness simulation results for MR fluid-elastic 1 mount at.5 Amp current increments Figure 5-8: Stiffness simulation results for MR fluid-elastic 2 mount at.5 Amp current increments Figure 5-9: Stiffness simulation results for MR fluid-elastic 3 mount at all current settings Figure 5-1: 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 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 Figure 5-12: Damping simulation results for MR Fluid-Elastic 1 mount at full range of current settings Figure 5-13: Damping simulation results for MR Fluid-Elastic 2 mount at full range of current settings Figure 5-14: Damping simulation results for MR Fluid-Elastic 3 mount at full range of current settings Figure 6-1: Automotive friendly design for an MR fluid-elastic mount Figure A-1: Top plate schematic of three plate mold Figure A-2: Middle plate schematic of three plate mold Figure A-3: End plate schematic of three plate mold Figure A-4: Lower housing base and core schematic to magnetic system xiii

14 Figure A-5: Upper housing schematic to magnetic system Figure A-6: Spacer schematic to lower housing in magnetic system Figure A-7: Upper-pole plate schematic for metal-elastic case mount Figure A-8: Magnetic-pole plate schematic for metal-elastic case mount Figure A-9: Lower housing test fixture schematic for Roehrig Dynamometer...17 Figure A-1: Upper Housing Test Fixture for Roehrig Dynamometer...17 Figure A-11: Elastic case mount chronology from initial case half mount to finalized MR Fluid-Elastic mount in a full elastic case Figure A-12: Paraphernalia readied for manufacturing an elastic case mount Figure A-13: Polyurethane in a degassing chamber under 28inHg to remove air Figure A-14: De-molding the half cases of the mount from the 3-plate mold 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..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 Figure A-18: Universal jig used to secure elastic case and position fluid syringe for MR fluid injection into the empty case cavity Figure D-1: First generation mold housing and plugs used for molding the lower section of an elastomeric case...24 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...25 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. 26 Figure D-6: First generation electromagnet housing schematic...27 xiv

15 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...28 Figure D-8: Second generation Electromagnet Aluminum Frame also known as Iteration 1 in Chapter Figure D-9: Second generation electromagnet flanged core also known as Iteration 1 in Chapter Figure D-1: Second generation electromagnet coils for flanged core with 21 AWG, 23 AWG, and 24 AWG magnet wire at 5, 75, and 1 turns, respectively.21 Figure D-11: Testing second generation electromagnet on elastic case mount with MRF- 128 which is a 28% by volume ferrous particle fluid...21 xv

16 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] Table 3-3: Polyurethane metal-elastic and elastic casing dimensions with internal cavity dimensions for the specified insert Table 3-4: Mount naming nomenclature for abbreviations and legends 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.25 Amp Table 4-2: Comparative stiffness and RMS-Error obtained from force-time and forcedisplacement analysis Table 4-3: Stiffness magnitude of metal-elastic case mounts at all current settings 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...1 Table 4-6: Equivalent damping results for passive elastic case mount air, rubber, steel and aluminum at all current settings...11 Table 4-7: Stiffness magnitude comparison for MR fluid-elastic and passive mounts at settings of., 1., 2. Amp Table 4-8: Equivalent damping comparison for MR fluid-elastic and passive mounts at settings of., 1., and 2. Amp...12 Table 5-1: Damping model and exponential coefficient values for MR fluid-elastic 1, 2, and 3 mounts Table A- 1: Bill of Materials without cost estimates for mount and magnet system and manufacture Table B-1: Passive mount damping analysis results for the air, rubber, steel, and aluminum inserts xvi

17 Table B-2: MR fluid-elastic mount damping analysis results for MRE s and blank MRE Table B-3: MR fluid-elastic mount and passive mount damping analysis comparison chart Table B-4: MR Fluid-elastic mount Stiffness Analysis Results for MRE s and blank MRE Table B-5: Passive mount stiffness analysis results for the air, rubber, steel, and aluminum inserts at.5 Amp current indexing Table B-6: MR fluid-elastic and passive mount stiffness analysis comparison chart Table B-7: MR Fluid-elastic mount parameters from force-amplitude and displacement modeling analysis at, 1, and 2-Amp current settings...18 Table B-8: MR Fluid-elastic mount error comparison between force-amplitude F /X and force-displacement Kx, sampled at, 1, 2-Amp for MR fluid-elastic mounts 1, 2 and Table B-9: Nominal transfer function parameters used to simulate the results in section xvii

18 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. 1

19 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 noisevibration-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. 2

20 1.3 Objectives The primary objectives of this research are to: 1. provide further evaluation and analysis of magnetorheological (MR) fluidelastic 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. 3

21 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 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 1-2 microns in size [7]. 4

22 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 2-4% 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 1 ms. when a magnetic field is applied [9, 1]. 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 5

23 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 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: [2]. MR fluid in valve mode with an applied magnetic field, adapted from 6

24 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: [2]. MR fluid in shear mode with an applied magnetic field, adapted from 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 [2]. The magnetic field is aligned axially with the applied force to create chains of the ferrous magnetic particles [2, 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 [2]. 7

25 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]. Magnetic field Figure 2-6: Ferrous particle aggregation in squeeze mode operation after experiencing a compressive load, adapted from [22]. 8

26 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. To Engine Q Decoupler To Chassis (a) (b) 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 9

27 [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, magnetorheological 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 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 [3] and adapted in Figure 2-8b with 1 mt field [3]. 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. 1

28 No Applied Field Field Direction Figure 2-8: (a) Zero field curing, and (b) 1 mt field curing of polyurethane magnetorheological elastomer with carbonyl-iron particles, adapted from [3]. 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 6% 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 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 11

29 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. MR Fluid Elastic Casing Figure 2-9: [35]. Magnetorheological fluid-elastomer study by Wang, adapted from 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 2-1. 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. 12

30 Flux Path Piston MR Fluid Figure 2-1: 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 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 The MR fluid in this mount is energized with an applied magnetic field to increase the stiffness of the mount. Here, Ahn 13

31 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. x(t) MR Fluid Coil Flux Path Core 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 [4, 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. 14

32 Inertia Track Elastomeric 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 crosssectional view in Figure 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. Elastic Piston Flux Path Coil Magnetic Pole Height MR Fluid 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 15

33 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 biodynamic 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 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 fk k = (2.2) x At static loading, this method is quite useful to recover the actual stiffness 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 x(t) + c x (t) (2.3) where c eq is the equivalent damping coefficient and x (t) 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 eq 16

34 hysteresis loop can be measured as the energy dissipated, Δ E. Additionally, the damping coefficient can then be determined from this dissipated energy ΔE ceq = (2.4) 2 πω X 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 stressstrain relationship. The energy dissipated for the stress-strain lissajou is 2 Δ E = π kβ X (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 kβ ceq = (2.6) ω 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 forcedisplacement lissajou. This is useful in interpreting the degree of each element present within a system such as the MR fluid mount 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, 17

35 as described by Dorf, must be linear. Force could then be F () t = k x() t + b (2.7) k 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 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 x() t = X sin(2 π f + ρ) + X (2.8) where X is the displacement amplitude, f is the input frequency, ρ is the input phase, and 18

36 X is the displacement offset. A resultant oscillatory force may then take the form Ft () = F sin(2 π f+ ϕ) + F (2.9) where F is the force amplitude, f is the output frequency, φ is the output phase, and F 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 19

37 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]. 2

38 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. 21

39 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 Ni = H L (3.1) the number of turns N and current i are related to the sum of the material magnetic intensity H n and material length L n. 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. n n 22

40 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. Magnet Upper Housing Spacer Metal-Elastic Case Mount Coil-Bobbin Magnet Lower Housing Figure 3-1: Isometric view of mount and magnetic system design. 23

41 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 metalelastic case is shown in Figure 3-2c which is followed by a discussion of design constraints. (a) Removed Elastic Fluid Cavity (b) Inserted Pole Plates (c) Magnetic-Pole Plate 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.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.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 24

42 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. Test Fixture MR Fluid Extruded Lower Housing Core Test Fixture Upper Housing Air Gap Bulge Space Spacer Coil Bobbin Figure 3-3: Cross-sectional view of empty metal-elastic casing and magnetic system with test fixtures. 25

43 One of the major parameters incorporated into the mount design is the height of the fluid cavity gap. This dimension was set to.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 1% 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 metalelastic 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 8 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 26

44 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 axissymmetric 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. 27

45 Upper Housing MR Fluid Coil Field Lines Lower Housing (a) (b) 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, 14, and 145, the B-H curves were determined using the model [1 exp( 1.97 μ )] μ B = Φ H + H (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 μ equals 4π1-7 [49]. Therefore, Φ is set to.22,.32,.4, and.45 based on the ferrous iron percentage making up the MR fluid. Next, the magnetic field intensity is increased from to 6 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 [5-52]. 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 8 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-14, and 197 kamp/m in MRF

46 MRF-122 MRF-14 3 Amp MRF-132 MRF B, Tesla H, Amp/m Figure 3-5: B-H curves for MRF-122, MRF-132, MRF-14, 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.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.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. 29

47 22 % 2. (a) Bottom Middle Top B, T Diameter, in. (b) 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. 3

48 32 % (a) Bottom Middle Top B, T Diameter, in. (b) Figure 3-7: Simulated (a) Flux density for mount system and (b) magnetic flux magnitude for MRF-132 with 3 Amps of current supplied to the electro coil. 31

49 4 % 2. (a) Bottom Middle Top B, T Diameter, in. (b) Figure 3-8: Simulated (a) Flux density for mount system and (b) magnetic flux magnitude for MRF-14 with 3 Amps of current supplied to the electro coil. Switching to the material properties of MRF-145, the flux density within the MR fluid gap is approximately 1. 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. T. In the middle of the fluid gap, however, the magnetic flux decreases slightly at the circumference. 32

50 45 % 2. (a) Bottom Middle Top B, T Diameter, in. (b) Figure 3-9: Simulated (a) Flux density for mount system and (b) magnetic flux magnitude for MRF-14 with 3 Amps of current supplied to the electro coil. Next, the yield stress of the various MR fluids is generated using Carlson s yield stress model YS.. = C 2717 Φ TANH(.633 H) (3.3) where C equals 1. 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 3-1. The yield stress is then 33

51 represented in Figure 3-11 for MRF-122, 132, 14, and 145 fluids. The yield stresses achieved in the fluid are depicted by a marker on each yield curve in Figure These yield stresses are 68 kpa for MRF-145, 59 kpa for MRF-14, 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 MRF122 MRF14 MRF132 MRF145 3 H, Amp/m Diameter, in. Figure 3-1: Magnitude of magnetic field intensity at the center of the fluid gap in the mount with various MR fluids. 1 9 MRF-122 MRF-132 MRF-14 MRF Amp 8 7 Yield Stress, Pa H, Amp/m 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

52 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 3 kamp/m to 197 kamp/m. The magnitude of magnetic flux B entering the fluid cavity is plotted in Figure 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.5 Amp up to 3. Amp, the flux magnitude increases from.3 T to 1. 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 Amp 1. Amp 1.5 Amp 2. Amp 2.5 Amp 3. Amp B, T Diameter, in. 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.5 Amp to 3. Amp in.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 35

