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A kinematically-coupled magnetic bearing calibration fixture
A kinematically-coupled magnetic bearing calibration fixture Tony Poovey, Mike Holmes, and David Trumper Precision Engineering Laboratory, University of North Carolina, Charlotte, NC, USA A calibration fixture has been developed that measures the force of an electromagnetic actuator as a function of coil current and target separation gap. The measured force characteristics are used to linearize the dynamics of magnetic bearings controlled by these actuators. The calibration fixture mechanical design and test procedure are described. An extremely stiff gothic arch ball-and-groove coupling fabricated from silicon carbide is used along with a lightweight target platen constructed from composite materials to place the resonant frequency of the calibration fixture at 1.8 kHz. This permits the characterization of magnetic bearing elements at frequencies up to 1 kHz. Experimental results from force testing a magnetic bearing element are presented along with the frequency response of the calibration fixture and suggestions for further research. We also present the design of an associated linear current amplifier designed specifically for I0 w noise and a high negative current slew rate.
Keywords: force testing; magnetic bearings; kinematic couplings
Introduction
© 1994 Butterworth-Heinemann
(NASA) annular momentum control device. 4 This design is only useful for forces up to 45 newtons (N). Furthermore, the selection of strain gauge-based load cells limits its use to low-frequency force measurement due to the low stiffness of these load cells. Knight et al. developed a calibration fixture that measures the deflection of a long rod subjected to the attractive force of a magnetic journal bearing element in order to measure forces up to 400 N. ~ This method of force measurement is only suitable for quasistatic forces, due to the relatively low-frequency resonance of the bearing mass on the long shaft, and is not capable of micron-scale positioning accuracy. The magnetic bearing calibration fixture described in this article attempts to achieve high resonant frequencies through the use of kinematic design principles. Additionally, the actuator air gap is determined with micron-scale precision. The electromagnet bearing elements designed for our linear magnetic-bearing positioning stage 1operate at separation gaps ranging from 10 to 650/~m with forces as large as several hundred newtons. In addition, wide-bandwidth force authority is required from the actuators so that the magnetic bearing that they control can achieve positional accuracies on the order of nanometers. Therefore, the calibration fixture resonance needs to be located at as high a frequency as possible in order to allow broadband
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99
Magnetic bearings represent a promising approach for achieving positioning accuracies on the order of nanometers in precision motion control stages. 1 In these stages, an accurate model of the actuator force as a function of coil current and air gap is required. This model is used within the bearing controller to linearize the actuators in real time. 2 Classical magnetic circuit theory can accurately model the magnetic bearing actuator force characteristics at low flux densities. However, at higher flux densities the actuator force characteristics deviate from first-order theory due to saturation. Therefore, experimental measurement is required. This article describes the development and testing of a magnetic bearing calibration fixture designed for such experimental measurement of the actuator force characteristics. 3 There are other magnetic bearing calibration fixtures currently in use. Groom and Poole developed a device for characterizing actuators for use on the National Aeronautics and Space Administration
Address reprint requests to David Trumper, Mechanical Engineering Department, Massachusetts Institute of Technology, 77 Massachusetts A venue, Cambridge, MA 02139, USA.
Poovey, Holmes, and Trumper: Calibration fixture characterization of the actuator frequency response. To address these needs, the magnetic bearing calibration fixture described herein has been developed with the following objectives: • Measure the electromagnet actuator force from minimum to maximum core flux density at separation gaps ranging from 10 to 1,000/zm • Support a peak force capability on the order of hundreds of newtons • Allow scaling to arbitrary force levels by an appropriate selection of materials, coupling geometry, and positioning hardware • Demonstrate the capability to study actuator torques where the torque vector lies in the plane of the pole face • Permit the adjustment of the target separation gap as well as the rotation of the target relative to the actuator • Use metrology components that are independent of the load-carrying components in order to accurately measure the target separation gap in addition to rotations about vectors lying in the plane of the actuator pole face • Allow testing of a variety of actuator sizes interchangeably • Measure actuator frequency response through several kilohertz This report details the means by which these goals have been achieved and is organized as follows. First, the mechanical components of the calibration fixture are described. This is followed by a description of the design of a linear power amplifier, which is used to energize the magnetic bearing element. Next, a brief description of the testing procedure is presented along with the system control and data acquisition scheme. This is followed by a summary of the most significant design improvements implemented during the development of the calibration fixture. Finally, the results of force testing a magnetic bearing actuator are given along with suggestions for improving the performance of the system.
Mechanical components The magnetic bearing calibration fixture consists of an aluminum fixture body and platform, three adjustable platen support columns with ball-andgroove kinematic couplings, three capacitance probes, and a target platen as shown in Figure 1. The electromagnet housed in the body is made from E-shaped nickel-iron alloy blanks laminated and wound with 400 turns of #26 wire and epoxied into a mounting cartridge. The electromagnet cartridge is mounted inside the calibration fixture body with its pole faces exposed upward toward the target platen. A laminated nickel-iron target is attached to the underside of the target platen such that it is attracted toward the electromagnet. Various-sized actuators and targets can be accommodated 100
through changing adapter plates in the body and the platen, respectively. Located at 120° intervals around the perimeter of the fixture body are the three support columns, each consisting of a kinematic ball/groove coupling mounted on top of a piezoelectric load cell. The hemisphere mounted to the top of each !oad cell makes two points of contact in each of three grooves; therefore, the three couplings provide ex actly six constraints on the six platen degrees of free. dom. This type of kinematic coupling, commonly ~eferred to as a Maxwell mouuL has the ability to provide submicron positional repeatability if prop erly designed. 6 The micrometers are used to adjust the height of the platen above the electromagnet, whereas the load cells measure the attractive force between the electromagnet and the target platen. Three capacitance probes mounted at the top p e r i m eter of the fixtu re body are used to sense the distance between the electromagnet and three datum pads on the target platen. These probes have a repeatabil ity of better than 1/zm and linearity errors on the on der of 1/zm out of a total range of ! m m An interesting feature of the design is that the mechanical accuracy of the fixture is determined entirely by a set of drilling and boring operations that can be accomplished while working from o~iv the top side of the fixture body and thus with ,-, single machine set-up. These operations are the boring of the holes for micrometers, capacitance probes, and the pins that locate the actuator in the housing. It is only these dimensions that are critical because the iron actuator [arge[ located on the, platen is made wider than the pole faces of the actuator. Thus, the body-referenced location and magnitude of the actuator force are indifferent to lateral motions of the target ,within some iimited range, and thereby to positioning errors of the kinematic mounts or target, which are less than ~,he excess size of the target. This means that the platen need not be tightly toleranced because all tight t o i erances are constrained within a single part and are readily established. The capability for kinematic adjustment of the target platen position allows the measurement of both force and torque as a function of target separa.. tion gap. Torque can be generated as a result of the pitch and roll rotation of the target platen with respect to the electromagnet, which can occur when a magnetically suspended object rotates. This carl be particularly important when dealing with electromagnets that have dimensions that are a significant fraction of the characteristic dimension of the sus. pended body. In the present case, because the actu~ ator dimensions are small relative to the anticipated suspension dimension, the torque term is not studied.
