Special Issue Article

Some fundamental aspects ofself-levitating sliding contact bearingsand their practical implementations

MA Atherton, C Mares and TA Stolarski

Abstract

In this study, fundamental aspects and mechanisms of acoustic levitation together with governing equations are presented

first. Then, the acoustic levitation phenomenon is considered as a new way to design air suspension systems capable of

self-levitation. A particular emphasis is laid on journal bearings and their specific geometrical configuration. A practical

feasibility of using acoustic levitation to separate contacting surfaces is supported and illustrated by results of experi-

mental testing of a number of prototype devices.

Keywords

Self-levitation, squeeze-film action, journal bearing, linear bearing, experimental validation

Date received: 10 July 2013; accepted: 26 November 2013

Introduction

In non-contact bearings, there is no direct physicalcontact between interacting surfaces. Consequently,these bearings have no wear, almost no friction(except for fluid drag) and they can achieve higheraccuracies than conventional rolling-element bear-ings. Fluid film bearings are the most popular non-contact bearings, whereby a thin film of fluid (liquidor gas) is used to separate the two surfaces. The load-carrying capacity is derived from the pressure withinthe lubricating film, which can be generated by themotion of the bearing surfaces (self-acting or hydro-dynamic/aerodynamic bearings) or by external pres-surisation (hydrostatic, or aerostatic) or squeezemotion (squeeze film acoustic bearing, the mainfocus of this paper), or by a combination of theseactions. In addition to fluid film bearings, magneticbearings are another type of non-contact bearing,which support a load using magnetic levitationforce, without the need for a lubricant or separatingmedium.

Besides existing types of non-contact bearings,acoustic levitation has been used more generally as ameans of non-contact suspension of particles. Anacoustic wave can exert a force on objects immersedin the wave field and these forces are normally weakbut they can become quite large when using high fre-quency (ultrasonic) and high intensity waves. Whenthese forces are large enough to suspend substances

against gravity force then this phenomenon is calledacoustic levitation, and since the sound waves usedare often in the ultrasonic frequency range (higherthan 20 kHz), it is more often called ultrasoniclevitation.

Ultrasonic levitation was first used for levitatingsmall particles by creating a standing wave fieldbetween a sound radiator and a reflector, namelystanding wave ultrasonic levitation. Standing wavetype ultrasonic levitators with various features weredesigned for applications in different scientific discip-lines such as container-less material processing andspace engineering.1 Another well-known type of ultra-sonic levitation is squeeze film ultrasonic levitation,typically where a flat surface is brought to a confor-mal radiation surface which vibrates at high fre-quency. The basic theory underlying the operatingprinciple of ultrasonic levitation together with exam-ples of design embodiments of sliding contacts utilis-ing squeeze-film ultrasonic levitation are presented inthis paper.

Department of Mechanical Engineering, School of Engineering and

Design, Brunel University, Uxbridge, UK

Corresponding author:

TA Stolarski, Department of Mechanical Engineering, School of

Engineering and Design, Brunel University, Kingston Lane, Uxbridge,

Middlesex UB8 3PH, UK.

Email: mesttas@brunel.ac.uk

Proc IMechE Part J:

J Engineering Tribology

2014, Vol. 228(9) 916–927

! IMechE 2014

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DOI: 10.1177/1350650113517110

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Fundamentals of ultrasonic levitation

Standing wave levitation

The standing wave levitation phenomenon was firstobserved in Kundt’s tube experiment,2 in whichsmall dust particles moved towards the pressurenodes of the standing wave created in a horizontal(Kundt’s) tube. Multiple reflections between an ultra-sonic radiator and a solid, flat or concave reflector,generate a standing wave with equally spaced nodesand anti-nodes of the sound pressure and velocityamplitude. Solid or liquid samples with effective diam-eters less than a wavelength can be levitated belowthese pressure nodes and so the axial suspension ofthe sample is an effect of the sound radiation pressurein a standing wave. In addition, when combined witha Bernoulli vacuum component, the sound wave canlocate the samples laterally as well.3

The first detailed theoretical description of stand-ing wave levitation was given by King,4 which wasextended by Hasegawa and Yosioka5 to include theeffects of compressibility. Embleton6 adopted King’sapproach to fit to the case of a rigid sphere ina progressive spherical or cylindrical wave field.Westervelt7–9 derived a general expression for theforce owing to radiation pressure acting on anobject of arbitrary shape and normal boundaryimpedance, and also showed that a boundary layerwith a high internal loss can lead to forces that areseveral orders of magnitude greater than those pre-dicted by the classical radiation pressure theory.

