ANALYTICAL AND EXPERIMENTAL INVESTIGATIONS OF HYBRID AIR FOIL BEARINGS A Thei by MANISH KUMAR Submitted to the Office of Graduate Studie of Texa A&M U

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1 ANALYTICAL AND EXPERIMENTAL INVESTIGATIONS OF HYBRID AIR FOIL BEARINGS A Thei by MANISH KUMAR Submitted to the Office of Graduate Studie of Texa A&M Univerity in partial fulfillment of the requirement for the degree of MASTER OF SCIENCE Augut 2008 Major Subject: Mechanical Engineering

2 ANALYTICAL AND EXPERIMENTAL INVESTIGATIONS OF HYBRID AIR FOIL BEARINGS A Thei by MANISH KUMAR Submitted to the Office of Graduate Studie of Texa A&M Univerity in partial fulfillment of the requirement for the degree of MASTER OF SCIENCE Chair of Committee, Committee Member, Head of Department, Daejong Kim Alan B. Palazzolo Hae-Kwon Jeong Denni O Neal Augut 2008 Major Subject: Mechanical Engineering

3 iii ABSTRACT Analytical and Experimental Invetigation of Hybrid Air Foil Bearing. (Augut 2008) Manih Kumar, B.Tech., Indian Intitute of Technology Chair of Adviory Committee: Dr. Daejong Kim Air foil bearing offer everal advantage over oil-lubricated bearing in high peed micro-turbomachinery. With no contact between the rotor and bearing, the air foil bearing have higher ervice life and conequently leer tandtill between operation. However, the foil bearing have reliability iue that come from dry rubbing during tart-up/hutdown and limited heat diipation capability. Regardle of lubricating media, the hydrodynamic preure generated provide only load upport but no diipation of paraitic energy generated by vicou drag and the heat conducted from other part of the machine through the rotor. The preent tudy i a continuation of the work on hybrid air foil bearing (HAFB) developed by Kim and Park, where they preent a new concept of air foil bearing combining hydrodynamic air foil bearing with hydrotatic lift. Their experimental tudie how that HAFB ha uperior performance compared to it hydrodynamic counterpart in load capacity and cooling performance. In thi article, the bearing tiffne and damping coefficient of HAFB are calculated uing a linear perturbation method developed for HAFB. The tudy focue on circular HAFB with a ingle continuou top foil upported by bump foil. The reearch alo

4 iv include a parametric tudy which outline the dependence of the tiffne and damping coefficient on variou deign parameter like upply preure ( P ), feed parameter ( Γ ), excitation frequency (ν ), and bearing number (Λ). Furthermore the preent reearch alo include experimental invetigation of HAFB with bump foil a compliant tructure. In the firt phae of the experimental reearch a high peed tet facility wa deigned and fabricated. The facility ha the capability of running up to 90,000 RPM and ha an electric motor drive. Thi article give detailed decription of thi tet rig and alo include data acquired during the commiioning phae of the tet rig. The tet rig wa then ued to meaure the load capacity of HAFB.

5 v ACKNOWLEDGEMENTS I would like to thank my committee chair, Dr. Kim, and my committee member, Dr. Palazzolo, Dr. Hung and Dr. Jeong, for their guidance and upport throughout the coure of thi reearch. My thank alo go to my friend and colleague at the Turbomachinery Lab at Texa A&M Univerity. Thi project i upported by the Texa Engineering Experiment Station, Turbomachinery Laboratory at Texa A&M Univerity and Turbomachinery Reearch Conortium.

6 vi NOMENCLATURE A b : Effective area that one bump cover A 0 : Reference orifice curtain area, A0 = π o C : Nominal clearance p a : Ambient preure d C P : Non-dimenional preure, P= p p a P 0 : Non-dimenional zeroth order preure p : Supply preure C b : Non-dimenional damping coefficient of elatic foundation, C b cbcω = p A a b c b : Bump damping h : Film thickne H : Non-dimenional film thickne, h H = C H X, Y : Non-dimenional perturbed film thickne gradient in X and Y k : Ratio of pecific heat for air K b : Non-dimenional tiffne coefficient of elatic foundation, k b : Bump tiffne m r : Rotor ma ṁ : Ma flow rate R g : Ga contant of air T : Ga temperature C d : Dicharge coefficient K b = kbc p A a b

7 vii 12 RgTm Ṁ : Non-dimenional ma flow rate, M µ = 2 3 pa C Γ : Feed parameter, Γ = 12µ Cd A0 g p C a 3 R T P X, Y : Non-dimenional perturbed preure gradient in X and Y R : Bearing radiu u U : Non-dimenional bump deflection, U = C u : Bump deflection Z : Non-dimenional axial coordinate, z Z = R Greek H : Non-dimenional perturbed film thickne P : Non-dimenional perturbed preure field cbω η : Structural lo factor, η = k ω : Rotor peed ω : Excitation frequency µ : Air vicoity Λ : Bearing number, 6µω Λ= p a b R C 2 ω ν : Excitation frequency ratio (EFR), ν = ω θ : Circumferential coordinate, τ : Non-dimenional time, τ = ωt x θ = R

8 viii TABLE OF CONTENTS Page ABSTRACT... iii ACKNOWLEDGEMENTS...v NOMENCLATURE...vi TABLE OF CONTENTS... viii LIST OF FIGURES...x LIST OF TABLES...xiv 1 INTRODUCTION Scope of the preent reearch LITERATURE REVIEW ON AIR FOIL BEARINGS DETERMINATION OF FORCE COEFFICIENTS Decription of hybrid air foil bearing Solution methodology Reult and dicuion DESIGN AND FABRICATION OF HIGH SPEED TEST RIG Requirement from the tet tig General layout and deign of the tet rig Decription of major tet rig component Spindle bearing Tet ection Electric motor drive Rotordynamic analyi of the rotor Commiioning of tet rig...50

9 ix Page 5 EXPERIMENTAL RESULTS Decription of prototype HAFB Experimental etup Tet reult Tet Tet Tet Tet Tet 5 & Tet Tet CONCLUSIONS AND FUTURE WORK Concluion from analytical tudie Concluion from experimental tudie Future work...84 REFERENCES...86 APPENDIX A...90 APPENDIX B...93 APPENDIX C...99 APPENDIX D APPENDIX E APPENDIX F VITA...109

10 x LIST OF FIGURES Page Figure 1: Hydrodynamic air foil bearing...2 Figure 2: Schematic decription of circular HAFB and coordinate ytem for analyi...11 Figure 3: Meh defined for analyi...12 Figure 4: Grid cheme...18 Figure 5: Zeroth order preure profile, Λ=1.25, tatic load 60N...19 Figure 6: Firt order perturbed preure profile (P X ), Λ=1.25, tatic load 60N...19 Figure 7: Predicted direct tiffne coefficient v. feed parameter ( Γ ) with increaing upply preure, Λ= Figure 8: Predicted journal eccentricitie v. feed parameter ( Γ ) with increaing upply preure, Λ= Figure 9: Predicted attitude angle v. feed parameter ( Γ ) with increaing upply preure, Λ= Figure 10: Predicted cro-coupled tiffne coefficient v. feed parameter ( Γ ) with increaing upply preure, Λ= Figure 11: Predicted direct damping coefficient v. feed parameter ( Γ ) with increaing upply preure, Λ= Figure 12: Predicted cro-coupled damping coefficient v. feed parameter ( Γ ) with increaing upply preure, Λ= Figure 13: Predicted direct tiffne coefficient v. excitation frequency ratio with increaing upply preure, Λ= Figure 14: Predicted cro-coupled tiffne coefficient v. excitation frequency ratio with increaing upply preure, Λ=

11 xi Page Figure 15: Predicted direct damping v. excitation frequency ratio with increaing upply preure, Λ= Figure 16: Predicted direct tiffne v. excitation frequency ratio with increaing upply preure, Λ= Figure 17: Predicted journal eccentricitie v. bearing number (Λ) with increaing upply preure, Γ = Figure 18: Predicted direct tiffne v. bearing number (Λ) with increaing upply preure, Γ= Figure 19: Predicted cro-coupled tiffne v. bearing number (Λ) with increaing upply preure, Γ = Figure 20: Predicted direct damping v. bearing number (Λ) with increaing upply preure, Γ = Figure 21: Predicted cro-coupled damping v. bearing number (Λ) with increaing upply preure, Γ = Figure 22: Tet rig...36 Figure 23: Spindle ball bearing, Source: GMN Bearing [21]...38 Figure 24: Bearing preload diagram...39 Figure 25: Tet ection...40 Figure 26: Electric motor drive - Motor...42 Figure 27: Electric motor drive Stator...43 Figure 28: Electric motor drive - Cooling jacket...44 Figure 29: Rotor model and rotor...45 Figure 30: Undamped critical peed map at MCOS = 50,000 RPM...46 Figure 31: Critical peed etimation...47 Figure 32: Mode hape plot...49

12 xii Page Figure 33: Commiioning of tet rig...51 Figure 34: Proximity probe arrangement...52 Figure 35: FFT at 10,000 and 20,000 RPM...53 Figure 36: FFT at 30,000 and 40,000 RPM...54 Figure 37: Grinding of tet ection...55 Figure 38: Hybrid air foil bearing (HAFB)...58 Figure 39: Tet facility...59 Figure 40: Tet1: Load capacity tet at 10,000 RPM with upply preure of 80 pi and 8 SCFH air flow...62 Figure 41: Top foil condition after Tet 1; 10,000 RPM with upply preure of 80 pi and 8 SCFH air flow...63 Figure 42: Top foil wear after Tet 2; 20,000 RPM with upply preure of 80 pi and 8 SCFH air flow...65 Figure 43: Load capacity tet at 20,000 RPM with upply preure of 80 pi and 14 SCFH air flow...67 Figure 44: Top foil wear after Tet 3; 20,000 RPM with upply preure of 80 pi and 14 SCFH air flow...68 Figure 45: Load capacity tet at 15,000 RPM with upply preure of 80 pi and 14 SCFH air flow...70 Figure 46: Top foil after Tet 4; 15,000 RPM with upply preure of 80 pi and 14 SCFH air flow...71 Figure 47: Top foil after Tet 5; 25,000 RPM with upply preure of 80 pi and 14 SCFH air flow...72 Figure 48: Top foil wear after Tet 6; 35,000 RPM with upply preure of 80 pi and 14 SCFH air flow...73 Figure 49: Load capacity tet at 25,000 RPM with upply preure of 80 pi and 14 SCFH air flow...74

13 xiii Page Figure 50: Load capacity tet at 35,000 RPM with upply preure of 80 pi and 14 SCFH air flow...75 Figure 51: Top foil wear after Tet 7; 25,000 RPM under hydrodynamic condition...77 Figure 52: Load capacity tet at 25,000 RPM Hydrodynamic operation...78 Figure 53: Comparative tudy at 10,000 RPM...81 Figure 54: PTC characteritic, Source: Elektromachinen und Antriebe AG [22] Figure 55: Calibration curve of proximity probe Figure 56: Two halve of the forming jig Figure 57: Bump foil geometry Figure 58: Forming jig with mandrel and bump foil Figure 59: Forming jig with mandrel and top foil...108

14 xiv LIST OF TABLES Page Table 1: Bearing parameter Simulation...10 Table 2: Spindle bearing parameter...38 Table 3: Electric motor drive parameter...41 Table 4: Critical peed...48 Table 5: Prototype bearing parameter...57 Table 6: Operating parameter: Tet Table 7: Operating parameter: Tet Table 8: Operating parameter: Tet Table 9: Operating parameter: Tet Table 10: Operating parameter: Tet 5 & Table 11: Top foil temperature...77 Table 12: Summary of bearing load capacitie...79 Table 13: Operating parameter Tet Table 14: Vernier pecification, Source : Newport Corporation [24] Table 15: Bearing parameter in [16] Table 16: Reult comparion...103

15 1 1 INTRODUCTION Air/Ga foil bearing have hown tremendou promie in the field of high-peed micro to mid-ized turbomachinery. Compared to roller element bearing, air foil bearing circumvent the need of oil lubrication circuit and complex eal making the ytem le complicated and more environmentally friendly. Becaue of leer number of part required to upport rotating machinery and no lubrication/eal ytem, air foil bearing have higher reliability. Conequently air foil bearing require leer cheduled maintenance reulting in higher ervice life and low operating cot. Air foil bearing have been uccefully deployed in many turbomachinery application. Air Cycle Machine (ACM) ued in Environmental Control Sytem (ECS) of aircraft ue air foil bearing. ECS with air foil bearing in Boeing 747 aircraft have demontrated a robut ervice life with Mean Time Before Failure (MTBF) exceeding 100,000 hour [1]. Other application include rotary flow compreor, micro-turbine [2] and oil-free turbocharger [3]. Air Foil bearing, however, have reliability iue that tem from the wear caued by dry rubbing during tartup and top. Thee bearing alo have limited heat diipation capability of paraitic heat generated within the turbomachinery. The reaon behind the low diipation i the low heat capacity of air. Another diadvantage of air foil bearing i that they have low load capacity a compared to roller or oil bearing. Low vicoity of air i the reaon behind the limited load capacity. Thi thei follow the tyle of Journal of Tribology.