53 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 Amp 1. Amp 1.5 Amp 2. Amp 2.5 Amp 3. Amp B, T Diameter, in. 2. (a) Amp 1. Amp 1.5 Amp 2. Amp 2.5 Amp 3. Amp 1.25 B, T Diameter, in. (b) Figure 3-13: Simulated flux magnitude plot using MRF-14 in FEMM for mount system at the (a) center of the fluid gap and at the (b) upper-pole plate boundary. 36

54 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 1.3 Ohms for the 8 turn coil with 24 AWG magnet wire. This resistance would require approximately a 3 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.3 to 1. T with a current input of.5 to 3. 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 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 1 turns supplied with 3. 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 37

55 diameter of in. and a height of.4375 in. The fluid cavity diameter is in. and the height is.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. Upper Shield MRF Flanged Magnet Core Coil (a) (b) 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. 38

56 Upper Shield MRF Flanged Magnet Core Coil (a) Figure 3-15: Magnetic system iteration-2 (a) model and (b) simulation contour plot of lower fluid cavity boundary, in FEMM software. (b) 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. 39

57 Upper Shield MRF Magnet Core Coil (a) (b) Figure 3-16: Magnetic system iteration-3 (a) model and (b) simulation contour plot of lower fluid cavity boundary, in FEMM software. 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.83 T as seen in Figure 3-17b. 4

58 Upper - pole plate MRF Coil Magnet -pole plate Magnet Core (a) Figure 3-17: Magnetic system iteration-4 (a) model and (b) simulation contour plot of lower fluid cavity boundary, in FEMM software. (b) 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 1 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.73 T as shown in the flux density plot of Figure 3-18b. 41

59 Upper - Housing MRF Spacer Coil Magnet -pole Magnet Core (a) Figure 3-18: Magnetic system iteration-5 (a) model and (b) simulation contour plot of lower fluid cavity boundary, in FEMM software. 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 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 (b) 42

60 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 74-3) was selected with a hardness of 3 Shore A (3 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 3 durometer polyurethane insert. Last, two metal inserts made of 118-Steel and 661- Aluminum are used. 43

61 Table 3-3: Polyurethane metal-elastic and elastic casing dimensions with internal cavity dimensions for the specified insert. P.U. Casing with Type of Insert Mount Mount Insert Insert Height Diameter Height Diameter inch inch inch inch MRF Air D Polyurethane Steel Aluminum As noticed in Table 3-3, the insert selection contained the following materials: MRF- 145, air, 3 durometer polyurethane, 118-Steel, and 661-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 118 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 661 Aluminum insert was used to counter the previous hypothesis of the 118 Steel networking with applied magnetic field. Several passive elements such as AIR and 3 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 3-D Polyurethane STE 118 Steel ALU 661 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 44

62 12L14 Steel. Additionally, an etching primer (SEM#39693) was used prior to the layer of epoxy primer as added insurance 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. Bottom Plate Mid Plate Top Plate O-Ring Figure 3-19: Three plate mold for manufacturing elastic mounts. Next, a vacuum pump and degassing chamber are needed as depicted in Figure 3-2. The pump used in the fabrication is rated at 3. 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 45

63 polyurethane early in the pot life as well as the MR fluid, but a more viscous elastomer may require higher vacuum. Bell Jar Figure 3-2: Vacuum Pump and Bell Jar. Vacuum Pump 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 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. Disposable cup Scales Beaker Syringe Figure 3-21: PolyTek TM polyurethane (Parts A and B), scales, and dispensing syringe. 46

64 Polyurethane is mixed by weight ratio and requires an accurate set of scales. The illflowing 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 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. Catalyst Part B Resin Part A Figure 3-22: Dispensing Polyurethane components by weight. Mixed Degassing Degassed Figure 3-23: Mixing polyurethane, degassing polyurethane, and degassed polyurethane processes. 47

65 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 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 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. Excess polymer material Sight Window Filling Syringe Injecting 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 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. 48

66 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 661 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 661 Aluminum (ALU), 118 Steel (STE), Air (AIR), and 3 D polyurethane (RUB) insert mounts. Aluminum Air Steel Rubber Figure 3-26: Elastic casing mounts with 661 aluminum, air, 118 steel, and solid 3 D polyurethane (rubber). 49

67 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.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.125-in. Sanded Chamfer 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 5

68 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. Emboss 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 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 metalelastic 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. 51

69 Upper Half Lower Half Parting Line 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 3-3, is filled with MRF-145 fluid and allows the fluid to drain in the internal cavity of the casing. Funnel Figure 3-3: 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, 52

70 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. Entrapped Air 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 The dry weight of the metal-elastic casing was 111.g 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. 53

71 Plug 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 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. software. The hardware within the EMA measures input displacement and velocity, while the load cell measures force. The linear actuator has a resolution of mm and able to produce harmonic inputs up to 1 Hz. Additionally, the Interface brand loadcell is able to measure forces of up to 2 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 54

72 power supply is a GW Instek GPS-233 DC power supply and has a current resolution of.1 Amp and Voltage resolution of.1 V. Figure 3-33: Roehrig-EMA Shock Dynamometer and Desktop Computer running Shock 6. 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 2 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. and zero the Load Cell 55

73 Unlock Clamps and lower Crossbar Allow Crossbar to load approximately 1 N on Mount and lock Clamps Fasten Circuit Leads to the coil Load test profile in Shock 6., run a warm-up session, run test and acquire results Crossbar Clamps Load Cell Test Fixture Test Fixture Circuit Leads Lower Housing Linear Actuator 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 1 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. 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 56

74 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 to 1 mm over 3 sec. is used for the soft core mounts and a ramp input from to.5 mm over 3 sec. is used for the solid core mounts as illustrated in Figure The current setting was varied from to 2 Amp at.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) Displacment, mm mm Ramp.5mm Ramp Time, sec. 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.5 mm is used for each sinusoidal test with the displacement bounds of to 1. mm as shown in Figure The solid core mounts denoted with an asterisk use a reduced amplitude of.25 mm and compression range of to.5 mm. The primary reason for reducing the displacement across the solid core mount is to protect the 57

75 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.5 Amp for all tested frequencies. Additional current increments of.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.5 mm to.25 mm mm Sine at 1 Hz.25mm Sine at 1 Hz Displacement, mm Time, sec. Figure 3-36: Sine displacement input for dynamic testing at 1 Hz on shock dyno. 58

76 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.8 T of magnetic flux density to the desired fluid cavity at 2. Amp and 1. T at 3. 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. 59

77 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 displacementamplitude 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. 6

78 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 Fk = 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 MRE 1,QST,.A Linear Model,Kx Nonlinear Region Discarded Region Model,Kx Displacement, mm Figure 4-1: Force-displacement plotting method example on a MR fluid-elastic mount with MRF-145 fluid. 61

79 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 N i= 1 ( d m ) 2 ε = (4.2) where d i is the data point and m i 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 i i k2 k sg.. = 1% (4.3) k where k 2 is the stiffness at 2. Amp, and k is the stiffness at. 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.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 295 to 363 N/mm, respectively. Therefore, from off-state to activated state at 2. 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 3 N/mm stiffer than MRE 1, but the stiffness at 2. Amps of current is comparable. 62

80 MRE 1,QST,.A Linear Model,Kx MRE 1,QST,.5A Linear Model,Kx Displacement, mm Displacement, mm MRE 1,QST,1.A Linear Model,Kx MRE 1,QST,1.5A Linear Model,Kx Displacement, mm Displacement, mm MRE 1,QST,2.A Linear Model,Kx Kx,.A Kx,.5A Kx,1.A Kx,1.5A Kx,2.A Displacement, mm (a) Displacement, mm MRE 2,QST,.A Linear Model,Kx MRE 2,QST,.5A Linear Model,Kx Displacement, mm Displacement, mm MRE 2,QST,1.A Linear Model,Kx MRE 2,QST,1.5A Linear Model,Kx Displacement, mm Displacement, mm MRE 2,QST,2.A Linear Model,Kx Kx,.A Kx,.5A Kx,1.A Kx,1.5A Kx,2.A Displacement, mm (b) Displacement, mm 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 metalelastic test specimens. The force-displacement plots for MR fluid-elastic 3 (MRE 3) and 63

81 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 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.-2. 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. 64

82 MRE 3,QST,.A Linear Model,Kx MRE 3,QST,.5A Linear Model,Kx Displacement, mm Displacement, mm MRE 3,QST,1.A Linear Model,Kx MRE 3,QST,1.5A Linear Model,Kx Displacement, mm Displacement, mm MRE 3,QST,2.A Linear Model,Kx Kx,.A Kx,.5A Kx,1.A Kx,1.5A Kx,2.A Displacement, mm (a) Displacement, mm MRE 3B,QST,.A Linear Model,Kx MRE 3B,QST,.5A Linear Model,Kx Displacement, mm Displacement, mm MRE 3B,QST,1.A Linear Model,Kx MRE 3B,QST,1.5A Linear Model,Kx Displacement, mm Displacement, mm MRE 3B,QST,2.A Linear Model,Kx Kx,.A Kx,.5A Kx,1.A Kx,1.5A Kx,2.A Displacement, mm (b) Displacement, mm 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.,.5, 1., 1.5, and 2. Amp. 65

83 Table 4-1: Static stiffness values for MR fluid-elastic mounts and passive mounts with air, rubber, steel, and aluminum inserts at an index.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 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,.5, 1., 1.5, and 2. Amps. The force-displacement analysis for the dynamic test data uses the same technique as presented in for linear approximation and modeling. A hysteretic content, however, is contained within the dynamic testing. Therefore, an example plot with hysteresis is 66

84 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 MRE 1,2.A,1hz Region,F(x) Model,Kx Discarded Region Loading Un-Loading Displacement, mm Figure 4-4: Force-displacement plotting method example with hysteretic content. Next, the equivalent damping coefficient is calculated as ΔE ceq = (4.4) 2 πω X where Δ E 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. 67

85 In the metal-elastic case testing, additional current settings at.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.5 mm for the oscillatory input, and current increments of.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 MRE 1,.A,1hz Region,F(x) 4 MRE 1,.5A,1hz Region,F(x) 3 2 Model,Kx 3 2 Model,Kx Displacement, mm Displacement, mm MRE 1,1.A,1hz Region,F(x) 4 MRE 1,1.5A,1hz Region,F(x) 3 2 Model,Kx 3 2 Model,Kx Displacement, mm Displacement, mm MRE 1,2.A,1hz Region,F(x) Model,Kx Displacement, mm (a) A.5A 1.A 1.5A 2.A Displacement, mm Figure 4-5: (continue) 68

86 5 5 4 MRE 2,.A,1hz Region,F(x) 4 MRE 2,.5A,1hz Region,F(x) 3 2 Model,Kx 3 2 Model,Kx Displacement, mm Displacement, mm MRE 2,1.A,1hz Region,F(x) 4 MRE 2,1.5A,1hz Region,F(x) 3 2 Model,Kx 3 2 Model,Kx Displacement, mm Displacement, mm MRE 2,2.A,1hz Region,F(x) Model,Kx Displacement, mm (b) A.5A 1.A 1.5A 2.A Displacement, mm MRE 3,.A,1hz Region,F(x) 4 MRE 3,.5A,1hz Region,F(x) 3 2 Model,Kx 3 2 Model,Kx Displacement, mm Displacement, mm MRE 3,1.A,1hz Region,F(x) 4 MRE 3,1.5A,1hz Region,F(x) 3 2 Model,Kx 3 2 Model,Kx Displacement, mm Displacement, mm MRE 3,2.A,1hz Region,F(x) Model,Kx Displacement, mm (c) A.5A 1.A 1.5A 2.A Displacement, mm Figure 4-5: (continue) 69

