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Rotary-axial spindle design for large load precision machining applications Usman, Irfan-ur-rab 2010

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Rotary-Axial Spindle Design for Large Load Precision Machining Applications  by Irfan-ur-rab Usman B.A., University of British Columbia, 2008 B.A.Sc., University of British Columbia, 2008  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF  Master of Applied Science in THE FACULTY OF GRADUATE STUDIES (Mechanical Engineering)  The University Of British Columbia (Vancouver) October 2010 c Irfan-ur-rab Usman, 2010  Abstract Normal stress electromagnetic actuators can be used as both an axial bearing and an in-feed motor in precision machine tool applications that require only millimeterrange axial stroke, such as silicon wafer face grinding or meso-machining. The rotary cutting stage may be integrated with the axially-feeding stage in a rotary-axial architecture. This typology allows the use of independent rotary and axial actuators acting on a single moving mass, rather than an axial actuator moving an entire rotary motor assembly in the feed direction as in typical machine tool architectures. Non-collocated resonances are therefore minimized and thrust and radial stiffness is increased through the elimination of intermediate lateral and thrust bearings, and achievable closed loop positioning performance is improved. This thesis presents the working principle, design, and analysis of radiallybiased electromagnetic bearing/actuators for large load precision rotary-axial spindle applications, and the integration of such an actuator in a full scale prototype to be used as a silicon wafer face grinder. The experimental results indicate that the rotary-axial spindle with radially-biased thrust bearing/actuator is capable of achieving less than 7 nm resolution over a 1.5 mm axial stroke, a worst case load capacity of approximately 5000 N and a best case load capacity of over 8000 N, with rotary-axial coupling of less than 30 nm axial error at 3000 rpm.  ii  Table of Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  ii  Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  iii  List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  vi  List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  vii  Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  xv  1  Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1  Next Generation of Silicon Wafer Manufacture for the Semiconductor Industry . . . . . . . . . . . . . . . . . . . . . . . . . . .  1  1.2  Rotary-Axial Spindle Architecture . . . . . . . . . . . . . . . . .  5  1.3  Non-Contact Axial Thrust Bearings/Actuators for the Rotary-Axial Spindle Architecture . . . . . . . . . . . . . . . . . . . . . . . .  6  1.3.1  Fluid Thrust Bearings and Actuators . . . . . . . . . . . .  6  1.3.2  Lorentz Force Electromagnetic Actuators . . . . . . . . .  8  1.3.3  Normal Force Electromagnetic Actuators . . . . . . . . .  9  Full Scale Rotary-Axial Spindle for Silicon Wafer Grinding . . . .  16  Radially-biased Actuator Design . . . . . . . . . . . . . . . . . . . .  20  2.1  Lumped Parameter Modeling . . . . . . . . . . . . . . . . . . . .  21  2.1.1  Infinite Permeability Model . . . . . . . . . . . . . . . .  23  2.1.2  Finite Permeability Model . . . . . . . . . . . . . . . . .  27  2.1.3  Magnetic Material Selection . . . . . . . . . . . . . . . .  31  1.4 2  1  iii  2.1.4  3  Parameters . . . . . . . . . . . . . . . . . . . . . . . . .  33  2.2  Finite Element Modeling . . . . . . . . . . . . . . . . . . . . . .  39  2.3  Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  46  Rotary-axial Spindle Mechanical Design, Manufacture and Assembly 48 3.1  Spindle Design . . . . . . . . . . . . . . . . . . . . . . . . . . .  50  3.1.1  Rotary-axial Spindle Layout . . . . . . . . . . . . . . . .  50  3.1.2  Axial Metrology . . . . . . . . . . . . . . . . . . . . . .  53  3.2  Shaft Assembly . . . . . . . . . . . . . . . . . . . . . . . . . . .  57  3.3  Hydrostatic Bearing . . . . . . . . . . . . . . . . . . . . . . . . .  61  3.4  Brushless Motor Housing . . . . . . . . . . . . . . . . . . . . . .  64  3.5  Radially-biased Actuator Manufacture . . . . . . . . . . . . . . .  64  3.5.1  Stator Coil Assemblies . . . . . . . . . . . . . . . . . . .  64  3.5.2  Stator Bias Magnet Assembly . . . . . . . . . . . . . . .  74  Actuator and Shaft Installation . . . . . . . . . . . . . . . . . . .  81  3.6.1  Shaft Installation . . . . . . . . . . . . . . . . . . . . . .  81  3.6.2  Front Coil Assembly and Armature . . . . . . . . . . . .  81  3.6.3  Ball Target Installation and Alignment . . . . . . . . . . .  83  3.6.4  Bias Magnet Installation . . . . . . . . . . . . . . . . . .  83  3.6.5  Rear Coil Installation . . . . . . . . . . . . . . . . . . . .  85  3.6.6  Probe Holder and Probe Installation and Alignment . . . .  87  Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  89  Static Load Characterization Experimental Results . . . . . . . . .  91  4.1  Hydrostatic Bearing Static Radial Stiffness Characterization . . .  91  4.1.1  Experimental Setup . . . . . . . . . . . . . . . . . . . . .  92  4.1.2  Radial Stiffness Performance . . . . . . . . . . . . . . . .  92  Radially-biased Actuator Axial Load Characterization . . . . . . .  95  4.2.1  Experimental Setup . . . . . . . . . . . . . . . . . . . . .  95  4.2.2  Initial Results . . . . . . . . . . . . . . . . . . . . . . . .  96  4.2.3  Magnetic Material Characterization . . . . . . . . . . . . 103  4.2.4  Revised FEM Model . . . . . . . . . . . . . . . . . . . . 120  3.6  3.7 4  Initial Design of Radially-Biased Thrust Bearing/Actuator  4.2  iv  4.3 5  Dynamic Motion Characterization Experimental Results . . . . . . 127 5.1  5.2  5.3 6  Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125  Motion Control Results . . . . . . . . . . . . . . . . . . . . . . . 127 5.1.1  Axial Motion Control . . . . . . . . . . . . . . . . . . . . 127  5.1.2  Rotary Control . . . . . . . . . . . . . . . . . . . . . . . 132  Rotary-axial Coupling . . . . . . . . . . . . . . . . . . . . . . . 134 5.2.1  Initial Results . . . . . . . . . . . . . . . . . . . . . . . . 136  5.2.2  Error Modeling . . . . . . . . . . . . . . . . . . . . . . . 141  5.2.3  Sources of Rotary-Axial Coupling . . . . . . . . . . . . . 149  Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158  Conclusions and Future Work . . . . . . . . . . . . . . . . . . . . . 160  Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165  v  List of Tables Table 1.1  Large Scale Rotary-Axial Spindle Target Performance Specifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  4  Table 2.1  Radially-Biased Thrust Bearing/Actuator Design Parameters .  47  Table 4.1  Magnetic Characterization Ring and Integrator Circuit Specifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111  Table 4.2  Large Scale RAS Achieved Static Load Performance Specifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126  Table 5.1  Large Scale RAS Achieved Dynamic Performance Specifications 159  vi  List of Figures Figure 1.1  a) Semiconductor manufacturing process flow adapted from [1, p.1298]; b) in-feed silicon wafer back grinding adapted from [1]. . . . . . . . . . . . . . . . . . . . . . . . . . . . .  Figure 1.2  2  Spindle architectural comparison. a) Stacked stage architecture; b) rotary-axial architecture. . . . . . . . . . . . . . . . .  4  Figure 1.3  Rotary-axial spindle topology. . . . . . . . . . . . . . . . . .  5  Figure 1.4  Working principle of fluid piston actuator. . . . . . . . . . . .  7  Figure 1.5  Lorentz force generation for a cylindrical conductor with current density J in a uniform applied magnetic field B. . . . . .  Figure 1.6  Basic normal stress force generation for a magnetic material in a uniform applied magnetic field through surface A. . . . . . .  Figure 1.7  11  Axially biased reluctance actuator (force generation areas shown in dashed red lines and below in red in top view of armature).  Figure 1.9  9  Horseshoe type reluctance actuator (force generation areas shown in top view of armature, below). . . . . . . . . . . . . . . . .  Figure 1.8  8  13  Radially biased electromagnetic axial actuator (force generation areas shown in dashed red lines and below in red in top view of armature). . . . . . . . . . . . . . . . . . . . . . . .  15  Figure 1.10 Proof of concept rotary-axial spindle, built using a modified Precitech aerostatic bearing spindle [2]. . . . . . . . . . . . .  17  Figure 1.11 Full scale rotary-axial spindle prototype design. . . . . . . . .  17  Figure 2.1  Working principle of radially-biased thrust bearing/actuator. .  21  Figure 2.2  Reluctance models of RBT actuator. a) Bias flux reluctance model; b) excitation flux reluctance model. . . . . . . . . . . vii  24  Figure 2.3  Detail of RBT actuator schematic, showing closed surface of armature over which Maxwell’s stress tensor is evaluated. . .  Figure 2.4  25  Reluctance models of RBT actuator assuming finite permeability of stator and armature material. a) Bias flux reluctance model; b) excitation flux reluctance model. . . . . . . . . . .  28  Figure 2.5  RBT actuator dimensions for finite reluctance calculation. . .  29  Figure 2.6  Somaloy 500 powdered iron magnetic characteristics from [3]. a) Initial BH curve, measured by manufacturer to 30 kA/m and extrapolated to 100 kA/m. Saturation flux density Bsat is read from this plot as approximately 2 T. b) Relative permeability µr . 32  Figure 2.7  Maximum shear and normal stress locations on armature due to axial loading.  Figure 2.8  . . . . . . . . . . . . . . . . . . . . . . . .  35  Radial loading on armature due to centripetal acceleration during rotary-axial operation. . . . . . . . . . . . . . . . . . . .  36  Finite element mesh of the RBT actuator using [4]. . . . . . .  40  Figure 2.10 Flux density distributions in finite element analysis using [4] .  41  Figure 2.9  Figure 2.11 Finite element model predictions of force-position characteristics of RBT actuator for various excitation currents, with a worst case load capacity of 5860 N for a 1.2 A excitation at z=750 µm. . . . . . . . . . . . . . . . . . . . . . . . . . . .  42  Figure 2.12 Finite element model predictions of force-current characteristic of RBT actuator at 0 µm, showing worst case nonlinearity and linear fit. . . . . . . . . . . . . . . . . . . . . . . . . . .  43  Figure 2.13 Force coefficient kZ evaluated over symmetric range of -750 µm to 750 µm, using datasheet initial BH curve. . . . . . . .  44  Figure 2.14 Force coefficient kI evaluated over symmetric range of -1.2 A to 1.2 A, using datasheet initial BH curve. . . . . . . . . . . . Figure 3.1  45  Rotary-axial spindle architectural options. Starting from tool end (bottom of page)a) Radial bearing, rotary motor, radial bearing, axial actuator; b) radial bearing, axial actuator, rotary motor; c) radial bearing, rotary motor, axial actuator. . . . . .  viii  49  Figure 3.2  Full scale rotary-axial spindle overall detailed design and completed full scale RAS prototype. . . . . . . . . . . . . . . . .  Figure 3.3  Eccentricity induced artifact axial error using flat target on end of shaft with perpendicularity error. . . . . . . . . . . . . . .  Figure 3.4  50 53  Axial control loop showing feedback disturbance from eccentricity induced artifact error, zd with controller C(s) and axial plant P(s). . . . . . . . . . . . . . . . . . . . . . . . . . . . .  Figure 3.5  54  Eccentricity induced artifact axial error, due to probe and ball target misalignment from shaft rotation axis. . . . . . . . . .  55  Figure 3.6  Ball target artifact showing 10 nm surface roundness. . . . . .  56  Figure 3.7  Shaft assembly overall mechanical design. . . . . . . . . . . .  57  Figure 3.8  Brushless motor rotor shaft attachment. . . . . . . . . . . . .  58  Figure 3.9  RBT actuator armature shaft attachment. . . . . . . . . . . .  59  Figure 3.10 Hydrostatic bearing mechanical design. . . . . . . . . . . . .  61  Figure 3.11 Hydrostatic bearing fluid porting solution. . . . . . . . . . . .  62  Figure 3.12 Brushless motor housing mechanical design. . . . . . . . . .  65  Figure 3.13 RBT actuator mechanical design, showing three main sub assemblies and axial probe, mount and bumper stop. . . . . . .  66  Figure 3.14 RBT actuator coil assemblies, mechanical design. a) Rear coil assembly; b) front coil assembly. . . . . . . . . . . . . . . . .  67  Figure 3.15 Freestanding coils. a) Single coil of 2x39 turns (78 turns); b) two coils hard wired in parallel. The two coils in parallel configuration is the basic coil unit for flexibility in wiring (with 16 terminals controlling 8 coil sets for each coil assembly.) . .  69  Figure 3.16 Coil assembly electromagnetic components. a) Stator segments; b) coil set for one assembly. . . . . . . . . . . . . . . . . . .  69  Figure 3.17 Manufacturing steps for rear coil assembly. . . . . . . . . . .  70  Figure 3.18 Manufacturing steps for rear coil assembly. . . . . . . . . . .  71  Figure 3.19 Design of bumper stop and probe mount. . . . . . . . . . . .  73  Figure 3.20 RBT actuator bias magnet assembly. a) Bias magnet assembly, mechanical design; b) single magnet segment showing magnetization direction.  . . . . . . . . . . . . . . . . . . . . . . .  75  Figure 3.21 Possible radial magnetization setup. . . . . . . . . . . . . . .  75  ix  Figure 3.22 RBT actuator bias magnet assembly. a) Prepping polycarbonate spacer by masking edges so epoxy will be easy to remove; b) laying epoxy in the spacer pockets; c) trimming excess epoxy by simply breaking the masking off; d) untrimmed and complete spacers. . . . . . . . . . . . . . . . . . . . . . . . . . .  77  Figure 3.23 RBT actuator bias magnet assembly manufacture. a) Epoxying magnets in place on top of magnetic block with non-magnetic stainless steel block in between, using aluminum jig to ease installation; b) compressing bias magnet assembly using second non-magnetic stainless steel block and jig bolts; c) removing cured bias magnet assembly using load bolts threaded through magnetic stainless steel bolt until magnetic attraction decreases.  . . . . . . . . . . . . . . . . . . . . . . . . . . .  78  Figure 3.24 RBT actuator bias magnet assembly finishing work. a) Removing excess epoxy and steel shims that squeezed out through fluid pressure from the epoxy as it cured; b) complete bias magnet assembly after grinding. . . . . . . . . . . . . . . . .  79  Figure 3.25 Installation of shaft assembly. . . . . . . . . . . . . . . . . .  82  Figure 3.26 Locking shaft assembly in axial direction. . . . . . . . . . . .  82  Figure 3.27 Installation of front coil assembly onto brushless motor housing. a) Front coil assembly lowered onto brushless motor housing; b) RBT actuator armature installed on shaft. . . . . . . .  83  Figure 3.28 Installation of front coil assembly onto brushless motor housing. a) Front coil assembly lowered onto brushless motor housing; b) RBT actuator armature installed on shaft. . . . . . . .  84  Figure 3.29 Installation of bias magnet assembly onto front coil assembly. a) Schematic procedure for lowering bias magnet assembly used threaded rod screwed into M10 threaded holes on the bias magnet housing; b) installation of bias magnet assembly; c) bias magnet assembly after centering using feeler gauges. .  x  85  Figure 3.30 Installation of rear coil assembly onto bias magnet assembly. a) Installation of rear coil assembly using engine hoist and threaded rods through M8 threaded holes on the rear coil housing; b) detail view of lowering procedure; c) rear coil assembly installed and centered. . . . . . . . . . . . . . . . . . . . . .  86  Figure 3.31 Installation of brass bumper stop, ball target locknut, axial probe mount and showing lateral alignment of probe. . . . . .  88  Figure 3.32 Full scale RAS installation at UBC. . . . . . . . . . . . . . .  90  Figure 4.1  Radial stiffness characterization of hydrostatic bearing, experimental setup. . . . . . . . . . . . . . . . . . . . . . . . . . .  93  Figure 4.2  Radial stiffness equivalent model. . . . . . . . . . . . . . . .  93  Figure 4.3  Measured tool end shaft stiffness, and derived hydrostatic bearing stiffness using both a rigid shaft and an elastic shaft. . . .  94  Figure 4.4  Setup for actuator axial load characterization. . . . . . . . . .  96  Figure 4.5  Raw axial load characterization measurements. . . . . . . . .  97  Figure 4.6  Measured force-position characteristic, zero and 0.45 A excitation current, compensated for shaft weight, hysteresis and axial probe misalignment. Compared to FEM prediction using datasheet initial BH curve. . . . . . . . . . . . . . . . . . . .  98  Figure 4.7  Experimental force-current load lines. . . . . . . . . . . . . .  99  Figure 4.8  Force coefficient kZ , experimental versus FEM prediction using datasheet initial BH curve. . . . . . . . . . . . . . . . . . 100  Figure 4.9  Force coefficient kI , experimental versus FEM prediction using datasheet initial BH curve. . . . . . . . . . . . . . . . . . . . 101  Figure 4.10 Initial BH curve measurement, numerical integration of secondary voltage method. . . . . . . . . . . . . . . . . . . . . . 104 Figure 4.11 Exact field intensity distribution inside ring shaped core (windings not shown). . . . . . . . . . . . . . . . . . . . . . . . . 105 Figure 4.12 Simulated initial BH curve, with inset showing averaging effect of flux density estimation. . . . . . . . . . . . . . . . . . 106 Figure 4.13 Extraction of initial BH curve from measured BH hysteresis loops, Somaloy 500. . . . . . . . . . . . . . . . . . . . . . . 107 xi  Figure 4.14 Initial BH curve measurement, analog integration of secondary voltage method. . . . . . . . . . . . . . . . . . . . . . . . . . 108 Figure 4.15 Somaloy 500 test ring construction for initial BH curve measurement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 Figure 4.16 Experimental initial BH curve test setup. 4130 test ring is shown.110 Figure 4.17 Initial BH curve of Somaloy 500, datasheet values versus measured using online integration. . . . . . . . . . . . . . . . . . 111 Figure 4.18 Relative permeability of Somaloy 500, datasheet versus measured values. . . . . . . . . . . . . . . . . . . . . . . . . . . 112 Figure 4.19 Frequency sweep of relative permeability µr at low applied field intensity with constant amplitude. . . . . . . . . . . . . 113 Figure 4.20 Comparison of BH hysteresis curves at 100 mHz (online integration) and 500 Hz (offline numerical integration). . . . . . . 114 Figure 4.21 Magnet flux density measurement set ups, a) free air and b) in situ (installed on front coil assembly). . . . . . . . . . . . . . 115 Figure 4.22 Magnet flux density measurements, free air and in situ (installed on front coil assembly). . . . . . . . . . . . . . . . . . 116 Figure 4.23 Finite element result showing flux density magnitude |B| using FEMM [4]. . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 Figure 4.24 Comparison of FEM air gap flux density prediction and experimental air gap flux measurement, for varying input remanence in the FEM. . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 Figure 4.25 Simulation results for radial force on armature from radial displacement of armature, using constant permeability 3D finite element analysis. . . . . . . . . . . . . . . . . . . . . . . . . 119 Figure 4.26 Force-position characteristics, experimental and corrected FEMM predictions. . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 Figure 4.27 Force coefficient kZ from initial FEMM predictions, corrected FEMM predictions and experimental measurements. . . . . . 122 Figure 4.28 Force coefficient kI from initial FEMM predictions, corrected FEMM predictions and experimental measurements. . . . . . 123 Figure 4.29 Corrected FEM prediction, showing worst case load capacities at -750 µm and 750 µm. . . . . . . . . . . . . . . . . . . . . 124 xii  Figure 4.30 Force-current characteristic at z = 0 µm, showing worst case nonlinearity, for FEM prediction using datasheet initial BH curve, FEM prediction using measured initial BH curve, and experimentally measured actuation load at the neutral position. 125 Figure 5.1  Axial control loop block diagram, showing force and feedback disturbances. . . . . . . . . . . . . . . . . . . . . . . . . . . 128  Figure 5.2  Axial plant measurement. . . . . . . . . . . . . . . . . . . . 129  Figure 5.3  Axial motion control. a) Controller frequency response function, b) negative loop transmission, c) dynamic stiffness as derived from plant, controller and negative loop transmission measurements. . . . . . . . . . . . . . . . . . . . . . . . . . 130  Figure 5.4  Axial regulation error. . . . . . . . . . . . . . . . . . . . . . 132  Figure 5.5  Axial motion tracking results. a) Sine wave reference with 1.4 mm stroke (19.5 nm rms error) at 0.1 Hz, b) Triangle wave with 1.2 mm stroke (16.3 nm rms error) at 0.1 Hz. . . . . . . . 133  Figure 5.6  Axial regulation error, at 500 rpm and 3000 rpm. . . . . . . . 135  Figure 5.7  Time domain FFT of axial regulation error, 500 rpm and 3000 rpm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136  Figure 5.8  Axial motion regulation at 3000 rpm with additional 300 nm shift, plotted in spatial domain versus shaft rotation angle. The 9 cpr component can clearly be seen as the dominant error detected at this speed, with a peak-to-valley amplitude of 30 nm; the electrical hash noise component occurs at a much higher spatial frequency, and has approximately 100 nm peakto-valley amplitude. . . . . . . . . . . . . . . . . . . . . . . . 137  Figure 5.9  Axial motion regulation at various rpm with additional 300 nm shift, plotted in spatial domain versus shaft rotation angle. . . 138  Figure 5.10 Axial motion regulation spatial domain FFT. . . . . . . . . . 139 Figure 5.11 Axial regulation rms error and 9 cpr component of axial error versus rotational speed. . . . . . . . . . . . . . . . . . . . . . 140 Figure 5.12 Motion induced artifact axial error due to ball artifact lateral motion during rotation.  . . . . . . . . . . . . . . . . . . . . 142 xiii  Figure 5.13 Error feedback disturbance function, e/zd . . . . . . . . . . . . 143 Figure 5.14 Axial error comparison. a) 9 cpr component of error plotted versus predicted error with 30 nm disturbance on feedback signal, b) force disturbance rejection frequency response function.  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144  Figure 5.15 a) Predicted axial force disturbance amplitude required to produce measured 9 cpr axial error (based on (Equation 5.9)) for each rotational speed; b) mean torque command to brushless DC motor for each rotational speed. . . . . . . . . . . . . . . 148 Figure 5.16 Brushless motor stator, showing slant of coil geometry. . . . . 149 Figure 5.17 Radial shaft motion leading to eccentricity shift of target artifact; maybe due to brushless motor cogging and/or bias magnet radial force on the armature. . . . . . . . . . . . . . . . . . . 150 Figure 5.18 Bias magnet radial force estimation. 18 cpr component is visible, but not 9 cpr. . . . . . . . . . . . . . . . . . . . . . . . . 151 Figure 5.19 Motor stator eccentricity schematic diagram. . . . . . . . . . 152 Figure 5.20 Gap measurements between brushless motor stator and rotor. a) Gap measurement for two measurements, where Measurement 2 is taken with the shaft rotated by 180 degrees from the rotor position of Measurement 1; b) repeatability between measurements. . . . . . . . . . . . . . . . . . . . . . . . . . 153 Figure 5.21 Variation in gap between brushless motor stator and rotor on shaft. Measurement 2 is taken with the shaft rotated 180 degrees from the rotor position of Measurement 1. . . . . . . . . 154 Figure 5.22 a) Schematic showing permanent magnet positioning; b) representative radial flux density Bn and representative radial force Fn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 Figure 5.23 Ball target centering, showing radial gap variation with additional 3 µm mean. . . . . . . . . . . . . . . . . . . . . . . . 156  xiv  Acknowledgments I would like to begin by thanking Allah. I want to thank my parents for (among other things) putting up with the ridiculous hours of a graduate student. I want to thank them also for always encouraging me to work on the things that interested and excited me. Without their love and support I would not have been able to complete this work. I would like to thank Dr. Xiaodong Lu for the enormous support and guidance he has provided me with since I was an unproven third year undergraduate in his lab. I am very grateful that he accepted me as a temporary summer researcher and then encouraged me to continue in academia by providing me with many opportunities to attend conferences and visit industrial companies, as well as offering me the chance to work on incredibly challenging and engaging research. He always sets a great example for me with his intellect, creative abilities and absolute dedication to the art and science of precision engineering. I would also like to thank the entire team that worked on the rotary-axial spindle project: Benito Moyls, Matthew Paone, Kris Smeds, Thomas Huryn and Jeffrey Abeysekera at UBC, Dr. Alex Slocum and Gerald Rothenhofer at MIT, and our as yet still unnamed sponsor company in Japan. I want to especially thank Matthew Paone for his work on both the small rotary-axial spindle prototype as well as the full scale rotary-axial spindle, on which he was the driving force behind the electronics and control implementation, as well as the setup in the final installation in Japan. I also want to especially mention the work and support of Kris Smeds, who built the 2 kW power amplifier used to drive the full scale rotary-axial spindle, and who has always been an enabler of creative research solutions. Working with Kris on both the large scale and small RAS prototypes has proven to elicit my best xv  work through discussion and critique. This is their work as much as it is mine: without each member of the team working to the best of their ability, the spindle would never have lifted off. My colleagues in the Precision Mechatronics Laboratory, Arash Jamalian, Richard Graetz, Darya Amin-Shahidi, Niankun Rao and others who were not directly part of the RAS project have always been there to provide often brilliant insight into research problems. Working with them has been a privilege and a pleasure. Finally, I would like to extend my warmest thanks and appreciation to Yoyo Hoi Yiu Au for his always helpful (and ever present) criticism and wit, and to Daniel Gordon Fritter for taking many of the better pictures of the full scale rotary-axial spindle.  xvi  Chapter 1  Introduction 1.1  Next Generation of Silicon Wafer Manufacture for the Semiconductor Industry  The current state of the art in semiconductor manufacturing involves constructing devices on a 300 mm diameter silicon wafers using a photolithography process [5]. The International Technology Roadmap for Semiconductors states that to remain on track for Moore’s law will require the industry overall to achieve at least a 30% cost reduction and a 50% cycle time improvement by 2012 [5, p.13]. Increasing productivity through parallel processing is one of the major enablers of Moore’s law [5]. One of the proposed solutions is enlarging the industry standard wafer diameter to 450 mm. The rationale for jumping to larger wafer diameters (instead of, for example, speeding up current productions cycles while retaining the current wafer dimensions) is to decrease the manufacturing cost per unit area of wafer by maximizing the number of devices produced per operation. In addition to increasing wafer diameter, flatness tolerances on the wafer are increasing with time. The trend of local flatness tolerances (total flatness for a relatively small area of the wafer) for silicon wafers over several decades indicate an increase in local flatness tolerances [1]. Tighter tolerances on flatness will be critical in the production of smaller devices with higher transistor densities. According to Pei et al., global flatness tolerances (total thickness variation, or TTV) have been less critical since the advent of the 300 mm wafer because lithography machines 1  Figure 1.1: a) Semiconductor manufacturing process flow adapted from [1, p.1298]; b) in-feed silicon wafer back grinding adapted from [1].  