Gorkhov10 presented a very different approach toKing, using a simple method to determine the forcesacting on a particle in an arbitrary acoustic field, inwhich the velocity potential was represented as sumof an incident and a scattered term. Barmatz11

applied Gorkhov’s method to derive the generalisedpotential and force expressions for arbitrary stand-ing wave modes in rectangular cylindrical andspherical geometries. Lierke12 gave an overview ofstanding wave acoustic levitation based on long-term research and development activities of theEuropean and the US space agencies. Xie andWei13 studied the acoustic levitation force on disksamples and the dynamics of large water drops in aplanar standing wave, by solving the acoustic scat-tering problem through incorporating the boundaryelement method.

The theoretical approach of King4 is presentedbelow for understanding the basic working principlesof standing wave levitation. Assuming that the work-ing fluid is adiabatic and barotropic, the equations ofmotion can be written as

�Du

Dt¼ �

@p

@x, �

Dv

Dt¼ �

@p

@y, �

Dw

Dt¼ �

@p

@zð1Þ

where DDt ¼

@@tþ u @

@xþ v @@yþ w @

@z is the absolutederivative, � the density of the medium, p the

pressure and (u, v,w) the Cartesian velocitycomponents.

By defining $ ¼ dp�� the equation of motion can

be rewritten as

Du

Dt¼ �

@$

@x,Dv

Dt¼ �

@$

@y,Dw

Dt¼ �

@$

@zð2Þ

When the motion is non-rotational, the velocity com-ponents can be expressed in terms of the velocitypotential �

ðu, v,wÞ ¼ �r� ð3Þ

In the case where air is the medium, the velocitypotential can be obtained from the approximatelinear wave equation

r2� ¼1

c2@2�

@t2ð4Þ

which is simplified from the exact differential equationfor � using a second-order approximation. The equa-tion of continuity is

@�

@tþ@

@xð�uÞ þ

@

@yð�vÞ þ

@

@zð�wÞ ¼ 0 ð5Þ

and the pressure variation can then be derived fromequation (5) as

�p ¼ p� p0 ¼ p0@�

@tþ1

2

�0c2

@�

@t

� �2

�1

2�0q

2 ð6Þ

where �0 is the density of the surrounding medium, �is the velocity potential, c is the sound speed in air andq is the velocity amplitude equal to

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiu2 þ v2 þ w2p

.Detailed derivation of equation (6) can be found inKing.4 The time-averaged acoustic pressure on a rigidbody can be calculated by integrating the acousticpressures acting on each surface element of thebody. In the case of a plane standing wave, the vel-ocity potential �s can be expressed as4

�s ¼ Aj j cos kh cos!t ð7Þ

where jAj is the amplitude of the velocity potential,k¼ 2p/� is the wave number and h is the positionwhere the sphere, which is assumed to be small, islocated. The acoustic radiation force on a rigidsphere can be calculated as

F ¼ �5

6��0 Aj j

2ðkRsÞ3 sinð2khÞ ð8Þ

with Rs being the radius of the sphere.Bucks and Muller14 presented the first experimen-

tal setup for positioning of small samples in acousticstanding waves, in which a small particle was trapped

Atherton et al. 917

at a position slightly below the pressure nodes of thestanding wave between a radiator and a reflector.Wang et al.15 presented an acoustic chamber for pos-itioning of molten materials in an extreme tempera-ture gradient. Whymark16 proposed an acousticlevitator for positioning of materials in space usinga single source of sound that achieved fine positioncontrol by adjusting the reflector. Lierke17 presentedan acoustic levitator for positioning materials samplesin mirror furnaces when processing in space. Trinh18

presented a compact acoustic levitation device formicrogravity studies of fluid dynamics and materialsscience. This classic structure was later modified toachieve better performance. Otsuka et al.19 used astepped circular vibrating plate as a radiator for pro-ducing high-intensity ultrasound fields which was dif-ferent from conventional piston-like vibration sourcesin using the flexural vibration mode of the plate withtwo nodal rings to achieve higher vibration amplitude.The stepped plate has a concave channel with fixeddepth equal to the half sound wavelength in air. Thisspecial design makes the concave and convex blocksvibrate in the counter phase so that the ultrasoundpropagating in the air is modulated in the samephase. As a result, a narrow, intensive, high direc-tional ultrasound beam is obtained. Xie and Wei20