16 2 The air foil bearing conit of a top foil and compliant elatic foundation which utain the applied load and provide tructural tiffne and damping. The compliant tructure can alo accommodate mialignment and ditortion of the haft. One of the mot commonly ued compliant tructure i a corrugated bump foil. Air foil bearing with bump foil a complaint tructure i hown in Figure 1. Hydrodynamic preure i generated when the haft drag the air between the rotor urface and the top foil. Becaue of the hydrodynamic preure the rotor i elevated and compliant tructure deform elatically. Rotor Bearing Sleeve Bump Foil Top Foil Figure 1: Hydrodynamic air foil bearing

17 3 1.1 Scope of the preent reearch The preent tudy i a continuation of the work done by Kim and Park [4], where they adopt time-domain orbit method to invetigate the rotordynamic performance of a rigid rotor in cylindrical mode. Ideally air foil bearing (and mot air bearing) are betuited for high peed rigid rotor (with large haft diameter) operating below their firt bending critical peed. However, air foil bearing are often conidered with flexible rotor with locally large haft diameter in region where the bearing are located. Adoption of time-domain non-linear orbit imulation to thee flexible rotor upported on air foil bearing require enormou computational time and thu i not practical. For general ynchronou rotordynamic vibration analye, tiffne and damping coefficient of the air foil bearing can be ued with commercial rotordynamic oftware. A pointed out earlier, non-linear time domain rotordynamic analye on a flexible rotor upported by air foil bearing require intene computational effort. A a preliminary deign tep, uage of bearing tiffne and damping coefficient with commercial rotordynamic oftware can reduce the computational time and provide quick deign guideline of whole rotor-bearing ytem. In thi article, the bearing tiffne and damping coefficient of HAFB are calculated uing a linear perturbation method developed for HAFB. The firt phae of the tudy focue on circular HAFB with a ingle continuou top foil upported by bump foil (page 11). The thei alo include a parametric tudy which outline the dependence of the tiffne and damping coefficient on variou deign parameter like

18 4 upply preure ( P ), feed parameter ( Γ ), excitation frequency (ν ), and bearing number (Λ). The above mentioned parametric tudy give u only a theoretical inight into the HAFB. To completely undertand the characteritic of thee bearing or any other air bearing an experimental invetigation i very important. One of the impediment in doing that i the requirement of a facility which can give the capability to tet the air bearing under moderate to high peed operation. The econd phae of thi tudy addree thi iue and center on the deign and fabrication of uch a tet rig. Following the fabrication of the tet rig, the HAFB wa teted for the load capacity at variou operating peed. The load capacity tudy followed the procedure outlined in [4].

19 5 2 LITERATURE REVIEW ON AIR FOIL BEARINGS Extenive reearch in air foil bearing have been made over the pat three decade. One of the firt work on the analytical ide wa done by Hehmat et al [5]. They olved the Reynold equation numerically to find the preure profile, film thickne and load capacity. They alo evaluated the effect of variou tructural, geometric and operational variable on the performance of the air foil bearing. Ku and Hehmat [6] preent a theoretical model of corrugated bump foil trip conidering frictional force between the bump foil and the bearing houing and alo between the bump foil and the top foil. They alo included local interaction force, variable load ditribution and different bump geometrie in the invetigation. They howed that higher frictional coefficient between the top foil and the bump foil can help in achieving efficient Coulomb damping and higher tiffne. Their follow up paper [7], preented the experimental verification of the model. Peng and Carpino [8] calculated the tiffne and damping coefficient of an elatically upported ga foil bearing. For their tructural model they ued a thin and extendable material a foil urface. The model neglected any bending and membrane effect and inertia of foil wa alo neglected. The Reynold equation to obtain preure and film thickne wa olved uing finite element method. Dynamic coefficient were olved uing perturbation method where Reynold equation wa linearized to yield force coefficient. Their reult howed that the compliance of the bearing at relatively low peed primarily depend on the hydrodynamic ga film. But, at high peed the tiffne

20 6 of the hydrodynamic ga film become very large and hence the compliance i due to the underlying elatic foundation. Later, Carpino [9] alo developed a finite element perturbation approach to predict foil bearing rotor dynamic coefficient. Han et al [10] tudied the characteritic of air bearing with external preurization. Their analyi involved determination of force coefficient uing perturbation analyi and a parametric tudy to ee the dependence of thee coefficient on bearing ize, external preure and number of upply retrictor. The tudy alo involved theoretical calculation to predict the rotor orbit and wa verified with experimental invetigation. Dellacorte and Valco [11] introduced a imple Rule of Thumb to etimate the load capacity of air foil bearing. The rule empirically related the load capacity of the bearing to the bearing ize and operating peed uing data available in the literature and from the experiment done by the author. Radil et al [12] tudied the dependence of load capacity of air foil bearing on the radial clearance. They howed that air foil bearing have an optimum radial clearance, below which thermal run-away can occur in the bearing which lead to ga film rupture. Above the optimum value the load capacity of the bearing i reduced. Wilde and San André [13] did a comparative tudy involving rotordynamic prediction and tet repone of a three lobed hybrid ga bearing. The bearing wa termed a hybrid becaue it wa both hydrotatic, from external preurization, and hydrodynamic in nature. They howed that by increaing the external preurization the critical peed can be hifted but the effective damping of the bearing i decreaed. The

21 7 meaurement done by them alo howed that whirl ratio decreaed with increae in upply preure. Peng and Khonari [14] developed a thermo-hydrodynamic model to analyze air foil bearing. The temperature ditribution on the top-foil of the bearing wa evaluated by olving coupled Reynold equation and Energy equation. The analyi model developed by them incorporated the compreibility of air and temperature dependence of air vicoity. The numerical reult were verified with the exiting experimental data and a comparative tudy of the thermal performance of olid walled bearing and foil bearing wa alo conducted. More recently Song and Kim [15] developed a new kind of compliant elatic foundation made of commercially-available compreion pring. They did analytical and experimental tudie to determine the performance of thi new air foil bearing. The analytical tudie involved tiffne calculation of pring under lateral loading and the reult were validated with experimental invetigation. Further in analytical tudie a computational model wa developed uing time-domain orbit imulation that could predict limit cycle behavior encountered in air foil bearing. They howed that a with any other air bearing with elatic foundation; their bearing could uppre the vibration at critical peed but not the onet of intability. Experimental invetigation revealed the poibility of large load capacity with appropriate cooling. In ubequent tudie, Kim [16] conducted parametric tudie on two different type of air foil bearing, circular and three-pad, and invetigated the dependence of rotor dynamic tability on the ditribution of tiffne and damping of the compliant

22 8 urface. The tudy howed that rotordynamic characteritic are more enitive to the overall bearing geometry rather than tiffne and damping ditribution within the elatic foundation. The author compared the reult from linear tability analyi and orbit imulation and found different onet peed of intability from the two method. The dicrepancy between the two method wa attributed to the limitation of linear tability analyi in the tability prediction. Kim and Park [4] developed air foil bearing with external preurization. The complaint tructure of the bearing developed by them had compreion pring arranged axially and wa imilar in contruction to the bearing in [15]. Their bearing wa both hydrodynamic and hydrotatic in nature and hence wa a Hybrid air foil bearing (HAFB). External preurization wa provided through the bump foil and top foil to the rotor urface. Four external feed tube were ued for thi purpoe. The tudy included both numerical analyi and experimental invetigation. The numerical invetigation wa concerned with the evaluation of preure profile and film thickne of the bearing under hybrid operation. Coat-down imulation for the bearing were alo performed. The imulation howed that hybrid operation increaed the onet peed of intability a compared to hydrodynamic operation. Their experimental invetigation dealt with the etimation of load capacity and tarting torque of hybrid air foil bearing. They howed that load capacity of the bearing increaed under hybrid operation and alo the frictional drag aociated during tartup wa reduced coniderably.

23 9 3 DETERMINATION OF FORCE COEFFICIENTS* 3.1 Decription of hybrid air foil bearing A chematic of the propoed bearing i hown in Figure 2. The bearing hown ha a ingle continuou top foil and a two trip bump foil. External preurization i upplied through four feed tube which directly dicharge air through the top foil to the bearing clearance. Circumferential arrangement of the feed tube i hown in Figure 2(b). The feed tube are located at θ = 72, 166, 247, and 341. The purpoe of the unymmetrical placement of the feed tube i to put the orifice on top of the bump a decribed in Figure 2. Table 1 give the parameter of the bearing ued during the imulation, the bump tiffne wa calculated uing the formula for free-free cae preented by Iordanoff [17]. * Reprinted with permiion from Parametric Studie on Dynamic Performance of Hybrid Airfoil Bearing by Kumar, M., and Kim, D., Journal of Engineering for Ga Turbine and Power, 130, Copyright 2008 by ASME.

