Univerzitet u Ni{u

Transcription

Univerzitet u Ni{u

A SERIES OF EXTRAORDINARY AND UNIQUE BOOKS RECOMMENDED BY MPI
Dr. Milan Đ. Radmanović, Dr. Dragan D. Mančić
DESIGN AND MODELING OF
THE POWER ULTRASONIC
TRANSDUCERS
Published 2004 in Switzerland by MPI
198 pages, Copyright © by MPI
All international distribution rights exclusively reserved for MPI
Book can be ordered from:
MP Interconsulting
Marais 36
2400, Le Locle
Switzerland
[email protected]
Phone/Fax: +41- (0)-32-9314045
email: [email protected]
http://www.mpi-ultrasonics.com
http://mastersonic.com
TRANSDUCERS
University of Niš
Faculty of Electronics
⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯
Milan Đ. Radmanović, Dragan D. Mančić
TRANSDUCERS
Edition: Monographies
⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯
2004.
M. Radmanović, D. Mančić
DESIGN AND MODELING OF THE POWER ULTRASONIC TRANSDUCERS
Publisher:
Faculty of Electronics in Niš
PO Box 73, 18000 Niš
http://www.elfak.ni.ac.yu
Reviewers:
Prof. Vančo Litovski, Ph.D.
Professor at Faculty of Electronics in Niš,
Prof. Stojan Ristić, Ph.D.
Professor at Faculty of Electronics in Niš
Editor in Chief: Prof Dragan Drača, Ph.D.
Technical Editor: Reader Dragan Mančić, Ph.D.
By enactment of Scientific-Educational Board of Faculty of Electronics in Niš, No.
1/05-042/04-002 from February 17th 2004, this manuscript is approved for printing.
ISBN 86-80135-87-9
CIP - Cataloguing in publishing
National Library of Serbia, Belgrade
621.373:534-8
RADMANOVIĆ, Milan Đ.
Design and Modeling of the Power Ultrasonic Transducers /
Milan Đ. Radmanović, Dragan D. Mančić. – Niš: Faculty of Electronics,
2004 (Niš: M Kops Centar). - 198 pp.: graph. presentations, tables; 24 cm. (Edition Monographies / [Faculty of Electronics, Niš])
On front page top.: University of Niš. Number of copies: 300. - Bibliography: pp. 189-198. Registry.
ISBN 86-80135-87-9
1. Mančić, Dragan D.
a) Ultrasonic transducers
COBISS.SR-ID 113288972
Monography is published with financial support of Ministry of Science and
Environmental Protection and Hydroelectric Plant “Đerdap 1”- Kladovo.
Reprinting or copying of this book is not allowed without permission in
writing from the publisher.
Number of copies: 300
⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯
Printed by: M KOPS CENTAR, Ni{
PREFACE
Field of power ultrasonic technique, which represents an important field of
industrial electronics, in recent two decades experienced very swift and dynamic
development. An intensive development concerns as design and construction of
new ultrasonic devices, as well as broadening of application fields of power
ultrasound in many industrial branches and processes (mechanical, electric, and
chemical industry). Aside with appearing of new applications of ultrasound, new,
more perfect sandwich transducers are designed and developed, and numerous
scientific papers appeared, in which are treated different aspects of power
ultrasonic technique, especially different electromechanical models by which is
obtained design and optimization of ultrasonic transducers.
In this monograph firstly is performed systematization of different existing
procedures and methods for modeling of power ultrasonic transducers. Besides
that, new procedures of modeling, design, and optimization of power ultrasonic
transducers are presented, based on previously realized original models of
piezoceramic and metal rings. Thus is completed design of a sandwich transducer
as a unique system, consisted of piezoceramic rings, emitting and reflecting metal
ending, as well as of central bolt. Basic idea of the authors was to help with
realized models to the designers of new ultrasonic systems, due to the fact that
currently there is no literature from this field in Serbian.
Original results, presented in this monograph, are product of several-yearresearch in the field of power ultrasound in the Laboratory for energetic
electronics and control of electroenergetic transducers in the Faculty of
Electronics in Niš, wherefrom originated over 50 scientific papers from this field.
Concrete results, presented here, are part of one master thesis and one doctoral
dissertation, realized in the frame of research in this field.
