Forward and aft thrust bearings located on the centerline hold therotor in axial equilibrium, while a journal bearing around the rotor periphery holds the rotor in radial equilibrium.. F
Trang 1microfabricated rotor (a cylinder of density ρ, diameter D, and length L) compared to the pressure (p)
acting on its projected surface area This can be expressed as a non-dimensional load parameter:
from which it can be seen that the load parameter decreases linearly as the device shrinks For example,the load parameter due to the rotor mass for the MIT Microengine, which is a relatively large MEMSdevices (measuring 4 mm in diameter and 300 microns deep), is approximately 103 The benefits of thisscaling are that orientation or the freely suspended part becomes effectively irrelevant and that unloadedoperation is easy to accomplish In addition, since the gravity loading is negligible, the primary forces thatone needs to support are pressure-induced loads and in a rotating device loads due to rotational imbal-ance This last load is very important and will be discussed in more detail in connection to rotating lubri-cation requirements The chief disadvantage of the low natural loading is that unloaded operation is oftenundesirable (in hydrodynamic lubrication where a minimum eccentricity is required for journal bearingstability), and in practice, gravity loading is often used to advantage Therefore a scheme for applying anartificial load needs to be developed This is discussed in more detail later in the chapter
11.2.2 Applicability of the Continuum Hypothesis
A common concern in microfluidic devices is the appropriateness of the continuum hypothesis as thedevice scale continues to fall At some scale, the typical inter-molecular distances will be comparable tothe device scales and the use of continuum fluid equations becomes suspect For gases, this is measured
by the Knudsen number (Kn) — the ratio of the mean free path to the typical device scale Numerous experiments [Arkilic et al., 1997, 1993; Breuer et al., 2001] have determined that non-continuum effects become observable when Kn reaches approximately 0.1 and that continuum equations become meaning- less (the “transition flow regime”) at Kn of approximately 0.3 For atmospheric temperature and pressure,
the mean free path of air is approximately 70 nm Thus, atmospheric devices with features smaller thanapproximately 0.2 microns will be subject to non-negligible non-continuum effects In many cases, suchsmall dimensions are not present, and the fluidic analysis can safely use the standard Navier–Stokes equa-tions (this is the case for the microengine)
Nevertheless, in applications where viscous damping is to be avoided (for example in high-Q
resonat-ing devices such as accelerometers or gyroscopes) the operatresonat-ing gaps are typically quite small (perhaps afew microns), and the gaps serve as both a physical standoff and a sense-gap where capacitive sensing isaccomplished In such examples one must work with the small dimension, and in order to minimize vis-cous effects, the device is packaged at low pressures where non-continuum effects are evident For smallKnudsen numbers, the Navier–Stokes equations can be used with a single modification — the boundarycondition is relaxed from the standard non-slip condition to that of a slip-flow condition where the velocity at the wall is related to the Knudsen number and the gradient of velocity at the wall:
(11.2)where σis the tangential momentum accommodation coefficient (TMAC) which varies between 0 and 1.Experimental measurements [Breuer et al., 2001] indicate that smooth native silicon has a TMAC ofapproximately 0.7 in contact with several commonly used gases
Despite the fact that the slip-flow theory is valid only for low Kn, it is often used incorrectly with great
success at much higher Knudsen numbers Its adoption beyond its range of applicability stems primarilyfrom the lack of any better approach short of solving the Bolzmann equation or Direct Simulation MonteCarlo (DSMC) computations [Beskok and Karniadakis, 1994; Cai et al., 2000] For many simple geome-tries, the “extended” slip-flow theory works much better than it should and provides quite adequate results[Kwok et al., 2005] This theory is demonstrated in the sections on Couette and squeeze-film dampinglater in the chapter
Trang 211.2.3 Surface Roughness
Another peculiar feature of MEMS devices is that the surface roughness of the material used can become
a significant factor in the overall device geometry MEMS surface finishes are quite varied, ranging fromatomically smooth surfaces found on polished single-crystal silicon substrates to the rough surfaces left