24 10 Table 1: Bearing parameter Simulation Parameter Bearing diameter, 2R Bearing axial length, L Value 38.1 mm 38.1 mm Nominal clearance, C 32 µm Bump tiffne per unit area 4.7 GN/m 3 Top foil thickne 100µm Orifice Size (Diameter) 0.5 mm

25 11 Feed tube Top Foil Orifice (a) Schematic decription of HAFB θ Y X (b) Coordinate ytem for analyi Figure 2: Schematic decription of circular HAFB and coordinate ytem for analyi

26 Solution methodology The olution methodology followed in thi paper i baed on Finite Volume method. Figure 3 how the grid cheme for the control volume and the dynamic ma balance. ( ṁ ) z in i, j+ 1 i 1, j i, j ( ṁ x) ( ṁ ) in ( ṁ ) z in x ṁ out i+ 1, j z i, j 1 x Figure 3: Meh defined for analyi In Figure 3, ṁ i the air ma flow rate through the orifice. Alo, ṁ and x ṁz from the claical formulation of Couette-Poieuille flow are defined a 1 p p p hrω (1) 3 mx = h + z 12µ RgT x RgT 2 1 p p (2) 3 mz = h x 12µ RgT z

27 13 In the above equation x i a local coordinate attached on the bearing urface along the circumferential direction, z i a coordinate in axial direction, h i a film thickne, p i preure, µ i vicoity of air, R g i the ga contant of air and T i the temperature of upplied air. The dynamic ma balance of the control volume under tranient condition give ( m m ) m ( m m ) ( ρ ) x z ( ) d V d ph x+ z + in x+ z = = (3) out dt R T dt Subtituting value from (1) and (2) in the above equation we get the Reynold Equation for compreible fluid with hydrotatic upply g 1 3 p Rω 1 3 p ( ph) RgTm ph + ph + ph + = x 12µ x 2 z 12µ z t x z (4) Non-dimenionalizing Eq. (4) yield (ee Appendix A for more detail). M 3 P 3 P PH PH + + =Λ PH + Λ PH ( ) 2 ν ( ) θ Z θ θ Z θ θ τ (5) where p x P=, θ =, P R a Z z =, R τ = ω t, h H =, bearing number C 6µω Λ= p a R C 2, ω excitation frequency ratio ν =, and ω. M = µ R Tm 12 g 2 3 pa C. ω i the excitation frequency. See Figure 2 for more detail on the coordinate ytem ued. Auming the flow through the orifice a an ientropic proce, the ma flow rate of the compreible fluid for the choked and un-choked condition are given by Un-Choked: P P k ( k 1) 2 > = k+ 1

28 14 2/ k ( k+ 1)/ k 2k P P M S =ΓP H (6) k 1 P P 1/2 Choked: P P k ( k 1) 2 < = k+ 1 1/2 1/( k 1) k 2 M S =ΓP H 2 (7) k+ 1 k+ 1 where 12µ Cd A0 RgT Γ = i a feed parameter, P 3 i the upply preure, k i the ratio p C a of pecific heat for air, Cd i a dicharge coefficient. In the feed parameter, A 0 i the reference orifice curtain area defined a A0 = π doc, where d 0 i the orifice diameter. For the perturbation analyi, equation of motion for the elatic foundation correponding to the computational finite domain hould be developed. For implicity, the inertia of the elatic foundation i neglected and it i further aumed that each elatic foundation upport the correponding top foil independently. Then the equation of motion of elatic foundation become du = + (8) pab kbu cb dt where k b and c b are the effective tiffne and vicou damping coefficient of the elatic foundation, and A b i effective area for the elatic foundation. Note the tiffne per unit area in Table 1 i k / A. Auming the motion of elatic foundation i b b inuoidal in normal operating condition, the equivalent vicou damping coefficient can be found from tructural damping model through tructural lo factor, i.e.,

29 15 cω k b η=. For the preent imulation, the tructural lo coefficient of 0.25 i ued for b every elatic foundation. The choen tructural lo factor i from the empirical reult of a well-deigned bump foil bearing [18], [19] and [20]. Writing the bump dynamic equation in non-dimenional form yield du = + ν (9) τ P KbU Cb d where K b k C b b = and Cb pa Ab a c Cω = are non-dimenional bump tiffne and damping p A b coefficient, repectively. Linearizing Eq. (5) and (9) yield the zeroth and firt order equation (ee Appendix B for detail) Zeroth Order: 1/2 3 P0 3 P0 2k ΓP P H + P H + f ( H, P ) θ θ z z k 1 θ Z =Λ [ P0 H0] + 2Λν [ P0 H0] θ τ (10) Firt Order: 3 Pα 3 Pα P0 H0 + P0 H 0 θ θ Z Z k ΓP f ( P, H ) f ( P, H ) P a + Pα + + gα k 1 Z θ P P ( ) 0, H H 0 P0, H K 1 0 b + ηi 2 P a 3 P0 3H0 P0 + gα + H0 Pα θ Kb( 1+ ηi) θ 2 P a 3 P0 + 3H0 P0 + gα + H 0 Pα Z Kb( 1+ ηi) Z P a P a =Λ P0 + gα + Pα H0 + 2Λ ν ip0 P0 + gα + Pα H 0 θ Kb( 1+ ηi) Kb( 1+ ηi) (11)

30 16 where α = X, Y and g X = coθ, gy = inθ. Note that PX and PY are complex number with real and imaginary part. The value of f ( H0, P 0), f ( P, H ) P P, H 0 0 and f ( P, H ) H P, H 0 0 for choked and un-choked condition are given below. Choked f ( H, P ) = H (12) f ( P, H ) P P, H 0 0 = 0 (13) f ( P, H ) H P, H 0 0 = 1 (14) Un-choked 1 2 ( k+ 1) 2 k k 0 P0 P f ( H0, P0 ) = H 0 P P (15) 1 2 k k+ 1 2 k k k 1 1 k P 0 P 0 k k k = H0 P0 P0 P0, H 2 0 f ( P, H ) P P P k P k P (16) 1 2 k+ 1 2 k k 0 P0 f ( P, H ) P = H P0, H P 0 P (17) Once the zeroth order equation i olved for equilibrium preure profile and film thickne ( P 0, H 0 ), the firt order equation i olved to get the perturbed preure. The

31 17 perturbed preure profile i then ued to find the frequency-dependent tiffne and damping coefficient. kxx kxy W K Re( ) co Re( )co 0 xx K xy W0 R Px θ Py θ k yx k = yy C K yx K = yy C 2L Re( Px )inθ Re( Py )inθ (18) and cxx cxy W C Im( )co Im( ) co 0 xx Cxy W0 R Px θ Py θ cyx c = yy C Cyx C = ω yy 2 Im( Px )in Im( Py )in Cω L θ θ (19) A mentioned earlier, both the zeroth order and firt order equation were olved uing finite volume method with under relaxation. The grid ize ued for the numerical analyi wa 104 in the circumferential direction and 14 in the axial. The grid independency tudy of the numerical method followed in the preent paper wa done in [4]. The convergence criteria for the preure wa P P max 5 10 n+ 1 n i, j i, j 6 n P i, j, where n i the iteration index. The equilibrium poition of the rotor wa found uing orbit imulation which are detailed in [16]. For the zeroth order olution, 1-D analytical beam model developed by Kim and Park [4] i adopted to conider top foil agging effect under preure. The 1-D beam model ue the computational grid cheme hown in Figure 4. Note that between the elatic foundation, three computational grid point are aigned to accurately capture the effect of top foil agging. Further detail regarding thi model can be found in [4]. Figure 5 how the preure profile obtained by numerically olving the zeroth

32 18 order equation. The peak in the preure profile are due to hydrotatic feed line, the highet of which correpond to the loaded region of the bearing. Top Foil Strip Axial (Z) Top Foil Circumferential (θ) Computational Node Elatic foundation Figure 4: Grid cheme In the firt order olution, the preure and film thickne olved in the zeroth order olution are ued a input. The firt order olution ue the ame computational grid a hown in Figure 4 but the tiffne per unit area (Table 1) and correponding equivalent damping are aigned to each computational grid point. The perturbed preure in X-direction obtained from the firt order equation i hown in Figure 6.

33 19 P P θ Z S1 Z Figure 5: Zeroth order preure profile, Λ=1.25, tatic load 60N P X Px θθ Z S1 Z Figure 6: Firt order perturbed preure profile (P X ), Λ=1.25, tatic load 60N

34 Reult and dicuion The bearing tiffne and damping coefficient of HAFB were calculated with variou feed parameter ( Γ ), excitation frequencie (ν ), upply preure, and bearing number (Λ). Note that the feed parameter and bearing number are directly proportional to the orifice diameter and rotational peed, repectively. Firtly, effect of variou feed parameter on the bearing coefficient wa invetigated for different upply preure at fixed bearing number of Λ= 1.25 (rotor peed of about 30,000 rpm). For all the imulation the bearing i under a tatic load of 60N in X-direction (ee Figure 2). Figure 7 depict the predicted ynchronou direct tiffne coefficient veru the feed parameter. In general, the direct tiffne decreae with increae in either the feed parameter or the upply preure. At very low feed parameter the tiffne value for all the preure converge to a ingle value which correpond to the hydrodynamic cae. The decreae in tiffne with upply preure or feed parameter can be explained from journal eccentricity and attitude angle a hown in Figure 8 and Figure 9. Figure 9 preent the trend in attitude angle veru the feed parameter for different upply preure. Higher upply preure or increae in feed parameter decreae the attitude angle. At relatively high value of the upply preure and feed parameter the attitude angle i negative. Thi can be attributed to the fact that with higher preure we have more hydrotatic thrut on the rotor. Now with more thrut and with the preent arrangement of the feed tube (Figure 2), epecially the 2 nd tube in the direction of increaingθ, the rotor move into the forth quadrant correponding toθ and

35 21 hence we get negative attitude angle. The mall attitude angle in HAFB i very beneficial in term of reducing cro-coupled tiffne and reultant hydrodynamic intability. Figure 10 how predicted ynchronou cro-coupled tiffne which contribute to detabilizing force in the ga bearing. In general, the cro-coupled tiffne decreae with upply preure. For all preure, tiffne value are decreaing for Γ < 0.8, after which they increae gradually. Figure 10 how predicted ynchronou cro-coupled tiffne which contribute to detabilizing force in the ga bearing. In general, the cro-coupled tiffne decreae with upply preure. For all preure, tiffne value are decreaing for Γ < 0.8, after which they increae gradually Kxx (MN/m) P = 3 P = 4 P = 5 P = Feed Parameter (Γ ) Figure 7: Predicted direct tiffne coefficient v. feed parameter ( Γ ) with increaing upply preure, Λ=1.25

36 ex/c P = 3 P = 4 P = 5 P = Feed Parameter (Γ ) Figure 8: Predicted journal eccentricitie v. feed parameter ( Γ ) with increaing upply preure, Λ= Attitude Angle (deg) P = 3 P = 4 P = 5 P = Feed Parameter (Γ ) ` Figure 9: Predicted attitude angle v. feed parameter ( Γ ) with increaing upply preure, Λ=1.25

37 Kxy (MN/m) P = 3 P = 4 P = 5 P = Feed Parameter (Γ ) Figure 10: Predicted cro-coupled tiffne coefficient v. feed parameter ( Γ ) with increaing upply preure, Λ=1.25 Figure 11 and Figure 12 how the direct and cro-coupled damping coefficient, repectively. Both damping coefficient increae lightly with increae in feed parameter. The variation i not much for the direct damping coefficient, but ignificant for the cro-coupled damping for Γ < 1. The damping coefficient how a converging trend at higher feed parameter for all the upply preure. The cro-coupled damping croe over at Γ = 1.2 a hown in Figure 12. Thi can be attributed to the fact that higher the preure the earlier the cro-coupled damping become inenitive to increaing feed parameter. The cro over happen becaue damping in the cae of P = 6 become inenitive to feed parameter much earlier a compared to P = 3.