87 5 5 4 MRE 3B,.A,1hz Region,F(x) 4 MRE 3B,.5A,1hz Region,F(x) 3 2 Model,Kx 3 2 Model,Kx Displacement, mm Displacement, mm MRE 3B,1.A,1hz Region,F(x) 4 MRE 3B,1.5A,1hz Region,F(x) 3 2 Model,Kx 3 2 Model,Kx Displacement, mm Displacement, mm MRE 3B,2.A,1hz Region,F(x) Model,Kx Displacement, mm (d) A.5A 1.A 1.5A 2.A Displacement, mm 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.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.-2. Amps at.5 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 forcedisplacement lissajous have thin profiles. The coefficient of equivalent damping for these mounts is shown later in the results section. 7

88 5 5 4 AIR,.A,1hz Region,F(x) 4 AIR,.5A,1hz Region,F(x) 3 2 Model,Kx 3 2 Model,Kx Displacement, mm Displacement, mm AIR,1.A,1hz Region,F(x) 4 AIR,1.5A,1hz Region,F(x) 3 2 Model,Kx 3 2 Model,Kx Displacement, mm Displacement, mm AIR,2.A,1hz Region,F(x) Model,Kx Displacement, mm (a) A.5A 1.A 1.5A 2.A Displacement, mm RUB,.A,1hz Region,F(x) 4 RUB,.5A,1hz Region,F(x) 3 2 Model,Kx 3 2 Model,Kx Displacement, mm Displacement, mm RUB,1.A,1hz Region,F(x) 4 RUB,1.5A,1hz Region,F(x) 3 2 Model,Kx 3 2 Model,Kx Displacement, mm Displacement, mm RUB,2.A,1hz Region,F(x) Model,Kx Displacement, mm (b) A.5A 1.A 1.5A 2.A Displacement, mm Figure 4-6: (continue) 71

89 5 5 4 STE,.A,1hz Region,F(x) 4 STE,.5A,1hz Region,F(x) 3 2 Model,Kx 3 2 Model,Kx Displacement, mm Displacement, mm STE,1.A,1hz Region,F(x) 4 STE,1.5A,1hz Region,F(x) 3 2 Model,Kx 3 2 Model,Kx Displacement, mm Displacement, mm STE,2.A,1hz Region,F(x) Model,Kx Displacement, mm (c) A.5A 1.A 1.5A 2.A Displacement, mm ALU,.A,1hz Region,F(x) 4 ALU,.5A,1hz Region,F(x) 3 2 Model,Kx 3 2 Model,Kx Displacement, mm Displacement, mm ALU,1.A,1hz Region,F(x) 4 ALU,1.5A,1hz Region,F(x) 3 2 Model,Kx 3 2 Model,Kx Displacement, mm Displacement, mm ALU,2.A,1hz Region,F(x) Model,Kx Displacement, mm (d) A.5A 1.A 1.5A 2.A Displacement, mm 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

90 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 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 x() t = X sin( ω t+ ρ) + X (4.5) where X is the displacement amplitude, X 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 Ft () = F sin( ω t+ ϕ) + F (4.6) where F is the force amplitude, F 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 F [1 + sign( X + X sin( ω t + ρ) X )] 2 SAT SAT = (4.7) 73

91 where X SAT 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 [ ω ϕ ] Ft () F sin( t ) F [1 + sign( X + X sin( ω t + ρ) X SAT )] 2 = + + (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 F SM.. = (4.9) X 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. 74

92 5 4 Loading 3 2 UnLoading 1-1 X SAT -2-3 No Saturation MRE 1,2.A,1hz -4 Model,F(t) No Saturation Time, s 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,.5, 1., 1.5, and 2. 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. 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 75

93 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 MRE 1,.A,1hz Model,F(t) 4 MRE 1,.5A,1hz Model,F(t) Time, s Time, s MRE 1,1.A,1hz Model,F(t) 4 MRE 1,1.5A,1hz Model,F(t) Time, s Time, s MRE 1,2.A,1hz Model,F(t) Time, s (a) Time, s.a.5a 1.A 1.5A 2.A Figure 4-8: (continue) 76

94 5 5 4 MRE 2,.A,1hz Model,F(t) 4 MRE 2,.5A,1hz Model,F(t) Time, s Time, s MRE 2,1.A,1hz Model,F(t) 4 MRE 2,1.5A,1hz Model,F(t) Time, s Time, s MRE 2,2.A,1hz Model,F(t) Time, s (b) Time, s.a.5a 1.A 1.5A 2.A MRE 3,.A,1hz Model,F(t) 4 MRE 3,.5A,1hz Model,F(t) Time, s Time, s MRE 3,1.A,1hz Model,F(t) 4 MRE 3,1.5A,1hz Model,F(t) Time, s Time, s MRE 3,2.A,1hz Model,F(t) Time, s (c) Time, s.a.5a 1.A 1.5A 2.A Figure 4-8: (continue) 77

95 5 5 4 MRE 3B,.A,1hz Model,F(t) 4 MRE 3B,.5A,1hz Model,F(t) Time, s Time, s MRE 3B,1.A,1hz Model,F(t) 4 MRE 3B,1.5A,1hz Model,F(t) Time, s Time, s MRE 3B,2.A,1hz Model,F(t) Time, s (d) Time, s.a.5a 1.A 1.5A 2.A 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.5 mm for the soft core elastic case mounts and.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. 78

96 5 5 4 AIR,.A,1hz Model,F(t) 4 AIR,.5A,1hz Model,F(t) Time, s Time, s AIR,1.A,1hz Model,F(t) 4 AIR,1.5A,1hz Model,F(t) Time, s Time, s AIR,2.A,1hz Model,F(t) Time, s (a) Time, s.a.5a 1.A 1.5A 2.A RUB,.A,1hz Model,F(t) 4 RUB,.5A,1hz Model,F(t) Time, s Time, s RUB,1.A,1hz Model,F(t) 4 RUB,1.5A,1hz Model,F(t) Time, s Time, s RUB,2.A,1hz Model,F(t) Time, s (b) Time, s.a.5a 1.A 1.5A 2.A Figure 4-9: (continue) 79

97 5 5 4 STE,.A,1hz Model,F(t) 4 STE,.5A,1hz Model,F(t) Time, s Time, s STE,1.A,1hz Model,F(t) 4 STE,1.5A,1hz Model,F(t) Time, s Time, s STE,2.A,1hz Model,F(t) Time, s (c) Time, s.a.5a 1.A 1.5A 2.A ALU,.A,1hz Model,F(t) 4 ALU,.5A,1hz Model,F(t) Time, s Time, s ALU,1.A,1hz Model,F(t) 4 ALU,1.5A,1hz Model,F(t) Time, s Time, s ALU,2.A,1hz Model,F(t) Time, s (d) Time, s.a.5a 1.A 1.5A 2.A 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 forceamplitude method. The MR fluid-elastic mounts at increased current settings showed less 8

98 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 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 forceamplitude 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 forceamplitude 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 E = e + e (4.1) 2 2 F / X F( t) x( t) where e F(t) is the normalized RMS-error in the force F(t), and e x(t) is the normalized RMSerror 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. Amps. Upon inspection, the error is typically lower in the forceamplitude 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. 81

99 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., 1., and 2. Amps. Results from the metal-elastic case are shown in Figure 4-1 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 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 to 2 Amp. Similar processing error was found in the remainder of the fluid filled metal-elastic case mounts. 82

100 5 4 F /X=286-N/mm, rmse = 5% MRE 1,.A,1hz Model,F(t) 5 4 MRE 1,.A,1hz Region,F(x) K=2695-N/mm, rmse = 5% Model,Kx Time, s Displacement, mm 5 4 F /X=4192-N/mm, rmse = 1% MRE 1,1.A,1hz Model,F(t) 5 4 MRE 1,1.A,1hz Region,F(x) K=3268-N/mm, rmse = 16% Model,Kx Time, s Displacement, mm 5 4 F /X=5852-N/mm, rmse = 13% MRE 1,2.A,1hz Model,F(t) 5 4 MRE 1,2.A,1hz Region,F(x) K=3883-N/mm, rmse = 18% Model,Kx Time, s Displacement, mm Figure 4-1: 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 83

101 mount results. The final section makes a comparison at current settings of, 1., and 2. Amps for all the mounts of this study 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 -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 Hz. Additionally, the range for the stiffness magnitude is -1, N/mm, while the range for the equivalent damping coefficient is -165 Ns/mm. Remembering that the metal-elastic case was tested at.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 U F / X F / X f f N Amp f = N f = SM.. = 35 1 N f = F / X f Amp Amp (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 84

102 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 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 5% 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. Stiffness Magnitude, N/mm MRE 1, F /X,.-A MRE 1, F /X,.25-A MRE 1, F /X,.5-A MRE 1, F /X,.75-A MRE 1, F /X,1.-A MRE 1, F /X,1.25-A MRE 1, F /X,1.5-A MRE 1, F /X,1.75-A MRE 1, F /X,2.-A 1 Figure 4-11: (continue) Frequency, Hz (a) 85

103 Damping, Ns/mm MRE 1,Ceq,.-A MRE 1,Ceq,.25-A MRE 1,Ceq,.5-A MRE 1,Ceq,.75-A MRE 1,Ceq,1.-A MRE 1,Ceq,1.25-A MRE 1,Ceq,1.5-A MRE 1,Ceq,1.75-A MRE 1,Ceq,2.-A Frequency, Hz (b) 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 43% when the current is set to 2 Amps. Therefore, the same statements can be made about this mount as were made for MRE 1. 86

104 Stiffness Magnitude, N/mm MRE 2, F /X,.-A MRE 2, F /X,.25-A MRE 2, F /X,.5-A MRE 2, F /X,.75-A MRE 2, F /X,1.-A MRE 2, F /X,1.25-A MRE 2, F /X,1.5-A MRE 2, F /X,1.75-A MRE 2, F /X,2.-A Frequency, Hz (a) Damping, Ns/mm MRE 2,Ceq,.-A MRE 2,Ceq,.25-A MRE 2,Ceq,.5-A MRE 2,Ceq,.75-A MRE 2,Ceq,1.-A MRE 2,Ceq,1.25-A MRE 2,Ceq,1.5-A MRE 2,Ceq,1.75-A MRE 2,Ceq,2.-A Frequency, Hz (b) Figure 4-12: MR fluid-elastic 2 mount (MRF-145) (a) stiffness F /X, and (b) damping Ceq results obtained from analysis. 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 87

105 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 17% as the current was set to 2. 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. Stiffness Magnitude, N/mm MRE 3, F /X,.-A MRE 3, F /X,.25-A MRE 3, F /X,.5-A MRE 3, F /X,.75-A MRE 3, F /X,1.-A MRE 3, F /X,1.25-A MRE 3, F /X,1.5-A MRE 3, F /X,1.75-A MRE 3, F /X,2.-A Frequency, Hz (a) Figure 4-13: (continue) 88