Linear power amplifier The characterization of the electromagnetic actuators requires a current source capable of driving the APRIL t994 VOL i6 NO :
Poovey, Holmes, and Trumper: Cal/bration fixture Target Platen 165 mm
Kinematic Coupling
Load Cell
Micrometer Head
Figure 1
Solid model representation of the calibration fixture
electromagnets into saturation. The current source must also allow rapid reduction of the coil current to zero. The linear power amplifier shown in Figure 2 has been designed for low noise characteristics as well as high negative current slew rate capability. Circuit protection is provided by fuse F1 and the flyback network. Fuse F1 is used to protect the coil from overheating in the event that the current loop goes out of control (e.g., Q2 develops a drain to source short circuit) and the flyback network allows coil current to continue when O2 is turned off suddenly, thereby protecting Q2 from excessive drain to source voltages. The circuit elements which comprise the flyback network are enclosed by a dashed line in Figure 2. The function of this network is to act as the series combination of a diode and a high-power zener. ? In earlier work, where the power amplifier used only a diode for the flyback network, insufficient power amplifier negative current slew rate capabilities were found to contribute to instabilities of the magnetic suspension. 8 This compound zener replaces the earlier diode in order to provide larger flyback voltages, thereby increasing the achievable negative current slew rate. The function of the elements of the compound PRECISION ENGINEERING
zener are described as follows. Diode D3 prevents current from flowing through the flyback network under normal operating conditions when the drain of Q2 is more negative than the +40-V power supply. This diode also serves to protect the base-emitter junction of Q1 from reverse breakdown. Under conditions when the drain of Q2 is more positive than 40 V, the combination of D1, D2, R4, and Q1 acts as a 40-V zener. This compound-zener circuit has the advantage that the majority of the flyback energy is dissipated in Q1, which is a 2N3055 power transistor mounted on a heat sink. This transistor is capable of dissipating on the order of 100 W, whereas it is difficult to find similarly rated zener diodes. Approximately 0.5 mA must flow through R4 before Q1 will turn on. For this to occur, given the choice of zeners D1 and D2, the drain of Q2 must be approximately 40 V more positive than the +40-V coil power supply. This condition will only occur when the attempted coil current negative slew rate is greater than about 40/L, where L is the coil inductance. Expressed another way, the amplifier is linear for coil voltages bounded by +40 V, even though the current applied by the amplifier to the coil is of necessity unipolar. Thus, the character101
Poovey, Holmes, and Trumper." Calibration fixture
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istics of this amplifier are well matched to the application because the variable reluctance actuators apply force that is independent of the sign of the drive current and thus do not require a bipolar cur rent source. However a bipolar voltage source is required in order to allow rapid current changes. This amplifier is capable of applying the required bipolar voltages to the electromagnet coil while requiring only a single unipolar 40-V power supply and only one high-power control element. Due to its simplicity, the amplifier uses only several dollars of parts, as compared with a power op-amp-based design, which would cost between $50 and $100. The power op-amp design also requires an additional 40-V power supply costing hundreds of dol. lars. When one considers that a typical magnetic bearing system may require on the order of 12 or 16 channels of amplification, the economic advan tages of our design are significant. The snubber capacitor C7 was added to the circuit and empirically tuned to eliminate oscillations of the loop at a frequency of several hundred kilohertz. An analytical investigation of these oscit lations has not been undertaken, but they clearly result from the interaction between the coil impedance, the transistor dynamics, and the dynamics of the op-amp current loop. In comparison with the earlier switching amplifier design, 8 as well as with the earlier simple diode flyback, 2 the negative current slew rate capability is greatly improved. In addition, the switching amplifier generates noise difficult to shield. This noise is especially undesirable in high-precision systems; thus a linear amplifier has been chosen. The capa102
bilities of this amplifier can be ~eadsly scaled ~r~ w:.,q! age or current for other magnetic bearing systems simply by choosing an appropriate power FET fo~~ Q2.
Test procedure in a real-time control appiicat~o~, ;t is desirable h:_; linearize the nonlinear electromagnet force-current separation-gap relationship in order to determine the coil current required to obtain the desired actL~a tor force at a given gap. 8 The data collected fro~., the calibration fixture via data acquisition hardware and test software allow the creation of a two-dimensional look-up table that permits such a lineariza tion, The process of acquiring these data is described as follows. A block diagram for the system is shown ~ Figure 3. The user first initializes the actual air gap between the electromagnet face and target using a software-assisted procedure described later. Then, through a digital-to-analog converter, the test software outputs a sinusoidal control voltage that oscii-lates between 0 and 10 V. This sine wave is fed to the power amplifier, which outputs a current pro portional to the input voltage. The power amplifier output drives the electromagnet, which applies a corresponding force to the target platen. The force f(I, G) is a function of input current /and target separation gap G. The load cells transduce this force into a proportional charge, which is amplified and converted to a voltage bythe charge amplifier. Both the electromagnet current and the load cell output voltage are logged into the '386-based computer APRil !994 VOL i 6 NQ 2
Poovey, Holmes, and Trumper: Calibration fixture
Figure 3
Block diagram of electromagnet testing system
via analog-to-digital converters for storage. After data are taken for one separation gap, the gap is adjusted to a new value and the test procedure is repeated. Next the acquired data are used to build a two-dimensional look-up table of force versus current at the designated target separation gaps. The look-up table is stored in the form of an ASCII file for ease of manipulation by third-party software. The accuracy of the look-up table is verified via another test routine that uses the look-up table to apply a ramp of desired force to the actuator using the same test configuration as depicted by Figure 3. The result of this test is a plot of desired force versus actual force in which the degree of linearity determines how accurately the actuator force can be controlled. This verification routine sequences through many iterations, allowing the user to change the air gap to obtain data that test the accuracy of the look-up table over a desired range of operating air gaps. It is desirable to have the data table resident in memory yet separate from the test routine because this allows many different applications to access the data table independently. To accomplish this, one application program allocates memory for the data table, points user interrupt vector 0x65 at the data table, and then terminates and stays resident (TSR). Another application program opens and interprets the ASCII data files created by the characterization software and fills in the data table. Next the data table is available to any application program and can be refreshed at any time. Because the capacitance probes and the micrometer shafts are not axially aligned, motions of a single micrometer couple into all three probe position readings. The task of adjusting the three micrometer head displacements to obtain a specific target separation gap via the three capacitance probe readings is simplified through real-time coordinate transformations (1-2) described below. 2 2
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Figure 4 illustrates the relative location of the three micrometer heads at positions X 1, X 2, and X 3, and the three capacitance probes at locations C1, C2, and C3. In operation, the bearing is limited to milliradian scale rotations. We therefore only characterize the actuators for small rotations. Under the assumption of small rotations, the micrometer head displacements can be related to the capacitance probe readings by the linear transformations shown in Equations (1) and (2). In Figure 4, a is the distance between C1 and X 2 and is approximately 70 mm. The indicated lever arms are used to calculate the motions at C1, C2, and C3 due to the incremental extension of micrometer X 1. This yields the first column in the matrix in Equation (1) and the first row of the matrix in Equation (2). The remaining columns and rows are developed similarly. To assist in obtaining the desired air gaps, the transformation in Equation (2)is performed by the '386based computer in real-time on the capacitance probe readings in order to map the position error measured at the capacitance probes into equivalent displacements of the micrometer heads. The results of the transformation X~, X2, and X3 are displayed as three bar graphs on the computer monitor. On these bar graphs, the micrometer adjustments do not interact, thus greatly simplifying the adjustment of a desired air gap. The bar graphs are linear for errors between plus and minus 1 /zm and are logarithmic for plus and minus larger magnitude errors up to 1 mm. Thus, the error can be adjusted to null with submicron resolution, but is on-scale even for very large magnitudes. The test software uses one DT-2823 data acquisition board for the measurement of the three load cell signals and for control of the actuator current. 9 Another DT-2823 data acquisition board is used to measure the three capacitance probe position signals. The computer running the test software and containing the DT-2823 boards is a 20-MHz 80386 103
Poovey, Holmes, and Trumper: Calibration fixture X!