2  are currently unable to process the whole wafer in a single step. However, global flatness tolerances will surely increase in importance as lithography technology catches up with wafer size, and the trend in TTV will follow local flatness tolerances. One of the critical process steps in semiconductor manufacturing is silicon wafer back grinding (Figure 1.1a) where a wafer which has devices printed on one side is thinned to final thickness by grinding the blank side of the wafer. Only single-sided grinding topologies may be used for this operation [1]. One of the most widely used single sided grinding methods is in-feed grinding shown in Figure 1.1b: a grinding wheel is fed in the axial direction onto a wafer that is offset from the rotational axis of the tool by the radius of the grinding wheel. Both tool and wafer rotate about their own axes: because the contact length between wheel and wafer is constant, the total flatness tolerance is very good. How does increasing the diameter of the wafer workpiece affect the back grinding process? The target size for the next generation of silicon wafers requires a 50% increase in diameter and a 225% increase in surface area relative to the current standard. As global flatness error increases linearly with diameter, and cutting process forces increase with area, the current generation of silicon wafer face grinders are not sufficiently stiff or strong enough to maintain the global flatness tolerances required to manufacture usable 450 mm silicon wafers within reasonable process times, especially as the TTV requirements increase. In order to achieve both high flatness tolerances and reasonable process times, an aggressive set of tentative performance requirements for the wafer grinding spindle are formulated as a research target, shown in Table 1.1. The most stringent requirement is the axial motion accuracy and the worst case axial load capacity of 5000 N over the entire stroke of 1.5 mm. Large load capacities and high motion bandwidth do not typically go together, since large loads are generally associated with large, massive devices and high motion bandwidth is associated with a small moving mass; therefore both large load capacity and compactness are required in the next generation silicon wafer grinder.  3  Table 1.1: Large Scale Rotary-Axial Spindle Target Performance Specifications Parameter  Target Specification  Minimum Axial Load Capacity Axial Stroke Axial Motion Accuracy Axial Motion Bandwidth (-3 dB) Static Controlled Axial Stiffness Cutting Speed  5000 N 1.5 mm <10 nm 1000 Hz ∞ N/µm 3000 RPM  Figure 1.2: Spindle architectural comparison. a) Stacked stage architecture; b) rotary-axial architecture.  4  Figure 1.3: Rotary-axial spindle topology.  1.2  Rotary-Axial Spindle Architecture  The wafer back grinding operation, like many precision machine tool applications, requires two degree of freedom motion, with a rotary cutting stage and an in-feed positioning stage. Typical machine architectures stack 1DOF motion stages serially in the machine structural loop in order to achieve multiple degrees of freedom. This type of architecture limits the total stiffness of the machine tool due to serial addition of stiffness elements (as shown in Figure 1.2a) and also limits the motion bandwidth due to non-collocated resonances and the need to drive the mass of the entire rotary stage. Total cost may increase due to the number of components, often with duplicate sets of bearings to support each stage. More bearings imply a larger total bearing surface area which leads to greater heat generation and thermal growth errors in the frame of the machine. Many machine tools of this type (e.g. milling and face grinding machines) have an open ‘C-frame’ design to easily accommodate the stacked stage architecture (as in [6]), resulting in a machine that is particularly susceptible to thermal growth error. Increasing the stiffness performance of a single component will typically not have a drastic effect on overall machine stiffness because the critical limitation is the serial connection of components. A better architectural option is integration of both rotary and axial stages  5  into a single body acting on the shaft assembly (Figure 1.2b). For applications where the in-feed stroke is relatively small, as in silicon wafer face grinding operations or meso-machining, the rotary stage may be integrated with the z-stage in the rotary-axial architecture (Figure 1.3). This topology allows the use of decoupled actuators acting on a single moving mass, instead of a separate axial actuator acting on two masses connected via stiffness elements as in the stacked stage architecture. Coupling between motion stages is all but eliminated as the line of action of the axial actuator is coincident with the axis of shaft rotation, and there are no intermediate stiffness elements between each actuator and the shaft. Structural modes are improved by the elimination of both intermediate stiffness elements between the moving shaft assembly as well as the reduction in the inertia seen by the axial stage (since only the shaft assembly is now moving instead of an entire rotary stage including all bearings). This architecture greatly improves the achievable axial dynamic positioning performance, which is required for better disturbance rejection over a wide frequency range. Heat generation is minimized with the elimination of the thrust bearing between motion stages as well as the lateral support bearings for the separate axial stage. In order to implement this architecture, a non-contact axial actuator is required in order to retain both rotary and axial degrees of freedom while eliminating a separate z-stage.  1.3  Non-Contact Axial Thrust Bearings/Actuators for the Rotary-Axial Spindle Architecture  The key functional requirements for a non-contact axial actuator in the rotary-axial spindle architecture are: 1) large load density, in order to keep the design compact and minimize limiting structural modes; 2) efficient power dissipation properties to minimize heat generation; 3) good linearity between input control effort and output load and position; and 4) an axial stroke on the order of millimeters.  1.3.1  Fluid Thrust Bearings and Actuators  Fluid thrust bearings are very popular for machine tool design because of the inherently high load capacity and very good error motion properties due to the averaging effect of fluid separation [7]. Fluid thrust actuators have proven potential for large 6  Figure 1.4: Working principle of fluid piston actuator. load capacities with small error motion. However, long stroke piston-type actuators (as shown in Figure 1.4) have limited motion bandwidth since in order to move across a large stroke the actuator must deal with large fluid inertance. In addition, the bulk modulus of water is 2.2 GPa compared to that of most steel alloys at 160 GPa; this implies that the achievable stiffness of the fluid actuator is low relative to a mechanical type bearing. Another type of fluid actuator is the thin film bearing/actuator. Hirayama et al. in [8] has developed a precision fluid axial thrust bearing/actuator with micrometer stroke and nanometer accuracy. This design achieves an axial resolution of less than 2 nm and a controlled axial stiffness of 50-70 N/µm, but only over a small stroke of 20 µm: small stroke is a fundamental limitation of the thin film type bearing/actuator. The small thickness of the fluid film implies a very large velocity gradient across the gap, meaning viscous forces are quite high leading to very high 7  heat generation at grinding cutting speeds.  1.3.2  Lorentz Force Electromagnetic Actuators  Figure 1.5: Lorentz force generation for a cylindrical conductor with current density J in a uniform applied magnetic field B. Another type of non-contact actuator is the Lorentz force actuator, or voice-coil motor (VCM). The principle of Lorentz force generation is illustrated in Figure 1.5 for a cylindrical conductor of length L and diameter D with current density J in a uniform applied magnetic field with flux density B, generating a force on the conductor normal to both the current density and magnetic field vectors, in the direction n. The force generated has the magnitude Fn = BJ π4 D2 L. The force scales linearly with applied field and current density. For a given external field density, the force is limited by thermal dissipation of the conductor which determines the maximum current density that can be applied. Lorentz force actuators have been implemented in a magnetically levitated spindle for electro-discharge machining [9], a rotary-linear axis for meso-machining [10], and a high-speed drilling spindle with radial air bearings [11]. Such actuators have demonstrated millimeter to centimeter range axial motion with highly linear output force to input current characteristics over the entire stroke. However, the low force density of the Lorentz actuator limits the load capacity, disturbance rejection capacity and the axial motion bandwidth of such a spindle. To achieve the load capacity required for large load precision machining with a Lorentz force actuator would require a very large device with significant heat generation and limiting 8  structural modes.  1.3.3  Normal Force Electromagnetic Actuators  Figure 1.6: Basic normal stress force generation for a magnetic material in a uniform applied magnetic field through surface A. Another category of non-contact actuator is the normal stress or reluctance magnetic actuator, whose principle characteristic is the development of magnetic force oriented normal to the armature load generation surfaces. Normal stress may be induced on a plate made of a permeable magnetic material by the presence of an external magnetic field with flux density B normal to the surface A in the direction n as shown in Figure 1.6. The total force generated on the surface in the normal direction n is a function of the square of the flux density and may be calculated using Maxwell’s stress tensor [12] as Fn =  B2 A 2µo  (1.1)  assuming uniform flux density across the area A. Actuation on the plate is achieved through external variation of the magnetic field acting through the surface. Because the actuation pressure is proportional to the square of the applied flux density, the 9  inherent load density of the normal stress actuator is much higher than the Lorentz force motor. Based on the intrinsically larger load densities relative to voice coil motors, a more detailed review of normal stress actuators is presented in this section. Horseshoe Type Reluctance Actuator The horseshoe type magnet shown in Figure 1.7 is the most common normal stress bearing type utilized in machine tools ([13],[14],[15],[16]); in fact, it is the most common type of electromagnetic bearing in all such applications. Axial force is induced by applying excitation current through either the top or bottom set of excitation coils; this induces a magnetic flux through the stator core which generates force on the armature at the areas shown in red. The primary disadvantage to this type of electromagnetic device as an axial actuator is the highly non-linear force-current and force-position characteristics. Typically, a relatively large biasing current through both sets of coils is used to help linearize the actuator around a working point to minimize force-current nonlinearity. This method works adequately for position regulation but the actuator still suffers from large variations in controller gain and load capacity for an axial stroke that is on the same order as the air gap between stator and armature. Achieving a bi-directional actuator also requires two power amplifier channels in order to independently control the current through each coil set, increasing drive circuit complexity and overall cost. In addition to nonlinear force-current-position characteristics, the horseshoe actuator does not generate force across the excitation coil slot. This means the armature surface is only partially utilized and implies that it is larger than it needs to be to generate a given load, lowering the natural frequencies of structural modes in the armature moving assembly and thus degrading achievable dynamic performance of the spindle. One possible solution is to decrease the coil slot width relative to the armature diameter, but the practical limit to the minimum width of the coil slot is the flux shorting between inner and outer stator branches which degrades load capacity (any flux that is thus shorted does not contribute to the flux on the force generating areas of the armature). In addition, for a given magnetomotive force NI (where N is the number of coil turns in each slot and I is the excitation  10  Figure 1.7: Horseshoe type reluctance actuator (force generation areas shown in top view of armature, below). current through those turns), reducing the coil slot width will force the actuator to become longer, leading to more leakage flux across the coil slot and also poor structural dynamics. Historically, the horseshoe type actuator has been popular for non-contact thrust applications for two reasons: 1) complexity of the horseshoe actuator versus a per11  manent magnet biased actuator is lower (though its driving circuitry may be more complex), and 2) the popularity and widespread use of the stacked stage architecture has meant that combination non-contact thrust bearing/actuators were not in demand. It is expected that the horseshoe type actuator will remain popular for many applications, such as radial position regulation in 5DOF systems [9], but for the rotary-axial spindle architecture it does not maximize axial performance. Axially-biased Reluctance Actuator To improve on the the linearity of the horseshoe type actuator, a permanent magnet can be used to bias the device rather than an excitation current, as shown in Figure 1.8 where the bias flux exits the stator and enters the armatures in the axial direction. Biasing electromagnetic actuators with permanent magnets has become more popular as the cost of high strength rare earth magnets has decreased over time [17]. An example of an axially biased thrust actuator has been developed for a high speed energy storage flywheel [18]. The bias magnet lowers overall power consumption because the excitation current is used to essentially modify the existing bias flux rather than provide all of the flux generation. Because of the large reluctance across the permanent magnet compared to the total air gap in the device, the total bias flux is essentially invariant with armature position. The excitation coils provide a single flux through the stator and armature that adds to the bias flux on one face and subtracts from the bias flux on the other face; thus the natural subtraction of forces leads to a linearized net load on the armature. The excitation coils do not see a changing inductance because the total reluctance is the summation of both air gap reluctances whose total does not change. Because only one set of coils is necessary to control the actuator in both directions, only one power amplifier channel is required, reducing cost and complexity of the driving circuitry relative to the horseshoe actuator (though not the cost and complexity of the actuator itself). A disadvantage of the axially-biased actuator is the requirement for two armatures, which increases complexity and will diminish the structural performance of the moving assembly. In addition, neither of the two armature force generation surfaces is fully utilized, as there is no force generation across the coil gap.  12  Figure 1.8: Axially biased reluctance actuator (force generation areas shown in dashed red lines and below in red in top view of armature).  13  The minimum coil width is again limited by two factors: 1) the required NI will force the device to become longer inducing poor structural modes in the moving assembly, and 2) reducing coil slot width will increase leakage flux, degrading load performance. Radially-biased Reluctance Actuator The radially-biased normal stress actuator was first developed for the automotive industry, where it was used for a binary valve design for engines which utilized its auto latching features [19], and was later combined with radial bearings for 5DOF control of a high speed energy storage flywheel [20]. Later, it was used in [21] to develop an ultra-fast tool servo with accelerations in excess of 500 g. The radiallybiased thrust bearing/actuator topology was developed for use with the rotary-axial architecture in a proof of concept prototype in [2]. The radially biased topology has the bias flux enter the armature in the radial direction, and exit the top and bottom surfaces where the flux returns to the magnet via the stator core. The radial biasing accomplishes many of the same features as axial biasing. It has lower power generation through utilization of permanent magnet flux for partial force generation. Highly linear force generation is achieved for several reasons. First, the total axial air gap is invariant with armature position, leading to an excitation flux that is independent of the axial position of the armature. Second, while the bias flux on each axial pole face will vary with position, the bias flux entering the armature is relatively constant because the reluctance across the magnet is much larger than the reluctance of the air gaps. Third, excitation current generates flux in the same direction on both axial pole faces of the armature, causing differential net flux generation on both armature surfaces as the excitation flux adds to the bias flux on one side and subtracts from the bias flux on the other side. Finally, the net force on the armature is the result of the natural subtraction of the force generated on the top and bottom surfaces, adding another layer of the differential principle to linearize the actuator. One of the drawbacks of the radial biasing is the requirement for good radial support, as the armature is radially unstable due to negative magnetic stiffness between the bias magnet ring and the armature. However, this negative stiffness is typically several orders of magnitude  14  Figure 1.9: Radially biased electromagnetic axial actuator (force generation areas shown in dashed red lines and below in red in top view of armature). lower than the typical radial support stiffness from fluid journal bearings. In addition to the benefits of the axially biased actuator, the radially biased actuator fully utilizes the entire force generation surface of the armature. This implies that for a given load capacity, the radially biased actuator armature will be smaller than a horseshoe actuator or an axially-biased actuator. This will significantly improve structural dynamics of the moving assembly and improve overall spindle performance by increasing the achievable motion bandwidth of the system using a closed loop position control scheme. Because of the relatively high load density of the radially biased actuator, its highly linear force-current-position characteristic, and its potential for very high load capacity, the radially-biased actuator  15  is a very good choice for use as a thrust bearing/actuator for the rotary-axial spindle architecture.  1.4  Full Scale Rotary-Axial Spindle for Silicon Wafer Grinding  A prototype rotary-axial spindle using the RBT actuator (Figure 1.10) was developed as a proof of concept by replacing a mechanical roller ball thrust bearing on a commercially available air bearing spindle with the RBT actuator [2]. This prototype achieved an axial load capacity of 600 N and an axial motion bandwidth of 2.6 kHz with a moving assembly of 2 kg (this device is hereafter referred to as the small RAS prototype), as well as a stroke of 1 mm and an axial resolution of 4.6 nm (the electrical noise floor of the axial displacement probe). With such promising results from the proof of concept prototype, a full scale rotary-axial spindle was designed and manufactured by a team of students from UBC and MIT. The team at MIT (Dr. Alex Slocum and Gerald Rothenhofer) developed the fluid bearing design based on [22]. The team at UBC was comprised originally of 5 undergraduate students: Benito Moyls, Kris Smeds, Thomas Huryn, Jeff Abeysekera and this author. In addition, Matthew Paone was a Master’s student who worked on both the small scale RAS and the full scale RAS and presented initial results for the full scale RAS in [23]. This author graduated and worked on completing and testing the full scale RAS for graduate work, having developed the electromagnetic design and much of the mechanical design as an undergraduate student. The full scale RAS was built with cooperation and support from the industrial sponsor company, and was sent to their work site in Japan. The full scale RAS design is shown in Figure 1.11. The initial load characterization and motion control results were achieved in a test setup at UBC. This thesis presents the design, manufacture and experimental characterization of a radially-biased electromagnetic machine as a magnetic bearing/actuator in a precision rotary-axial spindle for silicon wafer face grinding. Its contributions include: • The design and manufacture of a radially-biased thrust bearing/actuator (RBT 16  Figure 1.10: Proof of concept rotary-axial spindle, built using a modified Precitech aerostatic bearing spindle [2].  Figure 1.11: Full scale rotary-axial spindle prototype design. 17  actuator) based on a novel electromagnetic actuator presented in [21] and a prototype rotary-axial spindle presented in [2]. • The integration of the RBT actuator with a hydrostatic radial bearing designed by Gerald Rothenhofer and Alex Slocum to produce a full scale rotary-axial spindle prototype for silicon wafer face grinding of next generation 450 mm wafers. • Experimental static load characterization of the RBT actuator, and a subsequent magnetic material investigation to validate the design process. • Investigation of rotary-axial coupling and discovery of a disturbance on the axial position feedback signal due to position sensing target motion from brushless motor cogging that is independent of true shaft axial motion. This disturbance is part of the basic limitations on accuracy for rotary-axial spindle architectures that measure axial position from the end of the spindle. It is posited that this motion is amplified by the radially unstable property of the RBT actuator which exaggerates any radial error motion initiated by any source whatsoever. The thesis is structured as follows: Chapter 1 introduces the rotary-axial spindle concept and discusses the overall justification for designing and manufacturing the rotary-axial spindle for silicon wafer grinding. An overview of normal stressed actuators used for thrust support and axial actuation in machine tool spindles is presented. Chapter 2 presents the design procedure for sizing the RBT actuator using lumped parameter analysis to obtain initial dimensions and choose magnetic materials, and finite element analysis to refine the design by modeling nonlinear material properties and non-ideal actuator behaviour. Chapter 3 details the overall mechanical design of the full scale rotary-axial spindle prototype and the detailed manufacture and installation of the RBT actuator. Chapter 4 presents experimental static load characterization of the radial fluid bearing and the static axial load characterization of the RBT actuator. Discrepan-  18  cies between the predicted and measured load characteristic of the RBT actuator are investigated. Chapter 5 presents the axial positioning performance and overall closed loop dynamic performance of the spindle. Rotary-axial coupling is measured and its sources investigated. Chapter 6 concludes the thesis with an overview of presented results and a discussion of the lessons learned from the full scale RAS prototype. Future work on the prototype is outlined.  19  Chapter 2  Radially-biased Actuator Design The goals of the RBT actuator design are to achieve a 1.5 mm axial stroke and a minimum 5000 N axial load capacity over the entire working range. In addition, a 1 kHz motion bandwidth is desired in order to obtain better disturbance rejection at high frequency; thus magnetic material dynamic response should be considered in material selection. In order to achieve the design goals of the RBT actuator design, actuator dimensional and material parameters are set initially with a lumped parameter analysis and later refined using a finite element analysis. The first section of this chapter goes over lumped parameter analysis of the RBT actuator assuming both an infinite permeability model, which is used to initially size the actuator, and a finite linear permeability model, which informs the choice of soft magnetic material used to construct the armature and stator core. In the second section of this chapter, the lumped parameter analysis is used to provide a starting point for the finite element analysis which models non-ideal behaviour such as leakage flux and magnetic saturation. The finite element model (FEM) is used to size the final actuator to achieve a minimum axial load capacity of at least 5000 N given the required stroke of 1.5 mm.  20  Figure 2.1: Working principle of radially-biased thrust bearing/actuator.  2.1  Lumped Parameter Modeling  Lumped parameter modeling does not provide an exact analysis of the RBT actuator, but can provide useful quantitative and dimensional intuition into actuator design. The overall RBT actuator working principle is presented in Figure 2.1. As first discussed in Section 1.3.3, the RBT actuator is constructed of four main com21  ponents: a C-shaped stator of soft magnetic material, a set of excitation coils that fit in the stator slots, a radially magnetized permanent magnet, and a disk shaped armature made of the same material as the stator and mounted on a non-magnetic shaft. The entire design is ideally axisymmetric about the rotational axis of the shaft. This symmetry insures that as the shaft rotates, no eddy currents are generated due to a changing magnetic field seen by the moving armature. Two axial air gaps, zo + z and zo − z, are formed between armature and stator pole faces, where zo is the mean air gap. The total air gap, 2zo , is invariant with armature axial position, z. The permanent magnet provides a radial bias flux to the armature which exits in opposing axial directions from the armature pole faces, as shown by the solid lines. The excitation coils provide flux that enters and exits the armature from the same axial direction, as represented by the dashed lines. Flux from the excitation coil is opposed by the biasing flux from the permanent magnets on one side of the armature (−z face of the armature), and adds to the biasing flux on the other side of the armature (+z face of the armature). Normal forces are generated on each side of the armature, proportional to the square of the net flux density present on each face. As the excitation flux cancels out a portion of the biasing flux on the −z face of the armature and adds to the biasing flux on the +z face of the armature, the normal force between armature and stator pole is weaker on the −z face of the armature and stronger on the +z face. Thus a net force on the armature in the +z direction is produced by the natural subtraction of normal forces on each pole face. With no input excitation, the armature has only two stable positions, locked on either the +z or −z face of the stator. Ideally, no net radial force on the armature is generated by the bias flux when the armature is centered; however, the radial positioning of the armature is inherently unstable due to an equivalent negative magnetic stiffness between bias magnet and armature. This negative magnetic stiffness should intuitively be small, as the permanent magnet itself comprises a very large air gap in itself (since the permeability of the bias magnet is nearly identical with that of air). Any small radial displacement of the armature will not lead to large changes of flux along the circumferential surface of the armature. Later, a 3D finite element analysis is carried out to confirm this intuition. The soft magnetic material used to construct the stator and armature is first 22  assumed to have infinite permeability in order to establish the ideal static forceposition-current relations. A second lumped parameter model is used to determine what minimum relative permeability can be used without significantly lowering the load capacity relative to an infinitely permeable material. Counter-intuitively, having a highly permeable material is not desirable in the final design unless the material is also a very good insulator to prevent eddy currents. Material choice will be discussed in a section below.  