106 Damping, Ns/mm MRE 3,Ceq,.-A MRE 3,Ceq,.25-A MRE 3,Ceq,.5-A MRE 3,Ceq,.75-A MRE 3,Ceq,1.-A MRE 3,Ceq,1.25-A MRE 3,Ceq,1.5-A MRE 3,Ceq,1.75-A MRE 3,Ceq,2.-A Frequency, Hz (b) Figure 4-13: MR fluid-elastic 3 mount (MRF-145) (a) stiffness F /X, and (b) damping Ceq results obtained from analysis. The last metal-elastic case mount up for discussion is MRE 3B. With no fluid, this mount preformed passively as seen in Figure 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 1 Ns/mm of damping at 1 Hz and approximately 7 N/mm of stiffness magnitude at all frequencies to the results of MR fluid-elastic 3 filled with fluid. 89

107 Stiffness Magnitude, N/mm MRE 3B, F /X,.-A MRE 3B, F /X,.25-A MRE 3B, F /X,.5-A MRE 3B, F /X,.75-A MRE 3B, F /X,1.-A MRE 3B, F /X,1.25-A MRE 3B, F /X,1.5-A MRE 3B, F /X,1.75-A MRE 3B, F /X,2.-A Frequency, Hz (a) Damping, Ns/mm MRE 3B,Ceq,.-A MRE 3B,Ceq,.25-A MRE 3B,Ceq,.5-A MRE 3B,Ceq,.75-A MRE 3B,Ceq,1.-A MRE 3B,Ceq,1.25-A MRE 3B,Ceq,1.5-A MRE 3B,Ceq,1.75-A MRE 3B,Ceq,2.-A Frequency, Hz (b) Figure 4-14: Blank metal-elastic case MRE 3B (a) stiffness F /X, and (b) damping Ceq results obtained from analysis. 9

108 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 metalelastic 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. 91

109 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. 92

110 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 5%. The stiffness evaluation quotient for MRE 2 was 57% and the damping evaluation quotient was 43%. The stiffness evaluation quotient for MR fluid-elastic 3 was 46% and the damping evaluation quotient was 17%. 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 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 1-1, 2, and 3 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 5 N/mm and the damping maximum of 4 Ns/mm. Moreover, the elastic air insert mount s average stiffness is only 2 N/mm softer than the metal-elastic case mount. 93

111 Stiffness Magnitude, N/mm AIR, F /X,.-A AIR, F /X,.5-A AIR, F /X,1.-A AIR, F /X,1.5-A AIR, F /X,2.-A Frequency, Hz (a) AIR,Ceq,.-A AIR,Ceq,.5-A AIR,Ceq,1.-A AIR,Ceq,1.5-A AIR,Ceq,2.-A Damping, Ns/mm Frequency, Hz (b) Figure 4-15: Passive mount with air insert (a) stiffness F /X, and (b) damping Ceq results obtained from analysis. 94

112 The solid rubber mount results are plotted in Figure The stiffness for RUB, as in Figure 4-16a, remained consistent around 23 N/mm. The damping showed a slight decay from 2 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. Stiffness Magnitude, N/mm RUB, F /X,.-A RUB, F /X,.5-A RUB, F /X,1.-A RUB, F /X,1.5-A RUB, F /X,2.-A 2 1 Figure 4-16: (continue) Frequency, Hz (a) 95

113 RUB,Ceq,.-A RUB,Ceq,.5-A RUB,Ceq,1.-A RUB,Ceq,1.5-A RUB,Ceq,2.-A Damping, Ns/mm Frequency, Hz (b) Figure 4-16: Passive mount with 3 D rubber insert (a) stiffness F /X, and (b) damping Ceq results obtained from analysis. As you may recall, the steel mount with a 118-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 49 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 6 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. 96

114 Stiffness Magnitude, N/mm STE, F /X,.-A STE, F /X,.5-A STE, F /X,1.-A STE, F /X,1.5-A STE, F /X,2.-A Frequency, Hz (a) STE,Ceq,.-A STE,Ceq,.5-A STE,Ceq,1.-A STE,Ceq,1.5-A STE,Ceq,2.-A Damping, Ns/mm Frequency, Hz (b) Figure 4-17: Passive mount with 118 steel insert (a) stiffness F /X, and (b) damping Ceq results obtained from analysis. 97

115 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 5 N/mm. Additionally, the damping is plotted in Figure 4-18b and has an exponential decay trend that starts around 6 Ns/mm. Therefore, the behavior of either the ALU or STE mount can be used as an upper bound for the MRE mounts. Stiffness Magnitude, N/mm ALU, F /X,.-A ALU, F /X,.5-A ALU, F /X,1.-A ALU, F /X,1.5-A ALU, F /X,2.-A 2 1 Figure 4-18: (continue) Frequency, Hz (a) 98

116 ALU,Ceq,.-A ALU,Ceq,.5-A ALU,Ceq,1.-A ALU,Ceq,1.5-A ALU,Ceq,2.-A Damping, Ns/mm Frequency, Hz (b) Figure 4-18: Passive mount with 661 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 2 N/mm for the rubber mount. With a solid insert, the elastic compressive strength is greatly affected and more than doubles to approximately 5 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. 99

117 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 2 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. 1

118 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 5 N/mm and can be used as an upper bound in the comparative study. 11

119 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 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 4-2a at 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 4-2b. 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 metalelastic 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. 12

120 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 fluidelastic mount has a large capacity for damping at low frequency which ranges from 3-16 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. 13

121 Stiffness Magnitude, N/mm MRE 3B, F /X,.Amp AIR, F /X,.Amp Frequency, Hz (a) Stiffness Magnitude, N/mm MRE 3B, F /X,2.Amp AIR, F /X,2.Amp Frequency, Hz (b) Figure 4-19: Comparing stiffness magnitude of a metal-elastic case (MRE 3B) to an elastic case (AIR) mount at (a) Amps and (b) 2 Amps of current. 14

122 Stiffness Magnitude, N/mm MRE 1, F /X,.Amp MRE 3B, F /X,.Amp Frequency, Hz (a) Stiffness Magnitude, N/mm MRE 1, F /X,2.Amp MRE 3B, F /X,2.Amp Frequency, Hz (b) Figure 4-2: Comparing stiffness magnitude of a metal-elastic case (MRE 3B) to a MR fluid-elastic mount at (a) Amps and (b) 2 Amps of current. 15

123 Stiffness Magnitude, N/mm MRE 1, F /X,.Amp RUB, F /X,.Amp Frequency, Hz (a) Stiffness Magnitude, N/mm MRE 1, F /X,2.Amp RUB, F /X,2.Amp Frequency, Hz (b) Figure 4-21: Comparing stiffness magnitude of a solid elastic case (RUB) to a MR fluid-elastic mount at (a) Amps and (b) 2 Amps of current. 16

124 6 Stiffness Magnitude, N/mm MRE 1, F /X,.Amp STE, F /X,.Amp Frequency, Hz (a) Stiffness Magnitude, N/mm MRE 1, F /X,2.Amp STE, F /X,2.Amp Frequency, Hz (b) Figure 4-22: Comparing stiffness magnitude of an elastic case with steel insert (STE) to a MR fluid-elastic mount at (a) Amps and (b) 2 Amps of current. 17

125 6 Stiffness Magnitude, N/mm MRE 1, F /X,.Amp ALU, F /X,.Amp Frequency, Hz (a) Stiffness Magnitude, N/mm MRE 1, F /X,2.Amp ALU, F /X,2.Amp Frequency, Hz (b) Figure 4-23: Comparing stiffness magnitude of an elastic case with aluminum insert (ALU) to a MR fluid-elastic mount at (a) Amps and (b) 2 Amps of current. 18

126 16 14 MRE 3B,Ceq,.-A AIR,Ceq,.-A 12 Damping, Ns/mm Frequency, Hz (a) MRE 3B,Ceq,2.-A AIR,Ceq,2.-A 12 Damping, Ns/mm Frequency, Hz (b) Figure 4-24: Comparing damping of a metal-elastic case (MRE 3B) to an elastic case (AIR) mount at (a) Amps and (b) 2 Amps of current. 19

127 16 14 MRE 1,Ceq,.-A MRE 3B,Ceq,.-A 12 Damping, Ns/mm Frequency, Hz (a) MRE 1,Ceq,2.-A MRE 3B,Ceq,2.-A 12 Damping, Ns/mm Frequency, Hz (b) Figure 4-25: Comparing damping of a metal-elastic case (MRE 3B) to a MR fluidelastic mount at (a) Amps and (b) 2 Amps of current. 11

128 16 14 MRE 1,Ceq,.-A RUB,Ceq,.-A 12 Damping, Ns/mm Frequency, Hz (a) MRE 1,Ceq,2.-A RUB,Ceq,2.-A 12 Damping, Ns/mm Frequency, Hz (b) Figure 4-26: Comparing damping of a solid elastic case (RUB) to a MR fluid-elastic mount at (a) Amps and (b) 2 Amps of current. 111

129 16 14 MRE 1,Ceq,.-A STE,Ceq,.-A 12 Damping, Ns/mm Frequency, Hz (a) MRE 1,Ceq,2.-A STE,Ceq,2.-A 12 Damping, Ns/mm Frequency, Hz (b) Figure 4-27: Comparing damping of an elastic case with steel insert (STE) to a MR fluid-elastic mount at (a) Amps and (b) 2 Amps of current. 112

130 16 14 MRE 1,Ceq,.-A ALU,Ceq,.-A 12 Damping, Ns/mm Frequency, Hz (a) MRE 1,Ceq,2.-A ALU,Ceq,2.-A 12 Damping, Ns/mm Frequency, Hz (b) Figure 4-28: Comparing damping of an elastic case with aluminum insert (ALU) to a MR fluid-elastic mount at (a) Amps and (b) 2 Amps of current. 113

131 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 fluidelastic 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. Amp in Figure 4-29a and b for stiffness and damping, respectively. At. 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. Amp. Stiffness Magnitude, N/mm MRE 1, F /X,.Amp MRE 2, F /X,.Amp MRE 3, F /X,.Amp MRE 3B, F /X,.Amp AIR, F /X,.Amp RUB, F /X,.Amp STE, F /X,.Amp ALU, F /X,.Amp 2 1 Figure 4-29: (continue) Frequency, Hz (a) 114

132 Damping, Ns/mm MRE 1,Ceq,.-A MRE 2,Ceq,.-A MRE 3,Ceq,.-A MRE 3B,Ceq,.-A AIR,Ceq,.-A RUB,Ceq,.-A STE,Ceq,.-A ALU,Ceq,.-A Frequency, Hz (b) Figure 4-29: Comparative (a) stiffness F /X, and (b) damping Ceq results obtained at.-amps from force-amplitude and force-displacement analysis, respectively. Previously, the stiffness magnitude of MRE 3B in Figure 4-29a, an empty metalelastic 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 fluidelastic mounts at the. 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. Amp, the mount stiffness and damping results are then configured in Figure 4-3a 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 115

133 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. Stiffness Magnitude, N/mm MRE 1, F /X,1.Amp MRE 2, F /X,1.Amp MRE 3, F /X,1.Amp MRE 3B, F /X,1.Amp AIR, F /X,1.Amp RUB, F /X,1.Amp STE, F /X,1.Amp ALU, F /X,1.Amp Frequency, Hz (a) Damping, Ns/mm MRE 1,Ceq,1.-A MRE 2,Ceq,1.-A MRE 3,Ceq,1.-A MRE 3B,Ceq,1.-A AIR,Ceq,1.-A RUB,Ceq,1.-A STE,Ceq,1.-A ALU,Ceq,1.-A Frequency, Hz (b) Figure 4-3: Comparative (a) stiffness F /X, and (b) damping Ceq results obtained at 1.-Amps from force-amplitude and force-displacement analysis, respectively. 116