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with an 80387 coprocessor. This computer is running the DOS 5.0 operating system and the test software is written utilizing Microsoft C version 6.0 because this compiler supports in-line assembly programming. Care has been taken to maximize processor efficiency during critical times in order to ensure data integrity and timing accuracy. This has been accomplished by using hardware direct memory access (DMA) to transfer data readings di m rectly into processor memory. Also, all interfacing software to the DT-2823 boards and all core functional test codes are written in assembly language. Finally, to allow faster execution, all math functions are performed by programming the 80387 coprocessor directly. By developing the software in this way, acquisition rates of up to 10 kHz can be achieved. This sample rate translates into a test waveform output frequency of up to 10 Hz while simultaneously collecting 4,000 data points per half cycle of the sine wave. Thus, the data at a given separation gap can be acquired in about 0.1 second. This speed proves to be more than sufficient to allow rapid actuator characterization.
Calibration fixture improvements The calibration fixture has undergone a number of revisions, with each design change adding some incremental improvement to the performance of the system. The original prototype uses a 28-ram thick aluminum platen supported by three ball-andgroove kinematic couplings. Each coupling consists of a pair of 5-mm diameter by 13-mm long polished steel cylinders arranged parallel to one another and resting on a 5-mm diameter precision ground steel ball. The cylinder pair is recessed into the underside of the platen to form a convex V-groove, which couples to the ball secured to the top of the load cell. The frequency response of the original prototype exhibits a resonant peak at about 630 Hz, which limits the useful range of frequency testing to well below the desired 1 to 2 kHz range. Through a systematic evaluation of component stiffnesses throughout the calibration fixture, the dominant compliance has been traced to the ball/cylinder ki104
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Figure 5 A one degree-of-freedom mass-sprmcj system
nematic couplings. A possible source of the compli ance is the use of epoxy for mounting the bali ano cylinders. The cylinders are epoxied into the r e cessed pockets, and it is possible that a smaii amount of epoxy may have remained between the walls and the cylinders, thus preventing solid metalto-metal contact. The same situation could exist with the ball mounted to the !oad cell. This effect also results in significant mechanical hysteresis, which manifests itself as apparent force hysteresis in the electromagnet characteristics. To eliminate this problem, the kinematic coupling has been rede signed to eliminate hysteresis and to increase its stiffness and thereby increase the resonant fre quency of the calibration fixture
Kinematic coupling design The kinematic couplings behave as springs collnected between the platen and the load cells. Fur thermore, we have shown experimentally that the couplings form the dominant compliance. Thus, if the fixture is bolted to a large mass, the system can be modelled as a single mass-spring system as shown in Figure 5. During testing, the electromagnet applies a sinusoidatly varying force to the mass of the target platen M. This mass, combined with the stiffness of the couplings K, sets the resonant frequency of the calibration fixture f,~ as i
:k
The goal is to design a coupling stiff enough to allow a high system resonant frequency, but that APRIL 1994 VOL 16 NO i
Poovey, Holmes, and Trumper: Calibration fixture
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Figure 6 Gothic arch kinematic coupling Figure 7 Carbon fiber platen with gothic arch kineretains enough point contact to preserve the kinematic support configuration. The original prototype ball/cylinder coupling has been analyzed using a kinematic coupling design program that calculates stresses at the contact interface and predicts deflections. 1° The analysis program predicts deflections consistent with the resonant frequencies measured experimentally. The deflection due to contact between two elastic bodies can be predicted using Hertz theory. The area of contact area between two bodies depends on the applied force, equivalent radius of the coupling, and equivalent elastic modulus of the coupling. 11 As the contact area increases for a given load, the contact pressure decreases, resulting in less stress and deflection, and thereby a stiffer coupling. To increase this area of contact, the original ball and cylinder design has been replaced with a monolithic gothic arch design of a form commonly used in ball bearing races. Here, the groove radius is slightly larger than the ball radius, thus resulting in an elliptical contact area that is considerably larger than the contact area of a ball on a cylinder. By increasing both the ball and groove radii of curvatures, the equivalent radius of the system is increased, which increases the stiffness of the coupling. As shown in Figure 6, the revised coupling design uses a gothic arch V-groove with a 12.7-mm radius resting on a hemisphere of 9.53-mm radius. Hot-pressed silicon carbide with a modulus of elasticity of 415 GPa is used for both the upper and lower halves of the coupling. The large radii of curvatures present at the contact interface keep stresses to approximately 20% of the maximum allowable contact stress of the material under maximum loading. The upper half of the coupling is bonded to the underside of the platen using a high-strength epoxy, whereas the lower half uses high-strength epoxy along with a mounting thread to securely fasten it to the top of the load cell.
Platen mass reduction To this point, the description of improving the system resonant frequency has focused on increasing the stiffness of the kinematic coupling. However, it is clear that the natural frequency can also be PRECISION ENGINEERING
matic couplings
increased by reducing the mass of the system. For example, a reduction of the platen mass to 75% of its original value would increase the natural frequency by the square root of 4/3, or approximately 1.15. The original 28-mm thick solid aluminum target platen with a mass of 1.6 kg has thus been replaced with a lighter target platen constructed from carbon fiber with a foam core. In this lighter platen, 42 plies of BASF G40-800/BMI "pre-preg" carbon graphite fiber are laminated to each face of a 37-mm thick core of rigid polystyrene foam to form an extremely stiff but lightweight sandwich construction with a mass of 0.75 kg. Composites are strongest when the fibers are aligned with the load; therefore, the plies are assembled with a 0°/ 60o/-60 ° orientation between plies to align with the coupling locations of the platen. Due to the nonconductive nature of the new platen, it is necessary to attach three metallic capacitance probe targets to the underside of the platen. The redesigned platen with the electromagnet target cartridge, gothic arch couplings, and capacitance probe targets is shown in Figure 7.
System equivalent mass reduction For initial experiments, the calibration fixture was bolted to an optical table and modelled as a singlemass/spring system. However, this approach has been abandoned for the following reasons. If there are any vibrations transmitted through the table top, they interact with the system dynamics and complicate the frequency response. Additionally, any modes of the support table appear as modes in the actuator frequency response. These interactions can be eliminated and the effective mass of the system can be reduced by isolating the calibration fixture base from the table top with a compliant element such as a foam pad. This essentially isolates the platen from the table dynamics and changes the system model from a single mass/ spring system to a double mass/spring system as illustrated in Figure 8. 105
Poovey, Holmes, and Trumper: Calibration fixture
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Figure 8 A double-mass/spring dynamic model ot the calibration fixture. Here, M~ represents the platen mass, M 2 the mass of the remainder of the fixture, and K the coupling stiffness 160 140 120 tO0 f - f
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Figure 10 Commanded force versus measurec~ force at each target separation gap
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Force curves for an electromagnetic bearing element with 400 turns of #26 wire at target separation gaps ranging from 20 to 650/~m
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Given a platen mass of 0.75 kg and a fixture mass of 4.5 kg, the equivalent mass of the system Meq is 0.643 kg, which theoretically increases the system resonant frequency by 8% over the case where the fixture is rigidly attached to a massive table. Most importantly, modes of the table top now have no effect on the frequency response of the calibration fixture, as evidenced by the pure secondorder response shown in Figure 11, which shows the measured transfer function from incremental actuator current to incremental actuator force.
