Dynamic characteristics of ultra-precision aerostatic bearings

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Adv.Manuf.(2013)1：82-86DOI 10.1O07/s40436-013-0013-6Dynamic characteristics of ultra·precision aerostatic bearingsXue-Dong Chen·Jin-Cheng ZhuHan ChenReceived：3 October 2012/Accepted：12 November 2012/Published online：13 M arch 2013### Shanghai University and Springer-Verlag Berlin Heidelberg 2013Abstract With high acceleration and ultra-precisionrequirements，the design of aerostatic bearings has beengradually focused on their dynamic performances.In thispaper，the dynamic stiffness and damping coefficients ofaerostatic bearings are investigated.Due to compressibilityof the gas，the dynamic characteristics of aerostatic bear-ings show nonlinear frequency dependence.Particularly，their nonlinear dvnamic behaviors are quite remarkable forultra.precision aerostatic bearings with smal1 air gapheights and high supply pressure。

Keywords Aerostatic bearing·Dynamic characteristicsDynamic mesh1 IntroductionRecently，aerostatic bearingsultra-precision machine toolsare increasingly utilized inand semiconductor manu-facturing equipments.W ith increasing requirements suchas higher acceleration and smaler vibration，traditionalstatic characteristics，such as load carrying capacity andstatic stiffness，are no longer sufficient in the design ofultra-precision aerostatic bearings.Instead，dynamic per-formances become increasingly critical and should beconsidered in the design of ultra-precision aerostaticbearings。

Due to squeeze film effects，aerostatic bearings showcomplex dynamic behaviors when they are subjected toX.-D.Chen( 1.J.-C.Zhu ·H.ChenState Key Laboratory of Digital Manufacturing Equipment andTechnology，Huazhong University of Science and Technology，W uhan 430074，Peoples Republic of Chinae-mail：chenxd###mail.hust.edu.cn垒Springerdynamic impact loads.The dynamic instability of the airbearing，e.g.，self-excited vibration (pneumatic hammer”)，was considered in early studies1].For high speedmoving applications，some new contiguration designs wereproposed to improve dynamic stiffness and dampingcoemcients of aerostatic bearings2-41.There were alsoqualitative analyses of the compressible thin air films in gaslubricated components.which indicated the dependence oftheir dynamic characteristics on the excitation frequency[5]。

Generally，analytical solutions of the dynamic pressuredistribution in aerostatic bearings，especially those withcomplex geometrical configurations，are impossible toobtain.Numerical techniques，such as finite diferencemethod(FDM)[4and finite element method(FEM)[6，have been adopted in the study of dynamic characteristicsof aerostatic bearings.In this Paper，a dynamic meshmodeling technique using commercial CFD soflware isproposed to investigate dynamic characteristics of multi-restrictor aerostatic bearings with shallow recesses.Thefocus of this work is on the influences of excitation mag-nitude and frequency on the dynamic stiffness and dampingcoe珩cients of aerostatic bearings with diferent air gapheights and supply pressure values。

2 Aerostatic bearing and theoretical modelA circular pad aerostatic thrust bearing is studied，as shownin Fig.1. The bearing pad has an outer diameterd2 77 mm and four supply orifce restrictors equallyspaced along a circumference of 50 mm diameter.A11 re-strictors are identica1.and each of them has a shallowrecess used to improve static perform ances of the bearing。

The orifice diameter is 0.3 mm.The diameter and theDynamic characteristics of aerostatic bearings 83Fig.1 Sketch of the aerostatic bearingdepth of each cylindrica1 recess are d1 5 mm andD 0.1 5 mm.respectively.The small air gap height onthe order of 10 gm is considered，so that high enoughstiffness of the bearing can be achieved in ultra precisionapplications。

Pressure distilbution of the air flowing in the bearinggap can be determined from the Reynolds equation asf01ows/，z.3 p)昌(p 考)12#q5 12 昙( ，(1)where P is the air pressure in the gap，P the air density，dynamic viscosity of the air.q the mass flow rate，andKronecker delta(at orifce inlet， 1；otherwise， ： 0)。

