181
Copyright by Daejong Kim 2004
Copyright
by
Daejong Kim
2004
Design and Fabrication of Sub-Millimeter Scale Gas Bearings
with Tungsten-Containing Diamond Like Carbon Coatings
by
Daejong Kim, B.S.; M.S.
Dissertation
Presented to the Faculty of the Graduate School of
The University of Texas at Austin
in Partial Fulfillment
of the Requirements
for the Degree of
Doctor of Philosophy
The University of Texas at Austin
May, 2004
The dissertation Committee for Daejong Kim certifies that this is the approved version of the following dissertation:
Design and Fabrication of Sub-Millimeter Scale Gas Bearings
with Tungsten-Containing Diamond Like Carbon Coatings
Committee:
Michael D. Bryant, Supervisor
Frederick F. Ling
Wenjin Meng
Li Shi
Shaochen Chen
Dedication
To my parents and family who endlessly loved and supported me
v
Acknowledgements
I am grateful to all the people who were with me throughout the PhD program. Most
special appreciation goes to my advisor, Dr. Michael D. Bryant, for helping me and
guiding me to accomplish this work. I was very fortunate to have him as an advisor. He
has always motivated me, gave emotional support to overcome every difficulty
encountered during the work and led me to be a good researcher. Another special
appreciation goes to Dr. Frederick F. Ling, co-advisor, for leading me to this exciting
area and all his emotional support. I appreciate Dr. Wenjin Meng, Louisiana State
University for valuable advice and support for coating. I also appreciate Dr. Shaochen
Chen and Dr. Li Shi for their valuable advice and help to finish my PhD work.
Special thanks go to Yohannes Desta, Yoonyoung Jin, and Dr. Jost Goettert for their
help during the fabrication of gas bearings. Additionally, I thank Don Artieschoufsky,
Curtis Johnson, Danny Jares and John Pedrachine for their technical support for
machining parts and other staffs in the department. Partial support from NSF is also
acknowledged.
Special thanks go to Byungsoon Kim, Myungjin Kim and their family, who shared
happy and exciting life in Austin. Thanks to office mates, Jihoon, Sanghoon, Huijie and
other Korean graduate students in the department. Our family will miss all the wonderful
memories of neighbors and families in Brackenridge apartments. Finally, this work was
not possible without endless love and emotional support of my wife Donghee, my
children, Yushin and Edward, and my parents, brothers and sisters in Korea.
vi
Design and Fabrication of Sub-Millimeter Scale Gas Bearings with
Tungsten-Containing Diamond Like Carbon Coatings
Publication No._____________
Daejong Kim, PhD
The University of Texas at Austin, 2004
Supervisor: Michael D. Bryant
Micro gas bearings of sub millimeter size (diameter of 500µm and length of
300µm) with micron clearances were designed and fabricated through X-ray lithography
and Ni electroplating. Details of the fabrication processes for the micro gas bearings are
described. Procedures to make X-ray mask and technical issues are discussed. Static and
dynamic hydrodynamic characteristics of the micro gas bearings were assessed using
Molecular Gas Lubrication (MGL) theory. Fabricated micro gas bearings had lower
stability than plain circular gas bearings due to lower load capacity and poor gas
damping. Improved bearing designs having non-symmetrical step geometry with deep
axial groove were suggested. These bearings were predicted to have much higher load
capacities and dynamic stabilities than the fabricated micro gas bearings.
Micro gas bearings were tested via an air jet-driven micro turbine made of SU-8,
press-fitted onto the shaft. A self-aligning assembly technique for the micro gas bearings
(using capillary action of photo resist) was developed and proved to be very effective.
Successful operation of micro gas bearings at 60,000 rpm was demonstrated under
limited operating conditions.
Tribological characteristics of Ni micro bearings and amorphous tungsten
hydrocarbon (W-C:H) coated micro bearings were investigated in dry friction mode,
vii
using a newly designed micro tribo tester. 900nm thick uniform conformal coatings were
deposited onto the inner surface of micro bearings. Chemical and mechanical
microstructures were studied via X-ray Photoelectron Spectroscopy (XPS), Raman micro
spectroscopy. Wear rates, mechanical and material properties, and other tribological
characteristics of W-C:H coated Ni micro bearings were investigated and compared to
those of uncoated Ni micro bearings.
Uncoated Ni micro bearings, as deposited and annealed at 800oC, experienced
severe wear and appeared inadequate for tribological applications. Micro bearings with
low tungsten-containing (5% wt) hydrocarbon had higher wear resistance than micro
bearings with high tungsten-containing (11% wt) hydrocarbon. During the wear test of
the W-C:H coated micro bearings, a transfer layer formed on the counter steel shaft even
under very small contact pressure, leading to low steady state friction and high wear
resistance.
viii
Table of Contents
List of Tables .................................................................................................................... xii
List of Figures .................................................................................................................. xiii
Chapter 1 Introduction .........................................................................................................1 1.1 Reliability Issue on MEMS...................................................................................1 1.2 Micro Gas Bearings ..............................................................................................2
1.2.1 Operating Principle of Gas Bearings ...........................................................2 1.2.2 Past Research on Micro Gas Bearings .........................................................4
1.3 Mechanism of Stiction of Two Surfaces in MEMS..............................................6 1.3.1 Capillary Force ............................................................................................6 1.3.2 Van der Waals Force....................................................................................8 1.3.3 Electrostatic Force .......................................................................................9
1.4 Review of Nano Scale Friction Study via AFM .................................................10 1.5 Past Studies on Coatings for MEMS devices .....................................................14
1.5.1 Chemisorbed Monolayer............................................................................15 1.5.2 Hydrocarbon Based Coatings ....................................................................16
1.6 Overview of Dissertation ....................................................................................18
Chapter 2 Fabrication Processes of Micro Gas Bearings...................................................19 2.1 Introduction.........................................................................................................19 2.2 Fabrication Processes of Micro Gas Bearings ....................................................20 2.3 X-ray Lithography on PMMA ............................................................................21
2.3.1 Substrate Preparation .................................................................................21 2.3.2 Material Selection for X-ray Mask ............................................................22 2.3.3 Fabrication Processes of X-ray Mask ........................................................24 2.3.4 X-ray Exposure and Development.............................................................26
2.4 Post Processes .....................................................................................................26 2.4.1 Electroplating.............................................................................................26 2.4.2 Processes to Form Thrust Bearings............................................................31
ix
2.4.3 Sacrificial Layer Etching to Release Bearings...........................................32
Chapter 3 Static Performance of Micro Gas Bearings.......................................................35 3.1 Theory .................................................................................................................35
3.1.1 Journal Bearing ..........................................................................................39 3.1.2 Thrust Bearing ...........................................................................................43
3.2 Numerical Method ..............................................................................................46 3.3 Static Analysis ....................................................................................................47
3.3.1 Journal Bearing ..........................................................................................47 3.3.2 Thrust Bearing ...........................................................................................54 3.3.3 Rotational Friction Factor..........................................................................54
Chapter 4 Dynamic Performance of Micro Gas Bearings .................................................56 4.1 Approach.............................................................................................................56
4.1.1 Scheme for Numerical Integration.............................................................58 4.1.2 Stability Analysis: Threshold Speed..........................................................59
4.2 Discussion on the Whirl Instability.....................................................................64 4.2.1 Quasi-Stable Behavior of Stepped Bearings..............................................64 4.2.2 Threshold Rotor Mass................................................................................67
4.3 Design Improvement of Stepped Micro Gas Bearings .......................................69 4.3.1 Static Analysis of Improved Design ..........................................................70 4.3.2 Dynamic Analysis of Improved Design.....................................................73 4.3.3 Four-stepped Bearings with Axial Grooves...............................................76
4.4 Discussions from Stability Analyses ..................................................................84 4.5 Feasibility Study of Meso Scale Gas Bearing.....................................................84
4.5.1 Fabrication Processes of Meso Scale Gas Bearings ..................................84 4.5.2 Applications of Meso Scale Gas Bearings.................................................85 4.5.3 Three-Dimensional Imbalance Response ..................................................88
Chapter 5 Testing of Gas Bearings ....................................................................................97 5.1 Issues on Gas Bearing Tests ...............................................................................97 5.2 Testing of Micro Gas Bearings .........................................................................99
5.2.1 Assembly of micro gas bearings ..............................................................100
x
5.2.2 Test Results..............................................................................................102
Chapter 6 Tribological Study of Micro Bearings ..........................................................112 6.1 Introduction.......................................................................................................112 6.2 Coating Process of W-C:H on Micro Sleeve bearings......................................115
6.2.1 Process conditions of W-C:H coatings ....................................................116 6.2.2 Material Properties of W-C:H coatings ...................................................117
6.3 Micro Tribo Tester ............................................................................................119 6.4 Results and Discussions....................................................................................123
6.4.1 Nickel Bearings........................................................................................123 6.4.2 W-C:H Coated Micro Bearings ...............................................................127
6.5 Summary and Conclusions of the Experiments ................................................132
Chapter 7 Future Work and Conclusions.........................................................................133 7.1 Future Works ....................................................................................................133
7.1.1 Hydrostatic Meso Scale Gas Bearing ......................................................133 7.1.2 Meso Scale Foil Gas Bearing...................................................................134 7.1.3 New Software for Performance Analyses of Gas Bearings .....................136 7.1.4 Advanced Testing Method of Gas Bearings ............................................137 7.1.5 Improvement of Micro Tribo Tester ........................................................139 7.1.6 Tribological Characteristics at Various Environments ............................139 7.1.7 Other Suggestions ....................................................................................139
7.2 Contributions and Conclusions of Dissertation ................................................140
Appendix A Discretization of the Gas Film Equation .....................................................143
Nomenclature...................................................................................................................149
References........................................................................................................................155
Vita ...............................................................................................................................162
xi
List of Tables
Table 2.1: Chemical composition of Ti oxidation solution............................................22
Table 2.2: Chemical composition of PMMA bonding epoxy ........................................22
Table 2.3: Chemical compositions of G-G developer and rinse ....................................26
Table 2.4: Chemical composition of Ni sulfamate solution and the process condition.29
Table 3.1: Numerical resolution for various grid schemes ( Λ=1, θR =-15o) .................48
Table 4.1: ω* and m* for stepped bearing with axial grooves (θS/θP =0.5, θR=0o) ....74
Table 4.2: ω* and m* for different θS/θP, the number of steps (θR=0o, ε=0.6, Λ=1) ...74
Table 4.3: Threshold speed ω* and rotor mass m* for four-stepped gas bearings with
axial grooves; C=1µm, step height=1µm, θS/θP =0.333 (θS =30o) ...............79
Table 4.4: Design parameters of LSU rotor ...................................................................88
Table 4.5: Design parameters of HDD rotor ................................................................88
Table 5.1: Load capacity and threshold rotor mass of plain micro gas bearings .........108
Table 5.2: Load capacity and threshold rotor mass of stepped micro gas bearings.....108
Table 6.1: Chemical composition and mechanical properties of 900nm thick W-C:H
coating.........................................................................................................117
Table 6.2: Wear rates of Ni bearings ...........................................................................124
Table 6.3: Wear rates of W-C:H coated micro bearings..............................................128
xii
List of Figures
Figure 1.1: Stiction failure of surface micro machined micro cantilever beam array
[Maboudian and Howe, 1997] ........................................................................2
Figure 1.2: Failure of poly silicon joint by surface friction [Tanner, 2000] .....................2
Figure 1.3: Operating principles of gas journal bearings ..................................................3
Figure 1.4: Micro fabricated micro gas bearing test rig [Lin, 1999].................................5
Figure 1.5: Capillary condensations around spherical asperity.........................................7
Figure 1.6: Surface energy balance at interface of three media ........................................7
Figure 1.7: Deformation of elastic sphere on rigid surface [Israelachvili, 1992]............13
Figure 1.8: Friction coefficients of 250nm thick DLC on glass: (a) Friction coefficient-
tip radius (b) Friction coefficient-applied load (contact pressure) [Bandorf et
al, 2003] ........................................................................................................14
Figure 1.9: Friction coefficient in (a) vacuum / ambient (b) dry N2 / ambient [Donnet et
al 1994] : Test conditions :contact pressure 1GPa ,sliding speed 1.7mm/s,
Ra=0.28nm, Steel ball on DLC.....................................................................17
Figure 2.1: Micro gas bearing design, the heights of steps and recesses are exaggerated,
and the overall diameter is 2mm: (a) Dimensions of journal bearings, (b)
Dimensions of thrust bearings ......................................................................19
Figure 2.2: Overall fabrication processes of micro gas bearings: (a) X-ray lithography,
(b) Ni plating/polishing, (c) SU-8 photolithography, (d) Ni plating/polishing,
(e) Photolithography/etching, (f) Releasing..................................................21
Figure 2.3: Absorption contrast of 3µm Ti membrane mask versus Au thickness with
various PMMA thicknesses ..........................................................................23
Figure 2.4: SEM image of patterned 13µm thick SPR: (a) overall image of bearings, (b)
near perfect vertical sidewall ........................................................................25
Figure 2.5: SEM images of Ti-membrane X-ray mask with 8µm thick Au absorber: (a)
Top view, (b) 2µm step in journal bearings..................................................25
Figure 2.6: Ti membrane X-ray mask with Au absorber.................................................25
Figure 2.7: Reaction in the Ni electroplating bath ..........................................................27
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Figure 2.8: Contamination of Ni electro deposit surface by nickel sulfide particles,
shown as small black dots on white Ni electro deposit ................................28
Figure 2.9: Non uniform current density at the edge of patterned plating mold cause non
uniform deposit thickness [Judy, 1996] ........................................................30
Figure 2.10: High magnification SEM image of bearing sidewall: cavity formed by air
bubbles attached to PMMM..........................................................................30
Figure 2.11: SEM images of bearing surface after different sacrificial layer etching time:
(a) Bearing surface after 2 days release (× 2K), (b) Bearing surface after 2
weeks release (× 2K).....................................................................................33
Figure 2.12: Higher magnification of Figure 2.11: (a) Bearing surface after 2 days release
(× 10K), (b) Bearing surface after 2 weeks release (× 10K).........................33
Figure 2.13: Micro gas bearing: (a) SEM images of overall bearing, (b) SEM images of
2µm steps on journal bearings before processing thrust bearings, (c)
Magnified image of 2µm step.......................................................................34
Figure 2.14: Surface morphology of bearing surface measured from AFM (scan length
10µm×10µm) and SEM: (a) AFM topography, (b) High magnification SEM
images ...........................................................................................................34
Figure 3.1: Journal bearing in normal operating condition .............................................36
Figure 3.2: Comparison of shear stress factor by Couette flow for fully diffusive walls;
database from Kang [Kang, 1997] and first order slip model.......................43
Figure 3.3: Grid scheme for control volume method ......................................................47
Figure 3.4: Stepped bearing configuration ......................................................................48
Figure 3.5: Static performance of stepped micro gas bearings (θR =5o) .........................50
Figure 3.6: Static performance of plain micro gas journal bearings ...............................50
Figure 3.7: Pressure profiles (Λ=1, ε=0.4) of fabricated stepped gas bearing ................51
Figure 3.8: Load parameters and attitude angles of stepped micro gas journal bearing at
ε=0.8 for different θR ....................................................................................52
Figure 3.9: Load parameters of stepped micro gas journal bearing at ε=0.6 and ε=0.98
for different θR ..............................................................................................53
xiv
Figure 3.10: Circumferential pressure profile at the bearing center (Λ=0.2, θR= 25o)......53
Figure 3.11: Direction of load capacity vector F for the conditions in Figure 3.10 with
ε=0.98, Λ=0.2, and θR= 25o. ζX = 0.0318 and ζY = 0.0156............................54
Figure 3.12: Load capacity as a function of step height for thrust bearings .....................55
Figure 3.13: Non-dimensional rotational friction of micro gas bearing (θR =5o for stepped
journal bearings, step height 3µm for thrust bearings) .................................55
Figure 4.1: Journal bearing operating at an equilibrium point with eccentricity e, where
F is bearing reaction force represented as integral term in Equation (4.1)...57
Figure 4.2: Converging orbit from origin (stepped gas journal bearing, Λ=0.6, ε0=0.6,
ω*=0.4, C=1µm) .........................................................................................61
Figure 4.3: Diverging orbits Plain bearing: (a) Λ=1, ω*=1.6, ε0=0.6, (b) Stepped
bearing, Λ=3, ω*=2.7, ε0=0.8 .......................................................................62
Figure 4.4: Converging orbit (stepped gas journal bearing, Λ=2, ω*=1.5, ε0=0.8) ..62
Figure 4.5: Threshold speed map for plain gas bearings.................................................63
Figure 4.6: Threshold speed map for stepped gas bearings.............................................63
Figure 4.7: Stabilizing motion of journal to (εX,εY) =(0.694,0.547) by small disturbance
at (εX0,εY0)=(0.8,0), with Λ=0.6, ω*=0.9 ......................................................65
Figure 4.8: Motion of journal to (εX,εY) =(0.694,0.547) from origin for static loading
corresponding to (εX0,εY0)=(0.8,0) with Λ=0.6, ω*=0.9. The angle between
external load and eccentricity vector is attitude angle ..................................65
Figure 4.9: Pressure profile at (εX,εY)=(0.694,0.547) ......................................................66
Figure 4.10: Pressure profile at (εX,εY)=(0.8,0.0) ..............................................................66
Figure 4.11: Self sustained vibration by perturbation, 50 cycles, ε=0.8, Λ=1, ω*=1.7,
stepped gas journal bearing...........................................................................66
Figure 4.12: Stability chart of plain gas journal bearing given as non-dimensional
threshold speed m*........................................................................................68
Figure 4.13: Stability chart of stepped gas journal bearing given as non-dimensional
threshold speed m*........................................................................................68
xv
Figure 4.14: Axial grooves formed at the beginning of every step. Axial grooves supply
gas with ambient pressure to minimize high negative pressure and decrease
attitude angles ...............................................................................................69
Figure 4.15: Static performance for stepped gas journal bearings with axial grooves (θR=
0o): (a) Λ=1, (b) Λ=10...................................................................................71
Figure 4.16: Load parameter and attitude angle of six-stepped bearings without axial
grooves for various step configurations (θR=0o, ε=0.6, and Λ=1) ................72
Figure 4.17: Load parameter and attitude angle of four-stepped bearings without axial
grooves for various step configurations for (θR=0o, ε=0.6, and Λ=1) ..........72
Figure 4.18: Orbit for 50 cycles for four stepped bearing without axial grooves, θS/θP
=0.333, Λ=1, ε=0.6, step height 2µm, and ω*=6 (m*=1.5022, m=0.12g)....75
Figure 4.19: Four-stepped micro gas bearings with axial grooves....................................76
Figure 4.20: Static performance of four-stepped micro gas journal bearings with axial
grooves: (a) C=1µm, step height 2µm, (b) C=1µm, step height 1µm ..........77
Figure 4.21: Pressure profiles (Λ=1, ε=0.4) of four-stepped micro gas bearings with axial
grooves..........................................................................................................78
Figure 4.22: Orbit for 100 cycles of four-stepped gas bearings with axial grooves, Λ=1,
ε=0.6, and ω*=15: (a) Orbit for step height 1µm, (b) Orbit for step height
2µm ...............................................................................................................80
Figure 4.23: Orbit for 400 cycles of four-stepped gas bearings with axial grooves, Λ=1,
ε=0.6, step height 2µm, and ω*=40. Very slow chaotic motion initiates and
does not converge or diverge ........................................................................81
Figure 4.24: Orbit for four-stepped gas bearings with axial grooves, Λ=5, ε=0.6, ω*=5,
and step height 1µm: (a) Orbit for total 200 cycles, (b) Orbit for last 10
cycles.............................................................................................................82
Figure 4.25: Orbit for 200 cycles of four-stepped gas bearings with axial grooves, Λ=5,
ε=0.8, and step height 1µm: (a) Orbit for ω*=1, (b) Orbit for ω*=5, (c) Orbit
for ω*=10, (d) Orbit for ω*=20 ....................................................................83
xvi
Figure 4.26: Conceptual figure of meso scale gas bearing with spiral grooved thrust
bearings, D=L=2mm.....................................................................................85
Figure 4.27: Laser scanner: (a) Assembled unit, (b) Rotor. The X-Y-Z is a fixed reference
frame and x-y-z is a rotational coordinate attached to rotor center ...............87
Figure 4.28: HDD spindle rotor: The X-Y-Z is a fixed reference frame and x-y-z is a
rotational coordinate attached to rotor center. y(Y) axis is into the plane....87
Figure 4.29: Rotor with angular rotation and eccentricity: The X-Y-Z is a fixed reference
frame with origin at the center of bearing and x-y-z is a rotational coordinate
attached to rotor center..................................................................................89
Figure 4.30: Three-dimensional imbalance response of HDD rotor, 50 cycles: (a) Orbit at
Z=0, (b) Stable rotor from initial misalignment (ψX0=0.2, ψY0=0.2),
Imbalance force: 0.56N, Speed: 15,000 RPM ..............................................93
Figure 4.31: Three-dimensional imbalance response of HDD rotor, 50 cycles: (a) Orbit at
Z=0, (b) Stable rotor from initial misalignment (ψX0=0.2, ψY0=0.2),
Imbalance force: 0.033N, Speed: 15,000 RPM ............................................93
Figure 4.32: Rotor orbits of Laser scanner rotor, 30 cycles; (a) 100,000rpm (b) 200,000
rpm (c) 300,000 rpm (d) 400,000rpm (e) 440,000rpm (f) 500,000 rpm (g)
600,000 rpm (h) 700,000 rpm (i) 800,000 rpm (j) 860,000 rpm...................95
Figure 4.33: Frequency spectrum of Laser scanner rotor orbit; (a) 660,000 rpm (b)
680,000 rpm (c) 700,000 rpm (d) 800,000 rpm ............................................96
Figure 4.34: Trace of misalignment of Laser scanner rotor at 860,000 rpm from initial
misalignment.................................................................................................96
Figure 5.1: Description of rotor running at certain eccentricity under given external
load................................................................................................................98
Figure 5.2: Macro scale gas bearing tester [Wilde and Andres, 2003] ...........................98
Figure 5.3: Meso scale turbo jet simulator supported by foil gas bearings [Heshmat,
2003] .............................................................................................................99
Figure 5.4: Photo of gas bearing tester; (a) Description of gas bearing tester (b) Photo of
assembled gas bearing tester (c) Capacitance sensor and drive shaft with
press-fitted SU-8 turbine.............................................................................101
xvii
Figure 5.5: Assembly procedures of micro gas bearings; (a) Pre-assembly (b) Photo
resist dispensing (c) Soft baking (d) Anchoring to base (e) Release .......102
Figure 5.6: Photonic sensor signal measuring rotor speeds; (a) Supply pressure 35psi
(b) Supply pressure 50psi............................................................................102
Figure 5.7: Impulse turbine with open air jet ................................................................103
Figure 5.8: Simulated speed of dental drill turbine .......................................................105
Figure 5.9: Vibration signal from bearings with 2.3mg rotor; (a) Vibration signal from
plain bearing, speed 8890 rpm (b) Vibration signal from step bearing ......106
Figure 5.10: SEM images after operation for 1 hour with repeated start/stops every 5 to
10 minutes...................................................................................................107
Figure 5.11: New gas bearing tester with machined shroud to drive micro turbine; (a)
Shroud (b) Enlarged image of circled region in (a) (c) Photo of assembled
new gas bearing tester.................................................................................109
Figure 5.12: Photonic sensor signal measuring rotor speeds with new gas bearing tester
shown in Figure 5.11, supply pressure 35psi, plain gas bearing.................110
Figure 5.13: Simulated orbit of 2.3mg rotor supported by plain gas bearing at 60,000 rpm
with combined load of rotor mass and preload (7.4mg) .............................111
Figure 5.14: SEM image of plain gas bearing surface after test with new test rig shown in
Figure 5.11 ..................................................................................................111
Figure 6.1: Ni micro bearing: (a) Photograph of micro sleeve bearings, (b) SEM image
of Ni micro sleeve bearing surface before coating, the lower arc is the inner
diameter.......................................................................................................113
Figure 6.2: Microstructure of sidewall of electroplated Ni, as deposited and annealed
[Hemker et al, 2001; Cho et al, 2003].........................................................115
Figure 6.3: Fixture to coat W-C:H coatings on the micro bearings: (a) 1mm thick
stainless steel sheet with holes to hold micro bearings (b) Conformal coating
process of bearing surfaces .........................................................................116
Figure 6.4: Photo of W-C:H coated micro bearings......................................................117
Figure 6.5: Raman spectrum of new W-C:H.................................................................118
Figure 6.6: 900 nm thick uniform coating thickness on the bearing surface ................118
xviii
Figure 6.7: SEM images of W-DLC2 coated bearing surface: (a) Low magnification
SEM image of W-DLC2 coated surface (b) High magnification SEM image
of W-DLC2 coated surface (×40K) ............................................................119
Figure 6.8: Micro wear tester: (a) Overview, (b) Detailed view within dotted circle in
(a), (c) Photo of micro wear tester ..............................................................121
Figure 6.9: Micro friction tester: (a) Schematic diagram, (b) Principle of friction
measurement (side view), (c) Photo of friction tester.................................122
Figure 6.10: Wear characteristics of as-deposited Ni bearings: (a) Top view, worn
materials moved along the axial direction and accumulated at the thrust
surface of the bearing (b) Inner bearing surface after test ..........................125
Figure 6.11: Wear characteristics of annealed (at 800oC for 1h) Ni bearings: (a) Top
view, worn materials moved along the axial direction and accumulated at
the thrust surface of the bearing as multiple layers (b) Inner bearing
surface after test ..........................................................................................126
Figure 6.12: SEM images of shaft tested against (a) As-deposited Ni bearing (b)
Annealed Ni bearings. No noticeable wear was observed..........................126
Figure 6.13: RPM-time relation of the W-C:H coated micro bearings ...........................128
Figure 6.14: SEM images of W-C:H coated micro bearing surfaces after wear test: (a) W-
DLC1 coated micro bearing (b) W-DLC2 coated micro bearing ...............128
Figure 6.15: Voltage signal from capacitance sensor and converted rotational angle of
horizontal bar attached to bearing holder with W-DLC1 coated bearing...130
Figure 6.16: Evolution of friction coefficient of W-C:H coated micro bearing..............130
Figure 6.17: Raman spectrum on the wear scar on the steel shaft tested with W-DLC2
coated bearing .............................................................................................131
Figure 6.18: SEM images of shaft tested against W-DLC2 coated bearing after 2hour
continuous wear test....................................................................................131
Figure 7.1: Conceptual design of hydrostatic micro gas bearing ..................................134
Figure 7.2: Detail fabrication processes of hydrostatic micro gas bearings..................135
Figure 7.3: Principle of foil gas bearings ......................................................................135
xix
Figure 7.4: Fabrication process of micro stamping mold for foil; (a) X-ray lithography
(b) Stacking (c) Solid micro stamping mold (d) Coated micro stamping
mold ............................................................................................................136
Figure 7.5: Gas bearing tester using reaction turbine....................................................138
Figure 7.6: Auxiliary devices for gas bearing tester in Figure 6.3: (a) Radial impulse
turbine (b) Centrifugal pump ......................................................................138
Figure A.1: Control volume (Figure 3.3 is repeated) ..................................................143
1
Chapter 1
Introduction
The first part of this chapter addresses reliability issues of micro electro
mechanical systems (MEMS). The second part introduces the gas bearing, reviews
operating principles of gas bearings and past research on micro gas bearings. Various
surface forces causing stiction failures in MEMS devices and past nano scale tribological
studies via Atomic Force Microscopy (AFM) are reviewed. Finally, past tribological
studies on MEMS surfaces are reviewed, and the dissertation is overviewed.