On this occasion authors express their gratitude to the reviewers, Prof Vanča
Litovski, Ph.D. and Prof Stojan Ristić, Ph.D. on their useful suggestions and notes.
Niš, January 2004
Authors
LIST OF SYMBOLS
a, ai
adB
A, Ai, Aij
Apmax
b, bi
B, Bi, Bij
E
D
cij , cij
cij
p
cij
c
C
C0, C0’
Ci, Cn, CS
Cin
d
dij
D
D, Dz
D0
eij
p
e31
EY
EYr, EYz
E
Εr, Εθ
f
fr
fa
fi
Fi
FJ, FY
Gn
h
hij
H(s)
I
Ja, Jb, Ya, Yb
Ji, Yi
k, kr, kz
k0, kn, kY
keff
kp, k31, k33
outer radii of consisting elements of transducers ............................ (m)
attenuation ..................................................................................... (dB)
constants
maximal attenuation in band-pass ................................................. (dB)
inner radii of consisting elements of transducers ............................ (m)
constants
2
piezoelastic constants ................................................................ (N/m )
2
coefficients of elasticity ............................................................. (N/m )
2
constants ................................................................................... (N/m )
-1
damping factor (internal friction) .............................................. (kg s )
constant
capacitances of compressed ceramic ................................................(F)
capacitances .....................................................................................(F)
normalized capacitances
diameter of consisting element of a transducer ............................... (m)
piezoelectric constants ................................................................ (m/V)
constant
dielectric displacement tensor and component of dielectric
2
displacement tensor ....................................................................(C/m )
2
amplitude of dielectric displacement .........................................(C/m )
2
piezoelectric constants ...............................................................(C/m )
2
constant ......................................................................................(C/m )
2
Young’s modulus of elasticity .................................................. (N/m )
2
seeming moduli of elasticity ..................................................... (N/m )
electric field tensor....................................................................... (V/m)
components of electric field tensor ............................................. (V/m)
frequency ...................................................................................... (Hz)
resonant frequency ........................................................................ (Hz)
antiresonant frequency .................................................................. (Hz)
values of function in point i in simplex algorithm
forces on external surfaces of elements .......................................... (N)
constants
amplification of amplitude of oscillation
half-thickness of consisting element of a transducer ...................... (m)
piezoelectric constants ................................................................ (V/m)
transfer function
current ............................................................................................. (A)
constants
Bessel’s functions of i-th order, first and second rank
-1
wave (characteristic, eigenvalue) numbers ................................... (m )
-1
wave (characteristic, eigenvalue) numbers .................................. (m )
effective electromechanical coupling factor
one-dimensional static coupling factors
List of symbols
ka, kb, ki
k’
l, li, l’, l’’
le, lr, lp, l0
Li, Ln
Lin
mg
nS, nSi
Ni
N’
Pi
Q
r, θ, z
rm1, rm2, ri
constants
-2
constant of spring (spring stiffness)........................................... (kg s )
lengths of consisting elements of transducers.................................. (m)
lengths of consisting elements of transducers ................................. (m)
inductances ..................................................................................... (H)
normalized inductances
mass ............................................................................................... (kg)
coupling factors of oscillations
constants
transfer relation of a transformer
2
areas of consisting elements of transducers .................................. (m )
Q factor
polar cylindrical coordinates
distances in radial direction
(radii on circular surfaces) .............................................................. (m)
constants
R’, Ri’
Rp, Rg, Rn, Ri resistance ......................................................................................... (Ω)
normalized resistances
Rin
R’, L’, C’, G’ distributed line parameters
2
E
D
piezoelastic constants ............................................................... (m / N)
sij , sij
s
Laplace’s operator
relative strain tensor and components of relative mechanical
S, Spq
strain in point
t
time .................................................................................................. (s)
mechanical stress tensor and components of
T, Tpq
2
mechanical stress in point .......................................................... (N/m )
o
temperature ....................................................................................( C)
Tp
displacement vector and components of displacement vector of ring
u, ui
and disk points ................................................................................ (m)
-1