enhanced the standing wave acoustic levitation forceby properly curving the surface and enlarging thereflector and thus high density material, e.g. tungsten,was successfully levitated for the first time usingstanding wave ultrasonic levitation. Recently Xieand Wei21 reported the successful levitation of smallliving animals such as an ant, ladybird and small fishwith a standing wave acoustic levitator and theirexperiments showed that the vitality of the small ani-mals was not impaired during levitation.

All the standing wave levitation systems presentedabove possessed the classic radiator-reflector config-uration and so applications of such configurationwere limited to the levitation of small particleswhose dimension does not exceed the wavelength ofthe imposed sound wave. Moreover, the levitationforce obtainable from this configuration is very lim-ited and therefore modifications and improvementsare needed before standing wave ultrasonic levitationcan be applied in non-contact suspension systemssuch as linear and rotational bearings.

Squeeze-film ultrasonic levitation

Salbu22 reported a levitation system for objects withflat surfaces that used magnetic actuators to excitetwo conforming surfaces oscillating next to eachother to generate a positive load-supporting force.In 1975, Whymark16 reported that a brass planardisk of 50mm in diameter and 0.5mm in thicknesswas levitated extremely close to a piston vibrationsource driven harmonically at a frequency of20 kHz. These levitation effects reported by Salbu

and Whymark were named as squeeze-film levitationand also called near-field acoustic levitation.

A schematic diagram of squeeze-film levitationsystem usually consists of two parallel plates sepa-rated by an air film of thickness h0. The time-averagedmean pressure in the gap has a value which is higherthan the surroundings, which is caused by the second-order effects possessed by the rapidly squeezed andreleased gas film between two plane surfaces. Twodistinct properties distinguish this type of levitationfrom standing-wave levitation. First, the reflector isno longer needed; instead, the levitated object itselfacts as an obstacle for the free propagation of theultrasonic wave-front. Second, the gap between radi-ation source and the levitated object must be muchsmaller than the sound wavelength in air. Thus,instead of a standing wave, a thin gas film is formedbetween the radiator and the levitated object, which israpidly squeezed and released. A simple model intro-duced by Wiesendanger23 is presented here to demon-strate the basic idea of how squeeze-film levitationworks. The leaking and pumping at the boundary isneglected in this model, thus only the trapped gas thatis rapidly squeezed and released is considered. Thetotal mass of air in a fixed volume remains constant,resulting in

pVn � phn ¼ const ð9Þ

where p represents the pressure, V the volume of thetrapped gas, h the gap distance for a one-dimensionalcontact and n the polytropic constant (n¼ 1 for iso-thermal condition, n¼ k& 1.4 for an adiabatic condi-tion of air). The relation between pressure andlevitation distance is nonlinear, which leads to a dis-torted pressure pðtÞ resulting from the imposed peri-odic gap distance h(t). The gap distance (air filmthickness) oscillates harmonically around a equilib-rium position, h0, i.e.

hðtÞ ¼ h0ð1þ " sin!tÞ ð10Þ

in which ! is the angular frequency of the oscillation," the excursion ratio ð" ¼ a0=h0Þ. The excursion ratiodenotes the ratio of the vibration amplitude over themean gap distance, where a0 is the vibration displace-ment amplitude. The mean pressure under isothermalconditions (n¼ 1) can be expressed as in23

�p ¼p0h02�

Z2�

0

1

hðtÞdð!tÞ ¼

p0ffiffiffiffiffiffiffiffiffiffiffiffiffi1� "2p ð11Þ

It can be easily seen that mean pressure �p exceeds theambient pressure, p0. The harmonic motion of theradiating surface produces a non-harmonic pressure

918 Proc IMechE Part J: J Engineering Tribology 228(9)

oscillation whose mean value is not equal to the quasi-static value, p0. The positive mean pressure �p is largerthan the ambient pressure, p0, which is a qualitativeconfirmation of the existence of a levitation pressure.However, to obtain a quantitative value of the levita-tion pressure, more sophisticated models are neededwhich take into account the boundary conditions suchas the pressure release at the edge of the gap. Such amodel of squeeze-film levitation can be established byfollowing two different routes: acoustic radiationpressure theory and gas film lubrication theory. Thefirst one modifies the acoustic radiation pressuretheory according to the different physical conditionsin squeeze-film levitation. The second one starts fromthe theory of gas film lubrication since the workingprinciple is actually similar.24,25 Gas film lubricationhas been investigated for many years in micro-mechanical systems commonly by solving Reynoldsequation.

