Investigation of Non-contact Bearing Systems
Based on Ultrasonic Levitation
zur Erlangung des akademischen Grades eines
DOKTORS DER INGENIEURWISSENSCHAFTEN (Dr.-Ing.)
der Fakult¨
at f¨
ur Maschinenbau
der Universit¨
at Paderborn
genehmigte
DISSERTATION
von
M. Sc. Su Zhao
aus Liaoning, VR China
Tag des Kolloquiums: 19.03.2010
Referent: Prof. Dr.-Ing. J¨
org Wallaschek
Korreferent: Prof. Dr.-Ing. habil. Ansgar Tr¨
achtler
Abstract
Non-contact bearings based on acoustic/ultrasonic levitation are investi-
gated in this thesis. Both standing wave type and squeeze film type ultra-
sonic levitation are investigated theoretically and experimentally.
The conventional standing wave type ultrasonic levitation has not found
technical applications in non-contact bearings due to the fact that it has
very limited load capacity and can only levitate elements which are smaller
than the sound wavelength. In this thesis work, a new configuration of
standing wave levitation is presented which is able to levitate large planar
object at a position of multiple times of a half wavelength of the sound wave.
The theoretical model for the proposed levitation system is established and a
prototype system is constructed accordingly. A CD is successfully levitated
with the proposed system at a height of half a wavelength. A levitation
force of 1 N is measured at the position of half wavelength.
Squeeze film type ultrasonic levitation is investigated theoretically to find
the crucial design parameters and to improve the levitation capacity. Two
analytical models based on acoustic theory and fluid dynamics are presented
and compared. The governing fluid dynamics equation is solved numerically
to obtain precise pressure distributions. Based on the theoretical investi-
gation, a novel non-contact journal bearing is developed for suspension of
a solid steel spindle with diameter of 50 mm. The maximum load capac-
ity of 51 N (6.37 N/cm2) is obtained which is considerably larger than the
previously reported squeeze film bearings whose load capacities are usually
within a few Newton (less than 1 N/cm2).
Zusammenfassung
Im Rahmen dieser Arbeit werden ber¨
uhrungslose Lagerungen mit akustis-
cher Ultraschall-Levitation entwickelt und vorgestellt. Dabei werden sowohl
Stehwellen-, als auch Squeezefilmlevitation theoretisch und experimentell
untersucht.
Bisher wurde die Ultraschall Levitation mit stehenden Wellen aufgrund der
begrenzten Lastaufnahme und der auf die halbe Wellenl¨
ange begrenzten
Objektgr¨
oße nicht in technischen Anwendungen als Lagerung appliziert.
In dieser Arbeit wird ein neuer Aufbau vorgestellt, mit dem große ebene
Objekte im Abstand einiger halber Wellenl¨
angen levitiert werden k¨
onnen.
Neben theoretischen Betrachtungen wird ein Prototyp aufgebaut, an dem
bei einem Abstand einer halben Wellenl¨
ange eine Levitationskraft von 1N
gemessen werden kann. Damit kann erfolgreich eine CD levitiert werden.
Zur Auslegung und Optimierung der Squeezefilmlevitation werden zwei an-
alytische Modelle, die einerseits auf der Akustik und anderseits auf der
Fluiddynamik beruhen, vorgestellt und verglichen. Das akustische und das
numerisch ausgewertete fluiddynamische Modell werden anschließend mit
Hilfe von Messungen validiert. Basierend auf den Simulationen wird ein
neuartiges ber¨
uhrungsloses Lager f¨
ur eine Welle mit einem Durchmesser von
50 mm entworfen. Mit dem neuartigen Aufbau kann eine maximale Last von
51N (6,37 N/cm2) aufgenommen werden und damit eine wesentlich h¨
ohere
Traglast als mit vorhergehenden Squeezefilmlagern, deren Last auf wenige
Newton (weniger als 1 N/cm2) beschr¨
ankt war, erreicht werden.
Acknowledgements
This thesis represents the conclusion of my research from 2006 to 2009. I
joined the International Graduate School Dynamic Intelligent System (IGS)
in Universit¨
at Paderborn in April 2006. After one year stay, I followed my
supervisor to Leibniz Universit¨
at Hannover and continued my study in the
Institute for Dynamics and Vibration Research. During all these years, I
have received help from so many people, and to them I would like to express
my gratitude. I offer my apologies in advance if I miss to mention anyone
who has help me. Also, after struggling long enough with the accuracy of
my English in this document, I reserve the right to use a more conversa-
tional tone in this section.
First of all, it would have not been possible for me to live and study in
Germany without the fellowship from IGS of Universit¨
ar Paderborn. Here
I would like to thank Prof. Steffen for recruiting me and to Dr. Arnold
Hueck-Stiftung for the fellowship. I am also thankful to my supervisor
Prof. Wallaschek for continuing supporting me after the 3-year IGS pro-
gram.
On the faculty side, I’d first like to thank my thesis supervisor Prof. Wal-
laschek for offering me the possibility to work in a liberal environment and
giving me the freedom to conduct my research in a independent way. I am
also deeply thankful to my thesis committee, Prof. Sextor, Prof. Tr¨
achtler
and Prof. Schmid, for taking their time reviewing my work.
On the non-faculty side, I would like to thank all my colleagues in MUD and
IDS. They have made me feel really comfortable at work. In particular, I’d
like to thank Jens Twiefel and Wiebold Wurpts for their extensive review
and insightful advices of my draft thesis. Also, I am indebted to Walter
Littmann and Tobias Hemsel for guiding me into the topic of ultrasonic
levitation and helping me building up the first experimental setup. The CD
levitation system will never exist without Walter’s inspiration. He has also
given me valuable suggestions on improving the transducers on my bearing.
I am also thankful to my team leader Jens Twiefel for his constant guid-
ance in the lab and on the design of my bearing; David Oliva Uribe, Florian
Schiedeck and Sebastian Mojrzisch for the helps with the electronics; Xu
and Minghui for taking care of everything when I am not there; Marcus
Neubauer for teaching me how to use LATEX and many other things; and
Andreas Hohl, Sasa Mihajlovic for sharing the tea time.
Off campus, I’d like to thank Timo and Hua for socializing me and helping
me to adapt to the German life; ”Ma homies”(Erik, Martin, Roy, Fran,
Chengyee) for providing a solid foundation of friendship from which I draw
much strength; and the BiB-dorm guys (Annett, Christina, Vincenzo, Sven,
Melisa...) for making it feel like a family; my Chinese community (Mo Lao
Shi, Lao Ge, Minghui 2D, Han Xu&Xiaohui&Budou, Yang Lei, An Lu,
Monster Bin, Little Ma, Big Ding, Yu Zuo, Zhou Li&Shizhou, Little Bai,
Little Wang, Shui Ni, Huanhuan, Lulu, Shao Ye, Wu Tao, Sister Ma, Lao
Yu, Wu Hai, Wu Hao, Guo Chao, He Meng&Lao He...) for making here
feel like home.
Last, but certainly not least, I am grateful to Yingmei for beautifying many
diagrams in this thesis. I’d like to thank her and my parents for their
constant love, support, understanding and encouragement, as well as their
patience during this process.
Hannover, April 2010
Contents
1 Motivation 1
2 State of the art 3
2.1 Non-contact bearings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.2 Ultrasonic levitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2.1 Standing wave type . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2.2 Squeeze film type . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.3 Non-contact bearings using squeeze film ultrasonic levitation . . . . . . 16
3 Research objective and thesis outline 23
3.1 Research objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.2 Thesis outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
4 Basic theory on acoustics 27
4.1 Linear theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4.1.1 Elastic waves in fluids . . . . . . . . . . . . . . . . . . . . . . . . 27
4.1.2 Equations of linear acoustics . . . . . . . . . . . . . . . . . . . . 28
4.1.3 Acoustic energy density and intensity . . . . . . . . . . . . . . . 29
4.1.4 Atmosphere absorption of sound wave . . . . . . . . . . . . . . . 30
4.2 Nonlinear theory - acoustic radiation pressure . . . . . . . . . . . . . . . 31
5 Piezoelectric ultrasonic transducers 35
5.1 Piezoelectric actuator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
5.1.1 Piezoelectric effect . . . . . . . . . . . . . . . . . . . . . . . . . . 35
5.1.2 Piezoelectric actuators . . . . . . . . . . . . . . . . . . . . . . . . 36
5.1.3 Lumped parameter model . . . . . . . . . . . . . . . . . . . . . . 37
iii
CONTENTS
5.2 Langevin type ultrasonic transducers . . . . . . . . . . . . . . . . . . . . 41
5.2.1 The half-wavelength-synthesis . . . . . . . . . . . . . . . . . . . . 42
5.2.2 Dimensioning method . . . . . . . . . . . . . . . . . . . . . . . . 42
5.2.3 Performance criteria . . . . . . . . . . . . . . . . . . . . . . . . . 44
5.3 Driving method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
5.3.1 Self oscillating circuit . . . . . . . . . . . . . . . . . . . . . . . . 47
5.3.2 Phase-locked-loop (PLL) controller . . . . . . . . . . . . . . . . . 47
6 Standing wave ultrasonic levitation 51
6.1 A configuration for large planar objects . . . . . . . . . . . . . . . . . . 51
6.2 Modeling the proposed levitation system . . . . . . . . . . . . . . . . . . 53
6.2.1 Flexural vibration mode of the radiator . . . . . . . . . . . . . . 53
6.2.2 Sound beam in the acoustic near-field . . . . . . . . . . . . . . . 54
6.2.3 Increased absorption due to nonlinear effects . . . . . . . . . . . 56
6.2.4 Modeling the sound field . . . . . . . . . . . . . . . . . . . . . . . 57
6.3 Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
6.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
7 Suspension of large planar objects using ultrasonic standing waves 63
7.1 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
7.1.1 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . 63
7.1.2 Levitation force measurement . . . . . . . . . . . . . . . . . . . . 64
7.1.3 Levitating a compact disc . . . . . . . . . . . . . . . . . . . . . . 67
7.1.4 Sound field visualization . . . . . . . . . . . . . . . . . . . . . . . 68
7.2 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
7.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
8 Squeeze film ultrasonic levitation 73
8.1 Modeling based on acoustic theory - acoustic radiation pressure . . . . . 73
8.2 Modeling based on fluid mechanics - solving the Reynolds equation . . . 76
8.2.1 Approximate solution of the Reynolds equation for large squeeze
number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
8.2.2 Solving the Reynolds equation numerically . . . . . . . . . . . . 79
8.3 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
iv
CONTENTS
8.3.1 Experimental validation . . . . . . . . . . . . . . . . . . . . . . . 80
8.3.2 Crucial parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 85
9 An non-contact journal bearing based on squeeze film ultrasonic lev-
itation 89
9.1 Design of the proposed bearing . . . . . . . . . . . . . . . . . . . . . . . 90
9.1.1 The Langevin ultrasonic transducer . . . . . . . . . . . . . . . . 91
9.1.2 The spindle-bearing system . . . . . . . . . . . . . . . . . . . . . 95
9.2 Testing the prototype bearing . . . . . . . . . . . . . . . . . . . . . . . . 95
9.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
10 Summary and outlook 99
Bibliography 101
v
CONTENTS
vi
1
Motivation
Non-contact suspension of objects (with rotational or linear motion) has significant
advantages in many situations. Being non-contact, the systems can be operated at
much higher speeds than using conventional mechanical bearings. Also, there will not
be problems such as overheating and wear of the bearing components any more. Thus,
high precision and high speed of motion can be achieved.
Classical non-contact bearings such as air bearings and magnetic bearings are al-
ready being used in several practical applications. However, a continuous supply of a
large volume of clean air is required for the air bearings, which leads to high running
cost. And, the requirement of an external pump excludes this type of bearing from
certain applications. Magnetic bearings can not be used for magnetically sensitive con-
figurations due to the strong magnetic flux. It is therefore of great interest to find other
concepts for realizing non-contact suspension which can resolve these problems.
Acoustic/ultrasonic levitation has been found to be a promising alternative solu-
tion. It has drawn great attention in the last decades and shown good potentials
to overcome some of the shortcomings of the existing non-contact suspension methods.
Non-contact bearings based on acoustic/ultrasonic levitation use air as lubricant. They
share the advantages of classic air bearings. However, in acoustic/ultrasonic levitation
the load-carrying pressure is generated internally by means of high frequency mechani-
cal vibrations. Therefore, external supply of pressurized air is not needed anymore. The
bearing can be as compact as two conforming surfaces. These distinct characteristics
1
1. MOTIVATION
make it suitable for certain situations where air or magnet bearings are not applica-
ble. In the current thesis, theoretical and experimental investigations on non-contact
bearings based on acoustic/ultrasonic levitation are performed.
2
2
State of the art
2.1 Non-contact bearings
A bearing is a device that supports and guides one machine component with respect
to others in such a way that prescribed relative motion can occur while the forces asso-
ciated with machine operation are transmitted smoothly and efficiently. Bearings may
be classified broadly according to the motions they allow and according to their prin-
ciple of operation as well as by the directions of applied loads they can handle. Here
we classify bearings by whether or not a direct mechanical contact exists between the
bearing and the supported member. In this way, bearings are classified into to contact
and non-contact bearings.
Contact bearings are the most common used bearings such as plain rubbing bear-
ings, ball bearings, and roller bearings, in which the load is supported by means of
direct physical contact. These bearings have been developed since centuries and have
very wide range of applications. Due to mechanical contact between rigid bodies, these
bearings often suffer from wear, heat generation, vibration and noise generation, espe-
cially in high speed operations.
In non-contact bearings, there is no direct physical contact between the bearing and
the load. In consequence, these bearings have no wear, nearly no friction (except for
fluid drag) and they can achieve higher accuracies. Fluid film bearings are the most
popular non-contact bearings. They use a thin film of fluid (liquid or gas) to separate
3
2. STATE OF THE ART
the two surfaces. The load-carrying capacity is derived from the pressure within the lu-
bricating film and can be generated by the motion of the machine elements (self-acting
or hydrodynamic/aerodynamic bearings) or by external pressurization (hydrostatic, or
air bearings) or squeeze motion (squeeze film acoustic bearing, the main focus of this
research work), or by a combination of these actions. Besides fluid film bearings, mag-
netic bearings are another type of non-contact bearing. They support a load using
magnetic levitation force. Therefore, no lubricant or medium is needed to perform the
levitation.
Non-contact bearings are frequently used in high load, high speed or high precision
applications where ordinary rolling element bearings have short life or high noise and
vibration. In the following part of this section, some typical non-contact bearings will
be introduced and their strength and weakness will be discussed.
•Hydrodynamic bearing
In hydrodynamic bearings, the bearing rotation sucks the fluid onto the inner surface
of the bearing, forming a lubricating wedge under or around the shaft. They are suited
for improving rotational accuracy and enhancing quietness and robustness. These bear-
ings have excellent damping properties due to squeeze film damping. The fabrication
of hydrodynamic bearings is simple compared to dry-rubbing bearings, which leads
to low cost for production. Hydrodynamic bearings have been successfully applied in
computer hard drive industry to replace the conventional ball bearings. Hydrodynamic
bearings rely on bearing motion to maintain the load-carrying pressure. They may
suffer on high friction and short life at low speeds or during starts and stops.
•Hydrostatic bearing
Hydrostatic bearings use a pressurized liquid film to support the load. They rely on
an external pump to constantly supply the pressurized liquid in the bearing clearance
to maintain the film. Hydrostatic bearings have excellent damping as well as very high
stiffness. They are widely used for high precision and high speed machine tools such
4
2.1 Non-contact bearings
as precision milling machine, precision manufacturing center. The design of a hydro-
static bearing is more complicated than that of a hydrodynamic bearing. It requires
the precise adjustment of a number of parameters including pad geometry, restricted
size, supply pressure, and journal bearing clearance to optimize performance.
•Aerodynamic bearing
Aerodynamic bearings are analogous to hydrodynamic bearings. One major difference
is that the lubricant is gas (normally air) which is compressible. Air also has a much
lower (more than 1000 times less) viscosity than even the thinnest oils. Therefore the
viscosity induced friction force is much smaller. Aerodynamic bearings have very dis-
tinct advantages and disadvantages. They allow extremely high operating speed and
can be operated in extremely low or high temperatures. The lubricant is ample, clean
and does not contaminate the surfaces and the surroundings. However, similar to hy-
drodynamic bearings, the load capacity is dependent on the relative speed at which the
surface moves and therefore at zero speed, the bearing supports no load. Aerodynamic
bearings have much lower load capacity compared to hydrodynamic bearings with sim-
ilar size. Another weakness is that they generally have poor stability that can produce
destructive contact between rotational and stationary components.
•Aerostatic bearing
Aerostatic bearings (often referred as air bearing) are similar to hydrostatic bearings,
and use a thin pressured gas film to carry the loads. They need an external pressure
source to create the air film. This type of bearing shares many of the advantages of the
aerodynamic bearing. In addition, it supports load at zero speed. There are two types
of air supply mechanism in aerostatic bearings, orifice type and porous carbon/metal
type. In orifice type, the air is fed through the orifices into the bearing clearance. As air
escapes from the orifice it expands and so its pressure drops as it flows across the face
of the bearing resulting in variances of pressure in the air gap. In porous carbon/metal
type, the air pressure drops as it flows through the porous layer. Even pressure then
bleeds from the entire bearing face resulting in a more uniform pressure in the air gap.
Porous carbon/metal type aerostatic bearings have advantages. They are more damage
5
2. STATE OF THE ART
tolerant, have higher air film stiffness and are naturally stable.
Aerostatic bearings are employed in many high speed and high precision machines
doing work inside a micrometer or in nanometer range. They are also suitable for in-
dustries such as food productions or pharmaceutical productions where oil lubrication
is a problem.
•Electromagnetic bearing
An electromagnetic bearing positions and supports a moving shaft using magnetic forces
without mechanical contact. Magnetic bearings allow the highest rotating speed among
all kinds of bearings since they are non-contact and require no lubricant at all. They
work in extreme conditions such as vacuum as well. With position feedback control
system, an electromagnetic bearing can be operated as an active bearing system which
offers dynamic stiffness and error compensation abilities by varying the amount of cur-
rent in the coils of the magnet system.
