PROCEEDINGS of the23rd International Congress on Acoustics
9 to 13 September 2019 in Aachen, Germany

Numerical Analysis of the Nonlinear Restoring Force Based onNear-field Acoustic Levitation
Yuanyuan LIU1; Minghui SHI2; Kheirollah SEPAHVAND1; Steffen MARBURG1
1

Department of Mechanical Engineering, Technical University of Munich, 85748 Garching, Germany
2

College of Mechanical and Vehicle Engineering, Hunan University, 410082 Changsha, ChinaABSTRACT
The stability of the floating object is a key factor in near-field acoustic levitation, which has an extensive
application prospect in manufacturing of micro-electromechanical system parts. This study presents anumerical study of the restoring force based on the theory of gas film lubrication. Finite difference scheme
is used to solve the Reynolds equation considering the movement of reflector and to obtain the air pressure
distribution. After coordinate transformation, the restoring force which acts on the levitator can be acquired.
An experimental rig is constructed to measure the restoring force with different eccentricity. The
experimental results show that the restoring force increases with the increase of eccentricity. The numericalresults match well with experimental results.
Keywords: Acoustic levitation; Eccentricity; Restoring force

1. INTRODUCTION
Recently, near-field acoustic levitation (NFAL) technology has been rapidly developed. Comparing withthe traditional non-contact levitation technology, such as air cushion (1) and magnetic system (2), NFAL
system is a very compact system, which don’t need an additional air compressor to provide high pressure airand don’t produce an undesirable magnetic field. However, the manufacture and installation error willproduce the acoustic viscous force caused by the gradient of the sound field in the fluid, which will affect thestability of floating object (3). Owing to the existence of the viscous force, if the floating object has a littleeccentricity, it will push the object toward center, which is also called restoring force. In 2000, Koike et. al(4) used the block-spring model to evaluate the stability and calculated acoustic viscous forces in twodirection based on the theory of the near-boundary acoustic streaming. Li et. al (5) pointed out that the blockspring system model can only treat small amplitude oscillation because of its linear approximation. They
used Stokes micro-continuum theory and acoustic radiation potential method to solve the restoring force.
However, the corresponding experimental results were not given in that paper.
This paper introduces a suggested method to acquire the restoring force when centers between radiatorplate and reflector plate are misaligned. At first, an updated Reynolds Equation which considering themovement of reflector is derived based on the gas film lubrication. After coupled the gas film thickness andReynolds Equation, the pressure distribution can be acquired by using finite difference methods (FDM).
Secondly, the shear force which produced by pressure gradient can be decomposed into reflector. In themeantime, the experimental setup has been built to verify the relationship between eccentricity and restoring
force.

2. GOVERNING EQUATION
2.1 Reynolds Equation
For a symmetrical model, which means centers of radiator and reflector are concentric, the reflector
will keep equilibrium in horizontal direction. However, in this study, the centers between radiatorplate and reflector plate are misaligned. The distance between them is called eccentricity and isdefined as e, as shown in Figure 1. In this condition, the reflector will do a damping oscillation in thehorizontal direction until it stops at the equilibrium position. In order to simplify the numerical model,
1yuanyuan.liu@tum.de4769

the effect of inclination will be neglected in this study.

Figure 1 – Misaligned squeeze film model in the cylindrical coordinate
Some assumptions are introduced to simplify the momentum equations. First of all, since the radius of
the plate L is considerably greater than the mean thickness of the air film h0, the pressure gradient in the film
thickness direction can be neglected (6). Secondly, this squeeze film is assumed as isothermal film (7).
Meantime, the fluid inertia force and body force compared with the viscous force can be neglected in thelower Reynolds number (8). Therefore, the momentum equations in the r- and θ-direction reduced to (9)wpw 2 vr
§ wvr ·¸   Pa 2wrwz© wt ¹,w 2v
1 wp§ wv ·U¨ T ¸  Pa 2Tr wTwz
© wt ¹U¨(1)where vr and vT are the airflow velocity in the r- and θ-direction, Pa is the air dynamic viscosity,and U , p are the density and pressure of the air in the squeeze film, respectively. Assuming norelative slip between the air and the bottom surface of the reflector, for laminar flow, velocity profilein the r- and θ-directions derived from equation (1) are (10)vrz wpzUz 2 2zh u uz hh
2 Pa wr
6 Pa hvTU zrzwp
zrz2  h2z  h M Mh2 Pa r wT
6 Pa hz(2)ˈwhere u and M are the reflector relative horizontal displacement and rotational angle, and h is thefilm thickness. Combined the airflow velocity expression and the continuity equation, the Reynolds
equation in cylindrical coordinate system can be expressed asw phw rphw §1 w § 3 wp ·3 wp ·
 6 Pa u
¨ rph
¸
¨ ph
¸ 12 Pa r
wr ©
wr ¹ r wT ©
wT ¹wtwrpU hw pph u w rp 6PaM
wrwT23M w pU hwT23.

(3)

2.2 Boundary Conditions
Since the reflector will do a damping oscillation in the horizontal direction, the connection line between
O and O1 can be set as x-axis, as shown in Figure 2. The whole solving domain : contains two parts:squeeze domain :s and no squeeze domain :n . The domain : is decided by the radius L and closed byboundary Γ r L . The squeeze domain boundary * s which represented by heavy line in Figure 2 canbe expressed by * s r

L2  (e sin T )2  e cos T d L .

Therefore, the thickness of the air film in squeeze domain can be expressed byh h0  [0  Z r  sin wt ,where [0 and Z r

are the maximum vibration amplitude and mode shape of the radiator in Figure 1,respectively. The pressure in no squeeze domain always equals to the ambient pressure, which meansp :spa .