134 Presenting the final current of 2. 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 fluidelastic 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. Stiffness Magnitude, N/mm MRE 1, F /X,2.Amp MRE 2, F /X,2.Amp MRE 3, F /X,2.Amp MRE 3B, F /X,2.Amp AIR, F /X,2.Amp RUB, F /X,2.Amp STE, F /X,2.Amp ALU, F /X,2.Amp 1 Figure 4-31: (continue) Frequency, Hz (a) 117

135 Damping, Ns/mm MRE 1,Ceq,2.-A MRE 2,Ceq,2.-A MRE 3,Ceq,2.-A MRE 3B,Ceq,2.-A AIR,Ceq,2.-A RUB,Ceq,2.-A STE,Ceq,2.-A ALU,Ceq,2.-A Frequency, Hz (b) Figure 4-31: Comparative (a) stiffness F /X, and (b) damping Ceq results obtained at 2.-Amps from force-amplitude and force-displacement analysis, respectively. Further mount comparison is implemented by tabulating the stiffness magnitude in Table 4-7. Herein, this table shows the stiffness magnitude results from., 1., and 2. Amp current supply. MR fluid-elastic 1 and MR fluid-elastic 3 have similar stiffness magnitude values at. Amp with MR fluid-elastic 2 having the higher stiffness magnitude. At 1. and 2. 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. 118

136 Table 4-7: Stiffness magnitude comparison for MR fluid-elastic and passive mounts at settings of., 1., 2. 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. 119

137 Table 4-8: Equivalent damping comparison for MR fluid-elastic and passive mounts at settings of., 1., and 2. 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. Amp an MRE behaves like a solid rubber mount and at 2. 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. 12

138 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 to 2. 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 5%, 43%, and 17%. 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. Amp, the MR fluid-elastic mounts had a stiffness magnitude comparable to the rubber mount which has a solid construction of polyurethane. At

139 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. 122

140 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 fluidelastic mounts in this study. Nominal parameters for the transfer function are found using a nonlinear optimizer in Matlab 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 F at the speed of operation ω, the use of an isolation device can reduce the force transmitted to the platform F T. The force generated by the machinery and the force transmitted to the foundation define the ratio of transmissibility TR as 123

141 F T TR = (5.1) F With the ratio of transmissibility defined as a target reduction, the force generated by an oscillatory input is f () t = Mxt () + c xt () + kxt () (5.2) where M is the mass of the machine, c eq 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 eq f () t = c x () t + kx() t (5.3) T eq Upon conversion to the Laplace domain, the force transmitted F(s) and input X(s) transfer function is Fs () ceqs k X() s = + (5.4) Furthermore, the equation of motion for the complete system has a relationship of output X(s) to input F (s) which is combined with equation (5.4) to define the transmissibility as Fs () Xs () Fs () TR = = (5.5) X () s F () s F () s Building on the aforementioned approach, the transfer function for modeling the metal-elastic case mount is Num() s F TFMRE = K = (5.6) Den () s X 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 fluidelastic 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 TF K s + 2ζωn s+ ωn MRE = (5.7) s αβ s + β where K is the gain, ζ is the nominal zero damping ratio, ω n is the nominal zero frequency, α is the nominal pole damping ratio, and β is the nominal pole frequency. This 124

142 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. Stiffness Magnitude, N/mm MRE 1, F /X,2.Amp Zero Pole Frequency, Hz 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 TR() s = s K s + 2ζω s+ ω + 2αβ s + β s + 2ζω s+ ω 2 2 n n n n Ms + K s αβ s + 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 TF MRE in equation (5.7) in a black box approach. Therefore, the transmitted force prior to taking the magnitude would be 125

143 F T 2 2 s + 2ζωn s+ ωn K s αβ s + β = F s + 2ζωn s+ ωn Ms + K s αβ s + β (5.9) where F 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 Nominal Parameter Results and Comparison This section discusses the methods used to process the nominal parameters for the proposed transfer function model TF MRE 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 F J = norm TF X (5.1) 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 -2. 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, KI = I + 257; MRE 2, KI = I + 36; (5.11) MRE 3, K = I ; I 126

144 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 K I MRE 1 MRE 2 MRE Gain, K Current, A 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, ζ =.8315 I ; α =.3394 I ; I MRE 2, ζ =.8153 I ; α =.2593 I ; I MRE 3, ζ =.6154 I ; α =.2397 I ; I I I I (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 127

145 parameters increasing with current is suggestive of increasing the parametric damping within the isolator MRE 1 MRE 2 MRE 3 Zero-Damping Ratio, ζ Figure 5-3: Current, A (continue) (a) 128

146 MRE 1 MRE 2 MRE 3 Pole-Damping Ratio, α Current, A (b) 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 ω n 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 ω n and β is quasi-linear for all three MREs and is also modeled as a function of current as follows: MRE 1, ωni, = I ; βi = I ; MRE 2, ωni, = I ; βi = 2.17 I + 15.; (5.13) MRE 3, ω =.53 I ; β =.169 I ; ni, 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. I 129

147 MRE 1 MRE 2 MRE 3 14 Zero-Frequency, ω n Figure 5-4: Current, A (a) (continue) 13

148 MRE 1 MRE 2 MRE 3 14 Pole-Frequency, β Current, A (b) 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 TF 2 2 ( I + 3.1) ( s 6.2 ( I 4.4) ( I + 1.5) s+ 14 ( I 4.4) ) MRE1 = s 2.4 ( I 4.7) ( I + 4.4) s+ 12 ( I 4.7) (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. 131

149 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 TF MRE. 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 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 Zero/Pole Damping Ratio ζ/α MRE 1.2 MRE 2 MRE Current, A Figure 5-5: Non-parametric damping ratio relationship, ζ/α, at each current setting for MR fluid-elastic mount models. 132

150 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 kz P ω = (5.15) m where k Z-P is the nominal stiffness of either the zero or pole, and m Z-P is a normalized mass. By alteration, the Zero/Pole stiffness relationship k R is defined as k R Z P ω = (5.16) β 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. 2 nz, 2 P MRE 1 MRE 2 MRE 3 Zero/Pole Stiffness Ratio, ω 2 n /β Current, A Figure 5-6: Non-parametric stiffness ratio relationship, ω 2 n /β 2, at each current setting for MR fluid-elastic mount models. 133

151 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 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 to 35 Hz and for the range with 2 to 7, N/mm. To reduce congestion, current increments at.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 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. 134

152 Stiffness, N/mm MRE 1, F /X,.-A MRE 1,TF,.-A MRE 1, F /X,.5-A MRE 1,TF,.5-A MRE 1, F /X,1.-A MRE 1,TF,1.-A MRE 1, F /X,1.5-A MRE 1,TF,1.5-A MRE 1, F /X,2.-A MRE 1,TF,2.-A Frequency, Hz Figure 5-7: Stiffness simulation results for MR fluid-elastic 1 mount at.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

153 Stiffness, N/mm MRE 2, F /X,.-A MRE 2,TF,.-A MRE 2, F /X,.5-A MRE 2,TF,.5-A MRE 2, F /X,1.-A MRE 2,TF,1.-A MRE 2, F /X,1.5-A MRE 2,TF,1.5-A MRE 2, F /X,2.-A MRE 2,TF,2.-A Frequency, Hz Figure 5-8: Stiffness simulation results for MR fluid-elastic 2 mount at.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. 136

154 Stiffness, N/mm MRE 3, F /X,.-A MRE 3,TF,.-A MRE 3, F /X,.5-A MRE 3,TF,.5-A MRE 3, F /X,1.-A MRE 3,TF,1.-A MRE 3, F /X,1.5-A MRE 3,TF,1.5-A MRE 3, F /X,2.-A MRE 3,TF,2.-A Frequency, Hz 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 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. 137

155 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 TF Error yf x f = abs 1% (5.17) x f where y f is the simulated stiffness magnitude value, x f 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 to 5% for each.25 Amp current increment. The maximum discrete error that occurred at each current is shown in Figure 5-1a, 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. 138

156 5 4.5 MRE 1 TF Max Error MRE 1 TF Mean Error Error, % Current, Amps (a) MRE 2 TF Max Error MRE 2 TF Mean Error Error, % Current, Amps (b) Figure 5-1: (continue) 139

157 MRE 3 TF Max Error MRE 3 TF Mean Error Error, % Current, Amps (c) Figure 5-1: 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. 14

158 Error, % -5-1 MRE 1,TF-Error,.-A MRE 1,TF-Error,.25-A MRE 1,TF-Error,.5-A MRE 1,TF-Error,.75-A MRE 1,TF-Error,1.-A MRE 1,TF-Error,1.25-A MRE 1,TF-Error,1.5-A MRE 1,TF-Error,1.75-A MRE 1,TF-Error,2.-A Frequency, Hz 15 (a) 1 5 Error, % -5-1 MRE 2,TF-Error,.-A MRE 2,TF-Error,.25-A MRE 2,TF-Error,.5-A MRE 2,TF-Error,.75-A MRE 2,TF-Error,1.-A MRE 2,TF-Error,1.25-A MRE 2,TF-Error,1.5-A MRE 2,TF-Error,1.75-A MRE 2,TF-Error,2.-A Frequency, Hz (b) Figure 5-11: (continue) 141

159 Error, % -5-1 MRE 3,TF-Error,.-A MRE 3,TF-Error,.25-A MRE 3,TF-Error,.5-A MRE 3,TF-Error,.75-A MRE 3,TF-Error,1.-A MRE 3,TF-Error,1.25-A MRE 3,TF-Error,1.5-A MRE 3,TF-Error,1.75-A MRE 3,TF-Error,2.-A Frequency, Hz (c) 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 C eq results. 142

160 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 C ( C b f ) Mount = Cae (5.18) where C a is damping model coefficient, C b 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 C a s current dependence for each mount as follows: MRE 1, C = 133 I + 51, ai, MRE 2, C = 116 I + 68, ai, MRE 3, C = 83.6 I ai, (5.19) With little variation in the exponential coefficient, C b = -.6 and the damping model for MR fluid-elastic 1 is CMRE1 (133 I 51) e (.6 f ) = + (5.2) where I is the input current, and f is the oscillatory input frequency. Therefore, equation (5.2) 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. 143

161 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.5 Amp at all currents. Beyond 4 Hz and above.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. Est. Damping, Ns/mm MRE 1,Ceq,.-A MRE 1,Model,.-A MRE 1,Ceq,.25-A MRE 1,Model,.25-A MRE 1,Ceq,.5-A MRE 1,Model,.5-A MRE 1,Ceq,.75-A MRE 1,Model,.75-A MRE 1,Ceq,1.-A MRE 1,Model,1.-A MRE 1,Ceq,1.25-A MRE 1,Model,1.25-A MRE 1,Ceq,1.5-A MRE 1,Model,1.5-A MRE 1,Ceq,1.75-A MRE 1,Model,1.75-A MRE 1,Ceq,2.-A MRE 1,Model,2.-A Frequency, Hz 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 The exponential 144