Electromagnetic bearings are increasingly used in industrial machines such as com-
pressors, turbines, pumps, motors and generators. Electromagnetic bearings are also
used in high-precision instruments. Unfortunately, electromagnetic bearings have some
severe disadvantages. Magnetic force is an attractive force. It decreases with greater,
and increases with smaller gap. This makes a magnetic bearing unstable from its
nature. A control system must be installed to maintain the stable suspension. Contin-
uous power supply is needed for most electromagnetic bearings. Therefore, some kind
of back-up bearing is typically needed in case of power or control system failure. The
magnetic flux leakage during operation can affect the sensitive electronics around the
bearings.
Beside the existing types of non-contact bearings, ultrasonic levitation has been
found as an alternative way to realize non-contact suspension. The basic theory under-
lying the operating principle of ultrasonic levitation will be discussed in the following
section.
6
2.2 Ultrasonic levitation
2.2 Ultrasonic levitation
An acoustic wave can exert a force on objects immersed in the wave field. These forces
are normally weak, but they can become quite large when using high frequency (ul-
trasonic) and high intensity waves. The forces can even be large enough to suspend
substances against gravity force. This technique is called acoustic levitation. Since the
sound waves used are often in the ultrasonic frequency range (higher than 20kHz), it
is more often called as ultrasonic levitation.
Ultrasonic levitation has been firstly used for levitating small particles by creating
a standing wave field between a sound radiator and a reflector, namely standing wave
ultrasonic levitation. Standing wave type ultrasonic levitators with various features
were designed for applications in different scientific disciplines such as containerless
material processing and space engineering (53). Another well-known type of ultra-
sonic levitation is squeeze film ultrasonic levitation. It happens when a flat surface is
brought to a conformal radiation surface which vibrates in high frequency. In the fol-
lowing part of this section, both types of ultrasonic levitation will be reviewed in detail.
2.2.1 Standing wave type
Standing wave levitation phenomenon was first observed in Kundt’s tube experiment
(41) in 1866, that small dust particles moved towards the pressure nodes of the standing
wave created in a horizontal Kundt’s tube. A typical setup for standing wave levitation
is shown in Fig. 2.1. As a result of multiple reflections between an ultrasonic radiator
and a solid, flat or concave reflector, a standing wave with equally spaced nodes and
anti-nodes of the sound pressure and velocity amplitude will be generated. Solid or
liquid samples with effective diameters less than a wavelength can be levitated below
the pressure nodes. The axial suspension of the sample is an effect of the sound radia-
tion pressure of a standing wave. Combining with a Bernoulli vacuum component, the
sound wave can locate the samples laterally as well (11).
The first detailed theoretical description of standing wave levitation was given by
King (28) in 1934. King’s work was extended by Hasegawa and Yosioka (19) to include
7
2. STATE OF THE ART
+
+
–
–
Pressure Velocity Piston sound source
Reflector
z z
g
z
n
λ/2
Heavy
sphere
Force
Stable
Unstable
Figure 2.1: Distribution of sound pressure, air particles’s velocity and levitation force in
a standing wave type levitation system(53)
the effects of compressibility. Embleton (12) adopted King’s approach to fit to the
case of a rigid sphere in a progressive spherical or cylindrical wave field. Westervelt
(57;58;59) derived a general expression for the force owing to radiation pressure acting
on an object of arbitrary shape and normal boundary impedance. Westervelt showed
that a boundary layer with a high internal loss can lead to forces that are several orders
of magnitude greater than those predicted by the classical radiation pressure theory.
A very different approach compared to King was presented by Gor’kov (16), who
presented a simple method to determine the forces acting on a particle in an arbitrary
acoustic field. The velocity potential was represented as sum of an incident φin and
a scattered term φsc. Barmatz (4) applied Gor’kov’s method to derive the general-
ized potential and force expressions for arbitrary standing wave modes in rectangular
cylindrical and spherical geometries. Lierke gave an overview of standing wave acoustic
levitation (31) based on long term research and development activities for the Euro-
pean and the US space agencies. Xie and Wei (64) studied the acoustic levitation force
on disk samples and the dynamics of large water drops in a planar standing wave, by
solving the acoustic scattering problem through incorporating the boundary element
method.
8
2.2 Ultrasonic levitation
For understanding the basic working principles of standing wave levitation, the
theoretical approach of King (28) will be discussed shortly in the following. Assuming
that the fluid being considered is adiabatic and barotropic, the equations of motion can
be written as:
ρDu
Dt =−∂p
∂x, ρDv
Dt =−∂p
∂y, ρDw
Dt =−∂p
∂z (2.1)
where
D
Dt =∂
∂t +u∂
∂x +v∂
∂y +w∂
∂z
is the absolute derivative, ρthe density of the medium, pthe pressure, and (u, v, w)
the Cartesian velocity components. By defining ˜ω=dp/ρ the equation of motion can
be rewritten as:
Du
Dt =−∂˜ω
∂x ,Dv
Dt =−∂˜ω
∂y ,Dw
Dt =−∂˜ω
∂z (2.2)
When the motion is irrotational, the velocity components can be expressed in terms of
the velocity potential φ,
(u, v, w) = −∇φ(2.3)
In case of air as the medium, the velocity potential can be obtained from the approxi-
mate linear wave equation
∇2φ=1
c2
∂2φ
∂t2(2.4)
which is simplified from the exact differential equation for φusing a second order
approximation. The equation of continuity is:
∂ρ
∂t +∂
∂x (ρu) + ∂
∂y (ρv) + ∂
∂z (ρw) = 0 (2.5)
and the pressure variation can then be derived from Eq. (2.5) as:
∆p=p−p0=p0
∂φ
∂t +1
2
ρ0
c2∂φ
∂t 2
−1
2ρ0q2(2.6)
where ρ0is the density of the surrounding medium, φis the velocity potential, cis
the sound speed in air and qis the velocity amplitude equal to √u2+v2+w2. Detail
derivation of Equ. 2.6 can be found in Ref. (28). The time averaged acoustic pressure
on a rigid body can be calculated by integrating the acoustic pressures acting on each
surface element of the body.
9
2. STATE OF THE ART
In the case of a plane standing wave, the velocity potential φscan be expressed as
(28):
φs=|A|cos kh cos ωt (2.7)
where, |A|is the amplitude of the velocity potential, k= 2π/λ is the wave number,
and his the position where the sphere, which is assumed to be small, is located. The
acoustic radiation force on a rigid sphere can be calculated as:
F=−5
6πρ0|A|2(kRs)3sin (2kh) (2.8)
with Rsbeing the radius of the sphere.
B¨
ucks and M¨
uller in 1933 (8) presented the first experimental setup for positioning
of small samples in acoustic standing waves. A small particle was trapped at a position
slightly below the pressure nodes of the standing wave between a radiator and a reflec-
tor. In 1974, Wang et. al. (55) presented an acoustic chamber for positioning of molten
materials. The chamber was used for positioning in an extreme temperature gradient.
In 1975, Whymark (60) proposed an acoustic levitator for positioning of materials in
space using a single source of sound. Fine control of position could be obtained by
adjusting the reflector. In 1983 Lierke (32) presented an acoustic levitator for posi-
tioning the materials samples in mirror furnaces in space processing. In 1985, Trinh
(52) presented a compact acoustic levitation device for studies in fluid dynamics and
materials science in microgravity. This classic structure was later modified to achieve
better performances. Otsuka et. al. (40) used a stepped circular vibrating plate as
the radiator which can produce high intensity ultrasound fields. Different from conven-
tional piston-like vibration sources, this approach used the flexural vibration mode of
the plate with two nodal rings to achieve higher vibration amplitude. The stepped plate
has a concave channel with fixed depth of half sound wavelength in air. This special
design makes the concave and convex blocks vibrate in the counter phase so that the ul-
trasound propagating in the air is modulated in the same phase. As a result, a narrow,
intensive, high directional ultrasound beam is obtained. In 2001, Xie and Wei (63) en-
hanced the standing wave acoustic levitation force by properly curving the surface and
enlarging the reflector. High density material like tungsten (ρ= 18.92g/cm3) was suc-
cessfully levitated for the first time using standing wave ultrasonic levitation. Recently
in 2006, Xie and Wei reported the successful levitation of small living animals such as
10
2.2 Ultrasonic levitation
Squeezed gas film
L
0
h
0
a0 0
( ) sinh t h a t
Z
z
x
y
Figure 2.2: Schematic diagram of squeeze film levitation system
ant, ladybug, and little fish with a standing wave acoustic levitator (62). Their exper-
iments showed that the vitality of the small animals was not impaired during levitation.
All the standing wave levitation systems presented above possessed the classic
radiator-reflector configuration. The applications of such configuration were limited
to the levitation of small particles whose dimension does not exceed the wavelength
of the imposed sound wave. Moreover, the levitation force can be obtained from this
configuration is very limited. Therefore, modifications and improvements are needed
before standing wave ultrasonic levitation can be applied for non-contact suspension
systems such as linear and rotational bearings.
2.2.2 Squeeze film type
In 1964, Salbu (47) reported a levitation system for objects with flat surface. Salbu
used magnetic actuators to excite two conforming surfaces oscillating next to each other
to generate a positive load supporting force. In 1975, Whymark (60) reported that a
brass planar disk of 50 mm in diameter and 0.5 mm in thickness was levitated extremely
close to a piston vibration source driven harmonically at a frequency of 20 kHz. The
levitation effect reported by Salbu and Whymark is named as squeeze film levitation
and also called near field acoustic levitation.
A schematic diagram of squeeze film levitation system is shown in Fig. 2.2. The
time-averaged mean pressure in the gap has a value which is higher than the surround-
ing, caused by the second-order effects possessed by the rapidly squeezed and released
gas film between two plane surfaces. Two distinct properties distinguish this type of lev-
11
2. STATE OF THE ART
itation from standing wave levitation. First, the reflector is no longer needed; instead,
the levitated object itself acts as an obstacle for the free propagation of the ultrasonic
wave-front. Second, the gap between radiation source and the levitated object must be
much smaller than the sound wavelength in air. Thus, instead of a standing wave, a
thin gas film is formed between the radiator and the levitated object, which is rapidly
squeezed and released.
A simple model introduced by Wiesendanger (61) is presented in the following to
demonstrate the basic idea of how the squeeze film levitation works. The leaking and
pumping at the boundary is neglected in this model. Only the trapped gas which is
rapidly squeezed and released is considered. Thus the total mass of air in a fixed volume
remains constant, resulting in
pV n∼phn= const. (2.9)
where prepresents the pressure, Vthe volume of the trapped gas, hthe gap distance
and nthe polytrophic constant (n = 1 for isothermal condition, n=k≈1.4 for
adiabatic condition and air). The relation between pressure and levitation distance is
nonlinear, which leads to a distorted pressure p(t) resulting from the imposed periodic
gap distance h(t). Considering the case shown in Fig. 2.2, the gap distance oscillates
harmonically around a equilibrium position h0, i.e.
h(t) = h0(1 + ǫsin ωt) (2.10)
in which ωis the angular frequency of the oscillation, ǫthe excursion ratio (ǫ=a0/h0).
The excursion ratio denotes the ratio of the vibration amplitude over the mean gap
distance, where a0is the vibration displacement amplitude. The mean pressure under
isothermal condition (n= 1) can be expressed as (61)
¯p=p0h0
2πZ2π
0
1
h(t)d(ωt) = p0
√1−ǫ2(2.11)
It can be easily seen that a mean pressure ¯pwhich exceeds the ambient pressure p0is
obtained. The result is illustrated in Fig. 2.3. The harmonic motion of the radiating
surface produces a non-harmonic pressure oscillation whose mean value is not equal
to the quasi-static value p0. The positive mean pressure pwhich is larger than the
12
2.2 Ultrasonic levitation
8
6
7 DIE SQUEEZEFILMLEVITATION
p
pmax
p > p0
p0
pmin
t
t
h0
h - h
0δh + h
0δ
h
ph = const.
κ
Abbildung 18: Entstehen des mittleren Drucks
h
p
i
! p
0
nach Gleichung 8101:
str;mt wieder von au?en zurAck:. Demnach wArde der ReDektor zwar anfFnglich
angehob en werden, dann ab er wieder absinken, bis er den Schwinger b erAhrt.
Dies wird in der RealitFt jedo ch nicht beobachtet.
Der Grund hierfAr muss in str;mungsmechanischen EJekten liegen, die sich im
Luftspalt zwischen Schwinger und ReDektor selbst, sowie am Rand des oszillieren-
dem Spalts und der Umgebung abspielen. Rein intuitiv k;nnte man prognostizie-
ren, dass sich in dem oszillierenden Spalt ein Gleichgewicht zwischen der ein- und
ausstr;menden Gasmenge einstellt. Dab ei entsteht insgesamt unter dem ReDektor
eine Art ODruckp olsterP , auf dem der ReDektor OschwimmtP . Diese Vermutung Ab er
das physikalische Funktionsprinzip der SqueezeUlmlevitation muss allerdings mit
Hilfe der Str;mungsmechanik untermauert werden. Aus diesem Grund werden im
nun folgenden Abschnitt die im Spalt auftretenden Gasstr;me genauer analysiert.
7.2.2 Die Poiseuille-Str/mung
Wie im vorangegangenen Abschnitt dargestellt wurde, wirkt auf den ReDektor
durch die Bewegung des Schwingers bei der sehr einfachen Mo dellvorstellung einer
thermo dynamisch p olytrop en ZustandsFnderung ein mittlerer Druck
h
p
i
, der den
Figure 2.3: The non-harmonic pressure oscillation caused by a harmonic motion (61)
ambient pressure p0is clearly shown.
Fig. 2.3 shows qualitatively the existence of a levitation pressure. However, to ob-
tain a quantitative result of the levitation pressure, more sophisticated models should
be build which take into account the boundary conditions such as the pressure release
at the edge of the gap. The model of squeeze film levitation can be established by
following two different routes: acoustic radiation pressure theory and gas film lubrica-
tion theory (36). The first one modifies the acoustic radiation pressure theory (will be
discussed in Chap. 4) according to the different physical conditions in squeeze film lev-
itation; the second one starts from the theory of gas film lubrication since the working
principle is actually similar. Gas film lubrication has been investigated for many years
in micro-mechanical systems commonly by solving Reynolds equation (18).
In 1996, Hashimoto et. al. (21) derived a simplified equation for the radiation
pressure in squeeze film acoustic levitation from the acoustic radiation pressure theory
presented by Chu and Apfel (10). Chu and Apfel calculated the Rayleigh radiation
pressure in an ideal gas on a perfectly reflecting target as:
p=hP−P0i=1 + γ
21 + sin(2kh)
2kh hEi(2.12)
13
2. STATE OF THE ART
Here, E is the energy density which can be expressed as
E=α2
0/4ρ0ω2/sin2kh(2.13)
in which krepresents the wave number, γa specific heat ratio, ωthe angular velocity of
the wave, a0the vibration amplitude and hthe distance between vibration source and
target. In squeeze film levitation, the levitation distance is very small compared to the
wavelength of sound in the free field. It ranges from several to several tens micrometers,
therefore sin kh ≈kh. Eq. (2.12) was simplified to a linear equation for the radiation
pressure in squeeze film levitation:
Π = 1 + γ
4ρac2a2
0
h2(2.14)
The radiation pressure Π in squeeze film levitation is reversely proportional to the
square of the levitation distance and proportional to the square of the vibration ampli-
tude a0. Hashimoto did experiments to verify Equ. (2.14). The experimental results of
maximum levitation force were 25 percent lower than the calculation results from eq.
(2.14). The author supposed that this discrepancy might be due to the finite dimension
of the surfaces and the non-uniformity of the amplitude of the radiation surface.
In 2001, Wiesendanger (61) followed the gas film lubrication theory (18;36) and
resolved the general Reynolds equation both analytically and numerically to achieve
quantitative results for the levitation forces. In 2002, Nomura and Kamakura (39)
theoretically and experimentally examined the squeeze film acoustic levitation. By
numerically solving the basic equations of a viscous fluid by means of MacCormack’s
finite-difference scheme, viscosity and acoustic energy leakage were included in the
model. In 2003, Minikes (38) studied the levitation force induced by pressure radiation
in gas squeeze films. He investigated the flow induced by vibrations perpendicular to
a flat surface and by a flexural wave propagating parallel to the surface. For the first
case, numerical and second order analytical perturbation solutions were compared and
proved to be in good agreement to each other. For the second case, a modified Reynolds
equation was derived to obtain the pressure distribution and the velocity profile in the
film for determining the reaction forces. Later in 2006, Minikes examined the validity
of the pressure release boundary condition and the isothermal assumptions by a CFD
scheme (37). By comparing his results to a one-dimensional analytical solution, the
14
2.2 Ultrasonic levitation
Figure 2.4: Configuration of the transportation system (20)
author found that the levitation force reduced to a half when the energy leakage near
the edges of the levitated object was taken in to account. This indicates that the
assumption of pressure release at the boundaries, implied in the Reynolds equation, is
inadequate in cases where the driving surface is sufficiently larger than the levitated
surface.
The important application of squeeze film levitation is to develop non-contact linear
and rotational bearings. This is the major concern of the present thesis and will be dis-
cussed in detail in a separated section later. Besides non-contact bearings, squeeze film
levitation was also used for non-contact transportation systems. In 1998, Hashimoto
(20) presented a non-contact transportation system using flexural traveling waves. Fig.
2.4 shows the configuration of the experimental setup. An aluminum plate is connected
to two longitudinal transducers and is driven in a flexural vibration mode by one of
the transducers. In order to obtain a traveling wave, one transducer acts as vibration
source and the other one as a receiver. The transportation speed can be changed by
controlling the vibration amplitude. Parallel combination of the systems can provide
transportation of wide objects. In, 2000, Amano et. al. (3) proposed a squeeze film
15
2. STATE OF THE ART
levitation system for large-size planar objects. The presented system was an extension
of Ueha’s transportation system by synchronizing multiple sets of transportation sys-
tem according to the size of the objects to be transported.