1.1 Reliability Issue on MEMS
In physical systems, the surface area to volume ratio is inversely proportional to
L, the characteristic dimension of the system [Drexler, 1992]. Therefore, as a 3-D
structure is miniaturized, surface effects become very important in characterizing the
behavior of micro scale systems. Common failures of IC-processed MEMS are stiction,
during the release and operation in humid environment [Maboudian et al, 2000;
Mastrangelo, 1997; Ashurst, 2001]. In Figure 1.1, the low two micro machined
cantilevers failed via stiction to the bottom surface.
In the case of micro mechanical systems with a relative sliding contact, friction
and wear at the contact surface are critical factors for reliability and long-term
performance [Bhushan, 1996; Tanner, 2000; Beerschwinger, 1997]. Figure 1.2 shows
catastrophic failure of a polysilicon joint fabricated through surface and bulk micro
machining.
Bearing systems in micro rotating machinery should have sufficient load carrying
capacity, low friction and high wear resistance for continuous operation. Small bearing
friction can cause severe power loss.
2
Figure 1.1 Stiction failure of surface micro machined micro cantilever beam
array [Maboudian and Howe, 1997]
Figure l.2 Failure of poly silicon joint by surface friction [Tanner, 2000]
1.2 Micro Gas Bearings To avoid wear and failure, frictionless bearings are needed. In practice, micro gas
bearings come close to this ideal.
1.2.1 Operating Principle of Gas Bearings Figure 1.3 explains the operating principle of gas journal bearings. The clearance
C between bearing and shaft was exaggerated. Pressure within the bearing clearance, to
support an external load, is generated via two mechanisms. One is a wedge effect
between the two converging surfaces. When a bearing is stationary and the journal shaft
rotates at a constant speed ω with eccentricity e, the converging wedge (at θ=0o ~180o) is
3
formed. The pressure builds between the two converging surfaces to maintain gas flow
continuity at the inlet (θ=0o) and outlet (θ=180o), assuming no side leakage. When there
is side leakage (that is, for actual bearings), the pressure from the wedge effect is smaller.
The other mechanism is the squeeze film effect. When the gas film in the bearing
clearance is squeezed suddenly, escaping gas molecules feel resistance from surrounding
gas molecules, and a positive pressure is generated. The pressure profile in a journal
bearing causes the journal shaft to orient with attitude angle φ with respect to the load
direction. The main advantages of gas bearings over high precision ball bearings are
frictionless operation, negligible thermal degradation, and ultra high precision. Macro
scale gas journal bearings have been used for high precision and high speed rotating
machinery for decades.
Chapters 3 and 4 present a detailed mathematical model, to estimate the pressure
distribution in the bearing clearance, and numerical analyses of the static and dynamic
performance of the micro gas bearings.
Figure 1.3 Operating principles of gas journal bearings
4
1.2.2 Past Research on Micro Gas Bearings Extensive research on micro gas bearings was performed at Massachusetts
Institute of Technology (MIT). Lin [Lin, 1999] demonstrated a test rig, with maximum
design speed of 2.4M rpm, shown in Figure 1.4. Deep Reactive Ion Etching (DRIE)
process was used to separate a rotor from bearing, resulting in a bearing clearance (C) of
about 10µm. The rotor diameter (D) and bearing length (L) were 4mm and 300µm
respectively, which resulted in L/D ratio of 0.0075 and C/D ratio of 0.003. Piekos &
Breuer [Piekos & Breuer, 1999], at MIT, performed hydrodynamic analyses of the micro
gas bearings embedded in the test rig. Orr Jr [Orr Jr, 2000] constructed a macro scale gas
bearing test rig with similar bearing geometry with the DRIE-processed micro gas
bearings, to validate the hydrodynamic analyseis of [Piekos & Breuer, 1999]. Frechette et
al [Frechette et al, 2001] demonstrated an electrostatic induction motor supported by the
DRIE-processed micro gas bearings.
The gas bearing test rig [Lin, 1999] was operated both in hydrostatic and
hydrodynamic modes. In the hydrostatic mode, compressed air was blown into the
bearing clearance from thrust side (along axial direction, see Figure 1.4), to generate a
fully developed journal bearing sump flow, and to levitate the rotor in the radial
direction1. Maximum speed was 110,000 rpm, far below the initial design speed (2.4M
rpm) in hydrostatic mode. In a hydrodynamic mode, 70,000 rpm was achieved for only
0.5 sec (total 500 rev). By continuous improvements (precision, supply pressure, etc) of
the test rig, they reached maximum 296,000 rpm in a hydrostatic mode [Lin, 1999].
In the macro scale, gas bearings are fabricated as separate components and
assembled to the system. However, in micro scale devices, assembly procedures are
known to be very tedious and time consuming. This led other researchers in micro gas
bearings to pursue single material monolithic structures. Problems of the silicon micro
gas bearings were, among others, use of limited micro machining process (DRIE), which
led to relatively a large bearing clearance, comparable to clearances of macro scale gas
1 In common hydrostatic gas bearings, air is supplied along the radial direction through small orifices or capillary tubes formed at the bearing surface.
5
bearings with rotor diameters 50-100mm. Whirl instability due to the large bearing
clearance, and catastrophic failure of the silicon rotor upon touch-down to bearing (made
of silicon, too) resulted.
Rotor
Figure 1.4 Micro fabricated micro gas bearing test rig [Lin, 1999]
This work pursued new approaches. Micro gas bearings were fabricated as a
single component through X-ray lithography and electroplating. A stainless steel gauge
pin machined to ultra precision (±0.25µm), served as a journal shaft. Simple and very
easy assembly processes using self-aligning concepts (such as surface tension or
precision dowel pins) were developed. Detailed specifications and fabrication processes
will be discussed in Chapter 2.
6
1.3 Mechanism of Stiction of Two Surfaces in MEMS As discussed earlier, one of the failure mechanisms of IC-processed MEMS
devices is stiction caused by strong surface forces, generated during release and/or
operation of the devices. Stiction also causes high friction in MEMS devices. In this
section, various surface forces are reviewed, and methods to minimize these surface
forces and friction are suggested.
1.3.1 Capillary Force When a microscopic asperity is close to a flat surface in a humid environment as
in Figure1.5, water vapor undergoes capillary condensation and forms a liquid meniscus.
The Kelvin radius of the meniscus rk is defined [Israelachvili, 1992] as
21
21
rrrr
rk += (1.1)
where r1 and r2 are radii of the meniscus curvature in principle directions. In a spherical
meniscus, r1=r2=rK. In a cylindrical meniscus, r2=∞. The Kelvin radius, derived from
equilibrium of the chemical potential across the meniscus surface, is given by
[Israelachvili, 1992] as
)100/log(%ˆ RHTR
r LVk
νγ= (1.2)
where, RH is relative humidity (%), γLV and v are surface tension and molar volume of
water, respectively, and R is the ideal gas constant.
The adhesion force associated with liquid condensation arises from pressure
differences across the meniscus is given by [Israelachvili, 1992]
7
θγπ cos4 LVcap RF = (1.3)
Here, θ is the water contact angle on the solid surface, determined from a force balance
at the interface between water droplet, the vapor phase and the solid surface as in
Figure1.6. The force balance equation is called Young’s equation [Meyer et al, 2000],
given as
SVSLLV γγθγ =+cos (1.4a)
or
LV
SLSV
γγγ
θ−
=cos (1.4b)
r1
R
x y
φ
θ
θ r1cosθ r1
(a) (b)
Figure 1.5 Capillary condensations around spherical asperity
γSV
γSL
γLV
θ
Figure1.6 Surface energy balance at interface of three media
When a sphere and a flat surface are separated by a distance d, the capillary
attractive force can be derived by minimizing the surface free energy, keeping constant
liquid volume, as
8
drRr
dyyRdF
K
LVKLVcap +
=+
=θ
θγπθγπ
cos2cos8
cos4)(2
(1.5)
From equation (1.5), the capillary force depends on the water contact angle θ, and
the relative humidity of the environment.
When the surface energy of solid γSV exceeds the solid-liquid interfacial energy
γSL, i.e., θ <90 o, the surface is hydrophilic and attractive. When γSV is smaller than γSL,
i.e., θ >90o, the surface is hydrophobic, making the capillary force repulsive. Minimizing
the surface energy γSV and increasing the water contact angle θ for a given relative
humidity can reduce capillary attractive forces.
1.3.2 Van der Waals Force Van der Waals force originates from dipole-dipole interaction between molecules.
There are three kinds of dipole-dipole interactions: permanent dipole-permanent dipole
(between polar molecules, called Keesom type); permanent dipole-induced dipole (Debye
type); and induced dipole-induced dipole (called London dispersion force). The
interaction energy W(r) between two molecules (dipoles) can be represented as
[Israelachvili, 1992],
6)(rCrW −= (1.6)
for all the three types of dipole-dipole interactions. Here, C is a constant for given
temperature, depending on the type of dipole interaction and r is distance between the
molecules.
Adhesion forces by van der Waals interaction between macroscopic particles
result from integration of the interaction energy, given by equation (1.6), over all atoms
or molecules that constitute the two surfaces. For a macroscopic sphere of radius R and a
9
flat surface separated by d, the adhesion force due to the van der Waals interaction can be
derived as [Israelachvili, 1992]
d
HRdFwdw 6)( = (1.7)
Here, H is called Hamaker constant, a property depending on the materials of the two
surfaces. The adhesion force per unit area between two flat surfaces becomes
36)(
dHdfwdw π
= (1.8)
Equations (1.8) and (1.9) are not valid for d=0 because the interaction potential,
equation (1.6), accounts for only attraction energy when the distance is larger than
intermolecular or inter atomic distance.
1.3.3 Electrostatic Force
When electric field is applied between two surfaces, static surface charges build
up on the surface and electrostatic forces arise by Coulom attraction. When a sphere and
a flat surface are close compared with the radius of the sphere, the electrostatic attractive
force is given by [Arai et al, 1996]
εεσπσ
0
221)( RdFelec = (1.9)
Here, σ1 andσ2 are charge densities of the two surfaces,ε0 is the permittivity of free space,
and ε is dielectric constant of the media between the sphere and flat surface. For two flat
surfaces, the electrostatic attraction per unit area becomes
10
20
21
4)(
ddfelec επε
σσ= (1.10)
From the above, van der Waals forces are unavoidable. Electrostatic force is
generated only when an electric field is applied. However to minimize electrostatic
attractions after an electric field is removed, the charges trapped on the surface should be
quickly dissipated. Highly conductive materials help dissipate surface charge. Reducing
surface energy of the material can minimize capillary attractive forces in humid
environments. By choosing adequate materials and surface modification processes, it is
possible to reduce stiction failures and friction in MEMS devices.
1.4 Review of Nano Scale Friction Study via AFM Dry friction without liquid lubricants can be classified into adhesion-controlled
and load-controlled friction [Meyer et al, 2000]. In adhesion-controlled friction, the load
is usually small enough for the contact deformation to be elastic (usually with no wear).
The friction force depends on the interfacial properties such as surface energy and
presence of boundary lubricants. In load-controlled friction, the friction force is less
sensitive to interfacial properties or environmental factors such as humidity, and depends
more on bulk mechanical properties such as hardness, Young’s modulus, and toughness.
High stress, concentrated at the local contact, causes wear and damage to the surface
layers. The dry friction force at the macro scale can be interpreted as a summation over
the adhesion-controlled and load-controlled friction force.
Very light loads usually characterize working conditions of MEMS surfaces.
Therefore, studies on tribological characteristics of MEMS surface employed AFM or
Surface Force Apparatus (SFA) to simulate single asperity contact under very light load
[Lu and Komvopoulos, 2001; Schwarz et al, 1997; Enachescu et al, 1998]. Typical nano
tribological studies using AFM had contact load of 10~100µN, sliding speed of ~µm/s,
11
and tip radius of 20~100nm. In this section, issues on nano scale study on adhesion-
controlled friction (under elastic contact), are investigated.
Modern theory of adhesion-controlled friction employs Bowden and Tabor’s
adhesion model [Meyer et al, 2000], in which the friction force
Rf SAF = (1.11)
is proportional to real contact area AR and the interfacial shear strength S, dependent on
the interfacial conditions such as surface energy and lubrication. The dependence on load
is contained in the real contact area AR. Nano tribological studies of different carbon
based coatings [Schwarz et al, 1997; Lu and Komvopoulos, 2001] and diamond and WC
interfaces [Enachescu et al, 1998], showed that S is nearly constant, and the friction force,
between a single asperity and a smooth surface at the atomic level, is proportional to the
contact area predicted by Johnson, Kendal and Roberts (JKR) or Derjaguin, Muller and
Toporov (DMT) contact model (explained later). They used accurately manufactured
nano scale tips (17~58nm radius) to simulate single spherical asperity contact.
A simple contact model to estimate the contact area for single asperity contact is
Hertzian [Johnson, 1985], in which the deformation of two spherical surfaces is assumed
elastic. The contact area follows from elastic theory is given by
3/2
=
KFR
A eeπ (1.12)
where 1
2
22
1
21 11
34
−
−+
−=
EEK
νν (1.13)
12
is an effective elastic modulus based on Young’s modulus E1, E2 and Poison’s ratio ν1, ν2
for the contacting surfaces and Fe is an external force. In the foregoing, an effective
radius Re of the two contacting curved surfaces becomes
21
21
RRRRRe +
= (1.14)
If surface energy effects and elastic deformations are considered, the JKR model
predicts contact area well for materials with high surface energy and low stiffness
[Schwarz et al, 1997; Enachescu et al, 1998]. The DMT model accurately predicts the
contact area for materials with low surface energy and high stiffness.
The contact area between two spheres, when adhesion force Fa is considered, is
given by, from equation (1.12), 3/2)(
+
=K
FFRA aeeπ (1.15)
Here 2)3(63 adeeadeadea WRFWRWRF πππ ++= for JKR model
eada RWF π= for DMT model
where Wad is work of adhesion, given by [Israelachvili, 1992],
1221 γγγ −+=daW (1.16)
Here, 1γ and 2γ are surface energy of two surfaces and 12γ is an interfacial surface energy
between surface 1 and surface 2. When two surfaces consist of the identical material,
12γ=daW . The contact area A increases with the surface energy of the solid surface. As
seen in Figure 1.7, in the presence of surface energy, two surfaces tend to adhere even
without external force.
13
Figure 1.7 Deformation of elastic sphere on rigid surface [Israelachvili, 1992]
Putting the contact area predicted by JKR or DMT model into the adhesion-
controlled friction equation (1.11) yields
( ) 3/23/2aeef FFCRF += , where
3/2KSC π
≡ (1.17)
Because C is constant (S is constant as assumed earlier), the friction force depends on the
radius of asperity, the surface energies of the two surfaces, and external force. The
friction force depends on the AFM tip radius, surface energies of the two surfaces, and
external force. If surface energies are very small, Fa can be neglected and Ff is
proportional to R2/3.
Another noticeable fact is that the equation (1.17) is only true for very small
sliding speed, where relaxation time of the atoms experiencing elastic deformation during
the scanning of AFM tip is much smaller than the inter atomic distance divided by the
scanning speed. Figure 1.8 shows the tendency of nano scale friction, where nano scale
friction coefficients of DLC coatings on a glass wafer increased, as AFM tip radius and
applied load (contact pressure) increased [Bandorf et al, 2003].
Tip radius, surface energy, and scanning speed-dependent behavior of friction
forces makes comparison of measured nano scale friction coefficients, to friction
coefficients measured by other methods, difficult, especially if the test environment and
tip radius are not known exactly, or the tip is not perfectly spherical.
14
Surfaces of micro mechanical systems possess multiple nano meter scales
asperities and follow fractal geometry characterized by self-affinity over a wide range of
length scales [Ling, 1990]. Even if the load is very small and the macroscopic contact
pressure is far below the elastic limit of the materials, some local asperities will
plastically deform because of concentrated loads on those spots. Therefore, wear can be
induced even under an extremely small load, which complicates investigation of
frictional behavior of actual MEMS surface and also makes it difficult to estimate a real
friction from nano scale friction via AFM.
In this work, direct characterization of the working surface of the micro bearings
(sidewalls), under similar working conditions of the micro bearings, was pursued via
specially designed tribo tester. Detailed investigation and test results are presented in
Chapter 6.
(a) Friction coefficient-tip radius (b) Friction coefficient-contact pressure
Figure 1.8 Friction coefficients of 250nm thick DLC on glass: (a) Friction
coefficient-tip radius (b) Friction coefficient-contact pressure [Bandorf et al, 2003]
1.5 Past Studies on Coatings for MEMS devices In this section, past research on tribological coatings for MEMS devices,
chemisorbed monolayers on hydrophilic silicon oxide surfaces, and DLC coatings are
reviewed.
15
1.5.1 Chemisorbed Monolayer The chemisorbed monolayer, well known for nano scale thin anti-stiction coatings,
modifies hydrophilic oxide surfaces to hydrophobic surface by long chain molecules with
non-polar end groups. The attractive force an atom has for shared electrons in a molecule
is its electro negativity. The polarity of end groups forming long chain molecules is
determined by the difference of electro negativities of constituent atoms. The larger the
difference, the larger the polarity of the end group. End groups such as –OH (hydroxyl), -
COOH (carboxyl), -SH (thiol), -SiCl3 (trichlorosilane) with strong polarity, can anchor
the molecule to the surface. Non-polar end groups such as -CH3 (methyl), -CF3
(trifluoromethyl) can decrease surface energy and form hydrophobic end groups.
Octadecyl-Trichloro-Silane (C18H37SiCl3) monolayer (OTS) coatings [Maboudian
and Howe, 1997; Ashurst, 2001], have -CH3 end groups to lower surface energy.
Compared to physisorbed Langmuir-Blodgett monolayers formed by weak van der Waals
bonds, OTS molecules have strong covalent bonds to the oxide surface and show high
durability even under large load [Berman et al, 1998; Ruhe et al, 1993; DePalma and
Tillman, 1989]. The chain length of one OTS molecule determines the coating thickness.
The OTS has 18 carbons, rendering a thickness of 2.5nm.
Friction coefficient of OTS coating on SiO2 wafer, measured via pin-on disc test,
was 0.07 under various loads and sliding speeds [Bhushan et al, 1995; DePalma and
Tillman, 1989]. Xiao et al [Xiao et al, 1996] and Israelachvili [Israelachvili, 1992]
showed that attractive van der Waals force between molecules plays an essential role in
ensuring good close packing and self-organization of the molecules. They also showed
that alkylsilane molecules having -CH3 end group should have more than 10 carbon
atoms to have high packing density and low friction.
Perfluorodecyltrichlorosilane (CF3(CF2)7(CH2)2SiCl3) monolayers (FDTS) seems
to be more promising compared to OTS [Srinivasan et al, 1998]. FDTS has -CF3 end
groups, with lower surface energy (6mJ/m2) than OTS (22 mJ/m2). FDTS shows more
stable characteristics at high temperature. OTS degraded above 150oC but FDTS could
16
withstand up to 400oC in air. Electrical properties are also important reducing permanent
stiction in MEMS device. FDTS has much lower resistivity than OTS, which leads to less
surface charge accumulation and low stiction. A FDTS coated cantilever actuator showed
far superior performance over 40 million cycles without failure compared to an OTS
coated actuator [Srinivasan et al, 1998].
Teflon-like fluorinated hydrocarbon coatings (FC) consisting of CFx chains,
grown commonly by plasma polymerization [Yasuda, 1985], also show very low surface
energy of 7mJ/m2 and high durability in wear tests (collapse cycles of cantilever) even at
400oC [Mastrangelo, 1997]. Periodic contact tests with an actuated cantilever showed 108
collapse cycles at 150oC without any degradation of 20nm thick FC films [Mastrangelo,
1997].
Despite very low surface energy and high reliability, during repeated collapse
cycles in certain conditions, sliding wear resistance and long-term reliability of these
chemisorbed monolayer coatings are questionable, especially in harsh environments.
1.5.2 Hydrocarbon Based Coatings
Carbon based films such as graphite, polycrystalline diamond, amorphous hydro
carbon (a-C:H or DLC), and metal containing hydrocarbon (Me-C:H) are widely used for
low friction surfaces in macro scale applications. Various techniques have studied
microstructure, mechanical properties, and tribological characteristics of these films
[Koskinen et al, 1998; Donnet. and Grill, 1997; Erdemir et al, 2000; Liu et al, 1997; Shi
et al, 2000; Cao et al, 2001].
In the macro-scale pin-on disc tests, steel or ceramic balls are rubbed against flat
surface with DLC coatings. Adsorbates such as oxygen or moisture affect the result
significantly, especially at low speed tests [Donnet et al 1994]. Typical friction
coefficients versus time in ultra high vacuum (UHV), in dry nitrogen and ambient are
shown in Figure 1.9. Initially the friction coefficient is high, but after a break in period,
lower steady state values appear in UHV and dry nitrogen environments.
17
Feasible theory for low friction in UHV or inert environment is graphitization of
the sp3 carbon structure (diamond phase) during the initial break in period suggested by
several researchers [Liu et al, 1996; Liu et al, 1997, Donnet et al, 1994; Koskinen et al,
1998]. Hydrogen within DLC is released from sp3 carbon at about 450oC, which leads to
collapse of sp3 carbon to sp2 carbon. Frictional heating and sliding-induced strain energy
in the surface layer seems to cause extensive graphitization [Liu et al, 1996].
(a) (b)
Figure 1.9 Friction coefficient in: (a) Vacuum / ambient (b) Dry N2 / ambient [Donnet et al 1994] ; Test conditions :contact pressure 1GPa ,sliding speed 1.7mm/s,
Ra=0.28nm, Steel ball on DLC
Recent research on highly hydrogenated DLC [Erdemir et al, 2000] revealed the
importance of hydrogen to very low friction in vacuum or dry nitrogen environment.
Erdemir [Erdemir et al, 2000] suggested the passivation of dangling bonds of surface
carbon by continuously released hydrogen atoms from bulk DLC, and strong repulsion
between the surface hydrogen atoms, to be the main cause of ultra low friction.
Despite promising tribological characteristics and proven performance of DLC in
macro scale applications, direct applications to MEMS have been limited. Beerschwinger
et al [Beerschwinger et al, 1995] measured friction of DLC coating on flat silicon surface,
via surface micro machined small friction tester. Bandorf et al [Bandorf et al, 2003]
showed that DLC on soft polymer surface had better wear resistance than DLC on silicon
wafer. Mousinho et al [Mousinho et al, 2003] demonstrated microstructures made with
DLC film, deposited by RF magnetron sputtering. Cao et al [Cao et al, 2003] coated Ti-
18
containing DLC (Ti-C:H) on Ni micro mold insert, fabricated by X-ray lithography and
electroplating. However, applications of DLC coatings to the sidewall of micro scale
mechanical parts, that experience sliding contacts (bearings, gears, etc), have not been
reported.