by different etching processes The effects of these topologies can be important in several areas for vice performance Probably the most important effect is the way in which the roughness can affect struc-tural characteristics such as crack initiation, yield strength, etc., although this will not be explored in thischapter Secondly, the surface finish can affect fluidic phenomena such as the energy and momentumaccommodation coefficient, and consequently, the momentum and heat transfer Lastly, the surface char-acteristics (not only the roughness, but also the surface chemistry and affinity) can strongly affect itsadhesive force This is not treated in detail in this discussion, although it is mentioned briefly at the end
microde-of the chapter in connection with tribology issues in MEMS
11.3 Governing Equations for Lubrication
With the proviso that the continuum hypothesis holds for micron-scale devices (perhaps with a modifiedboundary condition), the equations for microlubrication are identical to those used in conventionallubrication analysis and can be found in any standard lubrication textbook [Hamrock, 1984] We presentthe essential results here, but the reader is referred to more complete treatments for full derivations and
a detailed discussion of the appropriate limitations
Starting with the Navier–Stokes equations, we can make a number of simplifying assumptions priate for lubrication problems These are itemized here:
appro-Inertia: The terms representing transfer of momentum due to inertia may be neglected This arises
because of the small dimensions that characterize lubrication geometries and MEMS in particular
In very high speed devices such as the MIT Microengine, inertial terms may not be as small as onemight like, and corrections for inertia may be applied However, preliminary studies suggest that thesecorrections are small [Piekos, 2000]
Curvature: Lubrication geometries are typically characterized by a thin fluid film with a slowly varying
film thickness The critical dimension in such systems is the film thickness, and this is assumed to
be much smaller than any radius of curvature associated with the overall system This assumption isparticularly important in rotating systems where a circular journal bearing is used Assuming that
the radius of the bearing R is much larger than the typical film thickness c (i.e., c/R 1) greatly
simplifies the governing equations
Isothermal: Because volumes are small and surface areas are large, thermal contact between the fluid and
the surrounding solid is very good in a MEMS device In addition, common MEMS materials are goodthermal conductors For both these reasons, it is safe to assume that the lubrication film is isothermal.With these restrictions, the Navier–Stokes equations, the equation for the conservation of mass, andthe equation of state for a perfect gas may be combined to yield the Reynolds equation [Reynolds, 1886],written here for two-dimensional films:
Trang 3shearing velocity u b, and a characteristic unsteady frequency ω In addition, gas rarefaction can be porated for low Knudsen numbers by assuming a slip-flow wall boundary condition:
11.4 Couette-Flow Damping
The viscous damping of a plate oscillating in parallel motion to a substrate has been a problem of dous importance in MEMS devices, particularly in the development of resonating structures such asaccelerometers and gyros The problem arises because the proof mass, which may be hundreds of microns
tremen-in the lateral dimension, is typically suspended above the substrate with a separation of only a few microns
A simple analysis of Couette-flow damping for rarefied flows is easy to demonstrate by choosing amodel problem of a one-dimensional proof mass (i.e., ignoring the dimension perpendicular to the platemotion) This is shown schematically in Figure 11.1
The Navier–Stokes equations for this geometry reduce to:
in which only viscous stresses due to the velocity gradient and the unsteady terms survive This can besolved using separation of variables and employing a slip-flow boundary condition [Arkilic and Breuer,1993] yielding the solution the drag force experienced by the moving plate:
6µu b l2
p a h2 min
Trang 4is a Stokes number, representing the balance between unsteady and viscous effects, and D Ris a correctiondue to slip flow at the wall:
D R 2Knβ(sinhβ sinβ) 2Kn2β2(coshβ cosβ) (11.9)
A typical MEMS geometry might have a plate separation of one micron and an operating frequency of