2.3 Non-contact bearings using squeeze film ultrasonic
levitation
To overcome the difficulties that are inherent to the existing non-contact bearings,
novel concepts for non-contact bearing are consistently of great interest. As one of the
promising alternative solutions, squeeze film levitation has been widely investigated for
building non-contact linear and rotational bearings. In principle, squeeze film bearing
should have most of the advantages of aerostatic bearings. Instead of pressurized air
fed through orifice or porous carbon/metal in aerostatic bearings, the load-carrying air
film is generated by high frequency vibrations and the corresponding squeeze actions
between two surfaces. External pressurized air supply is no longer needed. This feature
allows the bearing interface to be as simple as two plain surfaces. The additional effort
needed in this kind of bearing is to bring in high frequency vibration to the bearing
surfaces. Several prototype non-contact suspension and transportation systems based
on squeeze film levitation have been reported in the last decades.
In 1964, Salbu(47) first described the concept of constructing a non-contact bearing
using squeeze film action. Salbu used magnetic actuators to generate the oscillation and
the operating frequency was in the audible range, therefore the bearing was extremely
loud. In the later publications on squeeze film levitation, piezoelectric transducers in
various shapes were commonly used to generate the squeeze action effectively. Several
designs of squeeze film bearings using bulk piezoelectric ceramics can be found in early
U.S. patents filed in 1960s, invented by Warnock (56), Farron (14), Emmerich (13).
These designs used bulky piezoelectric materials to create uniform vibration amplitude
over the entire bearing surfaces. Therefore the transducers were rather massive and
required high power to generate sufficient vibration amplitude. Scranton (48) suggested
using bending piezoelectric elements to excite a flexural vibration mode of the bearing.
This led to a very compact system design and much lower power dissipation. However,
16
2.3 Non-contact bearings using squeeze film ultrasonic levitation
Figure 2.5: Linear slider bearing: a) complete system with glass rail; b) bottom view of
the slider with five bearing elements (61)
in Scranton’s patent, only the basic concept is sketched, and there was no concrete
implementation presented.
Wiesendanger (61) developed a linear guide using disc shape piezoelectric bending
elements. The transducers are placed in the sliding part. The carriage which can move
freely in a V-shaped rail made of two glass plates is shown in Fig. 2.5 (a). Five disk-
shaped piezoelectric bending elements are mounted on the carriage as shown in Fig. 2.5
b. These elements directly constitute the bearing surface, resulting in a highly compact
overall design. Wiesendanger also presented a rotational bearing using a tube shape
piezoelectric bending element, as shown in Fig. 2.6. Both devices were operated in a
resonant frequency in the ultrasonic range. Non-contact suspension was successfully
realized. Since the vibration of a bending element is sensitive to load, the vibration
amplitude decreases when load is applied on the surface. Therefore, the load capacity
of such kind of bearing is always rather limited.
In 2003 Ide et. al. (26) presented a linear bearing based on squeeze film levitation.
To hold the lateral position of the slider, a beam with an “L”-shaped cross-section was
used as a guide rail, and a slider of the same cross-section is levitated by ultrasonic
bending vibrations excited along the beam. Fig. 2.7 shows the configuration of the
proposed non-contact linear bearing. A flexural traveling wave is excited along the
“L”-shaped beam, and the slider with grooves of the same cross-section is levitated
17
2. STATE OF THE ART
Figure 2.6: Squeeze film bearing, with (right) and without (left) rotor (61)
Figure 2.7: A non-contact linear bearing using L-shaped vibrating beams (26)
and moved.
In a further study by Koyama et.al (29), the profile of the sliding table was re-
designed using FEM for high levitation and transportation efficiency (see Fig. 2.8).
Two bolt-clamped Langevin transducers are used to excite the flexural vibration mode
of the sliding guide. The levitation force and levitation capacity were measured to be
4800 N/m2and 2500 N/µm/m2respectively for a vibration amplitude of 1 µm and a
levitation distance of 2.2 µm.
Yoshimoto (65) introduced the use of elastic hinges in squeeze film bearing design
and presented a squeeze-film gas bearing with elastic hinges for linear motion guide.
Stolarski (51) followed Yoshimoto’s idea and developed a self-lifting linear air contact
using elastic hinges (see Fig. 2.9). Both Yosimoto (65) and Stolarski (51)’s devices
18
2.3 Non-contact bearings using squeeze film ultrasonic levitation
Screw nut
Transducer
40
152
11
Screw penetrating
the alumina guide
sliderUnit: mm
8
313 17 11 9
PZT Support plate
Figure 1 Illustration of the ultrasonically levitated sliding table made of
Figure 2.8: Illustration of the ultrasonically levitated sliding table made of alumina
ceramics designed by the FEM (29)
Figure 2.9: Layout of the linear motion bearing (51)
19
2. STATE OF THE ART
Figure 2.10: Squeeze film bearing using elastic hing, deformed state (17)
were operated at several kHz, out of the resonance of the piezoelectric transducer or
the bearing structure. Therefore loud noise can be heard during operation. The ob-
tained vibration amplitudes were also small (few micrometers). This leads to a strict
machining tolerance of the bearing and a limited load capacity. In 2007, Yosimoto (66)
introduced a new design of the earlier presented linear motion guide (66). The bearing
plate was excited to vibrate in a bending mode at ultrasonic frequency by two piezoelec-
tric transducers placed on top of the plate. The newly designed bearing was found to
be quiet and have higher load capacity up to 10 N. Ha (17) presented an aerodynamic
journal bearing capable of self-lift using squeeze film pressure. The basic principle was
the same as Yoshimoto (65) and Stolarski(51)’s design, using elastic hinges and piezo-
electric actuators to deform the bearing surface (see Fig. 2.10). The difference was that
the bearing has a cylindrical inner surface. During the start and stop, the squeeze film
pressure was generated by oscillating the bearing clearance to lift up the spindle; when
the spindle reached sufficiently high rotation speed, a three-lobe clearance was formed
by deforming the bearing’s inner surface using bias voltage applied to the transducers.
The bearing could then work as a normal aerodynamic bearing. The load capacity of
this bearing was found to be limited at 2.18N, with bearing clearance and frequency of
0.45 µm and 1400 Hz.
According to the model and experimental result presented by Hashimoto et. al.
20
2.3 Non-contact bearings using squeeze film ultrasonic levitation
(21), squeeze film ultrasonic levitation can provide considerably high levitation capacity
of about 7 N/cm2, which is comparable to common air bearings which normally have
load capacity in the range of 10 to 20 N/cm2. However, in practice, the load capacity
is restricted by many factors such as the performance of the ultrasonic transducer, the
energy efficiency, the limitation of the dimension and so on. The previously reported
squeeze film bearings all had very limited load capacities which are much lower than
the load capacity measured by Hashimoto et. al. (21) with a piston vibration source.
Up to date, the maximum load capacity per unit area of a squeeze film bearing was
achieved by the linear air bearing presented by Yosimoto (66), which is about 1 N/cm2.
The load capacity of the previous squeeze film bearings is inadequate for most practical
applications. Therefore, it is of great importance to improve the load capacity of the
squeeze film bearings.
21
2. STATE OF THE ART
22
3
Research objective and thesis
outline
3.1 Research objectives
The goal of the present thesis is to provide a guideline of how to apply ultrasonic lev-
itation technique to develop non-contact suspension systems with high load capacity.
As discussed in the preceding chapter, ultrasonic levitation has good potential to be
applied in non-contact bearing systems. Squeeze film type ultrasonic levitation has
already been applied to develop non-contact bearings by several researchers. However,
the load capacity achieved by the existing prototypes is still low and inadequate for
many applications which require high load capacity. In the current thesis, squeeze film
type ultrasonic levitation will be investigated theoretically to find the crucial design
parameters and to improve the levitation capacity. The theoretical model will be vali-
dated using experimental results. A novel non-contact journal bearing will be developed
based on the theoretical investigation, which aims on high load capacity. The proposed
bearing should have comparable performance as other types of non-contact bearings
such as conventional air bearing.
Surprisingly enough, standing wave type ultrasonic levitation has not found tech-
nical applications in non-contact bearings. This is mainly due to the fact that the
conventional radiator-reflector structure for levitation of small particles has very lim-
ited load capacity and can only levitate elements which are smaller than the sound
23
3. RESEARCH OBJECTIVE AND THESIS OUTLINE
wavelength. In this thesis work, we present a configuration which levitates a planar
object which corresponds to the “reflector” in the conventional configuration by form-
ing a standing wave field. In other words, the reflector is the levitated object. The
configuration will be able to levitate large planar objects at certain positions at which
a standing wave is formed by the radiator and the levitated planar object. Such a
levitation system can be applied to build non-contact bearings for applications which
require low load capacity but very high separation distance. The theoretical model for
the proposed levitation system will be established. A prototype system will be devel-
oped to validate the proposed method. To the best of the author‘s knowledge, this is
the first detailed investigation of levitation systems of this type.
3.2 Thesis outline
Chap. 2discusses the state of the art of ultrasonic levitation and its application in
non-contact suspension. Chap. 4and 5provide the fundamentals of acoustics theory
and the basics of piezoelectric ultrasonic transducer. The knowledge is essential to
understand the context in the further chapters about ultrasonic levitation systems.
Chap. 6and 7focuse on the standing wave type levitation system. In Chap. 6the
conventional standing wave levitation is reviewed first, and followed by a novel con-
figuration which improves the levitation capacity and make it suitable for suspending
large planar objects. A prototype system for suspension of large planar objects using
the proposed configuration will presented in Chap. 7.
Chap. 8and 9focuse on the squeeze film type levitation. The basic working prin-
ciple of squeeze film levitation is presented in Chap. 8. Mathematical models for
predicting the levitation force will be presented. The proposed models are validated
using experiment results. Once again, a prototype system will be presented in Chap.
9. The prototype system is a non-contact journal bearing for rotating elements such a
machine spindle. Design and performance of the proposed bearing will be investigated
and presented in this chapter.
24
3.2 Thesis outline
To conclude, a summary of the achieved results and an outlook on future work is
given at the end.
25
3. RESEARCH OBJECTIVE AND THESIS OUTLINE
26
4
Basic theory on acoustics
The study of high-power acoustic effects on acoustic levitation requires certain knowl-
edge of acoustics and ultrasonics.Therefore, the basic principles of ultrasonics including
the propagation of low and finite amplitude waves in fluids will be introduced in this
chapter. Nonlinear effects such as radiation pressure and increased absorption will also
be discussed.
4.1 Linear theory
4.1.1 Elastic waves in fluids
To describe the wave motion, one has to establish the relationship between the distur-
bance (i.e. the displacement of the medium particles from their equilibrium positions),
time, and distance from the source of the oscillations. Sound waves traveling in gases
(air) represent an alternating flow and obey the laws of hydrodynamics. If viscosity
and thermal conductivity are neglected at the outset, a complete set of hydrodynamic
equations can be written as (2;46):
ρD~v
Dt =−∇p+~gρ (4.1)
Dρ
Dt +∇·(ρ~v) = 0 (4.2)
f(P, ρ, T) = 0 (4.3)
27
4. BASIC THEORY ON ACOUSTICS
in which, prepresents the pressure, ρthe medium density, ~v the velocity vector, Tthe
absolute temperature, and ~g the acceleration vector due to gravity. The convective
time derivative D
Dt is a derivative taken with respect to a moving coordinate system,
which is given by D
Dt =∂
∂t +~v ·∇ (4.4)
where ∇is the gradient operator.
Equ. 4.1 is called Euler equation which describes the motion of particles subjected
to a pressure gradient and gravity forces. Equ. 4.2 is known as the equation of conti-
nuity which is valid if there are no discontinuities in the medium, as in the absence of
cavitation. Equ. 4.3 is the equation of state, which provides a mathematical relation-
ship between the state functions associated with the matter. Its specific form depends
on the material properties and the given physical conditions. In many applications of
acoustics, it is these equations that are used as a starting point. Due to the convective
terms, Equ. 4.1 to 4.3 are nonlinear in the unknowns. The nonlinear equations can be
linearly approximated for the case of low-amplitude waves.
4.1.2 Equations of linear acoustics
In the case of low amplitude acoustic waves, changes in density ρ′and pressure pare
small as compared with ρ0and P0. In other words:
P−P0
P0
=
p
P0≪1 (4.5)
ρ−ρ0
ρ0
=
ρ′
ρ0≪1 (4.6)
The ambient fluid velocity (such as wind in air) may be neglected if it is much less
than the sound speed c. The effects of the gravitational force can usually be neglected.
Analogous arguments can be made for the thermal conductivity and viscosity. By
neglecting gravity and assuming the ambient fluid velocity is zero, the ambient pressure
is then constant. In such cases, Equ. 4.2 can be linearized as
∂ρ
∂t +ρ0∇·~v = 0 (4.7)
and the Euler equation becomes
ρ∂~v
∂t =−∇p(4.8)
28
4.1 Linear theory
A single partial differential equation for the acoustic part of the pressure can be obtained
by taking the time derivative of Equ. 4.7 and then using Equ. 4.8 to re-express the
time derivative of the fluid velocity in terms of pressure. The resulting equation is
∇2p−1
c2
∂2p
∂t2= 0 (4.9)
which is the general wave equation of linear acoustics.
A solution of the wave equation that plays a central role in many acoustical concepts
is that of a plane traveling wave. The mathematical representation of a plane wave is
such that all acoustic field quantities vary with time and with one Cartesian coordinate
(usually taken as z) only. For a plane wave traveling in the positive z-direction at a
velocity cand a constant angular frequency ω, the acoustic pressure disturbance can
be expressed as
p=A0cos [k(z−ct)] = A0cos(kz −ωt) (4.10)
in which, k=ω/c is the wave number. In complex representation, the pressure can be
written as
p= Re A0ej(kz−ωt)(4.11)
where Re indicates the real part of the expression.
4.1.3 Acoustic energy density and intensity
In a plane progressive harmonic wave both pressure and velocity are in phase. Potential
and kinetic energy are transported by the wave without transportation of mass. The
potential energy density Vdue to compression of the fluid and the kinetic energy density
Kof the wave can be expressed as:
V=1
2
1
ρ0c2p2(4.12)
K=1
2ρ0v2(4.13)
The total energy density Eis then
E=V+K=1
2
1
ρ0c2p2+1
2ρ0v2(4.14)
29
4. BASIC THEORY ON ACOUSTICS
The acoustic intensity ~
Iis defined as
~
I=p~v (4.15)
For a plane wave, the kinetic energy density and the potential energy density are
the same, and the total energy density is given by
E=1
ρ0c2p2(4.16)
4.1.4 Atmosphere absorption of sound wave
When a sound wave travels through air, a proportion of the sound energy is converted to
heat. There are losses due to heat conduction, shear viscosity and molecular relaxation.
The air absorption becomes significant at high frequencies and at long range (46). For
a plane wave, the pressure |ˆp|at a distance zfrom a position where the pressure is |ˆp0|
is given as
|ˆp|=|ˆp0|e−αz (4.17)
The attenuation coefficient αfor air absorption depends on frequency, humidity, tem-
perature and atmospheric pressure and may be calculated using the equations given in
Refs. (5;6),
α=f2(1.84 ×10−11 T
T0ps0
ps+T
T0−5/2
0.01278exp(−2239.1/T)
frO +f2/frO
+ 0.1068exp(−3352/T)
frN +f2/frN (4.18)
where frepresents the frequency of the wave, psthe atmospheric pressure, ps0 the
reference atmospheric pressure, Tthe atmospheric temperature in K, T0the reference
atmospheric temperature, frO and frN the relaxation frequencies of the molecular oxy-
gen and nitrogen. The relaxation frequencies frO and frN can be calculated as
frO =ps
ps0 24 + 4.04 ×104cw
0.02 + cw
0.391 + cw(4.19)
frN =ps
ps0 T0
T1/2 9 + 280cw×exp (−4.17 "T0
T1/3
−1#)! (4.20)
30
4.2 Nonlinear theory - acoustic radiation pressure
Figure 4.1: Sound absorption coefficient in air (dB/100 m) versus frequency/pressure
ratio for various relative humidities (in percent) at 20◦C (5)
respectively, where cwis the concentration of water vapor in percent. The relation
between cwand relative humidity hris
cw=hr(psat/ps0)/(ps/ps0) = hr
psat
ps
(4.21)
where the saturated vapor pressure psat is given by
log10
psat
ps0
=−6.8346 T01
T1.261
+ 4.6151 (4.22)
where T01 = 273.16 K is the triple-point temperature.
Employing the above formulas, the sound absorption coefficient in air versus fre-
quency/pressure ratio was calculated for various percent relative humidity at 20◦C by
Bass (5). The results are plotted in Fig. 4.1.
4.2 Nonlinear theory - acoustic radiation pressure
A body immersed in a sound field is known to experience a steady force that is called the
acoustic radiation pressure. This force plays an important role in acoustic levitation.
Acoustic radiation pressure is owing to the relative motion of the body and the fluid
elements in the medium. Acoustic radiation pressure was first studied by Rayleigh (44)
31
4. BASIC THEORY ON ACOUSTICS
in 1902 as an acoustic counterpart of electromagnetic waves. The radiation pressure on
the object in the sound field varies with the frequency of the vibrations and equals zero
in a linear approximation. A second order approximation is needed to obtain a non-zero
pressure, which is small compared with the sound pressure amplitude. Rayleigh found
that the acoustic radiation pressure on a perfectly reflecting surface due to a normally
incident plane sound wave in an ideal gas is
pra =γ+ 1
2hEi(4.23)
where γis the ratio of the specific heats of the gas, and hEithe time averaged energy
density of the standing wave formed by the incident and reflected waves. From these
basic equations, the subject has been investigated by many researchers with differ-
ent results and had been continued to be associated with a lot of confusion for a few
decades, mainly because the phenomenon involves a subtle nonlinear effect. Chu noted
that the problem has to be very carefully posed in order to get a unique answer (9). In
this work, the approach of Lee and Wang (30) is adopted to explain the principles of
acoustic radiation pressure.
Euler equation for an ideal fluid is written as
∂~v
∂t + (~v ·∇)~v =−∇P
ρ(4.24)
Here P=P0+pis the total total pressure, in which P0is the ambient pressure and
pthe acoustic pressure. Since sound oscillation is irrotational, one can write ~v =∇φ,
where φis the velocity potential. Then Equ. 4.24 can be written as
∇∂φ
∂t +1
2|∇φ|2=−∇P
ρ(4.25)
The first law of thermodynamics gives (46)
dw=TdS+ dP/ρ (4.26)
where Tis the temperature, and Sand ware the entropy per unit mass and the enthalpy
per unit mass of the fluid respectively. Assuming that the motion is adiabatic, one can
have dw= dP/ρ, or ∇w=∇P/ρ. Thus Equ. 4.25 can be integrated once in space to
give
w=−∂φ
∂t −1
2|∇φ|2+C′(4.27)
32
4.2 Nonlinear theory - acoustic radiation pressure
where C′is constant in space but can depend on time.