When the air gap height is excited with a small distur-bance as hhoAhh0hi (、/-1， l《 ho)，the resulting pressure in the air film is also disturbed asPpO△pp0plej due to the well-knownsqueezefilm effects”.Based on the perturbation theory。the per-turbed form of f11 can be writen as2(岳( 岳 。△Ap )昌( 昌P0 )))24 癌 3(岳( 荽) h。2 8 p20" k) (2) j24#(h。AP Op。co1Comparing(2)with its incompressible counterpart(3)，itis apparent that the pressure and gap height variation isnonlinear due to the additional nonlinear term on the righthand side of(2)。

Furthermore，the solution of(2)can be represented as Re(co，po，h0)JI )where Re denotes the real part，and Im denotes the imaginarypart.Apparently，they are nonlinearly frequency-dependentand can be influenced by the initial conditions，Po and ho。

From (4)，the complex stiffness of the aerostatic bearingcan be calculated as re/co，p0，h0)dAj-/Im(co，po，h0)dAwhere the real part denotes the stiffness coeficient K，while the imaginary part is the product of the dampingcoeficient C an d excitation frequency co，i.e。

Re/co，p0 ho)dA， c Im(o，po，ho)logdAAccordingly，with the effects of air compressibility，thedynamic coefficients K and C of aerostatic bearings arealso nonlinearly frequency-dependent.Moreover，they willvary with different air gap heights and supply pressures。

3 Numerical simulation and validationThe commercial CFD software ANSYS-Fluent is adoptedto numerically study the dynamic characteristics Of theaerostatic beating.According to symmetry of the geometryand boundary conditions.only 1/8 of the 3D air flow field inthe aerostatic bearing is considered，and the computationalmeshes are shown in Fig.2.There are totally 135.5 10hexahedral control volumes in the computational domain。

Pressure inlet and pressure outlet boundary conditions arespecified at the orifice inlet and the bearing periphery，respectively.The two radial boundaries are symmetric，andadiabatic boundary conditions are used on the solid walls。

Since the Knudsen number based on the air film height is nomore than 0.0 1，slip effects can be ignored[7]and no-slipboundary conditions are used on the solid walls.Thedynamic mesh modeling approach is employed to solve forthe time.dependent pressure field in the aerostatic bearing。

In this study，the botom wal1 of the air gap is specified as asinusoidally moving boundary (displacement excitation)，and the adjacent control volumes can deform accordingly。

Airfilm l 0惫。 O，rifce№ ess叠00-0- 0誓§lI l-00000 00誊j0≮1。

Fig.2 Computational model of the aerostatic bearing鱼Springer盟劬∞ 。 告、/ (苦旦舐 0 / 3旦 -/， 、X.-D.Chen et alFig.3 Test setup琵弓趸∽ Airgapheighth/I.tmFig.4 Static load varies with air gap heightIn order to validate the CFD model of the aerostatic bear-ing，a static load experiment was performed，and the test setupis displayed in Fig.3.Figure 4 shows the variation ofthe staticload with the air gap height，in which it can be seen that thesimulation results agree with the experimental data very wel1。

4 Results and discussionFrom the theoretical analysis，the pressure variation in theaerostatic bearing is related to the excitation frequency ofthe air gap height，and so is the dynamic load.Figure 5plots the numerically computed variations of the bearingload in ten periods under three different excitations.whoseamplitudes are identica1 while frequencies are 10.100 and1，000 Hz.From Fig.5，it can be seen that the amplitude ofthe dynamic load increases significantly with increasingexcitation frequency，and the phase of the response curvealso shifts with the excitation frequency.This conclusion isalso consistent with that in the squeeze film dampinganalysis。