1.6 Overview of Dissertation This chapter briefly reviewed reliability of past micro rotating machinery
including benefits of micro gas bearings and current research. Various surface forces, and
related failures were introduced. Nano scale friction studies via AFM, and limitations
interpreting the results were addressed. Also reviewed were tribological studies on
chemisorbed monolayers for oxide surfaces and hydrocarbon (DLC) for macro scale
applications.
Chapter 2 will present fabrication processes for micro gas bearings. Procedures to
make an X-ray mask and technical issues will be discussed. Finally, SU-8 lithography,
electroplating principles, and other issues will be addressed.
Chapter 3 and 4 will investigate static and dynamic performance of the fabricated
micro gas bearings using Molecular Gas Lubrication (MGL) theory and improved
designs will be suggested in terms of load capacity and dynamic stability. Fabrication
methods of meso scale gas bearings, via X-ray lithography and precision assembly
techniques will be proposed. Performance analyses of the meso scale gas bearings will be
presented in terms of three-dimensional imbalance response.
Chapter 5 will give test results of micro gas bearings with a simple air jet driven
turbine configuration. Technical issues involved in the testing will be also addressed.
Chapter 6 will present tribological characteristics of tungsten containing
hydrocarbon (W-C:H) coated micro bearings, compared with uncoated micro bearings.
Finally, chapter 7 will suggest future work and summarize contributions and conclusions
of the dissertation.
19
Chapter 2
Fabrication Processes of Micro Gas Bearings
In this chapter, detailed fabrication processes of micro gas bearings are described.
Procedures to make X-ray mask and technical issues are discussed. Finally SU-8
lithography, electroplating principle, and other issues are addressed.
2.1 Introduction
Figure 2.1 shows the configuration and dimensions of fabricated micro gas
journal bearings with thrust pads. The journal bearings have several evenly distributed
recesses 2µm deep along the circumferential direction, to control inherent whirl
instability [Cheng and Pan 1965; Castelli and Elrod, 1965]. Thrust bearings with 4 thrust
pads, 3µm high with an angular width of 30o, were integrated with the journal bearing.
Radial grooves with an angular width of 20o and a 100µm depth were formed at the end
of every thrust pad to present atmospheric pressure to the beginnings and ends of thrust
pads. The inner and outer diameters of the thrust pads are 0.6mm and 1.5mm respectively.
2µm step
300µm
φ 500µm
20o
30o
3µm step
φ 1.5mm
φ 600µm
100µm deep groove
Figure 2.1 Micro gas bearing design, the heights of steps and recesses are
exaggerated, and the overall diameter is 2mm
20
2.2 Fabrication Processes of Micro Gas Bearings
The design specifications of the micro gas bearings require a high degree of
circumferential roundness and limit the deviation from a perfect vertical sidewall to
within 0.5µm, to maintain the nominal clearance of 1µm between the bearing and shaft.
The bearing surfaces must have a surface roughness less than 50nm to ensure
hydrodynamic operation. To meet these very strict geometrical tolerances, unique
fabrication processes were developed that combined X-ray lithography, electroplating,
and wet etching.
The sequence of fabrication processes (shown in Figure 2.2) to make the micro
gas bearings (Figure 2.1) include the following steps: (a) bonding of 2mm thick PMMA
(polymethylmethacrylate) sheet atop a TiO sacrificial layer and fly cut to a 320µm thick
layer, followed by X-ray exposure and developing with a G-G solution; (b) Ni
electroplating and polishing down to the PMMA layer; (c) spin coating of a 100µm-thick
SU-8 layer, followed by UV lithography to pattern a mold for electroplating thrust pads;
(d) a second Ni electroplating and polishing down to the SU-8 layer; (e) a photo resist
spin, photolithography, and wet etching of the Ni to make 3µm steps; (f) removal of all
photo resists, and a sacrificial layer etching to release the bearings. The detail procedures
of each process are explained in the following sections.
21
(a) X-ray lithography (b) Ni plating/polishing (c) SU-8 photolithography
(d) Ni plating/polishing (e) Photolithography/etching (f) Releasing
Figure 2.2 Overall fabrication processes of micro gas bearings: (a) X-ray lithography, (b) Ni plating/polishing, (c) SU-8 photolithography, (d) Ni
plating/polishing, (e) Photolithography/etching, (f) Releasing
2.3 X-ray Lithography on PMMA
2.3.1 Substrate Preparation
A layer of 2µm thick Ti was evaporated onto a 1mm thick Alumina ceramic wafer,
which becomes a substrate for subsequent processes. The Ti was partially oxidized in
sodium hydroxide and hydrogen peroxide solutions at 65 °C to improve adhesion of the
PMMA sheet. The partially oxidized Ti layer has very low electrical resistivity and serves
as a seed layer for subsequent Ni plating (Figure 2.2b) and also as the sacrificial layer
when making the final Ni bearing in Figure 2.2(f). The detailed composition of the Ti
oxidation solution is given in Table 2.1.
22
A commercially available 2mm-thick PMMA sheet of CQ grade2 was bonded on
top of the ceramic substrate with the partially oxidized Ti layer. Detailed information on
bonding epoxy for the PMMA sheet is given in Table 2.2
Table 2.1 Chemical composition of Ti oxidation solution
Chemical Quantity Hydrogen peroxide (H2O2) 30% solution 2.04g Sodium hydroxide (NaOH) 2.0g DI water Up to 100ml
Table 2.2 Chemical composition of PMMA bonding epoxy
Chemical Amount Purpose MEMO3 0.10g Resist adhesion promoter BPO4 0.15g Catalyst as hardener (radical builder) DMA5 0.10g Polymerization Initiator MMA6(85%)/PMMA(15%) 10.0g Bulk resin
To reduce the stress of the bond resin and increase the molecular weight, >1%
BPO was used to minimize the monomer content below 0.5% [Madou, 2002]. The
prepared epoxy resin was placed in the vacuum oven for 3 minutes to remove air bubbles
in the resin. The PMMA sheet was bonded under constant pressure (20psi) for 8 hours to
allow complete polymerization of the bonding epoxy. The bonded PMMA sheet was cut
to 350µm thickness with a fly cutter.
2.3.2 Material Selection for X-ray Mask
To make high aspect ratio structures using X-ray lithography, the Au absorber
must be thick to increase contrast and to minimize fluorescence from the Au absorber
2 CQ grade is a particular cast sheet of PMMA for X-ray lithography that is additive-free and is the highest molecular weight grade of PMMA available. It contains a minimum UV absorber. 3 MEMO: 3-Methacryloxypropyltrimethoxysilane 4 BPO: Benzoyl peroxide 5 DMA: Dimethylaniline 6 MMA: Methylmethacrylate Monomer
23
itself. A 3µm Ti film was chosen as an X-ray transmitting membrane because of the good
adhesion property of the thick photo resist for Au plating, and the moderate radiation
stability. In the X-ray lithography, contrast refers to absorption contrast, defined as the
ratio of the total dose at the bottom of the resist under the X-ray transparent membrane, to
the total dose below the X-ray absorber. Figure 2.3 plots the absorption contrast for the
3µm Ti membrane mask for various PMMA thicknesses versus the Au absorber thickness,
calculated by dose simulation software from the Center for Advanced Microstructures
and Devices (CAMD), Louisiana State University; here the bottom dose was 2500J/cm3
with a 175µm thick Beryllium window and a 9.5µm thick Al filter.
An acceptable contrast for a process with PMMA combined with a G-G developer
is known to be above 40 to produce good lithographic patterns [Desta et al, 2003]. In this
work, 8µm thick Au was chosen to obtain contrasts of 71. The maximum contrast was
compromised by limitations on the maximum thickness of SPR 220-7 (a positive tone
photo resist from Shipley, Inc. patterned for Au plating on the 3 µm Ti membrane) that
can produce good vertical profiles. The detailed process conditions for the lithography of
SPR 220-7 is explained in the next section.
Figure 2.3 Absorption contrast of 3µm Ti membrane mask versus Au
thickness with various PMMA thicknesses
24
2.3.3 Fabrication Processes of X-ray Mask
The fabrication processes of the Ti membrane mask are as follows: 3µm Ti was
sputter deposited on to a Si wafer. A 50nm layer of Au was evaporated onto the Ti as a
seed layer for Au electroplating. To improve the adhesion of Au to Ti, 10nm Cr was used
as a bonding layer between the Ti and Au layer. Shipley’s SPR 220-7 was the positive
photo resist used for patterning the Au electroplating mold, where an e-beam photo mask
with critical dimension 0.1µm was used.
Lithographic conditions were carefully optimized to obtain a nearly vertical
sidewall. The process conditions of SPR 220-7 were as follows: it was spin-coated for 40
seconds at 1500 rpm to obtain a 13µm thick layer. To minimize cracking and a build-up
of internal stresses, a soft bake was performed on a contact hot plate in a two-step
process: for 150 seconds at 100oC, followed by 240 seconds at 115oC. The dose was
optimized as 420mJ/cm2 for best contrast and vertical sidewall, using a broadband
(350~550nm) Oriel UV station in CAMD. Any attempt to post-exposure bake resulted in
extensive cracking all over the resist, regardless of baking conditions for the resist thicker
than 10µm. The developer was a 1:1 mixture of MF 321 and MF 322. The developing
condition was very critical to achieve a high-quality photo resist pattern. Even a slight
over-development resulted in rough-edged profiles and blunted corners. Moderate
agitation and careful time control were very important. Figure 2.4 shows the SEM images
of the patterned photo resist.
After the SPR 220-7 was patterned, 8µm Au was electroplated at a current density
of 1mA/cm2 and at a bath temperature of 43oC, resulting in a plating rate of 4µm/hr. SEM
images of the X-ray mask in Figure 2.5 show very smooth circumferential profiles, which
are critical realizing very smooth surface roughness on the final Ni structure. After the
Au was electroplated, the backside of the silicon wafer was etched by 40% KOH at 65oC
to suspend the Au pattern on the Ti membrane. Figure 2.6 shows Ti membrane mask
fabricated through the processes described above.
25
(a) Overall image of bearings (b) Near perfect vertical sidewall
Figure 2.4 SEM image of patterned 13µm thick SPR: (a) overall image of bearings, (b) near perfect vertical sidewall
(a) Top view (b) 2µm step in journal bearings
Figure 2.5 SEM images of Ti-membrane X-ray mask with 8µm thick Au absorber: (a) Top view, (b) 2µm step in journal bearings
Figure 2.6 Ti membrane X-ray mask with Au absorber
26
2.3.4 X-ray Exposure and Development
The X-ray source was XRLM3 at CAMD with a 175µm thick beryllium window.
An additional 9.5µm thick Al filter was inserted into the beam line to remove low-energy
photons and to decrease top to bottom dose ratio. The minimum bottom dose below the
Ti membrane was chosen as 2500J/cm3, and the top to bottom dose ratio was 4.84. The
dose below the Au absorber was 34J/cm3.
PMMA was developed in a G-G solution at room temperature to minimize
cracking and to improve the contrast between the exposed and unexposed areas
[Pantenburg et al, 1998]. Higher development temperature can increase development rate
but G-G solution can also develop unexposed area. Table 2.3 shows the chemical
compositions of the G-G developer and rinse solution. One cycle consists of 20 minutes
developing and 40 minutes rinse. Each cycle develops 100µm.
Table 2.3 Chemical compositions of G-G developer and rinse Developer (for 1L solution)
2-(2-Butoxyethoxy)ethanol 600mL Morpholine 200mL 2-aminoethanol 50mL DI water 150mL
Rinse (for 1L solution)
2-(2-Butoxyethoxy)ethanol 800mL DI water 200mL
2.4 Post Processes
2.4.1 Electroplating
The basic procedure of Ni electroplating is similar to other electroplating
processes. In Figure 2.7, the Ni anode is dissolved in a Ni sulfamate solution, where it
replenishes the Ni2+ ions. In the cathode, which is the substrate to be Ni plated, the Ni2+
ions react with two electrons and are deposited on the cathode in a metallic form. Since
27
3-5% of the current is consumed by reduction of H+ ions in the cathode, cathode
efficiency is 95-97%, depending on the solution conditions [Bari, 1994; Judy, 1996].
The reactions at the cathode and anode are as follows:
Cathode reaction
Ni 2+ + 2e → Ni (main reaction) (2.1a)
2H+ + 2e → H2 (2.1b)
Anode reaction
Ni → Ni 2++ 2e (main reaction) (2.2a)
OH- → ½ O2+ H + +2e (2.2b)
Ni 2+
H+OH -
Ni 2+
Ni
2e
Cathode (substrate
Anode
2eH2
Ni
O2
NH2SO3 -
Figure 2.7 Reaction in the Ni electroplating bath
Under normal operating conditions, there should be no hydroxyl ions (OH-)
dissolved from the water, and the dissolution efficiency of the Ni anode should be 100%.
When there are too many hydroxyl ions (OH-), oxygen is evolved at the anode (Equation
2.2b) instead of the Ni dissolution. In that case, the anode stops dissolving, Ni2+ ions are
exhaused quickly at the cathode, and electroplating stops. To keep the anode efficiency at
100%, chloride ions are added to the Ni sulfamate solution, or an activated Ni anode is
used instead of pure Ni [Bari, 1994]. The activated nickel , which contains about 0.025%
sulfur, is commercially availlable [Online reference 1]. The sulfur forms insoluble black
28
nickel sulfide residue on the Ni anode surfaces, and the residues should be retained in a
polyprophylene anode bag to prevent contamination on the electroplated Ni surface, as
shown in Figure 2.8. The nickle sulfide also removes unwanted copper impurities from
the plating solution.
Figure 2.8 Contamination of Ni electro deposit surface by nickel sulfide particles,
shown as small black dots on white Ni electro deposit
Table 2.4 shows the chemical composition of the nickel sulfamate solution and
the process conditions. To adjust the pH, diluted sulfamic acid (H3NO3S) was used for
pH>4 and diluted sodium hydroxide (NaOH) was used for pH<3.5. Small amount of
sodium lauryl sulfate was added as a wetting agent (surfactant) to reduce the surface
tension of the Ni sulfamate solution and to permit it to penetrate into small
microstructures.
Faraday’s Law for the Ni plating is expressed by the following equation [Bari,
1994]:
tIFnM
me
cη= (2.3)
, where m is mass of electroplated Ni (g), M is the molecular weight of Ni (58.7g/mol), ne
(=2) is the number of elecrons involved in the reduction of Ni, F is Faraday’s constant
(96488 C/mol), ηc is cathode efficiency (0.95-0.97), t is time, and I is total current.
29
If the total area to be electroplated is A(cm2) and the thickness is d(µm), Equation
(2.3) becomes
tJFnM
de
c
ρη
= . (2.4)
Here ρ is the density of Ni (=8.9g/cm3) and J is the current density in mA/cm2. Plugging
the values into equation (2.4) gives
Jdd
JMFn
tcc
e
ηηρ
871.925,2== (2.5)
Assuming ηc =1, the total time to electroplate 300µm thick Ni is about 24 hours
(~0.2µm/min). Because the current density is not uniform at the edges of the structures as
sown in Figure 2.9, the patterned PMMA structure was over plated for 30 hours and
polished down to 300µm (Figure 2.2b) for subsequent processes to form the thrust
bearing.
Table 2.4 Chemical composition of Ni sulfamate solution and the process conditions
Component Amount per 1L final solution Nickle sulfamate(Ni(NH2SO3)2) 50% aqueous solution 450ml (78g of Ni2+ ions/L)
Sodium Lauryl Sulfate 1g Boric Acid (H3BO3) 37.5g DI water to 1L final volume pH 3.7 Temperature 50oC Current density 10mA/cm2
30
Plating mold
Seed layer
Figure 2.9 Non uniform current density at the edge of patterned plating mold cause
non uniform deposit thickness [Judy, 1996]
Important factors, easily overseen in Ni electroplating, are hydrogen evolution in
a cathode (3~5% of total current), and nucleation of air bubbles on a PMMA surface,
when cathode is immersed in electroplating bath. Because electroplating is very fast
process, small air or hydrogen bubbles are easily trapped and form a micro scale cavity
after electroplating is finished. Figure 2.10 is a high magnification SEM image of bearing
surface with cavity formed by air bubble nucleated right after substrate was immersed in
a plating bath. Increasing wetting agent or fast stirring of electroplating solution can
remove air or hydrogen bubble effectively.
Figure 2.10 High magnification SEM image of bearing sidewall: cavity formed by
air bubbles attached to PMMM
31
2.4.2 Processes to Form Thrust Bearing
A 100µm thick layer of negative photo resist, SU-8 2100 from MicroChem, Inc.,
was spin-coated on top of the polished Ni-PMMA structure (shown in Figure 2.2b) to
pattern the second Ni plating mold for the thrust pads. Spin conditions were 3000 rpm for
30 seconds. SU-8 2100 was selected as a plating mold because of its unique capability to
coat more than 100µm thick layer with a single spin and its high sensitivity to UV (250-
450nm).
The spin coated SU-8 had 10 hours of hold time before the soft bake to relax
stress built up during the spin. Soft baking was performed at 95oC for 30minutes on a
contact hot plate that was partially sealed by a cover to maintain a uniform temperature
over the thickness of the SU-8 layer. During the soft baking, the temperature was
changed slowly to minimize shrinkage and thermal shock. A KARL SUSS UV aligner
was used for the UV exposure and for alignment of the location of the thrust pads with
respect to the journal bearing center. The dose was optimized as 400mJ/cm2 for the best
vertical sidewall. SU-8 is cross-linked in two steps. During the exposure, strong acid is
formed, and the acid initiates a photoreaction to cross link the polymer during the post
exposure bake (PEB). The PEB was performed for 25minutes at 96 °C. Slow cooling to
room temperature after the PEB is very important to prevent resist cracking. Overnight
cooling is recommended for best results. The unexposed area was developed with SU-8
developer from MicroChem, Inc. and rinsed by isopropyl alcohol (IPA).
Before the second Ni layer was electroplated to form the thrust pads, the exposed
Ni surface (polished surface at step (b) in Figure 2.2) was etched by diluted HNO3 for 5
minutes to remove a natural Ni oxide layer on the surface. With the oxide layer, the
adhesion of the second Ni structure on the first Ni structure was poor, and the second Ni
structure was easily peeled off. Ni was electroplated again within the patterned SU-8
mold and polished down to the SU-8 (Figure 2.2d).
Positive i-line (365nm) photo resist AZ 4330 was spin coated on the SU-8 and Ni
structure (Figure 2.2d), to process the final 3µm recesses on the thrust pads. The nitric
acid-based nickel etchant TFB from Transene, Inc. (etch rate of 0.18µm/min at 25oC) was
32
used to etch the final 3µm recesses. The widely used sulfuric acid based etchant TFG
from the same company reacted with the Ni surface to form presumably black nickel
sulfide, preventing further etching.
2.4.3 Sacrificial Layer Etching to Release Bearings
After steps in thrust bearing were formed, SU-8 and PMMA were removed in a
Dynasolve 165 and chlorobenzene, respectively. The final releasing step of the bearings
from the substrate was performed in diluted (~10%) HF, to etch TiO sacrificial layer on
ceramic wafer. Ni does not dissolve in HF because chemical activity of Ni is very close
to H [Whitten et al, 2004]. Ti and TiO are dissolved very fast in even highly diluted HF.
Etching time depends on the thickness of TiO and size of the structures. Thicker TiO
layer allows better diffusion of diluted HF, leading to shorter release time. Release time is
very important achieving good sidewall roughness.
Figure 2.11 compares SEM images of sidewall generated by different sacrificial
layer etching time; (a) and (b) are surfaces generated by sacrificial layer etching for 2
days and 2 weeks, respectively. Figure 2.12 is a high magnification image of Figure 2.11.
Small cavity in Figure 2.12(a) was formed by an air bubble nucleated on a PMMA
surface when electroplating began. Both samples had the identical process conditions (X-
ray dose, PMMA development and electroplating conditions, etc), except the release time.
Even if activity of Ni with H is negligible, long etching time increase surface roughness
of final Ni microstructure significantly. To minimize etching of Ni during the sacrificial
layer etching, Ti layer should be thick enough to allow diluted HF to diffuse very quickly
or sonic agitation is recommended. Figure 2.13 is SEM images of the final micro gas
bearings with thrust pads, with releasing time of 2 days. The surface roughness of the
bearings was measured by the D-3100 Atomic Force Microscope (AFM) from Digital
Instruments, Inc. in a tapping mode. Instead of the inside surfaces of the bearing, the
outer edges were measured because, in principle, their lithographic conditions are exactly
the same as the inside surfaces, and, in practice, this surface is accessible. Figure 2.14
shows the morphology of the bearing surfaces of dimensions 10µm × 10µm. Both Ra and
33
RMS values were below 20nm. The surface profiles do not show the noticeable
orientation that is typical in X-ray lithography patterns using graphite mask [Coane,
2000], which means very smooth circumferential profiles of the Au absorber in the
fabricated Ti membrane X-ray mask.
(a) After 2 days release (× 2K) (b) After 2 weeks release (× 2K)
Figure 2.11 SEM images of bearing surface after different sacrificial layer
etching time: (a) After 2 days release (× 2K), (b) After 2 weeks release (× 2K)
(a) After 2 days release (× 10K) (b) After 2 weeks release (× 10K)
Figure 2.12 Higher magnification of Figure 2.11: (a) After 2 days release (×
10K), (b) After 2 weeks release (× 10K)
34
(a) Overall bearing (b) 2µm steps on journal bearings
(c) Magnified image of 2µm step
Figure 2.13 SEM images of micro gas bearing: (a) Overall bearing, (b) 2µm steps on journal bearings before processing thrust bearings, (c) Magnified image of 2µm step
(a) AFM topography (b) High magnification SEM images
Figure 2.14 Surface morphology of bearing surface measured from AFM (scan
length 10µm×10µm) and SEM: (a) AFM topography, (b) High magnification SEM images
35
Chapter 3
Static Performance of Micro Gas Bearing
In the Chapter 3, Reynolds Equations for gas journal and thrust bearings, based on
Molecular Gas Lubrication (MGL) theory, are introduced. Numerical analyses results on
static performance and rotational friction factor of the micro gas journal and thrust
bearings are presented.
3.1 Theory Figure 3.1 shows a schematic diagram of the micro gas journal bearing in the
normal operating conditions with eccentricity e of the journal center from the bearing
center. In the figure, ω is the rotational speed of the journal, and θ is the circumferential
coordinate with the same direction as ω. C is the radial clearance between the bearing and
journal. φ is the attitude angle of the journal, which is the angle between the direction of
load capacity F and eccentricity e.
Upon rotation of the journal shaft, the gas between two surfaces of a gas bearing
is pressurized by the squeeze film effect. This generates a hydrodynamic force, lifting the
journal immediately after the initial slip. At steady state, there is no contact between the
two surfaces, and friction approaches zero.
The governing equation for a gas journal bearing was derived from the general
Navier Stoke’s equation, with the assumptions of a thin gas film, negligible curvature
effect, and ideal gas under isothermal equilibrium. Here the local gas film thickness h is
much smaller than the bearing length L and diameter D. The result, Reynolds Equation
for gas film pressure p, was derived using a no slip boundary condition of gas at the
bearing surfaces [Cameron, 1966].
36
)(12)(6)()( 33 pht
phx
Uzpph
zxpph
x ∂∂
+∂∂
=∂∂
∂∂
+∂∂
∂∂ µµ (3.1)
In the foregoing, x is the direction of linear speed U (=Dω/2), z is the direction of side
flow, and µ is the viscosity of the lubricating gas. The ideal gas law was used to include
the compressible flow effect to change a density to pressure.
x
θ
Χ
ω
e
y r
YYXX iFiFF +=
φ
C(1+ε)
C(1-ε)
Y
Figure 3.1 Journal bearing in normal operating condition
For a micro gas bearing, the slip at the surface should be considered, because the
molecular mean free path of gas molecules is not negligible compared with the bearing
clearance C. An important parameter in a gas kinetic theory is Knudsen number, defined
as Kn = l/h, where l is a molecular mean free path derived from the gas kinetic theory
[Kennard, 1938]7.
pNdTR
A22
ˆ
πλ = (3.2)
The concept of gas kinetic theory was first introduced to a rarefied gas film by
Burgdorfer [Burgdorfer, 1959]. He adopted Maxwell’s wall slip analysis [Kennard, 1938]
7 The mean free path is defined as the average distance that a molecule travels between successive collisions.
37
and modified the Poiseuille flow component, using a slip flow velocity boundary
condition for a small Knudsen number (Kn << 1). He assumed the slip is proportional to
the velocity gradient dzdv /0 at the wall. Then the slip velocity us at the wall can be
written as
dzdv
u as0ζ= (3.3)
Maxwell derived the Knudsen layer thickness ζa based on the tangential
momentum transfer ratio of molecules with molecular mean free path λ:
λλα
αζ aca ≡
−
=22 (3.4)
Here, c is an adjustable parameter with values between 0.491 and 0.499 (such that very
nearly 2c ≈ 1).
