Modelling Influence of Poisson ’ s Contraction on Squeeze Film Levitation of Planar Objects

Poisson's contraction effect is utilised to generate squeeze film levitation and is investigated through a computer model and validated experimentally. A finite element model (ANSYS) is verified by the experimental testing of five different plate designs. Each plate is subjected to uniaxial plain stress by an arrangement of two hard piezoelectric actuators bounded to the bottom of the plate and driven with DC and AC voltages. It is observed that a pulsating dimple or crest shape along the longitudinal axis in the central area of the plate is created due to Poisson's contraction, generating a squeeze-film between the plate and a levitated object. The separation distance between the floating planar object and the plate is analysed using computational fluid dynamics (ANSYS) and experimentally measured by a laser sensor. A good agreement has been found between model predictions and experimental results.


A. Background
Frictionless or non-contact transportation is essential for manufacturing ultra-precision products in micro-fabrication and nanotechnology, for example, silicon wafers and integrated circuits; where tiny flaws, infinitesimal scratches and contamination must be avoided.It is also indispensable for precision positioning and manipulation of light planar objects.
Techniques such as air-bed flotation and magnetic levitation, have been utilised practically for frictionless, non-contact production-line transportation.However, air-beds require huge amounts of clean air at high cost due to exterior equipment, e.g.piping and pumps, and magnetic levitation only works for ferrous material.Moreover, magnetic levitation cannot always be utilised for delicate objects due to strong magnetic flux.Consequently, there is considerable interest in developing new techniques for free suspension of delicate objects 1,2,3 .
The literature reveals that non-contact levitation, without using externally pressurised air, comprises three main approaches, namely: standing wave acoustic levitation (SWAL), nearfield acoustic levitation (NFAL) and squeeze film levitation (SFL).However, the literature is often confused between these three approaches 4,5,6 .SWAL has a different working principle to the NFAL method, whereas the former is only utilised for tiny particles, the latter can be used for a heavier floating object.However, the principle underpinning both methods (SWAL and NFAL) is based on acoustic radiation pressure.In contrast, the SFL method is fundamentally based on the Bernoulli principle and usually it is modelled using the Reynolds equation, requiring a closing gap in the direction of flow in order to create average pressure above ambient.The main parameter of a fluid flowing in the direction of a converging gap is its viscosity.Closing the gap is subsequently followed by its opening and the whole cycle resembles a pumping action.It should be noted that in some studies 7,8 it is assumed that there is a similarity between NFAL and SFL but in reality there are significant differences regarding their lifting force phenomena.

B. Poisson's contraction as a source of SFL
The overall aim of the paper is to demonstrate that the Poisson's contraction effect created in a plate clamped at both ends and under the plane stress conditions can be the source of squeeze film levitation for light planar objects.Therefore, there is a potential to induce the SFL mechanism without operating the plate at resonant frequencies of vibration.Exciting an arrangement of piezoelectric actuators at many cycles per second produces periodic forces elongating a thin plate which in turn results in a uniaxial plain state of stress and formation of a dimple in the direction perpendicular to that of the applied uniaxial stress (Fig 1).
Pulsating with controlled frequency the dimple squeezes the air in the gap between the plate and the object, producing a sustained levitating force (Fig 2).In other words, this is not a conventional vibrating plate where the vibration modes deform the plate most at frequencies determined by plate geometry, material and boundary characteristics.Therefore, the term 'oscillation' rather than 'vibration' is used here to avoid confusing oscillation of localised deformation due to the Poisson's effect with that of modal shapes resulting from plate vibration.

II. THEORETICAL MODEL
A computer model of a system comprising of the plate together with floating object is presented and executed using the finite element method (ANSYS Workbench).