38 P = 6 Cxx (KN/m) P = 5 P = 4 P = Feed Parameter (Γ ) Figure 11: Predicted direct damping coefficient v. feed parameter ( Γ ) with increaing upply preure, Λ= Cyx (KN/m) P = 4 P = 5 P = 6 P = Feed Parameter (Γ ) Figure 12: Predicted cro-coupled damping coefficient v. feed parameter ( Γ ) with increaing upply preure, Λ=1.25

39 25 Figure 13 depict the predicted direct tiffne coefficient veru the excitation frequency ratio (ν) at increaing upply preure. Stiffne coefficient increae for ν < 1 but how converging trend at high excitation frequency ratio. In general at low ν, lower preure give higher tiffne value but the variation i not much with the upply preure at higher frequencie. Cro-coupled tiffne veru ν i hown in Figure 14. The cro-coupled tiffne i rather high at ν<1 with rapid decreae with ν. However, the cro-coupled tiffne i almot independent of the ν for ν > 1. Kxx (MN/m) P = P = 5 6 P = 3 P = Excitation Excitation Frequency Frequency Ratio Ratio (ν (ω/ω) =ω /ω) Figure 13: Predicted direct tiffne coefficient v. excitation frequency ratio with increaing upply preure, Λ=1.25

40 Kxy (MN/m) P = 3 P = 4 P = 5 P = Excitation Frequency Ratio Ratio (ν (ω/ω) =ω /ω) Figure 14: Predicted cro-coupled tiffne coefficient v. excitation frequency ratio with increaing upply preure, Λ=1.25 Frequency-dependency characteritic of direct and cro-coupled damping coefficient are hown in Figure 15 and Figure 16, repectively. Both damping coefficient how a decreaing trend with increae in ν at low ν and are almot frequency-independent at higher excitation frequencie. The damping coefficient how a converging trend toward null value at high ν. The lo of damping i accompanied by large direct tiffne coefficient at high ν (Figure 13) howing a typical hardening effect of ga bearing.

41 P = 6 P = 5 Cxx (KN/m) P = 4 P = Excitation Frequency Ratio (ν =ω Excitation Frequency Ratio (ω/ω) /ω) Figure 15: Predicted direct damping v. excitation frequency ratio with increaing upply preure, Λ=1.25 Cyx (N/m) P = 6 P = 3 P = 4 P = Excitation Frequency Ratio (ν =ω Excitation Frequency Ratio (ω/ω) /ω) Figure 16: Predicted direct tiffne v. excitation frequency ratio with increaing upply preure, Λ=1.25

42 28 Figure 17 how the predicted non-dimenional journal eccentricity veru the bearing number (Λ) for increaing upply preure at feed parameter Γ = 0.6. In general, the non-dimenional eccentricity decreae with increae in either the bearing number or upply preure. The variation in eccentricity with upply preure i large at low bearing number (low rotational peed). At high bearing number, the variation decreae and non-dimenional eccentricity how a converging trend. 1 P = 4 P = ex/c P = 6 P = Bearing Number (Λ) Figure 17: Predicted journal eccentricitie v. bearing number (Λ) with increaing upply preure, Γ =0.6 Figure 18 and Figure 19 how the direct and cro-coupled tiffne coefficient, repectively. The direct tiffne coefficient increae rapidly at low bearing number but how converging trend at high bearing number. In general, lower upply preure give

43 29 higher tiffne value and thi can be attributed to the fact that eccentricity increae with decreaing the upply preure (Figure 17). Cro-coupled tiffne decreae rapidly with bearing number, and in general, higher preure give lower cro-coupled tiffne value P = 4 P = 3 Kxx(MN/m) P = 6 P = Figure 18: Bearing Number (Λ) Predicted direct tiffne v. bearing number (Λ) with increaing upply preure, Γ=0.6

44 P = 3 P = 4 Kxy(MN/m) P = 5 P = Bearing Number (Λ) Figure 19: Predicted cro-coupled tiffne v. bearing number (Λ) with increaing upply preure, Γ =0.6 Figure 20 depict the direct damping coefficient with increaing bearing number for variou upply preure. The direct damping coefficient decreae rapidly for Λ < 1 after which a converging trend i oberved. Direct damping value are almot independent of variation in upply preure. Cro-coupled damping (Figure 21) how imilar variation with bearing number, although in thi cae they change with the upply preure.

45 P = 5 P = 6 Cxx(KN/m) P = P = Figure 20: Bearing Number (Λ) Predicted direct damping v. bearing number (Λ) with increaing upply preure, Γ =0.6 Cyx(KN/m) P = 3 P = P = 5 P = Bearing Number (Λ) Figure 21: Predicted cro-coupled damping v. bearing number (Λ) with increaing upply preure, Γ =0.6

46 32 Benchmark reult for the preent analyi are included in Appendix E. There a limiting cae of zero feed parameter i compared with the reult by Kim [16].

47 33 4 DESIGN AND FABRICATION OF HIGH SPEED TEST RIG The deign and fabrication of a tet rig which could evaluate the performance of variou ga bearing at high peed wa undertaken. Thi ection decribe thi unique tet rig and it capabilitie. 4.1 Requirement from the tet tig The baic requirement of thi new tet rig wa to experimentally determine the capabilitie of foil bearing at moderate to high peed. The tet rig wa primarily enviaged to provide the capability to gather the following information 1. Load capacity of air foil bearing at variou peed 2. Thermal run way of the bearing at variou peed 3. Frictional bearing torque generated during tartup, hutdown and at high peed operation. 4. Stiffne and damping coefficient of the bearing at variou peed. Beide the experimental data that the tet rig could furnih, the following feature of the tet rig were alo deired 1. The tet rig hould have the capability of accommodating air foil bearing of different ize. It i deired that in order for the tet rig to tet bearing of different ize only a minimal portion of the tet rig hould be changed or replaced. The above requirement warrant the eparation of the drive mechanim from the tet ection. The capability to tet bearing of different ize hould be provided with the

48 34 ue of appropriate adapter which can be eaily attached and removed for different bearing. 2. The tet rig hould run on an electric motor drive. The other option beide the electric motor drive wa air powered drive uing impule turbine. The later option wa ruled out a it would have made the tet rig extremely noiy. The air powered tet rig would have alo required very tight tolerance and alignment for the impule turbine. 3. Since the tet rig wa deigned to evaluate the thermal performance of the air bearing, the generation and tranfer of paraitic heat from the driving mechanim hould be minimized. The major ource of heat in the electric drive train i uually the motor therefore an appropriate cooling mechanim for the motor would be required. The cooling mechanim hould have the capability of uing both air and water a coolant. 4. The primary function of the tet rig wa to meaure tatic performance (load capacity and frictional torque) therefore the journal hould be rigidly upported. 5. The tet hould have a loading mechanim which could provide external load to the air foil bearing. 6. Depending on the choice of upport for the rotor, the upport ytem may require appropriate lubrication, ealing and a preload mechanim.

49 General layout and deign of the tet rig Baed on the requirement lited in the previou ection the author came up with a deign of the tet rig hown in Figure 22. Decription of individual component hown in the figure i given below: 1. Electric motor: Specially fabricated 4kW electric motor, with maximum peed of 90,000 RPM. 2. Motor tator: Stator for electric motor. 3. Cooling jacket: In order to utain uch high peed the electric motor require cooling jacket to diipate heat. 4. Spindle bearing. High peed pindle ball bearing with ceramic ball from GMN, Germany. 5. Bearing inert: Inert to upport pindle bearing. 6. Houing: Aluminum houing to upport rotor, motor and bearing inert. 7. Oil jet lubrication etup: Oil jet lubrication for pindle bearing. 8. Lip eal: Hydraulic-cylinder ealing with Buna-N O-ring. 9. End plate: Plate to hold lip eal. 10. Wave pring waher compreion type: Wave pring to provide axial pre-load to pindle bearing 11. Rotor: 20mm/12 haft. 12. Tet ection: Removable tet ection over which foil bearing i inerted. 13. Hybrid air foil bearing: Propoed hybrid air foil bearing for 1.5 haft. 14. Foil bearing houing: Houing to hold foil bearing.

50 Thermo-Couple Oil Inlet Cooling Fluid IN OUT Proximity Probe Oil Inlet External Load Oil Drain Figure 22: Tet rig 36

51 Decription of major tet rig component Thi ection give a detailed decription of major component of the tet rig. The rotordynamic characteritic of the pindle contructed i in the following ection Spindle bearing Given the moderate to high peed operation and rigid upport requirement of the tet rig, pindle bearing were ued. Spindle bearing are angular contact bearing in which force are tranmitted from one raceway to other under a pecific contact angle. To further increae the maximum achievable peed (limiting peed) and the ervice life, pindle bearing with ceramic ball were ued. Spindle bearing require adjutment againt a econd bearing and thi arrangement hould be under a permanent axial load, the preload. The arrangement can either have a pring preload or rigid preload. Spring preload are uitable for high peed application and are inenitive to thermal expanion of the rotor or the bearing houing. Rigid preload though are eaier to implement have lower limiting peed a compared to pring preload. For the preent tet rig, bearing arrangement with pring preload wa ued. Spring preload wa provided uing wave pring waher and tainle him were ued to provide appropriate compreion to thee pring in order to get the required preload. The pecification and the decription of ball bearing ued are given in Figure 23 and Table 2. Two et of preloaded bearing were ued on either ide of the motor a hown in the Figure 22. The free body diagram howing the preload force on the bearing i hown in Figure 24.

52 38 Figure 23: Spindle ball bearing, Source: GMN bearing [21] Table 2: Spindle bearing parameter Parameter Value Unit Decription d 20 mm Bore diameter D 42 mm Outer diameter B 12 mm Width ingle bearing r min 0.6 mm Chamfer r min 0.3 mm Chamfer open ide (pindle bearing) D w 6.35 mm Ball diameter Z 13 piece Ball complement m kg Weight of bearing d mm Outer diameter inner ring d mm Land inner ring, open ide d k 31.4 mm Cage bore d m 31 mm Pitch circle diameter D mm Bore outer ring D mm Bore outer ring (open ide) n rpm Speed value C 8400 N Dynamic load rating

53 39 Table 2: Continued Parameter Value Unit Decription C N Static load rating F v 120 N Preload (Medium) F amax 387 N Lift off force (Medium) C ax 37 µm Axial rigidity (pair) (Medium) F f 300 mm Minimum pring preload α 0 15 Contact angle 240N 120N 120N 240N Figure 24: Bearing preload diagram Tet ection The tet ection which will be intalled on the tet rig i hown in Figure 25. Note the tet ection i a two piece aembly. The outer hell of the tet ection i where the air bearing will be intalled. Due to numerou tartup and coat-down, the outer hell of the tet ection will be ubject to wear. An aembly intead of a ingle piece tet

54 40 ection give the flexibility to diaemble the outer hell and have it ground and coated for lating ue. The a-built dimenion of the tet ection (with the dimenion of the propoed bearing) hould render a radial clearance of 30~40µm. It i intereting to note that with the preent arrangement, the tet ection will act a an overhang impeller. Figure 25: Tet ection Electric motor drive The decription of the electric motor drive choen for the tet rig i hown in Figure 26 and Figure 27. The motor wa purchaed from Elektromachinen u. Antriebe AG, Switzerland [22]. The motor i a 2-pole, aynchronou, high-peed and medium frequency motor. The motor ha a wound tator and a raw rotor. Further decription of the motor i given in Table 3. Becaue of the mall ize of the rotor additional balancing ring which were tacked on either ide of the rotor were ued for balancing. The electric motor drive require a cooling mechanim with either water or air a a coolant. The

55 41 cooling jacket deigned for the preent drive i hown in Figure 28. The cooling jacket ha four circumferential groove for the flow of the coolant. Clearance cut on the top and the bottom of the jacket provide the paage of coolant from one circumferential groove to another. The cooling jacket alo ha groove for the O-ring which provide ealing of the coolant. Table 3: Electric motor drive parameter Element Value Unit Motor Speed 89,000 rpm Frequency 1500 Hz Power 2.8 kw Peak Power 7 kw Voltage 380 V Current 6.5 A Stator Inulation Cla F Maximum Permiible Heating 120 K Coolant Temperature 20 C Coolant Water, Air Rotor Circumferential Speed m/ Material of Squirrel Cage Copper, Ring enforced Material of Shaft Magnetic