Test results An electromagnetic bearing element has been characterized using the redesigned calibration fixture. The force-versus-current characteristics are shown 106
in Figure 9. Note that actuato~ saturation i~nuts fi-~l~ maximum achievable force and that this limit ~:,~ lower at larger air gaps. This is due to the fact that saturation occurs first in the back of the E core as a result of leakage flux. Figure 10 shows the results of an experimeni that verifies our ability to linearize the actuator if real time. In this experiment, the data of Figure :~ are used as a look-up table to determine the coil current required to generate a desired force, given a known air gap. The actual force is recorded in this experiment as the desired force is varied sinusoi dally over one cycle. The unity*slope line in Figure 10 shows that the linearization is successful, The nonlinearities present in Figure I0 occur only ~ separation gaps less than 50 ~m and are believe{~ due to small bending errors it~ the target mounting fixture on the platen. These effects are currenti~, under investigation The calibration fixture has been tested using a swept-sine frequency response to determine the resonant frequency of the system and to investi gate the stiffness of the kinematic couplings, The resonant frequency of the system is approximately 1.8 kHz, as shown in Figure 11. The predicted resonant frequency, based on an analytical prediction of the coupling stiffness, is 2.3 kHz. Thus, there is a factor of about 1.6 difference in system stiffness between the analytical and experimental results. This discrepancy may be due to form errors in the gothic arch coupling or to additionai sources of compliance in the structural loop. These sources are currently under investigation. However, the current resonant frequency is high enough to permit frequency testing of actuators to approximately 1 kHz
Suggestions for further work The kinematic coupling anaiysis program permits the investigation of the effects of higher coupling APRIL 1994 VOL t6 NO 2
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preload on the stiffness of the couplings. The calibration fixture currently uses three 25-mm diameter butyl O-rings stretched adjacent to each coupling. These O-rings provide 66 N of preload per coupling. This limit is imposed only by the convenience of using O-rings, and much higher preloads are possible through the use of springs. Another alternative would be to install a bolt through the center of the kinematic coupling such that it clamps the two coupling halves together without disturbing the contact points. The bolt could then be torqued to apply a much higher preload force than possible with O-rings or springs. This would require a slight modification to the lower kinematic coupling in order to prevent applying torque to the load cell and possibly damaging it. Other possibilities include the use of a C-clamp device over each coupling. The preload limit (50% of allowable contact stress) for the present silicon carbide coupling is calculated to be about 690 N or 155 pounds force per coupling. The coupling analysis program predicts that, under this preload, a system resonant frequency of 2.9 kHz is attainable from the present design. One important principle learned from the investigation of Hertz contact stress theory is that the radius of curvature of the kinematic coupling has a strong effect on the stiffness. The gothic arch coupling design is limited to easily obtained geometrical shapes such as the 9.53-mm radius hemisphere. PRECISION ENGINEERING
/
Figure 12 A proposed kinematic coupling design. Note that the centers of curvature need not necessarily be within the confines of the component
Larger radius hemispherical shapes have been avoided due to the excessive height build-up that would result from using larger radii. If geometrical restrictions are relaxed such that shapes having large radii with centers of curvature lying outside the boundaries of the component are considered, the kinematic coupling design shown in Figure 12 will be advantageous. The lower half ofthe coupling has large radii R1 and R2 in two orthogonal planes, and the upper half resembles a V-groove of slightly larger radius. 107
Poovey, Holmes, and Trumper: Calibration fixture
Conclusions
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Figure 13 Top view of a novel design for a reduced mass target platen. The platen profile would consist of an I-section, which would arch from a m a x i m u m at the center down to a m i n i m u m height at the perimeter
Another self-imposed geometrical constraint could be lifted to allow the design of a lighter target platen. The disk shape platen was chosen for its ease of fabrication. However, this shape could be replaced with a triangular design that eliminates the excess material between the coupling locations as illustrated in Figure 13. For example, the platen could be made from a high specific stiffness material, such as silicon carbide, or a m e t a l - m a t r i x c o m posite that could be cast to near net shapes, thus reducing expensive machining. Also, by casting the platen to a desired shape, structurally efficient I-sections could be used and the profile could be arched much as in bridge design to optimize the section area at the points of m a x i m u m moment. In doing so, the capacitance probe location would necessarily be moved inboard of each kinematic coupling support. Although the calibration fixture resonant frequency has been increased significantly from the original design, its response is very lightly damped as evidenced by the 30-dB resonant peak shown in Figure 11. In order to obtain accurate test results nearer to the fixture resonant frequency, damping needs to be added to flatten the response curve preceding the resonance peak. This could be ac~ complished by designing a mass-damper to be placed within the fixture base. The mass-damper would consist of a mass enclosed within a chamber of viscous fluid. This system could then be tuned to damp the resonant peak at 1.8 kHz. This could then permit accurate testing at frequencies to within a few hundred hertz of the resonant frequency. Alternately, a viscoelastic material could be used to bridge the gap between the two halves of the fixture and thereby damp the resonance. 108
An improved magnetic bearing calibration fixture has been built and tested using redesigned kinematic couplings, a reduced mass target platen, and other changes described above. Frequency response re. suits show a system resonant frequency of 1,8 kHz that compares with a predicted resonant frequency of 2 kHz using a kinematic coupling analysis spreadsheet. Form errors in the geometry ofthe gothic arch coupling most likely limit the stiffness of the coupling desig n, and efforts are underway to resolve this mat ter. In addition, a stray compliance is present withir, the structural loop of the system, which further rc duces the system stiffness. We have also presented the design of a cost-effective power amplifier with improved current slew rate capabilities. The present calibration fixture design provides. satisfactory data and performance for charactedz ing magnetic bearing actuators However, a number of additional improvements have been suggested during the course of testing. By extension of the ideas presented herein, the present instrument ca~,. be scaled in force and size in order to design fixtures capable of characterizing a wide variety of electro-magnetic actuators with force capabilities from frac.. tions to thousands of newtons, Such a calibration, fixture should prove useful to others working witP magnetic bearings and with actuators in other preci, sion motion control s y s t e m s
References ! Trurnper, D. L. and Queen, M. ~.~ Precision magnetic su,-, pension linear bearing," presented at the NASA International Symposium on Magnetic Suspension Technology, Hampton, VA, August 19-23, 1991 2 Trumper, D. m "Magnetic suspension techniques for precision motion control." Ph.D. thesis, Massachusetts institute of Technology, Cambridge, MA, 1990 3 Poovey, T. L. "A kinematically coupled magnetic bearmt.i test fixture," Master's thesis, University of North Caroline Charlotte, NC, 1992 4 Groom, N J. and Poole, W. L. Description of a magneb~ bearing test fixture," NASA TM-89081 [N87-20478i. 5 Knight, J. D., Xia, Z. and McCau!, E B "Forces in magneb,~: journal bearings: nonlinear computation and experimental measurement," presented at the Third International Sympo slum on Magnetic Bearings, Alexandria, VA, July 29-31 1992 6 Slocum, A. H. and Donmez, A 'Kinematic couplings fo~ precision fixturing--Part 2: Experimental determination of repeatability and stiffness," Prec/sion Eng, 1988, 10, 115-122 7 Pease, R. A. Troubleshooting ,qnaiog Circutls S[oneham MA: Butterworth-Heinemann, !991, p. 72 8 Trumper, D. L., Sanders, J. C.. Nguyen, T H. and Quee~ M. A. "Experimental results in nonlinear compensation of a one degree-of-freedom magnetic suspension," presented at the NASA International Symposium on Magnetic Suspension Technology, Hampton, VA, August 19-23, 1991 9 Data Translation Incorporated, !00 Locke Drive, Marlboro, MA, 01752 10 Slocum, A. H. "'Design of tmee-groove kinematmc co~ plings," Precision Eng 1992, 14, 67-76 11 Slocum, A. H, Precision Machine Design, Englewood Cliffs, NJ: Prentice-Hall, 1992, pp. 231-233