162 model is able to replicate the empirical results as long as the current is lower than.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. Est. Damping, Ns/mm MRE 2,Ceq,.-A MRE 2,Model,.-A MRE 2,Ceq,.25-A MRE 2,Model,.25-A MRE 2,Ceq,.5-A MRE 2,Model,.5-A MRE 2,Ceq,.75-A MRE 2,Model,.75-A MRE 2,Ceq,1.-A MRE 2,Model,1.-A MRE 2,Ceq,1.25-A MRE 2,Model,1.25-A MRE 2,Ceq,1.5-A MRE 2,Model,1.5-A MRE 2,Ceq,1.75-A MRE 2,Model,1.75-A MRE 2,Ceq,2.-A MRE 2,Model,2.-A Frequency, Hz Figure 5-13: Damping simulation results for MR Fluid-Elastic 2 mount at full range of current settings. Lastly, the model is simulated for the damping in MR fluid-elastic 3 as shown in Figure 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. 145

163 Est. Damping, Ns/mm MRE 3,Ceq,.-A MRE 3,Model,.-A MRE 3,Ceq,.25-A MRE 3,Model,.25-A MRE 3,Ceq,.5-A MRE 3,Model,.5-A MRE 3,Ceq,.75-A MRE 3,Model,.75-A MRE 3,Ceq,1.-A MRE 3,Model,1.-A MRE 3,Ceq,1.25-A MRE 3,Model,1.25-A MRE 3,Ceq,1.5-A MRE 3,Model,1.5-A MRE 3,Ceq,1.75-A MRE 3,Model,1.75-A MRE 3,Ceq,2.-A MRE 3,Model,2.-A Frequency, Hz Figure 5-14: Damping simulation results for MR Fluid-Elastic 3 mount at full range of current settings. 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.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 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. 146

164 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 nonparametric 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 147

165 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 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 (5.2) 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 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.75 Amp of input current may not be estimated with confidence using this model. 148

166 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. 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. 149

167 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 1% of its static height. Therefore, the height of the fluid cavity was set to.1875 in. and the height of the mount was set to.4375 in. The sidewall of the casing was designed to a thickness of.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 3 durometer rated polyurethane rubber from PolyTek (PolyTek 74-3). 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 metalelastic 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. 15

168 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 mm and therefore, testing was limited to.5 mm amplitude for the MR fluid-elastic mounts. The lower resolution of.25 mm was used for the elastomer mounts with metal inserts to prevent damaging the actuator and the 2 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 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 5%, 43%, and 17%, 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 151

169 than the rubber mount, but activated with 2. 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 3 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 152

170 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.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.5 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. 153

171 If additional consultation is needed you may contact the author at 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.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 3 turn coil with 24 AWG magnet wire for compliance with a 12 V power supply. The magnitude of magnetic flux 154

172 at the center of the fluid gap is.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 155

173 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 156

174 References 1. Miller, L.R., Ahmadian, M., Active Mounts - A Discussion of Future Technological Trends, Proceedings of Internoise-92. Toronto, Canada, Rabinow, J., The magnetic fluid clutch. AIEE Trans. 67, p. 138, Lange, U., Richter, L., Zipser, L., Flow of Magnetorheological Fluids, Journal of Intelligent Material Systems and Structures, Vol. 12, p , 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, 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 Lord Corporation, LORD Product Selector Guide, Ahn, Y.K., Yang, B., Ahmadian, M., Morishita, S., A Small-sized Variabledamping Mount Using Magnetorheological Fluid, International Journal of Vehicle System Dynamics, Vol. 16, p , Carlson, J.D., Weiss, K.D., A growing attraction to magnetic fluids, Machine Design, Vol. 66, No. 15, p , 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 , General Motors Corporation, Magnetic Selective Ride Control, Audi, Audi Magnetic Ride Suspension, 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. 8-87,

175 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 Song, X., Ahmadian, M., Southward, S.C., An Adaptive Semiactive Control Algorithm for Vehicle Suspension Systems, Proceedings of IMECE. Washington, D.C., Ahmadian, M., Vahdati, N., Transient Dynamics of Semiactive Suspensions with Hybrid Control, Journal of Intelligent Material Systems and Structures, Vol. 17 (No.2), p , 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., 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. 1 (No. 9), p , Lord Materials Division, Designing with MR Fluids, Engineering Note, p. 1-5, Carlson, J.D., Jolly, M.R., MR fluid, foam and elastomer devices, Mechatronics, Vol. 1, p , Mazlan, S.A., Ekreem, N.B., Olabi, A.G., The performance of magnetorheological fluid in squeeze mode, Smart Materials and Structures, Vol. 16, p , 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, York, D., Wang, X., Gordaninejad, F., A New MRF-E Vibration Isolator, Journal of Intelligent Material Systems and Structures, Vol. 18, p ,

176 24. Christopherson, J., Jazar, G.N., Dynamic behavior comparison of passive hydraulic engine mounts. Part 1: Mathematical analysis, Journal of Sound and Vibration, Vol. 29, p , 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 , 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, Vahdati, N., Ahmadian, M., Variable Volumetric Stiffness Fluid Mount Design, Shock and Vibration, Vol. 11, p.21-32, 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 , 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. 1, p , Boczkowska, A., Awietjan, S.F., Wroblewski, R., Microstructure-property relationships of urethane magnetorheological elastomers, Smart Materials and Structures, Vol. 16, p , Zhou, G.Y., Complex shear modulus of a magnetorheological elastomer, Smart Materials and Structures, Vol. 13, p , 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 , Gong, X.L., Zhang, X.Z., and Zhang, P.Q., Fabrication and characterization of isotropic magnetorheological elastomers, Polymer Testing, Vol. 24, p ,

177 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 , Wang, X., Gordaninejad, F., Hitchcock, G., A magneto-rheological fluid elastomer vibration isolator, Smart Structures and Materials, Proc. of SPIE Vol. 576, p , Gordaninejad, F., Fuchs, A., Wang, X., Hitchcock, G.H., and Xin, M., Magnetorheological fluid encased in flexible materials for vibration control, U.S. Patent 6,971,491 B1, 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 27 IDETC/CIE, Las Vegas, Nevada, Ahmadian, M., Ahn, Y.A., Performance Analysis of Magneto-Rheological Mounts, Journal of Intelligent Material Systems and Structures, Vol. 1, p , Delphi Chassis and Steering Systems, Delphi Magneto-Rheological (MR) Powertrain Mount, p. 1-4, 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, Tewani, S.G., Bodie, M., Kiefer, J., Long, M., Walterbusch, J., Magnetorheological-Fluid Hydraulic Mount, Delphi Technologies, Inc., U.S. Patent 7,118,1 B2, Vahdati, N., Ahmadian, M., Single Pumper Semi-active Fluid Mount, Proc. of ASME 23 IMECE 23, International Mechanical Engineers Conference, Washington, D.C, 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. 539, p ,

178 44. Inman, D.J., Engineering Vibration, Third Edition, Pearson Education, Inc., Upper Saddle River, New Jersey, 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, Dorf, R.C., Bishop, R.H., Modern Control Systems, Ninth Edition, p. 38-4, Prentice-Hall, Inc, Upper Saddle River, New Jersey, Burchett, B.T., Layton, R.A., An Undergraduate System Identification Experiment, American Control Conference, Portland, OR, USA, Meeker, D., Finite Element Method Magnetics (FEMM), Foster-Miller, Inc., Carlson, J. D., Mr Fluids and Devices in the Real World, International Journal of Modern Physics B, Vol. 19, p , Lord Corporation, MRF-122 Product Bulletin, Lord Technical Data, Lord Corporation, MRF-132 Product Bulletin, Lord Technical Data, Lord Corporation, MRF-14 Product Bulletin, Lord Technical Data, Lord Corporation, Lord Magneto-Rheological Fluids, Lord Product Selector Guide, PolyTek Development Corp., Redwood Plastics Corp., Hardness Comparison Chart, Redco Polyurethane Brochure, Ogata, K., System Dynamics, Third Edition, p , Prenctice Hall, Saddle River, New Jersey, Lord Corporation, Cosmo Corporation, A.P.W Company Inc., 6. Belmont Machining, Inc., (54)

179 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. 162

180 Table A- 1: Bill of Materials without cost estimates for mount and magnet system and manufacture. 163

181 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. 164

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

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

184 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 nonmagnetic 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-1 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. 167

185 Figure A-5: Upper housing schematic to magnetic system Figure A-6: Spacer schematic to lower housing in magnetic system. 168

186 Figure A-7: Upper-pole plate schematic for metal-elastic case mount. Figure A-8: Magnetic-pole plate schematic for metal-elastic case mount. 169

187 Figure A-9: Lower housing test fixture schematic for Roehrig Dynamometer. Figure A-1: Upper Housing Test Fixture for Roehrig Dynamometer. 17

188 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. Case Half Un-Prepped Prepped Filled Elastic Case Case Elastic Case 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 Put polyurethane mixture in degas chamber as seen in Figure A Remove polyurethane and pour into syringe 6. Plug syringe into the mold and dispense liquid rubber 171

189 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 1. 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 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. 172

190 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. 173

191 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. 174

192 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. 175

193 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 176

194 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. 177

195 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 fluidelastic case mounts. Table B-4: MR Fluid-elastic mount Stiffness Analysis Results for MRE s and blank MRE

196 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.5 Amp current indexing. Table B-6: MR fluid-elastic and passive mount stiffness analysis comparison chart. 179

197 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, 1, and 2-Amp current settings. 18

198 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, 1, 2-Amp for MR fluid-elastic mounts 1, 2 and

199 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

200 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 forcedisplacement 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 = 2; %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'}; 183

201 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_1mm_','MRE_2_DST_1mm_','MRE_3_DST_1mm_',... 'MRE_3b_DST_1mm_','AIR_DST_1mm_','RUB_DST_1mm_',... 'STE_DST_5mm_','ALU_1_DST_5mm_'}; loadermount = loadermounts{mount}; elastomer = {'MRE_','MRE_','MRE_','MRE_','AIR','RUB_','STE_','ALU_'}; %Model elastomers = fileholes; if mount <=4 current = {'','25','5','75','1','125','15','175','2'}; ampleg = {'.-A','.25-A','.5-A','.75-A','1.-A','1.25-A',... '1.5-A','1.75-A','2.-A'}; L4 = {[' 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 = {'','5','1','15','2'}; ampleg = {'.-A','.5-A','1.-A','1.5-A','2.-A'}; L4 = {[' 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','1hz',... '2hz','3hz'}; freqfull = [1,2,3,4,5,6,7,8,9,1,2,3]; freqleg = {'1-Hz','2-Hz','3-Hz','4-Hz','5-Hz','6-Hz','7-Hz','8-Hz',... '9-Hz','1-Hz','2-Hz','3-Hz'}; nfiles = 12; else hzs = {'1hz','2hz','3hz','4hz','5hz','6hz','7hz','8hz','9hz','1hz',... '12hz','14hz','16hz','18hz','2hz','25hz','3hz','35hz'}; freqfull = [1,2,3,4,5,6,7,8,9,1,12,14,16,18,2,25,3,35]; freqleg = {'1-Hz','2-Hz','3-Hz','4-Hz','5-Hz','6-Hz','7-Hz','8-Hz',... '9-Hz','1-Hz','12-Hz','14-Hz','16-Hz','18-Hz','2-Hz','25-Hz',... '3-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 = 1; %x axis font size yfsize = 1; %y axis font size 184