56 42 Figure 26: Electric motor drive - Motor

57 43 Figure 27: Electric motor drive Stator

58 44 Clearance cut BOTTOM TOP 4 Circumferential groove Clearance cut Figure 28: Electric motor drive - Cooling jacket 4.4 Rotordynamic analyi of the rotor Lateral vibrational analyi of the rotor wa done to compute critical peed, mode hape and undamped critical peed map. Two method were ued for thi analyi. The firt method involved tranfer matrice and the econd wa baed on Finite Element Method (FEM). Since the rotor i upported on rigid ball bearing and there i no external damping the analyi wa done conidering undamped condition. Note, becaue there i clearance between the lip-eal and the rotor any tiffne and damping aociated with the eal wa not conidered. Both the above mentioned method require the haft to be modeled a erie of lumped mae and flexible ma-le beam. The

59 45 rotor model and the rotor of the preent tet rig are hown in Figure 29. The model i made of 39 tation with bearing at tation 12, 18, 29 and 35. With the preent arrangement of the rotor, the tet ection can be thought of a an overhang impeller and the motor a an impeller within the bearing pan. 1 Station 12 Station 18 Station 29 Station 35 Shaft Radiu (in) Axial Location (in) Figure 29: Rotor model and rotor

60 46 Uing FEM the undamped critical peed map of the rotor i hown in Figure 30. The map how the natural frequencie at a given operating peed with varying upport tiffne. The map i generated for the maximum continuou operating peed (MCOS) of 50,000 rpm. Note the above MCOS i when the tet rig i running without oil mit lubrication and the ball bearing have only greae lubrication. The MCOS under oil-mit lubrication will be much higher. The mode hown in the map are all forward mode. Firt Second Third Fourth E+00 1.E+01 1.E+02 1.E+03 Krpm 1.E+04 1.E+05 1.E+06 1.E+07 1.E+08 1.E+09 1.E+10 Bearing Stiffne (lb/in) Figure 30: Undamped critical peed map at MCOS = 50,000 RPM Note, the only mode that appear within the operating range (0~90,000 rpm) of the tet rig i the firt mode. It i intereting to note that the natural frequencie for all the mode are inenitive to high bearing tiffne value (>1e6 lb/in). Since the haft ha a rigid upport from the ball bearing, therefore conidering the higher bearing tiffne

61 47 value (1e6 ~ 1e10 lb/in) in the critical peed map the firt mode natural frequency i around 57,000 rpm. Figure 31 depict the critical peed etimation of the haft at ball bearing tiffne of ~1e6 lb/in [21]. The figure how the backward and forward natural frequency of the haft at variou operating peed. The critical peed by definition i the peed at which the pin frequency coincide with the natural frequency. Hence the interection of pin = natural frequency line with the natural frequency curve are the critical peed a hown in the figure. The critical peed from the tranfer matrix function method wa alo calculated, the reult obtained from both the method i hown in Table 4. Note, there i a very good agreement between the two method in the prediction of critical peed. Backward(1t) Forward(1t) Natural Frequency in RPM Spin=Natural frequency line Spin Frequency in RPM Figure 31: Critical peed etimation

62 48 Table 4: Critical peed Mode Whirl Direction Critical Speed FEM (RPM) Critical Speed Tranfer Matrix (RPM) 1t Backward t Forward The mode hape at the above forward critical peed i hown in Figure 32. Again we ee good agreement in the reult from the two method a hown in Figure 32 (b).

63 49 (a) Station Number Y X Tranfer Matrix FEM (b) Normalized Mode Shape Bearing Station Figure 32: Mode hape plot

64 Commiioning of tet rig The commiioning of the tet rig wa conducted after the aembly of all the part excluding the tet ection. The tet ection wa not aembled a it aembly would have precluded the poibility of increaing the preload on the ball bearing located near the tet ection (ee Figure 22). To monitor the health of the tet rig thermocouple where attached on the outer race of three of the four ball bearing to monitor their temperature. The outer race of the fourth bearing wa inacceible becaue of the electrical connection box intalled on the tet rig houing for the power upply of the motor. The temperature of the motor wa monitored through a thermitor which wa factory intalled in the tator element of the motor. The intalled thermitor wa of poitive temperature coefficient (PTC) type and the threhold reitance value which correpond to maximum operating temperature of motor wa 3990 Ω. The calibration chart of the thermitor i hown in Appendix C. The maximum allowable operating temperature of the bearing i limited by the retaining cage that hold the ball in place within the outer and the inner race and for the preent cae wa 120 C [21]. For the preliminary commiioning of the tet rig, only greae lubrication in the ball bearing and only air a the coolant for the motor wa ued. Figure 33 how the commiioning report of the tet rig at 30,000 and 40,000 RPM. During the tet bearing temperature and motor reitance wa monitored after every 2 minute. The data wa collected until aturation in the bearing temperature wa een. Note the irregularity in the reitance value obtained during the two tet can be attributed to the highly non-linear nature of the thermitor.

65 51 Bearing 1 Bearing 2 Bearing 3 Thermitor Temperature ( C) Time (min) (a) Commiioning at 30,000 RPM Reitence (Ω) Temperature ( C) Bearing 1 Bearing 2 Bearing 3 Thermitor Time (min) (b) Commiioning at 40,000 RPM Reitence (Ω) Figure 33: Commiioning of tet rig

66 52 To determine the poibility of haft bow and quantify it at the location of tet ection, proximity enor wa intalled a hown in Figure 34. The enor wa intalled on a manual linear tage equipped with a vernier micrometer. The proximity probe wa firt calibrated, ee Appendix D for the calibration chart. Proximity Probe Tet Section Linear Stage Figure 34: Proximity probe arrangement The haft wa rotated by hand to examine the maximum and the minimum voltage output from the proximity probe. The angular eparation between the point that gave thee voltage wa cloe to 180. Furthermore uing the voltage difference between thee point and the calibration chart the diplacement wa etimated a 20µm. Thi diplacement i the peak to peak diplacement due to the haft bow. Note, the zero to peak diplacement of haft of 10µm i coniderably high given the nominal clearance of the bearing i jut 25µm. Next, the level of vibration at variou operating peed wa acquired uing a data acquiition program developed in LabView. The FFT data obtained for 4 different peed are hown in Figure 35 and Figure 36.

67 Amplitude (µm) KRPM (a) Operating Speed: 10,000 RPM Amplitude (µm) KRPM (b) Operating Speed: 20,000 RPM Figure 35: FFT at 10,000 and 20,000 RPM

68 Amplitude (µm) KRPM (a) Operating Speed: 30,000 RPM Amplitude (µm) KRPM (a) Operating Speed: 40,000 RPM Figure 36: FFT at 30,000 and 40,000 RPM

69 55 The ignificant bow in the haft can be attributed to the rather cold mounting of the rotor element of the motor on to the haft. The cold mounting had to be performed a there wa limitation on the maximum temperature (300 C) up to which the rotor element could be heated. According to the motor vendor, the limitation wa due to the winding inide the rotor element. All the other part e.g. haft leeve, were pre fitted at temperature of around 500 C. The haft bow wa rectified by grinding the tet ection at low roll. The grinding operation i depicted in Figure 37. The haft wa rotated from the non-tet ection end of the tet rig, a hown, to nullify the haft bow at the tet ection. The grinding wheel wa travered over the whole pan of the tet ection. Grinder Wheel Tet Section Non-Tet Section Figure 37: Grinding of tet ection

70 56 5 EXPERIMENTAL RESULTS The preent ection decribe the experimental reult acquired uing the newly fabricated tet rig and the developed HAFB. A detailed overview of the HAB, experimental etup and the data acquiition apparatu i alo included. Appendix F include the fabrication of bump foil and top foil for the HAFB. 5.1 Decription of prototype HAFB The HAFB a decribed earlier ha four teel feed tube (OD: 0.05 / ID: ) for external preurization (ee Figure 38 (a)). The teel tube are connected to the urface of the top-foil uing ilicone rubber tubing (OD: / ID: 0.03 ) a hown in Figure 38 (a). The rubber tubing provide flexibility and i eaier to glue on to the curved urface of the top foil. For the load capacity meaurement of the HAFB, a thermocouple i glued uing epoxy to the back ide of the top foil a hown in Figure 38 (b). Note, the location of the thermocouple i exactly oppoite to the leading edge and i downtream of the econd feed tube in the direction of rotation. Bearing parameter before and after removal of haft bow i hown in Table 5.

71 57 Table 5: Parameter Prototype bearing parameter Value Bearing diameter, 2R Bearing axial length, L Nominal clearance, C (Before correction of haft bow) Nominal clearance, C (After correction of haft bow) inch inch inch inch Bump tiffne per unit area 4.7 GN/m 3 Top foil thickne Bump Foil Height inch 0.02 inch

72 58 (a) Steel Feed Tube Orifice Silicone Rubber Tube Bearing Houing (b) Location of Thermocouple Figure 38: Hybrid air foil bearing (HAFB)

73 Experimental etup The experimental etup of the tet i hown in Figure 39(a). The load i applied to the bearing uing a pulley ytem. A chematic depicting the loading mechanim i hown in Figure 39(b). The preent arrangement of the load mechanim enure that the load i applied evenly over the axial pan of the bearing. The air flow to the bearing feed tube i regulated uing an acrylic panel mount flow meter and the upply preure i meaured through an air preure gauge. The temperature data i read through a thermocouple diplay which i connected to the computer for data logging. Load Tet rig hroud Cooling air inlet Bearing houing Feed tube Oil drainage Bearing thermocouple Figure 39: Tet facility

74 60 (b) Pulley Drive Train Bearing and houing Load Figure 39: Continued 5.3 Tet reult Several tet were performed to determine the load carrying capacity of HAFB at variou operating peed. The tet condition, obervation and reult from load capacity tet at each peed follow Tet 1 The firt tet conducted on the prototype HAFB wa with the bow in the haft (ee ection 4.5). All the other operating condition are ummarized in Table 6. To determine the load carrying capacity of the bearing in the preent cae increaing load wa applied to the bearing and within each load application the temperature wa allowed to tabilize (Figure 40). A reported by Kim and Park [4] the threhold where the bearing

75 61 reache the load capacity there hould be a harp increae in temperature. Thi harp increae in temperature either correpond to rotor rubbing on the top foil urface or the thermal runaway of the bearing. In the preent cae the tet wa not concluded (ee Figure 40) a the wire ued in the loading mechanim napped after the load of N wa applied. Note, though the tet didn t finih and wa done when the haft wa bowed, till the load that the bearing wa able to utain before the loading mechanim failed far urpaed the previouly reported load capacity of N by Kim and Park [4] which wa alo at a higher peed (20,000 RPM). The better performance of the preent bearing in term of load carrying capacity can be attributed to the tiffer complaint tructure a compared to the bearing in [4]. Table 6: Operating parameter: Tet 1 Parameter Speed Supply Preure Air Flow Value 10,000 RPM 80 pi 8 SCFH The condition of the top foil after the tet wa completed i hown in Figure 41.

76 60 Wire napped N Temperature( C) N 31.8 N 97.5 N N N Time(min) Figure 40: Tet1: Load capacity tet at 10,000 RPM with upply preure of 80 pi and 8 SCFH air flow 62

77 63 Figure 41: Top foil condition after Tet 1; 10,000 RPM with upply preure of 80 pi and 8 SCFH air flow Tet 2 The econd tet wa performed with the ame bearing (a Tet1) but at a higher operating peed of 20,000 RPM. The operating parameter are ummarized in table below. Table 7: Operating parameter: Tet 2 Parameter Speed Supply Preure Air Flow Value 20,000 RPM 80 pi 8 SCFH In thi cae even a mall initial load of N to the bearing reulted in bearing failure. The bearing failure wa preceded with high vibration of the bearing and houing.