Dynamic stiffness and damping coefficients of theaerostatic bearing are plotted in Fig.6 as functions of theexcitation frequency，where the air gap height is h equal to1 0 gm and the air supply pressure is Ps 0.4 M Pa. 垒SpringerDimensionless timeFig.5 Variations of dynamic load with different frequenciesExcitation fequencyflI-IzFig.6 Dynamic coeficients versus frequency with different excitation amplitudes(h l0 lam，尸 0.4 MPa)Frequency dependence of the dynamic coefficients areclearly observed，ie with the increasing of excitationfrequency the dynamic stiffness K increases while thedamping coefficient C decreases.and this dependence isespecially stronger from 100 to 1.000 Hz.In addition.theexcitation amplitude has little influence on the dynamiccoefficients.Due to compressibility effects as mentioned inSect.2，in particular，the relationship between dynamiccoefficients and excitation frequency is nonlinear.In thelower frequency range(below 10 Hz1，the dynamic stiff-ness is asymptotically equal to the static stiffness.Other。

wise，effects of dynamic squeeze film become dominantand the dynamic stiffness of the bearing increases withincreasing excitation frequency.This increase slows downuntil an asymptotic value is reached above about1 0，000 Hz，which is also due to air compressibility.Simi-larly，the damping is relatively larger at lower frequenciesand is negligible beyond 1.000 Hz。

The influence of the air gaD height on dynamic char-acteristics of the aerostatic bearing is also quantitativelyinvestigated by numerical calculations.Three air gapheights 1 0，20 and 30 Um are considered。and the ratioZ 、勺 0-0-謇 .摹I.吕.s. 苦0l000 -口g∞Q-T-g ∞∞Q 甚∞Q-吕日uAQDynamic characteristics of aerostatic bearings 85Excitation frequencyflHz ExcitationequencyflHzFig·7 Dynamic stiffness with various air gap heights(Ps0-MPa) Fig.9 Dynamic stifness with various supply pressures( 1 0 um)Excitation frequencyflHz Excitation frequencyflHzFig.8 Damping with various air gap heights(Ps：0.4 MPa) Fig·10 Dam ping with various supplY pressures( 1 0 m)h1/h is kept as a constant in each case.W ith different airgap heights，dynamic stiffness and damping coefi cients ofthe bearing are plotted in Figs.7 and 8，respectively.FromFig.7，the dynamic stiffness reaches a maximum value at acertain air gap height in the low frequency range，whichcorresponds to the static stiffness of the bearing，while itbecomes larger at smaller air gap heights in the high fre-quency range.The dynamic stiffness also tends to beindependent of the excitation frequency at very smal gapheights.From Fig.8，the damping coeficient at a smalexcitation frequency(below 5 Hz)is larger at a 20 m gapheight than that at the other heights，and it declines faster toalmost zero at larger air gap heights。

The influence of the air supply pressure is illustrated inFigs.9 and 1 0.With the increase of supply pressure。

dynamic stiffness and damping coefi cients are increasingin the whole excitation frequency range.Moreover.varia-tion of these dynamic coefncients with the excitation fre-quency is larger at higher supply pressure values。

5 ConclusionsThis paper numericaly investigates dynamic stiffness anddamping characteristics of an ultra-precision aerostaticbearing under small air gap height perturbations.Nonlinearfrequency dependence of these dynamic characteristics isrevealed by a theoretical analysis，which is due to com-pressibility of the air.Furthermore。the numerical resultsdemonstrate their behaviors.ie the dynamic stiffnessincreases while the damping decreases with increasingexcitation frequency.At small bearing gap heights and highsupply pressure，stronger frequency-dependent behaviorsof dynamic characteristics of the bearing are observed。

From these results，it is further confirmed that the frequency-dependent dynamic characteristics should be takeninto account in the design of ultra-precision aerostaticbearings。

Acknowledgments This study was supported by the National BasicResearch and Development Program of China (Grant No。

2009CB724205)and the National Natural Science Foundation ofChina(Grant Nos.51121002，51175196)。

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