The pressure Pcan be expanded in Taylor series in was
P=P0+dP
dws,0
w+1
2d2P
dw2s,0
w2+··· (4.28)
where the subscript s, 0 means evaluated at constant entropy and at equilibrium. Since
(dw/dP)s= 1/ρ, we have then (dP/dw)s=ρ=ρ0, and
d2P
dw2s
=dρ
dws
=dρ
dPsdP
dws
=ρ
c2=ρ0
c2
0
(4.29)
making use of the elementary relation (dP/dρ)s=c2and letting all these quantities
take equilibrium values. Equation 4.28 becomes
P=P0+ρ0−∂φ
∂t −1
2|∇φ|2+C′+1
2
ρ0
c2
0−∂φ
∂t −1
2|∇φ|2+C′2
+··· (4.30)
By time averaging Equ. 4.30 and keeping up to the second order, we obtain
hP−P0i=−1
2ρ0∂φ2+1
2
ρ0
c2
0*∂φ
∂t 2++C(4.31)
Where Cis a constant in space and time. Since it is sufficient to keep up to second
order only, in the quadratic terms on the right side of Equ. 4.31 will be sufficiently
represented by its linear solution. Substituting ~u =∇φand ∂φ/∂t =−p/ρ0into Equ.
4.31, we have the mean Eulerian excess pressure as (30)
PE−P0=1
2
1
ρ0c2p2−1
2ρ0v2+C=hVi−hKi+C(4.32)
PE−P0is called Eulerian excess pressure because it is evaluated at a fixed point in
space, as opposed to the Lagrangian one. The mean Lagrangian excess pressure can
be derived from the Eulerian one using the relation between the Lagrangian and an
Eulerian quantities (30) as
PL−P0=hVi+hKi+C=hEi+C(4.33)
For 1-D case in general, if the material surface vibrates with the sound, the mean
pressure on it is Lagrangian. In the case of a rigid surface, the mean pressure on it
becomes Eulerian (30). The mean pressure on the material is the acoustic radiation
33
4. BASIC THEORY ON ACOUSTICS
pressure. The levitation force can be obtained by integrating the acoustic radiation
pressure over the object surface. However, in different types of levitation systems, dif-
ferent boundary conditions and calculation methods are needed. And, the considered
levitated objects are very unlike. This will be discussed in detail when the specific type
of levitation is concerned in the following chapters.
34
5
Piezoelectric ultrasonic
transducers
The essential requirement to realize any kind of application of high-intensity ultra-
sound, such as ultrasonic levitation, is the generation and transmission of mechanical
vibrations of a certain intensity and frequency. The devices which convert a particu-
lar type of energy (electrical or mechanical) into acoustic energy are called ultrasonic
transducers. To date, piezoelectric ultrasonic transducers are most widely used in ul-
trasonic technology. The performance of an ultrasonic levitation system will largely
depend on the ultrasonic transducer that is employed in the system. Therefore, the
understanding of the working principle and the basic design methods is important for
investigating ultrasonic levitation systems. This chapter is dedicated to explain the
basic modeling, design and driving methods of piezoelectric ultrasonic transducers, in
particular the Langevin type transducers.
5.1 Piezoelectric actuator
5.1.1 Piezoelectric effect
The piezoelectric effect was discovered by Jacques and Pierre Curie in 1880. They found
that if certain crystals were subjected to mechanical strain, they became electrically
polarized and the degree of polarization was proportional to the applied strain. The
Curies also discovered that these same materials deformed when they were exposed to an
35
5. PIEZOELECTRIC ULTRASONIC TRANSDUCERS
33
(a) (b)
Figure 5.1: Piezoelectric actuators. a), d33, b), d31
electric field. This is known as the inverse piezoelectric effect which is the fundamental
for the application of piezoelectric materials in actuators (27).
The electric field deforms a piezoelectric body in different directions with different
intensities. For simplicity, only the two most often used types of piezoelectric effects
for building piezoelectric actuators are show in Fig. 5.1, namely the longitudinal and
transversal deformation due to an applied electric field in the direction of polarization.
As piezoelectric materials are anisotropic, the polarization is done in one direction
which is normally given an index of 3, and the directions normal to that with index 1
and 2. When subject to an electrical field in the direction of polarization, piezoelectric
element which can deform freely exhibits a strain that can be written as
s=E3d3i(5.1)
in which srepresents the resulted strain, E3the applied electrical field and d3ithe charge
constant which has the dimension of m/V. For actuators using the longitudinal effect,
the piezoelectric element deforms primarily in the same direction as the applied electric
field. As of the transversal effect d31, the deformation is in the direction perpendicular
to the direction of applied electrical field. Fig. 5.1 shows the working principle of the
two types of piezoelectric effects.
5.1.2 Piezoelectric actuators
Piezoelectric actuators convert electrical energy into mechanical force and motion using
piezoelectric elements, most often piezoelectric ceramics. According to the electrical
36
5.1 Piezoelectric actuator
drive method, they can be divided into two categories: the resonant driven piezoelec-
tric actuators and the non-resonant driven piezoelectric actuators. The resonant driven
piezoelectric actuators include e.g. ultrasonic transducers (converters), ultrasonic mo-
tors and piezoelectric transformers. The non-resonant driven piezoelectric actuators
include various one-stroke actuators. Their operating frequency range is from quasi-
static up to about half of the first resonant frequency of the mechanical system.
Typical strain levels that can be obtained by piezoelectric materials are in the region
of 0.1%. For quasi-static applications, this strain occurres at field strength in the region
of 1000 V/mm. This maximum field strength is limited to approximately 75% of the
value of the coercive field. In order to increase the total displacement capability of a
piezoelectric actuator, stack and multilayer piezoelectric elements are often used. For
resonant driven actuators, another common way to increase the dynamic displacement
is to use amplifying mechanisms such as horns and boosters.
5.1.3 Lumped parameter model
The dynamic behavior of a linear system subjected to harmonic excitation can usually
be described with sufficient accuracy by superposing only a few of the eigenmodes. Each
eigenmode dominates the vibration behavior of the system in the range of the respective
resonant frequency. Therefore, if a piezoelectric system is driven in the range of one
of its resonant frequencies, its behavior can most often be described with reasonable
accuracy by a model with only one degree of freedom.
Based on electro-mechanical analogies the vibration behavior of a piezoelectric ac-
tuator operating in the vicinity of one of its resonant frequencies can be described by an
equivalent mechanical or electrical model as shown in Fig. 5.2 and 5.3 (33;54). In the
models, m,c,d,Cand Rare modal mass, modal stiffness, modal damping, electrical ca-
pacitance and electrical resistance, respectively. The electromechanical transformation
factor αdescribes the transmission ratio of electrical and mechanical quantities. The
input voltage and charge are represented by Uand Q. The modal displacement and
mechanical load are represented by uand F. According to these models, the dynamics
37
5.1 Piezoelectric actuator
of the system can be described by
m¨u+d˙u+cu =αU +F(5.2)
1
CQ−αu+R˙
Q−α˙u=U(5.3)
For harmonic excitation U(t) = Re[ ˆ
U·ejΩt], the relation between inputs and outputs
can be written as (22)
ˆ
I
ˆ
˙u=y11 y12
y21 y22ˆ
U
ˆ
F="jΩ
jRΩ+ 1
C
+jΩα2
−mΩ2+jdΩ+c
jΩα
−mΩ2+jdΩ+c
jΩα
−mΩ2+dΩ+c
jΩ
−mΩ2+jdΩ+c#ˆ
U
ˆ
F(5.4)
where ˆ
U,ˆ
I,ˆ
Fand ˆ
˙uare complex amplitudes, ˆ
I=jΩˆ
Qand ˆv=jΩˆu.Yis the transfer
matrix, which is also called the conductance matrix. The elements of the transfer
matrix are defined as
y11 =ˆ
I
ˆ
U=jΩ
jRΩ + 1
C
+jΩα2
−mΩ2+ jdΩ + c(5.5)
: the short-circuit input admittance
y21 =ˆ
˙u
ˆ
U=jΩα
−mΩ2+ dΩ + c(5.6)
: the short-circuit core admittance (forward)
y12 =ˆ
I
ˆ
F=jΩα
−mΩ2+ jdΩ + c(5.7)
: the short-circuit core admittance (backward)
y22 =ˆ
˙u
ˆ
F=jΩ
−mΩ2+ jdΩ + c(5.8)
: the short-circuit output admittance
Figure 5.4 shows the typical variation of the amplitude and phase of the shortcircuit
input admittance y11 as a function of frequency (Bode plot) for a piezoelectric actuator.
Figure 5.5 gives the corresponding locus of y11 in the complex plane (Nyquist plot).
Three pairs of characteristic frequencies can be identified in Fig. 5.5, namely the
series and parallel resonant frequencies fsand fp, the maximum admittance and mini-
mum admittance frequencies fmand fnas well as the resonant and anti-resonant fre-
quencies frand fa. Frequencies frand faare the frequencies at which the phase of the
39
5.2 Langevin type ultrasonic transducers
Central
bolt
Rear
cover
Piezoelectric
ceramics
Front
cover
Figure 5.6: Typical structure of a Langevin type ultrasonic transducer
admittance becomes zero. For weakly damped systems, fm≈fr≈fsand fn≈fa≈fp.
The model parameters can be calculated from measurement result of admittances.
Thus, once the electrical or mechanical admittance of a given piezoelectric system is
measured, the dynamic behavior of the system can be precisely described by the model
shown in Fig. 5.2 and 5.3 (15;22).
5.2 Langevin type ultrasonic transducers
For ultrasonic applications, piezoelectric transducers are the most widely used element
to generate ultrasonic waves. For power ultrasonic applications such as ultrasonic ma-
chining, ultrasonic wire bonding, as well as ultrasonic levitation, the working frequencies
are often between 20 kHz and 100 kHz, with high requirements on the power, efficiency
and vibration amplitude of the employed ultrasonic transducers. For such applications,
Langevin (or sandwich) transducers are often used. A Langevin transducer is composed
of head and tail masses, a central bolt for pre-stressing and piezoelectric ceramic rings
pressed in the middle. A typical Langevin type transducer is shown in Fig. 5.6.
Langevin transducers have several advantages. Because a mechanical pre-stress is
applied on the piezoelectric elements by means of a central bolt or peripheral sleeve,
the admissible dynamic stress amplitude and hence the maximum power intensity is
considerably increased. The mechanical contact between the parts is improved due to
the pre-stressing; hence the mechanical damping is decreased. The metal end sections
41
5. PIEZOELECTRIC ULTRASONIC TRANSDUCERS
Transducer Horn/Booster Tool
Figure 5.7: The principle of the λ/2-synthesis
are good heat-sinks, so that the transducer can be driven at higher vibration levels than
other types of ultrasonic transducers. As the manufacture of metal is much easier than
that of piezoelectric materials, more variations of shape and dimension of transducers
are available.
5.2.1 The half-wavelength-synthesis
Langevin transducer is often designed to have length equal to half of the wavelength
of the first longitudinal vibration mode of the transducer, often called as λ/2 vibration
mode. The reason is that a λ/2 transducer can be combined with other λ/2 parts
like boosters or tools to form a whole ultrasonic device without obvious changes of the
resonance compared with that of each part before synthesis (33). The advantage of the
λ/2-synthesis is that in the ideal case no forces act at the interfaces between individual
parts. Therefore, the boundary conditions of each part in the whole synthesized system
are the same as those of each part free at both sides. Under this condition, each part
(transducer, booster or tool) can be first developed according to the specified resonance
frequency and then they are synthesized into a whole device. Figure 5.7 describes the
λ/2-synthesis principle schematically.
5.2.2 Dimensioning method
To design a Langevin type ultrasonic transducer a lot of design parameters and aspects
have to be considered, such as working frequency, radial dimensions, amount of piezo-
ceramics, materials for front and rear covers, output power, impedance matching and
42
5.2 Langevin type ultrasonic transducers
mds
L3L2L1L5
L4
Rear cover PZT Front coverNodal plane
Figure 5.8: Design of the power ultrasonic transducer with nodal plane inside of the front
cover
so on. Many of these parameters are determined by the requirement of specific applica-
tions. The detailed design method can be found in Refs. 1;43;49. Here, a simple case is
demonstrated in which all parameters are fixed except the axial lengthsL1, L2···, L5.
A schematic diagram of the transducer is shown in Fig. 5.8. The nodal plane of the
transducer is designed to be within the front cover, so that the transducer can be mod-
eled as two individual 1/4 wavelength systems divided by the nodal plane. Consider the
longitudinal vibration of a thin bar with uniform cross section, the frequency equation
of the 1/4 wave length transducer on the left side of the nodal plane can be written as
(43):
z3
z2
tan k2L2tan k3L3+z3
z1
tan k1L1tan k3L3+z2
z1
tan k1L1tan k2L2= 1 (5.9)
The equation for the stepped horn on the right side of the nodal plane can be written
as:
tan k4L4tan k5L5=z4
z5
(5.10)
in which, zi, ki, Li, (i= 1,2,3,4,5) are the impedances, wave numbers and lengths of
each part of the transducer respectively.
zi=ρiciSi(5.11)
in which ρi, ci, Si, (i= 1,2,3,4,5) are the density, sound speed and area of cross-section
of each part of the transducer. Unknown lengths of the transducer parts can be calcu-
lated directly form the above two equations.
43
5. PIEZOELECTRIC ULTRASONIC TRANSDUCERS
5.2.3 Performance criteria
The performance of piezoelectric transducers is evaluated by different criteria according
to different applications. For the power ultrasonic transducers, the most commonly
performance criteria are collected here.
•Resonance frequency
The resonance frequency of the transducer should be equal to the specified working
frequency. Commonly, this resonance frequency is that of the first longitudinal vibration
mode.
•Input/output power
The input power pe(t) of the transducer is given as
pe(t) = U(t)·I(t) (5.12)
where U(t) and I(t) are the AC input voltage and current respectively. When the
system is not driven in a region around the resonant frequency, it will show a capacitive
behavior. The mean effective power is given by
Pe=1
TZT
0
U(t)I(t)dt (5.13)
In case of harmonic U(t) and I(t) the effective power results as
ˆ
Pe=1
2ˆ
Uˆ
Icos φe=ˆ
Pacos φe(5.14)
where ˆ
Pais the apparent power, φeis the phase difference between U(t) and I(t). The
apparent power determines the size of the power electric device used to drive transduc-
ers. Therefore, in order to reduce the size of the power electric device, the apparent
power should be minimized for a given output requirement such as a given amplitude
or mechanical power.
The mechanical output power of a transducer can be calculated as
pm(t) = F(t)·v(t) (5.15)
44
5.2 Langevin type ultrasonic transducers
The effective power for harmonic vibrations can be calculated by analogy with the
electrical one as
ˆ
Pm=1
2ˆ
Fˆvcos φm=ˆ
Pacos φm(5.16)
where ˆ
Pais the apparent power, φmis the phase difference between F(t) and v(t).
•Efficiency and power efficiency
The efficiency ηis defined as a ratio of the output mechanical energy to the consumed
electrical energy. For the harmonic vibration of the transducer, ηcan be calculated as
η=ˆ
Pm
ˆ
Pa
=ˆ
Fˆvcos φm
ˆ
Uˆ
Icos φe
(5.17)
The power efficiency λpcan be defined as the ratio between the mechanical energy
delivered and the electrical energy absorbed by the transducer. For the harmonic
vibration of the transducer, this results in
λp=ˆ
Pm
ˆpe
=ˆ
Fˆvcos φm
ˆ
Uˆ
I(5.18)
Obviously, the high possible efficiency and power efficiency are desirable for the
piezoelectric transducers.
•Mechanical quality factor
The mechanical quality factor Qmis a measure for the resonance rise of the piezoelectric
transducer. It can be derived from the 3dB bandwidth of the admittance at fsas
Qm=fs
∆fs(3 dB) =fs
f2−f1
(5.19)
The frequencies f1and f2are frequencies that correspond to the admittances that
are 3dB lower than the maximal admittance, respectively. A large mechanical Qm
corresponds to a large efficiency of the piezoelectric transducer and a large resonant
amplitude. The material and structural damping of the piezoelectric transducer mainly
determine the mechanical quality factor.
•Piezoelectric quality number
45
5. PIEZOELECTRIC ULTRASONIC TRANSDUCERS
According to Fig. 5.5, a piezoelectric quality number is geometrically defined as
M=Yr
Yc
(5.20)
where Yr=Ymax −Ymin is the diameter of the locus of y11 as shown in Fig. 5.5, in which
Ymax and Ymin are the values of y11 at the frequencies fmand fn, respectively. The
piezoelectric quality number is an important performance parameter. It presents the
extent of the phase rise or phase drop of the admittance functions and is appropriate
to classifying piezoelectric actuators concerning electrical behaviors. When M < 2,
the resonance and anti-resonance frequencies do not exist anymore, the transducer
can not be driven with zero reactive power. More apparent power is needed which
leads to a large power electric device. Therefore, the piezoelectric quality number of
the transducer used in ultrasonic bonding and machining should be larger than two.
Furthermore, the larger the value of M is, the better the phase reserve of the transducer.
When M > 2, it is assured that the resonance frequency exists and that the transducer
can be driven with zero reactive power even though the load damping may be large
(15).
5.3 Driving method
The ultrasonic transducers are resonance driven systems, therefore the driving fre-
quency must match the resonant frequency of the system. It is worth mentioning that
the term “resonant frequency” is originally a concept of free vibrations to describe
the frequency when the system has large vibration amplitude. For forced vibration,
“resonant frequency” is often misused as the frequency when the response amplitude
is maximal. It has become an accepted mistake. Therefore, we will also use the term
“resonant frequency” in this context to discuss the forced vibration of piezoelectric
transducers.