A simplified equation for the radiation pressure insqueeze-film acoustic levitation was derived from theacoustic radiation pressure theory presented by Chuand Apfel26 by Hashimoto et al.27 They calculated theRayleigh radiation pressure in an ideal gas on a per-fectly reflecting target as

p ¼ P� P0h i ¼1þ �

21þ

sinð2khÞ

2kh

� �Eh i ð12Þ

Here, E is the energy density which can be expressedas

E ¼a204

� ��0!

2

sin2ðkhÞ

� �ð13Þ

in which k represents the wave number, � a specificheat ratio, ! the angular velocity of the wave, a0 thevibration amplitude and h the distance between vibra-tion source and target. In squeeze film levitation, thelevitation distance is very small compared to thewavelength of sound in the free field. It ranges fromseveral to several tens micrometers, therefore in equa-tion (12) sin kh � kh was simplified to a linear equa-tion for the radiation pressure in squeeze filmlevitation

� ¼1þ �

4�0c

2 a20

h2ð14Þ

The radiation pressure, �, in squeeze film levitation isinversely proportional to the square of the levitationdistance and proportional to the square of the vibra-tion amplitude a0. Hashimoto et al. did experimentsto verify equation (14) and the results for maximumlevitation force were 25% lower than those calculatedfrom equation (14). The authors put this discrepancydown to the finite dimensions of the surfaces and thenon-uniformity of the amplitude of the radiationsurface.

Wiesendanger23 also used gas film lubricationtheory24,25 and solved the general Reynolds equationboth analytically and numerically to achieve quanti-tative results for the levitation forces. Nomura andKamakura28 theoretically and experimentally exam-ined squeeze-film acoustic levitation by numericallysolving the basic equations of a viscous fluid usingMacCormack’s finite-difference scheme and includingviscosity and acoustic energy leakage in the model.Minikes29 studied the levitation force induced by pres-sure radiation in gas squeeze films, investigating theflow induced both by vibrations perpendicular to aflat surface and by a flexural wave propagating paral-lel to the surface. For the first case, numerical andsecond-order analytical perturbation solutions werecompared and proved to be in good agreement toeach other. For the second case, a modifiedReynolds equation was derived to obtain the pressuredistribution and the velocity profile in the film inorder to determine the reaction forces. Later,Minikes30 examined the validity of the pressurerelease boundary condition and the isothermalassumptions through a CFD scheme. By comparinghis results to a one-dimensional analytical solution,Minikes found that the levitation force reduced to ahalf when the energy leakage near the edges of thelevitated object were taken in to account. This indi-cates that the assumption of pressure release at theboundaries, implied in the Reynolds equation, is inad-equate in cases where the driving surface is signifi-cantly larger than the levitated surface.

Squeeze-film ultrasonic levitationwith inertia effect

In depth analysis of the significance of inertia effect insqueeze-film pressure generation was carried out byStolarski and Chai,31 based on the Reynolds equationmodified to include an inertia effect. Therefore, inorder to avoid unnecessary repetition, only selectedequations are presented here as a comprehensivetreatment of the problem is given elsewhere.31 In thecase of a linear bearing configuration (discussed inmore details later), air film geometry can be repre-sented as in Figure 1. Cyclic movement e sinð!tÞ inthe vertical direction (z-axis) generates a characteristicair velocity given by w � e!, where w is flow velocityof air in z direction, e is the amplitude of vibration ofa squeeze surface produced by a piezoelectric actu-ator, and ! denotes angular velocity of the squeezesurface rotation. The order of magnitude of the aver-age flow rate Q due to squeeze action over all theboundaries can be determined, approximately, fromthe expression below (for derivation details seeStolarski and Chai31)

u �3Q

14loho¼

3wlob

14loho¼

3wb

14ho¼

3e!b

14hoð15Þ

Atherton et al. 919

Therefore,

w

u�

14e!ho3e!b

�hob� 5 0:001 ð16Þ

This value implies that the velocity in the vertical dir-ection (see Figure 1) is a small quantity of a higherorder. This inference is applied to the Navier–Stokesequation, given below in a simplified form by oneorder of magnitude