The surface accommodation coefficient of the wall α is the energy transfer ratio
defined by Knudsen [Kennard, 1938] as
wi
ri
EEEE
−−
=α (3.5)
In Equation (3.5) Ei denotes the energy of incident molecules, Er denotes the
energy of reflected molecules, and Ew denotes the energy of molecules that reflected
diffusively at the wall. In “diffusive” reflection, the time that a molecule stays at the wall
is long enough for the molecule to absorb sufficient kinetic energy of the wall
temperature and to reflect in a direction independent of the incident direction. In that case,
Er approaches Ew and α=1. When α is zero, the molecules are reflected “specularly” with
no change of their initial kinetic energy. The accommodation coefficient depends on the
surface conditions, such as material, surface roughness, adsorbates on the surface, etc.
38
[Kennard, 1938; Rettner, 1997]. Maxwell [Kennard 1938] assumed a fraction φ of the
incident molecules were reflected diffusively with α=1, and the rest were reflected with
α=0. With Maxwell’s assumptions, the total reflected energy will be Er=φEw+(1-φ)Ei .
Since from Equation (3.5), φ and α are governed by the same equation, the
accommodation coefficient represents the fraction of diffusive reflection.
From experiments [Rettner, 1997; Gabis et al, 1996] on the hard disk surfaces,
various real engineering surfaces have α=0.9 -0.95. In this work, the accommodation
coefficients at both walls are assumed to be 1.
Taking account of the wall slip effect, the Reynolds Equation was modified as
[Burgdorfer, 1959]
)(12)(6
))61(())61(( 33
pht
phx
U
zp
haph
zxp
haph
x
∂∂
+∂∂
=
∂∂
+∂∂
+∂∂
+∂∂
µµ
λλ
(3.6)
Fukui and Kaneko [Fukui and Kaneko, 1988] derived a Molecular Gas
Lubrication (MGL) model based on the linearized Boltzmann Equation8, to account for
the molecular slip effect in Poiseuille flow. They rigorously derived a Poiseuille flow
factor PQ , defined as the ratio of the real Poiseuille flow rate PQ in rarefied gas regions
to the continuum flow rate conQ . They created the numerical tabulation of the database
for the flow rate coefficient PQ . Kang [Kang, 1997] improved the database by correcting
PQ at very small Kn numbers and by taking into account of the different accommodation
coefficients at both wall boundaries. One of the key assumptions on the MGL model
from the linearized Bolzmann Equation is that the bulk flow speed is much smaller than
the molecular speed at a given temperature. In this case, the velocity distribution function
of the gas flow within the bearing clearance can be assumed to be almost the same as that
of isotropic equilibrium [Kang, 1997]. Therefore, when bulk flow velocity (∝ sliding
8 Called as the linearized BGK (Bhatnagar, Gross, Krook) model.
39
speed U of the journal) is comparable to the molecular speed at a given temperature; that
is, as the Mach number is approaches 1, MGL model fails to predict the correct pressure
profile, and direct molecular dynamic simulation or Direct Simulation Monte Carlo
(DSMC) should be used [Alexander, 1994] to calculate the direct momentum change
(pressure) at the wall.
3.1.1 Journal bearing In spite of the limitation of the MGL model explained above, a direct comparison
of the pressure profile from the MGL model with that from DSMC [Alexander, 1994]
and experimental results [Menon, 2000] confirmed the validity of the MGL model with
reasonable accuracy, up to Mach number 0.5 and a bearing clearance as low as 10 nm.
The Equation (3.1) can be non-dimensionalized as
)()()()( 33 PHPHZPPHQ
ZPPHQ PP τ
σθθθ ∂
∂+
∂∂
Λ=∂∂
∂∂
+∂∂
∂∂
(3.7)
where,
appP /= (3.8a)
Rx /=θ (3.8b)
RzZ /= (3.8c)
ChH /= (3.8d)
tωτ = (3.8e)
Ce /=ε (3.8f)
Here pa is atmospheric pressure and R is bearing radius. In the right-hand side of
Equation (3.7), bearing number Λ and squeeze number σ are defined as
26
=Λ
CR
pa
µω (3.9a)
40
212
=
CR
pa
µωσ (3.9b)
The Poiseuille flow factor PQ can be
1P =Q for the continuum model
PHKn
aQ a61P += for the first order slip model [Burgdorfer, 1959]
( )α;P KnfQ = for the MGL model [Fukui and Kaneko, 1988]
The non-dimensional gas film thickness H is given by
θεθε sincos1 YXH −−= (3.10a)
θεθε sincos1/ YXR CCH −−+= (3.10b)
for a bearing surface without steps and with steps, respectively. Using Equation (3.8), the
Knudsen number Kn can be represented as follows
PHKn
CPHPCHpNdRT
phNdRT
hKn a
aAA
===== 022 22
λππ
λ (3.11)
where λ0 (≅ 64nm) is a mean free path at atmospheric pressure and 20oC, and Kna is
defined as the characteristic Knudsen number, which is the ratio of λ0 to the bearing
clearance C.
Once the pressure field P(θ, Z) is obtained, the load carrying capacities FX and FY
and the dimensionless load parameters ζX and ζY along X and Y directions become
∫ ∫−= π θθθ20
/0
2 cos),(RLaX dZdZPRpF (3.12a)
∫ ∫−=π
θθθ2
0
/
0
2 sin),(RL
aY dZdZPRpF (3.12b)
41
RLpF
a
XX 2
=ζ (3.12c)
RLpF
a
YY 2
=ζ (3.12d)
The load parameters ζX and ζY are defined as a load capacity normalized with respect to
the projection area of the bearing multiplied by the ambient pressure. The total bearing
load parameter ζ becomes
22YX ζζζ += (3.13)
The attitude angle φ is the angle between the external load vector and the
eccentricity vector (see Figure 3.1), and is given by
2222
)(cosYX
X
YX
X
FF
FFeFe
ζζ
ζφ+
−=
+
−=
−−⋅
= (3.14)
The friction by air drag at the journal shaft can be calculated from the shear stress.
For fully diffusive reflection at both walls, the local velocity distribution function from
[Burgdorfer, 1959] in the x direction (Figure 3.1) becomes
( )λλλ
µ 2)()(
21 2
++
++−=h
yUhyydxdpu (3.15)
and the shear stress at the journal shaft (y=h) is given as
conccconppcphy
xy WWyu
,, ττττµτ +=+≡∂∂
==
(3.16)
42
Here, τp is the shear stress by Poiseuille flow, and τc is the shear stress by Couette flow.
Wp and Wc are the shear stress factors for Poiseuille flow and Couette flow, respectively.
The subscript “con” refers to continuum flow. Using gas kinetic theory, Kang [Kang,
1997] derived and tabulated numerically Wc and Wp for an arbitrary Knudsen number and
accommodation coefficients at both walls. Kang [Kang, 1997] showed that Wp is unity
when both walls have identical accommodation coefficients. The Couette shear stress
factor for a fully diffusive wall becomes
KnaPHPH
KnhhWc 221
12 +
=+
=+
=λ
(3.17)
Figure 3.2 compares the numerical tabulation Wc by Kang with that by the first
order slip model in Equation (3.17). The maximum error is about 10% at very high Kn
numbers. In this work, Equation (3.17) was used instead of the exact numerical tabulation
in Figure 4.2 to calculate the friction at the journal surface.
The friction torque by air drag becomes
∫ +∫=
∫ +∫∫ ∫ ==
π
ππ
θµ
θτττ
20 0
20 0
20 0
)2
(
)(
dzdRhUW
dxdphR
dzdRWRdART
cL
ccpLL
xyF
(3.18)
43
Figure 3.2 Comparison of shear stress factor by Couette flow for fully diffusive
walls; database from Kang [Kang, 1997] and first order slip model
Representing Equation (3.18) with non-dimensional parameters defined in
Equation (3.8), gives
∫ ∫
+= π θµ
θ20
/0
32
2dZd
CUR
HW
ddPH
RCpT RL ca
F (3.19)
By arranging terms, the non-dimensional rotational friction factor βJ for the journal
bearing becomes
∫ ∫
+Λ
== π θθωµ
β 20
/04
3 dZdH
WddPH
RCT RL cF
J (3.20)
3.1.2 Thrust Bearing The Reynolds Equation given in Equation (3.1) can be transformed to cylindrical
coordinates. The mass fluxes in cylindrical coordinates are
rphmr ∂
∂−=
µρ12
3
& (3.21a)
44
212
3 hUr
phm θθ
ρθµ
ρ+
∂∂
−=& (3.21b)
Inserting the mass flux into the continuity equation and adopting the ideal gas law gives
tph
rphU
rpph
rrpphr
rr ∂∂
+∂
∂=
∂∂
∂∂
+
∂∂
∂∂ )()(
2121
121 33
θθµθµθ (3.22)
where Uθ=rω. Non-dimensionalizing Equation (3.22) at steady state by
Rr
=η and ChH = (3.23)
yields
θη
θηθηη
η ∂∂
Λ=
∂∂
∂∂
+
∂∂
∂∂ )(33 PHPPHPPH (3.24)
Here r and the gas film thickness h of thrust bearing was non-dimensionalized using
radius R of the journal bearing and the journal bearing clearance C to get the same
bearing number for the same rotational speed.
Taking gas rarefaction effects into consideration, Equation (3.24) becomes
θη
θηθηη
η ∂∂
Λ=
∂∂
∂∂
+
∂∂
∂∂ )(33 PHPPHQPPHQ pp (3.25)
Integration of the pressure distribution over the thrust bearing surface area gives the load
capacity fT and load parameter ζT as
∫ ∫ −= π ηθηηθ20
//
2 )1),((RRRRaT
o
iddPRpf (3.26a)
45
)( 22ioa
TT RRp
f−
=π
ζ (3.26b)
where Ri and Ro are the inner and outer radii of the thrust bearings.
Rotational friction damping can be calculated the same way as for the journal
bearing. The flow velocity distribution in the θ direction considering first order slip
effects becomes
λλωλ
θµθ 2)())((
21 2
++
++−∂∂
=h
zrhzzr
pu (3.27)
Then, the shear stress at the bearing surface is
ccpz Wh
rrddph
zu
ττλ
ωµθ
µτ θθ +≡
++=
∂∂
=22
(3.28)
Integrating the shear stress over the thrust bearing surface, the total frictional torque
becomes
∫ ∫ +=
∫ +∫∫ ∫ ==
π
ππθ
θωµθ
θτττ
20
2
20
220
)2
(
)(
drdhrW
rddphr
drdWrdArT
RoRi c
ccpRoRi
RoRi zFT
(3.29)
Representing Equation (3.29) with non-dimensional parameters defined at Equation (3.9)
and (3.23) gives
∫ ∫
+= π ηθηωµ
θη2
0/
/
342
2dd
HCRW
ddPH
CRpT RRo
RRica
FT (3.30)
By rearranging Equation (3.30), the non-dimensional rotational friction factor βT for
thrust bearings becomes
46
∫ ∫
+
Λ== π ηθ
ηθ
ηωµ
β 20
//
3
4
3 ddH
WddPH
RCT RRo
RRicFT
T (3.31)
3.2 Numerical Method The dynamic gas film equation for the journal bearing, Equation (3.7), and the
static gas film equation for the thrust bearing, Equation (3.25), can be rewritten as
)(2 PHQJJ τ∂
∂Λ−=⋅∇ (3.32a)
0=⋅∇TT Q (3.32b)
, where
ZPPJi
ZPPHQiPPHQPHQ
∂∂
−+
∂∂
−Λ= 33θθ
ηθ ηη
θηη iPPHQiPPHQPHQ ppT
∂∂
−+
∂∂
−Λ= 33
Here, subscripts J and T denote journal and thrust bearing, respectively, and
),,( ηθ Zkik = are unit vectors along the θ, Z, and η directions. ∇J and ∇T are gradient
operators. Integrating Equation (3.32) over the control surface SP (shaded in Figure 3.3)
gives
PPSPPS JJ dSPHdSQ ∫∫∂∂
Λ−=∫∫ ⋅∇ )(2τ
(3.33a)
0=⋅∇∫∫ PS TT dSQP
(3.33b)
Applying the divergence theorem to the left-hand sides of Equation (3.33) gives
PPSPJPldSPHdlnQ ∫∫
∂∂
Λ−=⋅∫ )(2τ
(3.34a)
47
0=⋅∫ Pl TdlnQ
P
(3.34b)
where n is a unit normal vector along line lP, surrounding the control surface SP.
i-1,j i+1,j
i,j-1
i,j+1
i,j
i,j-1/2
i-1/2,j
i,j+1/2
∆θ
∆Z
θ
Z
lP SP
i+1/2,j+θQ−
θQ
+ZQ
−ZQ
SQ
Figure 3.3 Grid scheme for control volume method
The power law schemes [Patankar, 1980] based on the central difference method,
and the Gauss-Seidel iteration were used in the simulation. Details of the discretization of
each flux term and the detail numerical schemes are presented in Appendix A.
3.3 Static analysis
3.3.1 Journal bearings The bearing design parameters and gas properties for a static analysis are diameter
D=500µm, length L=300µm, bearing clearance C=1.0µm, viscosity µ=19.6×10-6 Ns/m2,
and molecular mean free path λo=64nm. Simulations assumed isothermal air at 20oC. The
step configuration on the journal bearings was such that θS/θP =0.5. Here, θS (=30o) is the
angle one step occupies, and θP (=60 o) is 2π divided by the number of steps (see Figure
3.4). In the analysis of stepped micro gas journal bearings, the angular location (θR in
48
Figure3.4) of the first step was varied with respect to the journal eccentricity (θR =25o, 15
o, 5 o, -5 o, -15 o, -25 o).
The accuracy of the numerical method for high eccentricities with the stepped
bearing geometry was checked for different numbers of grid points, along the
circumferential (n) and axial (m) directions, for Λ=1 and θR =-15o. Table 1 shows the
variations of load parameter ζ (Equation 3.13), and attitude angle φ (Equation 3.14) for
different grid schemes for ε=0.8 and ε=0.9, since numerical stability is more sensitive at
higher ε. Using the most dense grid, 336×36, as a reference, variations of ζ and φ for the
other grid schemes were within 2.3% for ε=0.8 and within 5.8% for ε=0.9. The grid
scheme 180×36 was used in the foregoing analyses.
θS
θP θ
Χ
ω
e
YYXX iFiFF +=
φ
C(1+ε)
C(1-ε)
Y θR
Figure 3.4 Stepped bearing configuration
Table 3.1 Numerical resolution for various grid schemes ( Λ=1, θR =-15o)
n×m 96×24 120×32 180×36 240×36 300×36 336×36
ε=0.8 0.032952 0.033219 0.033228 0.033496 0.033339 0.033424 ζ
ε=0.9 0.057738 0.058588 0.058444 0.059867 0.059344 0.060052
ε=0.8 80.7489 80.4778 81.6499 79.9762 80.8017 79.8424 φ
ε=0.9 78.025 77.0634 79.0161 75.2124 76.7797 74.7021
49
The specific film thickness 22bss hh σσ += , where σs and σb are RMS surface
roughnesses of the shaft and the bearing, respectively, and h is the local film thickness,
should be larger than 4 for full hydrodynamic lubrication [Pirro, 2001]. Assuming that
the bearings and shafts are fabricated by the same processes (surface roughness with
σs=σb=18.2nm), the minimum film thickness for hydrodynamic lubrication is 103nm,
which is equivalent to an eccentricity ~0.9 with a 1µm bearing clearance. Therefore, the
maximum eccentricity was limited to 0.9 in the analyses.
Figures 3.5 and 3.6 plot the load parameter ζ and attitude angle φ vs. bearing
number Λ for plain micro gas bearings and stepped micro gas bearings with θR=5o. In the
plain journal bearing, ζ monotonically increases for Λ<10, but tends to flatten for Λ>10
due to excessive side leakage by very high pressures generated on the bearing surfaces.
These saturation effects were not observed for the stepped journal bearings because the
maximum pressures were lower compared to those of plain journal bearings. The attitude
angle φ starts at 80-90o for small Λ and tends to decrease with increasing Λ for plain
bearings and step bearings. For a given eccentricity, the stepped bearings have smaller ζ
than plain gas bearings because repeated step geometry produced chain saw-like pressure
profile as shown in Figure 3.7. The chin saw-like pressure profile is generated because
gas is compressible fluid. In a gas or liquid lubricated bearing, high attitude angle results
in large negative cross stiffness and leads to whirl instability. Gas bearings should be
designed to minimize attitude angles at wide operating conditions.
50
Figure 3.5 Static performance of stepped micro gas bearings (θR =5o)
Figure3.6 Static performance of plain micro gas journal bearings
51
Figure 3.7 Pressure profiles (Λ=1, ε=0.4) of fabricated stepped gas bearing
A unique characteristic of stepped micro gas bearings is the variation of the
attitude angle φ and the load parameters ζ with the relative position (θR) of the journal
center and steps, when eccentricity is high. Figure 3.8 plots the load parameters ζ and
attitude angles φ vs. the bearing number Λ for eccentricity ε=0.8, with θR as a curve
parameter. The attitude angles φ vary over a wide range from 67 o to 107 o, depending on
θR, and the load parameters also change as a function of θR. This implies that, for a given
external load (load parameter), the journal center can have more than one steady state
position with different eccentricities and attitude angles. However, for small
eccentricities, for example, ε<0.6, the load parameters ζ and attitude angles φ were
almost independent of θR, as shown in Figure 3.9. Also shown in Figure 3.9 is the
simulation at extreme eccentricity (ε=0.98, which may not be a feasible hydrodynamic
region of the fabricated micro gas bearings), where the load parameters showed large
variations depending on θR and showed its maximum at –5<θR<5 over a wide range of
bearing numbers.
Figure 3.10 is a circumferential pressure profile at the bearing center with attitude
angles for ε=0.8 and 0.98. At ε=0.98, the attitude angle φ was larger than 150o, due to
52
high negative pressure built at the diverging gas film due to the adjacent trailing step. In
this case, the load capacity F (bearing reaction) is directed as in Figure 3.11. It should be
noted that even if the bearing reaction is directed toward the rotor displacement due to
very large attitude angles, the bearings should not be regarded as statically unstable. In
these cases, the external load can be applied in the opposite direction of the bearing
reaction force. The application of the negative pressurized bearing can be found in a hard
disc slider, where the servo actuator lifts the slider arm and the slider is designed such
that negative pressure is built to maintain a very thin gas film at steady state. At the same
token, it is possible to operate the bearings with high attitude angles if the bearings are
dynamically stable.
Figure 3.8 Load parameters and attitude angles of stepped micro
gas journal bearing at ε=0.8 for different θR
53
Figure 3.9 Load parameters of stepped micro
gas journal bearing at ε=0.6 and ε=0.98 for different θR
Figure 3.10 Circumferential pressure profile at the bearing center
(Λ=0.2, θR= 25o)
54
θΧ
ω e
φ
25o
Y
F
Figure 3.11 Direction of load capacity vector F for the conditions in Figure 3.10 with
ε=0.98, Λ=0.2, and θR= 25o. ζX = 0.0318 and ζY = 0.0156
3.3.2 Thrust Bearings
Figure 3.12, showing local maxima in ζT vs. step height, indicates the existence of
an optimal step height as a function of thrust bearing clearances CT for a given bearing
number Λ; this optimal step height (which produces the maximum load capacity) is a
weak function of the bearing number. The optimum ratio of bearing clearance to step
height falls between 1.2-1.4, which is similar to the optimal ratio (1.15) for a rectangular
step bearing with fixed pad [Cameron, 1966]. The step height of the thrust bearing was
chosen as 3µm to get as much load capacity as possible over a wide range of bearing
clearances.
3.3.3 Rotational Friction Factor
Rotational friction factors, calculated as shown in Figure 3.13 using a first-order
slip model, were almost constant over a wide range of bearing numbers, for given journal
positions or operational points for the thrust bearings. The simulations suggest that a
stepped gas journal bearing should have much lower rotational friction than plain gas
journal bearings.
55
Figure 3.12 Load capacity as a function of step height for thrust bearings
Figure 3.13 Non-dimensional rotational friction of micro gas bearing (θR =5o for
stepped journal bearings, step height 3µm for thrust bearings)
56
Chapter 4
Dynamic Performance of Micro Gas Bearings
In this chapter, dynamic performance of the micro gas bearing is investigated.
Orbit method is employed to simulate a numerical test rig, and to investigate whirl
instability. Improved bearing profiles with higher load capacity and dynamic stability are
suggested. Fabrication methods of meso scale gas bearings are proposed. Performance
analyses of the meso scale gas bearings in terms of two-dimensional imbalance response
are presented.
4.1 Approach We consider stability of micro gas bearings with an assumption of rigid rotor with
perfect alignment with the gas bearing along the axial direction. The equations of motion
for the journal shown in the Figure 4.1 are
∫ ∫ +−= π θθθ
ωε 2
0/
0 22
2
cos),(RL
a
eXaX Rp
FdZdZP
mCRp
&& , (4.1a)
∫ ∫ +−= π θθθ
ωε 2
0/
0 22
2
sin),(RL
a
eYaY Rp
FdZdZP
mCRp
&& , (4.1b)
where m is rotor mass, eF is the total external load ( 5.022 )( eYeXe FFF += ) including rotor
weight, and εX and εY are X and Y components of non-dimensional eccentricity vector ε.
The integral terms represent the components of bearing reaction force F. At equilibrium
operating conditions, the external load eF and bearing reaction F have the same
magnitude and opposite directions.
57
Χ ω
e
YYXX iFiFF +=
φ
Y
θ Fe
Χ ω
e
YYXX iFiFF +=
φ
Y
θ Fe
Figure 4.1 Journal bearing operating at an equilibrium point with eccentricity e, where F is bearing reaction force represented as integral term in Equation (4.1)
One approach to study the stability of the gas bearings linearizes the dynamic
equation (Equation 4.1) by small perturbations about a steady state position [Cheng and
Pan, 1965; Castelli and Elrod, 1965; Pan et al, 2003]. Corresponding stiffness and
damping matrix of the gas bearing are calculated by perturbation of the Reynolds
Equation (Equation 3.7) [Han et al, 1994; Hwang and Ono, 2003; San Andres and Wilde,
2000]. After stiffness and damping matrix are achieved, the bearing reaction force (the
integral term in Equation 4.1) can be decomposed into the reaction forces due to stiffness
and damping of the gas. Once the Jacobian matrix of the linearized equation has been
obtained, stability is assured if all eigenvalues of the characteristic equation of the
Jacobian matrix are negative.
In the other hand, the orbit method [Fuller, 1969; Piekos and Breuer, 1999; Jang
and Yoon, 2002] can accommodate arbitrary external loading patterns and disturbances
such as forced vibration, step-jump displacement, imbalance forces, various bearing
shapes and external loads. The orbit method traces the path of the journal center over
time, by solving journal dynamics (Equation 4.1) and the unsteady Reynolds Equation
(Equation 3.7) simultaneously, for given external forces and disturbances. Bearing
reaction forces at each time step are calculated by solving the unsteady Reynolds
58
Equation (Equation 3.7). The orbit method can accommodate molecular rarefaction effect
very easily because non-linear unsteady Reynolds Equation is solved at every time step.
Due to the high non-linearity of the Reynolds Equation, all the previous works
[Cheng and Pan, 1965; Castelli and Elrod, 1965; Han et al, 1994; Piekos and Breuer,
1999; San Andres and Wilde, 2000; Jang and Yoon, 2002; Pan et al, 2003; Hwang and
Ono, 2003] assumed continuum Poiseuille flow ( PQ =1 in Equation 3.7). For the micro
gas bearings with sub micron bearing clearances, the molecular rarefaction effects cannot
be neglected and the perturbation method can not be used unless the Poiseuille flow
factor PQ is expressed in a explicit function of pressure P and gas film thickness H.
Previous studies on the DRIE-processed micro gas bearings, using the orbit
method [Piekos and Breuer, 1999], didn’t include molecular rarefaction effect because of
the relatively large bearing clearance (12-14µm).
4.1.1 Scheme for Numerical Integration If four state variables are defined as,
1XX =ε , (4.2a)
2XX =ε& , (4.2b)
3XY =ε , (4.2c)
4XY =ε& , (4.2d)
then, the Equation (4.1) can be represented as four 1st order differential equations as:
21 XX =& (4.3a)
∫ ∫ +−= π θθθ
ω20
/0 22
2
2 cos),(RL
a
eXa
RpF
dZdZPmC
RpX& (4.3b)
43 XX =& (4.3c)
59
∫ ∫ +−= π θθθ
ω20
/0 22
2
4 sin),(RL
a
eYa
RpF
dZdZPmC
RpX& . (4.3d)
A fifth order Adams-Bashforth scheme [Nikravesh, 1988] as
)2511274261627741901(720
43211 −−−−+ +−+−∆
+= ni
ni
ni
ni
ni
ni
ni ffffftXX (4.4)
integrated equations (4.3). First five state variables in time domain were obtained by 4th
order Runge Kutta method. The advantage of Adams-Bashforth scheme over Runge
Kutta method is single evaluation of function at each time step instead of four
evaluations.
4.1.2 Stability Analysis: Threshold Speed A non-dimensional threshold speed9 [Cheng and Pan, 1965; Castelli and Elrod,
1965] 5.0
*
≡
e
crit
FCm
ωω , (4.5)
was used to predict the onset of whirl instability, where mcrit is the maximum allowable
rotor mass for stable operation, and external load Fe has the same magnitude but opposite
direction with bearing reaction force at steady state for given operating conditions (ε, Λ),
see Figure 4.1.
The procedure of stability analyses are as follows: 1) Interested operating point (ε, Λ or
ω) 10 for stability check is selected and corresponding bearing reaction force F is
9 The current definition of threshold speed should be distinguished from rotor critical speed, where damped rotor vibration is maximized before reaching the threshold speed.