The resonance frequency of every individual transducer varies slightly due to manu-
facturing tolerances. Moreover, the resonant frequency is also subject to change during
operation from change of load, temperature, input power and so on. Therefore, it is nec-
essary to implement a resonance tracking scheme that can adjust the driving frequency
during operation. This is especially important when the system has a high mechanical
46
5.3 Driving method
Filter
Piezo
actuator
Power
amplifier
Current sensor
i t
u t
Figure 5.9: Basic scheme of a self oscillating circuit
quality factor Qm. Two widely used resonance tracking methods, self oscillating circuit
and Phase Locked Loop (PLL) control, will be introduced here.
5.3.1 Self oscillating circuit
The basic idea of a self oscillating circuit is to make the whole circuit unstable at the
desired vibration frequency in such a way that the system develops a limit cycle. To
do so, two conditions must be satisfied: first, the feedback must be positive, i.e. the
total phase shift in the feedback loop should be zero. Second, the loop gain must be
greater than unity so that there is a net amplification within the loop. If these two
requirements are met at a certain frequency (eigenmode), a self-induced oscillation will
be sustained (61). If, however, the requirements are met at more than one frequency
(eigenmode), a filter that is placed in the feedback path will allow to pick the desired
one. The basic scheme of a self oscillating circuit is shown in Fig. 5.9.
There are some drawbacks of the self oscillating circuit which make it unsuitable for
certain applications such as squeeze film levitation. The efficiency of the system is
poor. Moreover, at high power levels and in applications where loading conditions may
vary strongly over time, the system becomes highly nonlinear, so that the conditions
for instability are not fulfilled any more. As a result, the system will fail to work.
5.3.2 Phase-locked-loop (PLL) controller
The PLL controller is a more robust alternative algorithm to operate piezoelectric
transducers in their resonant frequency. The block-diagram of the loop is shown in Fig.
5.10.
47
5. PIEZOELECTRIC ULTRASONIC TRANSDUCERS
Phase
detector
Filter
VCO Piezo
actuator
Power
amplifier
Current sensor
el
M
u t
i t
Figure 5.10: Block diagram of phase-locked-loop algorithm
The PLL consists of a phase detection, a filter, a voltage controlled oscillator (VCO),
and a current sensor. The piezoelectric actuator (the ultrasonic transducer) is driven
by a power amplifier that, in turn, is driven by a voltage-controlled oscillator. The
change in output frequency of the VCO from its initial value is proportional to the
applied voltage at the input. The inputs to the phase detector are the actual driving
voltage and current on the transducer. The phase difference between these two signals
must be controlled to be zero to maintain resonance. After the filter, the output of
the phase detector is a voltage proportional to the phase difference between the sig-
nals. This voltage is proportional to the error between the driving frequency and the
resonant frequency and is applied to the VCO to bring the driving frequency toward
resonance (50). The desired driving power level is obtain by adjusting the gain of the
power amplifier.
The PLL controller is more complicated than the self oscillating circuit, but it
provides more stable and reliable operation. Therefore it is more suitable for driving
piezoelectric transducers in resonance. However for a highly damped system, in which
the minimum phase value of the admittance between resonance and anti-resonance
does not fall below the zero-phase line, the PLL will not be able to find the resonant
frequency based on the phase measurements. Thus a PLL will also fail to function.
In such cases, a load-adaptive phase controller, like the one developed by Littmann
(34), can be used as an extension of the conventional PLL controller. The presented
APLL controller includes a transducer model and is able to estimate the phase of
the mechanical admittance (v/u), which always has a phase zero-crossing, based on
48
5.3 Driving method
the measurement of electrical admittance. Using the APLL controller, highly damped
systems can be controlled to work at the mechanical resonant frequency.
49
5. PIEZOELECTRIC ULTRASONIC TRANSDUCERS
50
6
Standing wave ultrasonic
levitation
Standing wave type levitation is commonly used for levitating of small particles. Thor-
ough theoretical basis and successful applications have been reported over the last
decades. However, the size of objects that can be levitated by standing wave levitation
is limited to be smaller than the sound wavelength (mm range). This characteristic
prevents standing wave ultrasonic levitation from many potential applications in which
suspension of larger, heavier objects is needed. In this chapter, a new configuration of
standing wave levitation will be introduced which has an improved levitation capacity
and has no restriction on the dimension of the objects anymore. The corresponding
theoretical model is established for the proposed configuration.
6.1 A configuration for large planar objects
The typical radiator-reflector type standing wave levitation system, as discussed in
Chap. 2, has very limited load capacity and strict restrictions on the size of the
levitated objects. Modifications and improvements are needed before standing wave
ultrasonic levitation can be applied for non-contact suspension systems.
In 2001, Reinhart (45) reported that while measuring the squeeze film levitation
force, additional peaks of levitation force at intervals of half wavelengths from the radi-
ation surface were also observed. These additional peaks had much smaller amplitudes
51
6. STANDING WAVE ULTRASONIC LEVITATION
Standing wave
Sound radiator
Levitated object
2
n
O
Figure 6.1: Schematic diagram of the proposed acoustic levitation system
(<0.1N) compared to the squeeze-film region. Reinhart concluded that they were
caused by the standing wave pattern and pointed out a possibility of levitating planar
objects at these points. However, no further explanation or experimental investigation
was reported afterwards.
In the following parts of this chapter, an acoustic levitation system will presented
which uses the levitation effect observed by Reinhart. The levitation happens when
planar objects are placed at distances of half wavelengths in front of a radiator. A
schematic diagram of the proposed levitation system is shown in Fig. 6.1. In this
configuration, the levitation effect is achievable using similar setup of standing wave
or squeeze levitation, but the working principle is different in many aspects. First, the
acoustic field is formed in between the sound radiator and the levitated object which
performs as a reflector as in classic standing wave levitation systems. The position of
the object to be levitated determines how the sound field is maintained. Second, the
size of the object to be levitated is not limited by the wavelength, because the object is
levitated above the complete standing wave field. In the classic standing wave levita-
tion, the small particles are trapped stably slightly under a pressure node or anti-node
of a standing wave field formed by a radiator and a reflector. Thus, the particles can
only be levitated if they are smaller than a sound wavelength. At last, as compared to
squeeze film levitation, the object can be levitated at positions of multiple times of half
wavelength above the radiator. A stable levitation happens when a standing wave is
formed between the radiator and the object. However, the squeeze film levitation hap-
pens only when the object is placed extremely close to the radiator. Thus no standing
wave is formed in the gap.
52
6.2 Modeling the proposed levitation system
6.2 Modeling the proposed levitation system
The intention of the proposed acoustic levitation system is to levitate planar objects
with relative large size (a few times of the sound wavelength in air). In order to obtain
a large sound radiation surface, a circular plate which vibrates in its flexural vibration
mode is chosen as the sound radiator. The sound field to be considered can be simplified
and described as following: the sound radiator (a circular disc) vibrates in its flexural
vibration mode and generates a sound beam in front of it. The sound beam propagates
forwards and is reflected by a rigid surface (the object to be levitated) placed perpen-
dicular to the sound beam at a distance of Laway from the radiator. An acoustic
wave field from multiple reflections is formed between two surfaces. The acoustic field
becomes a standing wave when Lis equal to multiple times of half wavelength of sound
(as shown in Fig. 6.1). Excessive pressure on the rigid surface is generated by the
acoustic field.
6.2.1 Flexural vibration mode of the radiator
The selected sound radiator is a circular plate with constant thickness. Therefore,
cylindrical coordinates (r, θ, z) can be used, where ris a radius from the center, θthe
angle of that radius, and za length in the direction normal to the plane of the radiator.
The plate equation has the following general solution for the transverse vibration mode
of a circular plate (7)
Z(r, θ, t) = aijJiλijr
a+bijIiλijr
acos iθ cos 2πft (6.1)
where Z(r, θ, t) is the displacement of the mid-surface of the plate, athe radius of the
circular plate, fthe natural frequency of the related mode shape. The subscripts iand
jare the number of nodal diameters and nodal circles (not counting the boundary)
respectively. Functions Jiand Iiare Bessel function and modified Bessel function of
the first kind relatively, of iorder. Parameters aij and bij are constants which are
determined to within an arbitrary constant by the boundary conditions and mode
number. The parameter λij is a dimensionless frequency parameter related to the
boundary conditions on the plate, the plate geometry and Poisson’s ratio. For mode
shapes with only nodal circles (i= 0) the displacement of the circular plate becomes
53
6. STANDING WAVE ULTRASONIC LEVITATION
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1
−1
−0.5
0
0.5
1
r/a
Z(r)
Figure 6.2: Vibration amplitude distribution on the radiator surface
axial symmetric. The displacement Zis then independent of θ. Equation 6.1 can be
simplified as
Z(r, t) = a0jJ0λ0jr
a+b0jI0λ0jr
acos 2πft (6.2)
The distribution of the vibration amplitude of the mid-surface of the plate can be then
given as
Z(r) = a0jJ0λ0jr
a+b0jI0λ0jr
a(6.3)
The vibration mode of a free circular plate with two nodal circles is shown in Fig. 6.2.
Values of λij of common mode shapes have been calculated and listed in Ref. (7), where
λ02 is given as 6.209. By applying the boundary conditions, a02 and b02 are determined
as 0.997 and 0.003 respectively.
6.2.2 Sound beam in the acoustic near-field
According to Ref. (49), the acoustic far-field is approximately reached when zexceeds
a2/λ, where ais the radius of the radiator, and λthe wavelength of sound. For the
proposed levitation system, the working frequency is about 20 kHz with wavelength
λbeing about 0.017 m. The radius of the sound radiator is 0.06 m. These makes
a2/λ = 0.208 m, which is about twelve times of λ. The investigated range of the
levitation distance Lis around one to a few times of half wavelengths. Therefore the
considered sound field is completely located in the acoustic near-field.
54
6.2 Modeling the proposed levitation system
A sound beam can be described in cylindrical coordinates with Bessel functions in
rand sinusoidal function in zdirection (30),
p(r, z, t) = A0J(r) cos(ωt −kz) (6.4)
where A0is the amplitude of acoustic pressure at r= 0, kthe wave number, ωthe
angular frequency of the wave, zthe distance from the vibration source and J(r) the
Bessel functions describing the radial distribution of the sound beam. For calculation
it is convenient to use a complex number representation. Equation 6.4 can be then
written as
p(r, z, t) = Re nA0J(r)ej(ωt−kz)o(6.5)
where Re indicates the real part of the expression. In the following Re will not be
mentioned for simplification. All the complex parameters are automatically considered
as their real parts.
Unlike a plane wave, in a sound beam described by Equ. 6.5, air particles move
in both rand zdirections. However, in the near-field, the acoustic field is basically
cylindrical, the oscillations in the rdirection are small compared to zdirection and have
very little contribution on the radiation pressure. Therefore, the velocity component
in rdirection is not taken into account in the present model. A sound beam with the
same diameter as the radiator is considered. Assuming that the acoustic pressure at
the surface of the radiator varies exactly according to the transverse movement of the
radiator, the Bessel functions in Equ. 6.5 can be replaced by Z(r). The sound beam
in near-field can be then expressed as:
p(r, z, t) = A0Z(r)ej(ωt−kz)(6.6)
The axial pressure at the center of a sound beam generated by a circular piston varies
in the near-field and may have one or more maximums occur along the axis when
ka > π. The pressure variations are extreme for a circular piston because of the
high degree of symmetry (49). For the circular plate with flexural mode as shown in
Fig. 6.2, the fluctuation of axial pressure amplitude at the center is reduced by the
nearly symmetric high and low pressure field on the plate. Therefore, the variation of
55
6. STANDING WAVE ULTRASONIC LEVITATION
the pressure amplitude A0is not as significant as the circular piston anymore, and is
considered as constant in the near-field for the present model. The particle velocity in
zdirection is expressed as
vz(r, z, t) = A0
ρ0cZ(r)ej(ωt−kz)(6.7)
6.2.3 Increased absorption due to nonlinear effects
The absorption coefficient αdiscussed in Sec. 4.1.4 is calculated with linear assump-
tions. It fits to sound waves with pressure amplitude much smaller than the ambient
air pressure (A0≪P0). However, in acoustic levitation systems, high intensity and
high acoustic pressure amplitude are normally involved. Propagation of such finite-
amplitude waves is accompanied with a variety of nonlinear effects whose intensity
depends on the amplitude of vibrations, such as waveform distortion, formation of
shock waves, increased absorption, nonlinear interaction, cavitation and sonolumines-
cence (46). The waveform distortion always exits in nonlinear waves. They come from
the generation of higher harmonics during propagation. The absorption increases dra-
matically for higher frequencies, therefore the distorted wave are absorbed more than
the harmonic waves. A relation between the absorption of finite and small amplitude
waves is given in Ref. (35) as:
α′
α= 1 + 3ω2v0
4αc21 + B
2Ae−2αz 1−e−2αz(6.8)
In which, α′represents the increased absorption coefficient of finite amplitude wave, v0
the air particle speed amplitude, zthe traveling distance. The nonlinearity parameter
B/A is parameter for characterizing the strength of the nonlinearity of a medium and
B/A = 0.40 for air (46). The increased absorption coefficient α′is now a function of
the particle speed amplitude v0and increases as v0gets higher. When standing wave is
formed, as a result of signified resonance, the air particle speed increases significantly.
As a result, α′obtained from Equ. 6.8 can be significantly larger than α.
By substituting the attenuation coefficient into Equ.6.6 we obtain,
p(r, z, t) = A0Z(r)e−α′z+j(ωt−kz)(6.9)
56
6.2 Modeling the proposed levitation system
6.2.4 Modeling the sound field
Consider the problem in which the sound wave described by Equ. 6.9 traveling in the
zdirection is reflected at z= 0 by a rigid fixed plane wall. The reflected wave is again
reflected at the radiation surface located at z=Land propagates toward the reflector.
The reflection of the wave goes on and on between two rigid surfaces until the wave is
totally absorbed by the air. The resulted sound field between the sound radiator and
the reflector is the summation of all the foregoing and reflected waves. The infinite
summation of such waves is proved to be equal to the summation of one foregoing and
one reflected wave with same amplitude (23). The resulted sound field can be described
as
p(r, z, t) = A0Z(r)·cos kz ·cosh α′z+ j sin kz ·sinh α′zejωt (6.10)
The corresponding acoustic velocity ~v in zdirection is expressed as
vz(r, z, t) = −A0
ρ0cZ(r)·cos kz ·sinh α′z+ j sin kz ·cosh α′zejωt (6.11)
Notice that only the near-field region is considered in this context, and the speed of air
particles in rdirection is ignored. So far, the representation of the entire cylindrical
sound field is obtained. The next step is to calculate the radiation pressure produced
by the obtained sound field.
Since the reflector surface is considered as rigid, the radiation pressure on it is
Eulerian (30). Equation 4.32 can be then applied to calculate the radiation pressure at
the reflecting surface generated by the sound field described by Equ. 6.10. Substituting
Equ. 6.10 and Equ. 6.11 in to Equ. 4.12 and Equ. 4.13, after time-averaging, hViand
hKiare obtained as
hVi=A2
0Z(r)2
4ρ0c2cos2kz cosh2α′z−sin2kz sinh2α′z(6.12)
hKi=A2
0Z(r)2
4ρ0c2sin2kz cosh2α′z−cos2kz sinh2α′z(6.13)
For a sound beam in free space there is no constrain to satisfy. Therefore, C= 0.
Substituting Equ. 6.12 and Equ. 6.13 into Equ. 4.32, the mean Eulerian excess
pressure is obtained as
PE−P0=hVi−hKi=Z(r)2A2
0
4ρ0c2cos 2kz sinh2α′z+ cosh2α′z(6.14)
57
6. STANDING WAVE ULTRASONIC LEVITATION
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
10−5
10−4
10−3
10−2
10−1
100
L/λ
pra
0/P0
Figure 6.3: Change of the radiation pressure at the center of the reflector with different
distances
Equation 6.14 describes the mean excess pressure at fixed points within the sound
field. For a certain sound field, the mean excess pressure varies at different (r, z)
positions. On the plane at z= 0 perpendicular to z, the mean pressure becomes the
acoustic radiation pressure pra on the object. The acoustic radiation pressure pra can
be obtained by setting z= 0 in Equ. 6.14 as
pra =Z(r)2A2
0
4ρ0c2(6.15)
The radiation pressure pra is not uniform in r-direction and its distribution follows the
function Z2(r)
6.3 Simulation results
A normal laboratory condition with temperature of 20 degree Celsius and relative
humidity of 30% is assumed for all the calculations in this section. The absorption
coefficient αis calculated as 0.58 using Equ. 4.18.
Figure 6.3 shows the radiation pressure changes with different distances between
the radiator and the reflector. In the figure, Yaxis is radiation pressure at the center of
the reflector (p0
ra) caused by the center (near r= 0) of the sound beam. The vibration
speed amplitude V0at the center of the radiator used for this calculation is 3 m/s. In
58
6.3 Simulation results
−1 −0.5 0 0.5 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
r/a
pra/pra
0
Figure 6.4: Radiation pressure distribution along the surface of the reflector
the figure, p0
ra is normalized by the atmospheric pressure P0and the levitation distance
is normalized by the sound wavelength. It can be seen that the radiation pressure is
normally very small compared to the atmospheric pressure. However, at intervals of
half wavelength, clear peaks happen due to the formation of a standing wave. When L
is close to zero, the squeeze film effect takes place, and the resulting pressure increases
quickly to a maximum value of 0.4 times of the atmospheric pressure. This corresponds
to V0= 3 m/s.
The distribution of the radiation pressure on a circular surface with r= 0.06 m (as
in the experiment setup presented in the preceding section) is shown in Fig. 6.4. The
radiation pressure pra is normalized by the radiation pressure at the center of the plate
p0
ra. The radial distribution of the amplitude of pra follows the square of the surface
vibration amplitude. At the nodal circles of the radiator, the radiation pressure drops
to zero. The levitation force is obtained by integrating pra along r.