@p

@x¼@

@z�@u

@z

� �;@p

@y¼@

@z�@v

@z

� �;@p

@z¼ 0 ð17Þ

In order to find out that the inertia effect in the ver-tical direction (air film thickness direction) is negli-gible compared to that in x- and y-axes direction,modified Navier–Stokes equations with inertia termsgiven below are used

�Du

Dt¼ � u

@u

@xþ v

@u

@yþ w

@u

@zþ@u

@t

� �ð18Þ

�Dv

Dt¼ � u

@v

@xþ v

@v

@yþ w

@v

@zþ@v

@t

� �ð19Þ

�Dw

Dt¼ 0 ð20Þ

Therefore, equations given by (17) can be replaced by

�Du

Dtþ@p

@x¼@

@z�@u

@z

� �ð21Þ

�Dv

Dtþ@p

@y¼@

@z�@v

@z

� �ð22Þ

@p

@z¼ 0 ð23Þ

Terms � DuDt and �Dv

Dt relate to non-linear flow velocityin equations (21) and (22) and cannot be integrated to

determine velocities u and v. The thickness of an airfilm developed within the squeeze contact is small, soaveraging the inertia effect across the film thickness ispermissible. Thus

�Du

Dt� �

1

h

Zh

0

Du

Dtdz

0@

1A ð24Þ

�Dv

Dt� �

1

h

Zh

0

Dv

Dtdz

0@

1A ð25Þ

Combining equations (24) and (25) with equations(21) and (22) results in

�@2u

@z2¼ �

1

h

Zh

0

Du

Dtdz

0@

1Aþ @p

@xð26Þ

�@2v

@z2¼ �

1

h

Zh

0

Dv

Dtdz

0@

1Aþ @p

@yð27Þ

@p

@z¼ 0 ð28Þ

The use of the idea of ‘averaged inertia’ results in anumber of equations in which the inertia emerges as amodification term. Mass flow rate per unit length ofthe contact can be estimated from

qn ¼�h3�

12�

@p

@n

� �þ�h2�2

12�

Zh

0

DunDt

dz

0@

1A ¼ q0n þ q1n

ð29Þ

The first term, q0n, denotes contents of the first bracketand the second, q1n, symbolises the expression in thesecond bracket. According to equation (29) the flowrate consists of the flow caused by the pressure gradi-ent, q0n, and the flow due to inertia, q1n. The Reynoldsequation can be arrived at by integration of the con-tinuity equation across the film thickness

@qx@xþ@qy@yþ@ ð�hÞ

@t¼ 0 ð30Þ

where qx and qy are mass flow rates per unit length inCartesian co-ordinate system.

Selecting an infinitesimal area � in the xy planeenclosed by the boundary �, one can further integrateequation (30) to obtain

Z�

qn d�þ

Z Z�

@ ð�hÞ

@td� ¼ 0 ð31Þ

In the above equation, qn symbolises mass flow ratenormal to boundary �. According to equation (31)

y

o

h

x

0h l0

b

Figure 1. Diagram showing air film dimensions.

920 Proc IMechE Part J: J Engineering Tribology 228(9)

the mass flow rate through the boundary � is equal tothe mass reduction rate in the domain �.Involving the inertia effect contained in equation(29) with equation (31) leads to a boundary integralequation of the form

Z�

q0n þ q1n� �

d�þ

Z Z�

@ ð�hÞ

@td� ¼ 0 ð32Þ

Examples of design embodimentsof self-levitating sliding contacts

This paper is concerned with the important applica-tion of squeeze-film levitation to developing non-contact linear and rotational bearings. A number ofexamples of possible practical implementation of thefundamental principles of squeeze-film ultrasonic levi-tation are presented and briefly discussed in thissection.