wave propagation velocities in radial and axial direction............(ms )
vr, vz
-1
wave propagation velocities at symmetric transducer ends .........(ms )
ve, vr
uzm, ue, ur maximal amplitudes of displacement .............................................. (m)
-1
point velocities on external surfaces of elements.........................(ms )
vi
-1
longitudinal wave propagation velocity.......................................(ms )
v0
compression (longitudinal) wave propagation velocity in infinite
vd
-1
medium.....................................................................................(ms )
equivolume (transversal) wave propagation velocity in infinite
vs
-1
medium.....................................................................................(ms )
-1
velocity of Rayleigh’s waves.......................................................(ms )
vR
electric voltages ............................................................................... (V)
V, Vg, Vi
amplitude of excitation electric voltages ........................................ (V)
V0
vectors of variables
xi
shape functions
xr, Xr
reactance ......................................................................................... (Ω)
Xn
reactances in T network of Mason’s model .................................... (Ω)
Xa, Xb
List of symbols
X1
Zi
zul
Zul
Zm, Zij
Zc, Zci
Ze,p,r, qn
zij
reactance in KLM model ................................................................. (Ω)
acoustic impedances ................................................................... (Rayl)
input electric impedance ................................................................. (Ω)
modulus of input electric impedance (Zul=adB) ............................ (dB)
-1
mechanical impedances ........................................................... (Nsm )
-1
characteristic impedances ........................................................ (Nsm )
relations of characteristic impedances
elements of impedance matrices
α, β
normalized wave numbers
αr, αr', βr, δr normalized wave numbers
αrm, αrn, αrm’ constants
α’
coefficient of reflection in simplex algorithm
α”
constant
β’
coefficient of contraction in simplex algorithm
γ’
coefficient of expansion in simplex algorithm
γ p=α p+β p function of line propagation
δ
normalized wall thickness of ring
∆
determinant of system
∆ J1, ∆ Y1
constants
∆f
band-pass ....................................................................................... (Hz)
S
ε33
relative dielectric constant of compressed piezoceramic ..............(F/m)
p
ε33
constant ........................................................................................(F/m)
ε
constant
ξ
normalized wave number (ξ=αr)
ζ’, ζ”
coefficients of radiation
λ, λi
wave lengths .................................................................................... (m)
2
λm, µ
Lame’s material coefficients ..................................................... (N/m )
λν
constant
υ, υi, υij
Poisson’s relations
νν
displacement shape factor
3
ρ, ρi
material densities ..................................................................... (kg/m )
p
σ
constant
φ
transformer ratio
ω, ωr, ωa
circular frequencies .................................................................... (rad/s)
Ω
normalized circular frequency
2
Laplace’s differential operator
∇
i, j
indexes: (1, 2, ...) and (e, p, r)
p, q
indexes (r, θ, z)
indexes (0, 1, 2, ...)
m, mi
N
indexes (0, 1, 2, ...)
n
indexes: (0, 1, 2, ...) and (e, r)
CONTENTS
1. INTRODUCTION ................................................................................. 1
1.1. BASIC APPLICATIONS OF POWER ULTRASOUND ..........................2
1.2. BASIC FORM OF SANDWICH TRANSDUCERS...................................5
1.3. INTRODUCTION INTO MODELING OF ULTRASONIC
TRANSDUCERS ...........................................................................................8
2. MODELING OF PIEZOELECTRIC CERAMIC
RINGS AND DISKS............................................................................ 15
2.1. ONE-DIMENSIONAL MODELS OF PIEZOCERAMICS ....................15
2.1.1. BVD Model of Piezoelectric Ceramics .................................................15
2.1.2. Mason’s Model of Piezoelectric Ceramics ...........................................22
2.1.3. Redwood’s Version of Mason’s Model of Piezoceramics ...................24
2.1.4. KLM Model of Piezoelectric Ceramics ................................................25
2.1.5. PSpice Models of Piezoelectric Ceramics.............................................27
2.1.6. Martin’s Model of Package Piezoceramic Transducers .....................30
2.2. THREE-DIMENSIONAL MODELS OF PIEZOCERAMIC
RINGS ..........................................................................................................31
2.2.1. Berlincourt’s Model of Piezoceramic Rings.........................................33
2.2.2. FEM Models of Piezoceramic Rings.....................................................33
2.2.3. Brissaud’s 3D Model of Unloaded Piezoceramic Rings......................38
2.2.4. Matrix Radial Model of Thin Piezoceramic Rings..............................39
2.2.5. Three-dimensional Model of a Piezoceramic Ring Loaded on All
Contour Surfaces....................................................................................41
Contents
2.2.5.1. Analytical Model ................................................................................42
2.2.5.2. Model of External Behaviour of Piezoceramic Ring ..........................44