2.1.1  Infinite Permeability Model  Since a well chosen soft magnetic material will have a relative permeability µr much higher than air, the lumped parameter model of the RBT actuator can usefully assume infinite permeability of the soft magnetic material in the first approximation. The dominant reluctance in the magnetic circuit is therefore the air gap reluctance. This analysis was first carried out for the RBT-type actuator in [21]. Figure 2.2a shows the reluctance model of the bias flux path. The reluctances across the air gaps are R1 = R2 =  zo +z µo A zo −z µo A  (2.1)  where A = π(D2PO − D2PI )/4 is the effective armature pole face surface area (that is, the area that generates normal axial force). The reluctance of the magnet, RPM , is very large compared to all other reluctance in the system, therefore the total bias flux ΦPM remains constant with armature position, z, and is approximately ΦPM = Br πDPM tPM  (2.2)  where DPM is the bias magnet inner diameter, tPM is the bias magnet thickness, and Br is the permanent magnet remanence. From basic circuit theory, the following nodal and branch relationships must hold: B1 A + B2 A = ΦPM B1 AR1 = B2 AR2  23  (2.3)  Figure 2.2: Reluctance models of RBT actuator. a) Bias flux reluctance model; b) excitation flux reluctance model. By solving (Equation 2.3), the flux density through each branch of the magnetic circuit can be determined as follows: B1 =  zo − z B zo  (2.4)  B2 =  zo + z B zo  (2.5)  and  where B is the mean bias flux density, which can be calculated as B = ΦPM /2A  (2.6)  It can be seen from (Equation 2.4) and (Equation 2.5) that the bias flux density on each face of the armature is a linear function of the armature position z. In the non-ideal RBT actuator, the bias magnetic flux through each face of the armature is limited by the magnetic saturation of the material, Bsat . In order to make full use of the material, the mean bias flux should be designed such that B is set to Bsat /2,  24  Figure 2.3: Detail of RBT actuator schematic, showing closed surface of armature over which Maxwell’s stress tensor is evaluated. so that the bias fluxes B1 and B2 range between 0 and Bsat . However, generally the remanence of the bias magnet, Br , will be lower than the saturation limit of the stator core material; this implies that at no point will the bias flux density on either face exceed Br . Therefore the mean bias flux is taken to be Br /2 for the particular case where saturation of the magnetic material is higher than the remanence of the bias magnet. Figure 2.2b shows the reluctance model for the excitation flux from the coils. Because the total reluctance of the air gaps R1 + R2 is invariant with armature axial position, the total excitation flux density is independent of z and can be calculated as B=  2NI NIµo = (R1 + R2 )A zo  (2.7)  where N is the number of coil turns in each coil slot, and I is the input excitation current through each turn. The total flux density on each face of the armature is the superposition of the bias flux and excitation flux, giving B1 = B1 − B =  zo − z NIµo B− zo zo  25  (2.8)  and B2 = B2 + B =  zo + z NIµo B+ zo zo  (2.9)  In order to fully utilize the magnetic material range, and the bias flux magnetization, as well as minimizing the size of the actuator, the flux density on each face of the armature should be able to achieve Bsat at any given axial position. This is only possible if the following is true: =  B max  NIµo zo  = Bsat  (2.10)  max  Therefore, ideally the number of turns in each coil slot should be set as N=  Bsat zo µo Imax  (2.11)  where Imax is the maximum allowable current in each coil turn (as determined by the dissipation ability of the chosen wire gauge of the coils). The net force generated on the armature can be calculated by the evaluation of the Maxwell stress tensor over a closed surface on the armature [12] as shown in Figure 2.3. The Maxwell stress tensor is constructed of the field intensity vectors → − H (A/m) surrounding and entering a given surface, as Ti j = µ0 Hi H j −  µ0 δi j Hk Hk 2  (2.12)  where δi j is the Kronecker Delta function, δi j =  1  f or i = j  0  f or i = j  (2.13)  and Hk · Hk is Hk · Hk = Hx2 + Hy2 + Hz2  (2.14)  To evaluate force on the armature, the stress tensor matrix [T ] is constructed as   Txx Txy Txz      [T ] =  Tyx Tyy Tyz  Tzx Tzy Tzz 26  (2.15)  where each element Ti j corresponds to the tensor for the area normal to the i’th direction with field in the j’th direction. The total force on the armature can thus be calculated as the surface integral of the stress tensor over the surface S as [T ] • d s  F=  (2.16)  S  Integrating and getting rid of null terms in the matrix, the final net force acting on the armature is determined to be 2  Tz · ds =  Fz = S  B22 − B21 2ABN 2AB A= z I+ 2µ0 zo µo zo kI  (2.17)  kZ  where kI is the current force coefficient of the actuator, and kZ is the position force coefficient, or negative magnetic stiffness. This derivation shows that in the ideal case with infinite permeability, no saturation and no leakage flux, the actuation force on the armature is a linear combination of excitation current I and axial position z. While the maximum theoretical load capacity on the armature is 2  sat A Fmax = ± B2µ , the maximum worst case load capacity occurs when z and I are at 0  maximum amplitude and opposite sign: Fmax,worst−case = kI (Imax ) + kZ (zmin )  2.1.2  (2.18)  Finite Permeability Model  To evaluate what material is suitable to make the armature and stator, a finite permeability model is constructed, assuming constant relative permeability and no saturation. Very highly permeable soft magnetic materials will be susceptible to higher eddy current generation, as Faraday’s law from Maxwell’s equations states [24, p.29]:  C  d → → − → − − E · d− c = − ∫ B·d → a dt S  27  (2.19)  Figure 2.4: Reluctance models of RBT actuator assuming finite permeability of stator and armature material. a) Bias flux reluctance model; b) excitation flux reluctance model. → − where a changing field B through an unclosed surface S induces an electrical field → − potential E on a closed path C. The implication is that any changing magnetic field will induce an eddy current in a conducting material that will in turn induce a second magnetic field that will act in opposition to the first. Since the magnitude of the counteracting field produced by the eddy current is a function of the permeability of the material, it is clear that infinite permeability will engender poor AC performance as any change in field (such as AC excitation flux or changing position z leading to changing bias flux) will induce large eddy currents. Since field from the eddy current loops opposes the generating flux, the net flux through the core decreases (lowering the effective relative permeability). This in turn reduces the available load capacity at high frequencies and therefore the ability of the axial position loop to reject dynamic force disturbances. In addition, the energy utilized in creating the eddy currents will be rejected through Ohmic heating of the material, leading to poor thermal performance of the actuator. Thus the minimum relative permeability that does not prove a flux choke is necessary for a good design. Choosing a material with low conductivity and high permeability is ideal, by insuring small eddy current generation and full flux penetration of the stator and 28  Figure 2.5: RBT actuator dimensions for finite reluctance calculation.  29  armature. Since the RBT actuator is an air gap reluctance dominated design, reducing the stator-armature total reluctance significantly below that of the total air gap reluctance produces marginal benefits since stator and armature reluctances are in series with the axial air gaps. Thus the optimum design procedure is to match the reluctance of the stator and armature to the total reluctance of the air gap, giving a minimum acceptable permeability. The effective magnetic permeability of the material thus remains relatively constant with frequency, and the load capacity is not adversely affected. Another design criteria for the material selection is minimum conductivity, again to reduce eddy current generation and improve AC and thermal performance. The finite reluctance model is constructed as in Figure 2.4. The lumped reluctance Rs is the series addition of the stator and armature reluctance for the top and bottom halves of the actuator. The reluctance Rs is calculated as Rs = 1 µo µr A  1 µo µr A  DSO  h1 + h1 + t2A + ∫  h1 + h1 +  tA 2  +  dr 2πrtT = DPI 1 Ln(DSO /DPI ) µo µr 2πtT  (2.20)  where h1 , h2 , tT , DAI and DSO are actuator dimensions shown in Figure 2.5. This expression is approximate and neglects the exact reluctance at sharp corners in the core. To match the stator and armature reluctance to the total air gap reluctance, the following relationship must hold: 2RS ≤ R1 + R2 =  2zo µo A  (2.21)  This reduces to a single expression for the minimum relative permeability as µr,minimum ≥  A zo  D ) h1 + h2 + t2A Ln( DSO PI + A 2πtT  (2.22)  This relative permeability is the minimum required prior to making the stator and armature dominant reluctance elements. In reality, the stator and armature reluctance will become a significant choke prior to this minimum permeability; therefore a somewhat higher µr is desirable to insure that R1 + R2 30  Rs . Therefore a safety  factor of approximately 3 is introduced to (Equation 2.22) so that the designed permeability in the operating region of the spindle should not fall much below µr = 3µr,minimum  (2.23)  While the magnetic permeability in the linear operating region of the core material determines the force coefficients of the actuator, the ultimate load capacity is determined by the saturation limit of the material, Bsat . Thus, the core material saturation limit should be as high as possible given all other constraints. In addition, the bias magnet flux is set by the magnet remanence Br , which forms the maximum bias flux 2B; all efforts should be made to choose a hard magnetic material that has a remanence approximately that of the saturation limit of the core material. This will provide a minimum power solution by maximizing the assistive role of the biasing flux in force generation.  2.1.3  Magnetic Material Selection  The popularity of laminate steels as magnetic core materials for transformers and many actuators stems from their ability to limit eddy current generation by insulating thin layers of highly permeable silicon steel to reduce the size and therefore the magnitude of the eddy current loop and thus the strength of the induced opposing field. Their primary disadvantages are difficulty to form complex core shapes, and anisotropic permeability limiting the flexibility of electromagnetic design of actuators (essentially rendering all flux paths two dimensional). Typically a laminate core will have to be larger than a uniform isotropic material with equivalent permeability in order to carry the same amount of DC flux, and its multi-layered design may create many limiting structural modes. Another class of materials are the soft magnetic composites (SMC). Soft magnetic composites can come in the form of powdered iron materials, such as the final chosen material for the RBT actuator, Somaloy 500 (Figure 2.6) manufactured by Hoganas [3]. This material is analyzed in [25], in which its isotropic properties are explored. The basic concept is the same as steel laminations: small iron particles are compressed together to form a solid material, and remain insulated from each other by a polymer coating. The advantages to using such a material are ease of 31  Figure 2.6: Somaloy 500 powdered iron magnetic characteristics from [3]. a) Initial BH curve, measured by manufacturer to 30 kA/m and extrapolated to 100 kA/m. Saturation flux density Bsat is read from this plot as approximately 2 T. b) Relative permeability µr . forming complex core shapes and relatively isotropic permeability which has the potential to greatly improve actuator structural performance by allowing the development of more compact actuators. The main disadvantage to general use are significantly lower initial and maximum µr compared to either steel laminates or solid core designs. Based on Figure 2.6a, the saturation flux density of Somaloy 500 is approximately 2 T, and from Figure 2.6b the initial and maximum relative permeability is 500. Evaluating (Equation 2.22) with initial design parameters from Table 2.1 shows that its initial permeability µr is greater than µr,minimum of 154 by over 3 times, meeting the criteria in (Equation 2.23).  32  The rare earth permanent magnet material N44SH is chosen for the bias magnet, due to its high nominal remanence flux density (1.325 T) and relative temperature stability with a maximum working temperature of 150◦ C [26], though this is dependent on final magnet geometry.  2.1.4  Initial Design of Radially-Biased Thrust Bearing/Actuator Parameters  The RBT design is driven by both electromagnetic constraints and mechanical design considerations. The critical electromagnetic design constraint is the minimum area required to produce a worst-case load capacity of 5000 N. The axial air gap zo is set to 1 mm rather than 0.75 mm (which would give an exact maximum stroke of 1.5 mm) to allow room to implement a soft stop mechanism for the armature. Slamming the armature into the stator pole faces may lead to excessive wear and and possibly a cracked armature or stator. Coil Sizing The maximum current Imax is limited by the amplifier driving capacity and the constant current rating of the chosen wire diameter of 19 AWG copper solid core wire. This gauge was chosen based on the maximum operating frequency of the current control loop of ten times the targeted axial motion bandwidth of 1 kHz. The skin depth at 10 kHz for copper is 0.65 mm from the skin depth equation for uniform isotropic materials from [24, p.442]:  δ=  2 2π f σ µo µr  (2.24)  where f is the operating frequency in Hz, σ is the material conductivity in S/m, µo is the permeability of free space, µr is the relative permeability of the material and δ is the skin depth at which the current density in the material falls to 1/e (37%) of the current density at the surface of the material due to induced field cancellation. Since δ for 10 kHz is larger than the radius of 19 AWG wire, there is full current penetration in the wire at the maximum operating frequency of the power amplifier used to drive the RBT actuator.  33  From (Equation 2.11), the ideal coil turn number based on a saturation flux density of 2 T and a maximum current of 1.2 A is 1326.3 turns. Limitations of the current driver architecture and the necessity of building a set of coils with an integer number of turns reduced this to the final value of 1248 turns in each slot. The coil slot solution is 39 turns wide by 32 turns deep; the slot width is oversized by several millimeters on each side to allow room for epoxying the coil to the stator. Armature Effective Area Having chosen the number of coil turns, the mean air gap zo and determined the maximum current through each coil, Imax , the worst case load capacity relation (Equation 2.18) may be used to determine the required effective armature pole area to produce 5000 N minimum load capacity: A=  Fmax,worst−case 2  2BN 2B zo Imax + µo zo zmin  (2.25)  where Imax is 1.2 A and zmin is -0.75 mm (the negative end of the permissible stroke). Armature Mechanical Design The mechanical design of the armature is driven by the requirement of minimizing the armature outer diameter DAO for a given armature thickness in order to maximize stiffness, while meeting the minimum effective area criterion from (Equation 2.25). The armature outer diameter is sized larger than the outer diameter of the stator pole, DPO in order to minimize leakage flux from the stator pole face to the radial outer surface of the armature. DAO should be designed larger than DPO by more than the mean axial air gap in order to minimize leakage flux from the stator axial pole face to the radial face of the armature (bypassing the force generation area). The following design rule should be followed: DAO − DPO > zo 2  (2.26)  The inner armature diameter DAI is determined by the space required to secured the armature to the shaft; in the case of the built full scale prototype, several M8 34  Figure 2.7: Maximum shear and normal stress locations on armature due to axial loading.  35  Figure 2.8: Radial loading on armature due to centripetal acceleration during rotary-axial operation. bolts were used. The armature may fail under three loading conditions: 1) shear at the inner pole diameter DPI , 2) normal force yielding in the axial direction, and 3) radial load due to centripetal force during operating at 3000 rpm. Each loading condition is evaluated separately. The maximum shear stress occurs at the inner pole face diameter on the armature assuming that the shaft attachment is designed to support the armature up to the inner stator pole diameter, as shown in Figure 2.7a. The dimension DPI (the inner diameter of the effective armature area) and the armature thickness tA are used to evaluate the maximum shear load based on the maximum axial load capacity: 36  Fmax ≤ Yshear SF πDPI tA  (2.27)  where Yshear is the shear yield strength of Somaloy 500, and SF is the safety factor. The maximum load capacity is determined as Fmax =  B2sat A = 12810N 2µo  (2.28)  Since the armature will be experiencing large magnitude fatigue loading without regular inspections, the safety factor SF is chosen to be 7. The maximum shear stress experienced by the armature is on the inner face of the effective armature area, and is 2.17 MPa. This is over 7 times lower than the shear yield strength Yshear of 16 MPa. The maximum normal stress seen by the armature is due to magnetic pressure on the effective armature pole surface as shown in Figure 2.7, and must meet the following relation: Fmax π 2 2 4 (DPO − DPI )  ≤ Y f SF  (2.29)  where Y f is the fatigue strength of Somaloy 500, 23 MPa. The maximum normal stress on the armature effective pole face area is 1.59 MPa, more than 14 times lower than the fatigue strength of the powdered iron material, and more than double the safety factor SF of 7. Another loading condition is the normal pressure exerted in the radial direction on the armature due to centripetal force during rotary-axial operation. The maximum radial pressure will occur at the outer diameter of the armature. Radial pressure may be derived as follows, based on an analysis of an element on the armature of infinitesimal thickness across an arc dθ and a radial distance dr, with height tA and under centripetal acceleration a as shown in Figure 2.8. The maximum pressure at the outer surface of the armature is  37  Pradial,max = =  ∫ rDAI 2  Fradial (r) Acirc (r)  DAO 2  a(r)ρdV  rdθtA  (2.30) DAO 2  where Fradial is the total radial force on the armature due to centripetal acceleration at a radial position r (where the centripetal acceleration is a = ω 2 r at a rotational speed of ω), Acirc is the area on the element that is normal to the direction of the centripetal acceleration, ρ is the mass density of the powdered iron material, and dV is the volume of the element. Given a mass density ρ of 7370 kg/m3 , a rotational speed ω of 314.15 rad/s (3000 rpm), (Equation 2.30) may be evaluated as Pradial,max = =  ω 2ρ  ∫ rDAI ω 2 r2 tA ρdθ dr 2  rdθtA  (D3AO −D3AI ) D 3( AO 2 )  DAO 2  (2.31)  At the operating speed of 3000 rpm, the radial pressure is 0.94 MPa. This pressure is approximately 16 times lower than the powdered iron flow pressure of 15 MPa, and thus meets the following criteria: Pradial,max ≤ Y f low SF  (2.32)  where Y f low is the flow pressure of the core material. Stator Core Dimensions Geometric constraints on other actuator dimensions stem from the need to prevent choke points along the flux path; this is achieved by making sure the minimum area available throughout the entire flux path (for both the bias flux and the excitation flux) is at least the same as the effective armature pole area, A = πDPM tPM . The top portion of the stator is limited by the relation πtT DPO ≥ A  38  (2.33)  and the outer diameter of the stator is limited by the relation π 2 2 (D − DWO )≥A 4 SO  (2.34)  The height of the armature, tA should be set slightly larger than the height of the bias magnet tPM (larger even than matching the pole face area of the magnet to the effective area of the armature would require) in order to minimize leakage flux shorting directly from bias magnet to stator and bypassing the armature. In the final design, the height of the armature is set 5 mm larger than the height of the bias magnet. The final design parameters after refinement from the finite element analysis are listed in Table 2.1.  2.2  Finite Element Modeling  The lumped parameter analysis was used to obtain initial design parameters and specify an acceptable soft magnetic material for the stator and armature. Finite element analysis is used to model non-idealities such as leakage flux and nonlinearities such as saturation, using the software package FEMM [4] which is specifically designed for low frequency 2D problems. The design parameters in Table 2.1 including all dimensions were finalized using the finite element model predictions in this section. The magnetic material characterization provided by the manufacturer of the powdered iron material [3] shown in Figure 2.6 was used as a material input to the model. The finite element model (FEM) is built as shown in Figure 2.9. The finite element analysis solves for 2D magnetic potential A for a system made up of first order triangular elements, using as inputs specified current densities, permanent magnet flux and defined permeability of core materials. The magnetic potential A is related to the magnetic flux density vector B as B = ∇×A and the following constitutive relationships are always observed:  39  (2.35)  Figure 2.9: Finite element mesh of the RBT actuator using [4].     ∇×H = J ∇·B = 0   B = µr µo H  (2.36)  where J is current density, µr is the relative permeability of the material defined for each element, µo is the permeability of free space, and H is the magnetic field intensity in A/m [27]. A variation on the iterative conjugate gradient solver is used based on [28]. The model is purely axisymmetric around the centerline shown (r=0). Coil  40  Figure 2.10: Flux density distributions in finite element analysis using [4] turns are input as 19 AWG magnet wire with a strand diameter of 0.912 mm. The bias magnet is assumed to be purely axisymmetric, though in later chapters this assumption is shown to be limited in the final design. The permanent magnet strength is input into the model in terms of the magnet coercivity given a linear BH relationship: magnet coercivity Hc is set equal to Br /µr,PM µr , where µr,PM is the relative permeability of N44SH neodymium iron boron, 1.028. For a remanence of 1.325 T, the coercivity is set to 1025682.4 A/m. FEMM considers the permanent magnet essentially a solenoid with a specified (typically very low) permeability. − − The source of the permanent magnet flux is a current sheet of density H (→ m ×→ n) c  − A/m2 , where Hc is the coercivity of the magnet, → m is the direction of magnetization → − of the magnet, and n is an outward normal vector at the edge of the permanent magnet [27]. The spherical boundary is set to an asymptotic boundary condition, approxi-  41  Figure 2.11: Finite element model predictions of force-position characteristics of RBT actuator for various excitation currents, with a worst case load capacity of 5860 N for a 1.2 A excitation at z=750 µm. mating the impedance of unbounded space by setting the magnetic potential at the boundary such that 0=  1 1 ∂A A+ µo R µo ∂ n  (2.37)  where R is the radius of the spherical boundary, A is the magnetic potential, and n represents the direction normal to the boundary. This boundary condition was determined as per [29]. Figure 2.10 shows several flux density maps of the RBT actuator for different operating points. The raw FEMM results have been modified for clarity by mirroring the axisymmetric finite element model, adding coil indicators and changing the linetypes. Figure 2.10a shows the flux distribution inside the actuator with the 42  Figure 2.12: Finite element model predictions of force-current characteristic of RBT actuator at 0 µm, showing worst case nonlinearity and linear fit. armature in the neutral position, z = 0 mm, and with no current excitation I = 0 A. Figure 2.10b shows the flux distribution for the same armature position but with -0.5 A excitation current. The excitation flux is able to cancel out essentially all of the flux on the +z side of the actuator, inducing a net force in the −z direction. Figure 2.10c shows the flux density distribution for the armature at z = 0.75 mm and zero excitation current. The air gap reluctance on the +z face of the armature is now much smaller than the air gap on the −z face of the armature, so essentially all of the bias magnet flux shorts through the +z branch of the stator. Figure 2.10d shows the armature still at z = 0.75 mm, but with an excitation current of -1 A. This excitation current is able to induce a flux that completely cancels out the bias flux shorting through the +z branch of the stator, and a large flux density is generated on the −z face of the armature. 