202 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,); 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,1); dispfix = min(disp); dispnorm = disp - dispfix; dis = smooth(dispnorm,1); N = length(dis); t = (:(N-1)) * 1/fsamp; xtime = 1/freq; pn = floor(xtime*fsamp); ptime = (:(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 == 5 185

203 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*7*x+b)+c','a*sin(2*pi*8*x+b)+c','a*sin(2*pi*9*x+b)+c',... 'a*sin(2*pi*1*x+b)+c','a*sin(2*pi*2*x+b)+c','a*sin(2*pi*3*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*4*x+b)+c','a*sin(2*pi*5*x+b)+c','a*sin(2*pi*6*x+b)+c',... 'a*sin(2*pi*7*x+b)+c','a*sin(2*pi*8*x+b)+c','a*sin(2*pi*9*x+b)+c',... 'a*sin(2*pi*1*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*2*x+b)+c',.. 'a*sin(2*pi*25*x+b)+c','a*sin(2*pi*3*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)/2.3 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 186

204 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) >=.1 figure(1+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;%.25; %.35; %xlow xth =.1; %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',[23-3]); 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*1; XFHolds(row,8) = NormDisRMSE*1; XFHolds(row,9) = NormKxRMSE*1; 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 187

205 sysparam(row,1) = 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*1; %RMSE/(maxF-minF) sysparam(row,14) = NormDisRMSE*1; %RMSE/(maxX-minX) sysparam(row,15) = NormKxRMSE*1; %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*1; XFHold(rowa,8) = NormDisRMSE*1; XFHold(rowa,9) = NormKxRMSE*1; if ampfullset == 5 %% %Figure1s F(t) Model dks = {'k',':c'}; dk = dks{1}; L1s = {['Data:',freqleg{i}],'Model,F(t)'}; figure(i+),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([ max(t) 5]), legend(l1s,'location','northeast','fontsize',lsize), %% %Figure2s K(x) Model L2s = {['Data:',freqleg{i}],'RegionModel','Model-K(x)'}; figure(i+2),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(l2s,'location','northwest','fontsize',lsize), ylabel('','fontsize',yfsize), xlabel('disp, mm','fontsize',xfsize), axis([ 1 5]) %% %Figure6s F(t)_Model vs F/x_Model L62s = {['Data:',freqleg{i}],'RegionModel','Model-K(x)'}; 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+6),subplot(3,2,spot),plot(xdata,force,dk,'linewidth',2), hold on, plot(xdata,fvecs,'m','linewidth',1),xlabel(''), ylabel('force, N','fontsize',yfsize),legend(L1s,'Location','NorthEast','fontsize',lsize), xlabel('time, s','fontsize',xfsize),axis([ max(t) 5]), 188

206 text(xtime*.1,4,['nrmse = ',num2str(round(normftrmse*1)),'%'],'fontsize',lsize), figure(i+6),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 legend(l62s,'location','northwest','fontsize',lsize), axis([ 1 5]),ylabel('Force, N','fontsize',yfsize),xlabel('disp, mm','fontsize',xfsize), text(.5,2,['nrmse = ',num2str(round(normkxrmse*1)),'%'],'fontsize',lsize) end %% %Figure8s,1,2-Amp F(t)_Model vs K(x)_Model L8s = {['Data:',freqleg{i}],'Model-F(t)'}; L81s = {['Data:',freqleg{i}],'RegionModel','Model-K(x)'}; if (amps==1) (amps==3) (amps==5) spot = 1+(amps-1);%(1+(1-1)/2 = 1, 1+(5-1)/2=3, 1+(9-1)/2 = 5 figure(i+8),subplot(3,2,spot),plot(xdata,force,dk,'linewidth',2), hold on, plot(xdata,fvecs,'m','linewidth',1),xlabel('time, s','fontsize',xfsize), ylabel('force, N','fontsize',yfsize),legend(L8s,'fontsize',lsize), axis([ max(t) 5]), text(xtime*.1,4,['nrmse = ',num2str(round(normftrmse*1)),'%'],'fontsize',lsize) figure(i+8),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, text(.5,2,['nrmse = ',num2str(round(normkxrmse*1)),'%'],'fontsize',lsize) xlabel('disp, mm','fontsize',xfsize), axis([ 1 5]), 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+),subplot(3,3,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([ max(t) 5]), legend(l1s,'location','northeast','fontsize',lsize), %% %Figure2s K(x) Model L2s = {['Data:',freqleg{i}],'RegionModel','Model-K(x)'}; figure(i+2),subplot(3,3,amps),plot(pdis,pforce,dk,'linewidth',2),hold on, plot(pdis(~linear),pforce(~linear),'m','linewidth',1),hold on, 189

207 plot(pdis(~linear),linmod,'--b','linewidth',1), legend(l2s,'location','northwest','fontsize',lsize), ylabel('','fontsize',yfsize), xlabel('disp, mm','fontsize',xfsize), axis([ 1 5]) L62s = {['Data:',freqleg{i}],'RegionModel','Model-K(x)'}; if (amps==1) (amps==5) (amps==9) spot = 1+(amps-1)/2;%(1+(1-1)/2 = 1, figure(i+6),subplot(3,2,spot),plot(xdata,force,dk,'linewidth',2), hold on, plot(xdata,fvecs,'m','linewidth',1),xlabel(''), ylabel('force, N','fontsize',yfsize),legend(L1s,'Location','NorthEast','fontsize',lsize), xlabel('time, s','fontsize',xfsize),axis([ max(t) 5]), text(xtime*.1,4,['nrmse = ',num2str(round(normftrmse*1)),'%'],'fontsize',lsize), figure(i+6),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 legend(l62s,'location','northwest','fontsize',lsize), axis([ 1 5]),ylabel('Force, N','fontsize',yfsize),xlabel('disp, mm','fontsize',xfsize), text(.5,2,['nrmse = ',num2str(round(normkxrmse*1)),'%'],'fontsize',lsize) end %% %Figure8s,1,2-Amp F(t)_Model vs K(x)_Model L8s = {['Data:',freqleg{i}],'Model-F(t)'}; L81s = {['Data:',freqleg{i}],'RegionModel','Model-K(x)'}; if (amps==1) (amps==5) (amps==9) spot = 1+(amps-1)/2;%(1+(1-1)/2 = 1, 1+(5-1)/2=3, 1+(9-1)/2 figure(i+8),subplot(3,2,spot),plot(xdata,force,dk,'linewidth',2), hold on, plot(xdata,fvecs,'m','linewidth',1),xlabel('time, s','fontsize',xfsize), ylabel('force, N','fontsize',yfsize),legend(L8s,'fontsize',lsize), axis([ max(t) 5]), text(xtime*.1,4,['nrmse = ',num2str(round(normftrmse*1)),'%'],'fontsize',lsize) figure(i+8),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, text(.5,2,['nrmse = ',num2str(round(normkxrmse*1)),'%'],'fontsize',lsize) xlabel('disp, mm','fontsize',xfsize), axis([ 1 5]), legend(l81s,'location','northwest','fontsize',lsize),xlabel('disp, mm','fontsize',xfsize) end elseif (ampfullset ~= 5) && (ampfullset ~= 9) ['Figure Printing to Screen is Off'] 19

208 end %% %I save XFHold at amp(1-9) or -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']; %% %Figure4 Compare Stiffness Vs Frequency % if amps==1 amps==3 amp==5 amp==7 amp==9 figure(4),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(l4,'location','southeast','fontsize',lsize), axis([ 35 7]) % 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([ ]) figure(42),plot(xfhold(:,1),xfhold(:,2),style1),hold on, xlabel('frequency, Hz','fontsize',14),ylabel('Stiffness, N/mm','fontsize',14), legend(ampleg,'location','southeast'),hold on, axis([ 35 7]) figure(43),plot(xfhold(:,1),xfhold(:,3),style1),hold on, xlabel('frequency, Hz','fontsize',14),ylabel('Stiffness, N/mm','fontsize',14), legend(ampleg,'location','southeast'),hold on, axis([ 35 7]) %% %Write Data to Files when current = 2. 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) 191

209 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(4-43) when current = 2. or amp == 9 pathname2 = ['C:\Documents and Settings\Administrator\My Documents\My Pictures\Chapter 5 Mount\',filehole]; cd(pathname2) f4 = ['-f',num2str(4)]; print(f4,'-r9','-dtiff','stiffness_amplitude_linear_vs_freq'), pause(8) % figure(4),set(gcf),close gcf; f41 = ['-f',num2str(41)]; print(f41,'-r9','-dtiff','phase_amplitude_freq') % figure(41),set(gcf),close gcf; pause(8) f42 = ['-f',num2str(42)]; print(f42,'-r9','-dtiff','stiffness_amplitude_vs_freq') pause(8) f43 = ['-f',num2str(43)]; print(f43,'-r9','-dtiff','stiffness_linear_vs_freq') %% %Export figures(1-98) when current = 2., or amp == 9 for i = 1:nfiles % %Figure1s pause(8) famp3x3 = ['-f',num2str(i)]; print(famp3x3,'-r3','-dtiff',['amp_ft_fit_',current{amps},'a_',freqleg{i}]) %% %Figure2s pause(8) fi2 =['-f',num2str(i+2)]; print(fi2,'-r3','-dtiff',['line_kx_fit_',current{amps},'a_',freqleg{i}]) pause(1) fi6 = ['-f',num2str(i+6)]; 192

210 print(fi6,'-r3','-dtiff',['ft_kx_',current{amps},'a_',freqleg{i}]) pause(6) fi8 = ['-f',num2str(i+8)]; print(fi8,'-r3','-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,.2,1.9,1.8,1.1,1.3,.7]; xts = (1+(amp- 1)/16)*[.38,.37,.36,.35,.34,.33,.32,.3,.28,.26,.25,.24,.23,.22,.21,.2,.2,.2]; xtl =.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,.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,,1.5,1.5,1.5,1,1.5]; end xts = (1+(amp- 1)/16)*[.38,.37,.36,.35,.34,.33,.32,.3,.28,.26,.25,.24,.23,.22,.21,.2,.2,.2]; xtl =.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,,2]; end xts = (1+(amp- 1)/16)*[.38,.37,.36,.35,.34,.33,.32,.3,.28,.26,.25,.24,.23,.22,.21,.2,.2,.2]; xtl =.3; elseif mount == 4 %MRE 3b xts = [.3,.3,.3,.3,.3,.3,.3,.3,.3,.3,.3,.3,.3,.3,.3,.3,.3,.3]; %all amps pts = [.3,.3,.3,.3,.3,.3,.3,.3,.3,.8,2,.3,.3,.3,.25,.3,.3,.3]; xtl =.3; elseif mount == 5 %AIR 193