10 kHz With these parameters, the Stokes number is very small (approximately 0.1), and the flow may beconsidered quasi-steady to a high degree of approximation In addition, the rarefaction effects, indicated
by D R, are also vanishingly small at atmospheric conditions
11.4.1 Limit of Molecular Flow
Although the slip-flow solution is limited to low Knudsen numbers, the damping due to a gas at highdegrees of rarefaction can be computed using a free-molecular flow approximation In such cases the friction factor on a flat plate is given by Rohsenow and Choi (1961)
whereγ is the ratio of specific heats and M is the Mach number It is important to recognize that the damping (and Q) in this case is provided, not only by the flow in the gap, but also by the flow above the
vibrating plate However, it is unlikely that the fluid damping provides the dominant source of damping
at such extremely low pressures More likely, damping derived from the structure (e.g., flexing of the port tethers, non-elastic strain at material interfaces, etc.) will take over as the dominant energy-lossmechanism Kwok et al (2005) compared the continuum, slip-flow, and free molecular flow models forCouette damping with data obtained by measuring the “ring down” of a tuning fork gyroscope fabricated
sup-by Draper Laboratories Figure 11.2 shows the measurements and theory confirming the functional
behavior of the damping as the pressure drops (Kn increases) and the unexpected accuracy of these rather simple models Although the trends are well-predicted, the absolute value of the Q-factor is still in error
by a factor of two, suggesting that more detailed computations are still of interest
Slip flow Molecular flow
FIGURE 11.2 (See color insert following page 10-34 ) Theory and measurements of Couette damping in a tuning
fork gyro (Kwok et al [2005]) Note that in the high Knudsen number limit, the free molecular approximation
pre-dicts the damping more closely, but that the slip-flow model, though totally inappropriate at this high Kn level, is not
too far from the experimental measurements
Trang 511.5 Squeeze-Film Damping
Squeeze-film damping arises when the gap size changes in an oscillatory manner squeezing the trappedfluid (Figure 11.3) Fluid, usually air, is trapped between the vibrating proof mass and the substrate result-ing in a squeeze film, which can significantly reduce the quality factor of the resonator In some cases thisdamping is desirable, but as with the case of Couette-flow damping, it is often parasitic, and the MEMS
designer tries to minimize its effects and maximize the resonant Q-factor of the device Common methods
for alleviating squeeze-film effects are to fabricate breathing holes (“chimneys”) throughout the proof masswhich relieve the build up of pressure and to package the device at low pressure Both of these solutions havedrawbacks The introduction of vent holes reduces the vibrating mass, necessitating an even larger structure,while the low-pressure packaging adds considerable complexity to the overall device development andcost Figure 11.4 illustrates a high-performance tuning fork gyroscope fabricated by Draper Laboratories
11.5.1 Derivation of Governing Equations
The analysis of the squeeze-film damping is presented in the following section The Reynolds equationsmay be used as the starting point However, a particularly elegant and complete solution was published
by Blech (1983) for the case of the continuum flow and was extended by Kwok et al (2005) to the case ofslip-flow and flows films with vent holes This analysis is summarized here
Trang 6The Navier–Stokes equations are written for the case of a parallel plate vibrating sinusoidally in a scribed manner in the vertical direction If we assume that the motion, subsequent pressure, and velocityperturbations are small, a perturbation analysis yields the classical squeeze-film equation derived byBlech, with an additional term due to the rarefaction:
Assuming small amplitude, harmonic forcing of the gap H 1 εsin T, and a harmonic response of
the pressure, we can derive a pair of coupled equations describing the in-phase (Ψ0) and out-of-phase(Ψ1) pressure distributions in the gap representing stiffness and damping coefficients, respectively:
m⬅σ/(1 6K) The solutions are achieved either by manual substitution of Fourier sine and
cosine series or by direct numerical solution The results for rectangular plates with no vent holes areshown in Figure 11.5
σ
1 6K
∂2Ψ1
FIGURE 11.5 (See color insert following page 10-34 ) Solutions to the squeeze-film equation for a rectangular plate.