2.2.5.3. Numerical Results...............................................................................47
2.2.5.3.1. Input Electric Impedance of Ppiezoceramic Ring .........................47
2.2.5.3.2. Frequent Spectrum of Piezoceramic Ring .....................................54
2.2.5.3.3. Effective Electromechanical Coupling Factor .............................59
2.2.5.3.4. Components of Mechanical Displacement of Piezoceramic Ring
Points .............................................................................................62
2.2.5.4. Experimental Results ..........................................................................68
3. MODELING OF METAL CYLINDRICAL RESONATORS ........ 75
3.1. ONE-DIMENSIONAL MODELS OF METAL
RESONATORS ...........................................................................................76
3.1.1. Analysis of Oscillation of Long Half-wave Resonators.......................76
3.1.2. Rayleigh’s Correction of Wave Propagation Velocity ........................78
3.1.3. Experimental Studies of Metal Cylinders Oscillation.........................79
3.1.4. Method of Seeming Elasticity Moduli ..................................................81
3.2. THREE-DIMENSIONAL MODELS OF METAL RESONATORS......84
3.2.1. Hutchinson’s Theory of Oscillation of Metal Cylinders .....................85
3.2.2. Finite Element Method .........................................................................89
3.2.3. Numerical Analysis of Axisymmetric Oscillation of Metal Rings......90
3.2.3.1. Determination of Frequency Equation................................................91
3.2.3.2. Analysis of Longitudinal Waves Ppropagation Velocity of
Metal Rings ........................................................................................97
3.2.3.3. Determination of Resonant Frequencies of Metal Rings ....................99
3.2.4. Three-dimensional Matrix Model of Metal Rings.............................106
3.2.4.1. Analytical Model ..............................................................................106
3.2.4.2. Numerical Results.............................................................................108
3.2.4.3. Comparison of Numerical and Experimental Results.......................111
4. MODELING OF POWER ULTRASONIC
SANDWICH TRANSDUCERS........................................................ 119
4.1. APPLICATION OF EXPERIMENTAL METHODS IN ULTRASONIC
TRANSDUCERS DESIGN.......................................................................124
4.1.1. Design of Sandwich Transducers by Trial and Error Method .........124
Contents
4.1.1.1. An Example of Transducer Design with Heavy Metal Reflector .....125
4.1.1.2. Experimental Results ........................................................................127
4.1.2. BVD Model of Ultrasonic Sandwich Transducers ............................130
4.1.2.1. An Example of Application of BVD Model in Design of the Circuit
for
Ultrasonic Transducers Adaptation ...................................................131
4.1.2.2. Numerical and Experimental Results................................................135
4.2. ONE-DIMENSIONAL MODELS OF PIEZOCERAMIC
ULTRASONIC SANDWICH TRANSDUCERS ....................................138
4.2.1. Langevin’s Equation............................................................................138
4.2.2. Equation of Half-wave (λ/2) Sandwich Transducers ........................143
4.2.3. Ueha’s Equation ...................................................................................147
4.2.4. General One-dimensional Model of Sandwich Transducers............148
4.2.5. Design of Ultrasonic Sandwich Transducers by Seeming Elasticity
Moduli ...................................................................................................154
4.3. THREE-DIMENSIONAL MODELS OF ULTRASONIC
SANDWICH TRANSDUCERS ..............................................................162
4.3.1. Numerical FEM and BEM Methods ..................................................162
4.3.2. Three-dimensional Matrix Model of Piezoceramic
Ultrasonic Sandwich Transducers......................................................163
4.3.3. Comparison of Numerical and Experimental Results.......................165
LITERATURE ......................................................................................... 189
1. INTRODUCTION
Using of ultrasonic transducers started in 1917, when Paul Langevin designed the
first piezoelectric sandwich transducer [1]. At the very beginning of its development,
basic application was in sonars. Soon after that, specific effects of power ultrasound on
different processes were described, which caused numerous scientific activities in
studying cavitation, dispersion, thickening, and chemical and biological effects of
ultrasound. Not until 1950, main applications of power ultrasound, like ultrasonic
cleaning and machine processing, cross from laboratory research to industrial
application. Ultrasonic industry experienced an intensive development, especially in
last two decades, as from aspect of manufacturing of adequate devices, as well as in
expanding the field of application of this technique, so that current areas of application
of ultrasound are very diverse [2]. Ultrasound acts in different ways on solid bodies,
liquids, and gases performing desired effects. Applied procedures distinguish with
speed, rationality, low environmental pollution, etc.