A. Statement of the problem
The system modelled consists of a floating object and a flat thin plate with a single layer of hard piezoelectric actuators bonded to the underside of the plate.The plate is firmly fixed at both ends, as shown in Fig 3, and can be deformed repeatedly to create a dimple shape or a concave form in the downward direction at the centre of the plate.The deformation can be affected by many mechanical and electrical parameters.The most significant factor is the Poisson's ratio coefficient of a material of the plate together with the supply voltage (DCoffset voltage and AC -amplitude voltage) to the piezoelectric actuators.The model presented in this paper is a simplified case of a centrally positioned pulsating dimple created by the Poisson's effect.Undoubtedly, propagation of the initial deformation (dimple) as an elastic wave will establish a complex pattern of dimples all over the plate.This aspect of the problem is currently under further studies by the authors.

B. Configuration of the plate
The geometry and main dimensions of the plate used for manufacturing are shown in Fig 3 .Various plate sizes and materials (Table I) have been theoretically and experimentally investigated.Two piezoelectric actuators (PZTs) are adequate for the design to produce a dimple shape of deformation.The actuators utilised were of rounded shape with 35 mm diameter and 0.8 mm thickness, and were bonded to the underside of the plate.

C. Finite Element Modelling (FEM)
A simulation of the plate's deformation was carried out using the ANSYS Workbench transient structural analysis.Commands were written using a trigonometric equation (square wave equation), implemented with ANSYS mechanical APDL, and exported into ANSYS Workbench.The purpose of this was to actuate the PZTs with an On/Off time (Fig 4) to make a central area of plate deforming as a dimple in a downward direction as a concave form.The frequency excitation of the plate used in the ANSYS transient structural analysis simulation is the same as that used in the experimental test, which is 1.6 kHz.
In this study, static and dynamic (transient) analyses have been carried out for various plate designs.The purpose of the static analysis was to calculate the static deformation and to recognise the deformed shape of the plate designs when the piezoelectric actuators are operating with a DC voltage only.The aim of the dynamic analysis was to calculate the dynamic response of the plate by powering the piezoelectric actuators by an AC voltage.A square wave voltage signal (Fig 4) has been used to drive the actuators.The purpose of driving the actuators by the square wave amplitude of voltage is to create distinct repeated contraction and elongation of the actuators.Consequently, the plate will respond generating a repetitive pocket or dimple shaped deformation in the downward direction at the central area of the plate.
The piezoelectric actuator has been modelled with a SOLID227 element, because this has 10 nodes with up to five degrees of freedom per node (UX, UY, UZ, VOLT and TEMP -for definition of terms see Nomenclature).The structural capabilities include elasticity, plasticity and large strain.It is a coupled-field element and is suitable for three dimensional (3-D)   analyses as well as being capable of modelling the piezoelectric actuator.The element of SOLID187 was chosen to mesh the plate.It is a higher order 3-D, with 10 nodes having three degrees of freedom at each node (translations in the nodal x, y, and z directions).Furthermore, piezoelectric elasticity stiffness and stress matrices have been included in the simulation throughout using the Piezo ACT (ANSYS Customization Toolkit) extension.I).The plate has been modelled with three different materials: aluminium, stainless steel and a polymer (HDPE).In addition, various sizes of plates have been simulated (Table I), where the effective length is 160 mm due to clamping at both ends.The boundary conditions applied are firmed boundaries on the both ends of the plate and 150 V difference on each side of the actuators.The finite element mesh for one of the plate designs (Design A1) is shown in Fig 5.
The piezoelectric elasticity stiffness matrix N/m 2 , stress matrix C/m 2 and the relative permittivity matrix that have been utilised in the simulation are as follows:

D. Structural dynamics
The purpose of the dynamic analysis (transient) is to identify the dynamic characteristics of a structure exposed to dynamic driving loads.In this prototype design, the dynamic driving forces stem from the piezoelectric actuators when they are powered by an AC voltage.This is to produce enough displacement or to realise maximum dynamic reaction and thus achieve a better levitation.