78 64 Thee vibration point to the evere tick-lip rubbing of the rotor on the bearing urface. The reaon for the rubbing wa found to be the haft bow which generate ignificant haft whirl (ee ection 4.5). The wear on the top foil after the bearing failure i hown Figure 42. Note, the damage on the top foil i uniform in the circumferential direction and i predominantly on one of the edge. All the ubequent tet were done with the low roll elimination of the haft bow a decribed in ection 4.5. Alo, during the teting it wa oberved that the rubbing between the top foil and rotor reult in localized welding of the two. Thi localized welding put a ignificant amount of train on the driving motor and damage the urface of the tet ection. Therefore in order to circumvent the damage an upper limit on the top foil temperature wa required when etimating the load capacity of the bearing. Thi upper limit will however not provide the ultimate load capacity but given the poibility of damaging the tet ection thi methodology wa adopted for further teting. The upper limit on the temperature wa etablihed in the ubequent tet.

79 65 Localized welding Orifice Leading edge Direction of rotation Trailing edge Figure 42: Top foil wear after Tet 2; 20,000 RPM with upply preure of 80 pi and 8 SCFH air flow Tet 3 Following the failure of the previou bearing, a new HAFB wa made and a mentioned earlier the haft bow wa removed by grinding the tet ection. The preent teting wa done at 20,000 RPM, the peed at which the previou bearing failed. Further, the air flow wa increaed to contribute in circumventing any bearing failure. The operating parameter are lited in Table 8. Table 8: Operating parameter: Tet 3 Parameter Speed Supply Preure Air Flow Value 20,000 RPM 80 pi 14 SCFH

80 66 The ame methodology for increaing the load on the bearing wa followed a in Tet 1. The reult from the preent teting i hown in Figure 43. The bearing wa teted up to 153N and the bearing failed at 159N. Note the top foil temperature at the bearing failure wa in exce of 70 C. Thi temperature wa etablihed a the temperature beyond which there i very high poibility of bearing failure. A compared to the load capacity tet done in [4] at the ame operating peed, the load capacity in the preent cae i much higher.

81 80 Temperature ( C) N N N N N N Bearing Failure N Time(min) Figure 43: Load capacity tet at 20,000 RPM with upply preure of 80 pi and 14 SCFH air flow 67

82 68 The bearing failure in thi cae alo reulted in localized welding and damage to the tet ection urface. See Figure 44 for the top foil wear after the tet. Here again the damage wa on one of the edge of the top foil and i circumferentially uniform. Localized Welding Orifice Leading Edge Direction of Rotation Trailing Edge Figure 44: Top foil wear after Tet 3; 20,000 RPM with upply preure of 80 pi and 14 SCFH air flow Tet 4 Since the prototype bearing failed during the previou tet, a new bearing wa contructed. The previou tet were conducted at 10,000 RPM and 20,000 RPM, but thi tet wa conducted at 15,000 RPM. Other operating parameter are lited in Table 9.

83 69 Table 9: Operating parameter: Tet 4 Parameter Speed Supply Preure Air Flow Value 15,000 RPM 80 pi 14 SCFH Reult of the tet conducted i hown in Figure 45. The harp increae in temperature initially (t=18 to t=20 min) wa due to udden increae in load (60.80 to 80.1 N). Subequently the bearing load wa decreaed (80.1 to N) and then the increment in load wa gradually applied. Note that the tet wa topped at about 80 C which i 10 higher than the temperature where the previou bearing failed. Alo at thi time the top foil wa experiencing a harp increae in temperature. Since the top foil temperature wa well beyond the temperature where the previou bearing failed and wa harply increaing it wa decided to top the tet to avoid any damage to the tet ection urface and preerve thi bearing for future teting. The load on the bearing ( N) before the harp increae in temperature (t <90 min) wa etablihed a the load capacity under the above mentioned operating condition. The top foil after the tet i hown in Figure 46, where only minor break-in rubbing mark were oberved.

84 90 Motor Stopped N N N Temperature( C) N 80.1 N N N N N N N N 21.1 N Figure 45: Time(min) Load capacity tet at 15,000 RPM with upply preure of 80 pi and 14 SCFH air flow 70

85 71 Figure 46: Top foil after Tet 4; 15,000 RPM with upply preure of 80 pi and 14 SCFH air flow Tet 5 & 6 The ame bearing a in Tet 4 wa ued for the preent tet. Except for the peed all the other operating parameter were kept the ame and are lited below. Table 10: Operating parameter: Tet 5 & 6 Parameter Speed Supply Preure Air Flow Value Tet 5 25,000 RPM Tet 6 35,000 RPM 80 pi 14 SCFH

86 72 To maintain conitency with the previou tet (Tet 4), in both the preent cae teting wa done up to a maximum top-foil temperature of 80 C. The wear on the top foil after the tet i hown in Figure 47 and Figure 48. Note the condition of top foil after the tet at 35,000 rpm i almot identical to the condition after the tet at 25,000 rpm. Figure 47: Top foil after Tet 5; 25,000 RPM with upply preure of 80 pi and 14 SCFH air flow

87 73 Figure 48: Top foil wear after Tet 6; 35,000 RPM with upply preure of 80 pi and 14 SCFH air flow Reult for the two cae are hown in Figure 49 and Figure 50. Note in Tet 5 (Figure 49) the lat bearing load reulted in a teady increae in top foil temperature, taking a conervative etimate the econd lat applied load (164.75N) wa etablihed a the load capacity. In Tet 6 (Figure 50) the lat applied load (202.23N) reulted in a teady temperature of around 80 C and hence the load capacity wa etablihed a N

88 Temperature( C) N N N N 82.4 N 60.8 N N N N 97.0 N N 69 N Motor Stopped Time(min) Figure 49: Load capacity tet at 25,000 RPM with upply preure of 80 pi and 14 SCFH air flow 74

89 90 Motor Stopped Temperature( C) N 69.0 N N N N N N N Figure 50: Time(min) Load capacity tet at 35,000 RPM with upply preure of 80 pi and 14 SCFH air flow. 75

90 Tet 7 The preent tet wa conducted with bearing under hydrodynamic operation to compare with the hybrid operation. Note, the preent bearing can be made to run hydrodynamically jut by hutting the air upply to the feed line. The rotor peed wa choen a 25,000 RPM. Top-foil wear after the tet i hown in Figure 51 and the reult in Figure 52. The lat load applied took the bearing on the verge of failure. At thi time the motor wa bogging down and the top foil temperature hot up to 110 C. To avoid any damage, the load to the bearing wa reduced and motor wa ubequently topped. Taking the lat load that reulted in a teady temperature of the top foil, the load capacity wa etablihed a N. It i intereting to note that the current load capacity i higher than the load capacity in Tet 5 where the bearing wa under hybrid operation with the ame rotor peed. Thi difference can be attributed to the fact that the load capacity in the preent cae i the ultimate load capacity which wa not the cae for the bearing in Tet 5. However it hould alo be noted that lat load applied to the bearing in Tet 5 reulted in the top foil temperature of around 80 C, the temperature at which the preent bearing wa about to fail. Thee reult indicate that the hydrotatic upply line have no contribution toward the load capacity, epecially when the bearing i heavily loaded. Table 11 how the top foil temperature at imilar load from the preent tet and Tet 5. The top foil temperature i lower in the cae of hybrid operation at each load indicating cooling effect from the hydrotatic air upply.

91 77 Table 11: Top foil temperature Load Hybrid Hydrodynamic (N) ( C) ( C) Figure 51: Top foil wear after Tet 7; 25,000 RPM under hydrodynamic condition The ummary of all the tet done i given in Table 12.

92 120 Motor Stopped Temperature( C) N 61.8 N N 99 N N N N N N N Time(min) Figure 52: Load capacity tet at 25,000 RPM Hydrodynamic operation 78

93 Table 12: Summary of bearing load capacitie Tet Speed (RPM) Clearance (µm) Air Supply (SCFH) Preure (pi) Shaft Bow Tet Completed Maximum Load Applied (N) Load Capacity (N) 1 10, Ye No , Ye No , No Ye , No Ye , No Ye , No Ye , No Ye

94 Tet 8 To acertain the effectivene of the upply line at low peed and under light load a comparative tet wa conducted. The bearing wa run under hydrodynamic and hybrid mode at 10,000 RPM and under 21.1N load, ee Table 13 for other parameter. Since the peed i low, the hydrodynamic preure generated will be low and hence one can ee the effectivene of the hydrotatic upply line in term of load utenance. The reult from the comparative tet are hown in Figure 53. Running the bearing under hydrodynamic mode reulted in thermal intability and evere vibration. The bearing wa conequently unloaded and the motor wa topped. In the hybrid cae the bearing didn t how any intability or vibration and the top foil temperature tabilized after ome time. Alo, the top foil temperature under hybrid condition wa ignificantly low. Thee reult indicate the uperior performance of the bearing under hybrid mode. Table 13: Operating parameter Tet 8 Parameter Speed Supply Preure (Hybrid) Air Flow (Hybrid) Value 10,000 RPM 80 pi 14 SCFH

95 Temperature( C) Motor topped Hybrid Time(min) Hydrodynamic Figure 53: Comparative tudy at 10,000 RPM 81

96 82 6 CONCLUSIONS AND FUTURE WORK 6.1 Concluion from analytical tudie Simulation how that feed parameter and upply preure affect the dynamic characteritic of air foil bearing. With the increae in either the upply preure or the feed parameter, the rotor center itelf and hence one ee a decreae in direct tiffne. Simulation how that the cro-coupled tiffne, which contribute a a detabilizing force, could be reduced by increaing either the upply preure or the feed parameter. There i a critical feed parameter ( Γ ) at which the cro-coupled tiffne i minimal. Direct damping, which dampen the vibration in the bearing, howed increaing trend with the upply preure and the feed parameter. The prediction demontrate the intabilitie in air bearing can be attenuated by modulating the upply preure. Frequency-domain analyi of the bearing coefficient howed expected trend. The direct damping howed marginal change with upply preure but howed rapid increae with increaing excitation frequencie. The damping converged to null value for all the preure for uper-ynchronou excitation. The lo in damping with high tiffne value for high frequency excitation i a typical hardening effect of ga bearing. In almot all the cae there are rapid decreae in cro-coupled tiffne and damping and the value how converging trend in uper-ynchronou regime. The trend one ee in bearing tiffne coefficient with increaing bearing number are baically the trend with increaing rotational peed. In general direct tiffne increae rapidly at low bearing number but howed converging trend at high

97 83 bearing number. It i intereting to note that the cro-coupled tiffne and direct damping decreae with increaing bearing number and cro-coupled tiffne i in fact negative for high preure. Note that the detabilizing force i proportional to K XY C ω XX and with the trend in cro-coupled tiffne and direct-damping, there i a poibility that backward whirl may be induced in the bearing. But thi backward whirl will alway be dominated by forward whirl generated by the imbalance. 6.2 Concluion from experimental tudie A new tet rig with high peed capability wa deigned, contructed and commiioned. With minimal cooling (air a coolant) of the motor and greae lubrication in the pindle ball bearing, the tet rig wa commiioned up to 45,000 RPM. Commiioning of the tet rig beyond thi peed may require better cooling and air-mit lubrication of the ball bearing. The tet rig wa deigned uch that both of the above mentioned enhancement can be implemented very eaily. Following the completion of the tet rig, experiment were conducted on the prototype HAFB. Several tet were conducted to determine the load capacity of the hybrid air foil bearing at variou operating peed. Noticeable enhancement in load capacity wa oberved a compared to the tet conducted by author in [4]. Major contribution in the enhancement of the load capacity came form the tiffer complaint tructure of the preent bearing a compared to the HAFB in [4]. An increae in load capacity wa oberved with increaing peed (ee Tet 4, 5&6). Thi trend i expected becaue higher peed reult in higher wedge effect which conequently increae the