For the proposed acoustic levitation system, it is interesting to know how the fre-
quency affects the levitation force. In order to show the relation, the maximum radi-
ation pressure (at r= 0) is calculated with different frequencies. For consistency, the
surface vibration velocity is set constantly as V0= 3 m/s. The vibration speed is an
indication of the output power of the radiator. Therefore, it can be considered that
59
6. STANDING WAVE ULTRASONIC LEVITATION
0 20 40 60 80 100
2
4
6
8
10
12
14
16
18
Frequency (kHz)
radiation pressure (kPa)
V0 = 3m/s
T = 293 K
hr = 30 %
Figure 6.5: Radiation pressure versus frequency, constant vibration speed
the output sound power is kept approximately constant. Fig. 6.5 shows the calculation
results. From the figure, one can see that the radiation pressure in the audible range
(≤20 kHz) increases quickly with decreasing frequency. However, in the low frequency
range (≤5 kHz), vibration speed of 3 m/s will require a very large displacement of the
radiator. This is often not feasible in practice and will be unbearably loud.
It has been discussed earlier that the absorption coefficient is an important parame-
ter which affects the radiation pressure. The absorption coefficient is sensitive not only
to the frequency, but also to the humidity (5;6). Therefore, the humidity should also
be considered when performing standing wave levitation. The influence of humidity
on the radiation pressure at different frequencies is shown in Fig. 6.6 as an extension
of Fig. 6.5. Similarly, the surface vibration velocity is set constantly as V0= 3 m/s.
It can be seen that the radiation pressure increases in general with lower frequency in
the audible range. However, for the ultrasonic frequency range (≥20 kHz), the radia-
tion pressure is rather not sensitive to the frequency. For very low humidity condition
(below 5%), the absorption coefficient αfor frequencies higher than 300 Hz decreases
severely as can be seen in Fig. 4.1 (5;6). Such a change of αresults in a boost of
radiation pressure at extremely low humidity. The results shown in Fig. 6.6 imply
that it is necessary to maintain the humidity of the laboratory environment to achieve
60
6.4 Conclusion
0 10 20 30 40 50
0
20
40
60
80
100
120
Frequency (kHz)
Radiation pressure (kPa)
0
1
100
40
80
10
20
60
Figure 6.6: Radiation pressure versus frequency at different humidities, with constant
vibration speed
stable performance of the proposed levitation system.
6.4 Conclusion
A one dimensional acoustic levitation system using ultrasonic standing wave has been
discussed in this chapter. To improve the levitation capability of a standing wave
levitation system, a different configuration is presented which is able to levitate large
planar object (much larger than the sound wavelength) at a position of multiple times of
a half wavelength of the sound wave (much higher than squeeze film levitation system).
A mathematical model for the system is developed based on acoustic theory. Nonlinear
absorption effect is taken into account in the model. The theoretical model gives
good insights of the influences of different parameters on the levitation force, such as
frequency and vibration mode.
61
6. STANDING WAVE ULTRASONIC LEVITATION
62
7
Suspension of large planar
objects using ultrasonic standing
waves
A prototype system will be presented in this chapter which employs the theoretical
analysis in the preceding chapter. The proposed system is aimed on suspension of large
planar objects at a high levitation distance (in cm range). The experimental setup and
the performance of the system will be discussed. Experimental results will be compared
with the theoretical results.
7.1 Experiments
7.1.1 Experimental setup
A schematic diagram of the proposed levitation system is shown in Fig. 7.1. A piezo-
electric Langevin type transducer driven in its first longitudinal mode (λ/2) at 20 kHz
is used to generate ultrasonic vibrations. A stepped horn (λ/2) is attached to magnify
the vibration amplitude of the transducer. An aluminum plate of diameter 120 mm is
used as sound radiator. The plate is screwed into the horn.
For matching one of the plate’s axial symmetric flexural modes of vibration to
the axial resonant frequency of horn and transducer, the plate-thickness is chosen so
that the corresponding natural frequency of the free-vibrating plate appears at 20
63
7. SUSPENSION OF LARGE PLANAR OBJECTS USING
ULTRASONIC STANDING WAVES
Vibration
plate
Piezoelectric
Transducer
Horn
Disc
λ/2
Figure 7.1: Schematic diagram of the experimental setup of the proposed levitation
system
kHz (33). Flexural modes of vibration with one, two, and three nodal circles have
been constructed by properly matching the thickness. Experimentally, a plate with
two nodal circles is found to be a good compromise between mechanical strength and
achievable vibration amplitude. The thickness of the plate is 8.55 mm. A result of
a Finite-Element calculation of the vibration mode at 20 kHz with two nodal circles
is shown in Fig. 7.2. The analytical result from Equ. 6.3 is plotted together with
the FEM result. The mode shape of both results agrees very well. After assembly,
the resonant frequency of the entire system (transducer, horn, and plate) appears at
about 19 kHz. An Adaptive Phase Locked Loop control algorithm (APLL) is used for
tracking the resonant frequency of the system during operation (34).
7.1.2 Levitation force measurement
An experimental setup is built as shown in Fig. 7.3 to measure the levitation force
produced by the presented disc levitation system. An aluminum plate with the same
diameter as the radiation plate is positioned in opposite to the radiator. This sound-
64
7.1 Experiments
mds
-1 0 1
-1
0
1
a / A0
r / R
FEM
analytical
Figure 7.2: Calculation of the flexural vibration mode of the circular plate with two nodal
circles (FEM and analytical results)
reflecting plate is mounted on a vertical linear stage through a load cell for being able
to measure the vertical force acting on the reflector directly. Using the linear stage,
the reflector may be positioned freely between the contact position and a distance of
about 40 mm above the radiator. A laser interferometer is installed to measure the
displacement of the reflector.
Measurements have been done with fixed voltage input of 50 V at a fixed frequency
of 19 kHz (input power about 40 W). The actual vertical position of the reflector and
the value of the levitation force are recorded simultaneously while the reflector slowly
approaches the radiator from above until both plates get into contact. Levitation force
versus distance between the plates is plotted in Fig. 7.4. Clear peaks of the levitation
force can be seen in Fig. 7.4 at intervals of half wavelengths. The peak-values at the
positions of half wavelengths increase with decreasing distance between the plates. The
amplitude at the λ/2-position is about 1 N. Multiple peaks are observed near the po-
sitions of half wavelengths. This is because the nonlinear behavior of such a intensive
sound beam. The uneven pressure distribution along the radial direction causes un-
equal sound speed. Therefore the standing wave is formed at slightly different positions
for different parts of the sound beam.
When the gap between the plates gets smaller than approximately 0.5 mm, the
squeeze-film region is reached and the levitation force starts to increase significantly. A
maximum levitation force up to 100 N is measured right before contact happens. The
65
7.1 Experiments
Figure 7.5: Stable levitation of a CD at half wavelength (λ/2) above the radiator. The
central pin (screwed into the flexural plate) is used only for centering the plate in radial
direction
squeeze film levitation effect will be discussed in Chap. 8.
7.1.3 Levitating a compact disc
A common compact disc (CD) is chosen as the object to be levitated. It has the same
diameter as the vibrating plate, with a thickness of 1.3 mm and a mass of 16 g. A
stable levitation state is observed when the input power reaches about 30 W (see Fig.
7.5). The CD then rests without any instable vertical motion above the flexural plate.
Maximum vibration amplitude of the excitation system occurs at the center of the flex-
ural plate and is about 25 µm at 19 kHz for this level of power (measured using a laser
vibrometer). It is worth mentioning that the CD in this arrangement rests at a position
slightly higher than half a wavelength (above the peak of the levitation force), where
the levitation force equals the gravity force of the CD. This is different compared to
common radiator-reflector-type systems, in which small particles are levitated at posi-
tions slightly below the pressure nodes of the standing wave. Stable levitation could
not be achieved at one wavelength or higher positions with the proposed setup due to
67
7. SUSPENSION OF LARGE PLANAR OBJECTS USING
ULTRASONIC STANDING WAVES
Figure 7.6: Interferometric measurement of a 2D sound field (68)
the quickly dropping levitation force.
7.1.4 Sound field visualization
Visualization of sound wave propagation and interaction with structures in the levita-
tion mechanism is important for understanding the underlying acoustics. The conven-
tional method to measure acoustic fields with microphones is difficult in the presented
system, because the measuring volume is limited, high resolution is required, and distur-
bances due to sensors are not acceptable. Recently, Zipser et al. (68), (67) introduced
a new method for planar visualization of acoustic sound waves in gases by means of a
scanning laser vibrometer. The principle of interferometric measurement of a 2D sound
field is shown in Fig. 7.6. The laser beam is projected though the sound field and
reflected by the rigid reflector. The refractive index and the density of air chance with
the pressure in the sound field. Therefore the optical length of laser also changes with
the pressure. The optical length change is measured by the scanning laser vibrometer
as if the rigid reflector has been moved. The measured movement can be re-modulated
back to pressure change, so that the sound field is obtained. With this high sensitive
method, multi-frequency, repetitive sound fields can be measured. This measurement
technique is applied to study the sound field excited by the disc-shaped radiator and
the interaction with the levitated objects. Since pictures of the qualitative distribution
of pressure are sufficient here, no complicated post-processing is needed.
In the first experiment, the sound propagation from the vibrating radiator is visu-
alized and shown in Fig. 7.7(a). Red color (darker color) indicates high pressure and
68
7.1 Experiments
Figure 7.7: Sound wave free propagation above the radiator. a) measurement result; b)
schematic diagram
green (brighter color) indicates low pressure; in an additional animated video the con-
tinuous propagation of the sound-wave has been visualized as schematically depicted in
Fig. 7.7(b). Distinctive high and low pressure regions can be seen above the radiation
surface within the first wavelength. Several interference patterns are observed starting
from the second wavelength, which are typical for the acoustic near-field of this kind of
radiators. The interfering regions coincide with the positions of the two nodal circles
on the surface of the radiating plate as shown in Fig. 7.7(b).
Figure 7.8 shows the results measured when the CD is levitated and rests in a stable
position slightly above half a wavelength. A clear standing wave pattern is seen in an
animated depiction. Leak of sound pressure from the center hole of the CD is also
observed. It can be stated that the levitated CD itself acts as a reflector for build-
ing the λ/2 standing wave between plate and the CD. The distinct high-low pressure
distribution follows exactly the vibration mode of the radiator, which proves that the
assumption made for the sound field model is acceptable.
69
7. SUSPENSION OF LARGE PLANAR OBJECTS USING
ULTRASONIC STANDING WAVES
Figure 7.8: Visualization of the standing wave field during stable levitation, operating
frequency 19 kHz
7.2 Results and discussion
The theoretical model to predict the radiation pressure has been established in Chap.
6. To use Equ. 6.15 to calculate the pressure, one needs first to know the acoustic
pressure amplitude A0. However, the acoustic pressure is difficult to measure in the
presented arrangement due to the limited space. On the other hand, the mechanical
vibration is much easier to obtain. The following arrangement is made to link the
radiation pressure with the mechanical vibration of the radiation.
The magnitudes of air particle velocity |ˆvr,z|at each point according to rand z
directions can be derived from Equ. 6.11 as
|ˆvr,z|=A0Z(r)
ρ0c·p(cos kz ·sinh α′z)2+ (sin kz ·cosh α′z)2
=A0Z(r)
ρ0c·psinh2α′z+ sin2kz (7.1)
The distance from the reflector to the sound radiator is given as L. Therefor, |ˆv0,L|
represent the air particle speed magnitudes directly on the center of the sound radiator,
which can be found by setting z=Land r= 0 in Equ. 7.1. Assuming that the air
particles on the radiation surface move in the same way as the surface itself, |ˆv0,L|is
equal to surface vibration speed amplitude at the center of the radiator, represented by
V0. Therefore, the acoustic pressure amplitude A0at r= 0 can be derived from Equ.
70
7.2 Results and discussion
0 5 10 15 20 25 30 35
0
1
2
3
4
5
6
7
Distance (mm)
Levitation force (N)
Measurement
Simulation
Figure 7.9: Levitation force versus levitation distance, experiment and calculation results
7.1 as
A0=V0ρ0c/psinh2α′L+ sin2kL (7.2)
Equ. 7.2 gives a relation between the mechanical vibration of the radiator and the
acoustic field. It can be seen that for a given distance Land the surface vibration
amplitude V0at r= 0, the acoustic pressure amplitude A0at r= 0 can be obtained.
Substituting Equ. 7.2 into Equ. 6.15, we find
pra =Z(r)2V2
0ρ0
4sinh2α′L+ sin2kL(7.3)
In Equ. 7.3, all the parameters except V0and α′are known for a given experiment
setup. The vibration amplitude V0can be measured experimentally during operation.
The increased absorption coefficient α′depends on αand the wave properties. The
acoustic radiation pressure can be calculated using Equ. 6.8 and Equ. 7.3. The levita-
tion force can then be obtained by integrating pra on the surface of the levitated object.
Results of calculated levitation forces for different Lvalues are plotted in Fig. 7.9.
The measured results are plotted together with the calculated ones. It can be seen
that the calculated levitation forces agree well with the measurements. However, the
magnitude at the position of half wavelength is about 5 time of the measured value.
71
7. SUSPENSION OF LARGE PLANAR OBJECTS USING
ULTRASONIC STANDING WAVES
The possible reasons of the higher calculated force are as following. First, the reflector
used in the experiment is never a perfect reflecting surface. Diffraction and absorption
always exist at the reflector. Secondly, in the experiment, the two plates are not per-
fectly parallel. Thus, the standing wave is not formed at one single distance, but in
a small range. This has been proved by the multiple peaks measured in experiment
discussed in Sec. 7.1. Therefore, the resulted levitation force is more distributed with
smaller amplitude compared to the simulation.
7.3 Conclusion
Experimental investigation on a non-contact suspension system for large planar objects
has been presented in this chapter. A CD is successfully levitated with the proposed
system at a height of half a wavelength. The levitation forces at different distances
have been measured. A force of about 1 N has been observed at the position of half
wavelength in front of the sound radiator.
The presented non-contact suspension system can be used for non-contact handling
of sensitive parts such as a silicon wafer. With the high levitation distance, the levitated
part can be processed on both sides without additional contact. The proposed system
can be also used as a non-contact thrust bearing when the load-capacity is not the
primary concern for the bearing. Another possible application is to use the system
as a non-contact actuator to give a certain actuation force on a planar surface from a
certain distance without mechanical contact.
72
8
Squeeze film ultrasonic levitation
The working principles of squeeze film levitation will be explained in this chapter.
As before, analytical models will be discussed first, since they provide insight to the
parameters that are crucial to the performance of a squeeze film levitation system. Two
analytical models based on acoustic theory and fluid dynamics will be presented and
compared. After that, the governing fluid dynamics equation is solved numerically to
obtain more precise results.
8.1 Modeling based on acoustic theory - acoustic radia-
tion pressure
Squeeze film levitation happens when a planar object is brought close to an oscillating
surface which vibrates at high frequency (often in ultrasonic range). The object will
experience a normal pressure and can be levitated above the vibrating surface at a
very small distance (normally a few to several tens µm). Unlike the levitation system
introduced in Chap. 6, the standing wave field does not exist anymore in squeeze film
levitation. Instead, a very thin air film with pressure varying according to the movement
of the radiation surface is to be considered. The physical model shown in Fig. 2.2 is
considered. Since the vibration of the plate in squeeze film levitation is piston-like, the
wave is considered as a plane wave. Moreover, the distance h0between two surfaces
is normally of order 10−5m, which is three orders smaller than the dimension of the
vibrating surface L(with order of 10−2m). Thus the lateral movement of the air is
73
8. SQUEEZE FILM ULTRASONIC LEVITATION
neglected, resulting in a closed boundary plane wave, which can be described as
p=A0cos(ωt −kz) (8.1)
where A0is the amplitude of acoustic pressure, kthe wave number, ωthe angular
velocity of the wave and zthe distance from the vibration source. With the above
assumptions, the system can be considered as an extreme case of the disc levitation
system discussed in Chap. 6when the object (reflector) is brought very close to the
radiator. Therefore it can be described in the same manner using the model of the disc
levitation system. Since the traveling distance of the wave in squeeze film levitation
is very small, the absorption effect can be neglected safely. Considering the multiple
reflection of the waves between two rigid surfaces, the resulting sound field can be
described similarly as Equ. 6.10 as
p=A0cos kz cos ωt (8.2)
The corresponding acoustic velocity vin zdirection is derived as
v=A0
ρ0csin kz cos(ωt −π/2) (8.3)
Similar as the procedure in Sec 6.2.4, according to Equ. 4.12 and 4.13,hViand hKi
can be derived using Equ. 8.2 and Equ. 8.3 as
hVi=p2/2ρ0c2=A2
0
4ρ0c2cos2kz (8.4)
hKi=ρ0hv·vi/2 = A2
0
4ρ0c2sin2kz (8.5)
Substituting Equ. 8.4 to 8.5 into Equ. 4.32, the mean Eulerian excess pressure is
obtained as
pra =hVi−hKi+C=A2
0
4ρ0c2cos 2kz +C=Ecos 2kz +C(8.6)
where Eis the energy density of the considered wave. For a closed plane wave sound
field, constant Cis derived from the conservation of mass as (30)
C=Γ−1
2E(8.7)
Therefore, we have
pra =Γ−1
2+ cos 2kzE(8.8)
74
8.1 Modeling based on acoustic theory - acoustic radiation pressure
where Γ is a specific heat ratio, Γ = 1.4 for air. To calculated the radiation pressure
using Equ. 8.8, one needs first to know the acoustic pressure amplitude A0. Similar as
Equ. 7.2, the acoustic pressure amplitude A0can be expressed in terms of the vibration
velocity amplitude V0as
A0=V0ρ0c/ sin kh0(8.9)
In which h0is the mean distance between the two plane surfaces. Substituting Equ.
8.9 in to Equ. 8.8, we find
pra =Γ−1
2+ cos 2kzV2
0ρ0
4 sin2kh0
(8.10)
Since zand and h0are in the order of 10−4m and kis in the order of 102,kz and
kh0are in the order of 10−2, the relations sin kh0≈kh0and cos 2kz ≈1 hold. Given
a0=V0/ω as the oscillating displacement amplitude, Equ. 8.10 can be then simplified
as
pra =Γ + 1
8
V2
0ρ0
k2h2
0
=Γ + 1
8
a2
0ρ0c2
h2
0
(8.11)
It can be seen that the radiation pressure pra in squeeze film levitation is reversely pro-
portional to the square of the levitation distance and proportional to the square of the
vibration displacement amplitude a0. The levitation force can be then obtained by inte-
grating pra along the levitated object. The physical boundary condition of Equ. 8.11 is
a0< h0to avoid contact. Therefore, pra reaches its theoretical maximum when a0=h0.