Squeeze-film levitation modelling
An air film of certain thickness that is squeezed between the two surfaces with relative displacement of amplitude (e) at a frequency (ω), in the direction of the air film thickness, is assumed ( Fig 6).The pressure at a point in the air-layer is controlled by the Reynolds equation 7 .So, the analysis of the squeeze-film dynamics is combined with the Reynolds equation which describes pressure within the air film and the equation of motion which governs the plate's movement.The Reynolds equation can be expressed in general form 8,9 as specified by: Where, is the air density, h is the squeeze-film height and μ is the fluid viscosity, is time.The time dependent term is identified as the "squeeze term" as it signifies The relative permittivity matrix is The PZT density is 7500 kg/m^3 squeezing movement of the fluid.In addition, , and , refer to surface velocity components of bottom and top surfaces in x and y direction, respectively.
Langlois 7 introduced a perturbation factor for pressure of the order of into the Reynolds equation and obtained the squeeze-film force which represents the one that develops because of the pressure variations (squeeze-film pressure and ambient pressure) for a given instant of time. ( The boundary conditions of squeeze-film for circular plates are assumed as The initial pressure between discs is atmospheric.At the outer periphery, Where, is the squeeze number. To explore the dynamic of squeeze-film approach, it is essential to take into consideration the coupled dynamics of the entire system (fluid and structural domains), which comprises the where represents the fluid flow force which is evolved because of pressure variation and is the weight of the floating object (circular plate) as employed in the simulation.The following expressions , can be used to make Eq.( 7) dimensionless.
Where, is initial clearance, is an ambient pressure, is the frequency of vibration.
By differentiation of Eq. ( 10) with respect to time yields; (11)   The levitating height in terms of motion source can be represented by sin , thus; sin Eq. ( 12) can be rewritten in normalised form as; sin By substituting Eq. ( 13) into Eq.( 9) will produce the following expression; Or Eq. ( 14) is a second-order nonlinear partial differential equation and its initial conditions 0 are given as 0 1 To understand the floating performance based on the behaviour of squeeze film, the equation of motion in addition to the Reynolds equation needs to be added and solved by means of numerical estimation 10,11 .The non-dimensional Reynolds equation can be modified into the dimensions (2-D) through provided that the pressure is constant in the Y direction.Newton's Second Law can be applied to represent the equation of motion for the floating object 11 .
Initially, the levitation force produced through the squeeze film action must be greater than the weight of the levitating object for it to be levitated from rest, until equilibrium is reached at the maximum separation gap.The non-contact separation between the plate and the floating object can be inspected depending on the equation of motion coupled to the Reynolds equation with state boundary conditions.

Squeeze-film action
Squeeze-film levitation can be generated when an air-layer is entrapped between the two interacting surfaces.In fact, average pressure and resulting load carrying capacity that is produced by the cyclic displacement at the layer produces a complete support for a lightweight object (Fig 6(a)).
When inertia and the tensional stresses in the fluid are negligible, the Reynolds equation is the main non-linear partial differential equation that defines the pressure distribution in the squeezed air film and can be evaluated through the Navier-Stokes and continuity equations.
Furthermore, two important, instantaneous and recurrent steps are behind the pressure distribution in the squeezed air film, which stem from the influence of the boundaries on the flow cycle.Hence, throughout the compression stage, the development in pressure forms an outward flow, while simultaneously, the air layer between the surfaces reduces, letting only a slight airstream to occur.In contrast, during the decompression period, the pressure falls, and the stream alters to begin an inward flow.At this stage, the air-gap rises, letting a superior flow quantity pass in at a low velocity 7 .

E. Model of squeeze air-film in ANSYS CFX
A simulation utilising ANSYS CFD is implemented to examine the performance of a levitated object (freely suspended rigid mass).It is assumed that only one degree of freedom in the vertical direction is allowed.The instantaneous position of the object is evaluated by the balancing its inertia, the fluid flow force made by pressure distribution and gravity.An air film that exists between a planar object (5g disc) and a flat, rectangular plate (