Linear motion bearing

Precision linear motion is required in a number ofpractical applications. One example is lithographywhere silicon chip to be engraved with a circuit hasto be repositioned with nano-metre accuracy. Figure 2depicts a linear bearing concept consisting of a spe-cially shaped frame and a guiding beam of squarecross-section. The geometry of the frame has beenselected to have the maximum levitating effect by pro-ducing deformation of its three sides by means of‘elastic hinges’.

Figure 3, being a model for finite element (FE)analysis, shows the geometry of the frame in moredetail. When in a deformed state within its elasticlimits, the frame creates converging gaps with a flatand un-deformable guide, hence facilitating forma-tion of an air film/pocket. When constant and

periodically changeable deformations of the frameare generated at a suitable frequency (around 5kHz) a squeeze-film mechanism comes into operationand a pressure is created that is able to separate theframe from the guide. The load capacity of such abearing is not very impressive (a few Newtons),nevertheless for special applications such as contact-less paper transportation and lithography, the forceis sufficient.

Modelling of the linear motion bearing (floatingframe) is based on the Reynolds equation appropri-ately modified to account for the specific geometry(see Figure 4).

@

@XH3P

@P

@X

� þ@

@ZH3P

@P

@Z

� ¼ �

@ ðPHÞ

@Tð33Þ

Figure 2. Schematic of the linear bearing configuration.

PZT actuators

Elastic hinges

Figure 3. Geometry details of the frame.

Atherton et al. 921

where � is the squeeze number. Equation (33) can beexpanded to give

PH3 @2P

@X2þH3 @P

@X

� 2þ3PH2 @P

@X

@H

@X

þ PH3 @2P

@Z2þH3 @P

@Z

� 2þ3PH2 @P

@Z

@H

@Z

¼ �P@H

@Tþ �H

@P

@Tð34Þ

Assuming constant film thickness along the z-axis,@H@Z ¼ 0 (see Figure 4), part of the equation (34) canbe omitted and solved using known numericalalgorithms.

Floating pad

Converging gap geometry necessary for the squeeze-film mechanism can also be achieved by the use of acontraction effect usually represented by Poisson’sratio characteristic for a given material. The arrange-ment for practical implementation of the contraction

effect combined with the squeeze-film mechanism isshown in Figure 5.

An array of PZTs is fixed to the back side of thecontracting plate and under the action of PZTs theplate deforms as shown in Figure 6.

The dimple is a result of the plate’s contraction inthe vertical direction due to its elongation in the hori-zontal direction (Poisson’s effect). Cyclic fluctuationof the dimple with appropriate frequency creates

0=∂∂Y

P

x

z

y

0=∂∂

Z

H

Air film under top surface of bearing

Figure 4. Discretisation of the problem.

contrac�ng plate

suppor�ng structure

constraining screws

Figure 5. Computer-created image of the floating pad testing

set-up.

PZT

elas�c hinges

circular bore(undeformed shell)

three-lobe bore(deformed shell)

magnitude of converging gap

Figure 7. Diagram showing geometry, ‘elastic hinges’ and

arrangement for PZTs.

holes for constrainingscrews

dimple produced byPZTs action

Figure 6. FE image of the plate in deformed state.

922 Proc IMechE Part J: J Engineering Tribology 228(9)

conditions necessary for the squeeze-film operationand generation of pressure at the interface betweenthe plate and an object to be levitated.

Modelling of the floating pad for analysis purposestarts with determining the modes of the plate’sdeformation. Only those modes of deformation pro-ducing geometry favourable for the squeeze-filmmechanism should be selected. An easy way to carryout structural deformation analyses is to use finiteelement analysis.

Once the required geometry of deforming plate issecured, the squeeze-film effect equations can beemployed to determine pressure in the air film andhence load capacity of the floating pad.The effectiveness of squeeze-film ultrasonic levitationcan be assessed from

r½H3P � rP� ¼ �@ ðPHÞ

@Tð35Þ

In fact equation (35) is the Reynolds equation in non-dimensional variables, which are defined as follows

X ¼x

l0, Y ¼

y

l0, H ¼

h

h0, P ¼

p

pa, T ¼ !0t,

� ¼12!0l

20

pah20

Here H is the separation distance, P is the pressure, Tis the time, l0 is the lateral dimension of the pad, h0 isthe mean air film thickness, pa is the ambient air pres-sure, is the viscosity of air and � is the squeezenumber. The squeeze number is an important factor

in finding the load capacity of a bearing operatingwith squeeze-film ultrasonic levitation. Low s meansthat the air is simply flowing through the contactwithout undergoing compression and decompression.High s signifies a relatively stationary air filmundergoing cyclic compression and decompressionand hence contributing to the load capacity of thecontact.