43  Figure 2.13: Force coefficient kZ evaluated over symmetric range of -750 µm to 750 µm, using datasheet initial BH curve. To determine the actuating force on the armature, a weighted stress tensor volume integral used, and is based on the methods introduced in [30] (as shown in [27]). Figure 2.11 shows the axial actuating force on the armature as determined by stress tensor method as a function of axial position z. The worst case load at maximum input current 1.2 A and minimum stroke position -750 µm is 5860 N. This worst case load was designed almost 1000 N over the specified requirement in order to account for magnetic material variation. The force for a given excitation current is highly linear as a function of position, with some nonlinearity present near the edges of the stroke. The nonlinearity for a given excitation current is the result of changes to the leakage flux path as the armature nears the stator face. When the armature is centered axially between the stator faces (neutral position), a  44  Figure 2.14: Force coefficient kI evaluated over symmetric range of -1.2 A to 1.2 A, using datasheet initial BH curve. certain amount of flux bypasses the force generation surface of the stator pole face and enters the inner diameter of the coil slot as can be seen in Figure 2.10a. As the armature nears the stator pole face, this flux begins to short through the force generation face of the stator (as seen in Figure 2.10c) leading to an increase in force on that surface of the armature. The maximum nonlinearity for any force-position characteristic in the excitation range -1.2 A to 1.2 A is less than 2% over the stroke of the actuator. The force coefficient kZ can therefore be adequately assumed as constant for a given excitation current and estimated using a linear fit using least squares estimation via the poly f it function in MATLAB. The force dependency on excitation current, on the other hand, has severe nonlinearities at the extremes of excitation current, as shown in Figure 2.12 for the armature at z = 0 µm. The maximum non-linearity is 35% for a 1.2 A excitation at  45  z = 0 µm, leading to an underestimation of force by 3500 N. This nonlinearity is due to saturation of the stator core and armature material. However, a linear relationship may usefully be assumed in the range -0.5 A to 0.5 A, and this coefficient can be used to estimate the plant model for the axial position loop for small signal characteristics. The force-current linearity is more critical to good actuator performance than the magnetic stiffness kZ because the lower kI represents a smaller gain between current command and actuator load, thus the predicted saturation will lower the axial motion performance for large force amplitudes. Nonlinearity due to saturation is apparent in Figure 2.13 and is due to increasing excitation flux pushing the material into a higher operating point on the BH curve: for increasing current, magnetic stiffness decreases as the material enters deeper into saturation. This variation in actuator magnetic stiffness should in part be compensated by closing the axial control loop via position feedback. Figure 2.14 shows that the force coefficient kI (as estimated for the linear range -0.5 A to 0.5 A) is relatively constant versus position; this implies that armature position will not cause large variation in the force-current relationship, and therefore that saturation and not leakage is the dominant limiting factor in actuator linearity. The total power loss in the actuator is a function of the series resistance of the coil wires and the eddy current heat loss. From the final design parameters, the total heat loss due to Ohmic heating of approximately 1300 meters of 19 AWG copper wire at a maximum current excitation of 1.2 A is only 28 W.  2.3  Summary  A lumped parameter analysis utilizing a reluctance based circuit solver method is used to provide starting dimensions and inform material choice for the RBT actuator. The radially-biased actuator is shown to be linear with position and input excitation current given an infinitely permeable magnetic core material with no saturation and assuming no leakage flux. The generated dimensions and magnetic material choices are fed to a static finite element model and the force-positioncurrent characteristic of the RBT actuator are analyzed within bounded excitation and stroke. The final design parameters are found in Table 2.1 and are implemented in  46  Table 2.1: Radially-Biased Thrust Bearing/Actuator Design Parameters Parameter  Specification  Axial gap, zo Axial stroke Stator pole inner diameter, DPI Stator pole outer diameter, DPO Effective armature area, A Armature inner diameter, DAI Armature outer diameter, DAO Coil winding outer diameter, DWO Magnet inner diameter, DPM Magnet thickness, tPM Armature thickness, tA Coil turns in each slot, N Maximum current, Imax Shaft diameter, DS Stator outer diameter, DSO Stator total height, hS Magnet remanence flux density, Br Mean bias flux density, B Saturation flux density, Bsat Power Dissipation, Ploss,Ohmic  1 mm 1.5 mm 75 mm 123 mm 7464 mm2 35 mm 126 mm 205 mm 127 mm 20 mm 25 mm 1248 turns 1.2 A 120 mm 240 mm 141 mm 1.3 T 0.65 T 2T 28 W  the mechanical design of the RBT actuator to provide a predicted worst case load capacity of 5860N at either end of the 1.5 mm stroke with a maximum current excitation of 1.2 A per coil turn.  47  Chapter 3  Rotary-axial Spindle Mechanical Design, Manufacture and Assembly This chapter presents the mechanical design, manufacturing and assembly of the full scale rotary-axial spindle. The RBT actuator mechanical design was done at UBC by this author. The target ball design was done by Thomas Huryn and machined by Professional Instruments Company [31]. Huryn also carried out the thermal analysis of the spindle. The hydrostatic radial journal bearing was designed at MIT by Gerald Rothenhofer and Alex Slocum based on the monolithic ceramic bearing design presented in [32]. Overall integration of the RBT actuator and hydrostatic journal bearing was done by this author and Benito Moyls. The majority of component machining as well as the hydrostatic bearing assembly manufacture was carried out by the industrial sponsor company. Final assembly, installation and alignment of the magnetic actuator on the full scale RAS was done at UBC by this author, Benito Moyls and Matthew Paone.  48  Figure 3.1: Rotary-axial spindle architectural options. Starting from tool end (bottom of page)a) Radial bearing, rotary motor, radial bearing, axial actuator; b) radial bearing, axial actuator, rotary motor; c) radial bearing, rotary motor, axial actuator.  49  Figure 3.2: Full scale rotary-axial spindle overall detailed design and completed full scale RAS prototype.  3.1 3.1.1  Spindle Design Rotary-axial Spindle Layout  The rotary-axial spindle concept may be implemented in several different configurations. The essential elements of the RAS concept are 1) fluid journal bearing for radial support, 2) integrated rotary motor, and 3) non-contact axial actuator for thrust support and axial actuation. All configurations that were considered for the final design have the journal bearing immediately adjacent to the tool end to maximize radial stiffness at the working point of the spindle. In Figure 3.1a, the axial actuator is separated from the brushless motor by a journal bearing. The advantage to this configuration is less radial flexibility at the RBT actuator due to the radial bearing acting on a point close to the axial armature. This should reduce radial deflection due to the neg50  ative stiffness which exists because of the magnetic attraction between armature and bias magnet. The primary disadvantage in this configuration is that bearing complexity and spindle length increase significantly as four separate sealing mechanisms are required: two above and below the motor, one at the tool end and one at the axial actuator end. Every other configuration requires only two seals, minimizing complexity (and thereby cost). Additionally, the journal bearings must be aligned concentrically for maximum radial stiffness and minimum error motion; this is exceedingly difficult to accomplish if the brushless motor assembly is in the way. In Figure 3.1b, the motor is on top of the actuator. The advantage to this configuration is that having the primary heat source, the brushless motor, above the axial actuation point means that thermal shaft growth due to motor heat will not affect the tool end of the shaft because the axial thrust bearing/actuator is between the thermal growth and the tool end. However, this advantage is only realized if the axial position sensing point on the shaft is also below the motor. If the axial position is sensed from above the motor, thermal growth will simply induce motion at the tool end anyway because the thermal growth will appear to the position sensor as true shaft motion at the tool tip, and thus the axial closed loop will respond to correct the “error”. Sensing nearer the tool end has its own problems: when not sensed as close to the axis of rotation as possible, the position feedback is sensitive to surface finish and form errors on the target surface. In addition, this configuration requires the assembly and alignment of the axial actuator prior to attaching the brushless motor, leading to less flexibility in the prototyping stage and a longer assembly process. In Figure 3.1c, the axial actuator is above the motor. The major advantage of this configuration is relative ease of assembly and alignment of the various components: the hydrostatic bearing and the brushless motor stator assembly can be built and installed independently of each other and the axial actuator, with radial alignment of the main assemblies taking place when they are bolted together. Nevertheless, the spindle will still suffer from axial errors induced by thermal shaft growth from the motor heating during operation. However, this should be minimized if the operator runs the spindle until thermal equilibrium is reached prior to machining. Another more serious drawback is the large radial flexibility at the 51  armature (and a slightly smaller radial flexibility at the brushless motor rotor) due to the large moment arm between the axial location of the armature and the radial journal bearings. The overall effect of this lower flexibility is a larger susceptibility to error motion from radial load disturbances from the brushless motor rotor and possibly the bias magnet acting on the armature. The consequences of this increased radial error motion is discussed in Section 5.2. This architecture is chosen for the full scale RAS prototype because of the relative simplicity of assembly and alignment of the journal bearing, brushless motor and RBT actuator. The full scale RAS is still a prototype, and maintaining the option of removing and repairing both axial and rotary actuators independently of the hydrostatic bearing was deemed the deciding factor. The final RAS design is shown in Figure 3.2. It consists of four main assemblies: 1) the hydrostatic bearing assembly, 2) the brushless motor assembly, 3) the RBT actuator assembly and 4) the shaft assembly. The hydrostatic bearings, the brushless motor and the RBT actuator are separate assemblies that bolt to each other axially: the motor assembly bolts to a flange on the hydrostatic bearing assembly, and the RBT actuator (in three separate sub-assemblies) bolts directly to the motor housing. Non-contact axial position feedback is provided by an ADE capacitive probe sensing a precision ball target attached to the top of the shaft. The ADE capacitive probe is calibrated using a spherical target ball with the same dimensions and sphericity as the ball target artifact. Rotary torque is provided by a 21 kW, 18 pole, 3 phase DC brushless motor from Motion Control Systems Inc [33], configured as direct drive. Rotary position feedback for speed control is provided by the Hall Effect sensors used for commutation of the brushless DC motor, processed by a custom algorithm to extract high resolution speed feedback [23]. The flange on the hydrostatic bearing housing is designed to act as an attachment point for the whole spindle to a worktable designed by Gerald Rothenhofer for the final configuration of the silicon wafer grinder; this worktable will provide gross axial motion for the RAS prior to machining, and hold the silicon wafer work piece below the tool. The flange also houses all fluid supply and vacuum drain ports for the hydrostatic bearing. For initial testing of the full scale RAS at UBC, the flange alone is used as the attachment point for the RAS to a steel framed table. An additional removable gib 52  flange will also be installed at the tool end of the hydrostatic bearing housing to secure the spindle to the worktable in the setup at the industrial sponsor company (gib not shown in figure). The overall length of the spindle is 1 m, with a 120 mm diameter shaft and total shaft assembly weight of 78 kg without the tool. Total spindle weight is close to 350 kg.  3.1.2  Axial Metrology  Figure 3.3: Eccentricity induced artifact axial error using flat target on end of shaft with perpendicularity error. One of the key lessons of the small RAS prototype was the effect of the axial position target artifact form error and probe radial alignment on rotary-axial 53  Figure 3.4: Axial control loop showing feedback disturbance from eccentricity induced artifact error, zd with controller C(s) and axial plant P(s). coupling. In [23, p.203], it is found that a flat position sensing target leads to a disturbance on the feedback position signal, zd , which is a function of the perpendicularity error on the flat target, γ (shown in Figure 3.3). True shaft motion z cannot be measured by the capacitive probe while the spindle is rotating because the target position read by the probe is a function of angular position θ . The feedback error due to probe misalignment ∆ and perpendicularity error angle γ is when the axial position sensing probe is modeled as a point measurement probe is zd = ∆ sin (γ) sin(θ )  (3.1)  Figure 3.4 shows the axial control loop with the disturbance on the feedback signal in place. The danger in disturbances to the feedback signal is that such errors exist at low frequencies but are masked because the controller aggressively counters deviations from zr + zd , rather than simply deviation from zr ; thus errors are induced at the tool end, but hidden until the workpiece is possibly ruined. Better controller design cannot compensate because more aggressive controllers will simply track the “new” reference signal even more closely and to a higher frequency. In order to minimize this type of apparent motion error due to probe misalignment and form error on the target, a precision ground ball target is used as the sensed artifact fixed to the end of the shaft. It should be noted that though all artifact error models assume a point measurement probe, in reality a capacitive probe measures 54  Figure 3.5: Eccentricity induced artifact axial error, due to probe and ball target misalignment from shaft rotation axis. the average gap over a finite area; this means that an averaging effect occurs and the effect of artifact error is somewhat mitigated. For an ideal rotary-axial spindle, the axis of rotation is coincident with the central axis of the capacitive probe as well as the center of the target ball. In this case, the gap measured between the capacitive probe and the target ball would be a direct measurement of shaft motion. In reality (Figure 3.5), the center of the ball has some eccentricity relative to the shaft rotational axis, ε, and rotates with the shaft. The central axis of the capacitive probe has a fixed offset from the axis of rotation, ∆. At some point in the rotation of the shaft, assuming no shaft motion in the axial direction, the capacitive probe will read a minimum gap D1 at position 1 (solid outline in Figure 3.5). When the shaft rotates 180 degrees from position 1 to position 2 (dashed outline in Figure 3.5) the capacitive probe will read a maximum gap of D2 . The apparent motion D2 -D1 is artifact error on the motion feedback signal and appears as a disturbance on the feedback signal. The maximum peak to valley value of eccentricity induced artifact error is shown to be  55  Figure 3.6: Ball target artifact showing 10 nm surface roundness.  zd =  ∆ε sin(θ ) R  (3.2)  where R is the radius of the ball target. Thus in order to minimize this apparent motion due to eccentricity and probe misalignment, the ball target eccentricity and the capacitive probe offset must be minimized, while the ball target radius should be maximized within reason (i.e., not forming limiting structural modes). The ideal target surface is axisymmetric about the rotational axis of the shaft to avoid rotary-axial coupling on the axial position sensor. Any deviation from axisymmetry due to form error is then the fundamental limit of axial precision. In order to achieve high axisymmetry and low form error, the ball target was manufactured to within 10 nm roundness error over a 25.4mm diameter (this is better than mirror finish surface is shown in Figure 3.6). To achieve less than a 10 nm eccentricity induced artifact error, the product ∆ε must be less than 63.5 µm2 . A reasonable ball target eccentricity ε achievable by a skilled technician is on the order of 1-2µm. Thus an acceptable probe offset ∆ can be no larger than 32µm for a ball eccentricity of 2µm.  56  3.2  Shaft Assembly  Figure 3.7: Shaft assembly overall mechanical design. The functional requirements for the shaft assembly are as follows: • Provide hydrostatic bearing surface for radial support of moving assembly. • Secure brushless motor rotor to shaft. • Secure RBT actuator armature to shaft. • Secure axial position sensing artifact to shaft. • Provide attachment point for tool. In addition, the shaft assembly should be designed such that its structural modes occur at high natural frequencies. In most spindle systems, the moving shaft assembly is the limiting factor in terms of achievable motion bandwidth due to its low dynamic stiffness relative to the spindle housing. The shaft must therefore be 57  Figure 3.8: Brushless motor rotor shaft attachment. as short as possible in order to maximize bending stiffness. The shaft was initially designed with a hollow center in order to reduce weight without unduly affecting shaft bending stiffness, thus increasing the natural frequency of the first bending mode of the shaft. However, machining time and complexity greatly increases with a hollow shaft, and limits attachment points for the tool. Shaft diameter should also be a compromise between stiffness (maximizing diameter) and mass (minimizing diameter). The final shaft diameter of 120 mm was determined in consultation with Gerald Rothenhofer and Alex Slocum, who designed the hydrostatic bearing, and the tool design team at the sponsor company. 58  Figure 3.9: RBT actuator armature shaft attachment. The shaft material was chosen to be AISI 304 stainless steel for strength, ease of machinability and uniform grade. Stainless steel was required due to the presence of water in the hydrostatic bearing, as well as AISI 304’s non-magnetic properties which ensure low magnetic flux leakage in the RBT actuator. The entire shaft assembly is shown in Figure 3.7. Attachment of the tool was designed to accommodate the specifications of the sponsor company, and consists of eight M10 threaded holes in a ground face at the full diameter of 120 mm, a 10 mm raised section with a ground outer bore surface and 75 mm outer diameter with eight M4 countersunk threaded holes. The hydrostatic bearing surface is 518 mm long, in order to accommodate both the hydrostatic bearings themselves as well as the vacuum drain system and lip seals. Details of the sealing system are provided in Section 3.3. The surface finish of the bearing surface on the shaft requires a maximum cylindrical form error of 1 µm as the gap between the shaft and the bearing surface is nominally 20 µm. 59  The brushless motor rotor attachment went through several design iterations. Three main concepts were considered: 1. Axial friction plate between rotor and shaft. While the simplest solution to implement, requiring only a locknut, the friction plate would be highly prone to slipping under load because of the relatively small area on the axial face of the shaft, making this option very high risk. 2. Key way. While slipping should not be an issue with a properly designed key way, a non-axisymmetric load is applied to the shaft when the key is installed, leading to deformation of the shaft and possible alignment issues during operation. 3. Wedge shaped collet. This option utilizes the larger cylindrical face of the shaft and rotor to provide a more positive friction based lock. Shrink-fitting (or interference fitting) the rotor onto the shaft was not feasible as the temperatures required for such an operation might damage the permanent magnets in the rotor. Remagnetizing the brushless rotor once installed onto the shaft would not be possible. A major consideration for the brushless motor attachment was removal in the event of damage to either shaft or motor. The final design shown in Figure 3.8 has the rotor attached to a cylindrical collet via a rotor retaining locknut threaded to the collet. The rotor collet is attached to the shaft via two sets of commercially available collet wedges which are axially compressed by an axial forcing ring and rotor locknut threaded to the shaft itself. This compression forces the wedge outwards in the radial direction, locking the rotor collet against the shaft. This system allows for the removal of the rotor and cylindrical collet without damaging the brushless rotor or shaft. The radially-biased actuator armature is attached as shown in Figure 3.9, using 6 M8 bolts passing through a flange on the ball target artifact and threading into the shaft directly. Initially, a #7 Moore taper was considered for attachment of the armature, because of the Moore taper self-centering and high stiffness properties. However, the lead time was prohibitively long for such a feature, and the self-centering features of the Moore taper were unnecessary for a prototype. The 60  final design uses the ball target lower flange as a loading element against the armature shaft. Armature centering was designed to be determined by tight machining tolerances and a small gap between the armature inner diameter and shaft outer diameter: the cylindricity tolerance of the inner bore of the armature is set to 10 µm, and the outer diameter runout is a maximum of 10 µm. The armature is held very tight against the shaft face itself, so essentially no additional manual centering will be possible. The same bolt pattern that secures the armature to the shaft also secures the ball target artifact as shown in Figure 3.9. A larger minimum radial gap between the ball target and shaft is implemented in order to allow room for manual centering during assembly. Manual adjustment should allow for very small achievable concentricity error. The target eccentricity of the ball center to shaft rotation axis is on the order of 1 µm in order to minimize eccentricity induced artifact error discussed in Section 3.1.2.  3.3  Hydrostatic Bearing  Figure 3.10: Hydrostatic bearing mechanical design.  61  Figure 3.11: Hydrostatic bearing fluid porting solution. The RAS fluid journal bearing must constrain the shaft in the radial direction while allowing motion in the axial and rotational directions; it must also constrain the shaft against bending deflections. A hydrostatic bearing was chosen over an aerostatic journal bearing (as originally implemented for the small RAS prototype) because of its lower input pressure operation and higher load capacity and stiffness due to the incompressibility of water. The fluid radial bearings are externally pressurized hydrostatic self-compensating bearings, as per [32] and were designed by Dr.Alexander Slocum and Gerald Rothenhofer at MIT. The mechanical design for integration of the bearings into a stainless steel housing was carried out by Benito Moyls, this author, and Thomas Huryn. There are two ceramic hydrostatic bearings which use the same externally pressurized water supply located between them as shown by the supply ports in Figure 3.10. The bearing pads are epoxied in place and aligned to at least 1 µm cylindricity error over 500 mm because the nominal gap between shaft and bearing pad is only 20 µm. The bearings are located near the tool end as the highest radial stiffness requirements are from force disturbances from the workpiece. Vacuum drain is at each end of the spindle, as shown in Figure 3.10. For redundant security against leakage in the event that the vacuum drain is insufficient during a malfunction, custom designed rotary lip seals are used to further isolate the bearing fluid  62  from the brushless motor and RBT actuator as well as the tool [34] and are held in place by retaining rings bolted to the hydrostatic bearing housing. In addition, positive air pressure is ported to the area between the lip seal and the vacuum drain, as another sealing feature. All fluid porting is drilled into the stainless steel housing. Lip seals alone are insufficient due to their susceptibility to breakdown at higher shaft speeds. Slip-on shaft sleeves could be attached to the shaft to provide a smooth surface for the lip seals to act against, but monitoring and replacing these sleeves would require significant service time during which the machine would be non-operational. These rotary lip seals provide marginal radial stiffness compared to the hydrostatic bearings themselves, and also a small axial stiffness component that causes a hysteresis effect which can be seen in the actuator force characterization in Section 4.2.2. The operational input pressure for the bearing is 1.75-1.8 MPa. The hydrostatic bearing cooling design incorporates a cooling jacket over top of a helical flow channel that revolves around the spindle housing. The cooling jacket is sealed and held in place by o-ring seals as shown in Figure 3.10. All fluid, including water for both the cooling jacket and hydrostatic bearing as well as air for the overpressure seal and the vacuum drain for the hydrostatic bearing, is ported through the flange as shown in Figure 3.11. The initial design had the cooling fluid exiting out of the lower end of the spindle housing; however, rerouting the cooling outlet to flow out of the flange was more ideal because all fluid lines are easily managed since they come out at the same level and are situated well away from the cutting tool. The expected heat generation due to fluid shear at operating speeds of 25003000 rpm is on the order of 1.5 kW. However, when run up to 3000 rpm, the measured power loss was 6 kW [23, p.249]: the additional heat generation is due to the rubbing of the rotary lip seals against the shaft. This is a serious concern for the final application of the full scale RAS as the cooling architecture is designed to dissipate only up to 3 kW of thermal power. For future iterations of the RAS architecture, some form of non-contact sealing (such as labyrinth sealing) should be implemented.  63  3.4  Brushless Motor Housing  The functional requirements for the brushless motor stator attachment are 1) provide adequate frictional force against slipping against peak torques of 150 Nm, and b) provide adequate frictional force to prevent axial motion. The brushless motor housing is designed to allow the installation of the brushless motor stator completely independent of other components in the full scale RAS. This was critical as the stator attachment method chosen was interference fitting: the stainless steel housing is heated by submersion in boiling water which raises its temperature by 80◦ C, then the motor stator is fit into the housing and the whole assembly is allowed to cool and radially compress the stator. The method utilizes no adhesives that can degrade over time, and no mechanical fasteners which would require additional components and may come loose over time due to vibration or thermal expansion/contraction cycles. Installation is greatly simplified compared to a spindle design in which the motor housing and hydrostatic bearing housing were a monolithic part. The housing is sized according to the application note from Motion Control Systems regarding shrink fitting of frameless motors into steel housings [35]. Thermal analysis was carried out by Thomas Huryn to verify design prior to manufacture.  3.5  Radially-biased Actuator Manufacture  The RBT actuator is manufactured in three major sub-assemblies: the front coil assembly, the bias magnet assembly and the rear coil assembly, as shown in Figure 3.13. A smaller sub-assembly that is comprised of the bumper stop, probe mount and probe is attached to the rear coil assembly.  3.5.1  Stator Coil Assemblies  Figure 3.14 shows the overall mechanical design of the two coil assemblies. Primary functional requirements for both assemblies include housing the powdered iron stator ring and coils, strain relieving the coil leads and providing mounting points to the brushless motor housing. Design for assembly is also a critical concern, as the attractive forces between the bias magnet assembly and the coil assem64  Figure 3.12: Brushless motor housing mechanical design. blies is on the order of several thousand Newtons. To aid in the placement of the bias magnet and rear coil assemblies, blind holes in the front coil assembly and threaded holes in the rear coil assembly were implemented. Details of the use of these DFA features are discussed in Section 3.6.4 and Section 3.6.5. In addition to housing the coils and stator ring, each coil assembly has several additional functional requirements. The rear coil assembly, as shown in Figure 3.14a, is used as the attachment point for the axial probe mount and bumper stop. The front coil assembly, as shown in Figure 3.14b, must also strain relieve the brushless motor stator cable. This is accomplished by milling a slot on the underside of the front coil assembly to allow room for the brushless motor cable to pass through the front coil housing. The housings for both coil assemblies are non-magnetic stainless steel (AISI 304). Magnetic steel would greatly complicate the assembly procedure and reduce the load capacity of the RBT actuator because of an increase in leakage flux.  65  Figure 3.13: RBT actuator mechanical design, showing three main sub assemblies and axial probe, mount and bumper stop.  66  Figure 3.14: RBT actuator coil assemblies, mechanical design. a) Rear coil assembly; b) front coil assembly.  