60
calculated and external load Fe, in opposite direction to the bearing reaction force F, is
applied numerically, to establish static equilibrium; 2) Select test rotor mass m and give
very small step displacement (perturbation) from the steady state position; 3) Perform
orbit simulation, to check whether the disturbed rotor comes back to the steady state
position (the rotor is stable) or diverges (rotor is unstable); 3) Repeat the procedure
increasing the test rotor mass m until rotor becomes unstable; 4) Upon detection of rotor
instability, the test rotor mass m becomes critical rotor mass mcrit and ω*, the threshold
speed, can be calculated from equation (4.5). In early stages of gas bearing research
[Cheng and Pan, 1965; Castelli and Elrod, 1965], ω* was accepted as a general indicator
of gas bearing performance. The advantage of using ω* instead of critical rotor mass mcrit
is that ω* can be also decided by varying ω, with fixed rotor mass m becauseω* is a non-
dimensional number. In actual gas bearing systems, rotor mass is usually fixed and rotor
dynamics over wide speed ranges are more of interest. Disadvantage of using ω* comes
when plotting ω* versus Λ (or equivalently ω), because rotational speed ω appears at
both axes.
To test the orbit program, journal center was initially positioned at the origin
under external load corresponding to eccentricity (0.6,0) and convergence of journal orbit
to the steady state position (0.6,0) was followed as in Figure 4.2. The stability analyses of
stepped bearings were performed for θR=0o, and external load Fe was oriented such that
the journal was initially on the X-axis. For plain bearings, external load was directed
along –Y and initial journal location was obtained for successive disturbance and orbit
simulation. A step displacement of 0.0005 (0.05%) of the steady state eccentricity was
applied, and the corresponding journal orbits were obtained. Figure. 4.3 shows exemplary
diverging orbits of plain and stepped bearings with whirl frequency 0.465ω and 0.341ω
respectively. Figure 4.4 shows converging orbits for stepped bearing.
10 Note
26
=Λ
CR
pa
µω and proportional to ω
61
Figure 4.2 Converging orbit from origin (stepped gas journal bearing, Λ=0.6,
ε0=0.6, ω*=0.4, C=1µm)
The threshold speed maps were constructed by varying the non-dimensional
group on the right hand side of Equation (4.5), to estimate the threshold ω* at which
orbits diverge. Figure 4.5 and Figure 4.6 plot the threshold speeds ω* vs. Λ, with
eccentricity ε as a curve parameter for plain and stepped bearings respectively. Because
of the trial and error based method, the accuracy of ω* values is ±0.05. The general
tendencies of Figure 4.5 are similar to results reported in [Cheng and Panl, 1965; Castelli
and Elrod, 1965], where a peak threshold speed ω* exists around Λ=0.8~2, depending on
the L/D ratio, which indicates a maximum damping. Stepped gas bearings show a more
distinguished peak of ω* around Λ=2, at eccentricity ε=0.8 as shown in Figure 4.6
62
(a) Plain bearing, Λ=1, ω*=1.6, ε0=0.6 (b) Stepped bearing, Λ=3, ω*=2.7, ε0=0.8
Figure 4.3 Diverging orbits Plain bearing: (a) Λ=1, ω*=1.6, ε0=0.6, (b) Stepped bearing, Λ=3, ω*=2.7, ε0=0.8
Figure 4.4 Converging orbit (stepped gas journal bearing, Λ=2, ω*=1.5, ε0=0.8)
63
Figure 4.5 Threshold speed map for plain gas bearings
Figure 4.6 Threshold speed map for stepped gas bearings
64
4.2 Discussion on the Whirl Instability
4.2.1 Quasi-Stable Behavior of Stepped Bearings
Stepped bearings can exhibit peculiar journal orbits for high ε and low Λ. Figure
4.7 shows a journal orbit (Λ=0.6, ω*=0.9) stabilizing at (εX,εY)=(0.694,0.547), after a
small step displacement (of 0.05% eccentricity) disturbed the system from its initial
steady state position, (εX0,εY0)=(0.8,0).
To support the phenomena, another orbit simulation was performed. Journal was
positioned at bearing center at t=0 and external load opposite to the load capacity
corresponding to position (0.8,0) or (0.694, 0.547) was applied and corresponding orbits
were obtained as in Figure 4.8. For given external load, the rotor stabilized at position
(0.694, 0.547) again.
These peculiar phenomena can be explained by the fact that there can be more
than one position (ε, φ, θR) of journal center that produce the same load capacity. From
the static analysis, the journal at (εX,εY)= (0.694,0.547) had the same load parameter
(ζ=0.019) as the journal at (εX0,εY0)=(0.8,0). Interestingly, for given external load, the
position (0.694,0.547) with φ=101.2o (see Fig. 4.8) is preferred to position (0.8,0) with
φ=63.8o. The load vector is tangent to the initial journal orbit, as should be. The location
(0.694,0.547) has θR of 21.76 o from step geometry. The stabilizing motions are
considered phenomena that minimize average pressure inside the bearing. The average
pressure at (0.694,0.547) was lower than at (0.8,0). Figure 4.9 and Figure 4.10 shows the
pressure profiles for the two locations. These stabilizing motions were negligible for Λ>
2 with ε =0.8 (Figure 3b), or ε < 0.8 with all Λ (Figure 4.4), due to smaller variation of
load capacities as discussed earlier.
Another interesting phenomena of step bearings are the initiation of self-sustained
confined orbit shown in Figure 4.11, when rotor is perturbed at equilibrium points for
certain operating conditions (ε=0.8 and Λ ≤1). These self-sustained vibrations did not
65
progress to half–frequency whirl and did not converge to an equilibrium point either
within 50 cycles. It seems there are certain range of ω* below actual threshold speed (, at
which orbit begin to diverge,) for these self-sustained vibration to persist.
For stepped bearings with ε=0.8 and Λ ≤1, the data points in Figure 4.6 should be
understood as quasi-stable positions that transit to more stable journal positions
minimizing average pressure inside the bearings or that maintain self-sustained confined
vibration.
Figure 4.7 Stabilizing motion of journal to (εX,εY) =(0.694,0.547) by small
disturbance at (εX0,εY0)=(0.8,0), with Λ=0.6, ω*=0.9
external load
eccentricity
Figure 4.8 Motion of journal to (εX,εY) =(0.694,0.547) from origin for static loading corresponding to (εX0,εY0)=(0.8,0) with Λ=0.6, ω*=0.9. The angle between external
load and eccentricity vector is attitude angle
66
Figure 4.9 Pressure profile at (εX,εY)=(0.694,0.547)
Figure 4.10 Pressure profile at (εX,εY)=(0.8,0.0)
Figure 4.11 Self sustained vibration by perturbation, 50 cycles, ε=0.8, Λ=1, ω*=1.7,
stepped gas journal bearing
67
4.2.2 Threshold Rotor Mass
As discussed in section 4.1.2, plotting threshold speed ω* versus Λ complicates
understanding physical meaning of threshold speed. Of interest is the maximum
allowable rotor mass for a given operating speed (Λ) and operating position (ε) of the
rotor. Following the definition in [Piekos and Breuer, 1999], equation (4.5) can be
rewritten as a non-dimensional threshold rotor mass11,
25
2
*72
*
Λ=
≡
ωζµ R
CL
pmm acrit . (4.6)
Once ω* is calculated for specific operating conditions (for given Λ and ζ (or Fe)), m*
can be calculated from equation (4.6) and is directly proportional to mcrit , maximum
allowable rotor mass.
Assuming steel, the rotor mass m for L=300µm, and D=500µm is 0.46mg, which
corresponds to m* =5.75×10-3 from Equation (4.6). Stability charts with m* vs. Λ are
plotted in Figure 4.12 and 4.13 for plain and stepped bearings respectively. The dashed
horizontal line marks the minimum mass that the rotor can have, if composed of steel
with the required geometry. The region under the dashed line pertains to a very small
mass, and is not a physically feasible operating condition for a steel rotor.
Fabricated stepped bearings without axial grooves have significantly lower m*
than plain bearings. From Figure 4.5 and 4.6, ω* ranges 0.6~4 for plain gas journal
bearings, and 0.3~2 for stepped gas journal bearings. Assuming ω* is the same order of
magnitude for plain and stepped gas journal bearings for a given Λ, the corresponding
threshold rotor mass m is proportional to load carrying capacities from Equation (4.5). 11 From definitions of load parameter (3.12) and bearing number (3.9), ζRLpF ae 2= ,
2
6
Λ
=RCpa
µω .
Plugging these into equation (4.5) and taking square at both sides,
ζζµζµω
225
2
5
2
222 *
72236* Λ
=Λ
=
Λ
≡ mRC
Lpm
Lpm
RCp acrit
a
crita , where 5
272*
≡
RC
Lpmm acrit
µ
68
Stepped micro gas journal bearings have significantly lower load carrying capacities, see
Figure 3.5 and Figure 3.6. Initially, 2µm step was formed to lower load capacity and
allow the bearing to run with high eccentricity. However, due to small load capacities, the
fabricated stepped gas journal bearings show significantly lower threshold rotor mass
than plain micro gas journal bearings.
Figure 4.12 Stability chart of plain gas journal bearing given as non-dimensional
threshold speed m*
Figure 4.13 Stability chart of stepped gas journal bearing given as non-dimensional
threshold speed m*
69
Micro turbines and engines are designed to be operating at over 1 million rpm
[Frechette et al, 2001b; Epstein et al, 1997] to generate useful power. Λ=10 corresponds
to about 1.3 million rpm. Even if inertia effects are very small, micro gas bearing systems
are highly unstable at high Λ, and require very high eccentricities for stability. Equivalent
rotor mass is the sum of the actual mass plus an additional mass that renders an inertial
load equivalent to the static load capacity. Calculated equivalent rotor mass, assuming
external load is applied only by rotor weight, is much bigger than the threshold rotor
mass (m or m*) even for plain gas journal bearings. This implies significant external load
is needed for the rotor with such a small threshold mass to obtain the necessary
eccentricities and thus stability.
4.3 Design Improvement of Stepped Micro Gas Bearings As discussed above, the fabricated micro gas bearings had low stability. To
improve the design, effects of various step configurations on the static and dynamic
characteristics were investigated. The first approach was to place a deep axial groove at
the beginning of every step, as shown in Figure 4.14. Axial grooves supply gas with
ambient pressure to minimize high negative pressure, shown in Figure 3.7 and decrease
attitude angles.
Χ
θ ω
e
axial groove
Y
Figure 4.14 Axial grooves formed at the beginning of every step. Axial grooves
supply gas with ambient pressure to minimize high negative pressure and decrease attitude angles
70
The second approach was the variation of the number of steps, step height, and
θS/θP without adopting axial grooves, where θS is the angle one step occupies, and θP is
2π divided by the number of steps as shown in the following figure.
4.3.1 Static Analysis of Improved Design
Figure 4.15 shows ζ and φ vs. step height, for Λ=1 and 10 and θR= 0o. Two grid
points occupied one axial groove (of width 17.5µm) in the simulation. While load
capacity is similar to step bearings without axial grooves for the same step height,
attitude angles are much lower (than step bearings without axial grooves), and minimize
at around 0.75~1.25µm, implying an optimal step height in terms of attitude angles and
cross stiffness.
Figure 4.16 and 4.17 show load parameter ζ and attitude angle φ vs. θS/θP for
micro gas bearings with six steps and four steps respectively, with θR=0o, ε=0.6, and Λ=1.
The load parameter ζ increased as θS/θP decreases and step height decreases for both six-
stepped and four-stepped bearings. The bearings with four steps had larger load
parameter ζ than the bearings with six steps (for the case of the same step height) due to
large bearing land area to build enough hydrodynamic pressure. While, the stepped
bearings with 2µm steps reveal sharp decrease of load capacities as θS/θP increases, the
stepped bearings with 1µm steps had lower sensitivity of load parameters to the θS/θP
ratio, due to relatively small step height. The variation of attitude angles φ is small
compared to the variation of load parameters, as step configuration changes. The
optimum θS/θP was not found in terms of ζ.
71
(a) Λ=1
(b) Λ=10
Figure 4.15 Static performance for stepped gas journal bearings with axial grooves (θR= 0o): (a) Λ=1, (b) Λ=10
72
Figure 4.16 Load parameter and attitude angle of six-stepped bearings without
axial grooves for various step configurations (θR=0o, ε=0.6, and Λ=1)
Figure 4.17 Load parameter and attitude angle of four-stepped bearings without
axial grooves for various step configurations for (θR=0o, ε=0.6, and Λ=1)
73
4.3.2 Dynamic Analysis of Improved Design Orbit simulations were performed for the stepped bearings with axial grooves.
Table 4.1 summarizes the effect of axial grooves on the ω* and m* for θS/θP=0.5. The
values in parenthesis represent the m* ratio of the investigated bearings to the fabricated
stepped bearing (θS/θP =0.5, 2µm steps, no axial grooves). Table 4.1 shows substantial
improvement of stability for low Λ by adopting axial grooves. Table 4.1 also presents
orbit simulations for 1µm step height (considered optimal) with axial grooves. Over the
entire region of operating conditions, threshold mass m* increased with improvements
more prominent at lower Λ. Despite improved stability by adopting axial grooves, overall
performance was inferior to the plain gas bearings, due to similar load capacities with
bearings without axial grooves. However, very high threshold mass (shown as italicized
bold-faced) larger than plain bearings (m*= 2.791 for ε=0.8) was observed at Λ=1, and
ε=0.8, implying very high damping. There were no noticeable stabilizing journal motions
at high eccentricities for step bearings with axial grooves.
Table 4.2 presents the effects of different θS/θP, number of steps, and step height
on theω* and m* for the bearings without axial grooves. As shown in Figure 4.16,
attitude angles of six-stepped bearings are almost the same for different step
configurations; ω* is almost the same (0.9~1.05), and threshold mass m* is directly
proportional to the load capacity (the first two columns of Table. 4.2). However,
noticeable improvement of stability was observed by reducing the number of steps to
four. The italicized bold-faced ω* and m* in Table 4.2, represent the bearings with higher
threshold mass than the plain bearings (m*=0.257 for ε=0.6), even if the load capacity
was lower than the six-stepped bearings with 1µm steps or plain bearings.
When the number of steps is reduced without axial grooves, self-sustained
confined orbit (similar to Figure 4.11 for six stepped bearing without axial groove) was
observed (Figure 4.18) even at low eccentricity (ε=0.6) when Λ≤1. However the
magnitude of the orbits were much smaller compared to the orbits for six-stepped
bearings (Figure 4.11).
74
Table 4.1. ω* and m* for stepped bearing with axial grooves (θS/θP =0.5, θR=0o) Step height 2µm
ε=0.5 ε=0.6 ε=0.7 ε=0.8
Λ=1 ω*=1.7
m*=0.02778 (3.02)
1.7 0.03869 (2.52)
2.1 0.08722 (2.39)
11 4.09752 (41.2)
Λ=10 1.5
0.00247 (0.68)
1.5 0.00320 (0.75)
1.5 0.00416 (0.80)
1.6 0.00636 (0.90)
Step height 1µm ε=0.5 ε=0.6 ε=0.7 ε=0.8
Λ=1 1.4
0.03451 (3.75)
1.6 0.06140 (3.99)
2.1 0.14960 (4.10)
10 5.36439 (53.93)
Λ=10 1.5
0.00461 (1.27)
1.4 0.00532 (1.25)
1.35 0.00657 (1.26)
1.5 0.01113 (1.58)
Table 4.2. ω* and m* for different θS/θP, the number of steps (θR=0o, ε=0.6, Λ=1)
Bearings with 6 steps Bearings with 4 steps 2µm steps 1µm steps 2µm steps 1µm steps
θS/θP =0.333 ω*=0.95
m*=0.02365 (1.54)
1 0.04059 (2.64)
6 1.50223
(97.71)
1.7 0.15047 (9.79)
θS/θP =0.5 1
0.01537 (1)
1.05 0.03147 (2.05)
3.5 0.30970 (20.14)
1.55 0.08630 (5.61)
θS/θP =0.667 0.9
0.00676 (0.44)
0.9 0.01656 (1.08)
1.3 0.01868 (1.22)
1.4 0.04370 (2.84)
75
Figure 4.18 Orbit for 50 cycles for four stepped bearing without axial grooves, θS/θP =0.333, Λ=1, ε=0.6, step height 2µm, and ω*=6 (m*=1.5022, m=0.12g)
Whirl instability is often understood as resonance phenomena between journal
motion and circumferential gas flow, at low eccentricity and high attitude angle. From the
orbit simulations for various step geometries, and calculation of circumferential flow
speed, threshold mass was dependent on the load capacity, attitude angle, and how
effectively circumferential gas flow is blocked. For six-stepped bearings, load capacities
are significantly lower than the plain bearings, due to small bearing land area to build
effective hydrodynamic pressure. By adopting axial grooves, attitude angles became
much smaller at low eccentricities, rendering low cross stiffness and improved stability,
even if load capacities did not increase. Four-stepped bearings had much higher load
capacities than six-stepped bearings, permitting higher stability. However, for the four-
stepped bearings with 1µm steps, higher attitude angles (even if load capacities were a
little higher) than the bearings with 2µm steps (Figure 4.17), and insufficient step height
to block the circumferential gas flow effectively (presumably) as 2µm steps, led to lower
threshold mass than the bearings with 2µm steps.
76
4.3.3 Four-stepped Bearings with Axial Grooves From the previous investigation on the dynamic performance of the micro gas
bearings with various step geometries (with six stepped bearing with axial grooves and
four-stepped gas bearings without axial grooves), the four-stepped micro gas bearing with
axial grooves in Figure 4.19 seems the most effective to increase stability.
Figure 4.19 Four-stepped micro gas bearings with axial grooves
Figure 4.20 presents load parameter ζ and attitude angle φ versus bearing number Λ, for
θS=30o and θR=0o. Both figures have clearance C=1µm. With axial grooves, the bearings
with 1µm step showed lower attitude angle than those with 2µm step. Without axial
grooves, the bearings with 2µm step showed smaller attitude angles as shown in the
Figure 4.17. Figure 4.21 shows pressure profile at the bearing center. Unlike the pressure
profile of six stepped bearing without axial grooves (shown in Figure 3.7), more direct
stiffness and stability can be expected.
Orbit simulation was performed for the bearings with the geometry of Figure
4.19, for ε=0.5 and 0.6. Figures 4.22 to 4.25 display orbits for various operating
conditions. Figure 4.22 with Λ=1, ε=0.6 has converging orbits with chaotic motion, when
ω* is very large (> 15). Notice the different orbits for the different step heights. As ω* is
increased to 40 (see Figure 4.23) with other conditions as in Figure 4.22(b), the orbit
shows chaotic motion without converging or diverging, even after 400 simulated cycles.
77
(a) C=1µm, step height 2µm
(b) C=1µm, step height 1µm
Figure 4.20 Static performance of four-stepped micro gas journal bearings with axial grooves: (a) C=1µm, step height 2µm, (b) C=1µm, step height 1µm
78
Figure 4.21 Pressure profiles (Λ=1, ε=0.4) of four-stepped micro gas bearings
with axial grooves
In Figure 24 (with Λ=5, ε=0.6, ω*=5, and step height 1µm), the orbit diverges
initially (by perturbation) but finally becomes a self-sustained stable orbit. This is true for
2.4<ω*<8.5, however, for ω*>8.5, the orbit diverges continuously and becomes unstable
with whirling motion. The magnitude of vibration in the self-sustained stable orbit
increased as ω* increased within its stable range. For both step heights (1µm and 2µm),
there was a range of ω* that initiated the self-sustained stable orbit for ε=0.5 and 0.6.
When eccentricity was increased to 0.8, with the same bearing number (Λ=5), journal
orbit became chaotic (Figure 4.25), instead of a self-sustained finite orbit. The chaotic
characteristics increased as ω* varied from 1 to 20. The chaotic motions along the Y
direction were confined within the same range, and only the amplitude of vibration along
the X-direction increased as ω* increased (see Figures 4.25). However, the journal orbits
did not diverge.
The exact boundaries of eccentricity ε and bearing number Λ, which initiate self-
sustained stable orbit or chaotic motions, are not clear from the limited number of orbit
79
simulations performed here. Because the ranges of chaotic motions are extremely small
(∆ε~10-3), all orbits were assumed stable because half-frequency whirl was never
observed in the simulations.
Instead of full threshold mass for all operating conditions, threshold speed and
rotor mass at ε=0.5 and ε=0.6, for Λ=1 and Λ=5, were calculated in Table 4.3. Compared
to Table 4.1 and 4.2, the four-stepped bearings with axial grooves appear to be much
more stable than other stepped bearings.
Table 4.3 Threshold speed12 ω* and rotor mass m* for four-stepped gas bearings with axial grooves C=1µm, step height=1µm, θS/θP =0.333 (θS =30o)
Λ=1 Λ=5
Step 1µm Step 2µm Step 1µm Step 2µm
ε=0.5 ω* =3.0 m* =0.2793
2.0 0.1048
3.5 0.0767
2.4 0.0278
ε=0.6 >1513 >10.3495
>15 >8.8486
8.5 0.6408
3.6 0.0892
12 The error of the threshold speed ω* in the Table 4.3 is ±0.2. 13 At ω* =15, the orbit was still stable
80
(a) Orbit for step height 1µm
(b)
(b) Orbit for step height 2µm
Figure 4.22 Orbit for 100 cycles of four-stepped gas bearings with axial grooves, Λ=1, ε=0.6, and ω*=15: (a) Orbit for step height 1µm, (b) Orbit for step
height 2µm
81
Figure 4.23 Orbit for 400 cycles of four-stepped gas bearings with axial
grooves, Λ=1, ε=0.6, step height 2µm, and ω*=40. Very slow chaotic motion initiates and does not converge or diverge
82
(a) Orbit for total 200 cycles
(b) Orbit for last 10 cycles
Figure 4.24 Orbit for four-stepped gas bearings with axial grooves, Λ=5, ε=0.6, ω*=5, and step height 1µm: (a) Orbit for total 200 cycles, (b) Orbit for last 10 cycles
83
(a) Orbit for ω*=1
(b) Orbit for ω*=5
(c) Orbit for ω*=10
(d) Orbit for ω*=20
Figure 4.25 Orbit for 200 cycles of four-stepped gas bearings with axial grooves, Λ=5, ε=0.8, and step height 1µm: (a) Orbit for ω*=1, (b) Orbit for ω*=5, (c) Orbit
for ω*=10, (d) Orbit for ω*=20
84
4.4 Discussions of Stability Analyses The foregoing dynamic stability analyses can locate feasible stable operating
points of the micro gas bearing. In a practical sense, a rotor running at a prescribed
eccentricity (high for stable operations), poses difficulties, especially for micro or even
meso scale applications. When the rotor-bearing configuration and operating speed are
given, the eccentricity should be chosen from the stability chart (such as Figure 4.13),
where threshold rotor mass should exceed the given rotor mass. The weight of the rotor
mass should generate the necessary eccentricity, otherwise, additional external load
should be applied. Problems involving additional loading include: difficulties controlling
bearing and rotor orientation in terms of attitude angle and step locations; unavoidable
and very large imbalance force, especially at high rotational speeds; and crash of the rotor
onto the bearing wall, at high eccentricity.
More amenable is an investigation of bearing performance under actual dynamic
load (rotor weight and imbalance force, generated by operating speed). If rotor and
bearing are initially not perfectly aligned, the initial misalignment can become a
disturbance. In the following sections, the fabrication method of meso scale (~mm) gas
bearings using X-ray lithography, and performance of the gas bearings in two application
areas will be investigated, in terms of three-dimensional imbalance response.
4.5 Feasibility Study of Meso Scale Gas Bearing In this section, new fabrication processes for the meso scale gas bearings are
suggested. Two application areas will be introduced and dynamic performance at a
systems level will be investigated.
4.5.1 Fabrication Processes of Meso Scale Gas Bearings Typical electron-beam lithography for initial optical mask to make a final X-ray
mask, provides a maximum resolution of 6nm and a usual beam spot size of 0.1µm.
Theoretically, several identical patterns within the resolution of the initial optical mask
85
can be transferred to the final parts. These parts are identical within the resolution of the
X-ray mask. The accuracy of pattern transfer from the optical mask to the X-ray mask
depends on process conditions.
The fabrication processes of the meso scale gas bearings are based on stacking
several identical parts, made by X-ray lithography. Due to the in-plane lithographic
technique process, there is no limit on the diameter of the bearings. Figure 4.26
conceptually depicts a meso scale gas bearing of 2mm length, with step geometry shown
in Figure 4.19, made by stacking or fusion bonding of 500 µm thick four identical
bearings. Both end faces have spiral grooves, to serve as thrust bearings. The spiral
grooves14 can be easily formed by single optical lithography and etching process on the
electroplated part.
Figure 4.26 Conceptual figure of meso scale gas bearing with spiral grooved
thrust bearings, D=L=2mm
4.5.2 Applications of Meso Scale Gas Bearings Potential applications of meso scale gas journal bearings include Laser scanner
[Ono and Hwang, 1994; Hwang and Ono, 1996], and hard disc drive (HDD) main spindle
bearings [Hwang and Ono, 2003; Jang and Yoon, 2002]. Other applications may include:
meso scale turbine for micro power generation, dental drills, miniature precision machine
tool, and micro motors. Polygon mirror attached to a high-speed spindle in Laser scanner
14 The spiral grooved thrust bearings have a seal region, which was not described in the Figure 4.25, before the spiral grooves reach journal bearing.