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(4)

(5)

The boundary Γ contact with the ambient air meansp *pa .

(6)

The pressure gradient at the center O is zero, which means
wp / wr(r 0, t ) 0 .

(7)

At the beginning, the reflector plate still placed in the radiator, the film thickness equals zero, and thepressure equals to atmosphere pressure. Thus,(8)p(r, t 0) pa , h(r, t 0) 0 .
Combined the finite difference methods with Newton-Raphson method, the pressure distribution within
the solving domain can be obtained (11, 12).

Figure 2 – Schematic diagram of the boundary condition
2.3 Motion Equations
Three types force which caused by squeeze film are applied on reflector: levitation force Fz , horizontal
thrust force Ft mu
u and rotational moment M t mL2M / 2 . The forces which calculated in the polar
coordinate O need to decomposed to polar coordinate O1 , as shown in Figure 3. Since the solving domain
and boundary conditions are symmetry about x-axis, the pressure distribution will be also symmetric about
the x-axis. Thus, the rotational moment M r will equals to zero and thrust force Ft points to positive xdirection. Shear stresses W zr and W zT are produced by air flow in r- and θ-direction acting on a surface
normal to the z-axis. In the coordinate O1 , both W zr and W zT have one component in r1- and θ1-direction,
respectively. Thus, the thrust force Ft can be expressed byFt2SL00³ ³ - >W cos( T  T )  W T sin( T  T )@  r cos( T )drdT .
zr1z11(9)The position of N in coordinate O1 can be expressed by
r1r 2  e2  2re cos S  T , T1 T  arcsin ª¬e  sin S  T / r1 º¼ .

(10)

For a Newtonian fluid, combined equation (2) with the shear stresses in the flow acting on a surface
normal to the z-axis in cylindrical coordinate, the expression for the shear stress on the bottom surface of thereflector are expressed asW zr | z h
W zT | z hwvr|wz z hwv
Pa T | z hwz

Pah wpu U huhu Pa h2 wr
3.
h wp
M r U hM r
 Pa
h2r wT3(11)4771

Equation (11) shows the shear stress mainly depends on pressure gradient and the relative movement ofreflector. The levitation force Fz is the integral of pressure in the solving domain, which is given byFz

2SL00

³ ³ ( p  p )  rdrdT .
a(12)Figure 3 – Geometric diagram for the resolution of shear forces

3. EXPERIMENTAL VALIDATION

Figure 4 – Schematic diagram of experimental test system
Based on the same transducer and radiator in reference (12), a new reflector plate is used to measure therelationship between eccentricity and restoring force. It made of AL2024 and has the same diameter asradiator which is 120 mm. The thickness and gravity of this plate are 3 mm and 0.9244 N, respectively.
Reference the experimental measurement method (13), a universal tilting indexing disc is used toapproximately measure the restoring force. A schematic diagram of the experimental setup is shown in Figure
4. It includes a universal tilting indexing disc, transducer and the corresponding control unit, as well as adigital single-lens reflex camera for monitoring the levitation performance of the plate. The transducer was
fixed on the universal tilting indexing disc, which can be tilted in a certain angle E by adjusting the controlsystem. Therefore, the eccentricity e is adjusted until the restoring force and the gravity component balanced
each other, which means Ft mg sin E .

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4. RESULTS
4.1 Pressure DistributionFigure 5 shows the pressure distribution in radiator coordinate O at the start time of one stable cycle.
The excitation amplitude, levitation force and eccentricity are 16 μm, 0.9244 N and 10 mm, respectively. The
radiator works in second order vibration mode and the excitation frequency is 19 kHz. Note that the shape ofpressure is corresponding with the mode shape in the squeeze domain :s . The reason for this is that thevibration mode shape is one boundary condition to accurate the gas film thickness. If the eccentricity equals
to zero, the pressure distribution will be symmetrical in squeeze domain as shown in reference (12) and thereflector will keep stable situation. In the non-squeeze domain :n , the pressure value equals to 1, whichmeans the pressure equals to atmospheric pressure.

Figure 5 – Pressure distribution in solving domain
4.2

Eccentricity

Figure 6 – Comparison of the restoring force versus eccentricity
In general, eccentricity is the most important factor which affect the restoring force distribution. Thus, theinfluence of the eccentricity e on restoring force is shown in Figure 6. The hollow and solid dot denote thenumerical and experimental result, respectively. Notably, the numerical results exhibit reasonably goodagreement with the experimental results in lower eccentricity. Since the calculation with the assumption of
the inclination of reflector is neglectful. In the lower eccentricity, this assumption is acceptable. However,with the increase of eccentricity, the inclination is a very important factor which affect the stability of reflector.
In two different excitation amplitude, the restoring force increases with the increase of eccentricity. It is notedthat the increase gradient will be decrease with the eccentricity increases. The same variation tendency has
also been reported in reference (3).

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5. CONCLUSIONSThis study presents a novel theoretical analysis method for calculating the restoring force based on gas
film lubrication theory. At first, an updated Reynolds Equation which considering the movement of reflector
is derived. Combined with the levitation height, the pressure distribution can be acquired by using Finite
Difference Methods. Secondly, by using the coordinate transformation, the restoring force acting on thereflector can be calculated. Numerical results show that the restoring force increases as the increasing ofeccentricity and the relationship between them is nonlinear. Meanwhile, the experimental setup which used
to prove this relationship is built. The numerical experimental results show good agreement with theexperimental numerical results, which means the method is effective for predict the stability of NFAL system.

ACKNOWLEDGEMENTS
The authors acknowledge the support from China Scholarship Council (CSC) (File No. 201808340068).
Meantime, the authors would like to acknowledge Dr. Kai Feng, Professor at Hunan University, China, forvaluable suggestions and support on the experiment.

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