The stiffness and damping coefficients are presented as functions of the modified squeeze number, which includes acorrection due to first-order rarefaction effects [Blech, 1983; Kwok et al., 2005]
Trang 711.5.2 Effects of Vent Holes
The equations as previously derived are made more useful by extending them to account for the presence
of vent holes in the vibrating proof mass In such cases the boundary condition at each vent hole is nolonger atmospheric pressure (Ψ0 Ψ1 0), but rather an elevated pressure proscribed by the pressuredrop through the “chimney” which vents the squeeze film to the ambient Kwok et al (2005) demonstratethat this can be incorporated into the previous model (in the limit of low squeeze number) by a modi-fied boundary condition for the squeeze-film equations for Ψ0:
This boundary condition has three components: a geometric component dependent on the plate
thick-ness t, length L x , hole size L h , and nominal gap size h0; a rarefaction component (here based on the holesize); and a time-dependent component — the squeeze numberσ Note that as the thickness of the platedecreases and the chimney pressure drop falls, the boundary condition approaches zero Similarly, as theopen area fraction of the plate increases (more venting), the boundary condition approaches that of theambient This boundary condition can be applied at the chimney locations and can accurately simulatethe squeeze-film damping of perforated micromachined plates
11.5.3 Reduced-Order Models for Complex Geometries
Most devices of practical interest have geometries that are too complex to enable full numerical tion of the kind described previously Reduced-order models are of great value in such cases Many suchmodels have been developed, including those based on acousto-electric models [Veijola et al., 1995] Inthe case of squeeze-film damping in the limit of low squeeze numbers, such models reduce to solution of
simula-a resistor network thsimula-at models the pressure drops simula-associsimula-ated with esimula-ach segment of the squeeze film This
is effectively a finite-element approach to the problem Instead of modeling a large number of elements,
as is generally the case in a numerical solution, a relatively small number of discrete elements can be used,
if higher-order solutions can be employed to connect each element together Kwok et al (2005) strate this approach and model the damping associated with an inclined plate with vent holes More com-plex numerical solution techniques based on boundary integral techniques have also been presented[Aluru and White, 1998; Kanapka and White, 1999] providing a good balance between solution fidelityand required computing power
demon-11.6 Lubrication in Rotating Devices
Rotating MEMS devices bring a new level of complexity to MEMS fabrication and to the lubrication siderations As discussed in the introduction, many rotors and motors have been demonstrated with dry-rubbing bearings, and the success of these devices is due to the low surface speeds of the rotors However,
con-as the surface speed increcon-ases in order to get high power densities, the dry rubbing bearings are no longer
an option, and true lubrication systems need to be considered An example of “Power-MEMS” ment is provided by a project initiated at the Massachusetts Institute of Technology in 1995 to demon-strate a fully functional microfabricated gas turbine engine [Epstein et al., 1997] The baseline engine,illustrated in Figure 11.6, consists of a centrifugal compressor, fuel injectors (hydrogen is the initial fuel,although hydrocarbons are planned for later configurations), a combustor operating at 1600 K, and aradial inflow turbine The device is constructed from single crystal silicon and is fabricated by extensiveand complex fabrication of multiple silicon wafers that are fusion bonded in a stack to form the complete
Trang 8device An electrostatic induction generator may also be mounted on a shroud above the compressor toproduce electric power instead of thrust [Nagle and Lang, 1999] The baseline MIT Microengine has atits core a “stepped” rotor consisting of a compressor with an 8 mm diameter and a journal bearing andturbine with a diameter of 6 mm The rotor spins at 1.2 million r/min.