Rotating motion bearing

Early attempts to design rotating motion bearingoperating on squeeze-film ultrasonic levitation prin-ciple utilised elaborate geometry for the bearingshell. An example is shown in Figure 7 and an FEmodel of the bearing shell with stress map for itsdeformed state is depicted in Figure 8.

Figure 8. FE model showing stress map of the bearing shell in deformed state. Three-lobe geometry can be clearly seen.

PZT with wiring

elastic hinges

Figure 9. Photograph of the bearing shell with ‘elastic hinges’.

One pound coin shown illustrates the physical size of the shell.

Atherton et al. 923

The photograph in Figure 9 shows an actual bear-ing made of bronze. It can be seen that there are anumber of small holes and cuts which, together, con-stitute ‘elastic hinges’ whose main function is to makethe bearing shell more elastic and deformable underthe action of the PZTs. It should also be noted thatthis bearing is an early design having quite thick wallsand operating at subsonic frequencies.

The latest design of bearing is simpler to make as itdoes not have such an elaborate array of holes andcuts. Figure 10 shows two possible configurations.

An FE model of the bearing of Design One (seeFigure 11) depicts the bearing’s shell in a deformedstate. Again, one can clearly see that the circular boreof the shell is transformed into a three-lobe shapeunder the action of the PZTs. In the case shown byFigure 11 the frequency of cyclic deformation was21.6 kHz.

Computer modelling of a rotating bearing operat-ing on the squeeze-film ultrasonic levitation principleis based on an appropriately modified Reynolds’equation given by equation (30). However, the verysmall thickness of the air film developed within the

bearing, warrants neglecting the inertia effect of thefilm across its thickness.31

Using geometry depicted in Figure 12, the air filmthickness can be represented by the followingexpression

h ¼ Cr þ xcg sin þ ycg cos þ cð, z, tÞ ð36Þ

The term, c(, z, t) on the right-hand side of equation(36) represents the contribution of cyclic deformationof the bearing’s shell to the formation of the airfilm. Equations (30) and (36) constitute fundamentalsof a rotating motion bearing operating on squeeze-film ultrasonic levitation. They can be solved numer-ically with the help of experimentally measuredelastic deformations of the bearing’s shell inducedby the PZTs.

anchoring bracket

PZT element

bearing shell

anchoring ridge

PZT element

wiring

bearing shell(a) (b)

Figure 10. Two examples of possible geometrical configurations of bearings operating on squeeze-film ultrasonic levitation principle:

(a) design One; (b) design Two.

xcg

ycg

θ

r+C

r+c

r

Piezo electric element

(per 120deg)

x

y

xcg

ycg

θ

r+C

r+c

r

Piezo electric element

(per 120deg)

x

y

Figure 12. Geometry of the bearing used to express air film

thickness analytically.

Figure 11. Finite element model of the shell in deformed

state.

924 Proc IMechE Part J: J Engineering Tribology 228(9)

Experimental confirmation of theconcepts

In order to demonstrate the feasibility of squeeze-filmultrasonic levitation preliminary experimental testingof design embodiments presented earlier was under-taken. Undoubtedly the most useful information, inview of potential practical applications, is the load-carrying capacity. Some selected results obtainedfrom preliminary experimental testing are presentedin this section.

Floating pad

Figure 13 shows pressure, synonymous with the load-carrying capacity, developed within the context of theconfiguration shown in Figure 5. The plate made ofaluminium was 200mm long, 100mm wide and had athickness of 1.9mm. The frequency used to excite theplate was 25.6 kHz. In this figure the separation dis-tance (amount of levitation of the object above thefixed and vibrating aluminium plate) is plottedversus the inverse of the root of the total mass ofthe levitating object. A clear linear relationship is pre-sent between the levitation distance and the inverseroot of the applied mass. This relationship appearsvery strong as the straight line fits rather well withall the data points, even with the points where theseparation distances are less than 5 mm.Additionally, experiments showed that the change ofplate material, plate vibration frequency, amplitudeand offset voltage results in different applied load

versus separation distance characteristic. Details con-cerning experimental techniques used to generateresults plotted in Figure 13 are given elsewhere.Details concerning experimental techniques used toplot results shown in Figure 13 are given inWoolliscroft.32