86
has rotational speed ranging from 30,000 rpm to 50,000 rpm, for Laser printer and photo
printer applications. Another application of Laser scanner is an optical low coherence
reflectometry (OLCR) [Szydlo et al, 1998], an imaging technique, for various science and
engineering field including biological, medical, material growth, optoelectronics, etc.
OLCR consists of a broadband light source and a coherent cross correlation light
detector. Measurements must be performed rapidly, to capture or freeze motion of
moving or live objects. Increasing image acquisition rate and light scanning speed are
crucial. Szydlo et al [Szydlo et al, 1998] demonstrated an air turbine driven gold plated
cubic mirror with rotational speed of 427,000 rpm, equivalent to a scan repetition rate of
28.5kHz. Heshmat [Heshmat, 2003] reported stable operation of mesoscopic turbine
simulator, at 700,000 rpm, using miniature foil gas bearings, made via conventional
precision machining.
Figure 4.27 depicts suggested configuration of Laser scanner, with hydrodynamic
gas bearing suggested in Figure 4.26. The simulation parameters, including dimensions,
are shown in Table 4.4. The rotor mass and moments of inertia were calculated from
solid models and material densities. The origin of the simulation’s coordinate system was
the center of the gas bearing.
To meet high demand for bigger information storage capacity, performance
requirements of hard disc drive (HDD) spindle bearings became more stringent. For
20Gb/in2 HDD, tracks per inch (TPI) is 50000, which requires a non-repeatable run out
(NRRO) of 30nm[Hwang and Ono, 2003]. For 10Gb/in2 HDD, the maximum allowed
NRRO is 0.5µm [Hwang and Ono, 2003]. Figure 4.28 depicts an HDD spindle rotor
consisting of two discs, two clamps, a permanent magnet rotor, and two spacers. Included
in Table 4.5 are the rotor mass and moments of inertia, calculated from solid models and
the material densities. Simulated gas bearing has diameter and length of 6mm and
rotational speed was fixed at 15,000 rpm.
87
Scanner Mirror
Bearing unitMotor x (X)
y (Y)
z (Z)
(a) Assembled unit (b) Rotor
Figure 4.27 Laser scanner: (a) Assembled unit, (b) Rotor. The X-Y-Z is a fixed reference frame and x-y-z is a rotational coordinate attached to rotor center
Spacer
Clamp Disc
Permanent magnet
Clamp
Shaft x (X)
z (Z)
Figure 4.28 HDD spindle rotor: The X-Y-Z is a fixed reference frame and x-y-z is a
rotational coordinate attached to rotor center. y(Y) axis is into the plane
88
Table 4.4 Design parameters of Laser scanner rotor
Rotor mass 0.3g Imbalance radius 0.1µm
Ixx = Iyy 1.05x10-9 kgm2 Izz 8.97x10-10 kgm2
Bearing diameter 2mm Bearing length 2mm
Bearing clearance 1µm Rotational speed 100,000-1,000,000rpm
Working temperature 20oC Air viscosity 1.79x10-5 Ns/m2
Molecular mean free path 64nm
Table 4.5 Design parameters of HDD rotor
Rotor mass 57.3g Imbalance radius 4µm
Ixx =Iyy 2.18x10-5 kgm2 Izz 4.25x10-5 kgm2
Bearing diameter 6mm Bearing length 6mm
Bearing clearance 1µm Rotational speed 15,000 rpm
Working temperature 50oC Air viscosity 1.92x10-5 Ns/m2
Molecular mean free path 70nm
4.5.3 Three-Dimensional Imbalance Response In this section, three-dimensional imbalance responses of Laser scanner and HDD
rotors, including gyroscopic effects, are investigated. In Equation (4.1), Fe can
accommodate any time varying external force. Assuming gravity is directed along x-
direction, with imbalance radius a, equation (4.1) for the systems shown in the Figure
4.27 and 4.28 becomes
89
∫ ∫
++−= π τωθθθ
ωε 2
0/
0 2
2
2
2 coscos),(RL
a
aX Rp
mamgdZdZPmC
Rp&& , (4.7a)
∫ ∫ +−= π τωθθθ
ωε 2
0/
0 2
2
2
2 sinsin),(RL
a
aY Rp
madZdZPmC
Rp&& . (4.7b)
Here, dot means derivative with respective to tωτ = . Figure 4.29 shows a rotor (at
eccentricity e) with angular rotational speed xϕ& and yϕ& along the x- and y-axis,
respectively. X-Y-Z is a fixed reference frame with origin at the bearing center and x-y-z
is a rotational coordinate attached to the rotor center. The moment due to imbalance force
becomes
jmazimazM uuu τωτω cossin 22 +−= (4.8)
where zu locates the imbalance mass along the z direction. i and j are unit vectors along
the x and y direction, respectively.
Xx
Y
Z
e
y
zω
yϕ&
Figure 4.29 Rotor with angular rotation and eccentricity: The X-Y-Z is a
fixed reference frame with origin at the center of bearing and x-y-z is a rotational coordinate attached to rotor center.
90
When there are instantaneous rotational speeds xϕ& and yϕ& , the angular momentum
vector G and instantaneous angular rotation vector Ω of the rotor with the spin speed ω,
become
kIjIiIG zzyyyxxx ωϕϕ ++= && (4.9)
kji yx ωϕϕ ++=Ω && (4.10)
respectively. The relation between the change of angular momentum and the total
external moment M due to the imbalance force and bearing reaction force, becomes
MdtGd
= (4.11)
By inserting equation (4.9) into equation (4.11), incorporating equation (4.10), the
equations describing the conical motion of the spinning rotor can be derived as,
uxiuxyzzyyyxxx MMIII +=+− ϕωϕωϕ &&&& (4.12a)
uyiuyxzzxxxyyy MMIII +=−+ ϕωϕωϕ &&&& (4.12b)
where Miux and Miuy are reaction moment from gas bearings calculated via integration of
moment by dynamics pressure. Note equation (4.12) pertains to the rotating coordinate
systems x-y-z, attached to the rotor. The equation should be referenced to the fixed X-Y-Z
coordinate.
The relation between instantaneous rotational speed about x-y-z coordinate and X-
Y-Z coordinate becomes, from [Nelson, 1976]
[ ]
Ω=
Y
Xy
x
Tψψ
ωψψ
&
&&
&
or [ ]
=
Ω−
ωψψ
ψψ y
x
Y
X T &
&
&
& 1 (4.13a)
91
[ ]
−
−=
XXY
XXY
Y
Tψψψ
ψψψψ
sin0coscoscos0sincos
01sin (4.13b)
where Ω is rotation speed of rotor with respect to fixed X-Y-Z coordinate. Using relation
(4.13), equation (4.12) becomes
uXRLRLaYzzXxx MdZdZZPRpII +∫ ∫=+ −
π θθθϕωϕ 20
2/2/
3 sin),(&&& (4.14a)
uYRLRLaXzzYyy MdZdZZPRpII +∫ ∫−=− −
π θθθϕωϕ 20
2/2/
3 cos),(&&& (4.14b)
Now external moment uXM and uYM are for fixed reference frame X-Y-Z, attached to
bearings, and bearing reaction moments are expressed as integration terms of pressure
field at the bearing surfaces. Due to the gyroscopic effect of the spinning rotor,
Xϕ& and Yϕ& generate moments along Y- and X- directions in addition to those by the
imbalance force and bearing reaction force. Non-dimensionalizing equation (4.14) using
RCϕψ = and τ =ω t,
+∫ ∫+−= − 3
20
2/2/2
4
sin),(Rp
MdZdZZP
CIRp
II
a
uXRLRL
xx
aY
xx
zzX
π θθθω
ψψ &&& (4.15a)
+∫ ∫−+= − 3
20
2/2/2
4
cos),(Rp
MdZdZZP
CIRp
II
a
uYRLRL
yy
aX
yy
zzY
π θθθω
ψψ &&& (4.15b)
Here, dot means derivative with respective to τ. Equation (4.15) with equation (4.7), can
simulate complete 3-D imbalance responses. In a strict sense, the imbalance forces, in
equation (4.7), should be transformed to those with respect to the fixed bearing
coordinates X-Y-Z. However due to very small bearing clearance, compared to bearing
radius or length, the numerical error is negligible.
92
To calculate pressure field at bearing surface when journal has eccentricities εX, εY
and misalignment angles ψX, ψY, local gas film thickness H should be expressed as a
function of εX , εY , ψX ,and ψY . Gas film thickness H for Reynolds Equation becomes
θψεθψε sin2
cos2
1
+−−−
+−+−= Z
RLZ
RLH XYYX (4.16)
In the simulations of Laser scanner, imbalance mass was assumed as 0.01%15 of
rotor mass, located on the outer surface of rotor shaft, giving imbalance radius of 0.1µm.
For HDD spindle rotor, imbalance mass was 0.13% of rotor mass, to give the imbalance
radius and imbalance force of 4µm and 0.56N, respectively, at 15,000RPM, which are
typical operating conditions of HDD spindles [Hwang and Ono, 2003; Jang and Yoon,
2002]. The air viscosity and molecular mean free path were adjusted according to the
working temperatures as shown in Tables 4.4 and 4.5.
To investigate conical stability of the gas bearings, the rotor was given initial
misalignment angles (ψX0, ψY0)=(0.2,0.2), and trace of the misalignment angles was
simulated. In the simulation, zu=0 (the location of imbalance force in axial direction) to
eliminate a moment by imbalance force in equation (4.8).
Figures 4.30 and 4.31 are orbits for the HDD spindle rotor with imbalance forces
of 0.033N and 0.56N, respectively. For the meso scale gas bearing (with dimensions
given in Table 4.5) to be used for an HDD spindle with area density of 20Gb/in2, the
imbalance force should be reduced below 0.033N.
Figure 4.32 and Figure 4.33 shows series of rotor orbits and their frequency
spectrum for Laser scanner at different rpm. As rotational speed increased, vibrations
increased and reached its maximum at around 440,000 rpm, which was a first critical
speed. At the critical speed, the phase angle of rotor with respect to imbalance force was
109.5o. Above the first critical speed, the bearing was stable until above 600,000rpm. At 15 The small imbalance was chosen to excite an imbalance force, which is within the limit of load capacity of the bearing when rotor passes through the first critical speed.
93
around 660,000 rpm, half-frequency whirling initiated, indicating the bearing’s maximum
allowable speed. Figure 4.34 shows trace of misalignment angles from initial disturbance
at 860,000 rpm.
From the dynamic analyses, the meso scale gas bearings had adequate load capacities
and dynamic stabilities, over wide operating ranges.
(a) (b)
Figure 4.30 Three-dimensional imbalance response of HDD rotor, 50 cycles: (a)
Orbit at Z=0, (b) Stable rotor from initial misalignment (ψX0=0.2, ψY0=0.2), Imbalance force: 0.56N, Speed: 15,000 RPM
(a) (b)
Figure 4.31 Three-dimensional imbalance response of HDD rotor, 50 cycles: (a) Orbit at Z=0, (b) Stable rotor from initial misalignment (ψX0=0.2, ψY0=0.2),
Imbalance force: 0.033N, Speed: 15,000 RPM
94
(a) 100,000 rpm (b) 200,000 rpm
(c) 300,000 rpm (d) 400,000 rpm
(e) 440,000 rpm (f) 500,000 rpm
Figure 4.32 Rotor orbits of Laser scanner rotor, 30 cycles; cont.
95
(g) 600,000 rpm (h) 700,000 rpm
(i) 800,000 rpm (j) 860,000 rpm
Figure 4.32 Rotor orbits of Laser scanner rotor, 30 cycles; (a) 100,000rpm (b) 200,000 rpm (c) 300,000 rpm (d) 400,000rpm (e) 440,000rpm (f) 500,000 rpm (g)
600,000 rpm (h) 700,000 rpm (i) 800,000 rpm (j) 860,000 rpm
96
(a) 640,000 rpm (b) 660,000 rpm
(c) 700,000 rpm (d) 800,000 rpm
Figure 4.33 Frequency spectrum of Laser scanner rotor orbit; (a) 640,000 rpm (b) 660,000 rpm (c) 700,000 rpm (d) 800,000 rpm
Figure 4.34 Trace of misalignment of Laser scanner rotor at 860,000 rpm from
initial misalignment
97
Chapter 5
Testing of Micro Gas Bearing
In this chapter, test results are presented, and technical issues involved in the test
are discussed.
5.1 Issues on Gas Bearing Tests Figure 5.1 depicts a stable rotor running at eccentricity e under a given external
load. For the rotor to run at the given eccentricity, several ideal conditions should exist:
the rotor mass should be smaller than the threshold rotor mass for dynamic stability; there
should be no imbalance force; and an external load (including rotor weight),
corresponding to the eccentricity, should be applied along correct direction. Among the
conditions listed above, the most difficult to fulfill is avoiding imbalance force,
practically impossible in rotating machinery. Imbalance force, which depends on the
imbalance radius and imbalance mass, is usually much larger than the static load, at very
high rotating speeds.
Figure 5.2 depicts the macro scale gas bearing test rig from [Wilde and San
Andres, 2003]. Two three-lobe hybrid gas bearings support the rotor, driven by an
electric motor located at the center of the rotor. Radial run out was measured using an
eddy current sensor. Figure 5.3 depicts the meso scale turbo jet simulator [Heshmat,
2003], using foil gas bearings of 6mm diameter. Here an impulse turbine, formed directly
on the rotor surface, with multiple nozzles around the shroud, drove the rotor. With
choked conditions at the nozzles, stable operation at 700,000rpm was reported. Silicon
micro turbine from MIT [Lin, C.C., 1999] also embedded centrifugal reaction turbine to
drive a rotor with 4mm diameter (see Figure 1.4).
98
θΧ ω
e
φ
θR
Y
Reaction
External load
Figure 5.1 Description of rotor running at certain eccentricity under given
external load
Figure 5.2 Macro scale gas bearing tester [Wilde and Andres, 2003]
99
Figure 5.3 Meso scale turbo jet simulator supported by foil gas bearings
[Heshmat, 2003]
The macro or meso scale gas bearing test rig could use an electric motor or an
impulse turbine as a drive, supplying direct torque to the rotor. In general, a gas bearing
tester should be very similar to the actual system, where the gas bearings will be used.
Drives for high-speed spindles or turbo compressor, which use gas bearings, should be
engineered very carefully, to match the embedded gas bearings. The static and dynamic
stability analyses in previous chapters give a sanity check for basic performance,
including maximum load and allowable rotor mass that a specific bearing can support.
5.2 Testing of Micro Gas Bearings For testing of the fabricated micro gas bearings, the small shaft (0.5mm)
complicates embedding of the electric drive, or formation of the impulse turbine on the
shaft directly. First test rig is shown in Figure 5.4. Here, small turbine with diameter
1.6mm made of SU-8 was assembled onto the shaft, made of bearing steel SAE 52100
with diametral tolerance ±0.25µm, using the large thermal expansion coefficient of SU-8
at 50oC to aid an interference fit. Total rotor mass was measured as 2.3mg per bearing.
100
Two identical circular nozzles with 0.3mm diameter drove the turbine at offset locations
to cancel radial force16 and deliver only torque. Capacitance sensor with 25nm resolution
from MTi, inc., which is the smallest commercially available sensor, measured vibrations
of the test bearing (not on rotor shaft), because direct measurement of shaft vibrations
was not possible due to very small size of the shaft.
The first criterion of successful operation at gas bearing mode, was near zero
vibration signal from the bearings, because once the rotor took off, the vibration of the
bearing would nearly vanish. The second criterion was SEM image of tested bearing. In a
gas bearing mode, the bearing surface should be almost free of scratch. To identify
scratch easily after the test, uncoated bearings were used.
Due to the uncertainty of air jet forces from nozzles and the imbalance force,
precise quantification of the total external load was not possible.
5.2.1 Assembly of Test Rigs To assemble two micro gas bearings with shaft, simple and effective self-aligning
method was developed as described in Figure 5.5. In step (a), two bearings are put onto
rotor (shaft and turbine assembly). Photo resist is dispensed in step (b), allowing photo
resist to seep into bearing clearance by capillary action. In step (c), residual photo resist is
wiped out and soft baking follows. Then two gas bearings are fixed to the bearing
housing structure using epoxy in step (d). Photo resist inside the gap is dissolved in
acetone to release the rotor in step (e). In step (d), the epoxy should be chemically
resistant to acetone.
16 Eliminating radial load could lead to instability because rotor eccentricity could be very small with only rotor weight.
101
Nozzle
Capacitance sensor
Nozzle
Nozzle
(a) Description of gas bearing tester
(b) Photo of assembled gas bearing tester
(c) Capacitance sensor and drive shaft with press-fitted SU-8 turbine
Figure 5.4 Photo of gas bearing tester with open air jet; (a) Description of gas bearing tester (b) Photo of assembled gas bearing tester (c) Capacitance sensor and
drive shaft with press-fitted SU-8 turbine
102
(a) (b) (c) (d) (e)
Figure 5.5 Assembly procedures of micro gas bearings; (a) Pre-assembly (b) Photo resist dispensing (c) Soft baking (d) Anchoring to base (e) Release
5.2.2 Test Results Preliminarily measured speeds17 of 2.3mg rotor with plain gas bearings were
around 8,890 rpm as shown in Figure 5.6, far below the expected speed. Rotational speed
was measured by counting the number of peaks in certain time interval or taking FFT of
the voltage signal.
(a) Supply pressure 35psi (b) Supply pressure 50psi
Figure 5.6 Photonic sensor signal measuring rotor speeds; (a) Supply pressure 35psi (b) Supply pressure 50psi
To investigate the very low speed, a simple turbine model was developed. From
Figure 5.7, using control volume method, torque by air jet becomes
17 Because the photonic sensor had much larger diameter than width of single turbine blade, the photonic sensor measured averaged oscillating signal with frequency of rotating speed of turbine, which is not perfectly symmetrical.
103
( )2ωρη jjnajeffjet ruarT −= (5.1)
where ηeff is an overall turbine efficiency considering all geometric factors of nozzle and
turbine blades, ρa is an air density, an is nozzle area, uj is air jet speed, rj is the radius of
air jet action, and ω is a rotational speed of the turbine. Windage friction torque is
αωρ ⋅⋅≅ jjbabw rrANT 2)( (5.2)
where α is a windage loss factor, Nb is number of blade, Ab is blade area. When α=1, all
the air in front of each blade is assumed stationary and windage loss is maximum.
ρaanuj
ω
rj
Figure 5.7 Impulse turbine with open air jet
Bearing friction torque bT is
( ) RWruaRRFTT Rjjnaeffbbb µωρηµµ 2221 +−==+ (5.3)
where µ is a friction coefficient at the bearing surface, Fb is a summation of two bearing
forces, WR is a rotor weight per bearing, and R is radius of the bearing. From torque
equilibrium,
104
21 bbwjet TTTT ++= (5.4)
Rearranging equation (5.4) for ω gives
( )( )
( )
( )0
12
2
122
22
2 =+−
+−
++−
−−
αµη
ρµ
ωµη
ωαµη
ωµηω
jn
bjeff
jna
Rmjeff
jn
bjeff
mjeff
raA
rR
raRW
rR
raA
rR
rR (5.5a)
or
( )( ) ( ) ( ) ( )
−+
−+
−
+
+−
−=
RrarA
RraA
rraRrRW
raArR
rR
jneff
jb
jneff
b
jjnamjeff
R
jn
bjeff
mjeff
µηα
µηα
ρωµηµ
αµη
ωµηω 121212
1
12 2
(5.5b)
where jjm ru /=ω . Equation (5.5) was plotted for a dental drill turbine (for given speed
400,000 rpm@ 35 psi, assuming air jet speed 200 m/s), as a function of friction
coefficient of the bearing, as shown in Figure 5.8. From given speed information, turbine
efficiency ηeff and windage loss factor α were estimated as 0.6 and 0.1, respectively.
Surprisingly, turbine speed is almost independent of bearing friction 18 until bearing
friction reaches 0.2~0.3 (typical friction coefficients of low friction coatings). From the
figure, turbine speed is a strong function of windage loss and turbine efficiency, and a
very weak function of bearing friction.
With, ηeff =0.6 and α =0.1, the speed of SU-8 micro turbines was estimated to
be around 900,000 rpm. However, as shown in Figure 5.6, measured turbine speed was
far below the expected speed. Even under choked conditions of the nozzles, the turbine
speed did not reach over 20,000 rpm. In an open air jet configuration without properly
designed shroud, negative pressure at trailing edge of the turbine generated presumably
18 However, it should be noted that thermal heat generation is directly proportional to friction coefficient, and bearing performance deteriorates very quickly at dry friction region if adequate cooling is not available in actual rotating machinery
105
high negative pressure, rendering windage loss factor α >> 1 and turbine efficiency ηeff
<<1, and prevented the turbine from rotating fast.
Figure 5.8 Simulated speed of dental drill turbine
Figure 5.9 contains vibration signals from plain and step gas bearings (not from
rotor), measured by capacitance sensor with 25nm resolution. For step bearing19, 2.3 mg
rotor appeared unstable, causing large vibration at the bearing. The amplitude of the
vibration for plain gas bearing was estimated about 0.3µm. To check whether the plain
gas bearings were operated as gas bearing or dry friction bearing, SEM images were
taken from the bearing operated for 1 hour with repeated start/stops every 5 to 10 minutes.
Figure 5.10 shows such a SEM image on the bearing surface after the test. The picture
shows slight wear at both ends of the bearing during the repeated start/stops.
19 Note the stepped gas bearings in this section are fabricated six stepped gas bearing without axial grooves
106
(a) Vibration signal from plain bearing, speed 8890 rpm
(b) Vibration signal from step bearing
Figure 5.9 Vibration signal from bearings with 2.3mg rotor; (a) Vibration
signal from plain bearing, speed 8890 rpm (b) Vibration signal from step bearing
107
Figure 5.10 SEM images after operation for 1 hour with repeated start/stops
every 5 to 10 minutes
Tables 5.1 and 5.2 show simulated load capacities and threshold rotor masses of
plain and stepped micro gas bearings at low speed ranges. The table predicts the plain gas
bearing to be stable with 2.3mg rotor, if operated at eccentricity above 0.5. From the
simulated results and visual inspection of the tested gas journal bearings, the plain micro
gas journal bearings appeared to have been operated at gas bearing mode. When two
nozzles are used (to minimize radial force to the bearings), bearing load becomes ideally
zero, leading to low eccentricity and high vulnerability of instability, from basic gas
bearing theory. However, from the test, the plain gas bearing was considered to be stable.
Possible reasons include remnant imbalance force of the SU-8 micro turbine or
unbalanced air jet force (probably due to different air jet speeds of two nozzles, slightly
different distance of nozzle center from bearing center, etc).
To improve turbine performance, a simple shroud was machined as in Figure 5.11.
The radius of the shroud was 1.7mm, rendering about 50µm clearance between the
turbine blades and shroud housing. Only one nozzle was used to preload bearings. One
side was an air input and the other side was connected to vacuum. The arrow in Figure
5.11(b) indicates the direction of air jet flow through the shroud. Top of the turbine was
not covered with shroud to measure the speed and bearing vibration. Supply pressure was
35 psi, identical to the pressure at previous tests, with air jet speeds of 223.7m/s. Air jet
108
speed was measured by strain gauge signals attached to a slender cantilever20, onto which
air jet momentum was directly applied. With the simple shroud, turbine speed increased
to 60,000 rpm as in Figure 5.12, indicating that the open air jet caused very poor turbine
performance, preventing fast rotation.
Table 5.1 Load capacity and threshold rotor mass of plain micro gas bearings
Threshold rotor mass (mg) Load capacity (mg)
RPM e=0.5 e=0.6 e=0.7 e=0.8 ε=0.5 ε=0.6 ε=0.7 ε=0.8
5,266 9.8 19.9 62.4 175.6 4.3 6.0 8.3 12.2
7,899 10.1 20.9 65.6 180.2 6.5 9.0 12.5 18.4
10,532 10.5 22.1 70.5 190.3 8.8 12.0 16.8 24.5
13,165 11.2 23.2 77.3 200.5 11.0 15.1 21.0 30.7
26,329 12.3 25.7 81.1 214.5 22.6 31.1 43.7 65.0
Table 5.2 Load capacity and threshold rotor mass of stepped micro gas bearings
Threshold rotor mass (mg) Load capacity (mg)
RPM ε=0.5 ε=0.6 ε=0.7 ε=0.8 ε=0.5 ε=0.6 ε=0.7 ε=0.8
5,266 0.28 0.58 1.6 3.3 0.6 0.8 1.2 1.8
7,899 0.3 0.62 1.8 3.7 0.9 1.3 1.8 2.8
10,532 0.35 0.75 1.9 4.2 1.3 1.7 2.5 3.8
13,165 0.38 0.82 2 4.5 1.6 2.2 3.1 4.8
26,329 0.4 0.9 2.1 4.7 3.3 4.5 6.4 10.2
20 For the cantilever, 2.4mm wide, 0.2mm thick, 45mm long stainless steel sheet was used.
109
(a) Shroud
(b) Enlarged image of circled region in (a)
(c) Photo of assembled new gas bearing tester
Figure 5.11 New gas bearing tester with machined shroud to drive micro turbine; (a) Shroud (b) Enlarged image of circled region in (a) (c) Photo of
assembled new gas bearing tester
110
From the turbine model, equations (5.1) to (5.5), preload by single air jet was
estimated as 7.4mg in horizontal direction. Using combined load of rotor mass and the
preload, orbit simulation were performed with imbalance radius of 5µm, as shown in
Figure 5.13. The imbalance radius was estimated from the accuracy (5µm) of stepper,
which fabricated the mask pattern for the SU-8 turbine. As seen from the orbit simulation,
tested plain gas bearing was predicted stable. The SEM image of bearing surface, shown
in Figure 5.14, after test for 1 hour with frequent start/stops every 5 to 10 minutes,
confirms the gas bearing operation. Slight scratch during the start/stops was observed on
the bearing surface.