FIGURE 11.6 (See color insert following page 10-34 ) Schematic of the MIT Microengine, showing the air path
through the compressor, combustor, and turbine Forward and aft thrust bearings located on the centerline hold therotor in axial equilibrium, while a journal bearing around the rotor periphery holds the rotor in radial equilibrium
Forward thrust bearing
Journal bearing Rotor
High pressure plenum
Low pressure plenum
Aft thrust bearing
Forward thrust bearing
Journal bearing
Rotor Main flow path
Axis of rotation
FIGURE 11.7 Illustrating schematic and corresponding SEM of a typical microfabricated rotor, supported by axialthrust bearings and a radial journal bearing
Trang 9Key to the successful realization of such a device is the ability to spin a silicon rotor at high speed in acontrolled and sustained manner The key to spinning a rotor at such high speeds is the demonstration
of efficient lubrication between the rotating and stationary structures The lubrication system needs to besimple enough to be fabricated but with sufficient performance and robustness to be of practical use in thedevelopment program and in future devices.Figure 11.7 illustrates a microbearing rig that was fabricated
to develop this technology The core of the rotating machinery has been implemented but without thesubstantial complications of the thermal environment that the full engine brings The rig consists of aradial inflow turbine mounted on a rotor and embedded inside two thrust bearings that provide axial sup-port A journal bearing located around the disk periphery provides radial support for the disk as it rotates
11.7 Constraints on MEMS Bearing Geometries
11.7.1 Device Aspect Ratio
Perhaps the most restrictive aspect of microbearing design is that MEMS devices are limited to rather low etches, resulting in devices with low aspect ratio Even with the advent of deep reactive ion etchers(DRIE) in which the ion etching cycle is interleaved with a polymer passivation step, the maximum prac-tical etch depth that can be achieved while maintaining dimensional control is about 500 microns Eventhis has an etch time of about nine hours, which makes its adoption a very costly decision In compari-son, typical rotor dimensions are a few millimeters The result is that microbearings are characterized by
shal-very low aspect ratios (Length/Depth, or L/D) In the case of the MIT microturbine test rig, the journal
bearing is nominally 300 microns deep while the rotor is 4 mm in diameter, yielding an aspect ratio of0.075 To put this in perspective, commonly available design charts [Wilcox, 1972] present data for val-
ues of L/D as low as 0.5 or perhaps 0.25 Prior to this work there was no data for lower L/D The
impli-cations of the low aspect ratio bearings are that the task becomes supporting a disk rather than a shaft.The low aspect ratio bearings do not have terrible performance by any standard The key features of
the low L/D bearings are:
The load capacity is reduced compared to conventional designs This is because the fluid leaking out ofthe ends relieves any tendency for the bearing to build up a pressure distribution For a given geom-etry and speed, a short bearing supports a lower load per unit length than its longer counterpart.The bearing acts as an incompressible bearing over a wide range of operation Pressure rises, whichmight lead to gas compressibility, are minimized by the flow leaking out of the short bearing.Incompressible behavior (without the usual fluid cavitation that is commonly assumed in incom-pressible liquid bearings) can be observed to relatively high speeds and eccentricities
11.7.2 Minimum Etchable Clearance
It is reasonable to question why one could not fabricate a 300 micron long “shaft”, but with a much
smaller diameter, to greatly enhance the L/D For example, a shaft with a diameter of 300 microns would result in a reasonable value for L/D This raises the second major constraint on bearing design by current
microfabrication technologies — that of the minimum etchable clearance
In the current microengine manufacturing process, the bearing and rotor combination is defined by asingle deep and narrow etch, currently 300 microns deep and about 12 microns in width No foreseeableadvance in fabrication technology will make it possible to significantly reduce the minimum etchableclearance, and this has considerable implications for bearing design In particular, if one were to fabricate
a bearing with a diameter of 300 microns in an attempt to improve the L/D ratio, the result would be a bearing with a clearance to radius c/R of 12/300, or 0.04 For a fluid bearing, this is two orders of magni-
tude above conventional bearings and has several detrimental implications
The most severe implication is the impact on the dynamic stability of the bearing The non-dimensional