APR!L 1994 VOL 16 NC,' ;,
Keywords: force testing; magnetic bearings; kinematic couplings
Introduction
© 1994 Butterworth-Heinemann
(NASA) annular momentum control device. 4 This design is only useful for forces up to 45 newtons (N). Furthermore, the selection of strain gauge-based load cells limits its use to low-frequency force measurement due to the low stiffness of these load cells. Knight et al. developed a calibration fixture that measures the deflection of a long rod subjected to the attractive force of a magnetic journal bearing element in order to measure forces up to 400 N. ~ This method of force measurement is only suitable for quasistatic forces, due to the relatively low-frequency resonance of the bearing mass on the long shaft, and is not capable of micron-scale positioning accuracy. The magnetic bearing calibration fixture described in this article attempts to achieve high resonant frequencies through the use of kinematic design principles. Additionally, the actuator air gap is determined with micron-scale precision. The electromagnet bearing elements designed for our linear magnetic-bearing positioning stage 1operate at separation gaps ranging from 10 to 650/~m with forces as large as several hundred newtons. In addition, wide-bandwidth force authority is required from the actuators so that the magnetic bearing that they control can achieve positional accuracies on the order of nanometers. Therefore, the calibration fixture resonance needs to be located at as high a frequency as possible in order to allow broadband
PRECISION ENGINEERING
99
Magnetic bearings represent a promising approach for achieving positioning accuracies on the order of nanometers in precision motion control stages. 1 In these stages, an accurate model of the actuator force as a function of coil current and air gap is required. This model is used within the bearing controller to linearize the actuators in real time. 2 Classical magnetic circuit theory can accurately model the magnetic bearing actuator force characteristics at low flux densities. However, at higher flux densities the actuator force characteristics deviate from first-order theory due to saturation. Therefore, experimental measurement is required. This article describes the development and testing of a magnetic bearing calibration fixture designed for such experimental measurement of the actuator force characteristics. 3 There are other magnetic bearing calibration fixtures currently in use. Groom and Poole developed a device for characterizing actuators for use on the National Aeronautics and Space Administration
Address reprint requests to David Trumper, Mechanical Engineering Department, Massachusetts Institute of Technology, 77 Massachusetts A venue, Cambridge, MA 02139, USA.
Poovey, Holmes, and Trumper: Calibration fixture characterization of the actuator frequency response. To address these needs, the magnetic bearing calibration fixture described herein has been developed with the following objectives: • Measure the electromagnet actuator force from minimum to maximum core flux density at separation gaps ranging from 10 to 1,000/zm • Support a peak force capability on the order of hundreds of newtons • Allow scaling to arbitrary force levels by an appropriate selection of materials, coupling geometry, and positioning hardware • Demonstrate the capability to study actuator torques where the torque vector lies in the plane of the pole face • Permit the adjustment of the target separation gap as well as the rotation of the target relative to the actuator • Use metrology components that are independent of the load-carrying components in order to accurately measure the target separation gap in addition to rotations about vectors lying in the plane of the actuator pole face • Allow testing of a variety of actuator sizes interchangeably • Measure actuator frequency response through several kilohertz This report details the means by which these goals have been achieved and is organized as follows. First, the mechanical components of the calibration fixture are described. This is followed by a description of the design of a linear power amplifier, which is used to energize the magnetic bearing element. Next, a brief description of the testing procedure is presented along with the system control and data acquisition scheme. This is followed by a summary of the most significant design improvements implemented during the development of the calibration fixture. Finally, the results of force testing a magnetic bearing actuator are given along with suggestions for improving the performance of the system.
Mechanical components The magnetic bearing calibration fixture consists of an aluminum fixture body and platform, three adjustable platen support columns with ball-andgroove kinematic couplings, three capacitance probes, and a target platen as shown in Figure 1. The electromagnet housed in the body is made from E-shaped nickel-iron alloy blanks laminated and wound with 400 turns of #26 wire and epoxied into a mounting cartridge. The electromagnet cartridge is mounted inside the calibration fixture body with its pole faces exposed upward toward the target platen. A laminated nickel-iron target is attached to the underside of the target platen such that it is attracted toward the electromagnet. Various-sized actuators and targets can be accommodated 100
through changing adapter plates in the body and the platen, respectively. Located at 120° intervals around the perimeter of the fixture body are the three support columns, each consisting of a kinematic ball/groove coupling mounted on top of a piezoelectric load cell. The hemisphere mounted to the top of each !oad cell makes two points of contact in each of three grooves; therefore, the three couplings provide ex actly six constraints on the six platen degrees of free. dom. This type of kinematic coupling, commonly ~eferred to as a Maxwell mouuL has the ability to provide submicron positional repeatability if prop erly designed. 6 The micrometers are used to adjust the height of the platen above the electromagnet, whereas the load cells measure the attractive force between the electromagnet and the target platen. Three capacitance probes mounted at the top p e r i m eter of the fixtu re body are used to sense the distance between the electromagnet and three datum pads on the target platen. These probes have a repeatabil ity of better than 1/zm and linearity errors on the on der of 1/zm out of a total range of ! m m An interesting feature of the design is that the mechanical accuracy of the fixture is determined entirely by a set of drilling and boring operations that can be accomplished while working from o~iv the top side of the fixture body and thus with ,-, single machine set-up. These operations are the boring of the holes for micrometers, capacitance probes, and the pins that locate the actuator in the housing. It is only these dimensions that are critical because the iron actuator [arge[ located on the, platen is made wider than the pole faces of the actuator. Thus, the body-referenced location and magnitude of the actuator force are indifferent to lateral motions of the target ,within some iimited range, and thereby to positioning errors of the kinematic mounts or target, which are less than ~,he excess size of the target. This means that the platen need not be tightly toleranced because all tight t o i erances are constrained within a single part and are readily established. The capability for kinematic adjustment of the target platen position allows the measurement of both force and torque as a function of target separa.. tion gap. Torque can be generated as a result of the pitch and roll rotation of the target platen with respect to the electromagnet, which can occur when a magnetically suspended object rotates. This carl be particularly important when dealing with electromagnets that have dimensions that are a significant fraction of the characteristic dimension of the sus. pended body. In the present case, because the actu~ ator dimensions are small relative to the anticipated suspension dimension, the torque term is not studied.