Linear motion bearing

Figure 14 illustrates the experimentally measuredload capacity of the linear motion bearing (expressedas air film thickness) with configuration and geom-etry as shown in Figure 3.34 A clear trend of filmthickness reduction with applied load is demon-strated. The expected input amplitude trend occurredwith larger input amplitudes sustaining a thicker airfilm under the same load. As with the floating padcontact, performance of the linear motion bearingdepends on the frequency of cyclic deformation,offset voltage and amplitude. Moreover, it wasfound during experimental testing that the bearingis quite sensitive to the surface finish of its squeeze-film generating areas. Details concerning experimen-tal techniques used to generate results plotted inFigure 14 can be gleaned from Woolliscroft33 andTaghipourfard.34

Rotating motion bearing

As an illustration of the performance of a rotatingmotion bearing utilising squeeze-film ultrasonic levi-tation the load-bearing capacity, experimentally

y = 1E-04x - 1E-05

0.0E+00

5.0E-06

1.0E-05

1.5E-05

2.0E-05

2.5E-05

3.0E-05

0.0E+00 5.0E-02 1.0E-01 1.5E-01 2.0E-01 2.5E-01 3.0E-01 3.5E-01 4.0E-01

(Applied load)^-0.5 [kg^-0.5]

Sep

arat

ion

dist

ance

[m]

Figure 13. Separation distance vs inverse root of the total mass of the levitation object. A linear relationship is found.

Atherton et al. 925

determined, is shown. Figure 15 depicts the relation-ship between applied load on the bearing and the sep-aration of the shaft from the bearing shell (air filmthickness). The continuous line represents load cap-acity predicted by a computer model and the filledcircles represent experimentally determined load cap-acity. The tested bearing of geometry shown in Figure10(a) was made of aluminium 50mm in length and30mm nominal bore diameter. The diametral clear-ance was 20 mm. A typical resonant frequency atwhich the bearing achieved the highest load capacitywas in the order of 30–35 kHz. Computer predictionwas obtained by solving numerically (Newton–Raphson iteration) a suitably modified Reynolds

equation. Although the load capacity is not huge,nevertheless it is real and measurable and might beof sufficient magnitude for certain specialist applica-tions. Details concerning experimental techniquesused to generate results plotted in Figure 13 can beobtained from Stolarski et al.35

Concluding summary

Self-levitating non-contact bearings (with rotationalor linear motion) offer significant advantages inmany applications. Due to its non-contact principle,the system can run at much higher speeds than con-ventional sliding contact bearings. Also, problemssuch as excessive temperature or wear of the bearingcomponents would be avoided. Squeeze-film ultra-sonic levitation can be accomplished in a number ofways through different geometries of bearing config-uration dependent on the type of motion desired.Load-carrying capacities of this new class of bearingsare, at the moment, rather low although there is scopeto increase this by means of careful selection of geom-etry, materials, frequency and amplitude of vibration,and the power of PZT actuators. Likewise, computermodelling of this type of bearings is a dynamicallyevolving area of research. More accurate performanceprediction, important at the bearing design stage for aparticular application, is achievable as more factorsgoverning the acoustic pressure generation are incor-porated into theoretical analyses. This paper hasdemonstrated the feasibility of squeeze-film ultrasoniclevitation to be used in a number of bearing embodi-ments depending on the requirements of the practicalapplication.

8

10

12

14

16

18

20

40 60 80 100 120 140 1600 20 180 200

Applied load [g]

Film

Thi

ckne

ss [µ

m]

60v Input Amp

50v Input Amp

40v Input Amp

Figure 14. Bearing load capacity results at 2600 Hz and 70 V off-set.

0 1 2 3 4 5 60

2

4

6

8

10

Load Capacity[N]

Sepa

ratio

n [m

m]

Figure 15. Comparison of computer predicted load capacity

(continuous line) with experimentally determined load capacity

(filled circles).

926 Proc IMechE Part J: J Engineering Tribology 228(9)

Funding

This work was partially supported by the Engineering and

Physical Sciences Research Council, UK (grant number EP/F04979X/1).

Conflict of interest

None declared.

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