211 xts =[.125,.125,.125,.125,.125,.125,.125,.125,.125,.125,.125,.125]; %all amps pts = [.3,.3,.3,.3,.3,.3,.3,.3,.3,2,2,.3,.3,.3,.25,.3,.3,.3]; xtl =.125; elseif mount == 6 %RUB pts = [.3,.3,.3,.3,.3,.3,.3,.3,.3,1,1.15,1.3,2,1.3,1.25,.3,.3,.3] xts = [.1,.1,.1,.1,.1,.1,.1,.1,.1,.1,.1,.1,.1,.1,.1,.1,.1,.1]; xtl =.1; elseif mount == 7 %STE xts = [.11,.11,.11,.11,.11,.11,.11,.11,.11,.11,.11,.11,.11,.11,.11,.11,.11,.11 ]; %A=.3,.25A=.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 =.11; elseif mount == 8 %ALU xts = [.11,.11,.11,.11,.11,.11,.11,.11,.11,.11,.11,.11,.11,.11,.11,.11,.11,.11 ]; %A=.3,.25A=.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 =.11; end xtlout = xtl; xtsout = xts(freq); ptsout = pts(freq); %%END OF FUNCTION program4fun 194

212 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 = 2; %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 for no print and 8 for full print wantprint = ; %-off, 1-on for all figures wantsave = ; %-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_1mm_','MRE_2_DST_1mm_','MRE_3_DST_1mm_',... 'MRE_3b_DST_1mm_','AIR_DST_1mm_','RUB_DST_1mm_',... 'STE_DST_5mm_','ALU_1_DST_5mm_'}; 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 = {'','5','1','15','2'}; currentinc = {'.Amp','.5Amp','1.Amp','1.5Amp','2.Amp'}; if i <=4 current = {'','25','5','75','1','125','15','175','2'}; ampleg = {'.-A','.25-A','.5-A','.75-A','1.-A','1.25-A',... '1.5-A','1.75-A','2.-A'}; L4 = {[' 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}]}; Li = {['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,',a mpleg{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}],

213 [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 = [,.25,.5,.75,1.,1.25,1.5,1.75,2.]; elseif i >4 current = {'','5','1','15','2'}; ampleg = {'.-A','.5-A','1.-A','1.5-A','2.-A'}; L4 = {[' 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}]}; Li = {['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,',a mpleg{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 = [,.5,1,1.5,2]; end if i == 5 hzs = {'1hz','2hz','3hz','4hz','5hz','6hz','7hz','8hz','9hz','1hz',... '2hz','3hz'}; freqfull = [1,2,3,4,5,6,7,8,9,1,2,3]; freqleg = {'1-Hz','2-Hz','3-Hz','4-Hz','5-Hz','6-Hz','7-Hz','8-Hz',... '9-Hz','1-Hz','2-Hz','3-Hz'}; nfiles = 12; else hzs = {'1hz','2hz','3hz','4hz','5hz','6hz','7hz','8hz','9hz','1hz',... '12hz','14hz','16hz','18hz','2hz','25hz','3hz','35hz'}; freqfull = [1,2,3,4,5,6,7,8,9,1,12,14,16,18,2,25,3,35]; freqleg = {'1-Hz','2-Hz','3-Hz','4-Hz','5-Hz','6-Hz','7-Hz','8-Hz',... '9-Hz','1-Hz','12-Hz','14-Hz','16-Hz','18-Hz','2-Hz','25-Hz',... '3-Hz','35-Hz'}; nfiles = 18; end if loadstorage == 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 == sa = 1+nfiles*(amps-1); %(1,19,37,etc ea = nfiles + nfiles*(amps-1); 196

214 [base] = program5base(mountnum,amps); XFHold(1:nfiles+1,1) = [;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) = [;Phasedata(sa:ea)]; Freqvec = [;Freqdata(sa:ea)]; StiffVec = XFHold(1:nfiles+1,2); magvecnormdbe=2.*log1(abs(stiffvec)); [ka,za,wa,aa,ba] = program5guess(mountnum,amps); x = [ka,za,wa,aa,ba]; [xx,costval] = fminsearch(@(x)program5fun(x,magvecnormdbe),x); 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 = 2.*log1(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); 197

215 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 = 1; %x axis font size yfsize = 1; %y axis font size msize = 5; %markersize lsize = 8; fonts = 1; 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']; style7 = [':',marker{amps},colors2{amps}]; %figures 7-77 xmin = ; xmax = 35; ymin = ; ymax = 1; %% Figure For Model With Data for each current = 36+2 = 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([ 2 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([ 2 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([ 2 8]),legend(Mount,'fontsize',lsize),hold on figure(6),plot(current,zetal,style1iq,'markersize',msize,'linewidth',lwide),hold on, xlabel('current, A','fontsize',fonts),ylabel(['Zeta,','\zeta'],'fontsize',fonts), axis([ 2 7]),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([ 2 5]),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([ 2 7]),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([ 2 5]),legend(Mount,'fontsize',lsize),hold on end %% Figure For Model + Data -2 Amp Comparison, each mount = 9 or 5 plots on 8 graphs figure(69+i),plot(freqvec,stiffvec,style7,'markersize',msize),hold on, 198

216 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]) %%.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 %%.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 5 199

217 % 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(4-43) when current = 2. 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,'-r9','-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,'-r15','-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','-r9', '-dtiff','damping_ratio_plot'),pause(3) print('-f58','-r9', '-dtiff','stiffness_ratio_plot'),pause(3) print('-f59','-r9', '-dtiff','gain_plot'),pause(3) print('-f6','-r9', '-dtiff','zeta_plot'),pause(3) print('-f61','-r9', '-dtiff','omega_plot'),pause(3) print('-f62','-r9', '-dtiff','alpha_plot'),pause(3) print('-f63','-r9', '-dtiff','beta_plot'),pause(3) end 2

218 %% Export Figure Data and Model Comparison if amps == 1%ampfull pathname9 = 'C:\Documents and Settings\Administrator\My Documents\My Pictures\Chapter 6 Mount\Comparison'; cd(pathname9),pause(1) print('-f78','-r9', '-dtiff','mount_data_amp'),pause(3) print('-f79','-r9', '-dtiff','mount_data_5amp'),pause(3) print('-f8','-r9', '-dtiff','mount_data_1amp'),pause(3) print('-f81','-r9', '-dtiff','mount_data_15amp'),pause(3) print('-f82','-r9', '-dtiff','mount_data_2amp'),pause(3) print('-f84','-r9', '-dtiff','mount_model_amp'),pause(3) print('-f85','-r9', '-dtiff','mount_model_5amp'),pause(3) print('-f86','-r9', '-dtiff','mount_model_1amp'),pause(3) print('-f87','-r9', '-dtiff','mount_model_15amp'),pause(3) print('-f88','-r9', '-dtiff','mount_model_2amp'),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 = [2586.8,2586.8,2845.7,336.9,3235.8,3396.2,3542.8, ,3687.6]; elseif mount == 2 BSTF = [ , , , , , , , , ]; elseif mount == 3 BSTF = [ , , , , , , , , ]; elseif mount == 4 BSTF = [ , , , , , , , , ]; elseif mount == 5 BSTF = [46,46,46,46,46]; elseif mount == 6 BSTF = [ , , , , ]; elseif mount == 7 BSTF = [ ,424.19,423.9,425.76, ]; elseif mount == 8 BSTF = [ , , , , ]; end base = BSTF(amps); %%END of program5base Function 21

219 Program5guess provides an initial starting point for fminsearch: function [k,z,w,a,b] = program5guess(mountnum,ampere) mount = mountnum; amps = ampere; if mount == 1 k = [ ; ; ; ; ; ; ; ; ]; zeta = [1.8,1.8,1.92,2.7,1.6,1.6,1.85,1.8,1.6]; a = [1.3,1.3,1.6,1.13,1.11,1.1,1.26,1.2,1.15]; wn = [1,1,1,1,9.8,9.8,9.29,8.8,8.89]; b = [11,11,11.7,1.97,1.48,1.48,1.79,1.33,1.48]; % x = [zeta(i),wn(i),a(i),b(i),k(i)]; elseif mount == 2 k = [337.9,3374.5,373.1,3992,445,4747,484,58,5156]; zeta = [1.56,1.687,1.659,1.954,1.894,2.12,2.295,2.235,2.15]; a = [1.393,1.47,1.394,1.528,1.37,1.376,1.51,1.432,1.277]; wn = [11.561,11.977,1.228,1.94,9.969,9.475,9.673,9.132,8.868]; b = [12.34,12.79,11.29,12.28,11.6,11.2,11.4,1.85,1.61]; elseif mount == 3 k = [ , , ,317.57,331.21, ,3577.4, ,399.32]; zeta = [1.5352;1.55;1.7471;2.25;2.45;2.9;3.25;3.65;4]; wn = [7.8835;9.5;9.9452; ;13.5;14.5;15.5; ;17]; a = [1.415;1.4358;1.5645;1.9229;2.611;2.32;2.5811;2.8693;3.1233]; b = [8.429;1.2944;11.972;13.46; ;16.4; ; ; ]; elseif mount == 4 k = [ , , ,317.57,331.21, ,3577.4, ,399.32]; zeta = [1.45;1.52;1.5729;1.59;1.62;1.6531;1.6936;1.7;1.73]; wn = [16.5;15.873;15;14.5;14;13;12.4;12.25;11.7]; a = [1.43;1.5416;1.6278;1.6219;1.6481;1.717;1.7512;1.7543;1.7647]; b = [19.51;18.818;17.168; ;16.216;15.767;14.565;14.413; ]; elseif mount == 5 k = [4,4,4,4,4]; zeta = [1.67;1.67;1.67;1.67;1.67]; wn = [4.56;4.56;4.56;4.56;4.56]; a = [1.6189;1.5847;1.6283;1.638;1.581]; b = [5.164;5.157;5.134;5.124;5.1452]; elseif mount == 6 k = [2485;2485;249;2515;2535]; zeta = [3.935;3.935;3.935;3.935;3.935]; wn = [9.622;9.622;9.622;9.622;9.622]; a = [3.7466;3.7466;3.7466;3.7466;3.7466]; b = [1.6453;1.6453;1.6453;1.6453;1.6453]; elseif mount == 7 k = [56,56,56,56,56]; zeta = [7.592;7.592;7.592;7.592;7.592]; wn = [1.817;1.817;1.817;1.817;1.817]; a = [7.4289;7.4289;7.4289;7.4289;7.4289]; b = [12.43;12.43;12.43;12.43;12.43]; elseif mount == 8 k = [56,56,56,56,56]; zeta = [7.592;7.592;7.592;7.592;7.592]; wn = [1.817;1.817;1.817;1.817;1.817]; a = [7.4289;7.4289;7.4289;7.4289;7.4289]; b = [12.43;12.43;12.43;12.43;12.43]; end k = k(amps); z = zeta(amps); w = wn(amps); a = a(amps); b = b(amps); %%END of program5guess Function 22

220 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 = [,1,2,3,4,5,6,7,8,9,1,12,14,16,18,2,25,3,35]; elseif long == 13 N = 13; Fin = [,1,2,3,4,5,6,7,8,9,1,2,3]; 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 = 2.*log1(magmodnorm); % Experimental data expdata = magdata'; % Calculates the cost of the minimizer costout = norm(magmod - expdata); %%END of program5fun Function 23

221 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. 24

222 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 threeplate 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 25

223 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 fluidelastic mount in an elastic case. N S 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. 26

224 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. 27

225 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 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-1. A test setup is shown in 28

226 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. 29

227 Figure D-1: Second generation electromagnet coils for flanged core with 21 AWG, 23 AWG, and 24 AWG magnet wire at 5, 75, and 1 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. 21