Linear power amplifier The characterization of the electromagnetic actuators requires a current source capable of driving the APRIL t994 VOL i6 NO :
Poovey, Holmes, and Trumper: Cal/bration fixture Target Platen 165 mm
Kinematic Coupling
Load Cell
Micrometer Head
Figure 1
Solid model representation of the calibration fixture
electromagnets into saturation. The current source must also allow rapid reduction of the coil current to zero. The linear power amplifier shown in Figure 2 has been designed for low noise characteristics as well as high negative current slew rate capability. Circuit protection is provided by fuse F1 and the flyback network. Fuse F1 is used to protect the coil from overheating in the event that the current loop goes out of control (e.g., Q2 develops a drain to source short circuit) and the flyback network allows coil current to continue when O2 is turned off suddenly, thereby protecting Q2 from excessive drain to source voltages. The circuit elements which comprise the flyback network are enclosed by a dashed line in Figure 2. The function of this network is to act as the series combination of a diode and a high-power zener. ? In earlier work, where the power amplifier used only a diode for the flyback network, insufficient power amplifier negative current slew rate capabilities were found to contribute to instabilities of the magnetic suspension. 8 This compound zener replaces the earlier diode in order to provide larger flyback voltages, thereby increasing the achievable negative current slew rate. The function of the elements of the compound PRECISION ENGINEERING
zener are described as follows. Diode D3 prevents current from flowing through the flyback network under normal operating conditions when the drain of Q2 is more negative than the +40-V power supply. This diode also serves to protect the base-emitter junction of Q1 from reverse breakdown. Under conditions when the drain of Q2 is more positive than 40 V, the combination of D1, D2, R4, and Q1 acts as a 40-V zener. This compound-zener circuit has the advantage that the majority of the flyback energy is dissipated in Q1, which is a 2N3055 power transistor mounted on a heat sink. This transistor is capable of dissipating on the order of 100 W, whereas it is difficult to find similarly rated zener diodes. Approximately 0.5 mA must flow through R4 before Q1 will turn on. For this to occur, given the choice of zeners D1 and D2, the drain of Q2 must be approximately 40 V more positive than the +40-V coil power supply. This condition will only occur when the attempted coil current negative slew rate is greater than about 40/L, where L is the coil inductance. Expressed another way, the amplifier is linear for coil voltages bounded by +40 V, even though the current applied by the amplifier to the coil is of necessity unipolar. Thus, the character101
Poovey, Holmes, and Trumper." Calibration fixture
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istics of this amplifier are well matched to the application because the variable reluctance actuators apply force that is independent of the sign of the drive current and thus do not require a bipolar cur rent source. However a bipolar voltage source is required in order to allow rapid current changes. This amplifier is capable of applying the required bipolar voltages to the electromagnet coil while requiring only a single unipolar 40-V power supply and only one high-power control element. Due to its simplicity, the amplifier uses only several dollars of parts, as compared with a power op-amp-based design, which would cost between $50 and $100. The power op-amp design also requires an additional 40-V power supply costing hundreds of dol. lars. When one considers that a typical magnetic bearing system may require on the order of 12 or 16 channels of amplification, the economic advan tages of our design are significant. The snubber capacitor C7 was added to the circuit and empirically tuned to eliminate oscillations of the loop at a frequency of several hundred kilohertz. An analytical investigation of these oscit lations has not been undertaken, but they clearly result from the interaction between the coil impedance, the transistor dynamics, and the dynamics of the op-amp current loop. In comparison with the earlier switching amplifier design, 8 as well as with the earlier simple diode flyback, 2 the negative current slew rate capability is greatly improved. In addition, the switching amplifier generates noise difficult to shield. This noise is especially undesirable in high-precision systems; thus a linear amplifier has been chosen. The capa102
bilities of this amplifier can be ~eadsly scaled ~r~ w:.,q! age or current for other magnetic bearing systems simply by choosing an appropriate power FET fo~~ Q2.
Test procedure in a real-time control appiicat~o~, ;t is desirable h:_; linearize the nonlinear electromagnet force-current separation-gap relationship in order to determine the coil current required to obtain the desired actL~a tor force at a given gap. 8 The data collected fro~., the calibration fixture via data acquisition hardware and test software allow the creation of a two-dimensional look-up table that permits such a lineariza tion, The process of acquiring these data is described as follows. A block diagram for the system is shown ~ Figure 3. The user first initializes the actual air gap between the electromagnet face and target using a software-assisted procedure described later. Then, through a digital-to-analog converter, the test software outputs a sinusoidal control voltage that oscii-lates between 0 and 10 V. This sine wave is fed to the power amplifier, which outputs a current pro portional to the input voltage. The power amplifier output drives the electromagnet, which applies a corresponding force to the target platen. The force f(I, G) is a function of input current /and target separation gap G. The load cells transduce this force into a proportional charge, which is amplified and converted to a voltage bythe charge amplifier. Both the electromagnet current and the load cell output voltage are logged into the '386-based computer APRil !994 VOL i 6 NQ 2
Poovey, Holmes, and Trumper: Calibration fixture
Figure 3
Block diagram of electromagnet testing system
via analog-to-digital converters for storage. After data are taken for one separation gap, the gap is adjusted to a new value and the test procedure is repeated. Next the acquired data are used to build a two-dimensional look-up table of force versus current at the designated target separation gaps. The look-up table is stored in the form of an ASCII file for ease of manipulation by third-party software. The accuracy of the look-up table is verified via another test routine that uses the look-up table to apply a ramp of desired force to the actuator using the same test configuration as depicted by Figure 3. The result of this test is a plot of desired force versus actual force in which the degree of linearity determines how accurately the actuator force can be controlled. This verification routine sequences through many iterations, allowing the user to change the air gap to obtain data that test the accuracy of the look-up table over a desired range of operating air gaps. It is desirable to have the data table resident in memory yet separate from the test routine because this allows many different applications to access the data table independently. To accomplish this, one application program allocates memory for the data table, points user interrupt vector 0x65 at the data table, and then terminates and stays resident (TSR). Another application program opens and interprets the ASCII data files created by the characterization software and fills in the data table. Next the data table is available to any application program and can be refreshed at any time. Because the capacitance probes and the micrometer shafts are not axially aligned, motions of a single micrometer couple into all three probe position readings. The task of adjusting the three micrometer head displacements to obtain a specific target separation gap via the three capacitance probe readings is simplified through real-time coordinate transformations (1-2) described below. 2 2
C2
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3
3 rx,1
2
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PRECISION ENGINEERING
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Figure 4 illustrates the relative location of the three micrometer heads at positions X 1, X 2, and X 3, and the three capacitance probes at locations C1, C2, and C3. In operation, the bearing is limited to milliradian scale rotations. We therefore only characterize the actuators for small rotations. Under the assumption of small rotations, the micrometer head displacements can be related to the capacitance probe readings by the linear transformations shown in Equations (1) and (2). In Figure 4, a is the distance between C1 and X 2 and is approximately 70 mm. The indicated lever arms are used to calculate the motions at C1, C2, and C3 due to the incremental extension of micrometer X 1. This yields the first column in the matrix in Equation (1) and the first row of the matrix in Equation (2). The remaining columns and rows are developed similarly. To assist in obtaining the desired air gaps, the transformation in Equation (2)is performed by the '386based computer in real-time on the capacitance probe readings in order to map the position error measured at the capacitance probes into equivalent displacements of the micrometer heads. The results of the transformation X~, X2, and X3 are displayed as three bar graphs on the computer monitor. On these bar graphs, the micrometer adjustments do not interact, thus greatly simplifying the adjustment of a desired air gap. The bar graphs are linear for errors between plus and minus 1 /zm and are logarithmic for plus and minus larger magnitude errors up to 1 mm. Thus, the error can be adjusted to null with submicron resolution, but is on-scale even for very large magnitudes. The test software uses one DT-2823 data acquisition board for the measurement of the three load cell signals and for control of the actuator current. 9 Another DT-2823 data acquisition board is used to measure the three capacitance probe position signals. The computer running the test software and containing the DT-2823 boards is a 20-MHz 80386 103
Poovey, Holmes, and Trumper: Calibration fixture X!
LEVERARM 1
LEVERARM 2
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Figure 4 Geometry relating capacitance probe readings to micrometer head motion.
with an 80387 coprocessor. This computer is running the DOS 5.0 operating system and the test software is written utilizing Microsoft C version 6.0 because this compiler supports in-line assembly programming. Care has been taken to maximize processor efficiency during critical times in order to ensure data integrity and timing accuracy. This has been accomplished by using hardware direct memory access (DMA) to transfer data readings di m rectly into processor memory. Also, all interfacing software to the DT-2823 boards and all core functional test codes are written in assembly language. Finally, to allow faster execution, all math functions are performed by programming the 80387 coprocessor directly. By developing the software in this way, acquisition rates of up to 10 kHz can be achieved. This sample rate translates into a test waveform output frequency of up to 10 Hz while simultaneously collecting 4,000 data points per half cycle of the sine wave. Thus, the data at a given separation gap can be acquired in about 0.1 second. This speed proves to be more than sufficient to allow rapid actuator characterization.