For an oscillating surface with uneven but known vibration amplitude distribution,
the squeeze levitation force can still be calculated using Equ. 8.11 by considering the
uneven vibration as a sum of finite number of small piston-like vibrations. Since the
lateral movement of the gas is neglected, the small piston vibrators can be treated indi-
vidually and the resulting mean levitation pressure is the average of the levitation forces
induced by all the small piston vibrators. Taking the flexural circular plate radiator
presented in Chap. 6as an example, the governing equation of surface amplitude dis-
tribution is Z(r) as presented in Sec. 6.2.1. The mean squeeze film levitation pressure
induced by the flexural vibration mode can be calculated as
pmean =1
RZR
0
Γ + 1
8
(A0Z(r))2ρ0c2
h2
0
dr=Γ + 1
8
A2
0ρ0c2
h2
0ZR
0
Z(r)2dr(8.12)
75
8. SQUEEZE FILM ULTRASONIC LEVITATION
where Ris the radius of the circular radiator, rthe radial position on the radiator, A0
the maximum vibration amplitude at the center of the radiator.
8.2 Modeling based on fluid mechanics - solving the Reynolds
equation
A sound wave is an oscillation of pressure. In squeeze film levitation, the wavelength
of the sound wave that would be generated by the oscillating surface is several orders
larger than the gap distance. Assuming that a sound wave is propagating in such a small
distance, the pressure gradient in the direction of propagation (normal to the surface)
will be extremely small. If the pressure gradient is small enough to be ignored, the
assumption of sound propagation does not hold anymore. In such a case, an acoustic
radiation pressure based on the propagation of sound wave seems not reflecting the
actual physical condition of squeeze film levitation. Alternatively, a model based on
fluid dynamics which considers the viscous lateral flow of gas in a narrow gap may give
a better insight to the physical condition and provide better simulation results for the
highly squeezed gas film.
The differential equation governing the pressure distribution of a fluid film between
two opposing surfaces is called Reynolds Equation. This equation was first derived
in a remarkable paper by Osborne Reynolds in 1886. The Reynolds Equation can be
derived from the Navier-Stokes equation and continuity equation or directly from the
principle of mass conservation. The detailed derivation of Reynolds Equation can be
found in literatures such as (18), therefore the general form of Reynolds Equation is
given directly here.
∂
∂x ρh3
12η
∂p
∂x+∂
∂y ρh3
12η
∂p
∂y=∂
∂x ρh(ua+ub)
2+∂
∂y ρh(va+vb)
2+∂(ρh)
∂t (8.13)
where, ρrepresents the density of the fluid, hthe gap distance, pthe pressure, ηthe
absolute viscosity, tthe time, uand vthe velocity in xand ydirections with subscrip-
tion aand bdenoting the upper and lower limits of the fluid film.
Considering the Reynolds equation valid for squeeze film levitation shown in Fig.2.2,
ua,uband va,vbare zero since the surfaces have no lateral movement. Equation 8.13
76
8.2 Modeling based on fluid mechanics - solving the Reynolds equation
can be simplified as
∂
∂x ρh3
12η
∂p
∂x+∂
∂y ρh3
12η
∂p
∂y=∂(ρh)
∂t (8.14)
The equation of state for a perfect gas is
p=ρ¯
Rtm(8.15)
where ¯
Rrepresents the gas constant and tmthe absolute temperature. Therefore we
have
ρ=p
¯
Rtm
(8.16)
Substituting this equation in to Equ. 8.14 yields
∂
∂x ph3
12η
∂p
∂x+∂
∂y ph3
12η
∂p
∂y=∂(ph)
∂t (8.17)
In practice, it is often sufficient to consider the 1-D case of Equ. 8.17, which can be
written as
∂
∂x ph3
12η
∂p
∂x=∂(ph)
∂t (8.18)
The following dimensionless parameters are defined,
P=p
p0
, H =h
h0
, X =x
L, T =ωt, σ =12ωηL2
p0h2
0
where σis named squeeze number, Lthe characteristic length of the gas film. Substi-
tuting above dimensionless parameters in the Equ. 8.18, we obtain (18)
∂
∂X PH3∂P
∂X =σ∂(PH)
∂T (8.19)
Equ. 8.19 is the second order partial differential equation that governs the time-
dependent, laminar, Newtonian, isothermal and compressible thin film flow.
In general, Equ. 8.19 can not be solved analytically except for some special cases
(38;47;61). It has to be solved numerically for the distribution of pressure P along X
and Y (23;37;39;51). The analytical and numerical solutions of Reynolds equitation
will be discussed in the following sections.
77
8. SQUEEZE FILM ULTRASONIC LEVITATION
8.2.1 Approximate solution of the Reynolds equation for large squeeze
number
For the case illustrated in Fig. 2.2, a flat surface oscillates in the normal direction
against a fixed flat wall parallel to it. At high squeeze number, the gap thickness
become much smaller than the length, the volume flow at the peripheries becomes very
small. Thus the system can be considered as a gas film in a closed channel being
compressed at one side and closed at the other. By assuming isothermal condition, the
mass conservation yields
PH = Ψ (8.20)
where Ψ is a constant. Equation 8.19 can be rewritten as
∂
∂X ΨH2∂(Ψ/H)
∂X =∂
∂X 1
2H∂(Ψ2)
∂X −Ψ2∂H
∂X =σ∂(Ψ)
∂T (8.21)
Since the gap is assumed as independent from length X, we have ∂H/∂X = 0. The
above equation becomes
∂
∂X 1
2H∂(Ψ2)
∂X =σ∂(Ψ)
∂T (8.22)
Under the above assumptions, by integrating both sides of Equ. 8.22 over a time period
yields (47)
Z2π
0
∂
∂X 1
2H∂(Ψ2)
∂X dT=σZ2π
0
∂(Ψ)
∂T dT= 0 (8.23)
Integrating Equ. 8.23 with respect to Xyields
Z2π
0
H∂(Ψ2)
∂X dT=C1(8.24)
Equ. 8.24 holds for all values of X. Since the system is symmetric, the pressure gradient
at the center of the gap should satisfy the condition ∂P/∂X = 0. Therefore, at X= 0,
the left hand side of Equ. 8.24 is equal to zero. In turn, C1is equal to 0 also. By
integrating Equ. 8.24 with respect to Xagain, we obtain
Z2π
0
HΨ2dT=Z2π
0
H3P2dT=C2(8.25)
C2can be found be applying the boundary condition of P= 1 at the peripheries of the
gap and setting H= 1 + ǫsin Tas
C2=Z2π
0
H3dT=Z2π
0
(1 + ǫsin T)3dT=π(2 + 3ǫ2) (8.26)
78
8.2 Modeling based on fluid mechanics - solving the Reynolds equation
Substituting Equ. 8.26 and H= 1 + ǫsin Tinto Equ. 8.25 yields
Z2π
0
HΨ2dT= Ψ2Z2π
0
(1 + ǫsin T) dT=π(2 + 3ǫ2) (8.27)
Ψ is found readily as Ψ = p1 + 3/2ǫ2, and we obtain Pas
P=Ψ
H=q1 + 3
2ǫ2
1 + ǫsin T(8.28)
The time and space averaged mean pressure in the gap can then be obtained as
¯
P=1
2πZ2π
0
PdT=1
2πZ2π
0q1 + 3
2ǫ2
1 + ǫsin TdT=s1 + 3
2ǫ2
1−ǫ2(8.29)
This approximate solution based on mass conservation gives a mean pressure along the
entire length of the gap and is independent of the squeeze number. It is suitable only
for conditions with high squeeze number (47).
8.2.2 Solving the Reynolds equation numerically
In order to obtain the pressure distribution along the gap, Equ. 8.19 is solved numeri-
cally using the finite difference method. Since the considered problem is symmetric in
X, only one half of the length L(from the center X= 0 to X= 1/2) is considered in
the calculation to reduce the computation cost. By applying a finite difference scheme
in X, one half of the length is divided into Nfinite lengths ∆X. Equation 8.19 then
takes the form of a set of ordinary normalized differential equations in time
σdPi(T)
dT=−σPi(T)
H
dH
dT+H2
Xi
1
2∆X2
hXi+1
2P2
i+1 −P2
i−Xi−1
2P2
i−P2
i−1i(8.30)
The subscript iindicates the grid coordinate along the xaxis. The initial pressure in
the gap at T= 0 is assumed to be equal to the ambient pressure p0. The boundary
condition at the edge of the plate is set as pressure release which means that the pressure
near the edge always approaches to p0. At the center of the plate (X= 0) there should
be no pressure gradient along the x-direction, so that the pressure gradient across the
complete gap is always differentiable and no local extreme exists. This means P0=P1
79
8. SQUEEZE FILM ULTRASONIC LEVITATION
at all time points. To sum up, the initial condition and boundary conditions are listed
as following:
Initial condition: P(X, T = 0) = 1
Boundary condition 1: P(X= 1/2, T) = 1
Boundary condition 2: ∂P(X= 0, T)
∂X = 0
Equ. 8.30 can be solved easily with mathematical tool such as MATLAB.
8.3 Results and discussion
By numerically solving Equ. 8.30, the pressure distribution along half of the gap length
can be obtained as a function of time. For a typical levitation system with L= 30
mm, a0= 5 µm, f= 20 kHz, pressure distribution for three different gap distances of
10 µm, 50 µm and 100 µm are calculated and shown in Fig. 8.1. The corresponding
squeeze numbers are 2425, 97 and 24 respectively. It can be seen that the pressure in
the gap oscillates with time according to the movement of the vibrating surface. The
peak values are higher than the absolute anti-peak values, which leads to a positive
time-averaged pressure. As specified in the boundary condition 1, the pressure at the
edge of the plate is always equal to atmosphere pressure. The pressure increases as
moving inside the gap. For high squeeze number (Fig. 8.1 (a)) the pressure increases
quickly to maximum and keeps constant along the length; for small squeeze number
(Fig. 8.1 (c)), the pressure increases gradually and reaches maximum at the center.
8.3.1 Experimental validation
In order to examine the accuracy of the calculation results from the models presented
above, a series of experiments are conducted and compared with the simulation results.
The squeeze film levitation force is measured with the same setting as when measur-
ing the acoustic radiation force. Fast increasing of levitation force has been measured
when the levitation distance gets smaller than 0.5 mm. A maximum levitation force of
28 N is obtained with input power of 50 W at frequency of 19 kHz. The mean levitation
distance his 63 µm before some regions get into contact. The diameter of the circular
80
8.3 Results and discussion
00.1 0.2 0.3 0.4 0.5
0
2
4
6
8
0.5
1
1.5
2
2.5
L
(a) σ = 2425, ε = 0.5
T
P
00.1 0.2 0.3 0.4 0.5
0
2
4
6
8
0.8
0.9
1
1.1
1.2
L
(b) σ = 97, ε = 0.1
T
P
00.1 0.2 0.3 0.4 0.5
0
2
4
6
8
0.9
0.95
1
1.05
1.1
L
(c) σ = 24, ε = 0.05
T
P
Figure 8.1: Numerical simulation results for the pressure distribution along the gap
varying with time
81
8. SQUEEZE FILM ULTRASONIC LEVITATION
0 0.2 0.4 0.6 0.8 1
10−2
10−1
100
101
102
103
Levitation force (N)
Squeeze number
Distance (mm)
101
102
103
Reynolds − numerical
Acoustic radiation
Reynolds − analytical
Measurement
Figure 8.2: Comparison of the measured and calculated levitation forces
radiator is 120 mm with vibration amplitude 40 µm at the center. The vibration am-
plitude distribution is shown in Fig. 6.2. Actual parameters of the presented levitation
system are used to calculate the pressure. The mean vibration amplitude along the
surface is 15 µm which is used for the simulation. The gap distance is varied to mimic
the situation when measuring the levitation force in the experiment. The squeeze film
levitation force is plotted in Fig. 8.2 together with calculation results obtain from the
acoustic radiation model (Equ. 8.12), the approximated (Equ. 8.29) and the numerical
(Equ. 8.30) solution of the general Reynolds equation. For the analytical solution of
Reynolds equation and the acoustic radiation pressure, the calculated mean pressure is
multiplied by the area of the radiator to get the mean levitation force. The distributed
pressure obtained from the numerical solution of Reynolds equation is integrated over
the surface to get the mean levitation force.
It can be seen that all models give the correct tendency for the levitation force
with respect to the levitation distance. At large distance (squeeze number<100) the
measured curve agrees well with the calculated result from the acoustic radiation model.
82
8.3 Results and discussion
The numerical result of Reynolds equation is much lower than the measurement. When
the gap distance gets smaller (higher squeeze number), the measured pressure increased
faster than the acoustic radiation model, and fits better to the numerical and analytical
results of Reynolds equation. The transition happens at the position with squeeze
number of about 100. The approximated solution of the Reynolds equation agrees
with the numerical solution well at high squeeze number, but is much higher in case of
low squeeze number. The different fitting of the curves at different levitation distance
ranges can be explained as following. In the 1-D model based on acoustic theory, the
gas is considered as inviscid and has no lateral movement. The wave is considered as
propagating and bouncing within a very small distance between two surfaces. These
assumptions are suitable when the gap distance h0is still comparable with the sound
wavelength λ(i.e. h0/λ > 10−2) and the vibration amplitude a0is small compared
to h0. When the gap distance become more than two orders smaller than the sound
wavelength and length of the gap L(resulting a high squeeze number), the squeeze
effect becomes dominant. At the edge of the gap, the gas will be squeezed out and
sucked in according to the oscillation of the surface. In the interior region, the gas
does not have enough time to be squeezed out in the short time cycle. Therefore, the
gas is just being pressurized and released according to the movement of the oscillating
surface. Such a physical condition is exactly what the Reynolds equation represents
for, therefore, as expected and observed in the Fig. 8.2, the fluid dynamic model fits
better to the experiment result at higher squeeze number.
In order to have a close look on the squeeze film levitation force, a modified experi-
mental setup is constructed as shown in Fig. 8.3. Instead of the flexural plate radiator,
aλ/2 aluminum cylinder with diameter of 50 mm is used to provide a piston-like vi-
bration. A plate is mounted on a force sensor through a ball joint to keep it parallel to
the vibrating surface during the measurement. The measurement is conducted with a
vibration amplitude of 10 µm on the surface at a frequency of 20 kHz. The gap distance
is slowly reduced from 300 µm by adjusting the vertical stage until the two surfaces
get into contact. A maximum levitation force of 115 N is measured at input power of
about 50 W. Similar as for Fig. 8.2, the measured force is plotted in Fig. 8.4 together
with the theoretical calculation results.
83
8. SQUEEZE FILM ULTRASONIC LEVITATION
Displacement
a
0
Ball joint
Vertical stage
Force sensor
Displacement
sensor
a
0
Ultrasonic
transducer
Horn
Figure 8.3: Experiment setup for measuring squeeze film levitation force
0 0.02 0.04 0.06 0.08 0.1 0.12
100
101
102
Distance (mm)
Levitation force (N)
Squeeze number
Reynolds − numerical
Acoustic radiation
Reynolds − analytical
Measurement
101 102
Figure 8.4: Comparison of the measured and calculated levitation forces
84
8.3 Results and discussion
As expected, the measurement results agree well with the solution of Reynolds
equation for the squeeze number higher than 100. At squeeze number lower than 100,
the measured result is closer to the calculated force from acoustic radiation pressure.
It can be concluded that for a squeeze film levitation system with high squeeze
numbers (>100), the levitation force can be predicted precisely using the Reynolds
equation. For very high squeeze numbers (>1000), the approximated analytical solu-
tion of Reynolds equation can be applied instead of solving it numerically, since they
agree to each other very well at this level of squeeze number. For the systems with low
squeeze numbers (<100), the acoustic radiation pressure model gives rather accurate
results for predicting the levitation force.
8.3.2 Crucial parameters
For a squeeze film levitation system, a high levitation capacity (force per unit area) is
always desirable. Therefore it is important to investigate how the various parameters
affect the levitation capacity. Based on the simulation and experimental results, two
parameters are found to have great influence on the load capacity of a squeeze film
levitation system which are the squeeze number, σand the excursion ratio, ǫ. The
significance of σand ǫwill be discussed in the following of this section.
•Squeeze number σ
If one takes a close look at the definition of the squeeze number (σ= 12ωηL2/p0h2
0),
it is easy to find that the squeeze number actually indicates how “intensive” the squeeze
action is. It contains three important parameters, namely the driving frequency ω, the
gap length Land the gap distance h0. When other parameters of a levitation system
are fixed, to increase the squeeze number, ωor L/h0can be increased. A higher ω
means a faster squeeze motion. If the amplitude is kept unchanged, higher ωmeans
more energy is brought into the gap. The ratio between Land h0indicates how big
the levitation surface relatively is. Fig. 8.5 shows the numerical calculation results of
pressure distribution in the gas film at different squeeze numbers. It can be seen from
Fig. 8.5 that at a lower squeeze number, the pressure increases slowly from the edge of
85
8. SQUEEZE FILM ULTRASONIC LEVITATION
−0.5 0 0.5
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
X
P
σ = 1
σ = 10
σ = 100 σ = 1000
Figure 8.5: Numerical result of mean pressure distribution in the gap at different squeeze
numbers
the gap and reaches peak at the center. When the squeeze number gets higher, the pres-
sure increases quickly to maximum and distribute more evenly along the gap. Further
increase of squeeze number does not help much to obtain higher levitation pressure any
more. Therefore, when designing a squeeze film levitation system, a minimum squeeze
number higher than 100 is necessary for building up the pressure efficiently.
As Salbu (47) concluded, the squeezed air film behaves as if it is incompressible at
the boundary and solely compressible in the interior. The interior area increases from
zero to the full length of the wall, while increasing the squeeze number. The interior
region behaves like a spring with pressure in phase with vibration displacement. The
exterior region behaves like a damper with pressure in phase with the vibration velocity.
•Excursion ratio ǫ
It is clearly seen from Fig. 8.2 and 8.4 that the levitation pressure increases with
reducing the gap distance while the vibration amplitude is kept constant. Such a
relation can be seen from Equ. 8.28 and 8.11 that the levitation force is roughly
proportional to the square of the excursion ratio ǫ.