67  Excitation Coils The excitation coils are built by winding 19 AWG copper magnet wire coated with thermally activated epoxy wrapped onto a prepared spindle mold; the spindle mold limits the total turns of the finished coil to 2 layers of 39 turns each. The coil is resistively heated to 200◦ C until the epoxy flows, and then the coil is allowed to cool within the mold. A finished coil as shown in Figure 3.15a. Figure 3.15b shows a coil nearly ready for installation, with two of the “pancake” type coils from Figure 3.15a in parallel. For flexibility in driving amplifiers, further coil connections are made outside of the coil housing, allowing the implementation of several different wiring configurations with 8 sets of leads from each coil assembly controlling 1248 turns (a full set is shown in Figure 3.16b prior to being wired for installation). The advantage of such flexibility is that with different wiring schemes (serial versus parallel), the entire design space of the actuator can be explored and is not as limited by particular amplifier limitations. Stator Ring The stator ring for both assemblies is comprised of inner and outer machined segments of Somaloy 500 as shown in Figure 3.16a. Twelve pieces of each type make up one stator ring for a coil assembly. The segmented design is due to size limitations of the powdered iron billets available from the manufacturer. For a production run of the full scale RAS, it is recommended that a monolithic stator ring be constructed for each coil assembly to minimize additional flux chokes introduced by the epoxy filled gap between segments and to stiffen the entire structure. The dimensions of the stator and number of coil turns are determined from the analysis in Chapter 2 and listed in Table 2.1. Coil Assembly Manufacture Both coils assemblies follow the same assembly procedure. The specific construction of the rear coil assembly is shown in Figure 3.17 to illustrate the process. The stator ring is constructed by epoxying 12 pieces each of the inner and outer segments into a stainless steel housing, as shown in Figure 3.17. Inner and outer stator segments are epoxied into the stainless steel housing using Hysol EA9360 68  Figure 3.15: Freestanding coils. a) Single coil of 2x39 turns (78 turns); b) two coils hard wired in parallel. The two coils in parallel configuration is the basic coil unit for flexibility in wiring (with 16 terminals controlling 8 coil sets for each coil assembly.)  Figure 3.16: Coil assembly electromagnetic components. a) Stator segments; b) coil set for one assembly.  69  Figure 3.17: Manufacturing steps for rear coil assembly.  70  Figure 3.18: Manufacturing steps for rear coil assembly. two part epoxy (chosen for its excellent metal bonding properties) [36]. As all of the axial load will be carried at the stator pole face, it is imperative to properly secure the stator ring to the stainless housing with good quality epoxy. Plastic shims of 100 µm nominal thickness are used to space the segments from each other and from the rear coil housing (Figure 3.17a) in order to prevent epoxy starvation as the segments are pressed into place. After the stator ring has cured (EA9360 has a five day cure time), a slot is milled in the stator ring to a pre-existing tapped hole in the coil housing for the coil leads to exit the housing as shown in Figure 3.17b. After this, the coils are ready to be installed: Loctite E-20NS is used to fix the coils to the stator ring and each other, because of its short cure time and because the coil surface should carry little load [37]. No spacing shims are required as the round geometry of the wire insures enough epoxy contact in the gaps between coils. As the coils are installed, a threaded cable strain relief is installed on the housing. Figure 3.17c shows the coil compression jig used to compress the coils while they cure: an aluminum plate is used to press down on three nylon bolts which in turn compress the coils axially. This is done to prevent the top coil surface from being exposed and shorted during grinding. Figure 3.17d shows the rear coil assembly after the E-20NS has cured. For good electromagnetic contact, the surface of the coil assembly must be ground flat; 71  in order to not load up the grinding wheel with the relatively soft E-20NS epoxy, the assembly is milled to within approximately 100 µm of final height tolerance (Figure 3.17e). Figure 3.17f shows the finished rear coil assembly after grinding. Both completed front and rear coil assemblies are shown in Figure 3.18. Axial Bumper Stop The function of the bumper stop and a locknut on the ball target is to act as a soft stop for the RBT actuator in either the top or bottom stable axial position. From Figure 3.19a it can be seen that no portion of the armature is ever in contact with either stator pole face because the bottom of the locknut and the top of the target ball flange are constrained in the axial direction by the inner flange of the brass bumper (Figure 3.19b). The maximum achievable axial stroke is determined by the gap between the inner flange and the locknut and lower flange of the ball target. The brass bumper is bolted to the stainless steel housing of the rear coil assembly. Axial Probe Mount The axial probe mount holds the capacitive probe through clamping force exerted on the cylindrical face of the probe. This is achieved by machining a thin slot to a hole slightly oversized for the probe as shown in Figure 3.19c. Two screws compress the slot edges together, reducing the diameter of the hole, thus securing the probe. The importance of the radial/lateral alignment of the probe centerline to the axis of rotation is discussed at length in Section 3.1.2. Since lateral alignment on the order of 10’s of microns is required, specific alignment features are incorporated into the rear coil housing, as shown in Figure 3.14a. These features are threaded holes that allow set screws to be used to position the probe mount relative to the spindle axis of rotation. The initial axial alignment method run the hydrostatic bearing and shift the axial probe laterally in two orthogonal directions until a minimum gap is read; this will be when the center of the probe is aligned with the center of the ball. Thus the lateral misalignment will be should be close to the same order as the ball target eccentricity. In [23], an online method of lateral  72  Figure 3.19: Design of bumper stop and probe mount.  73  probe alignment was introduced where the spindle axial loop and rotary loop are run simultaneously, and the probe shifted until the minimum axial error is output at high spindle rotational speed.  3.5.2  Stator Bias Magnet Assembly  The overall mechanical design of the bias magnetic assembly is shown in Figure 3.20a. The bias magnet assembly functional requirements are 1) fix the stator ring to the housing, 2) fix the magnet to the stator ring and 3) include features for assembly and integration with the front and rear coil housings. Because the magnet is slightly thinner than the armature (25 mm versus 27 mm), a 1 mm gap exists on the top and bottom faces of the centered magnet. Polycarbonate spacers are used to center the magnet during assembly, and offer a small amount of mechanical support after installation in the armature. The stator ring is comprised of 6 arc segments of powdered iron material due to size limitations on the Somaloy 500 billets; again, it is reccomended that a production full scale RAS be constructed with monolithic stator rings. The housing is non-magnetic stainless steel (AISI 304) and incorporates several blind holes and additional threaded holes for assembly (discussed in full in Section 3.6.4). Magnet Ring Design One possible setup that may be used to generate a true radial magnetization is shown in Figure 3.21, where a set of coils is used to produce a high field intensity in order to magnetize the permanent magnetic material of the bias magnet ring. In this setup, the bias magnet ring is a single piece. The primary problem with this setup is that it requires a very high magnetomotive force in order to generate the requisite field intensity across both the air gaps and the permanent magnet (which is essentially a large air gap since its relative permeability is close to that of air). This high current requirement means that constructing such a setup is prohibitively expensive in order to produce just a single (one-off) bias magnet for the full scale prototype. Instead of a true radial magnetization, an axisymmetric radial field is approximated by 18 separate piecewise linearly magnetized permanent magnet arcs made  74  Figure 3.20: RBT actuator bias magnet assembly. a) Bias magnet assembly, mechanical design; b) single magnet segment showing magnetization direction.  Figure 3.21: Possible radial magnetization setup.  75  of neodymium iron boron (grade N44SH) by a custom magnetics company using a commercially available magnetization setup. Each piece is magnetized as shown in Figure 3.20b. Theoretically, the smaller the angle each magnet arc subtends, the greater the number of arcs required and the closer the approximation to true radial magnetization. However, this does not take into account the fixed gap between segments required for epoxy: as the number of segments increase, the ration between magnet pole area and epoxy filled area on the inner bore of the magnet ring decreases to the point where total bias flux generation decreases, degrading load performance. In addition, smaller magnet segments are more prone to breakage during installation. As discussed in later chapters, this approximation, while apparently not limiting the load performance of the RBT actuator, has an effect on rotary-axial coupling. In addition, the armature will see some changing flux due to the bias flux ripple, leading to some eddy current generation. Spacer Construction The bias magnet assembly will be ground on both top and bottom surfaces for good electromagnetic contact, just as the front and rear coil assemblies were ground on the single contacting face. There is a danger that the polycarbonate spacers may load up the grinding wheel and ruin it. Therefore the spacers are designed with pockets which will be filled with Loctite E-20NS epoxy, which can be safely ground for small depths. Figure 3.22 shows the process for filling the epoxy pockets. First, the edges of the spacer are masked with a 50 µm thick plastic tape to prevent epoxy adhering to the outside surface (Figure 3.22a). Next, the pockets are filled with epoxy (Figure 3.22b). After the epoxy is laid into the pockets, the spacer is compressed between two plates to force the epoxy out and leave a relatively flat surface (not shown in figure). This is done to eliminate the milling step required for the coil assemblies prior to grinding: any elimination of a machining step in the construction of the bias magnet assembly minimizes risk of damage to the magnet and the machinist. Figure 3.22c shows the spacer after the epoxy has cured: the edges are shown being trimmed with a razor, though generally the excess can be pulled off with the 50 µm tape. Figure 3.22d shows a completed spacer next to an untrimmed spacer.  76  Figure 3.22: RBT actuator bias magnet assembly. a) Prepping polycarbonate spacer by masking edges so epoxy will be easy to remove; b) laying epoxy in the spacer pockets; c) trimming excess epoxy by simply breaking the masking off; d) untrimmed and complete spacers. Bias Magnet Assembly Manufacture The construction method for the bias magnet assembly is shown in Figure 3.23. Unlike the coil assemblies, the stator ring and the magnet ring are installed in the steel housing at the same time; Hysol EA9360 epoxy is used because of its relatively long working time of 1 hour. In the final arrangement, each magnet segment repulses its neighboring magnets because of like magnetization direction. In order to construct the ring, the entire assembly is built on top of a magnetic stainless steel block with a non-magnetic stainless steel block in between (as shown in Figure 3.23a). As each magnet is placed, the friction force due to the attraction between magnet and magnetic block is slightly greater than the repulsion force between magnets (at least until the magnets are very close together, at which point  77  Figure 3.23: RBT actuator bias magnet assembly manufacture. a) Epoxying magnets in place on top of magnetic block with non-magnetic stainless steel block in between, using aluminum jig to ease installation; b) compressing bias magnet assembly using second non-magnetic stainless steel block and jig bolts; c) removing cured bias magnet assembly using load bolts threaded through magnetic stainless steel bolt until magnetic attraction decreases. 78  Figure 3.24: RBT actuator bias magnet assembly finishing work. a) Removing excess epoxy and steel shims that squeezed out through fluid pressure from the epoxy as it cured; b) complete bias magnet assembly after grinding.  79  brute strength is used). Because of an angular tolerance that was too loose, it was feared that the gap between segments would be larger than 100 µm, which would lead to weaker epoxy sections. Steel shims (unfortunately magnetic) were utilized as fillers to reduce the nominal gap to an acceptable 100 µm. Initially every gap between magnets was filled with steel shims. After the stator ring has been epoxied into the housing, an aluminum hub approximately the diameter of the armature is placed in the center of the assembly (Figure 3.23a) and the magnet segments are slid into place. The purpose of the hub is prevent twisting of the magnet segments as they are repulsed by their neighboring permanent magnets: the arc is constrained against the hub and can only travel down the hub outer face, and is easily slid down into place. The final magnet placement is shown in Figure 3.23a: this was the most difficult moment of the entire RBT actuator manufacture, as the combined repulsive force of all 17 other magnet arcs made the last empty spot slightly too small for the magnet. Approximately half of the steel shims used to reduce the gap were removed, and the final magnet was able to slide in place. Magnetic flux measurements later showed the effect of the steel shims, as will be discussed in Section 4.2.3. After the magnets and stator ring are fixed to the housing with epoxy, a second non-magnetic stainless steel block is placed on top of the assembly and bolted through to the magnetic steel block (Figure 3.23b). This is done to minimize excess epoxy on the top and bottom surface, again to eliminate the necessity of milling away epoxy prior to grinding. When the epoxy is cured, the bias magnet assembly can be removed from the magnetic block by threading load bolts into M12 tapped holes in the magnetic bolt: the bolts push against the non-magnetic plate, which lifts the entire bias magnet assembly away from the magnetic block. When the force between the assembly and block decreases enough to move the magnet safely, the entire assembly is removed and placed inside a large wooden crate until it can be ground. Figure 3.24a shows the results of epoxy flow during curing: the viscous flow squeezed excess epoxy into the center of the assembly, taking with it some of the steel shims. This excess is carefully removed with razor and bent hacksaw blades, and finally sanded smooth. This process essentially ruins any roundness tolerance carried over from the close tolerances of the stator rings and magnets. Figure 3.24b 80  shows the completed bias magnet assembly after grinding.  3.6  Actuator and Shaft Installation  This section illustrates the installation process for the shaft assembly and RBT actuator. This installation was done at UBC by this author and Matthew Paone.  3.6.1  Shaft Installation  Figure 3.25 shows the hydrostatic bearing with motor housing installed placed in a steel framed table at UBC. The shaft assembly, without armature or ball target in place, is installed in the hydrostatic bearing. The shaft is lubed with detergent (safe for the filters on the hydrostatic bearing fluid lines) and the hydrostatic bearing is run with a bucket over the end of the spindle to remove excess water and return it to the drain tank. An engine hoist is used to maneuver the over 70 kg shaft into place; jamming the shaft against the hydrostatic bearing is a serious concern as there is only 40 µm of play between shaft and bearing. The brushless motor housing is also installed, and the magnetic attraction between the stator back iron and the rotor on the shaft also complicated installation. Figure 3.26a shows the installed shaft. There is no thrust support until the RBT actuator is installed and operational, so a locking mechanism for the shaft is installed at the tool end of the spindle, locking the shaft relative to the hydrostatic bearing housing (Figure 3.26b). In order to approximately center the brushless motor housing to the shaft rotational axis, the hydrostatic bearing is run to straighten the shaft in the spindle and feeler gauges are used to center the brushless motor housing to within 75 µm. This centering is not very accurate because 1) no rotation of the shaft and 2) based on inspection, the surface quality of the rotor and stator is questionable.  3.6.2  Front Coil Assembly and Armature  After the shaft is installed in the hydrostatic bearing and the brushless motor housing is centered, the front coil housing is lowered to the surface of the motor housing using an engine hoist (Figure 3.27a). The front coil assembly is centered relative to the shaft itself and fastened to the motor housing using M10 bolts. The centering of 81  Figure 3.25: Installation of shaft assembly.  Figure 3.26: Locking shaft assembly in axial direction.  82  Figure 3.27: Installation of front coil assembly onto brushless motor housing. a) Front coil assembly lowered onto brushless motor housing; b) RBT actuator armature installed on shaft. the coil assembly is limited by the segment misalignment on the inner bore of the stator ring. Better centering can be achieved in future iterations with a monolithic stator ring. The armature is installed as shown in Figure 3.27b.  3.6.3  Ball Target Installation and Alignment  The ball target artifact is installed without locking down the securing bolts (Figure 3.28a). In order to radially center the ball target, a temporary thrust bearing is installed in the form of a live center braced against a load plate, shown in Figure 3.28b. The brushless motor is brough online and the capacitive probe is mounted in the radial direction on a centering probe mount (Figure 3.28c). The primary difficulty encountered was the repeatability of the centering as the bolts securing the ball target and armature to the shaft were tightened. In addition, the heat generation from the rotary lip seals and the undersized chillers for the cooling water meant that thermal equilibrium was not reached in the UBC setup. The final measured eccentricity of the ball target is 1.5 µm.  3.6.4  Bias Magnet Installation  Based on finite element modeling, when the bias magnet is within 20 mm of the front coil assembly, the attraction between them will be over 2000 N because of 83  Figure 3.28: Installation of front coil assembly onto brushless motor housing. a) Front coil assembly lowered onto brushless motor housing; b) RBT actuator armature installed on shaft. the closing of one branch of the magnetic circuit shown in Figure 2.2a. In order to safely lower the bias magnet onto the front coil assembly, non-magnetic stainless steel threaded rods are screwed into M10 tapped holes in the bias magnet housing. The ends of the threaded rods are then placed on top of ball bearings placed at the bottom of blind holes in the front coil housing. The ball bearings provide essentially a point contact and allow the threaded rods to be turned even under very high axial loads. Anti-sieze is used to prevent galling between ball bearing and housing and threaded rod and bias magnet housing. The procedure is shown in Figure 3.29a: the threaded rods are “loosened”, thereby lowering the bias magnet  84  Figure 3.29: Installation of bias magnet assembly onto front coil assembly. a) Schematic procedure for lowering bias magnet assembly used threaded rod screwed into M10 threaded holes on the bias magnet housing; b) installation of bias magnet assembly; c) bias magnet assembly after centering using feeler gauges. assembly onto the front coil assembly in a controlled descent. Figure 3.29b shows the bias magnet installed and secured with M8 bolts to the front coil housing.  3.6.5  Rear Coil Installation  The rear coil assembly is maneuvered into position over the bias magnet assembly using an engine hoist. Similarly to the bias magnet installation, a high load develops when the rear coil stator face approaches the bias magnet. However, because  85  Figure 3.30: Installation of rear coil assembly onto bias magnet assembly. a) Installation of rear coil assembly using engine hoist and threaded rods through M8 threaded holes on the rear coil housing; b) detail view of lowering procedure; c) rear coil assembly installed and centered. almost all of the magnetic flux from the bias magnet is shorting through the front coil stator ring, the load on the rear assembly does not become appreciable until it is within 2 mm (where it quickly approaches 1200 N, based on finite element modeling). Nevertheless, the M10 threaded rod lowering method from Figure 3.29a is used again, though this time the blind holes in which the ball bearings are placed are in the bias magnet itself. Figure 3.30a shows the rear coil assembly being slowly lowered onto the rest of the RBT actuator. Figure 3.30b provides a detailed view of the procedure, including a last view of the armature. Figure 3.30c shows 86  the rear coil assembly fully installed and secured with 6 M10 bolts straight through to the brushless motor housing; 12 countersunk bolt holes are visible, but 6 are used for lowering the rear assembly and are afterwards filled with short M10 bolts for cosmetic effect.  3.6.6  Probe Holder and Probe Installation and Alignment  Once the rear coil assembly is secure, the brass bumper stop can be installed. Figure 3.31a shows the bumper stop in place and the ball target locknut threaded onto the ball target artifact. Figure 3.31b shows the locknut being tightened using a custom wrench. Figure 3.31c-d show the axial alignment of the probe. The axial alignment procedure for the capacitive probe is driven by two factors: 1) the required standoff for the probe (minimum distance from probe to target), and b) insuring that the probe and ball never contact each other. Any contact would permanently mar the 10 nm roundness of the ball target artifact, as well as perhaps damaging the capacitive probe. The procedure is as follows. The shaft is first allowed to rise to the top maximum position, so the bottom of the ball target flange is resting on the brass bumper. The axial probe is lowered until the readout is in range. Assuming that the nominal stroke of 1.5 mm is accurate, the shaft is lowered using the load plate shown in Figure 3.26b until approximately 500 /mum have been traversed (if the armature is shifted any closer to zero, it is possible it will snap to the bottom face). Assuming this is the approximate center of the stroke plus an additional 200 µm, the probe is moved axially until its readout is 200 µm. The probe is then safely axially aligned, though there will be some offset from the magnetic neutral point because of the error between the true stroke and the assumed stroke of 1.5 mm. After the axial position is set, the probe is secured by tigheting the clamping screws on the mount. Lateral probe adjustment is shown in Figure 3.31e, where custom long allen keys are used to turn a set of M3 and M5 set screws against the probe holder to adjust its lateral position relative to the center of the ball. The probe is adjusted laterally until a minimum gap between ball target and capacitive probe is read out. This is somewhat difficult as the sensitivity of the probe to lateral motion decreases as the probe approaches the true center of the ball. This initial static lateral probe  87  Figure 3.31: Installation of brass bumper stop, ball target locknut, axial probe mount and showing lateral alignment of probe. 88  centering is much improved upon by centering the probe while the axial and rotary loops are operatational and running at high speed [23]: as the shaft rotates, a peak to valley artifact error is read out and increases with rotational speed; the probe is shifted until this error is minimized. Figure 3.31f shows the probe mount, capacitive probe and bumper stop installed on the full scale RAS. The full scale RAS prototype installed in the steel frame table at UBC is shown in Figure 3.32.  3.7  Summary  The detailed mechanical design of each assembly in the full scale RAS was discussed and the final manufacturing process of the RBT actuator was illustrated. The assembled RBT actuator has an outer diameter of 315 mm and an overall height (excluding capacitive probe and mount) of 200 mm. The final spindle weight is 350 kg, with 78 kg for the moving shaft assembly and an overall height of 880 mm. While the design-for-assembly features of the full scale RAS proved adequate for a prototyping stage, several improvements are necessary should the RAS ever enter production. Manufacturing the bias magnet assembly in particular was a risk filled process, as the magnet segments required a lot of force to install and could have cracked under the load. Possibly a series of plastic spacers, in the shape and size of the magnet arcs, could be installed first then replaced one by one. The plastic spacers will help keep the magnets apart. Another area in which there could be improvement are the rotary lip seals. Non contact sealing solutions should be pursued to minimize frictional heating and power loss.  89  Figure 3.32: Full scale RAS installation at UBC.  90  Chapter 4  Static Load Characterization Experimental Results In this chapter, the static load characteristics of the hydrostatic bearing and the RBT actuator are experimentally measured. The radial stiffness at the tool end is directly measured, and the equivalent radial stiffness acting at the center of each bearing pad is derived. The measured tool end radial stiffness at the operating water pressure of 1.75 MPa is 157 N/µm. The force-current-position characteristic of the RBT actuator is measured, and found to differ significantly from the load behaviour predicted by finite element analysis in Section 2.2. This discrepancy is investigated and found to come primarily from a difference between the modeled and actual magnetic characteristics of the powdered iron material used for the stator and armature. The expected worst case load capacity for the RBT actuator is 4900 N.  4.1  Hydrostatic Bearing Static Radial Stiffness Characterization  The hydrostatic bearing static radial stiffness was characterized prior to the installation of the RBT actuator. The tool end radial stiffness is measured directly, and the equivalent radial stiffness of each hydrostatic bearing pad is derived from this measurement.  91  4.1.1  Experimental Setup  Figure 4.1 shows the experimental setup used to find the equivalent radial stiffness acting at the center of each fluid bearing pad. The spindle is placed on two wooden supports, horizontally in order to eliminate the need for a thrust bearing to keep the shaft from sliding out. A capacitance probe is mounted to the outer housing of the spindle and measures the change in distance x between housing and shaft, essentially the radial motion of the shaft. A load cell is placed between a car jack and the shaft to apply a radial load F. As the spindle assembly rests on two wooden supports and is vertically free to move, the maximum load that can be applied is limited by the weight of the spindle.  4.1.2  Radial Stiffness Performance  Figure 4.2 shows the equivalent model of measured radial stiffness as seen at the loading point, assuming that there exist equal radial stiffness elements Kb acting at the points shown in Figure 4.1. Of course, the load from the hydrostatic bearing is spread across the whole area of each bearing pad, but in order to determine radial stiffness at each end of the spindle this assumption is valid because it allows the equivalent stiffness elements to be reflected to any axial position on the shaft. Kb1 and Kb2 can be determined through the principle of superposition, by assuming the shaft and one bearing element infinitely stiff and reflecting all stiffnesses back to the measured point. The equivalent stiffnesses are as follows: 2  d2 d1  Kb1 = Kb  d2 d1 +d2  Kb2 = Kb  (4.1)  2  If the shaft is assumed to be infinitely stiff, the total equivalent stiffness of the model in Figure 4.2 is determined to be   Kmeasured = Kb   2  d2 d1 d2 d1  2  +  d2 d1 +d2 d2 d1 +d2  2     2  The derived radial stiffness Kb assuming infinite shaft stiffness is therefore  92  (4.2)  Figure 4.1: Radial stiffness characterization of hydrostatic bearing, experimental setup.  Figure 4.2: Radial stiffness equivalent model.  