One of the main problems in many fields of application of power ultrasound is
including of laboratory experiments directly into broad industrial application.
Intensive development of the ultrasonic technique enabled its breach into different
scientific, medical, technical, and technological areas. Field of power ultrasound
2
encompasses power densities in the radiation zone from several W/cm up to
2
several thousands W/cm . The most important applications of power ultrasound
that arose during its development for industrial demands are ultrasonic cleaning,
welding of metal and plastic, drilling and soldering. Power densities for particular
2
2
applications are, for example: 0.5÷3W/cm for ultrasonic cleaning, 10÷50W/cm
2
2
for welding of plastic, 10÷100W/cm for drilling, and 600÷6000W/cm for
welding of metals. The greatest score of industrial applications encompasses
frequency range from 20kHz to 1MHz, where the most often used frequency range
is 20÷60kHz. On emitting surfaces amplitudes of oscillation are from 1÷50µm [3].
However, besides the cited most frequent industrial applications, which will here
be the main subject of interest, there is also variety of other very important
applications of ultrasound (and not only of power ultrasound). Therefore, it is
necessary to present briefly their basic characteristics too, which was done in the
following exposure by brief review of almost all applications of ultrasound.
2
Introduction
1.1. BASIC APPLICATIONS OF POWER ULTRASOUND
Specific properties of ultrasound allow its multiple applications, which, in
spite of its broad spectrum, may be ranked into two principal fields. In the first one,
ultrasound is used for information obtaining, and besides the ultrasound sources
here are also used appropriate sensors. In the framework of the second field acting
of the ultrasound on matter is applied. A more subtle division may be done based
on the field of application: acting on solid bodies, acting on liquids and gases, and
application in medicine and biology [4], [5].
In using of power ultrasound in liquids [6], [7], [8], one of the oldest and most
spread applications is ultrasonic cleaning. Ultrasonic cleaning gives the best results in
cleaning of the relatively hard materials, as ceramic, glass, metal, and plastic, which
better reflect than absorb excitation ultrasonic field. Cleaning devices usually operate in
frequencies between 20÷50kHz with emitted ultrasonic power of 5÷25W per liter of
cleaning liquid. Attempts of overloading the vessel with ultrasonic energy leads to the
loss of cleaning effect in distant parts of the vessel due to the damping of the ultrasonic
power by cavitation zones formed near the vessel walls. Because the strength of the
cavitation impact is greater on 25kHz than on 50kHz, cleaning on lower frequencies is
more efficient. However, for finer parts (for example in electronic industry), is used
ultrasound on frequencies from 40÷50kHz. For cleaning of small, complicated objects
frequencies of several hundreds kHz [9], [10], [11], [12], [13], [14], [15] are used.
In ultrasonic processes of extraction, the most important effects are cavitation and
acoustic convection, whereat usually frequencies of around 20kHz are used, while for
some processes are used frequencies above 500kHz. Ultrasonic extraction processes
encompass manufacturing of floral perfumes, hop oil, sugar from sugarcane, chemicals
from plants, cod-liver oil, etc., by which are achieved great savings in material and time.
Using power ultrasound in various fluids, solid particles, as well as liquids, may be
undergone to the process of dispersion. Primary phenomena that lead to dispersion are
again cavitation and acoustic convection. These are processes at frequencies from
20÷60kHz and they are met in pharmaceutical industry, industry of paint and varnish and
preparation of specimens for electronic industry. Main advantages of application of power
ultrasound in emulsion derivation and homogenization of liquids relate to the ability of
blending some liquids that cannot be blended without using of additives. Ultrasonic
derivation of emulsions is used in food industry, pharmaceutical and cosmetic industry,
and especially in mixing of water with hot oil because of better economy and lower
pollution [16].