III. EXPERIMENTAL VALIDATION A. Configuration of the apparatus
The schematic layout of the experimental work is shown in   I).
As can be seen from The finite element model (Fig 5) and boundary conditions of the plate (Fig 3) are presented and the material properties are listed (Table Reynolds equation and the equation of motion utilising a force equilibrium.Based on the free body diagram (Fig 6(b)) the equation of motion can be expressed as, Fig 6) of various designs is simulated.In other words, different case studies are implemented to identify the factors that affect the floating performance.The air film is of thickness (20 µm) and it has the same area as the disc (35 mm, diameter) (Fig 6(b)).Three boundary conditions (named as Top, Bottom and Side) have been used to model the semi-cylindrical shape of the air-layer and to analyse the squeeze film levitation.The top boundary condition refers to the bottom surface of the floating object and represents the mass of the floating object.The bottom boundary condition denotes the top surface of the plate design and indicates the bottom plane of the air film that has been modelled as a deformed boundary.It is given an amplitude (displacement) acquired from the plate that will squeeze the air film in the normal direction (Fig 7).Finally, the side boundary condition has been modelled on the periphery of the air-layer as an open boundary, so the flow in this boundary is not restricted in any direction.In other words, the pressure value is identified (atmospheric pressure), but the direction of flow is unknown (Figure 6(b)).

Fig 8 .
The experimental apparatus comprises of an oscilloscope, DC power supply, AC power supply amplifier, signal generator, laser sensor, plate design rig and piezoelectric actuators.The piezoelectric actuators of type PZT-4 are bonded to the bottom of the plate and are connected in parallel.The experiment requires the signal generator to create a certain signal voltage.The output terminal of the signal generator is connected to the amplifier of type ENP-1-1U (Echo Electronics) as it is essential to increase the voltage input from the signal generator.The produces a stable 15 magnification for the AC and DC signal inputs.Its operational point can be regulated to gain the appropriate bandwidth for the experimental measurements.The oscilloscope is used to read and display the output signals of the signal generator and the amplifier.The laser triangulation sensor from the Micro-Epsilon Company (type ILD-2300-5) is vertically placed above the plate design so as to achieve sufficiently accurate measurements according to the guide setting provided by the company.The sensor captures the static and transient deformation of the plate design plus the levitation distance of the floating object (a disc of 5g) without any contact.The output measurement data from the laser sensor is processed by special software provided by the Micro-Epsilon Company.For example, it can capture the entire output data without averaging, whilst having the ability to view the average estimated data.These measurements enable observation of the dynamic response of the plate designs and the floating object behaviour.Additionally, these measurements are used to validate the static and dynamic deformations calculated from the theory of the coupled-field analysis.Moreover, in order to avoid sideways movement of the floating object and also to measure the normal separation, the disc is placed inside a suspended metal ring of inner diameter 5 mm larger than the disc diameter (Fig9).Four thin cords are stretched parallel to the ring and connected to a fixed support such that there is no contact between the ring and the plate design.In this way, the disc will only lift inside the ring in a normal direction (1-DOF) when the plate design is driven.The PZTs offset the plate with an appropriate DC voltage (up to 150 V) and the plate is then dynamically excited by a square-wave AC voltage (Fig4) applied to the PZTs.IV.RESULTS AND DISCUSSIONSimulations of one of the plate designs (Design A1), which show contour plots for the static (DC voltage) and transient (AC voltage) deformations respectively, are shown in (Fig10 and

Fig 11 )
Fig 11).Static and transient deformations for a lengthwise segment (Fig 12 & Fig 13) in the central area of the top surface of the plate have been obtained using ANSYS Workbench and also measured experimentally for all five designs (TableI).

Start Input Plate and Piezo material properties Create Plate and Piezoes Geometries Make mesh and Check Quality Yes No Applying Boundary Conditions ( e.g. Voltage). Update mesh Quality Solve Yes Post -Process (Dynamic Displacement) No End Extract Dynamic Displacement (.csv file) Start ANSYS CFX Create 3D circular Geometry (Air-Film) Make mesh and Check Quality Update mesh Quality No Yes Apply Initial Conditions, Boundary Conditions and Equation of Motion Run CFX Convergence Yes Post -Process (Dynamic Displacement and Pressure) End No FIG
. 7. Theoretical model flowchart.

Computer to Analysis Data Laser Sensor (Micro-Epsilon) Thin plate with PZTs attached
FIG. 8. Layout of the experimental work.

Table I :
Plates design configurations