Calibration fixture improvements The calibration fixture has undergone a number of revisions, with each design change adding some incremental improvement to the performance of the system. The original prototype uses a 28-ram thick aluminum platen supported by three ball-andgroove kinematic couplings. Each coupling consists of a pair of 5-mm diameter by 13-mm long polished steel cylinders arranged parallel to one another and resting on a 5-mm diameter precision ground steel ball. The cylinder pair is recessed into the underside of the platen to form a convex V-groove, which couples to the ball secured to the top of the load cell. The frequency response of the original prototype exhibits a resonant peak at about 630 Hz, which limits the useful range of frequency testing to well below the desired 1 to 2 kHz range. Through a systematic evaluation of component stiffnesses throughout the calibration fixture, the dominant compliance has been traced to the ball/cylinder ki104
v!
Figure 5 A one degree-of-freedom mass-sprmcj system
nematic couplings. A possible source of the compli ance is the use of epoxy for mounting the bali ano cylinders. The cylinders are epoxied into the r e cessed pockets, and it is possible that a smaii amount of epoxy may have remained between the walls and the cylinders, thus preventing solid metalto-metal contact. The same situation could exist with the ball mounted to the !oad cell. This effect also results in significant mechanical hysteresis, which manifests itself as apparent force hysteresis in the electromagnet characteristics. To eliminate this problem, the kinematic coupling has been rede signed to eliminate hysteresis and to increase its stiffness and thereby increase the resonant fre quency of the calibration fixture
Kinematic coupling design The kinematic couplings behave as springs collnected between the platen and the load cells. Fur thermore, we have shown experimentally that the couplings form the dominant compliance. Thus, if the fixture is bolted to a large mass, the system can be modelled as a single mass-spring system as shown in Figure 5. During testing, the electromagnet applies a sinusoidatly varying force to the mass of the target platen M. This mass, combined with the stiffness of the couplings K, sets the resonant frequency of the calibration fixture f,~ as i
:k
The goal is to design a coupling stiff enough to allow a high system resonant frequency, but that APRIL 1994 VOL 16 NO i
Poovey, Holmes, and Trumper: Calibration fixture
a
R o" 12.7 mm 5 0 mm
Rb- 9.53 mm
Figure 6 Gothic arch kinematic coupling Figure 7 Carbon fiber platen with gothic arch kineretains enough point contact to preserve the kinematic support configuration. The original prototype ball/cylinder coupling has been analyzed using a kinematic coupling design program that calculates stresses at the contact interface and predicts deflections. 1° The analysis program predicts deflections consistent with the resonant frequencies measured experimentally. The deflection due to contact between two elastic bodies can be predicted using Hertz theory. The area of contact area between two bodies depends on the applied force, equivalent radius of the coupling, and equivalent elastic modulus of the coupling. 11 As the contact area increases for a given load, the contact pressure decreases, resulting in less stress and deflection, and thereby a stiffer coupling. To increase this area of contact, the original ball and cylinder design has been replaced with a monolithic gothic arch design of a form commonly used in ball bearing races. Here, the groove radius is slightly larger than the ball radius, thus resulting in an elliptical contact area that is considerably larger than the contact area of a ball on a cylinder. By increasing both the ball and groove radii of curvatures, the equivalent radius of the system is increased, which increases the stiffness of the coupling. As shown in Figure 6, the revised coupling design uses a gothic arch V-groove with a 12.7-mm radius resting on a hemisphere of 9.53-mm radius. Hot-pressed silicon carbide with a modulus of elasticity of 415 GPa is used for both the upper and lower halves of the coupling. The large radii of curvatures present at the contact interface keep stresses to approximately 20% of the maximum allowable contact stress of the material under maximum loading. The upper half of the coupling is bonded to the underside of the platen using a high-strength epoxy, whereas the lower half uses high-strength epoxy along with a mounting thread to securely fasten it to the top of the load cell.
Platen mass reduction To this point, the description of improving the system resonant frequency has focused on increasing the stiffness of the kinematic coupling. However, it is clear that the natural frequency can also be PRECISION ENGINEERING
matic couplings
increased by reducing the mass of the system. For example, a reduction of the platen mass to 75% of its original value would increase the natural frequency by the square root of 4/3, or approximately 1.15. The original 28-mm thick solid aluminum target platen with a mass of 1.6 kg has thus been replaced with a lighter target platen constructed from carbon fiber with a foam core. In this lighter platen, 42 plies of BASF G40-800/BMI "pre-preg" carbon graphite fiber are laminated to each face of a 37-mm thick core of rigid polystyrene foam to form an extremely stiff but lightweight sandwich construction with a mass of 0.75 kg. Composites are strongest when the fibers are aligned with the load; therefore, the plies are assembled with a 0°/ 60o/-60 ° orientation between plies to align with the coupling locations of the platen. Due to the nonconductive nature of the new platen, it is necessary to attach three metallic capacitance probe targets to the underside of the platen. The redesigned platen with the electromagnet target cartridge, gothic arch couplings, and capacitance probe targets is shown in Figure 7.
System equivalent mass reduction For initial experiments, the calibration fixture was bolted to an optical table and modelled as a singlemass/spring system. However, this approach has been abandoned for the following reasons. If there are any vibrations transmitted through the table top, they interact with the system dynamics and complicate the frequency response. Additionally, any modes of the support table appear as modes in the actuator frequency response. These interactions can be eliminated and the effective mass of the system can be reduced by isolating the calibration fixture base from the table top with a compliant element such as a foam pad. This essentially isolates the platen from the table dynamics and changes the system model from a single mass/ spring system to a double mass/spring system as illustrated in Figure 8. 105
Poovey, Holmes, and Trumper: Calibration fixture
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Figure 8 A double-mass/spring dynamic model ot the calibration fixture. Here, M~ represents the platen mass, M 2 the mass of the remainder of the fixture, and K the coupling stiffness 160 140 120 tO0 f - f
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Figure 10 Commanded force versus measurec~ force at each target separation gap
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Force curves for an electromagnetic bearing element with 400 turns of #26 wire at target separation gaps ranging from 20 to 650/~m
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M1M2 Meq -
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Given a platen mass of 0.75 kg and a fixture mass of 4.5 kg, the equivalent mass of the system Meq is 0.643 kg, which theoretically increases the system resonant frequency by 8% over the case where the fixture is rigidly attached to a massive table. Most importantly, modes of the table top now have no effect on the frequency response of the calibration fixture, as evidenced by the pure secondorder response shown in Figure 11, which shows the measured transfer function from incremental actuator current to incremental actuator force.