Figure 5.12 Photonic sensor signal measuring rotor speeds with new gas bearing
tester shown in Figure 5.11, supply pressure 35psi, plain gas bearing
111
Figure 5.13 Simulated orbit of 2.3mg rotor supported by plain gas bearing at 60,000
rpm with combined load of rotor mass and preload (7.4mg)
Figure 5.14 SEM image of plain gas bearing surface after test with new test rig shown in Figure 5.11
112
Chapter 6
Tribological Study of Micro Bearings
In this chapter, tribological characteristics of tungsten containing hydrocarbon
(W-C:H) coated micro sleeve bearings are discussed and compared with uncoated Ni
micro bearings.
6.1 Introduction Micro gas bearings are ideal for micro rotating machinery. Nevertheless, wide
application of the micro gas bearings in micro systems is limited, due to the following
reasons: high rotational speed is required to ensure sufficient load capacity; aligning the
bearing with the rotor is difficult; extensive engineering is needed to design proper rotor-
bearing configuration, to guarantee stable operation; the gas bearings are vulnerable to
dust particles, and need a clean environment; and wear during repeated start-stop and/or
unexpected touch down onto bearing surface increases bearing clearance, leading to
smaller load capacity and changing bearing characteristics. Even with negligible friction
at normal operating conditions, initial high static friction can be problematic when the
actuator lacks enough power to overcome the static friction.
Dry friction bearings with very low friction and high wear resistance can be
alternatives to gas bearings, due to easy of use and assembly. As discussed in Chapter 1,
surface modification can reduce static friction and surface energies of sliding surfaces.
Nickel micro sleeve bearings, shown in Figure 6.1 with nominal inner diameters
ranging from 500µm to 506µm, and length of 300µm, were fabricated via the X-ray
lithography and electroplating processes introduced in Chapter 2. A uniform 900nm thick
W-C:H was coated on the bearing surfaces using an ICP assisted, hybrid CVD/PVD tool.
Mechanical properties of these coating were studied via nano indentation. Hydrocarbon
113
containing metal showed better adhesion [Meng, 2003] to metal surfaces than a-C:H, i.e.,
amorphous hydro carbon.
Very light loads usually characterize working conditions of MEMS surfaces.
Therefore, studies on tribological characteristics of MEMS surface employed AFM or
Surface Force Apparatus (SFA) to simulate a single asperity contact under very light load
[Lu and Komvopoulos, 2001; Schwarz et al, 1997; Enachescu et al, 1998]. Typical nano
tribological studies using AFM had contact load of 10~100µN, sliding speed of ~µm/s,
and tip radius of 20~100nm.
However, as discussed in detail in Chapter 1, tip radius, surface energy, and
scanning speed-dependent behavior of friction forces makes comparison of measured
nano scale friction coefficients, to friction coefficients measured by other methods,
difficult, especially if the test environment and tip radius are not known exactly, or the tip
is not perfectly spherical.
(a) Photograph of micro sleeve bearings (b) SEM image of Ni micro bearing
Figure 6.1 Ni micro bearing: (a) Photograph of micro sleeve bearings, (b)
SEM image of Ni micro bearings
Surfaces of micro mechanical systems possess multiple nano meter scales
asperities and follow fractal geometry characterized by self-affinity over a wide range of
length scales [Ling, 1990]. Even if the load is very small and the macroscopic contact
pressure is far below the elastic limit of the materials, some local asperities will
plastically deform because of concentrated loads on those spots. Wear can be induced,
114
even under extremely small load, which complicates investigation of frictional behavior
of actual MEMS surface, and makes estimating real friction from nano scale friction via
AFM difficult.
Despite promising tribological characteristics and proven performance of DLC in
macro scale applications, direct application to MEMS have been limited. Beerschwinger
et al [Beerschwinger et al, 1995] measured friction of DLC coatings on flat silicon
surfaces, via a surface micro machined small friction tester. Bandorf et al [Bandorf et al,
2003] showed that DLC on a soft polymer surface had better wear resistance than DLC
on silicon wafer. Mousinho et al [Mousinho et al, 2003] demonstrated microstructures
made with DLC film, deposited by RF magnetron sputtering. Cao et al [Cao et al, 2003]
coated Ti-containing DLC (Ti-C:H) on Ni micro mold insert, fabricated by X-ray
lithography and electroplating. However, applications of DLC coatings to the sidewall of
micro scale mechanical parts, that experience sliding contacts (bearings, gears, etc), have
not been reported.
Working surfaces of these micro bearings were characterized using a newly
designed micro tribo tester. Wear rates, mechanical and material properties, and other
tribological characteristics of W-C:H coated Ni micro bearings are presented and
compared to uncoated Ni micro bearings.
Electroplated Ni has unique sidewall characteristics and mechanical properties,
compared to bulk Ni, as shown in Figure 6.2. The yield stress and ultimate tensile
strength of a nickel sample, deposited via the identical process conditions as the Ni
bearings of this work, were 360MPa and 535MPa, respectively [Hemker et al, 2001; Cho
et al, 2003]. After annealing the electroplated Ni at 800oC for 1 hour, the yield stress and
ultimate tensile strength reduced to 180MPa and 200MPa, respectively [Hemker et al,
2001; Cho et al, 2003]. The microstructure of as-deposited Ni had a columnar grain
structure with 2~4µm size, but the grain size increased to about 24µm with an equitaxial
direction after annealing [Hemker et al, 2001; Cho et al, 2003].
Tribological characteristics of annealed Ni micro bearings are important because
of potential high temperature applications. Because the microstructures and mechanical
115
properties changed after annealing, the tribological characteristics of annealed Ni micro
bearings should differ from as-deposited Ni bearings. In this work, tribological
characteristics of three micro bearings (as-deposited and annealed Ni micro bearings,
tungsten hydrocarbon coated micro bearings) were investigated via a new micro tribo
tester.
800oC 1hr
Grain size 2~4 µm (columnar)
Grain size ~ 24µm (equitaxial)
As-deposited Annealed
Figure 6.2 Microstructure of sidewall of electroplated Ni, as deposited and annealed
[Hemker et al, 2001; Cho et al, 2003]
6.2 Coating Process of W-C:H on Micro Sleeve bearings To deposit a conformal W-C:H coating on the bearing inner surfaces, multiple
bearings were fixed within the holes of two thin stainless steel sheets, as depicted in
Figure 6.3. The stainless steel sheets were Ni plated, in a Ni sulfamate solution, to
prevent cross contamination during coating. The entire assembly with bearings was
placed in a plasma deposition chamber. Plasma reached both sides of the bearing
surfaces.
116
plasma species
plasma species
plasma species
plasma species
(a) 1mm thick stainless steel sheet (b) Conformal coating process
with holes to hold micro bearings of bearing surfaces
Figure 6.3 Fixture to coat W-C:H coatings on the micro bearings: (a) 1mm thick stainless steel sheet with holes to hold micro bearings (b) Conformal coating process
of bearing surfaces
6.2.1 Process conditions of W-C:H coatings The deposition of W-C:H thin films on the bearings was carried out in an ICP
assisted, hybrid CVD)/PVD tool, with a base pressure ~ 4.7×10−6 Pa. Details are given in
reference [Shi et al, 2000; Meng et al, 2000]. The sequences of the deposition are: 10
minutes etching in a 0.23 Pa Ar (99.999+%) plasma with 1000 W total ICP input power;
pure tungsten interlayer deposition in the PVD mode for 10 minutes, wherein two
magnetron sources sputtered two pure W cathodes (99.99+%); and W-C:H deposition in
the hybrid mode for 41 minutes, with the same W cathodes in an Ar/C2H2 ICP plasma at
a fixed flow rate ratio (10:1) of Ar/C2H2(99.99+%), to achieve a coating thickness of
about 1µm.
During the pure W interlayer, the tungsten cathode current was fixed at 0.5A. W-
C:H was deposited at the tungsten cathode current of 0.15A for the first series of micro
bearings, and 0.1A for the second series of micro bearings. The total ICP power and total
pressure were 1000W and ~0.23 Pa, respectively. The bias voltage applied to the bearing
was fixed at -100V during etching, and -50V during the interlayer and W-C:H
117
depositions. No intentional heating was applied to the bearing during the deposition.
Figure 6.4 is a photograph of W-C:H coated micro bearings
Figure 6.4 Photo of W-C:H coated micro bearings
6.2.2 Material Properties of W-C:H Coatings
Shown in Table 6.1 are X-ray Photoelectron Spectroscopy (XPS) analysis results
of the chemical compositions of the coatings and mechanical properties. In-plane
Young’s modulus E/(1-ν2) and hardness H of the W-C:H coatings deposited on the
silicon wafer were measured by nano indentation using Digital Instrument Dimension
3100 AFM. Indentation depth was carefully controlled to minimize the substrate effect.
The in-plane Young’s modulus was used to calculate contact pressure during the wear
test.
Table 6.1 Chemical composition and mechanical properties of 900nm thick W-C:H coating
Coating name
Chemical composition (W/C ratio)
W current Hardness H
In-plane Young’s modulus E/(1-ν2)
W-DLC1 11/89 0.15A 8.4 GPa 99 GPa W-DLC2 5/95 0.1A 11 GPa 68 GPa
Figure 6.5 shows a Raman spectrum of a new W-C:H coating, measured with a
Renishaw Micro Raman Microscope with 784nm He-Cd Diode Laser source and input
power 30W. Exposure time was 30 sec, to maximize signal to noise ratio. The diamond to
118
graphite phase intensity ratios for the two coatings were almost identical at 1.17, because
the flow rate ratio of Ar/C2H2 was the same for the two coatings. The microstructure of
the coating appears to be amorphous from the Raman spectrum. The SEM photos of
Figure 6.6 suggest that uniform conformal coatings with thickness of about 900nm were
achieved by the coating method of Figure 6.3. The SEM image of W-DLC2 coated micro
bearing is shown in Figure 6.7.
Figure 6.5 Raman spectrum of new W-C:H
Figure 6.6 900 nm thick uniform coating thickness on the bearing surface
119
(a) Low magnification SEM image of W-DLC2 coated surface
(b) High magnification SEM image of W-DLC2 coated surface (×40K)
Figure 6.7 SEM images of W-DLC2 coated bearing surface: (a) Low magnification SEM image of W-DLC2 coated surface (b) High magnification SEM image of W-
DLC2 coated surface (×40K)
6.3 Micro Tribo Tester
Micro bearings made via X-ray lithography have sidewalls with non-symmetrical
fractal geometry. These sidewalls as working surfaces may experience very low contact
pressure during operation. Direct characterization of the sidewall (as-made & coated)
120
appears to be the best way to assess the microscopic tribological characteristics (and
performance) of the micro bearings. A new micro tribo tester was developed to measure
friction and wear characteristics of the as-made, annealed and coated micro bearings
A miniature impulse turbine with 5mm diameter and 2mm thickness was
fabricated by stacking (on the same shaft) four separate 0.5mm thick electroplated Ni
turbines. SU-8 2100 from MicroChem, Inc., was a photo resist used for plating mold for
0.5mm thick Ni turbines. A micro nozzle with cross section of 0.2mm height and 1.8mm
width and pressurized air at 50 psi drove the turbine. A gauge pin of diameter 500µm
(±0.25µm) composed of SAE 52100 bearing steel (1% C, 1.5% Cr, 0.1% Si) with surface
roughness Ra 50nm served as a driving shaft. Figure 6.8 shows the wear tester for the
micro bearings. The turbine was assembled onto the shaft using epoxy. Two micro
bearings (identical to the test bearings) supported the rotor. Air jetting from the nozzle
applied torque to the turbine. The air jetting also applied an external force to the bearings,
which was measured via the strain gauge attached at the bottom of the cantilever. A
photonic sensor measured the rotational speed stroboscopically.
Maximum Hertzian contact pressure Pmax between the bearing and shaft is given
by [Johnson, 1985] 5.0
max 488.0
=
LRKF
Pe
e (6.1a)
CR
RRRR
R B
SB
SBe
2
≅−
= (6.1b)
122 1134
−
−+
−=
S
S
B
B
EEK
νν (6.1c)
Here L is the length of the bearing, RB and RS are radii of the bearing and shaft,
respectively, Re is equivalent radius, K is equivalent elastic modulus and C is the
clearance between the bearing and shaft.
Figure 6.9 shows the micro friction tester. An air driven dental drill drove a test
shaft. Two W-C:H coated micro bearings supported the test shaft, to permit ultra-
121
precision operation. The friction tester consisted of precision and non-precision parts. A
flexible rubber tube coupled the dental drill to the precision test shaft. The coupling
isolated radial vibrations from the dental drill. Moving the stage deflected the cantilever
(Figure 6.9a), which applied and controlled the external load to the test bearings. This
load was measured via a strain gauge attached to the cantilever. The bearing holder with
the test bearing was connected to the cantilever by a flexible cotton string (Figure 6.9b).
The configuration is similar to a macro scale pin-on disc machine. Here the circular steel
shaft and bearing, which replaced the pin and disc, simulated extremely small contact
stress.
Cantilever sensor tomeasure bearing force directly
Strain gauge
Air nozzle Test bearing Micro turbine
Counter shaft: SAE 52100 bearing steel Diameter=500µm; Ra=50nm
(a) Overview of micro wear tester Detailed view within dotted circle in (a)
(b) Photo of micro wear tester
Figure 6.8 Micro wear tester: (a) Overview, (b) Detailed view within dotted circle in
(a), (c) Photo of micro wear tester
122
Capacitance probe
Test bearing Dental drill
turbine Rubber tube
Micro Bearing
Moving stage
Strain gauge
Precision Non-precision
String
(a) Schematic diagram
L
TF
T
Test bearing with bearing holder
θ
φ
D
Capacitance probe to measure θ
Cantilever
String
(b) Principle of friction measurement (side view)
(c) Photo of friction tester
Figure 6.9 Micro friction tester: (a) Schematic diagram, (b) Principle of friction
measurement (side view), (c) Photo of friction tester
123
The kinetic friction coefficient µk of the bearing surface can be calculated from
the static equilibrium (see Figure 6.9b) between friction torque TF and the restoring
torque TR from the external force T,
RTTTLT kFR µφθ ===+ )sin( (6.2)
Giving
RL
k)sin( φθµ +
= (6.3)
In the foregoing, R is the radius of the test bearing (250µm), and deflection angle θ of the
bearing from the vertical (Figure 6.9b) was calculated by measuring the displacement of
the end of the horizontal bar, attached to the bearing holder, via the capacitance probe.
The capacitance probe was calibrated with a micrometer, to compensate errors due to its
non-perpendicular facing with the horizontal bar. The angle φ (Figure 6.9b) was
calculated by trigonometry after θ was measured.
6.4 Results and Discussions
6.4.1 Nickel Bearings
The elastic modulus of electroplated Ni under the plating conditions described
earlier was 163±14GPa [Hemker et al, 2001; Cho et al, 2003]. From this, the equivalent
elastic modulus K=134.5GPa, via equation (6.1c) and material properties of the bearing
and shaft. The initial clearances between the test bearing and shaft was1µm. Bearings
were tested for 1hour, and diameters were measured using SEM. The shapes of the
bearings after the test were not circular, and the diameters in the vertical and horizontal
direction were averaged to calculate wear volume. The wear rates, averaged from four
test bearings, are summarized in Table 6.2. Because of slightly different assembly
124
conditions of the wear tester for each test (i.e., location of nozzle relative to the turbine),
the load and speed were slightly different for each test.
The annealed Ni bearings had about three-fold higher wear rate than as-deposited
Ni bearings, likely due to smaller yield stress and ultimate tensile strength after
annealing.
Figure 6.10 and 6.11 show the bearing surfaces of as-deposited and annealed Ni
bearings after testing. The lower images magnify the enclosed rectangular frame. Worn
materials were transferred along the axial direction, by axial movements of the turbine
rotor within the clearances between the bearings and turbine. There materials
accumulated on the thrust surfaces. Electroplated Ni bearings were soft, and suffered
severe plastic deformation presumably due to high impacts of the turbine rotor on to the
bearing surface. Even if the initial bearing clearance was 1µm, high impact load could be
expected because of looser clearances caused by rapid wear of the Ni bearings. Figure
6.12 shows SEM images of steel shafts, tested against the as-deposited and annealed Ni
bearings. Both shafts had slight scratching marks without any noticeable wear.
Table 6.2 Wear rates of Ni bearings
Wear rate (×10-3 mm3/Nm)
Bearing load (µN)
Contact pressure (MPa)
Speed (RPM)
As-deposited Ni bearing 0.54~0.96 15500 5.8~29.1 8000~9000
Annealed Ni bearing 1.25~2.98 17250 4.9~27.2 9000~12000
125
(a) (b)
Figure 6.10 Wear characteristics of as-deposited Ni bearings: (a) Top view, worn materials moved along the axial direction and accumulated at the thrust surface of
the bearing (b) Inner bearing surface after test
126
(a) (b)
Figure 6.11 Wear characteristics of annealed (at 800oC for 1h) Ni bearings: (a) Top view, worn materials moved along the axial direction and accumulated at the thrust surface of the bearing as multiple layers (b) Inner bearing surface after test
(a) (b)
Figure 6.12 SEM images of shaft tested against (a) As-deposited Ni bearing (b) Annealed Ni bearings. No noticeable wear was observed
127
6.4.2 W-C:H Coated Micro Bearings Contact pressure Pmax of the W-C:H coated bearings, calculated via equations
(6.1), employed the measured in-plane elastic modulus E/(1-ν2) (to estimate K), see Table
6.1. Challenging for the wear test was detection of when the coating was completely
worn, exposing the bare underlying Ni surface to rub against the steel shaft. Due to the
very small bearing size and inaccessibility to surfaces, there was no direct way to monitor
the conditions of the coatings during tests. Unlike Ni micro bearings, the wear tests
monitored rotational speed and shaft vibration every five minutes. Upon a sudden change
of rotational speed, it was assumed that the bearing experienced a sudden change of
coating conditions. Figure 6.13 records rotational speed vs. testing time. Initial turbine
speeds were around 3500 rpm for both bearings. The speed increased slowly until
reaching steady state. Initial slow speed was considered as a break-in period, for typical
hydrocarbon coatings. The turbine speed was steady for certain periods, but increased
suddenly to over 10000 rpm, suggesting a dramatic change of coating conditions. At
these transitions, the coating was considered worn, exposing the underlying Ni surface to
wear. W-DLC1 coated bearing had shorter break-in period, lower steady state RPM, and
reached coating failure earlier than the W-DLC2 coated bearing.
Figure 6.14 shows SEM images of W-C:H coated micro bearings after the wear
test. Here wear of the underlying Ni was assumed to begin after the coating was
completely worn. At this time, the wear tests were stopped. The small particles are stray
epoxy particles that fell off the wear tester when the test bearings were disassembled.
Table 6.3 summarizes the test conditions and the wear rates. The total sliding distance
was estimated from turbine speed, integrated up to the sudden speed increase. The wear
rate of W-DLC2 (high W composition, see Table 6.1) coated micro bearings was about
32% smaller than W-DLC1 (low W composition) coated micro bearings.
128
Table 6.3 Wear rates of W-C:H coated micro bearings
Coating Wear rate (×10-5 mm3/Nm)
Bearing load (µN)
Contact pressure (MPa)
Speed (RPM)
W-DLC1 1.11~1.43 40461 6.84 3400~10910 W-DLC2 0.75~0.96 44668 6.27 3582~13000
Figure 6.13 RPM-time relation of the W-C:H coated micro bearings
(a) (b)
Figure 6.14 SEM images of W-C:H coated micro bearing surfaces after wear test: (a) W-DLC1 coated micro bearing (b) W-DLC2 coated micro bearing
129
Direct friction measurements of W-C:H coatings using the micro friction tester
shown in Figure 6.9, were not successful, due to large run outs of the test shaft,
transmitted from dental drill turbine despite flexible rubber coupling. Even very small
shaft vibrations caused severe stick slip and rotational vibration of the bearing holder.
Instead, the turbine was stopped every three minutes and the test shaft was rotated
manually in both clockwise and counter clockwise directions, at slow speed (~5mm/s).
While the test shaft was rotating, deflection angle θ of the bearing holder was measured
via the displacement of the horizontal bar attached to the bearing holder. Figure 6.15 is an
exemplary signal from the capacitance sensor and converted deflection angle θ, in
degrees. Once θ and φ (from trigonometry, Figure 6.9) were measured, the friction
coefficient was calculated using equation (6.3).
Figure 6.16 shows the evolution over time of the friction coefficients of W-C:H
coated micro bearings. From a relatively high initial value, friction coefficient reached at
very low steady state value of about 0.12. Although failure of the rubber coupling
prevented friction measurement over longer periods, a low steady state friction is
suggested by the steady RPM (Figure 6.13), and the Raman spectrum of the wear scar on
the steel test shaft (Figure 6.17), where a transfer layer of W-C:H coating formed. Figure
6.18 is a SEM image of the steel shaft ran against the W-DLC2 coated micro bearing. No
noticeable wear was observed. The shaft tested against the W-DLC1 coated bearing
showed a similar pattern, with no noticeable wear on the shaft.
Low friction of amorphous hydrocarbon coatings at steady state is usually
initiated by a transfer layer formed on the counter surface at very high contact pressure
[Liu et al, 1996; Liu et al, 1997, Donnet et al, 1994; Koskinen et al, 1998]. Even if the
contact pressure is very small during the wear test of the W-C:H coated micro bearings, a
transfer layer formed on the counter surface, leading to low friction and high wear
resistance of both bearing and shaft.
130
Figure 6.15 Voltage signal from capacitance sensor and converted rotational
angle of horizontal bar attached to bearing holder with W-DLC1 coated bearing
Figure 6.16 Evolution of friction coefficient of W-C:H coated micro bearing
131
Figure 6.17 Raman spectrum on the wear scar on the steel shaft tested with W-
DLC2 coated bearing
Figure 6.18 SEM images of shaft tested against W-DLC2 coated bearing after 2hour continuous wear test
132
6.5 Summary and Conclusions of the Experiments
Micro bearings, with and without W-C:H coatings, were introduced and tested. A
deposition process of W-C:H coatings on the micro bearings was developed and
mechanical properties and tribological response of these films were measured. For this, a
micro tribo tester, to directly characterize the sidewall of the micro bearings, was
designed. From this article, the following can be concluded: 1) Ni micro bearings (as
deposited and annealed), without coatings, experienced severe wear and appear
inadequate for tribological applications. 2) Uniform conformal W-C:H coatings, achieved
by the deposition process discussed, proved very efficient tribological coatings for micro
bearings. 3) W-DLC2 (with low W composition) coated micro bearings had higher wear
resistance than W-DLC1 coated micro bearings. 4) During the wear test of the W-C:H
coated micro bearings, a transfer layer formed on the counter steel shaft even under very
small contact pressure, leading to low steady state friction and high wear resistance.
133
Chapter 7
Future Work and Conclusions
In this chapter, future research related to this dissertation will be suggested, and
conclusions will be made.
7.1 Future Work This section suggests future research work to improve the micro scale 21 gas
bearings, and experimental methods to characterize gas bearing characteristics and
tribological performance.
7.1.1 Hydrostatic Meso Scale Gas Bearing Figure 7.1 suggests a conceptual design for hydrostatic meso scale gas bearings.
Upper and lower parts can be fabricated separately and fusion bonded together as
suggested in section 4.4.1. Details to form a gas channel are described in Figure 7.2. In
step (a), the negative image of gas channels are formed on a substrate using SU-8, and Cr
(or adhesion layer with good electrical conductivity) is evaporated on the structure to
serve as an electroplating seed layer for later electroplating. A PMMA sheet is bonded on
the structure using PMMA bonding epoxy (see Table 2.2) in step (b). Because negative
images of gas channels have been already formed on the substrate, thick epoxy is
required to avoid damage during bonding. X-ray lithography is performed on PMMA to
make negative image of final bearing part as in step (c), and electroplating and polishing
are followed in step (d), to complete the part as in step (e). Once final part is fabricated,
two identical parts can be stacked or fusion bonded to make a complete hydrostatic meso
scale gas bearings
21 Here, micro scale ranges from sub millimeter to millimeter scale.
134
7.1.2 Meso Scale Foil Gas Bearing
Foil gas bearings (Figure 7.3) have been known as very stable gas bearings
because foil can accommodate local film thickness variation and minimizes negative
pressure at diverging land, leading to minimum negative cross stiffness [Heshmat, 2000].