mass of the rotor depends on (c/R) raised to the fifth power [Piekos, 2000] Bringing the bearing into the center of the disk and raising the c/R by a factor of 10 results in a mass parameter increasing by a factor
of 105 This increase in effective mass has severe implications for the stability of the bearing
Trang 10These reasons and others not enumerated here make the implementation of an inner-radius bearing
less attractive Therefore, the constraint of small L/D is unassailable as long as one requires that the
microdevice be fabricated in situ If one were to imagine a change in the fabrication process such that therotor and bearing could be fabricated separately and subsequently assembled reliably, this situation would
be quite different In such an event, the bearing gap is not constrained by the minimum etch dimension
of the fabrication process, and almost any “conventional” bearing geometry could be considered andwould probably be superior in performance to the bearings discussed here Such fabrication could beconsidered for a “one-off ” device, but does not appear feasible for mass production, which relies on themonolithic fabrication of the parts Lastly, the risk of contamination during assembly — a common con-cern for all precision-machined MEMS — effectively rules out piece-by-piece manufacture and assemblyand constrains the bearing geometry as described
11.8 Thrust Bearings
Thrust bearings support any axial loads generated by rotating devices such as turbines, engines, ormotors Current fabrication techniques require that the axis of rotation in MEMS devices lie normal tothe lithographic plane This lends a significant advantage in the design and operation of thrust bearingsbecause the area available for the thrust bearing is relatively large as defined by lithography, while theweight of the rotating elements will be typically small due to the cube-square law and the low thicknesses
of microfabricated parts For these reasons, thrust bearings are one area of microlubrication where tions abound and problems are relatively easily dealt with
solu-Two thrust bearing options exist: (a) hydrostatic (externally-pressured) thrust bearings, in which thefluid is fed from a high-pressure source to a lubrication film, and (b) hydrodynamic, where the support-ing pressure is generated by a viscous pump fabricated on the surface of the thrust bearing itself (see
Figure 11.9) Hydrostatic bearings are easy to operate and relatively easy to fabricate These have been cessfully demonstrated in the MIT Microengine program [Frechette et al., 2005; Liu et al., 2003] Thethrust bearing in Figure 11.8 shows an scanning electron micrograph (SEM) of the fabricated device cutthough the middle to reveal the plenum, restrictor holes, and the bearing lubrication gap, which isapproximately 1 micron wide Key to the successful operation of hydrostatic thrust bearings is the accurate
suc-FIGURE 11.8 Close-up cutaway view of micro thrust bearing showing the pressure plenum (on top), the feed-holes,and the bearing gap (faintly visible) (SEM reprinted with permission of Lin et al [1999].)
Trang 11manufacture of the restrictor holes, maintenance of sharp edges at the restrictor exit, and careful control
of the dimension of the lubrication film In an initial fabrication run, the restrictor holes were fabricated
2 microns larger than specified While the bearing operated, its performance was well below its designpeak because of the off-design restrictor size Current specifications of the fabrication protocols controlthe restrictor size carefully, ensuring close to optimal operation
Hydrodynamic or spiral groove bearings (SGBs), illustrated in Figure 11.9, were first analyzed in detailforty years ago [Muijderman, 1966] but have not received much attention due to their low load capacitycompared to hydrostatic thrust bearings and due to complex manufacturing requirements
SGBs operate by using the rotor motion against a series of spiral grooves etched in the bearing to cously pump fluid into the lubrication gap This process creates a high-pressure cushion on which the rotorcan ride The devices typically have relatively low load capacity, which has limited their use in macroscopicapplications The load capacity becomes more than adequate at microscales due to favorable cube-square
vis-−0 15
−0 3
0.30
FIGURE 11.9 Schematic of hydrodynamic thrust bearings and predicted performance (stiffness in N/m vs axialeccentricity) for a typical spiral groove thrust bearing for use in a high speed MEMS rotor
Trang 12scaling Thus, they gain considerable advantage when compared with conventional hydrostatic thrustbearings as the scale and Reynolds number decreases In addition, the fabrication of the multitude ofshallow spiral features, which is an expensive task for a traditional SGB, is ideally suited for lithographicfabrication technologies such as MEMS.