86
8.3 Results and discussion
0 10 20 30 40 50
0.5
0.6
0.7
0.8
0.9
1
a0 (µm)
ε
(a)
0 10 20 30 40 50
0
2000
4000
6000
8000
10000
a0 (µm)
σ
(b)
Figure 8.6: Change of (a), excursion ration ǫand (b), squeeze number σwith respect to
the vibration amplitude a0.δ=h0−a0= 2 µm
0 10 20 30 40 50
0
100
200
300
400
500
600
700
800
a0 (µm)
Levitation force (N)
(a)
0.5 0.6 0.7 0.8 0.9 1
0
100
200
300
400
500
600
700
800
ε
Levitation force (N)
(b)
Figure 8.7: Change of levitation force with respect to (a), vibration amplitude a0and
(b),excursion ration ǫ.δ=h0−a0= 2 µm
87
8. SQUEEZE FILM ULTRASONIC LEVITATION
The levitation capacity of the system shown in Fig. 8.3 at different ǫvalues is
calculated to demonstrate the influence of ǫand a0. In the simulation, the vibration
amplitude a0is varied, while the gap distance h0is kept as always 2 µm higher than
a0. The excursion ratio and squeeze number are determined according to the given
vibration amplitude values (as shown in Fig. 8.6). The levitation capacity is calculated
from the numerical solution of Reynolds equation and plotted with respect to ǫand a0
in Fig. 8.7. It can be seen that the levitation capacity increases with higher excursion
ratio. The levitation capacity approaches infinite when the excursion ratio approaches
one.
From the definition of the excursion ratio, we can see ǫ= 1 if a0=h0. In reality
a0=h0means that at the highest position of the oscillation, the gap between the two
surfaces is zero. Such a situation is impossible to achieve in practice since there will
be always a small mean gap δ(δ=h0−a0) between two surfaces due to the surface
roughness and flatness error as well as the misalignment between two surfaces. The
excursion ratio ǫis then a0/(δ+a0). The minimum gap δis usually a constant finite
value for a certain system. For example, in the experiment shown in Fig. 8.3, contact
happens at a mean gap distance of 23 µm and vibration amplitude of 10 µm (ǫ= 0.45
and δ= 13 µm). In practice, ǫcan only approach one if δapproaches zero or a0is
much larger than δ. The minimum gap δcan be reduced only by higher manufacturing
accuracy. Higher a0can be achieved by proper design of ultrasonic transducers.
In conclusion, for a squeeze film levitation system, a sufficiently high squeeze number
(σ > 100) is essential to build up the pressure efficiently in the gap. This has to be
considered during the design phase. The maximum achievable levitation capacity of a
squeeze film levitation system is determined by the excursion ratio ǫ. The excursion
ratio ǫdepends on the vibration amplitude a0and δ. The mean gap δalways exists and
is constant for a specific system. It can only be minimized by better manufacturing
accuracy. Therefore, good surface finishing and form accuracy are important for a
squeeze film levitation system to achieve higher levitation capacity. A higher vibration
amplitude is helpful to achieve higher levitation capacity when δcan not be further
reduced. In other words, a higher a0makes the system performance less sensitive to δ.
88
9
An non-contact journal bearing
based on squeeze film ultrasonic
levitation
According to the discussion in Chap. 8, squeeze film levitation can provide consider-
ably high load-carrying capacity (levitation force per unit area) by properly designing
the system. It is suitable to be applied to develop non-contact suspension systems. As
discussed in Sec. 2.2.2, several linear bearings based on squeeze film levitation have
been developed with satisfactory performance. Rotational bearings have been rarely
investigated due to the non-flat surfaces and the requirement of multiple directional
supports. The existing squeeze film rotational bearings uses either bulk piezoelectric
materials (14;56) or flexural vibration mode of piezoelectric elements or flexural hinges
(51;61;66). They all suffer from very limited load capacity and are not suitable for
high-load applications.
In this chapter, an improved design of a non-contact squeeze film journal bearing
using Langevin ultrasonic transducers will be presented. The proposed bearing has a
much higher load capacity and is aimed for applications such as bearing for precision
machine tool spindle systems. The design and experimental results will be discussed in
this chapter.
89
9. AN NON-CONTACT JOURNAL BEARING BASED ON SQUEEZE
FILM ULTRASONIC LEVITATION
Figure 9.1: Schematic diagram of the proposed bearing system
9.1 Design of the proposed bearing
The proposed bearing consists of three piezoelectric transducers mounted on a housing,
in a circle with 120 degree between each other. The center lines of all the transducers
go through the rotation center of the spindle. A schematic diagram is shown in Fig.
9.1. Each transducer has a concave radiation surface which covers about 100 degrees of
a cylindrical surface. The three radiation surfaces form the bearing inner ring, which
has a diameter slightly larger than the spindle. When the transducers are driven to
vibrate in their first longitudinal resonance, squeeze film levitation effect takes place
between the inner bearing ring and the spindle surface. Repelling force is generated in
the air gap which suspends the spindle and automatically keeps it at the equilibrium
position. Active control of the transducers is not necessary, since the bearing is natu-
rally a stable system. However, the actual position of the spindle center is monitored
precisely during operation by two inductive sensors, which allows a detailed investiga-
tion of the system’s behavior. And, of course, by controlling the three transducers in
an appropriate way, an active bearing can be realized.
90
9.1 Design of the proposed bearing
9.1.1 The Langevin ultrasonic transducer
The piezoelectric transducers are the key parts for the ultrasonic bearing system. They
generate high frequency mechanical vibration for creating strong and stable squeeze
film levitation. The basic working principles and design methods of piezoelectric trans-
ducers have been discussed in Chap. 5. Although, commercialized standard piezo-
electric transducers are available, they can not fulfill the specific requirements such as
the radiation surface contour and the vibration amplitude distribution. Therefore the
transducers must be specially designed for the bearing system.
Previously reported designs of squeeze film levitation bearings often used bending
piezoelectric elements as the active part to generate the levitation effect. Such designs
are rather compact, but the vibration is very sensitive to the load. At high-load situa-
tion, the vibration amplitude is “pressed” down, which leads to a dramatic reduction of
levitation force. To avoid such problems, for the present application, a half wavelength
Langevin type piezoelectric transducer (as shown in Fig. 5.8) is selected as the vibra-
tion source. Langevin type transducers are known to be very stable when subjected to
load and have many distinct advantages as already discussed in Sec. 5.2.
For the present application, the desired working frequency of the transducer is 20
kHz at the first longitudinal vibration mode. The expected unloaded vibration am-
plitude at the radiation surface is 15 µm in normal operation state. The maximum
allowed output power of the transducer is designed to be 1000 W. The piezoelectric ce-
ramic rings used in this investigation are provided by PI Ceramic GmbH. in Germany,
namely PIC-181. PIC-181 is a modified lead zirconate - lead titanate material with an
extremely high mechanical quality factor and a high Curie temperature. This material
is destined for the use in high-power acoustic applications. Furthermore, the good tem-
perature and time stability of its dielectric and elasticity constants makes it suitable
for resonance-mode ultrasonic applications. The specification is listed in Table 9.1 (25).
In this application, a relatively large radiation surface (about 8 cm2) and high
vibration amplitude (up to 15 µm) are required. It requires very high input power
to achieve the required amplitude directly from the deformation of the ceramic rings
91
9. AN NON-CONTACT JOURNAL BEARING BASED ON SQUEEZE
FILM ULTRASONIC LEVITATION
Parameter Symbol Value Unit
Density ρ7800 kg/m3
Electromechanical coupling factor k33 0.66
Piezoelectric charge constant d33 265 10−12 C/N
Piezoelectric voltage constant g33 25 10−3Vm/N
Elastic constant CD
33 16.6 1010 N/m2
Mechanical quality factor Qm2000
Static compressive strength 600 MPa
Table 9.1: Specifications of PIC-181
without amplification. Therefore a horn shape transducer has been designed to amplify
the vibration amplitude. For a certain area of radiation surface, a larger cross-section
area of the piezoelectric ceramic ring will result in a larger amplification factor. Thus,
ceramic rings of 50 mm in outer diameter, 20 mm in inner diameter and 5 mm in
thickness are selected.
The required volume of the ceramic rings can be estimated from the expected max-
imum output power. For PIC-181, the power capacity is commonly said to range from
1.5 to 3 W/cm3kHz for high power applications (1). Considering the selected ceramic
rings, the volume of one ring is 8.3cm3. At the designed working frequency of 20kHz,
each ring can provide about 250 W to 500 W output power. Thus, in order to fulfill
the designed output power, four ceramics rings are mounted in each transducer. Of
course, higher power can be achieved by adding more ceramic rings in the transducer.
It will also result in higher vibration amplitude without increasing input power. How-
ever the amount of the ceramic rings is limited by the structure of the transducer, and
furthermore, the transducer will become more sensitive to the load with more ceramic
rings (42).
The material selected for the front and back covers are Titanium alloy and stain-
less steel. The selected combination provides the best balance of acoustic impedance
matching and mechanical strength.
92
9.1 Design of the proposed bearing
Figure 9.2: FEM simulation result of the ultrasonic transducer
At first, the dimension of the transducer is calculated as the example given in Sec.
5.2.2 with nodal plane located in the front cover. Lengths of each part are calculated
according to Equ. 5.9 and 5.10. A FEM model is built according to the obtained dimen-
sions by employing ANSYS software, since the Finite Element Method (FEM) can give
a reliable indication of the natural frequency and the vibration mode. By repeating the
modal analysis and modifying the dimensional parameters, the transducer is tuned to
match the designed natural frequency and vibration mode. A concave radiation surface
to match the spindle surface is also modified on the front cover. The FEM simulation
result of the transducer is shown in Fig. 9.2.
The transducers are manufactured according to the dimensions from FEM optimiza-
tion. After assembling, the frequency response of all three transducers was measured
using an impedance analyzer. The admittance and phase versus frequency diagram of
all three transducers are shown in Fig. 9.3. The result shows that the resonant and
anti-resonant frequencies at the desired vibration mode are close to 20 kHz and 20.8
kHz respectively. Despite slight differences among the three transducers, the measure-
ments match with the simulation results quite well. Under unloaded condition, the
vibration amplitude at the middle of the radiation surface is measured as 12 µm using
a laser interferometer when the input power is approximately 50 Watt.
93
9. AN NON-CONTACT JOURNAL BEARING BASED ON SQUEEZE
FILM ULTRASONIC LEVITATION
1.9 1.95 2 2.05 2.1 2.15 2.2
x 104
10−5
100
f [Hz]
|Yel| [A/V]
1.9 1.95 2 2.05 2.1 2.15 2.2
x 104
−100
−50
0
50
100
f [Hz]
phase(Yel) [°]
Transducer 1
Transducer 2
Transducer 3
Figure 9.3: Measured frequency response of the ultrasonic transducers
Figure 9.4: Prototype of the proposed squeeze film bearing
94
9.2 Testing the prototype bearing
9.1.2 The spindle-bearing system
A spindle-bearing system is constructed in the lab to investigate the behavior of the
presented squeeze film journal bearing. The realized prototype system is shown in Fig.
9.4. The system consists of a spindle supported by the presented bearing at one end
and a ball bearing at the other end. The distance between the two bearings is 150 mm.
Two eddy current displacement sensors are installed to measure the spindle position.
The diameter of the spindle is 49.94 mm which forms a mean bearing clearance of 30
µm with the bearing journal. Each transducer is driven in its resonant frequency. After
manually adjusting the input power of each transducer, the bearing system provides
steady, friction-free suspension to the spindle.
9.2 Testing the prototype bearing
The levitation forces at different gap distances in vertical direction are measured using
the eddy current sensor and a load cell. External load is applied on the spindle in the
gravity direction through the load cell. The maximum load is measured right before
contact happens between the bearing and the spindle. The results are plotted in Fig.
9.5 together with the calculated results from the presented mathematical models by
utilizing the actual parameter values. Constant vibration amplitude of 20 µm of the
transducer is maintained during the measurement. The measurement results agree very
well with the numerical results of Reynolds equation. A load-carrying force of 51 N
(6.37 N/cm2)is obtained when the gap distance is 28 µm at the bottom transducer.
Seeing the trend in Fig. 9.5, higher bearing force may be achieved by further reducing
the gap distance. But this is limited by the manufacturing accuracy of the bearing
surfaces. Some region starts to get in contact by further reduction of the gap.
The run-out error of the levitated spindle is measured when the spindle is stationary
at the beginning and starts rotating after some time. The results are shown in Fig.
9.6. It can be seen that the levitation is very stable when the spindle is stationary.
When the spindle is rotating, periodic error of a few micrometers is measured which is
subjected to the form error and surface roughness of the spindle. The run-out error of
the center of the spindle remains very small.
95
9. AN NON-CONTACT JOURNAL BEARING BASED ON SQUEEZE
FILM ULTRASONIC LEVITATION
20 30 40 50 60 70 80 90 100
0
20
40
60
80
100
120
Distance [µm]
Levitation force [N]
Reynolds − numerical
Reynolds − analytical
Acoustic radiation
Measurement
Figure 9.5: Load capacity versus levitation distance at constant vibration amplitude of
20 micrometers
0 1 2 3 4 5 6 7 8
−5
−4
−3
−2
−1
0
1
2
3
4
5
t [s]
Position [µm]
Figure 9.6: Spindle run-out error during levitation
96
9.2 Testing the prototype bearing
0 0,05 0,1 0,15 0,2 0,25
0
5
10
15
20
25
t [s]
Displacement [µm], Current [A]/20
Current
Displacement
Figure 9.7: Displacement response of the spindle center subjected to a current step
The load-carrying force of the proposed bearing is the average force from every
compress-release cycle between the bearing and the spindle. For driving frequency
of 20 kHz, the time that is needed for one compress-release cycle is 50 µs. When
the spindle is rotating, a lateral movement is generated between the bearing and the
spindle surfaces. The lateral movement has the line speed of the rotating spindle. For a
relatively high rotation speed of 10000 rpm, the line speed of spindle (50 mm diameter)
is 8.33 m/s. The spindle rotates 3 degree in one compress-release cycle, and the relative
lateral movement between two surfaces is 0.42 mm, which is much smaller compared to
the circumference of the bearing. Therefore, the spindle can be considered as staying
still within one compress-release cycle. We can conclude that the levitation force is not
sensitive to the rotation speed. The proposed bearing is suitable for both low and high
speed applications.
Figure 9.7 shows the displacement response of the levitated spindle subjected to a
current amplitude envelope step. The current amplitude is proportional to the vibration
velocity amplitude V0, which is in turn proportional to the levitation force according to
Equ. 8.11. Thus figure 9.7 is the step response of a mass-spring-damper system excited
by a force. This step response can be used in the design of a PID controller with a
classic method like “open loop method” (24) to determine the optimal parameter of
97
9. AN NON-CONTACT JOURNAL BEARING BASED ON SQUEEZE
FILM ULTRASONIC LEVITATION
the controller.
9.3 Conclusion
In this chapter, a squeeze film bearing has been developed which can provide a non-
contact support to rotating parts. The proposed bearing does not require external
pressurized air or liquid supply. The bearing is naturally stable since the squeeze film
levitation force is a repelling force, which increases as the distance is getting smaller.
The spindle is “pushed” from all around and held at the equilibrium position. A solid
steel spindle with diameter of 50 mm has been successfully levitated. The maximum
load capacity of 51 N is achieved at the vibration amplitude of 20 µm. This is consider-
ably larger than that of all other squeeze film bearing presented before (17;51;65;66)
whose load capacities are usually within a few Newton. Load capacity can be further
increased by increasing the vibration amplitude and by improving the surface quality.
The proposed bearing can be applied to support heavy rotors such as spindles in ma-
chine tools.
98
10
Summary and outlook
Ultrasonic levitation has been investigated theoretically and experimentally to con-
struct non-contact bearing systems. Both type of ultrasonic levitation, standing wave
and squeeze film type, are studied with two prototype systems.
For standing wave type ultrasonic levitation, a one dimensional acoustic levitation
system using ultrasonic standing wave is proposed with detailed theoretical analysis.
The proposed system is able to levitate large planar object (much larger than the sound
wavelength) at a position of multiple times of a half wavelength of the sound wave (much
higher than squeeze film levitation system). The proposed levitation system is suitable
for building non-contact bearings for applications which require low load capacity but
very high separation distance. The theoretical model for the proposed levitation system
is established. A prototype system is constructed accordingly. A CD is successfully
levitated with the proposed system at a height of half a wavelength. The levitation
forces at different distances are measured. A force of about 1 N is observed at the
position of half wavelength in front of the sound radiator.
Squeeze film type ultrasonic levitation has been investigated theoretically to im-
prove the achievable levitation capacity. Excursion ratio ǫand squeeze number σare
found to be the most crucial parameters that determine the levitation capacity. A good
surface finishing and form accuracy are also important for a squeeze film levitation sys-
tem to achieve higher levitation capacity. A higher vibration amplitude is helpful to
achieve higher levitation capacity when the mean gap δcan not be further reduced. The
99
10. SUMMARY AND OUTLOOK
theoretical model is validated using experimental results. A novel non-contact journal
bearing is presented based on the theoretical investigation, which aims on high load
capacity. The proposed bearing is actuated by Langevin type piezoelectric transduc-
ers. A solid steel spindle with diameter of 50 mm has been successfully levitated. The
maximum load capacity of 51 N (6.37 N/cm2) is achieved at the vibration amplitude of
20 µm. This is considerably larger than the previously reported squeeze film bearings
(17;51;65;66) whose load capacities are usually within a few Newton (less than 1
N/cm2). The achieved load capacity from the proposed bearing is already comparable
to a conventional air bearing of similar size, whose load capacity is usually between 10
to 20 N/cm2. The proposed bearing has very stable performance in both low and high
rotation speed.
At high rotational speed, an aerodynamic pressure can be generated in the air gap.
This aerodynamic pressure can give additional load capacity to the bearing system. The
influence on the squeeze film pressure caused by the lateral relative motion between the
spindle and the bearing surface during rotation should be studied in the future. The
presented bearing shows good dynamic behaviors. By implementing a feedback control
system, an active bearing with run-out error and external disturbance compensation
can be achieved.
100
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