Procedures of ultrasonic derivation of aerosols by cavitation and radiation
pressure have as a result drops of very small dimensions, which was used in
medical inhalators, metal pulverizing, and manufacturing of powder from cast
metal [17], [18]. Power ultrasound also affects the processes of particle
accumulation, where the main effects are radiation pressure, acoustic convection
and Bernoulli’s forces. Strong ultrasonic fields of standing waves lead to the fast
increase of micron and submicron aerosol particles [19], [20]. In application of
ultrasonic degassing, directional diffusion induced by power ultrasound leads to
growth of gas bubbles in the liquid in which exists ultrasonic field.
Microconvection around pulsing bubbles in stable cavitation leads to the transport
Introduction
3
of mass and assists diffusion of the dissolved gas. These mechanisms of degassing
for eliminating the gas from molten metals, glass, and products of food industry
lead to the decrease of their porosity. Also, using power ultrasound one may
increase the degree of liquid flow through porous mediums, which means that
industrial devices based on power ultrasound serve for increase of filtration degree
[21], [22].
In ultrasonic field evaporation of liquid can be increased, among other things,
because of acoustic convection and moving of the drying object. This effect is used
for drying powders sensitive to heat in food and pharmaceutical industry, where
lower temperatures are used for preventing damages that would arise on sensitive
materials. Usually are used frequencies ranging 12÷20kHz [23]. By using of
ultrasound one may improve heat transfer in distillation devices. Also, cooling in
cooling systems may be improved by using of ultrasound with frequency of 20kHz,
and power 1÷5kW, that prevents forming of crystal layers in the cooling system
pipes. The process of heat transfer is affected by acoustic convection and
occurrence of turbulence due to bubble pulsing (stable cavitation) in the ultrasonic
field in liquids. Also, using power ultrasound with frequency range 20÷35kHz,
without using of additives, may be removed foam arose during manufacturing of
food and pharmaceutical preparations [24].
An activity with fast development where ultrasound can be applied is
sonochemistry. Several types of reactions may be accelerated by ultrasound.
Reactions to which ultrasound cane be applied encompass metal surfaces, reactions
including powders or other granular materials, emulsion reactions, and
homogenous reactions. Here are characteristic four ultrasonic chemical processes:
(1) acceleration of conventional reactions, (2) reduction processes in water
solutions, (3) degradation of polymers, and (4) dissolution of organic solvents and
reactions in them [25], [26]. In the processes of crystallization, cavitation and
acoustic convection in molten metals strongly affect on forming of growing crystal
cores. In their presence occurs a structural refinement of ingots, and considering
improvement of structural and technological qualities of materials [27], [28], [29],
[30].
Beside the cited, by power ultrasound is possible to solder without leaking,
where in most cases is improved solder adhesiveness. Acting of cavitation at
ultrasonic soldering disturbs and destroys the oxide surface and acts onto the metal
base similarly as at ultrasonic cleaning. Advantages of ultrasonic soldering use are:
(1) elimination of leaking, (2) possibility of soldering different materials, (3)
elimination of cleaning operation after soldering, (4) greater resistance to corrosion
of soldered parts due to absence of leaking, and (5) improved soaking. The most
often used frequency range is from 20÷30kHz [31].
Application of power ultrasound on solid bodies encompasses, above
all, machine processing of material. There are two basic processes that are
spread in using of ultrasound in machine processing: (1) processing of hard
and fragile materials using abrasive emulsions for grinding, and (2) lathing,
milling, drilling, and extrusion with ultrasonic vibrations. The first process
is in use for a few years, but its application declines. The second process,
4
Introduction
which is in essence a fast abrasive process, has increasing industrial
application due to the appearing of more reliable tools. By application of
ultrasound the quality of cutting is substantially increased, which prolongs
the exploiting lifetime of tools and lessens the cutting. Effects arising by
acting of ultrasound are: (1) cavitation, which leads to permanent cleaning
of tools, (2) acoustic convection, which increases the cooling level, and (3)
effect of accelerating of abrasive particles at their acting on the processed
part.