93  Figure 4.3: Measured tool end shaft stiffness, and derived hydrostatic bearing stiffness using both a rigid shaft and an elastic shaft.   α   Kb = Kmeasured 2   α=  d2 d1  d2 d1  + 2  2  d2 d1 +d2  d2 d1 +d2  (4.3)  2  The derived radial stiffness for finite shaft stiffness is just as easily calculated, as Kb =  Kmeasured Ksha f t α Ksha f t − Kmeasured  (4.4)  Using COSMOSWORKS in Solidworks [REF], a static deflection model of the shaft was created and a finite element analysis carried out. The predicted shaft stiffness at the measured point is 971 N/µm. The measured stiffness and the derived radial stiffness of the hydrostatic bearing (with infinite and finite shaft stiffness) is shown for varying water supply pressures in Figure 4.3. The measured radial stiffness Kmeasured is the tool end radial stiffness directly. At the operating pressure of 1.75 MPa, the measured radial stiffness is 157 N/µm, with a minimum hydrostatic bearing stiffness at each pad of 410 N/µm assuming an infinitely stiff shaft, or a maximum of 586 N/µm assuming a  94  shaft stiffness of 971 N/µm.  4.2  Radially-biased Actuator Axial Load Characterization  The static axial load characterization of the RBT actuator installed in the full scale RAS was carried out at UBC in the setup shown in FIG. The results showed a large discrepancy between the measured load characteristic and the load characteristic predicted by the finite element model from Section 2.2. This discrepancy was investigated and found to stem from incorrect magnetic characterization of the powdered iron material Somaloy 500 used as the stator material, as well as an overestimation of the bias magnet flux remanence Br . The derived worst case load capacity is determined to be 5200 N, and the maximum load measured was 5000 N at less than half of maximum amplifier current capacity.  4.2.1  Experimental Setup  To measure the RBT actuator load characteristics, a load cell is placed between a loading plate and the spindle shaft as shown in Figure 4.4. The RBT actuator is driven by an eight channel, 2 kW power amplifier designed by Kris Smeds based on previous work in [21] and a 4 channel, 1kW amplifier assembled by Matthew Paone in [23] for the small RAS prototype. Details on its construction and implementation can be found in [23]. The ADE capacitance probe position signal is fed back to a custom I/O board to enhance effective axial resolution to 19 bits (originally designed by Xiaodong Lu and Matthew Paone [23]), and the control algorithm is executed on a dSPACE 1103. For the purposes of the initial load characterization, the custom I/O board was not implemented, as the 16 bit ADC resolution of the dSPACE 1103 was deemed sufficient for the long stroke load test. All control algorithms are designed using dSPACE Simulink blocks in MATLAB and all host interactions (starting, stopping, logging, etc) are within the dSPACE ControlDesk environment on a host PC. The armature position begins on the face nearest the tool end (towards bottom of Figure 4.4). The input excitation current is set to a constant positive DC value, driving the armature towards the tool end (+z direction). An opposing axial load 95  Figure 4.4: Setup for actuator axial load characterization. is gradually applied in the -z direction by tightening the loading nuts below the loading plate. The position, excitation current and the axial load applied on the shaft are measured simultaneously to produce the actuator force-current-position characteristic.  4.2.2  Initial Results  The load characterization was limited in stroke to between 750 µm (starting position) and a little over 0 µm (maximum stroke) because of the danger of the armature snapping to the -z face of the stator. Usable data is only obtainable from approximately 700 µm to over 0 µm, due to the initial dynamic loading of the shaft as the loading nut begins to be tightened. It proved to be impossible to make the initial takeup of the nut smooth enough that dynamic load spikes did not appear for the first 50 µm of travel. The maximum excitation current obtained was only 0.45 A, less than half of the maximum current of 1.2 A, due to an unfortunate incident in which the linear amplifier was overdriven. A previously undetected current choke point on the power amplifier circuit board blew itself off of the board and caused a cascade failure in which this author feared for his life. A new design iteration  96  Figure 4.5: Raw axial load characterization measurements. of the board was manufactured and integrated with the full scale RAS prototype, but only after it was shipped to the sponsor company for installation into the work table designed by Gerald Rothenhofer, and no new load characterization has been undertaken to date. Initial measurements shown in Figure 4.5 indicate a pronounced hysteresis behaviour. The most likely possibility is interference from the rotary lip seals discussed in Section 3.3. As shown in Figure 4.5, the average hysteresis loop is 300 N, therefore the rotary lip seals exert an average of 150 N in the axial direction opposing the motion of the shaft. The load cell also measures the shaft assembly weight of 765 N. Hysteresis and shaft weight are compensated for by obtaining the initial loading line up to the turnaround point (approximately at z = -20 µm), and subtracting both the average hysteresis value (150 N) and shaft weight (765 N). Axial probe misalignment is also evident from the loading line with zero exci97  Figure 4.6: Measured force-position characteristic, zero and 0.45 A excitation current, compensated for shaft weight, hysteresis and axial probe misalignment. Compared to FEM prediction using datasheet initial BH curve. tation current. At z = 0 µm and I = 0 A, the axial load should be zero. From the raw results, the axial loading line at z = 0 µm should be the sum of hysteresis and shaft weight, 915 N. This value occurs at z = 43 µm. In order to compare directly to the finite element model predictions from Section 2.2, the load lines must be shifted by 43 µm. Figure 4.6 shows the direct comparison of the experimental force-position characteristic at two different excitation currents with the finite element model predictions that use the datasheet initial BH curve provided in [3]. A significant discrepancy is immediately apparent. From the 0.45 A comparison, the experimental actuation force is approximately 1000 N lower than predicted by the finite element model. This indicates a large deviation in the predicted force-current coefficient, kI . 98  Figure 4.7: Experimental force-current load lines. The experimental force coefficients kI and kZ are measured using the same linear polyfit function discussed in Section 2.2 (as indicated in Figure 4.6 and Figure 4.7), and compared to the force coefficients predicted by FEM. The polyfit function is evaluated over the same stroke and excitation current as the experimental measurements, so that the linear fit is comparable. Figure 4.8 shows that the FEM overestimates kZ by 66%, worst case. Figure 4.9 shows that the FEM overestimates kI by approximately 25.4%, worst case. Both predicted force coefficients exhibit saturation behaviour that is not seen in the experimental results. The error on kI is of special concern as it directly limits the maximum achievable load capacity at an excitation of 1.3 A. There are several possibilities that may explain the deviation, none of which are mutually exclusive: • The load cell calibration is incorrect (wrong gain).  99  Figure 4.8: Force coefficient kZ , experimental versus FEM prediction using datasheet initial BH curve. • Both the current command to the linear power amplifier and the position feedback signal from the ADE capacitance probe have incorrect gains. Both gains must be incorrect in order for discrepancies to show up on both kZ and kI . • The actuator finite element model is incorrect. It is unlikely that all three gains on the sensed variables (force, position and current) would be incorrect enough to give us such a result. The load cell calibration was carried out using known weights verified by mass balance and spring scales as well as volumetric calibration (under the assumption that the steel weights had a uniform density of 7800 kg/m3 ). The maximum mass used to calibrate the load cell was less than 80 kg. However, the total calibration er100  Figure 4.9: Force coefficient kI , experimental versus FEM prediction using datasheet initial BH curve. ror in gain is a maximum of 5.5% underestimation based on the calibration record: a maximum of 738 N (verified by mass scale, spring scale, and volume measurement of the steel weights at mass density of 7800 kg/m3 ) was read as 697 N on dSPACE. The discrepancies observed exceed these by more than 4 times for kI and by 12 times for kZ . The ADE capacitance probe has a measured gain error of 0.01% according to the calibration chart that accompanied the sensor. The gain between the reference current command on the dSPACE and the linear power amplifier was not compared to an independent measurement of current in the RBT actuator coils prior to shipping the spindle to its final location; however, such a measurement was undertaken for the small scale RAS in which a maximum error of 7% underestimation was discovered between the current reference command and the actual current in the  101  actuator. The discrepancy appears to be the result of a combination of gain errors on the current control board. If the same magnitude of error is assumed in the case of the full scale RAS, the current command error still cannot explain a 25.4% error on kI . Therefore the immediate conclusion is that the finite element prediction is incorrect. This conclusion is bolstered by the evident uniformity of the measured force coefficients compared to the highly nonlinear behaviour of the predicted force coefficients. Both the predicted kZ and kI appear affected by saturation, whereas the measured coefficients have no discernible saturation effects over the excitation range. This suggests a fundamentally different magnetic response to both bias flux and excitation flux from the stator soft magnetic material than is expected, and not simply a pure gain error which would preserve the predicted saturation behaviour if it existed in the actuator. There are five ways that the finite element analysis may be incorrect: • Geometric deviation from the actual RBT actuator. This is not so much a case of incorrect FEM modeling as it is a case of poor manufacturing and assembly. Gross geometric deviation in the stator and armature is very unlikely as the form error on most parts was held to less than 50 µm. In order to produce such large deviations as have been observed, similarly large deviations in form must occur (e.g., the total air gap z0 must deviate from nominal by more than 10%, or more than 200µm). Nonetheless, it is concievable that the tolerance stackup for the total axial air gap may exceed this amount as not all components were measured during assembly. • Axisymmetric assumption in FEM. The finite element model assumes that the RBT actuator is axisymmetric, whereas in fact the bias magnet is piecewise linearly magnetized and composed of 18 separate magnets. This possibility will be investigated further in the next section. • The finite element model did not take into account hysteretic effects of the soft magnetic material. Despite having been chosen specifically for low coercivity and low eddy current generation, Somaloy 500 has some non-zero hysteresis. This effect will be investigated in the next section. 102  • The solver is incorrect. This is ruled out by modeling systems with exact analytical solutions, such as axisymmetric air coil solenoids. The boundary conditions and mesh characteristics are proven correct for static magnetic analysis many times over through various known models. • The magnetic material parameters fed to the model are incorrect. The soft magnetic material Somaloy 500 may either have some batch variation in production, or the machining process may have changed its magnetic properties. The equivalent bias magnet strength may also vary from the nominal remanence of 1.325 T assumed in both the lumped parameter analysis and the finite element model.  4.2.3  Magnetic Material Characterization  Initial BH Curve of Stator Soft Magnetic Material The first step in determining if the magnetic material parameters fed to the finite element model are correct is to measure the initial BH curve of a sample of Somaloy 500 from the same production batch as that used in the final RBT actuator. The standard methodology presented in [38] is followed in general, with noted exceptions. The basic idea is illustrated in Figure 4.10. A ring-shaped transformer is constructed with the soft magnetic material to be tested used as the core, and driven on the primary windings with an AC current i. This alternating current will produce an alternating magnetic field intensity H that is analytically determined directly from i because of the smooth ring shape of the core (the field generation is essentially axisymmetric about the center of the core ring). The exact field intensity as a function of radius is determined from Ampere’s law about any circular path of radius r, shown in Figure 4.11, as H (r) =  Np i (A/m) 2πr  (4.5)  where Np is the number of primary winding turns and i is the excitation current through the primary windings. A mean radius is determined to simplify calculations using Equation 4.5: 103  Figure 4.10: Initial BH curve measurement, numerical integration of secondary voltage method.  r=  ro − ri ∫ H(r) · r · dr ∫rrio 1r · r · dr = ro 1 = ∫ H (r) · dr ∫ri r · dr Ln rroi  (4.6)  Thus the estimated field intensity may be simplified as H = Np i  2π(ro −ri ) (A/m) Ln( rro )  (4.7)  i  The field intensity will produce a magnetic flux density B that will in turn drive a voltage on a set of secondary or pickup coils according to Faraday’s law. In order to determine flux density within the core, the secondary voltage must be integrated over time, as B=  VNS dt (T ) Ns A  104  (4.8)  Figure 4.11: Exact field intensity distribution inside ring shaped core (windings not shown). where VNS is the secondary voltage, Ns is the number of secondary coil turns, and A is the cross-sectional area of the core. This estimation assumes a uniform flux distribution within the core, so no averaging effects skew the measured initial BH curve. This integration may be done either offline, using numerical methods, or online, using an integrator circuit. Both methods are utilized in this study. The initial BH curve is extracted from the maximal points of a series of BH hysteresis curves of increasing magnitude, as shown in Figure 4.13. In order to obtain a high resolution, over 40 different magnitudes are obtained for the final measurement. 105  Figure 4.12: Simulated initial BH curve, with inset showing averaging effect of flux density estimation. A significant concern regarding the estimation of field intensity and flux density is the possibility of saturating the inner radius of the ring prior to saturating the outer radius. That is, because the exact field intensity (Equation 4.5) is inversely proportional to the radius, the inner edge will see a larger field intensity at all excitation current amplitudes. If the ring is too fat, the assumption of uniform flux density within the core no longer holds. Since the estimated flux density B is always a spatial average of the flux density distribution within the core, the actual B will be underestimated for the given estimated field intensity H. Therefore, in order to determine if the standard ring size specified in [38] was thin enough to adequately estimate the initial BH curve of Somaloy 500, the averaging effect of the test method was numerically simulated as follows. For a series of excitation currents, the exact field intensity distribution within  106  Figure 4.13: Extraction of initial BH curve from measured BH hysteresis loops, Somaloy 500. the core was simulated by evaluating (Equation 4.5) between the inner and outer radius. For each field intensity H thus calculated, a flux density B was calculated using the datasheet initial BH curve (Figure 2.6) as a conversion table. Finally, the estimate B was determined by spatially averaging B over the region ri → ro . In Figure 4.12, the inset shows the maximum error occurring during the transition phase between linear operation and saturation. The maximum error in estimation due to spatial averaging of the actual flux density is is 5 mT. This error is negligible compared to the estimated region of operation of the RBT actuator as well as compared to the saturation limit of the material (1.6 T), therefore it can be assumed that the core dimensions are thin enough to achieve approximately uniform flux density. An additional concern is the driving frequency of the primary current. It is necessary to extract the DC characteristics of the material in order to have a fair  107  Figure 4.14: Initial BH curve measurement, analog integration of secondary voltage method. comparison between finite element results and the load characterization test, which was essentially carried out under DC excitation with smooth, slow motion to emulate a static load at each position. This requires a driving frequency on the primary windings as low as possible in order to prevent eddy current losses which will affect the hysteresis path of the material. While the powdered iron material has been chosen specifically for its low eddy current generation over a wide band of frequencies (up to at least 50 kHz [3]), this behaviour cannot be assumed without independent verification.The IEC standard calls for numerical integration of the secondary voltage as shown in Figure 4.10. The numerical integration method suffers from poor signal to noise ratio at low frequencies (below 50 Hz) as the magnitude of the secondary voltage increases with driving frequency. The numerical method, when driven at below 50 Hz frequency and using a standard oscilloscope to log secondary voltage, cannot adequately measure flux densities at low field intensities (below 100 A/m). In order to establish a baseline standard for low frequency  108  Figure 4.15: Somaloy 500 test ring construction for initial BH curve measurement. behaviour at low field intensity, an online integrator circuit is constructed using an OP-27 [39], as shown in Figure 4.14. This online integrator is essentially a low pass single pole filter with a break frequency of 1.6 mHz. The flux density B is determined as B=  VINT RC (T ) Ns A  (4.9)  where R is the feedback resister (nominally 100M) and C is the feedback capacitor (nominally 1 µF). The exact product RC is determined through a frequency sweep of the integrator circuit (with VNS as input and VINT as output). The integrator circuit cannot obtain a usable signal much beyond 100 Hz for low field intensities 109  Figure 4.16: Experimental initial BH curve test setup. 4130 test ring is shown. as the integration attenuates the signal below the noise floor of the circuit. Thus, the offline numerical integration method is used for high frequency testing and the integrator circuit is used for approximately D.C. testing. A ring shaped sample of Somaloy 500 (shown in Figure 4.15a) is machined from a billet of the same batch as used to manufacture the RBT actuator for the full scale RAS. The ring dimensions are nominally the same as the dimensions specified in both the manufacturer’s datasheet [3] and the IEC standard [38]. The ring sample is wrapped in a primary coil winding of 20 gauge Litz wire after the core has been wrapped in 50 µm tape for insulation (as shown in Figure 4.15b). The primary coil windings must be wrapped as closely as possible to the core in order to limit alternate field intensity paths through the air (in other words, all magnetic flux must be produced within the core). Secondary coil windings are wrapped around the primary windings after the primary windings have been dipped in cyanoacralate as further insurance against shorting between primary and secondary coils (Figure 4.15c). Specifications for the characterization ring and integrator circuit are shown in Table 4.1. The experimental setup is shown in Figure 4.16. The primary windings are driven by a 1 kW linear amplifier from [23] in voltage control mode driven by a sinusoidal reference signal from an Agilent arbitrary waveform generator. The primary current i is measured using a Tektronix TCP202 current probe, and both VNS and VINT are measured simultaneously and logged on a Tektronics TDS 3034B 110  Figure 4.17: Initial BH curve of Somaloy 500, datasheet values versus measured using online integration. Table 4.1: Magnetic Characterization Ring and Integrator Circuit Specifications Parameter  Specification  Np Ns ri ro Ring Height Integrator Gain RC  233 turns 191 turns 20.03 mm 24.89 mm 5.07 mm 0.0946 ΩF 111  Figure 4.18: Relative permeability of Somaloy 500, datasheet versus measured values. oscilloscope. The maximum current driven through the primary coils is 14.8 A peak. For the driving frequency of 100 mHz, the sampling rate is set to 500 Hz with a window size of 20 seconds. Post processing for both numerical integration and online integration methods include filtering of the secondary voltage signal using a Butterworth filter with a 1 Hz break to eliminate high frequency electrical noise from the integrator circuit. The measured BH curve is shown in Figure 4.17. In terms of absolute difference, ∆B (difference between datasheet flux density and measured flux density), a maximum error of 0.3 T is found at 1671 A/m (within the operating region of the RBT actuator). Considering that the mean biasing flux is 0.65 T, and the maximum saturation limit for the powdered iron material is 1.6 T, this is a significant deviation. In addition to the initial BH curve, the relative permeability is calculated 112  Figure 4.19: Frequency sweep of relative permeability µr at low applied field intensity with constant amplitude. as µr,k+1 =  1 Bk+1 − Bk µo Hk+1 − Hk  (4.10)  where µo is the permeability of free space and k represents a given data point. Figure 4.18 shows that the initial relative permeability of the sample is only 163 which is merely 32% of the rated initial relative permeability of 500. The measured maximum permeability is only 337, only 67% of the datasheet maximum µr value of 500. These discrepancies are on the same order as the difference between the FEM load characteristics and experimental load tests. Somaloy 500 was also  113  Figure 4.20: Comparison of BH hysteresis curves at 100 mHz (online integration) and 500 Hz (offline numerical integration). magnetically characterized in [25], where the BH hysteresis loops agree well with the measured results in this section. The frequency response of the relative permeability is shown in Figure 4.19. This frequency response was carried out by applying a low amplitude current to the primary windings, in order to keep the output signal a linear function of the input current. Both offline integration and online integration methods were used to cross check the results. Beyond 10 kHz, the parasitic capacitance between primary and secondary windings dominates the pickup coil voltage (hence the increase in phase) therefore the actual permeability of the test sample could not be measured beyond this frequency. This response shows that the powdered iron material has very low eddy current generation, as expected. This is confirmed by observing that the measured hysteresis loops do not change with frequency at least up to 10kHz.  114  Figure 4.20 shows the direct comparison of BH hysteresis loops at 100 mHz and 500 Hz: as expected from Figure 4.19, there is negligible difference between the hysteresis curves at different frequencies. Bias Magnet Remanence Measurement and Compensation  Figure 4.21: Magnet flux density measurement set ups, a) free air and b) in situ (installed on front coil assembly). After the initial BH curve for the stator soft magnetic material has been measured, it is possible to conduct an in situ investigation of the bias magnet remanence. It was necessary to first characterize the soft magnetic material in order to have a meaningful finite element comparison with the experimental flux density measurements. Figure 4.21 shows the two experimental setups used to measure magnetic flux density from the bias magnet. In Figure 4.21a, the bias magnet flux density at the midline of the assembly is measured in free air. The rotary table is connected to a 360 line encoder (giving an angular resolution of 0.25◦ ), and an F.W. Bell Sypris Model 7010 Gaussmeter and  115  Figure 4.22: Magnet flux density measurements, free air and in situ (installed on front coil assembly). a transvere STM71 probe are used to measure the radial flux density. In Figure 4.21b, the bias magnet has been installed on the front coil assembly, and the armature and ball target artifact are in place. Flux density is again measured using the Gaussmeter, but all angular measurements had to be estimated because the ball target artifact is in the way of installing any kind of rotary jig for angular measurement, and the armature itself cannot turn once the bias magnet is installed until the RBT actuator axial loop is closed. In addition, due to the small air gap between armature and bias magnet (nominally 500 µm), the probe was inserted by hand at each location. Due to variations in magnet placement in the bias magnet assembly during installation, it proved difficult to insert the Gaussmeter probe at the edges of the magnet segments, hence only the center section of each magnet segment was measured. Figure 4.22 shows the measured flux densities in both free air and in situ. The 116  Figure 4.23: Finite element result showing flux density magnitude |B| using FEMM [4]. free air measurements indicate significant bias flux density ripple, with large decreases in flux density at the locations of the steel shims put in place due to the overlarge gap between magnet segments (as discussed in Section 3.5.2). The steel shims provide a low reluctance path for magnetic flux, hence the drop in radial flux density at those points. Once the bias magnet is installed on the RBT actuator and the armature is in place, the armature-stator loop should be a lower impedance path and hence the bias flux should even out. The flux ripple exhibited at the edges of each magnet segment are due to the change in mean magnetic flux path along the arc of the magnet. The mean return path at the corner of the magnet (from the inner face, magnetized north, to the outer face of the magnet, magnetized south) is at a minimum length and therefore a minimum reluctance, hence the increase in the flux density magnitude. Minimum flux density occurs in the center of the magnet because it has the longest mean return path for the flux exiting the inner face of the magnet. When installed in the RBT actuator, the variation in mean path at different points in the arc will be reduced as a percentage of the total return path via the armature and stator, hence the ripple should decrease. The mean flux den117  Figure 4.24: Comparison of FEM air gap flux density prediction and experimental air gap flux measurement, for varying input remanence in the FEM. sity for the free air measurement derived from just the center arc measurements is 0.4944 T, which is 97.4% of the total mean of 0.5075 T (which includes all ripple components). In addition, for the purpose of static axial load characterization, the flux density variation due to the piece-wise linear magnetization of discrete magnets arranged in a ring may be averaged out to provide a mean flux density that is equivalent to a purely radially magnetized bias magnet. This is valid because from the perspective of pure axial motion, the relevant flux is the total flux from the bias 118  Figure 4.25: Simulation results for radial force on armature from radial displacement of armature, using constant permeability 3D finite element analysis. magnet entering the armature. This total flux varies with position in almost exactly the same manner whether the bias magnet is assumed axisymmetric or not. If the shaft rotates, however, the variations in flux density will affect motion, if through nothing other than eddy current generation within the armature due to field fluctuations seen by the rotating armature. Magnetic cogging may be seen, if the shaft assembly has non-axisymmetric stiffness and the armature is non-axisymmetric. The mean flux density of the in situ measurement is 1.2545 T. For magnetic remanence estimation, this value is taken as the mean flux density in the air gap, though this will be an underestimate because it does not take into account any flux ripple that may still exist at the corners of each magnet arc. Underestimation of magnetic remanence will do nothing to prove that it is a real effect since it can only decrease the average predicted force coefficients, matching (perhaps illegitimately) the experimental results. However, if the in situ measurement is assumed to have 119  at least the same percentage of ripple as that of the free air measurement (which will be an overestimate), the mean measurement of the center of each arc is less than 3% below the average flux density in the air gap. Therefore it is legitimate to use the mean air gap flux density as the average flux density in the air gap in determining the actual magnetic remanence of the bias magnet. A finite element model of the air gap flux measurement is made in [4] as shown in Figure 4.23, with the corrected initial BH curve for Somaloy 500, and varying magnetic remanence. In Figure 4.24, the magnetic remanence Br is varied and the flux density in the gap measured. As expected, the flux density predicted by the FEM within the air gap is essentially uniform. By comparing the measured mean flux density with the predicted flux density in the air gap, an equivalent magnetic remanence of 1.2161 T is found, which is 7% lower than the nominal Br of 1.3 T. Since this value was determined through the mean of the measured air gap flux density, it compensates for actual deviation in magnetic remanence between magnets as well as geometric form error in the magnet segments and manufacturing and assembly variation. In order to determine the negative radial stiffness due to the interaction between bias magnet and armature, a magnetostatics 3D finite element analysis was conducted using a simple constant permeability model in COMSOL Multiphysics AC/DC package [40]. The stator core and armature were set with a constant relative permeability of 164, the initial permeability of the powdered iron material. The armature was centered between stator poles (z = 0) and radially displaced, and the net radial force evaluated on the armature. The results shown in Figure 4.25 indicate a negative radial stiffness of 0.66 N/µm. If the maximum relative permeability is used, this figure will be 1.3 N/µm (based on the fact that the maximum permeability is roughly double the initial permeability). This is relatively insignificant compared to the radial stiffness of the hydrostatic bearing supports.  