98 84 generated hydrodynamic preure. Comparion between hybrid and hydrodynamic operation at a relatively high peed (25,000 rpm) indicated no contribution from the hydrotatic upply line toward the load capacity epecially when the bearing i heavily loaded. However in the cae of hybrid operation the top foil temperature wa lower at a given load which can be attributed to the cooling provided by the hydrotatic upply line. Furthermore, the comparative tet at low peed (10,000 RPM) howed much better performance of the bearing under hybrid operation a compared to hydrodynamic operation. 6.3 Future work Zeroth order orbit imulation can be extended to three dimenional orbit imulation. Thee imulation include both rotor and bearing in the analyi and can include both cylindrical and conical mode. Furthermore the bearing code can be coupled with the rotordynamic code which can extend the analyi beyond the rigid mode to the bending mode. Thi approach will however be iterative and time conuming. On the experimental ide, the invetigation can be extended to higher peed. The experiment at higher peed will help in tudying the effect of haft bending on the bearing load capacity. It will alo be intereting to monitor not only the temperature but the vicou torque generated by the bearing during the load capacity tet. More thermocouple can be included along the bearing circumferential and axial direction to get a better picture of the efficiency of the hydrotatic effect. More temperature probe will alo be helpful in determining thermal gradient within the bearing.

99 85 The bearing it elf can be improved with better feed line, poibly replacing all the ilicone tube with teel tube. Improvement in the tet rig can include a pneumatic loading ytem. A pneumatic ytem with an electronic regulator will provide an accurate loading mechanim a compared to the preent ytem which utilize dead weight. The data acquiition ytem can alo be upgraded to a fater data logger.

100 86 REFERENCES [1] Agrawal, G. L., 1997, "Foil Air/Ga Bearing Technology - an Overview," in Proceeding of the International Ga Turbine & Aeroengine Congre & Expoition, Orlando, FL, ASME paper 97-GT-347. [2] Lemmen, J. J. M., Gaunie, N. V. N., Overdiep, J. J., Bo, K. H., and Bartholomeu, P. M. G., 2006, "Demontration of Captone Microturbine Including High Efficiency Heat Exchanger, Ga Safeguard Module and Natural Ga Compreor, Developed by Gaunie Engineering & Technology," 23rd World Ga Conference. [3] Mohawk, 2008(Feburary 15), "Miti - Foil Bearing Example Application - Compreor - Ga Turbine - Turbocharger - Motor," [4] Kim, D., and Park, S., 2006, "Hybrid Air Foil Bearing with External Preurization," in Proceeding of the International Mechanical Engineering Congre and Expoition, Chicago, IL. [5] Hehmat, H., Walowit, J. A., and Pinku, O., 1983, "Analyi of Ga-Lubricated Foil Journal Bearing," Journal of Lubrication Technology, 105, pp [6] Hehmat, H., and Ku, C.-P. R., 1992, "Compliant Foil Bearing Structural Stiffne Analyi: Part I Theoretical Model Including Strip and Variable Bump Foil Geometry," ASME Journal of Tribology, 114(2), pp

101 87 [7] Ku, C.-P. R., and Hehmat, H., 1992, "Compliant Foil Bearing Structural Stiffne Analyi Part II: Experimental Invetigation," ASME Journal of Tribology, 115(3), pp [8] Peng, J.-P., and Carpino, M., 1992, "Calculation to Stiffne and Damping Coefficient for Elatically Supported Ga Foil Bearing," ASME Journal of Tribology, 115, pp [9] Carpino, M., 1997, "Finite Element Approach to the Prediction of Foil Bearing Rotor Dynamic Coefficient," ASME Journal of Tribology, 119, pp [10] Han, D.-C., Park, S.-S., Kim, W.-J., and Kim, J.-W., 1994, "A Study on the Characteritic of Externally Preurized Air Bearing," Journal of American Society of Preciion Engineering, 16, pp [11] Dellacorte, C., and Valco, M. J., 2001, "Load Capacity Etimation of Foil Air Journal Bearing for Oil-Free Turbomachinery Application," STLE Tribology Tranaction, 43(4), pp [12] Radil, K., Howard, S., and Dyka, B., 2002, "The Role of Radial Clearance on the Performance of Foil Air Bearing," STLE Tribology Tranaction, 45(4), pp [13] Wilde, D. A., and Andre, L. S., 2003, "Comparion of Rotordynamic Analyi Prediction with the Tet Repone of Simple Ga Hybrid Bearing for Oil Free Turbomachinery," ASME Journal of Engineering for Ga Turbine and Power, 128, pp

102 88 [14] Peng, Z.-C., and Khonari, M. M., 2006, "A Thermohydrodynamic Analyi of Foil Journal Bearing," ASME Journal of Tribology, 128, pp [15] Song, J.-H., and Kim, D., 2007, "Foil Ga Bearing with Compreion Spring: Analye and Experiment," ASME Journal of Tribology, 129(3), pp [16] Kim, D., 2007, "Parametric Studie on Static and Dynamic Performance of Air Foil Bearing with Different Top Foil Geometrie and Bump Stiffne Ditribution," ASME Journal of Tribology, 129(2), pp [17] Iordanoff, I., 1999, "Analyi of an Aerodynamic Compliant Foil Thrut Bearing: Method of a Rapid Deign," ASME Journal of Tribology, 121, pp [18] Rubio, D., and André, L. S., 2005, "Structural Stiffne, Dry-Friction Coefficient and Equivalent Vicou Damping in a Bump-Type Foil Ga Bearing," ASME Journal of Engineering for Ga Turbine and Power, 129(2), pp [19] Ku, C. P., and Hehmat, H., 1994, "Structural Stiffne and Coulomb Damping in Compliant Foil Journal Bearing: Parametric Studie," STLE Tribology Tranaction, 37, pp [20] Salehi, M., Hehmat, H., and Walton, J., 2003, "On the Frictional Damping Characteritic of Compliant Bump Foil," ASME Journal of Tribology, 125, pp [21] GMN, 2007(March 15), "Bearing Catalog,"

103 89 [22] e+a, 2007(March 15), "2 Pole Type Overview," [23] INSTA, 2008(May 15), "Inta Control - Ptc Thermitor," [24] Newport, 2008(May 15), "High-Performance Low-Profile Ball Bearing Linear Stage," Serie-High-Performance-Low-Profile-B.pdf.

104 90 APPENDIX A NON-DIMENSIONALIZATION OF REYNOLDS EQUATION Reynold Equation for Hydrotatic Bearing i given by, ee ection 3.2 for derivation RgTm 1 3 U +. ph p = ( ph) + ph A 12µ 2 x t ( ) (20) In the above equation x i a local coordinate attached on the bearing urface along the circumferential direction, z i a coordinate in axial direction, h i a film thickne, p i preure, µ i vicoity of air, R g i the ga contant of air and T i the temperature of upplied air. In the above equation ṁ i the ma flow rate from the hydrotatic upply line and for ientropic procee under choked and unchoked condition i cae i given by Choked cae, P P k ( k 1) 2 > = k+ 1 1/2 1/( k 1) p k 2 m = ρ AoV 0 = Ao 2γ g (21) R k 1 k 1 gt + + 1/2 Unchoked cae, Let k ( k 1) P 2 < = P k+ 1 2/ k ( k+ 1)/ k p k P P m = ρ AoV 0 = A o 2γ g (22) R k 1 P gt P 1/2

105 91 p x z h P=, θ =, Z =, τ = ωt, H = (23) P R R C a Subtituting we get R Tm g A U +. PPa H C PP a = ( PPa HC) + ω PPa HC 12µ 2R θ τ ( ) (24) Expanding we get Now µ 2 RgTm12 R 3 P 3 P + PH + PH 2 3 APa C Z θ θ θ 2 6Uµ R 12ωµ R = ( PH ) Pa C θ Pa C τ ( PH) 2 2 6ωµ R 12ωµ R U = ωr, Λ=, σ, A x z θ ZR 2 = 2 = = Pa C Pa C 2 (25) (26) Where U, Λ, σ are rotor urface velocity, bearing number and queeze number repectively. Subtituting the above expreion in equation (25) we get 12µ RgTm P P + + =Λ + P C θ Z θ θ Z θ θ τ 2 3 a ( ) σ ( ) 3 3 PH PH PH PH (27) Let. M = µ R Tm 12 g P C 2 3 a and ince σ =2Λ we get. M 3 P 3 P PH PH + + =Λ PH + Λ PH ( ) 2 ( ) θ Z θ θ Z θ θ τ (28) The above equation i non-dimenional form of Reynold equation for hydrotatic bearing. Again uing equation (21) and (22) we get the following expreion for. M. Choked,

106 ( k 1) k 2 M S =ΓP H 2γ g (29) k+ 1 k+ 1 Unchoked, 1 2 ( k+ 1) 2 k k k P P M S =ΓP H 2γ g (30) k 1 P P where µ 12 Cd A0 RgT Γ = i a feed parameter, 3 pac P i the upply preure, k i the ratio of pecific heat for air, Cd i a dicharge coefficient. In the feed parameter, A 0 i the reference orifice curtain area defined a A0 = π doc, where d 0 i the orifice diameter. Equation (28) can alo be expreed in vector form a hown below. M θ Z 2 Λ ( PH) =. Q τ J (31) where P P = Λ + θ Z 3 3 QJ PH PH iθ PH iz (32)

107 93 APPENDIX B ZEROTH AND FIRST ORDER EQUATIONS The haft i perturbed harmonically about the teady tate poition a hown below e = e + e e, e = e + e e (33) iωst iωs t X X 0 X Y Y 0 Y Where ω i the excitation frequency. Thee perturbation will alo produce mall harmonic deflection in the bump foil along with the teady tate deflection u= u + u e + u e (34) 0 X iωst iωs t Y The film thickne for the air foil bearing uing the coordinate ytem hown in Figure 2, page 11 i given by Subtituting value from value from (33) and (34) we get h= C+ e coθ+ e inθ + u (35) X y h= C+ e coθ+ e inθ+ e e coθ + e e inθ 0 iωs t iωst X 0 Y 0 X Y ω + u + u e + u e X ω i St i St Y (36) Rearranging we get h= h + h e + h e (37) 0 X iωs t iωs t Y Where h = e coθ+ u X X X h = e inθ+ u Y Y Y (38) In non-dimenional form the above equation i given by H = H + H e + H e, U = U + U e + U e iτ iτ iτ iτ 0 X Y 0 X Y H = ε co θ+ U, X X X H = ε inθ+ U Y Y Y (39)