Figure 11.9 illustrates the bearing stiffness for a particular single-point design for the MIT microrotorrotating at design speed (2.4 million r/min) and supported by matched forward and aft spiral groove bear-ings The stiffness at full speed is quite impressive and superior to comparable hydrostatic bearings, butthe SGB do suffer at lower speeds since the bearing stiffness is roughly proportional to rotational speed.For this design, the lift-off speed (the speed at which the film can support the weight of the rotor and thepressure distribution associated with the turbine flow) is only a few thousand r/min, and the dry rubbingendured during startup will be minimal SGBs also have the strong advantage that the two matched spi-ral groove bearings, forward and aft, naturally balance each other with no supply pressures to maintain
or adjust, and the removal of the thrust bearing plena and restrictor holes considerably simplifies theoverall device fabrication This simplification allows for the use of two fewer wafers in the wafer-bondedstack, which is a considerable advantage from the perspective of manufacturing process cost and yield Ahybrid bearing consisting of both hydrostatic and hydrodynamic bearings has been recently successfullydemonstrated [Wong et al., 2004] up to a speed of approximately 450,000 r/min
11.9 Journal Bearings
Journal bearings, which are used to support radial loads in a rotating machine, have somewhat unusualrequirements in MEMS These requirements derive from the extremely shallow structures that are cur-rently fabricated Rotating devices tend to be disk-shaped, and their corresponding journal bearings arecharacterized by very low aspect ratios which are defined as the ratio of the bearing height to its radius
In addition, the minimum etchable gap allowed by current fabrication techniques results in a paradoxically
large bearing clearance — a 10 micron gap over a 2 mm radius rotor, or a c/R of 1/200 This bearing ance is large in comparison to conventional journal bearings, which typically have c/R ratios that are
bear-11.9.1.1 Static Journal Bearing Behavior
Figure 11.10 shows the static behavior of a MEMS journal bearing This figure presents the load capacity
ζand the accompanying attitude angle (the angle between the applied load and the eccentricity vector)
as functions of the bearing number and the operating eccentricity The geometry considered here is for a
low-aspect ratio bearing (L/D 0.075) typical of a deep reactive ion etched rotor such as the MIT
microengine The bearing number is defined as:
Trang 13where µis the fluid viscosity,ωthe rotation rate, p the ambient pressure, and R/c the ratio of the radius
to clearance For a given bearing geometry, Λ can be interpreted as operating speed
Several aspects of these results should be noted The load capacity is quite small when compared with
bearings of higher L/D This is because for very short bearings, the applied load simply squeezes the fluid
out of the bearing ends, and consequently it is difficult to develop any significant restoring force Thesame mechanism is responsible for the load lines being straight Straight load lines indicate that little
= 0.9
FIGURE 11.10 Static performance (eccentricity and attitude angle vs bearing number) for a journal bearing with
L/D 0.075 Notice that the load lines are almost constant (linear), indicating the absence of compressibility effects.