4.2.4  Revised FEM Model  Using [4], a revised FEMM model is built with the corrected initial BH curve of Somaloy 500 and the corrected magnetic remanence of the bias magnet. Figure 4.26 shows the much closer agreement between the corrected FEM predictions and ex-  120  Figure 4.26: Force-position characteristics, experimental and corrected FEMM predictions. perimental actuator load measurements. From Figure 4.27, the maximum error on kZ is now 16%, down from 66%. From Figure 4.28, the maximum error on kI is now 3.4%, down from 25.4%. The remaining error may be a result of changes to the powdered iron material at the ground faces of the stator assemblies: since the test sample was only turned on a lathe and not ground, the surface effects of grinding may play a more prominent role, possibly ’smearing’ the powdered iron particles together thus providing a high relative permeability at the affected surface. This possibility must be investigated further. Having confirmed that the load characterization of the RBT actuator is real, by comparing the measured results to a finite element analysis using measured magnetic material characteristics, it is possible to predict the overall actuator load performance despite the limited excitation range of the test. The average measured  121  Figure 4.27: Force coefficient kZ from initial FEMM predictions, corrected FEMM predictions and experimental measurements. kZ is 2.8 N/µm, and the average measured kI is 7362 N/A. If for the moment it is assumed no saturation takes place at the maximum current excitation, the worst case load capacity may be determined as Fminimumloadcapacity = kI · Imax + kZ · zmin  (4.11)  where Imax = 1.2 A, and zmin = -750 µm, giving a minimum load capacity of 6734 N across the 1.5 mm stroke. This is still above the specification required for the full scale RAS prototype. From Figure 4.29, the minimum achievable load capacity at -750 µm is 5090 N as predicted by the corrected FEM (thus illustrating the effect of saturation). If the predicted value of kI is overestimated by 3.4% (the maximum error found during experimental load testing), and kZ remains at the maximum 122  Figure 4.28: Force coefficient kI from initial FEMM predictions, corrected FEMM predictions and experimental measurements. (giving a worst case scenario), the minimum load capacity should be no less than 4900 N. This result is as conservative an estimate as possible using the measured magnetic material characteristics. One side benefit of the lower permeability of the powdered iron material is the lower field operating point of the magnetic circuit. Operating at a lower point on the BH curve reduces nonlinearities due to saturation. Figure 4.30 shows the comparison of the force-current characteristic at the neutral position between the datasheet driven model and the model utilizing the measured BH curve. While the maximum load is reduced, the maximum nonlinearity is now 25.6% down from the initially predicted 35%, leading to a maximum load overestimation of 2200 N instead of 3500 N as expected from Section 2.2. However, the effects of saturation must still be confirmed by further experimental load testing.  123  Figure 4.29: Corrected FEM prediction, showing worst case load capacities at -750 µm and 750 µm. The overall lesson from the detected mismatch between nominal magnetic characteristics and real magnetic characteristics is that measuring the batch variation of magnetic materials should play a significant role in any mass production of the RBT actuator. The performance specification appears to have been met in the case of the full scale prototype, but this is no guarantee another lot of Somaloy 500 will be as satisfactory. In this case, a 16% drop in the worst case load capacity (from 5860 N to an expected 4900 N) is reasonable. In addition, efforts should be made during the next iteration of the RAS to obtain a better in situ bias flux measurement, in order to determine the effect of magnet variation on the DC load characteristics of the RBT actuator with more certainty.  124  Figure 4.30: Force-current characteristic at z = 0 µm, showing worst case nonlinearity, for FEM prediction using datasheet initial BH curve, FEM prediction using measured initial BH curve, and experimentally measured actuation load at the neutral position.  4.3  Summary  In this chapter, experimental characterization of the radial bearing and the RBT actuator was carried out and the results analyzed. The axial load characteristics of the RBT actuator were underestimated in the design phase due to incorrect magnetic material characterization as an input to the finite element model used to size the actuator. The soft magnetic material Somaloy 500 was experimentally characterized and the initial BH curve was extracted and input into a revised finite element model. The final axial load characteristic indicates that the achievable minimum load capacity may fall below the 5000 N requirement. A summary of results is shown in Table 4.2.  125  Table 4.2: Large Scale RAS Achieved Static Load Performance Specifications Parameter  Target Specification  Minimum (Worst Case) Load Capacity Tool End Radial Stiffness  5000 N -  126  Achieved Specification 4900 N 157 N/µm  Chapter 5  Dynamic Motion Characterization Experimental Results 5.1  Motion Control Results  The original axial and rotary control of the full scale RAS was designed and implemented by Matthew Paone, as published in [23]. The following sections utilize data collected by Paone, some of which was published in [23]. Axial regulation error is measured as 7 nm rms, using an axial loop shaping controller giving a zero crossing frequency of 700 Hz with 31◦ phase margin. Minimum derived axial stiffness is 342 N/µm within the closed loop frequency range.  5.1.1  Axial Motion Control  The axial control of the RBT actuator is as follows. The custom I/O board converts the dSPACE 1103 reference signal into a 20 bit analog signal, which is then subtracted from the analog position feedback via an operational amplifier to produce an analog error. This analog error is then multiplied by a factor of 8 before being passed back to the 16 bit dSPACE ADC, thus increasing the effective resolution of the feedback signal by 3 bits, giving an equivalent ADC resolution of 19 bits. 127  Figure 5.1: Axial control loop block diagram, showing force and feedback disturbances. Detailed design and analysis of the custom I/O board may be found in [23]. The rated noise floor of the ADE probe is 10 nm over a range of 1.5 mm, which limits the real resolution to approximately 17 bits. This noise floor has been reduced to approximately 7 nm through proper shielding and grounding of the setup. For basic motion control evaluation, the axial control loop may be considered to be as shown in Figure 5.1, with a an axial reference zre f , loop-shaping controller C(s) and axial plant function P(s) which is measured from the axial current command Ire f to output position z. Plant Measurement The axial plant is defined as P(s) =  −e(s) Ire f (s)  (5.1)  Because of the control architecture which provides a very high resolution error signal (rather than a high resolution output signal z), the plant is measured using swept sinusoid inputs with the excitation current command Ire f as the input and the error signal e as the sinusoidal output. The use of the high resolution error signal  128  Figure 5.2: Axial plant measurement. enables the measurement of the plant response up to high frequencies where the output response is small. Figure 5.2 shows the small signal response of the axial plant. The frequency region from DC to 150 Hz is dominated by the axial stiffness of the rotary lip seals in the hydrostatic bearing. The total small signal axial stiffness in this region is a linear combination of the negative magnetic stiffness kZ (approximately 3 N/µm) and the positive spring of the lip seals (approximately 90 N/µm), hence the positive stiffness dominates. When the shaft motion is large enough, the lip seal will slide along the shaft and this spring effect will disappear. From 150 Hz to 1 kHz, shaft motion is dominated by the rigid body effect, with a -40 dB/decade slope and -180◦ phase. This is the region of interest from the point of view of motion control, as the loop transmission zero crossing must occur in this region for both stability and good performance. Beyond 1 kHz, shaft assembly structural modes limit the achievable motion bandwidth of the system. Controller The axial controller is composed of three primary components: 1) an integrator in the low frequency region to drive the DC error to zero, 2) lead-lag compensators in the frequency region from 150 Hz to 1 kHz, to advance the phase in the zero crossing region of the loop transmission, and 3) several notch filters and two low pass filters beyond 1 kHz to attenuate the limiting structural modes (preventing multiple zero crossings) and rolling off the high frequency response. The final controller is constructed as below: 129  Figure 5.3: Axial motion control. a) Controller frequency response function, b) negative loop transmission, c) dynamic stiffness as derived from plant, controller and negative loop transmission measurements.  130  1.6×10−4 s+0.2 3.5×10−5 s+1 2.3×10−4 s+0.2 2.0×10−4 s+0.2 · 4.6×10−5 s+1 4.0×10−5 s+1 2 +214s+1.1×108 s2 +364s+3.3×108 · s2s+21400s+1.1×10 8 s2 +36400s+3.3×108 s2 +5030s+6.3×108 s2 +3770s+9.9×108 · s2 +62800s+9.9×108 s2 +25100s+6.3×108 1 1 1.2×10−4 s+1 6.6×10−5 s+1  · C (s) = 1.36 1 + 1256 s · · · ·  (5.2)  The axial controller frequency response function is shown in Figure 5.3a. The negative loop transmission is evaluated to determine the designed stability margins. The negative loop transmission is defined as NegativeLoopTransmission = C(s)P(s)  (5.3)  Figure 5.3b indicates a zero crossing frequency of 700 Hz, with a 31◦ phase margin. The derived minimum dynamic stiffness represents the disturbance rejection of the closed loop, Dynamic Sti f f ness =  1 + P (s)C(s) P(s)  (5.4)  The minimum dynamic stiffness up to the zero crossing frequency is 342 N/µm at 122 Hz, as shown in Figure 5.3c. The global minimum is 100 N/µm at 2 kHz. From [23, p.240], hammer testing indicates that this flexibility at 2 kHz shows up as 315 N/µm from a disturbance impulse at the tool end. Regulation Results Utilizing the custom I/O board (elsewhere referred to as Nanozoom [23]), the effective resolution of the axial loop is 19 bits giving a motion resolution of 2.86 nm. Through careful shielding and grounding of the system, the base noise floor of the ADE capacitive probe was brought down to 7 nm rms. Figure 5.4 shows the regulation position error with 7.1 nm rms noise. This regulation error is recorded without the rotary amplifier connected, and is very close to the measured electrical noise floor of the ADE probe.  131  Figure 5.4: Axial regulation error. Tracking Results Figure 5.5 shows tracking performance of the axial loop. In Figure 5.5a a sinusoidal reference with a peak to valley amplitude of 1.4 mm at 0.1 Hz is tracked with an axial error for the whole stroke of 19.5 nm rms. Figure 5.5b shows triangle reference wave with peak to valley amplitude of 1.2 mm with a feedrate of 240 µm/s, significantly faster than required for silicon wafer face grinding. The axial error for the 10 second window is 16.3 nm rms.  5.1.2  Rotary Control  Once the axial motion loop was stabilized, and the RBT actuator could be used as a thrust bearing, the rotary motion loop could be designed. The full scale RAS implements a type of sensorless rotary control loop through the utilization of 10 Hall Effect sensors (HES) (originally designed for commutation of the MCS brushless motor) as position sensors. The typical method for high precision speed control is some form of encoder, but the RAS typology limits the available encoder options: a drum type encoder requires lengthening the shaft assembly thus weakening the structure, and a disk type optical encoder requires a fixed axial air gap to function. Details on the sensorless control algorithm may be found in [23, p.242-247]. The final rotary loop had a 10 Hz zero crossing frequency with 38◦ phase margin, and 132  Figure 5.5: Axial motion tracking results. a) Sine wave reference with 1.4 mm stroke (19.5 nm rms error) at 0.1 Hz, b) Triangle wave with 1.2 mm stroke (16.3 nm rms error) at 0.1 Hz.  133  has been tested up to 3000 RPM without dynamic balancing of the shaft assembly [23, p.246]. Speed ripple was found to be limited by the hydrostatic bearing pump, with a final value of 0.293 RPM.  5.2  Rotary-axial Coupling  With both axial and rotary motion loops stabilized, the true purpose of the RAS architecture may be explored through an analysis of rotary-axial motion coupling. In [23], Paone discusses the rotary-axial coupling effect in the full scale RAS prototype as purely an issue with ball centering and probe alignment, and concludes that better centering of the target artifact and better online probe alignment will reduce the axial motion error to less than the already achieved 40 nm peak to valley axial error at 2500 rpm [23, p.250]. While to an extent this is true, the analysis that follows argues that an additional accuracy limitation of the full scale RAS prototype is ball target artifact radial motion registering as a disturbance on the axial position reference signal, zr . The reference signal the axial closed loop attempts to track is zr + zd where zd is the feedback disturbance signal. The data show that this disturbance signal may be the result of the ball target artifact moving in the radial direction as a result of rotary motor cogging bending the shaft. Any motion of the ball target artifact independent of shaft axial motion cannot be compensated for by more aggressive loop control. The data show a 9 cycle per revolution pattern in the axial error signal that is consistent with a radial disturbance from the 9 pole-pair brushless motor. This 9 cpr component of the axial error increases with rotary speed, showing that the error is likely a disturbance on the feedback signal (just like static eccentricity induced artifact error) and not simply a force disturbance. Thus the radial error motion of the spindle at the ball target location is one part of the fundamental limit of accuracy of the rotary axial spindle concept; the other component is the eccentricity induced artifact error discussed in Section 3.1.2. In addition, the magnetic attraction between the armature and the bias magnet increases the magnitude of the radial disturbance initiated by motor cogging; therefore the magnitude of motion induced artifact error is increased due to an inherent property of the RBT actuator topology. The magnitude of this  134  Figure 5.6: Axial regulation error, at 500 rpm and 3000 rpm. increase is a matter for future study. Any additional disturbance on the feedback signal due to the eccentricity induced artifact error is difficult to measure, as it shows up as only a 1 cycle per revolution disturbance. The maximum 1 cpr disturbance temporal frequency occurs at 3000 rpm and is only 50 Hz. At this frequency, the axial controller easily tracks the disturbance signal and it will not appear on the error. From the available data, no 1 cpr component was found; higher rotational speeds and/or a less aggressive controller are required to determine the actual eccentricity induced artifact error. Data in the following analysis was collected by Matthew Paone with the full scale RAS prototype installed in the work table developed at MIT by Gerald Rothenhofer and Alex Slocum. All data is analyzed by this author.  135  Figure 5.7: Time domain FFT of axial regulation error, 500 rpm and 3000 rpm.  5.2.1  Initial Results  A new axial controller was required once the full scale RAS was installed in the worktable for silicon wafer grinding as the axial plant acquired additional structural modes from the frame of the worktable. A significantly less aggressive controller was implemented, with a first zero crossing frequency of 485 Hz, with a phase margin of 45.2◦ ; two additional zero crossings were unavoidable, with a minimum phase margin of 17.1◦ . The sampling rate for the axial controller is set to 80 kHz. From Figure 5.6, it can be seen that at 500 rpm the axial regulation error is 24.1 nm rms, over three times the noise floor of the capacitance probe. This additional error is in part due to the switching noise of the rotary amplifier which, requiring high current switching for operation, emits high amplitude electro-magnetic inter136  Figure 5.8: Axial motion regulation at 3000 rpm with additional 300 nm shift, plotted in spatial domain versus shaft rotation angle. The 9 cpr component can clearly be seen as the dominant error detected at this speed, with a peak-to-valley amplitude of 30 nm; the electrical hash noise component occurs at a much higher spatial frequency, and has approximately 100 nm peak-to-valley amplitude. ference in close proximity to the capacitance probe. The switching frequency will vary with rotational speed. At 3000 rpm, the axial regulation error increases to 46.4 nm rms; it is this increase with rotational speed that will be the focus of this section. Figure 5.7 shows the time-domain Fast Fourier Transform (FFT) of the axial regulation error for two different rotational speeds. There exists line noise at 50 Hz on both signals, with approximately equal amplitudes (note that this setup is 137  Figure 5.9: Axial motion regulation at various rpm with additional 300 nm shift, plotted in spatial domain versus shaft rotation angle.  138  Figure 5.10: Axial motion regulation spatial domain FFT. in Japan, where the main lines operate at 50 Hz, not 60 Hz). From the 3000 rpm signal, the dominant error component occurs at 450 Hz, with a second component at 14 kHz. If the axial regulation error at 3000 rpm is plotted versus the angular position of the shaft, as in Figure 5.8, a pattern emerges. (Note that the polar plot shows the axial error plus an additional 300 nm mean, in order to see both positive and negative components of error.) In the spatial domain it is clear that a low spatial frequency component is the dominant error mode, at 9 cycles per revolution (cpr), with an additional high frequency hash that is not necessarily correlated with a particular spatial frequency. The polar plot shows 99 revolutions at 3000 rpm. The axial errors for a range of rotational speeds are plotted in the spatial domain in Figure 5.9. At lower speeds, it is the hash that dominates the axial error.  139  Figure 5.11: Axial regulation rms error and 9 cpr component of axial error versus rotational speed. From the spatial FFT of the axial regulation errors (Figure 5.10) it is possible to observe that the 9 cpr component exists for all speeds, but is dominated by additional noise sources at lower speeds. For example, at 500 rpm, the 50 Hz line noise is approximately equal to the 9 cpr component. The 50 Hz component appears to be spatially significant within each data set, but shifts with speed. This illustrates that the line noise occurs at a constant temporal frequency, and the ratio of spatial to temporal frequencies is set by rotational speed as ftemporal =  1 RPM ∗ fspatial (Hz) 60  (5.5)  where fspatial is the spatial frequency in cycles per revolution and RPM is the rotational speed in revolutions per minute. A similar effect is seen in terms of the high  140  temporal frequency hash that occurs around 14 kHz, though this high frequency component shifts from 13600 Hz at 3000 rpm to 14500 Hz at 500 rpm. This high frequency component is likely related to rotary amplifier switching directly. This effect must be further explored, as a spatially corellated switching effect is expected at 180 cpr due to the commutation of the motor (10 Hall Effect sensors, with a 9 pole-pair motor, means 180 switching events per revolution). A good job of shielding and grounding the system may have eliminated this interference. Of interest is the fact that no 1 cpr component is detectable on any speed other than 3000 rpm, in which it is merely the 50 Hz line noise reflected to the spatial domain. As stated in the introduction to this section, this does not imply that it does not exist, but rather that it does not register on the axial error signal because the closed loop is easily able to track the low temporal frequency 1 cpr disturbance. Figure 5.11 shows that at operating speeds for the silicon wafer grinder (above 2000 rpm), the 9 cpr error component is dominating the total measured error. A clear upwards trend can be observed, with all speeds exhibiting some 9 cpr component. Below 1500 rpm, the 9 cpr component values obtained from the spatial FFT are less reliable as the error falls below the probe noise floor of 7 nm. In the next section, the source of the 9 cpr component is investigated.  5.2.2  Error Modeling  There are three models that may explain the observed increase in axial regulation error with speed: • As per Section 3.1.2, eccentricity induced artifact error due to target probe misalignment and axial ball target eccentricity leading to disturbance on the feedback signal. • Bending of the shaft leading to variation in the radial position of the ball target artifact independent of the shaft axial motion. This bending can be attributed either to variations in motor rotor magnetic field and back iron, or a combination of armature eccentricity and bias magnet flux segment variation. • Force disturbance on the shaft in the axial direction. This may be due to a 141  Figure 5.12: Motion induced artifact axial error due to ball artifact lateral motion during rotation. constant amplitude force disturbance that occurs at 9 cpr, or a speed dependent axial disturbance force. Artifact Error Models Section 3.1.2 shows that the dominant error mode for eccentricity induced artifact error is 1 cycle per revolution, since the error calculation finds the maximum error between points that are 180◦ apart. Assuming that the shaft exhibits no true axial or radial motion, but only pure rotation, the maximum error term (Equation 3.2) is proportional to the product of the axial probe lateral misalignment, ∆, and the ball eccentricity (offset from shaft rotational axis), ε. Any additional error in this model is due to the surface roughness of the ball target, which is a second order effect (and is very small regardless, as the sphericity of the ball is 10 nm). The spatial domain FFT of the axial regulation errors at any speed do not show any 1 cpr component, except for 3000 rpm where it is 50 Hz line noise reflected to the 142  Figure 5.13: Error feedback disturbance function, e/zd . spatial domain (since the 1 cpr component of 3000 rpm is at 50 Hz). Because the feedback disturbance due to the eccentricity induced artifact error occurs at such low frequencies at operating speeds, the axial control loop is able to track it very closely. Such a disturbance must exist simply as a function of geometry, but it is buried below the noise and cannot be measured from this set of data. In order to detect it, a less aggressive controller must be implemented and/or the spindle run at higher rotational speeds. Another type of error is illustrated in Figure 5.12. The best case scenario is considered in order to obtain the maximum bound of accuracy, with zero initial ball eccentricity and with the center of the probe coincident with the axis of rotation of the shaft. The ball begins in the undeflected position and the true axial position is read by the capacitive probe. Once the shaft begins to rotate, some lateral motion is  143  Figure 5.14: Axial error comparison. a) 9 cpr component of error plotted versus predicted error with 30 nm disturbance on feedback signal, b) force disturbance rejection frequency response function.  