Spring alloy Inconel X750® Ni Alloy is widely used to make foils. In Figure 7.3, the foil
should be designed to have highly non-linear stiffness and damping characteristics. Foil
can have non- uniform geometry along axial direction to trap working gas at bearing
center and maximize damping. Despite the high performance, high manufacturing cost
limited their wide application. For meso scale application, manufacturing process of foil
poses additional difficulty. Figure 7.4 suggests fabrication process of micro stamping
mold to process Inconel. In steps (a) and (b), multiple layers consisting micro stamping
mold are processed via X-ray lithography and stacked to make single solid micro
stamping mold as shown in steps (c) and (d). Stamped foils can be wrapped in cylindrical
form and fitted into bearings.
Due to line of sight process of lithography technique, arbitrary circumferential
profile in stamping mold can be processed to tailor stiffness and damping of foil structure.
Changing dimensions of each layer with sub micron resolutions can easily process non-
homogeneous foil structure along axial direction.
Figure 7.1 Conceptual design of hydrostatic meso scale gas bearing
135
(a) (b) (c)
(d) (e) (f)
Figure 7.2 Detail fabrication processes of hydrostatic meso scale gas bearings; (a) SU-8 pattern with seed layer (b) PMMA bonding (c) X-ray lithography on PMMA
(d) Electroplating (e) Release (f) Final Part
Figure 7.3 Principle of foil gas bearings [Heshmat, 2000]
136
(a) X-ray lithography (b) Stacking
(c) Solid micro stamping mold (d) Coated micro stamping mold
Figure 7.4 Fabrication process of micro stamping mold for foil; (a) X-ray
lithography (b) Stacking (c) Solid micro stamping mold (d) Coated micro stamping mold
7.1.3 New Software for Performance Analyses of Gas Bearings
Detailed numerical performance analyses are required for optimal design of the
proposed hydrostatic and foil meso scale gas bearings. For the foil gas bearings, new
software should be developed to solve gas lubrication and non-linear elastic deformation
of the foil structure simultaneously.
Due to highly non-linear characteristics of Reynolds Equation (with molecular gas
rarefaction effects) and complicated bearing geometry, orbit method was used in the
dynamic analyses. More time efficient perturbation methods should be developed to
estimate stiffness and damping coefficients of meso scale gas bearings.
137
In Chapter 4, three-dimensional dynamic analyses were performed for meso scale
applications, such as Laser scanner and HDD spindle. Complete dynamic analyses
including thrust bearings will better predict maximum allowable speed and performance
of the meso scale gas bearings.
7.1.4 Advanced Testing Method of Gas Bearings Difficulties in performance testing of micro scale gas bearings were pointed out in
chapter 5. To facilitate testing and performance investigation, under similar operating
conditions, advanced experimental testing method is suggested.
Figure 7.5 is a turbine-driven gas bearing tester. The shaded part in the right hand
side is a rotor composted of drive shaft, reaction turbine and centrifugal pump. The detail
configuration of reaction turbine and pump are described in Figure 7.6. The reaction
turbine and pump can be built by two-step lithography technique using X-ray or/and UV
lithography. The turbine can be driven with compressed air to test the bearing. The main
purpose of the proposed micro turbine is to measure maximum operating speed and long-
term reliability of the meso scale gas bearings (proposed in section 4.4), meso scale
hydrostatic and foil gas bearings (introduced in sections 7.1.1 and 7.1.2). Another
purpose is to investigate performance of the centrifugal pump as a reliable auxiliary
device, with very low power consumption (through gas bearing), for other types of power
generation system such as thermoelectric power generator and micro fuel cell. Past
research on thermoelectric power generator [Stark and Stordeur, 1999] and/or micro fuel
cell [Hahn et al, 2003] assume the existence of a reliable auxiliary device, with very low
power consumption, to supply pressurized air, hydrogen gas, or hydrocarbon
continuously to main power generation modules such as combustor, electrode-membrane
module, etc. Development of reliable auxiliary devices (with very low power
consumption) is challenging and very important for the micro power generation system to
work as an independent power generation unit.
138
Turbine nozzle
Pump housing
Gas bearing
Figure 7.5 Gas bearing tester using reaction turbine
(a) Radial impulse turbine (b) Centrifugal pump
Figure 7.6 Auxiliary devices for gas bearing tester in Figure 6.3: (a) Radial impulse
turbine (b) Centrifugal pump
Small size complicates performance testing of the gas bearings. Direct
measurement of states can be very difficult or costly: typically, only one or two states can
be observed with off-the-shelf or even specially developed sensors. This necessitates
models to interpret measured data. Detailed model of rotor dynamics including micro gas
journal bearing and thrust bearing, and simulations will be used to estimate internal states
and parameters that cannot be measured.
139
7.1.5 Improvement of Micro Tribo Tester Micro wear tester, developed in this work needs further improvement. External
load to the micro bearing and driving speed are determined by single air nozzle. Ideally,
rotational speed and external load to the test bearings should be controlled independently.
Micro electric drive with reliable flexible coupling would be a choice. External loading
can be controlled by air jet independently. Indirect in-situ monitoring technique of
coatings should be employed to decide a moment of coating failure accurately. Viable
techniques would include: monitoring transition of acoustic noise from total system;
monitoring vibration of bearing; measuring rotor run out with specially designed micro
capacitance sensor; combination of the suggested methods, etc.
7.1.6 Tribological Characteristics at Various Environments Limited number of fabricated micro bearings and a shortage of necessary
equipment prohibited further investigation of tribological characteristics of W-C:H
coated micro bearings in various environments, such as dry nitrogen, vacuum and high
temperature. To drive wear tester or friction tester in vacuum, electric drive should be
developed instead of air jet driven turbine.
7.1.7 Other Suggestions
In addition to the proposed future research, the following are suggested.
Statistical investigation on accuracy of dowel pin-based assembly for meso scale gas
bearings: suggested fabrication method of mesocale gas bearings is based on
assumption of exact pattern transfer and repeatability of X-ray lithography process.
However, uncertainties involved in every unit process conditions require statistical
approach to estimate assembly accuracy using dowel pins.
Investigation of tribological characteristics of different solid thin film coatings (such
as CrN, TiN, organic thin films, etc) for other MEMS and biomechanical systems.
140
Tribological characteristics of thin solid film on soft substrate, such as micro
structures made of SU-8
Improvement of electroplating conditions to minimize micro scale cavity due to
hydrogen reduction and air bubble
7.2 Contributions and Conclusions of Dissertation The purpose of this work was to develop highly reliable micro bearings for micro
rotating machinery. Main contributions and conclusions of this dissertation are
summarized as follows.
1. Reviewed in chapter1 were reliability issues concerning past micro rotating
machinery, and benefits and current research on the micro gas bearings. Various
surface forces arising in MEMS devices, and related failures were introduced,
with their origins. Nano scale friction studies via Atomic Force Microscopy
(AFM), and limitations interpreting results were addressed. Also reviewed were
tribological studies on chemisorbed monolayers for oxide surfaces, and
hydrocarbon (DLC) for macro scale applications.
2. New integrated fabrication process of micro gas journal bearings with thrust
bearings was developed using X-ray lithography and electroplating. In chapter 2,
detailed procedures to make an X-ray mask and other technical issues were
discussed. By precise control and optimization of critical process parameters for
X-ray mask fabrication, a very smooth sidewall (Ra and RMS below 20nm) with
a near perfect vertical sidewall could be achieved. Very long sacrificial layer
etching (for 2 weeks) in diluted HF for final release from substrate, increased
surface roughness dramatically. To achieve a very smooth bearing surface,
sacrificial layer etching time should be minimized.
141
3. Software to predict hydrodynamic performance of micro gas bearings, including
molecular gas rarefaction effects in sub micron bearing clearances, were
developed. The orbit method, which solves non-linear transient Reynolds
Equation and rotor dynamics in time domain, was very effective for predicting
whirl instability of the micro gas bearings, and for simulating rotor dynamics
under various loading patterns. Chapter 3 and chapter 4 investigated and
summarized hydrodynamic performance of the fabricated micro gas journal, and
thrust bearing, using the software. Fabricated micro gas bearings had lower
stability than plain circular gas bearings due to lower load capacity and poor
system damping characteristics of the original step geometry. Improved bearing
designs having non-symmetrical step geometry with deep axial grooves were
suggested. These bearings were predicted to have much higher load capacities and
dynamic stabilities than the fabricated micro gas bearings.
4. A fabrication process combining X-ray lithography and precision assembly
technique was suggested to make meso scale gas bearings for various
applications. The meso scale gas bearings, applied to Laser scanner and HDD
spindle bearings, were predicted to be very stable over wide operating conditions.
5. A self-aligning assembly technique for micro gas bearings (using capillary action
of photo resist) was developed and proved to be a very effective assembly method
for performance testing of the micro gas bearings. Chapter 5 presented the test
results of the micro gas bearings. Successful operations of micro gas bearings at
60,000 rpm were demonstrated via an air jet-driven turbine under limited
operating conditions. Due to a very small shaft diameter (500µm), employing an
electric drive or a direct formation of an impulse turbine on the shaft, was not a
viable method for the micro gas bearing test. A small micro turbine made of SU-
8, press-fitted onto the shaft, was an alternative.
142
6. Hybrid CVD/PVD tools with a specially designed bearing holder was very
effective for achieving 900nm thick uniform conformal W-C:H coatings on the
micro bearings. New micro wear and friction testers were designed to characterize
bearing sidewalls directly. Tribological characteristics of as-made Ni micro
bearings, annealed Ni micro bearings, and W-C:H coated micro bearings, were
measured with the micro wear and friction tester. Chapter 6 investigated and
summarized tribological characteristics of Ni micro bearings and amorphous
tungsten hydrocarbon (W-C:H) coated micro bearings in dry friction mode, using
the micro wear and friction tester. Chemical and mechanical microstructures were
studied via X-ray Photoelectron Spectroscopy (XPS), Raman micro spectroscopy
and nano indentation. Wear rates, mechanical and material properties, and other
tribological characteristics of W-C:H coated Ni micro bearings were investigated
and compared to uncoated Ni micro bearings. Uncoated Ni micro bearings, as
deposited and annealed at 800oC, experienced severe wear and appeared
inadequate for tribological applications. Micro bearings with low tungsten-
containing (5% wt) hydrocarbon had higher wear resistance than micro bearings
with high tungsten-containing (11% wt) hydrocarbon. During the wear test of the
W-C:H coated micro bearings, a transfer layer formed on the counter steel shaft
even under very small contact pressure, leading to low steady state friction and
high wear resistance.
143
Appendix A.
Discretization of the Gas Film Equation
Journal Bearings
i-1,j i+1,j
i,j-1
i,j+1
i,j
i,j-1/2
i-1/2,j
i,j+1/2
∆θ
∆Z
θ
Z
lP SP
i+1/2,j+θQ−
θQ
+ZQ
−ZQ
SQ
Figure A.1 Control volume (Figure 3.3 is repeated)
Performing integral of the left hand side in equation (3.33a), along the control
boundary lP, in Figure A.1, and discretizating of each flux terms using a central
difference scheme, the mass flux term through the each face surrounding control surface
SP in Figure A.1 becomes
( )
( ) ( )
jijiji
jijiji
jijijijijiji
jijiji
jijiji
ji
PPDPPF
ZPP
PHQZPP
H
ZPPHQPHQ
,,2/1,2/1
,1,2/1,2/1
,,1,2/1,,1,2/1
,,1,2/1
3P
,,1,2/1
,2/1
3P
22
2
2
++
−=
−−+=
∆∆
−−∆
+Λ=
∆
∂∂
−Λ=
+
+++
+
++++
++
++
+
+
θθ
θ
θ
θ
θ
(A.1a)
144
( )
( ) ( )
jijiji
jijiji
jijijijijiji
jijiji
jijiji
ji
PPDPPF
ZPP
PHQZPP
H
ZPPHQPHQ
,,2/1,2/1
,1,2/1,2/1
,1,,2/1,,1,2/1
,1,,2/1
3P
,,1,2/1
,2/1
3P
22
2
2
−+
+=
−−+=
∆∆
−−∆
+Λ=
∆
∂∂
−Λ=
−
−−−
−
−−−−
−−
−−
−
−
θθ
θ
θ
θ
θ
(A.1b)
( )
( )jijiZ
ji
jijiji
jiZ
PPD
ZPP
PHQZPPHQQ
,1,2/1,
,1,2/1,
3P
2/1,
3P
−−=
∆∆
−−=∆
∂∂
−=
++
++
+
+ θθ (A.1c)
( )
( )1,,2/1,
1,,2/1,
3P
2/1,
3P
−−
−−
−
−
−−=
∆∆
−−=∆
∂∂
−=
jijiZ
ji
jijiji
jiZ
PPD
ZPP
PHQZPPHQQ θθ
(A.1d)
where convection term ZHF jiji ∆Λ= ,, , diffusion term along θ direction,
( )θ
θ
∆∆
=ZPHQD jiji ,
3P, , and diffusion term along Z direction ( )
ZPHQD ji
Zji ∆
∆=
θ,
3P, .
Squeeze term PPS dSPH∫∫∂∂
Λ )(2τ
can be treated as source term Qs and integrated
over the control surface SP to get
( ) ( )[ ]1,,,,
,, 2)(2
)(2
−−∆
∆Λ∆≅∆∆
∆
∆Λ≅
∫∫∂∂
Λ=
nAVjijiAVjiji
AVjiji
PPSs
HPHPZZHP
dSPHQ
τθθ
τ
τ , (A.2)
where 1,
−njiP is the pressure at time grid n-1. Subscript AV denotes the average value.
Applying continuity equation on the control volume in Figure A.1, gives
145
0=+−+− −+−+sZZ QQQQQ θθ (A.3)
Inserting (A.1) and (A.2) into (A.3), and rearranging gives
0,1,1,1,1,,1,1,1,1,, =−+++++ −−++−−++ jijijijijijijijijijiji bPaPaPaPaPa (A.4)
where
( )
( ) 1,
1,,
2/1,1,
2/1,1,
,2/1,2/1
,1
,2/1,2/1,
,2/1,2/1
,2/1,2/1
,
,2/1,2/1
,1
2
2
222
2
−−
−−
++
−−
−
−+
−−
++
++
+
∆∆Λ∆
=
−=
−=
−−=
∆∆Λ∆
+++
−−
+=
−=
nji
nAVjiji
Zjiji
Zjiji
jiji
ji
AVjiZ
jiZ
ji
jiji
jiji
ji
jiji
ji
PHZb
Da
Da
DF
a
HZDD
DF
DF
a
DF
a
τθ
τθ
θ
θθ
θ
(A.5)
Using definition of Peclet number Pe [Patankar, 1980] for θ and Z direction, given as
θθ
ji
jiji D
FPe
,
,, = (A.6a)
Zji
jiZji D
FPe
,
,, = (A.6b)
and rearranging (A.5) gives,
146
( )
( ) 1,
1,,
2/1,1,
2/1,1,
,2/1,2/1,1
,,2/1,2/11,1,,1,1,
,2/1,2/1,1
2
21
2
21
−−
−−
++
−−−
−+−+−+
+++
∆∆Λ∆
=
−=
−=
+−=
∆∆Λ∆
+−+−−−−=
−−=
nji
nAVjiji
Zjiji
Zjiji
jijiji
AVjijijijijijijiji
jijiji
PHZb
Da
Da
PeDa
HZFFaaaaa
PeDa
τθ
τθ
θθ
θθ
(A.7)
The Scarborough criterion is a sufficient condition for the convergence of the
Gauss-Seidel iteration method:
<≤
=+++∑
=∑ −+−+
equation oneleast at for 1 equations allfor 1
,
1,1,,1j1,i
,
pointsneighber
ji
jijiji
ji a
aaaa
a
a
Patankar [Patankar, 1980] provided the basic rules of convergence of central
difference schemes. For a high bearing number Λ such that 2or ,, >Zjiji PePeθ , a basic
rule that all coefficients must always be positive, is violated. Patankar [Patankar, 1980]
suggested various hybrid convection-diffusion schemes to meet the Scarborough criterion
and to improve numerical stability.
Yum [Yum, 2002] showed the power-law scheme yielded the best stability for
large bearing numbers and coarse grids. Gauss-Seidel iteration method with power-law
scheme was used to get pressure profiles. A more general form of the discretization
equations can be written as
147
( )
( )( )[ ]
( ) 1,
1,,,2/1,2/1,
2/1,1,
2/1,1,
,/1,/1,2/1,1
,,2/1,2/1
1,1,,1,1,
,/1,2/1,1
2),0max(
2),0max(
−−−+
−−
++
−−−−
−+
−+−+
+++
∆∆Λ∆
++−=
−=
−=
+−=∆
∆Λ∆+−+
−−−−=
−=
nji
nAVjijijijiji
Zjiji
Zjiji
jsijsijiji
AVjijiji
jijijijiji
jsijiji
PHZPFFb
Da
Da
PePeADa
HZFF
aaaaa
PeADa
τθ
τθ
θθθ
θθ
(A.8)
where ( ) ( )
−≡
5
,, 5.01,0max jiji PePeA and jiP , is the latest value of jiP , in the
iteration.
Thrust Bearing
The same procedures adopted for the journal bearing are applied. Equation
(3.32b) is integrated over the control boundary lP, in Figure A.1, where Z is replaced to η.
( )
( ) ( )
jijiji
jijiji
jijijijijiji
ji
jijiji
jijijiji
ji
PPDPPF
PPPHQ
PPH
PPHQPHQ
,,2/1,2/1
,1,2/1,2/1
,,1,2/1,,1,2/1
,2/1
,,1,2/1
3P
,,1,2/1,2/1
,2/1
3P
22
2
2
++
−=
−−+=
∆∆
−−∆
+Λ=
∆
∂∂
−Λ=
+
+++
+
++++
+
++
+++
+
+
θθ
θ
θ
ηθη
ηη
ηθη
η
(A.9a)
148
( )
( ) ( )
jijiji
jijiji
jijijijijiji
ji
jijiji
jijijiji
ji
PPDPPF
PPPHQ
PPH
PPHQPHQ
,,2/1,2/1
,1,2/1,2/1
,1,,2/1,,1,2/1
,2/1
,1,,2/1
3P
,,1,2/1,2/1
,2/1
3P
22
2
2
−+
+=
−−+=
∆∆
−−∆
+Λ=
∆
∂∂
−Λ=
−
−−−
−
−−−−
−
−−
−−−
−
−
θθ
θ
θ
ηθη
ηη
ηθη
η
(A.9b)
( )
( )jijiji
jijiji
ji
PPD
PPPHQPPHQQ
,1,2/1,
,1,2/1,
3P
2/1,
3P
−−=
∆∆
−−=∆
∂∂
−=
++
++
+
+
η
η θη
ηθη
η (A.9c)
( )
( )1,,2/1,
1,,2/1,
3P
2/1,
3P
−−
−−
−
−
−−=
∆∆
−−=∆
∂∂
−=
jijiji
jijiji
ji
PPD
PPPHQPPHQQ
η
η θη
ηθη
η (A.9d)
where convection term ηη ∆Λ= jijiji HF ,,, , diffusion term along θ direction,
θη
ηθ
∆∆
=
jiji
PHQD
,
3P
, , and diffusion term along η direction ( )ηθηη
∆∆
= jiji PHQD ,3
P, .
Following the same argument for journal bearings without source term,
0,1,1,1,1,,1,1,1,1,, =−+++++ −−++−−++ jijijijijijijijijijiji bPaPaPaPaPa (A.10)
with
( )
( )[ ]
jijijiji
jiji
jiji
jsijsijiji
jijijijijijiji
jsijiji
PFFb
Da
Da
PePeADa
FFaaaaa
PeADa
,,2/1,2/1,
2/1,1,
2/1,1,
,/1,/1,2/1,1
,2/1,2/11,1,,1,1,
,/1,2/1,1
),0max(
),0max(
−+
−−
++
−−−−
−+−+−+
+++
+−=
−=
−=
+−=
−+−−−−=
−=
η
η
θθθ
θθ
(A.11)
149
Nomenclatures
Chapter 3
H Local gas film thickness
C Nominal clearance of journal bearing
L Bearing length
D Bearing diameter (=2R)
CT Nominal clearance of thrust bearing
Ri Inner radius of thrust pad
Ro Outer radius of thrust pad
e Journal eccentricity
ε Non-dimensional eccentricity (=e/C)
p Pressure
P Non-dimensional pressure (=p/pa)
pa Atmospheric pressure
U Surface velocity (U=Rω)
ω Angular velocity
µ Viscosity of gas
X, Y, Z Defined in Figure 3.1
x, y, z Defined in Figure 3.1
θ Angular coordinate (=x/R)
Z Non-dimensional axial coordinate (=z/R)
r Radial coordinate in thrust bearing
η Non-dimensional radial coordinate in thrust bearing (=r/R)
l Molecular mean free path of air
lo Molecular mean free path of air in atmospheric pressure
150
Kn Knudsen number (=l / h)
Kna Characteristic Knudsen number (=lo / C)
NA Abogadro’s number
d Diameter of gas molecules
T Absolute temperature (K)
us Slip velocity
ζa Knudsen layer thickness
α Surface accommodation coefficient of the wall
PQ Poiseuille flow rate in rarefied gas regions
conQ Poiseuille flow rate in continuum flow regions
PQ Poiseuille flow factor (= PQ / conQ )
Λ Bearing number
σ Squeeze number
H Non-dimensional local gas film thickness (=h/C)
CR Bearing clearance at steps
F Load capacity )( 22YX FF +=
ζ Load parameter )( 22YX ζζ +=
φ Attitude angle
u Local velocity distribution in x direction
τp Shear stresses by Poiseuille flow in rarefied region
τc Shear stresses by Couette flow in rarefied region
τp,con Shear stresses by continuum Poiseuille flow
τc,con Shear stresses by continuum Couette flow
Wp Shear stress factor for Poiseuille flow (=τp/τp,con )
Wc Shear stress factor for Couette flow (=τc/τc,con )
τxy Shear stress at wall of journal shaft
151
TF Friction torque of journal bearing
βJ Non-dimensional rotational friction factor of journal bearing
rm& Mass flux along r direction
θm& Mass flux along θ direction
FT Load carrying capacity of thrust bearing
ζT Load parameter of thrust bearing
TFT Friction torque of thrust bearing
βT Non-dimensional rotational friction factor of thrust bearing
τθz Shear stress at wall of journal shaft
uθ Local velocity distribution in θ direction
∇k Gradient operators (k=J for journal bearing, k=T for thrust bearing)
ki Unit vectors along the θ, Z, and η directions ),,( ηθ Zk =
SP Control surface
lP Line surrounding control surface SP
θs Angle of one recess
sh
Specific film thickness
σs RMS surface roughness of shaft
σb RMS surface roughness bearing
θR Location of first recess from the line of eccentricity
Chapter 4
eF Total external load (5.022 )( eYeXe FFF += )
Xk State variables (k=1,2,3,4)
ω* Non-dimensional threshold speed
m* Non-dimensional threshold mass
x,y,z Rotational coordinate attached to rotor center
X,Y,Z Fixed reference frame attached to bearing center
152
a Imbalance radius
ϕx Misalignment of journal along x
ϕy Misalignment of journal along y
uF Imbalance force
i , j Unit vectors along the -x and -y direction
uM Imbalance moment by imbalance force
zu Location of imbalance mass along the -z direction
G Angular momentum vector of journal
Ω Instantaneous angular rotation vector of journal
ψ Non-dimensional misalignment of journal
ψ0 Initial non-dimensional misalignment of journal
Chapter 5
jetT Torque by air jet
Tw Windage loss
Tb Friction torque at bearings
Fb Friction force at bearings
ηeff Turbine efficiency
ρa Density of air
an Nozzle cross section area
Nb Number of turbine blades
Ab Frontal area of one turbine blade
α Windage loss factor
uj Air jet speed
rj Distance of nozzle from bearing center
ω Rotational speed of turbine (rad/s)
153
R Radius of sotor shaft
µ Friction coefficient of bearing and rotor shaft pair
WR Rotor weight per bearing
Chapter 6 E Young’s modulus
ν Poissson’s ratio
H Hardness
K Equivalent Young’s modulus
Re Equivalent contact radius
C Clearance of test bearing
RB Radius of bearing
RS Radius of shaft
µk Kinetic friction coefficient of the bearing surface
TF Friction torque
TR Restoring torque
T External force
R Radius of test bearing
θ Deflection angle of test bearing
Appendix A +θQ Flow out of control volume in θ direction
−θQ Flow into control volume in θ direction +ZQ Flow out of control volume in Z direction
−ZQ Flow into control volume in Z direction
jiF , Convection term at grid point (i, j)
154
θjiD , Diffusion term along θ direction
ZjiD , Diffusion term along Z direction
Qs Source term integrated over the control surface
Pe Peclet number
155
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162
Vita
Daejong Kim was born in Puan, Korea, on December 02, 1969. He earned his Bachelor
of Science and Master of Science degree in Mechanical Design and Production
Engineering at the Seoul National University in 1991 and 1993, respectively. He entered
doctoral program in Mechanical Engineering at the University of Texas at Austin in
2000.
Permanent address: 1161 DeokLim-Ri, Jusan-Myun, Puan-Kun, Jeonbuk, 579-920,
Korea
This dissertation was typed by author.