108 94 The harmonic excitation will alo perturb the preure profile; the total non-dimenional preure i given by P= P + P= P + ε P e + ε P e = P + Σ ε P e (40) i τ i τ i τ 0 0 X X Y Y 0 α α α= X, Y Where P α P = ( α = X, Y ε α ). Note that PX and PY are complex number with real and imaginary part. Non-dimenional bump deflection equation i given by du = + ν (41) τ P KbU Cb d Subtituting value from equation (39) and (40) in the above equation we get P + ε P e + ε P e 0 iτ X X Y Y iτ iτ iτ iτ iτ ( 0 ) ν( ) = K U + U e + U e + C U ie + U ie b X Y b X Y (42) Uing relation from equation (39) and dropping the zeroth order term we get ε P + ε P X X Y Y ( ε coθ ε inθ) ( co in ) = K H + H b X X Y Y + Cνi H ε θ+ H ε θ b X X Y Y Separating the X and Y component we get ε P = K H ε coθ + Cνi H ε coθ Dividing by ( ) ( ) ( in ) ( in ) X X b X X b X X ε P = K H ε θ + Cνi H ε θ Y Y b Y Y b Y Y εα and uing H α ε α = H α ( α = X, Y ) we get ( coθ) ν ( coθ) ( inθ) ν ( inθ) P = K H + C i H X b X b X P = K H + C i H Y b Y b Y (43) (44) (45) Rearranging we get

109 95 PX PY + co θ = H X, + inθ = H K + Cνi K + Cνi b b b b Y (46) Introducing mall perturbation in equation (28) we get ( P P) 3 0+ ( P0 + P)( H0+ H) + θ θ 3 ( P0 + P) M ( H0+ H, P0 + P) ( P0 + P)( H0+ H) + = Z Z θ Z Λ ( P0 + P)( H0+ H) + 2Λ ( P0 + P)( H0+ H) θ τ (47) Expanding the above equation we get P P ( P0 P)( H0 H 3 H H0 3 HH 0) θ θ θ P0 P ( P0 + P)( H0 + H + 3 H H0+ 3 HH0) + + Z Z Z M ( H0+ H, P0 + P) = θ Z Λ Λ θ τ ( P H P H H P P H) 2 ν ( P H P H H P P H) (48) By further expanion and neglecting higher order term we get P0 P ( P0 H0 + 3P0 HH0 + PH0) + + θ θ θ P0 P ( P0 H0 + 3P0 HH0 + PH0) + + Z Z Z M ( H0+ H, P0 + P) = θ Z Λ Λ θ τ ( P H P H H P) 2 ν ( P H P H H P) (49) The perturbation in the ma flow term i done uing Taylor erie expanion. Let the ma flow be repreented by

110 k M S = 2 γ g ΓP f ( H, P) (50) k 1 Therefore 1/2 k M S( H+ H, P+ P) = 2 γ g Γ P f ( H + H, P+ P) (51) k 1 Uing Taylor erie expanion and neglecting higher order term we get f ( P, H ) f ( P, H ) f ( H0+ H, P0 + P) = f ( H0, P0 ) + P+ H P H P0, H0 P0, H0 (52) The function f and it derivate for the choked and unchoked condition i given by Choked, f ( H, P ) = H (53) f ( P, H ) P P0, H0 = 0 (54) f ( P, H ) H P0, H0 = 1 (55) Un-Choked 1 2 ( k+ 1) 2 k k 0 P0 P f ( H0, P0 ) = H 0 P P (56) 1 2 k k+ 1 2 k k k 1 1 k P 0 P 0 k k k = H0 P0 P0 P0, H 2 0 f ( P, H ) P P P k P k P (57)

111 k+ 1 2 k k 0 P0 f ( P, H ) P = H P0, H P 0 P (58) Uing equation (52) equation (49) become P0 P ( P0 H0 + 3P0 HH0 + PH0) + + θ θ θ P0 P ( P0 H0 + 3P0 HH0 + PH0) + + Z Z Z 1/2 ΓP k f ( P, H ) f ( P, H ) 2 γ f ( H, P ) + P+ H θ Z k P H g P0, H0 P0, H0 =Λ Λ + ( P0 H0 P0 H H P) 2 ν ( P0 H0 P ) 0 θ τ H + H P (59) Let the perturbation in preure and film thickne be repreented a iτ P= Σ ε P e, H = Σ ε H e (60) iτ α α α= X, Y α= X, Y α α Subtituting and rearranging uing the above perturbation expreion equation (59) can be written a The above will be true for any ( ) ε e iτ ( ) Zeroth Order term + Σ Firt Order term = 0 (61) α= X, Y α εα if and only if Zeroth Order term= 0 (62) and Firt Order term= 0 (63) The above condition lead to Zeroth and Firt order equation a hown below Zeroth order equation

112 P 0 3 P 0 Γ P k γ g 0 0 P H + P H + 2 f ( H, P ) θ θ z z θ Z k 1 = Λ ( P0 H0) + 2Λν ( P0 H0) θ τ (64) Firt order equation 2 3 P0 2 3 P0 ( 3P0 Hα H0 + Pα H0) + ( 3P0 Hα H0 + Pα H0) θ θ Z Z 2 3 Pα 2 3 Pα + ( 3P0 Hα H0 + Pα H0) + ( 3P0 Hα H0 + Pα H0) θ θ Z Z 3 Pα 3 P0 + P0 H0 + P0 H0 θ θ Z Z 1 2 ΓP k f ( P, H ) f ( P, H ) + 2γ g Pα + θ Z k 1 P P0, H H 0 =Λ ( P0 Hα + H0Pα ) + 2Λ ν ( P0 Hα + H0Pα ) θ τ P0, H0 H α (65) Uing relation between P α and H α from equation (46) in the above equation we get 3 Pα 3 Pα P0 H0 + P0 H0 θ θ Z Z k ΓP f ( P, H ) f ( P, H ) P a + Pα + + gα k 1 Z θ P P ( ) 0, H H 0 P0, H K 1 0 b + ηi 2 P a 3 P0 3H0 P0 + gα + H0 Pα θ Kb( 1+ ηi) θ 2 P a 3 P0 + 3H0 P0 + gα + H0 Pα Z Kb( 1+ ηi) Z P a P a =Λ P0 + gα + Pα H0 + 2Λ νip0 P0 + gα + Pα H0 θ Kb( 1+ ηi) Kb( 1+ ηi) (66) where α = X, Y and g = coθ, g = inθ. X Y

113 99 APPENDIX C PTC CHARACTERISTICS The characteritic of the poitive type thermitor are hown in Figure 54.

114 Figure 54: PTC characteritic, Source: Inta Control [23] 100

115 101 APPENDIX D CALIBRATION OF PROXIMITY PROBE Below curve how the calibration data of proximity probe intalled near the tet ection (ee Figure 34 on page 52) to meaure the haft bow. The x axi how the vernier reading and the y axi the correponding reading from the proximity probe. The pecification of the vernier cale i given in Table 14. Voltage (mv) y = x Vernier Reading (mm) Figure 55: Calibration curve of proximity probe Table 14: Vernier pecification, Source : Newport Corporation [24] Parameter Value Thread Pitch (mm) 0.5 Graduation (µm) 10 Vernier Graduation (µm) 1 Senitivity (µm) 1

116 102 APPENDIX E BENCHMARKING OF SOLUTIONS Thi ection decribe the benchmarking of the zeroth and firt order olution decribed in Section 3 with reult from Kim [16]. Kim [16] preent a parametric tudy involving hydrodynamic ga bearing where the compliant tructure i modeled by aigning average tiffne and damping value to individual computational grid point. Note that thi model differ from the model in the preent analyi ince tiffne and damping are only aigned to grid point that lie on bump. For comparion, ynchronou force coefficient were found under the limiting cae of zero feed parameter. Zero feed parameter repreent the cae of no external preurization and make the bearing hydrodynamic. Table 15 how the hydrodynamic bearing parameter lited in [16]. Reult from [16] and preent analyi are hown in Table 16 Table 15: Bearing parameter in [16] Parameter Bearing diameter, 2R Bearing axial length, L Value 38.1 mm 38.1 mm Nominal clearance, C 32 µm Bump tiffne per unit area 4.7 GN/m 3 Load on Bearing 30 N Structural Lo Factor 0.25

117 103 Table 16: Reult comparion Parameter Preent Cae Reference [16] % Variation e/c ~ 0.6 c xx (KN-/m) c yx (KN-/m) k xx (MN/m) k xy (MN/m) Note that Kim [16] pecifie a range of eccentricity and doe not include attitude angle. Comparion how that there i difference in the force coefficient value from the two analye, which may be attributed to the different modeling of bearing compliant tructure in the two analye. Alo, the preent analyi include agging effect of the top foil which i abent in [16].

118 104 APPENDIX F FABRICATION OF BUMP FOIL AND TOP FOIL Thi ection outline the procedure and intruction for the fabrication of top foil and bump foil. Alo included i the decription of jig ued for the fabrication of foil. Bump foil fabrication Bump foil form the compliant tructure in air foil bearing. Compliance from bump foil can help in accommodating mialignment, prevent high-preciion manufacturing and provide greater damping. Bump foil made and ued in the preent tudy are generation I bump foil. Thi mean that tiffne in the axial and circumferential direction are contant. For fabrication of bump trip a forming jig wa contructed in which the deired bump foil geometry wa machined on two halve of the jig die (ee Figure 56). Thi machining wa done uing wire EDM. The decription of the bump foil geometry i hown in Figure 57. The geometry wa provided by Foter Miller Corporation, a leading foil bearing manufacturer. The bump foil fabrication procedure i a follow: 1. Cut a 4 mil (0.004 ) thick tainle teel heet of ize 1.5 X Place the heet between the mating urface of the forming jig and align the two halve uing the dowel pin (ee Figure 56). Note that it i important to align the heet properly in jig (heet and jig hould have parallel edge). 3. Compre the two halve of the forming jig uing a hydraulic pre. Load the pre to a maximum force of 15 ton and bolt the jig.

119 Unload the hydraulic pre and place the forming jig in a furnace for heat treatment. Heat the forming jig to a temperature of 500 C for four hour. 5. Switch off the furnace but keep the forming jig inide until the furnace temperature reache room temperature. 6. Remove the jig from the furnace and place it under the hydraulic pre again. Load the pre to 15 ton force. 7. Remove the bolt and dowel pin and unload the pre to get the bump foil. 8. Retain 26 bump on the bump trip and cut away the ret. Clearance Hole for Bolt Hole for Dowel Pin Figure 56: Two halve of the forming jig

120 R Figure 57: Bump foil geometry Once the bump foil i formed a econd heat treatment i done to make the foil circular in hape. The bump foil i wrapped around a mandrel, placed inide a forming jig and heat treated to 400 C. See Figure 58 for detail on the mandrel and forming jig. Note, the above fabrication proce outline the contruction of a ingle trip bump foil. A bump foil may have 3-4 trip that are attached at one end. Thee type of bump foil provide enhanced axial compliance. To make uch a bump foil one can tart with a tainle teel heet with appropriate trip cut.

121 107 Bump foil wrapped around the mandrel 1.5 diameter mandrel Two-piece forming jig with bore. Figure 58: Forming jig with mandrel and bump foil Top foil fabrication Top foil make the mooth urface of the compliant tructure and i placed over the bump foil (ee Figure 1 on page 2). Top foil i made uing a 4 mil thick tainle teel heet of ize 1.5 X 5. The contruction of top foil i imilar to the heat treatment proce to make the bump foil circular. Here again the teel heet i wrapped around the mandrel and placed in the forming jig. One of the edge of the teel heet i bent and

122 108 i inerted in the gap between the two halve of the forming jig (ee Figure 59). The bent part i trimmed after the heat treatment and it goe inide a groove in the bearing leeve. The bent part lock in place the top-foil and the underlying bump foil circumferentially in the bearing leeve, ee Figure 2 on page 11 for detail. Bent part of the top foil inerted between the gap Top foil wrapped around the mandrel 1.5 Diameter mandrel Two-piece Forming jig with bore. Figure 59: Forming jig with mandrel and top foil