This is also indicated by the attitude angle, which remains close to 90 degrees except at very high eccentricities [Piekosand Breuer, 1998]
Trang 14compressibility of the fluid is taking place, which usually results in a “saturation” of the load parameter athigher values of the bearing number Again, this is because any tendency to compress the gas is alleviated
by the fluid venting at the bearing edge The behavior of the attitude angle, which maintains a high angle(close to π/2) over a wide range of bearing numbers and eccentricities, illustrates this point This value of theattitude angle corresponds to the analytic behavior of a Full-Sommerfeld incompressible short bearing [Orr,1999] This value is a good approximation for such short bearings at low to moderate eccentricities when theeccentricity remains below approximately 0.6 Below 0.6, compressibility finally becomes important Thisincompressible behavior is much more extensive than conventional gas bearings of higher aspect ratio andhas profound ramifications, particularly with respect to the dynamic properties of the system
11.9.1.2 Journal Bearing Stability
The stability of a hydrodynamic journal bearing has long been recognized as troublesome and is owed by the static behavior shown in Figure 11.10.The high attitude angle suggests that the bearing springstiffness is dominated by cross stiffness as opposed to direct stiffness Thus, any perturbation to the rotorwill result in its motion perpendicular to the applied force If this reaction is not damped, the rotor will enter
foreshad-a whirling motion This is precisely whforeshad-at is observed, foreshad-and gforeshad-as beforeshad-arings foreshad-are notorious for their susceptibility
to fractional-speed whirl The instability is suppressed by the generation of more damping and increaseddirect stiffness, both of which are obtained by increasing the loading and the static eccentricity of the rotor.Figure 11.11 shows a somewhat unusual presentation of the stability boundaries for a low-aspect ratio
MEMS journal bearing The vertical axis shows the non-dimensional mass of the rotor M苶 which is defined as:
M
(11.16)This is the “mass” which appears in the non-dimensionalized equations of motion for the rotor and it is
fixed for a given geometry Close inspection of Figure 11.11 indicates that M苶 does changes very slightlywith speed This is because of the elastic expansion of the rotor due to centrifugal forces, the variation inthe ambient pressure at different speeds, and temperature effects on viscosity The horizontal axis ofFigure 11.11 shows the bearing number, which can be interpreted as speed, for a fixed bearing geometry.The contours on the graph represent the stability boundary at fixed eccentricity Stable operation lies
above each line For a fixed M苶 at low bearing number (i.e., speed), a minimum eccentricity must be
FIGURE 11.11 Stability boundaries for a typical microbearing plotted vs bearing number (speed for a fixed
geom-etry) The dotted line represents an operating line for a microbearing which has almost constant M苶 (varying only due
to centrifugal expansion of the rotor at high speeds [Piekos, 2000])
Trang 15obtained to ensure stability As the speed increases, this minimum eccentricity remains almost constant(the lines are horizontal) until a particular speed at which the lines break upward, and the minimumeccentricity required for stability starts to drop as indicated byFigure 11.10, as Λ increases, the loadrequired to maintain a fixed eccentricity increases linearly due to the stiffening of the hydrodynamic bear-ing The key feature of this chart is that the minimum eccentricities are very high and suggest that stableoperation requires running very close to the wall This is troublesome The high eccentricities are driven
by high values of the mass parameter M苶 which is due to the relatively high value of the
clearance-to-radius ratio (c/R) and the short length L The low aspect ratio (L/D) also contributes to high minimum
eccentricities At high speeds, the problem becomes less severe, because the high speed allows the bearing
to generate sufficient direct stiffness Even at these points, the eccentricity is very high and might not bemanageable in practical operation
Orr (1999) has demonstrated on a scaled-up experimental rig that matches the microengine geometrythat stable high-eccentricity operation is possible for extended periods of time His experiments achieved46,000 r/min which, when translated to the equivalent speed at the microscale, correspond to approxi-mately 1.6 million r/min In order to accomplish this high eccentricity operation, he noted that the rotorsystem must (a) have very good axial thrust bearings to control axial and tipping modes of the rotor sys-tem, and (b) be well-balanced A rotor with imbalance of more than a few percent could not be startedfrom rest Piekos (2000) also explored the tolerance of the microbearing system to imbalance and foundthat it was surprisingly robust to imbalance of several percent His computations were achieved assumingthat the rotor was at full speed and then carefully subjected to imbalance In practice, the imbalance willexist at rest, and the rotor is stable at full speed but unable to accelerate to that point This “operating line”issue is discussed in more detail by Savoulides et al (2001) who explored several options for acceleratingmicrobearings from rest under both hydrodynamic and hydrostatic modes of operation
Figure 11.12 illustrates a convenient summary of the trade-offs for design of a hydrodynamic MEMSbearing This figure presents the variation of the low-speed minimum eccentricity asymptote, or worst-
case eccentricity, as a function of the mass parameter M 苶 and other geometric factors (L/D, clearance, c,
0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5
L/D 0 7
L/D 0 5