Test results An electromagnetic bearing element has been characterized using the redesigned calibration fixture. The force-versus-current characteristics are shown 106
in Figure 9. Note that actuato~ saturation i~nuts fi-~l~ maximum achievable force and that this limit ~:,~ lower at larger air gaps. This is due to the fact that saturation occurs first in the back of the E core as a result of leakage flux. Figure 10 shows the results of an experimeni that verifies our ability to linearize the actuator if real time. In this experiment, the data of Figure :~ are used as a look-up table to determine the coil current required to generate a desired force, given a known air gap. The actual force is recorded in this experiment as the desired force is varied sinusoi dally over one cycle. The unity*slope line in Figure 10 shows that the linearization is successful, The nonlinearities present in Figure I0 occur only ~ separation gaps less than 50 ~m and are believe{~ due to small bending errors it~ the target mounting fixture on the platen. These effects are currenti~, under investigation The calibration fixture has been tested using a swept-sine frequency response to determine the resonant frequency of the system and to investi gate the stiffness of the kinematic couplings, The resonant frequency of the system is approximately 1.8 kHz, as shown in Figure 11. The predicted resonant frequency, based on an analytical prediction of the coupling stiffness, is 2.3 kHz. Thus, there is a factor of about 1.6 difference in system stiffness between the analytical and experimental results. This discrepancy may be due to form errors in the gothic arch coupling or to additionai sources of compliance in the structural loop. These sources are currently under investigation. However, the current resonant frequency is high enough to permit frequency testing of actuators to approximately 1 kHz
Suggestions for further work The kinematic coupling anaiysis program permits the investigation of the effects of higher coupling APRIL 1994 VOL t6 NO 2
Poovey, Holmes, and Trumper." Calibration fixture A:FreQ
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Figure 11
preload on the stiffness of the couplings. The calibration fixture currently uses three 25-mm diameter butyl O-rings stretched adjacent to each coupling. These O-rings provide 66 N of preload per coupling. This limit is imposed only by the convenience of using O-rings, and much higher preloads are possible through the use of springs. Another alternative would be to install a bolt through the center of the kinematic coupling such that it clamps the two coupling halves together without disturbing the contact points. The bolt could then be torqued to apply a much higher preload force than possible with O-rings or springs. This would require a slight modification to the lower kinematic coupling in order to prevent applying torque to the load cell and possibly damaging it. Other possibilities include the use of a C-clamp device over each coupling. The preload limit (50% of allowable contact stress) for the present silicon carbide coupling is calculated to be about 690 N or 155 pounds force per coupling. The coupling analysis program predicts that, under this preload, a system resonant frequency of 2.9 kHz is attainable from the present design. One important principle learned from the investigation of Hertz contact stress theory is that the radius of curvature of the kinematic coupling has a strong effect on the stiffness. The gothic arch coupling design is limited to easily obtained geometrical shapes such as the 9.53-mm radius hemisphere. PRECISION ENGINEERING
/
Figure 12 A proposed kinematic coupling design. Note that the centers of curvature need not necessarily be within the confines of the component
Larger radius hemispherical shapes have been avoided due to the excessive height build-up that would result from using larger radii. If geometrical restrictions are relaxed such that shapes having large radii with centers of curvature lying outside the boundaries of the component are considered, the kinematic coupling design shown in Figure 12 will be advantageous. The lower half ofthe coupling has large radii R1 and R2 in two orthogonal planes, and the upper half resembles a V-groove of slightly larger radius. 107
Poovey, Holmes, and Trumper: Calibration fixture
Conclusions
F
A
Section A-A
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Figure 13 Top view of a novel design for a reduced mass target platen. The platen profile would consist of an I-section, which would arch from a m a x i m u m at the center down to a m i n i m u m height at the perimeter
Another self-imposed geometrical constraint could be lifted to allow the design of a lighter target platen. The disk shape platen was chosen for its ease of fabrication. However, this shape could be replaced with a triangular design that eliminates the excess material between the coupling locations as illustrated in Figure 13. For example, the platen could be made from a high specific stiffness material, such as silicon carbide, or a m e t a l - m a t r i x c o m posite that could be cast to near net shapes, thus reducing expensive machining. Also, by casting the platen to a desired shape, structurally efficient I-sections could be used and the profile could be arched much as in bridge design to optimize the section area at the points of m a x i m u m moment. In doing so, the capacitance probe location would necessarily be moved inboard of each kinematic coupling support. Although the calibration fixture resonant frequency has been increased significantly from the original design, its response is very lightly damped as evidenced by the 30-dB resonant peak shown in Figure 11. In order to obtain accurate test results nearer to the fixture resonant frequency, damping needs to be added to flatten the response curve preceding the resonance peak. This could be ac~ complished by designing a mass-damper to be placed within the fixture base. The mass-damper would consist of a mass enclosed within a chamber of viscous fluid. This system could then be tuned to damp the resonant peak at 1.8 kHz. This could then permit accurate testing at frequencies to within a few hundred hertz of the resonant frequency. Alternately, a viscoelastic material could be used to bridge the gap between the two halves of the fixture and thereby damp the resonance. 108
An improved magnetic bearing calibration fixture has been built and tested using redesigned kinematic couplings, a reduced mass target platen, and other changes described above. Frequency response re. suits show a system resonant frequency of 1,8 kHz that compares with a predicted resonant frequency of 2 kHz using a kinematic coupling analysis spreadsheet. Form errors in the geometry ofthe gothic arch coupling most likely limit the stiffness of the coupling desig n, and efforts are underway to resolve this mat ter. In addition, a stray compliance is present withir, the structural loop of the system, which further rc duces the system stiffness. We have also presented the design of a cost-effective power amplifier with improved current slew rate capabilities. The present calibration fixture design provides. satisfactory data and performance for charactedz ing magnetic bearing actuators However, a number of additional improvements have been suggested during the course of testing. By extension of the ideas presented herein, the present instrument ca~,. be scaled in force and size in order to design fixtures capable of characterizing a wide variety of electro-magnetic actuators with force capabilities from frac.. tions to thousands of newtons, Such a calibration, fixture should prove useful to others working witP magnetic bearings and with actuators in other preci, sion motion control s y s t e m s
References ! Trurnper, D. L. and Queen, M. ~.~ Precision magnetic su,-, pension linear bearing," presented at the NASA International Symposium on Magnetic Suspension Technology, Hampton, VA, August 19-23, 1991 2 Trumper, D. m "Magnetic suspension techniques for precision motion control." Ph.D. thesis, Massachusetts institute of Technology, Cambridge, MA, 1990 3 Poovey, T. L. "A kinematically coupled magnetic bearmt.i test fixture," Master's thesis, University of North Caroline Charlotte, NC, 1992 4 Groom, N J. and Poole, W. L. Description of a magneb~ bearing test fixture," NASA TM-89081 [N87-20478i. 5 Knight, J. D., Xia, Z. and McCau!, E B "Forces in magneb,~: journal bearings: nonlinear computation and experimental measurement," presented at the Third International Sympo slum on Magnetic Bearings, Alexandria, VA, July 29-31 1992 6 Slocum, A. H. and Donmez, A 'Kinematic couplings fo~ precision fixturing--Part 2: Experimental determination of repeatability and stiffness," Prec/sion Eng, 1988, 10, 115-122 7 Pease, R. A. Troubleshooting ,qnaiog Circutls S[oneham MA: Butterworth-Heinemann, !991, p. 72 8 Trumper, D. L., Sanders, J. C.. Nguyen, T H. and Quee~ M. A. "Experimental results in nonlinear compensation of a one degree-of-freedom magnetic suspension," presented at the NASA International Symposium on Magnetic Suspension Technology, Hampton, VA, August 19-23, 1991 9 Data Translation Incorporated, !00 Locke Drive, Marlboro, MA, 01752 10 Slocum, A. H. "'Design of tmee-groove kinematmc co~ plings," Precision Eng 1992, 14, 67-76 11 Slocum, A. H, Precision Machine Design, Englewood Cliffs, NJ: Prentice-Hall, 1992, pp. 231-233
APR!L 1994 VOL 16 NC,' ;,