144  introduced to the ball target center; here it is assumed that the moment arm of shaft bending is sufficiently long that no significant axial movement of the target ball center takes place. The distance ζ is the total change in apparent axial motion at the centerline of the probe, and is proportional to the disturbance on the feedback signal, zd . The motion induced artifact error, ζ , as a function of lateral motion η is found as ζ = R−  R2 − η 2  (5.6)  where R is the radius of the ball target artifact, 12.7 mm. The fundamental difference between the motion induced artifact error represented in (Equation 5.6) and eccentricity induced artifact error found in (Equation 3.2) is that motion induced artifact error can occur at spatial frequencies other than 1 cycle per revolution and is directly affected by spindle radial motion error at the target ball location. Estimation of Feedback Disturbance Since the system certainly has some axial probe misalignment and ball artifact eccentricity the total disturbance on the feedback signal is found as zd,amplitude = (R −  R2 − η 2 ) +  2∆ε R  (5.7)  where the first term is from the motion induced artifact error, and the second term is from the static eccentricity induced artifact error as discussed in Section 3.1.2. Since the 1 cpr component of axial error is found to be undetectable across all tested rotational speeds, the second term of total feedback disturbance is neglected for the moment and β ζ is taken as the whole artifact error for initial error estimation. As with eccentricity induced artifact error, motion induced artifact error is present but masked at low frequencies. Figure 5.1 shows the block diagram of the axial loop showing feedback and force disturbance on the loop. The frequency response of the axial error e to feedback disturbance, zd , is the feedback disturbance rejection function, and is calculated as  145  e (s) 1 = zd (s) 1 +C (s) P(s)  (5.8)  Since artifact error due to shaft bending may be interpreted as a feedback signal disturbance, a direct comparison of the feedback disturbance rejection function with experimentally measured axial errors will indicate if it is a significant effect. If the error is caused by a force disturbance, the force disturbance rejection function, shown below, must be compared to measured errors. e (s) 1 −P(s) = Fd (s) kI 1 +C (s) P(s)  (5.9)  Figure 5.13 shows the expected frequency response of the axial regulation error to a pure feedback disturbance. From this frequency response function it is clear why the 1 cpr component associated with eccentricity induced artifact error is significantly harder to detect than the 9 cpr motion induced artifact error. For example, at 3000 RPM, based on the feedback disturbance rejection function shown in Figure 5.13, the 9 cpr component at 450 Hz is 6.4 times smaller than the 1 cpr component at 50 Hz if the errors are of the same magnitude. If the feedback disturbance component ζ from radial motion is on the order of the total error of approximately 40 nm (the outer bound possible from the measured results), and the 1 cpr component of feedback disturbance is of approximately the same magnitude, then the detectable 1 cpr component will be 6 nm. Since this is below the noise floor of the probe, the 1 cpr component in this scenario is undetectable. In the frequency range of interest (rotational speeds 500-3000 rpm translate to 74.7-450 Hz 9 cpr disturbances) Figure 5.14a indicates exceptionally good match with a constant amplitude feedback disturbance of 30 nm. In Figure 5.14b, in the frequency range of interest the disturbance force rejection frequency response function indicates that any constant 9 cpr disturbance force would be attenuated as frequency increased. From this fit, it seems likely that the dominant source of the 9 cpr component is a 30 nm disturbance on the axial feedback signal. Because of the 9 cycle per revolution pattern, the error must be induced somehow by the brushless motor, which has 9 pole pairs and will always have some level of either torque or radial or axial force disturbance at 9 cpr.  146  For higher speeds, the larger detected 9 cpr error is a function of the inability of the axial loop to track the higher temporal frequency feedback disturbance. Thus the true shaft motion will be less than the detected error and therefore the true rotary-axial coupling at 3000 rpm is less than 30 nm. Based on (Equation 5.6), and assuming that the sensitivity of the axial measurement to lateral motion, β , is close to 1, the minimum radial motion required for a 30 nm feedback disturbance is calculated as follows: η= =  R2 − (R − ζ )2 12700µm2 − 12700µm − 30x10−3 µm  2  (5.10)  = 27µm This is excessive considering the measured radial ball eccentricity ε is 1.5 µm. As discussed in a later section, the radial disturbance force will be on the order of hundreds of Newtons, which is unlikely. During ball centering, no dominant 9 cpr component was detected in the radial direction, though an 18 cpr component was seen. However, the amplitude of radial position variation was much less than 1 µm. In addition, it is likely that the sensitivity β is significantly less than 1 when the ball target and probe are well centered. Therefore, though the model shows that radial motion will induce axial error and Figure 5.14a shows that a good match is found between experimental axial error measurement and predicted error from a constant feedback disturbance, the lateral motion model alone requires an unfeasible radial motion. Estimation of Axial Force Disturbance A constant amplitude 9 cpr force disturbance cannot explain the measured axial 9 cpr error, as shown by comparing the measured error trend (which appears to depend on the square of the rotational speed) and the force disturbance rejection function in Figure 5.15b (which attenuates error with increasing speed in the frequency region of interest). In order for the measured 9 cpr axial error to come from an axial disturbance force, the disturbance force Fd must increase with rotational speed, and therefore with temporal frequency. Based on the known disturbance force rejection frequency response shown in Figure 5.14b, the magnitude of 9 cpr 147  Figure 5.15: a) Predicted axial force disturbance amplitude required to produce measured 9 cpr axial error (based on (Equation 5.9)) for each rotational speed; b) mean torque command to brushless DC motor for each rotational speed. force disturbance that is required to produce the 9 cpr error measured for each rotational speed can be determined. From Figure 5.15a, the maximum required axial force at 3000 rpm is 17 N, with a cubic dependence on frequency (60 dB/decade slope on the magnitude, plotted versus frequency). As discussed in the next section, this cubic dependency on frequency implies that the source of the rotary-axial coupling error is not purely an axial force dependence. The axial force disturbance magnitudes required to produce the measured error are fairly low: a 9 cpr, 17 N disturbance force at 3000 rpm is sufficient to produce 40 nm of detected axial error. The most likely source for this error is the axial thrust force developed by the brushless motor as a consequence of the stator coil geometry. While the motor produces torque, an axial force is induced due to the slant angle of the coils and back iron slots (at 2 degrees, as shown in Figure 5.16). This indicates that the axial force should be proportional to the mean torque developed at the motor. Since the measured torque is not available, the torque command 148  Figure 5.16: Brushless motor stator, showing slant of coil geometry. is considered approximately equal to the mean motor torque in Figure 5.15b. Comparing the predicted 9 cpr axial force disturbance, which has a cubic dependence on rotational speed, and the mean torque command which has a linear dependence on rotational speed, it is obvious that any 9 cpr axial force disturbance cannot be wholly attributed to the axial thrust force generated during normal brushless motor operation.  5.2.3  Sources of Rotary-Axial Coupling  While (Equation 5.6) shows the effects of a constant amplitude 9 cpr radial displacement of the target ball artifact, it does not tell us the source of the radial disturbance motion. Whatever the ultimate cause of the artifact lateral motion, to produce the measured 9 cpr error it must entail the rotation of the shaft to produce some radial motion as a function of angular position (shown greatly exaggerated in Figure 5.17). Two possible sources are suggested: 1) the bias flux magnet, which has 18 magnet segments which may produce a radial disturbance force that varies 149  Figure 5.17: Radial shaft motion leading to eccentricity shift of target artifact; maybe due to brushless motor cogging and/or bias magnet radial force on the armature. with shaft angular position given a non-axisymmetric armature, and 2) the brushless motor, which has 18 poles and 9 pole-pairs. In order to produce a feedback disturbance as per the motion induced artifact error model, the dominant source must be capable of producing a radial disturbance force with a pattern of 9 cycles per revolution. The armature can produce radial motion as a function of rotational angle by itself only if two conditions are met: 1) the armature is non-axisymmetric, and 2) there exists bias flux ripple as a function of rotational angle θ . If the armature is axisymmetric, no amount of flux ripple from the bias magnet will elicit a radial force dependent on angular position: the shaft will merely always bend in one direction, towards the maximum net flux. The non-axis-symmetry is  150  Figure 5.18: Bias magnet radial force estimation. 18 cpr component is visible, but not 9 cpr. on the order of 20 µm, based on the tolerances demanded during manufacturing (see Section 3.2); as the nominal air gap between armature and bias magnet is 500 µm, this gives a possible variation of 4%. This may or may not be enough to produce detectable force variation as the shaft rotates, depending on how far out of alignment the individual magnet segments are compared to the armature out-ofround error. While the in situ magnetic flux density measurement shown in Figure 4.22 was unable to measure the bias flux ripple, it can be assumed that the spatial frequencies of flux variation are unchanged from the free air measurement shown in the same figure. Given the free air measurements, the reluctance based force between armature and bias magnet can be estimated, at least in terms of the spatial variation, by the following relationship: 151  Figure 5.19: Motor stator eccentricity schematic diagram.  Fradial ∝ B(θ )2 − B(θ − 180◦ )2  (5.11)  which describes net radial force generation on the armature as the square of radial flux density on one element subtracted from the square of radial flux density at a point 180◦ away. Figure 5.18 shows the radial force estimation versus rotation angle. While a clear 18 cpr component is visible (18 force spikes, 18 force zeros), no 9 cpr force disturbance is possible from this analysis. Therefore, the bias magnet alone cannot create the measured error motion, nor can it be the dominant source. The mechanism for radial force ripple from the permanent magnet brushless  152  Figure 5.20: Gap measurements between brushless motor stator and rotor. a) Gap measurement for two measurements, where Measurement 2 is taken with the shaft rotated by 180 degrees from the rotor position of Measurement 1; b) repeatability between measurements. motor rotor is variation in back iron and magnet geometry as a function of angle. From Figure 5.19, it can be seen that in order to exert a net radial force on the motor rotor, the brushless motor stator (including back iron) must have some radial eccentricity relative to the center of the rotor, δ , and some radial flux density entering the back iron. Only the flux components normal to the inner face of the stator back iron can produce radial force. In order to center the motor stator relative to the rotor on the shaft, the gap between the motor rotor and the inner bore of the motor stator was measured at 20 degree increments. The gap measurement was done using feeler gauges. Two sets of gap measurements were taken, with the second set of measurements taken 153  Figure 5.21: Variation in gap between brushless motor stator and rotor on shaft. Measurement 2 is taken with the shaft rotated 180 degrees from the rotor position of Measurement 1. after rotating the shaft 180 degrees relative to the original shaft position in the first measurement. This was done in order to cancel out any eccentricity of the permanent magnet rotor relative to the shaft rotational axis. Figure 5.20a shows the two raw gap measurements, and Figure 5.20b shows the repeatability of the gap measurement at each angular position to be approximately 10 µm. From the variation of the gap measurements from the mean gap dimension from each set of measurements, the gap variation amplitude is approximately 30 µm as shown in Figure 5.21. Therefore the estimated eccentricity is approximately 30 µm with an uncertainty of 10 µm. However, it should be noted that this measurement was between the outer aluminum housing of the permanent brushless rotor and the inner coils of the motor stator. The actual magnetic field variation was not measured.  154  Figure 5.22: a) Schematic showing permanent magnet positioning; b) representative radial flux density Bn and representative radial force Fn . 155  Figure 5.23: Ball target centering, showing radial gap variation with additional 3 µm mean. Installation error of the magnets, variation in remanence magnetization and eccentricity of the permanent magnet array center to the center of the motor rotor will provide additional error. The relatively small gap variation (30 µm for a 1 mm gap) means that the force variation cannot be large. During installation, the motor rotor and shaft were installed in the spindle without radial support, indicating fairly low radial negative stiffness between brushless motor stator and rotor. There are 9 permanent magnet pole pairs (as shown in Figure 5.22a) that provide a radial flux in the n direction that varies with angular position θ . The radial flux pattern is therefore 9 cpr as indicated by the representative radial flux Bn in Figure 5.22b. However, this translates to an 18 cpr radial force, Fn , since normal stress magnetic force generation is always attractive (in the same direction). This 156  is shown in Figure 5.22b: while the radial bias flux has a 9 cpr component, the force on the rotor due to the misaligned back iron of the stator is twice the spatial frequency spatial frequency of the magnetic pole-pairs at 18 cpr. The radial force ripple therefore should appear as 18 cpr in the shaft spatial domain. In order to minimize the ball target eccentricity, the ball was centered to the shaft rotational axis (prior to the installation of the bias magnet) by measuring the radial gap between the ball and a stationary capacitive probe while the shaft was rotated as shown in Figure 3.28c. Since the bias magnet was not installed at the time, this data will show purely the effects on radial motion of the brushless motor and hydrostatic bearings, as well as the live center used as a temporary thrust bearing. The ball centering results are shown in Figure 5.23 and illustrate an 18 cpr radial motion component. This is consistent with the normal stress generation model, and shows that the effect of radial motion due to the brushless motor radial disturbance force is not the dominant cause of the 9 cpr error. Figure 5.23 also shows that a 90 cpr component, which matches half the number of switching events per revolution (for 10 Hall Effect sensors and 9 pole-pairs, 180 current switching events are expected per revolution). The radial stiffness due to the hydrostatic bearings at the rotor is 79.6 N/µm; at the armature it is 34.4 N/µm. From this fact, two major problems are apparent with the radial motion theory: 1) the magnitude of radial force at the brushless motor rotor required to produce 27 µm of motion at the ball target is 1360 N, and 2) if the radial motion is due purely to a 9 cpr radial force variation with constant magnitude, the radial motion must decrease with increasing speed due to inertia. Radial motion will certainly cause axial motion error and appear as a feedback disturbance; however, the magnitude of the measured axial error precludes the rotor radial force from being the dominant source of rotary-axial coupling, and the fact that the measured lateral ball centering An interesting consequence of the radial negative magnetic stiffness between the armature and bias magnet is that any radial motion caused by brushless motor cogging is exaggerated by the attraction of the magnet to the armature. The sequence is as follows: the negative magnetic stiffness is a function of the square of the air gap distance; since the radial motion caused by the brushless motor cogging bends the shaft slightly, the air gap decreases, increasing the attractive load 157  between armature and magnet and bending the shaft even further. That is, the shaft bending forces the armature position to be slightly non-axisymmetric. Thus while it is most likely brushless motor cogging that dominates the radial shaft motion, the attractive force between armature and bias magnet may provide an additional component of radial error motion due to variations in radial bias flux and an initial displacement from brushless motor cogging. This additional error motion must exist, but its magnitude is to be determined by spindle radial error characterization with and without the bias magnet. This will have to be done with an additional thrust bearing. Whatever the source of radial motion of the ball target artifact, the conclusion is clear: the radial motion of the ball target artifact must be constrained much closer to the artifact itself. The ultimate cause of this disturbance is the length of the moment arm between shaft and possible sources of radial force. One solution that may be feasible for the next iteration of the RAS design is to shorten the moment arm between possible sources of radial disturbance and the fluid radial bearing, using an axially shorter brushless motor and a shorter, fatter version of the RBT actuator. Another solution is an additional radial fluid bearing between the actuator, motor and ball target. This is not ideal, since it will make the actuator assembly very complex compared to the current design. It would require an extra alignment step between the additional radial bearing and the fluid journal bearing supporting the tool end of the shaft, with the axial actuator in place. Shaft rotation is necessary for radial bearing alignment, but with the actuator in place the axial control loop must be operational in order for the shaft to rotate because there is no other thrust bearing in the system. In addition, the total shaft length will increase, lowering the frequency of the shaft structural modes, thereby limiting achievable axial motion bandwidth.  5.3  Summary  The closed loop motion control characteristics of the independent axial and rotary loops were discussed, and finally rotary-axial motion coupling was analyzed. The final performance specifications achieved by the full scale RAS are listed in  158  Table 5.1. One of the fundamental results found from rotary-axial coupling analysis is that one of the basic limitations of axial accuracy is spindle radial error motion. At this time, no conclusions can be reached with regards to the ultimate source of the rotary-axial coupling. The motion induced artifact error model which leads to a feedback disturbance, though correct, cannot be the dominant source of the error unless there is some amplifying effect that has not been considered that will significantly increase the magnitude of radial motion or the sensitivity of the probe to lateral motion. A pure axial force disturbance can be ruled out, as the only source of such a disturbance (axial thrust force generated by the motor) does not increase with speed fast enough to be the cause of the required force disturbance. Further investigation must be conducted. One useful test will be to measure the input currents to the three phase brushless motor and reconstruct the generated torque: this should be able to determine if the 9 cpr component of the generated torque increases with the cube of rotational speed instead of directly with rotational speed.  Table 5.1: Large Scale RAS Achieved Dynamic Performance Specifications Parameter  Target Specification  Axial Stroke Axial Motion Resolution (rms positioning noise) Axial Motion Bandwidth (-3 dB) Cutting Speed Static Controlled Axial Stiffness Rotary-axial Motion Coupling at 3000 RPM  1.5 mm 10 nm 1000 Hz 3000 RPM ∞ N/µm <10 nm  159  Achieved Specification 1.5 mm 7.1 nm 800 Hz 3000 RPM ∞ N/µm <30 nm  Chapter 6  Conclusions and Future Work A full scale rotary-axial spindle using a radially-biased thrust bearing/actuator for axial positioning has been developed and experimentally tested. The key performance results are: 5000 N measured load capacity, with a predicted worst case load capacity of 4900 N; 7 nm rms axial positioning error over an axial stroke of 1.5 mm running the axial drive only (due primarily to electrical noise on the feedback signal); axial positioning loop transmission zero crossing frequency of 700 Hz; minimum controlled dynamic stiffness of 340 N/µm (below the zero crossing frequency); and less than 30 nm of rotary-axial coupling error at an operating speed of 3000 rpm, due to shaft bending from an angular position dependent radial disturbance force from the brushless motor. Manufacturing of RBT Actuator From Chapter 3, the key lessons for constructing the RBT actuator include the need for better centering features for the actuator sub-assemblies, ball target artifact and axial probe. The limit in centering of the actuator coil sub-assemblies was the misalignment between stator segments; a monolithic stator ring can be turned to a much better cylindricity form error than is possible using manual alignment of individual stator segments. Another alternative is epoxying together multiple blocks of the powdered iron material together and machining the constructed composite as a whole stator ring. Centering the bias magnet ring is limited by the skewing between magnet seg160  ments due to repulsive forces between them as well as excess epoxy buildup on the inner bore of the magnet ring. This may be improved by applying a polymer layer to the aluminum hub used for magnet installation, with the combined diameter of the polymer and aluminum hub closely matching the required inner diameter of the bias magnet ring. With the polymer layer in place, the centering hub may be left in place while the epoxy cures and removed afterwards, minimizing the induced magnet skew and excess buildup by preventing intrusion into the inner bore area. There is still the issue of concentricity of the inner magnet bore and the enveloping stator ring: careful setup work must take place to insure they are concentric. The target ball centering was in large part limited by the non-repeatability of centering as the bolts were tightened to the shaft, as well as thermal fluctuations from day to day at the UBC setup. The lateral alignment of the axial probe was limited by the Coulomb friction developed between the probe mount and the rear coil housing, leading to jerky movement as the set screws were turned to position the mount. A more deterministic sliding bearing surface between the axial probe mount and the rear coil housing should be developed. An alternative is to simply isolate the capacitive probe from the spindle entirely: the capacitive probe may be mounted to a frame that is independent of the RAS but attached to the wafer holding table. This would have the advantage of measuring the relative motion between the work table and the shaft rather than the relative motion between the RAS and the shaft; this is one step closer to the ideal of measuring the relative motion between workpiece and tool. Static Load Capacity From Chapter 4, the RBT actuator analysis is generally well matched by experimental results. The initial finite element predictions were the result of incorrect inputs to the model rather than incorrect modeling. The dominant nonlinearities are the result of material saturation and not armature position, just as expected by the intuition developed from the lumped parameter analysis. In future iterations of the RBT actuator, efforts should be made to characterize the soft magnetic material for stator and armature prior to designing the actuator; this will be critical for any production run of the RBT actuator, and batch to batch variations between soft  161  magnetic material lots will have to be minimized. A better in situ measurement of the bias flux density variation should be obtained in order to help characterize the possibility of variation in radial load on the armature as a function of angular position, and a non-axisymmetric finite element analysis should be conducted to determine the precise effects of a segmented bias magnet. Contact sealing of the shaft should be eliminated in future versions of the RAS: the rotary lip seals provide a small axial disturbance force which is not very importance since the control loop will compensate, but they also greatly increase heat generation in the hydrostatic bearing. This additional heat generation greatly degrades performance unless a better cooling architecture is devised. Dynamic Error and Rotary-Axial Coupling From Chapter 5, the key lesson to be drawn from the discovery of motion induced artifact error is the necessity of minimizing the distance between radial support bearings and the axial position sensing point on the shaft in order to limit the radial motion of the axial position sensing point. Sensing the shaft axial position with a sensor that is collinear with the axis of rotation is in many ways ideal: the sensor is relatively insensitive to the rotational degree of freedom because variation in the sensed area is small, and the effects of rotation become a second order effect. Since it is impossible to measure the axial position at the tool end while remaining collinear with the axis of rotation without cutting a hole in the work piece and measuring upwards (whilst being covered with cutting fluid and chips), the shaft position must be sensed from the opposing end. However, the assumption of insensitivity to rotation assumes a shaft with infinite bending stiffness and infinitely stiff radial supports with no radial error motion as a function of angular position. In the real case, both finite bending stiffness of the shaft and finite radial support stiffness will allow radial disturbance forces to translate to radial motion of the axial position sensing point, and thus to a disturbance on the sensed axial position signal which is interpreted by the axial control loop reference command though it is a disturbance on the feedback signal. This error cannot be compensated for by more aggressive controller design, only hidden. Alternatives should be explored and a method found to measure the tool end axial position that will not introduce  162  even worse position sensing artifact error. In addition to the insight into the nature of axial error motion in rotary-axial architectures, the effect of radially biased electromagnetic actuators on existing radial error motion should be studied in further detail. The instability of the armature in the radial direction insures that it will increase any shaft error motion due to any source whatsoever; the variation of magnetic flux due to segmentation means that the radial stiffness itself will vary with angular position. This is a fundamental property of radially-biased actuators using a segmented bias magnet. It should be noted that this problem may easily be solved by utilizing a purely radially biased magnet (meaning no variation in radial stiffness due to bias flux variation with angular position), though the cost of producing low volume radially biased magnets may be prohibitive. The ultimate cause of the large magnitude of axial error needs to be investigated further. In particular, the generated torque must be analyzed in order to determine if an axial force disturbance due to the thrust force generated by the brushless motor can be the root cause of the measured 9 cpr axial error. This problem cannot be solved without a clearer understanding of its source. Future Work Future work on the full scale RAS prototype should undertake the following: • Calibrating the axial motion of the tool end of the shaft assembly. This should determine the amount of true rotary-axial coupling at the operating speed of 3000 rpm of both motion induced artifact error and static eccentricity induced artifact error. • Measuring the generated torque at various speeds. The torque command does not adequately reflect the generated torque. • A full range static load test, using the maximum capacity of the linear amplifier. This will experimentally measure the effect of material saturation at the maximum loads, and validate the expected worst case axial load. • Characterization of spindle error motion with and without the bias magnet in place. This may be possible given another thrust bearing being mounted to 163  the tool end of the shaft as was done during ball centering, and by removing the rear coil assembly. This will conclusively show the effect of radial biased actuators on spindle radial error motion. • A cutting test on a 450 mm diameter silicon wafer. This is the final test of the full scale RAS, and the ultimate determinant of the success of the design.  164  Bibliography [1] Z.J. Pei, Graham R. Fisher, and J. Liu. Grinding of silicon wafers: A review from historical perspectives. International Journal of Machine Tools and Manufacture, 48:1297–1307, 2008. → pages vii, 1, 2, 3 [2] X. Lu, M. Paone, I. Usman, B. Moyls, K. Smeds, G. Rothenhofer, and A.H. Slocum. Rotary-axial spindles for ultra-precision machining. CIRP Annals-Manufacturing Technology, 58(1):323–326, 2009. → pages vii, 14, 16, 17, 18 [3] Soft magnetic composites from Hoganas metal powders-SOMALOY 500, Hoganas Manual, 1997. → pages viii, 31, 32, 39, 98, 108, 110 [4] D.C. Meeker. Finite element method magnetics (femm), version 4.0.1 (02 april 2007 build). → pages viii, xii, 39, 40, 41, 117, 120 [5] ITRS. Itrs 2009 edition executive summary. Technical report, International Technology Roadmap for Semiconductors, 2009. → pages 1 [6] Robert Gerber. 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