Physical parameters that have influence on machine processing efficiency are: (1)
amplitude of vibrations, where the most often used amplitudes are from 10÷40µm at
frequency of 20 kHz, (2) frequency in range 10÷50kHz, whereat most of the ultrasonic
processing is done at 20kHz, (3) static pressure, (4) size of abrasive particles, (5)
concentration of abrasive material in the grinding emulsion, (6) circulation of the
grinding emulsion, and (7) material from which the tool is made.
Use of the power ultrasound during modeling of large pieces of material has not a
completed application, while in processing of small parts it is widespread [32], [33], [34],
and [35]. Advantages of using of ultrasound in shaping processes (extruding of pipes and
wires, continuous casting of steel bars, treating of metals, and squirting and casting of
plastic) among other things are: decreased force of extrusion 20÷40%, faster extrusion up to
40%, better precision, longer lifetime of tools, better quality of surface, thinner layer of
lubricating oil, possibility of producing different shapes like rectangular pipes with sharp
edges or pipes with large ratio of diameter vs. wall thickness, etc. These advantages should
enable decrease of stress in the processed part. Besides, interaction between ultrasound and
dislocation may lead to nonlinear change of the modulus of elasticity, which causes
softening and decrease of the tension stresses at increase of plastic stressing [36]. Also,
changes of friction coefficient and direction of the friction vector may affect the shaping
process. Here are used frequencies from 20÷30kHz and power ranging from 0.4÷14kW.
Welding of plastic materials can be done by power ultrasound. Welding of
metals is characterized with small amount of heat, since the welding temperature is
lower than the melting temperature of the metal. Distortions at ultrasonic welding
are very small.
Welding of metals is based on shear movements between the welded surfaces.
Welding of materials, which is usually impeded due to occurrence of oxide, is now
improved by abrasion of surfaces joined by process of ultrasonic welding. Joint is
efficiently realized by thin aluminum electrodes at frequencies from 19÷27kHz, at
2
ultrasonic field intensities from 1÷3kW/cm . For welding of wires and small
electronic components frequencies from 40÷60kHz and ultrasonic field intensities
2
ranging 5÷50W/cm [37], [38], [39] are used.
Welding of plastic is performed by moving of ultrasonic tool perpendicularly
at welding surface. Vibrational welding of plastic by shearing has advantages at
welding of thin films.
Most of the thermoplastic materials are suitable for ultrasonic welding, since
they have ability of transfer and absorption of vibrational energy, relatively low
melting temperature, and low thermal conductivity, so they can make local heated
Introduction
5
zones. Soft plastic, as well as rubberized porous plastics, have great reduction of
ultrasonic field and they are most suitable for the so-called welding in the close
zone, where the tool is in direct contact with the welded material. Welding in the
distant zone, where melting occurs at some distance from the contact with the
ultrasonic tool, is best performed at hard amorphous materials. Frequencies used
are in range from 10÷40kHz, with emitted power of welding tool between
2
100÷5000W. Field intensities in the welding zone are around 200÷1000W/cm ,
and welding times are from 0.1÷1s [40], [41], [42], [43], [44], and [45].
Ultrasonic testing of material fracture is in application for a few years (NDTnondestructive testing). It enables fast testing, estimate, and determination of the
fracture limit of the material. Besides that, this testing also provides information
about characteristics of crack propagation. Used frequencies are in range from
10÷60kHz (mostly 20kHz), with power around 1kW [46].
Cutting of hard materials can be performed using power ultrasound [47], as
well as splintering of crystals by direct impact of ultrasonic funnel, whereat is
eliminated accidental fracture [48]. Also, by using of power ultrasound one may
achieve increase of density at powder sintering.
In medicine, effects of power ultrasound are used in two principal areas, in
therapy, and in surgery. In therapy is used frequency range from 600kHz to 1MHz,
where most instruments operate at frequencies around 800kHz. In application of power
ultrasound in surgery, usually are used greater field intensities than in therapy [49],
[50].
1.2. BASIC FORM OF SANDWICH TRANSDUCER
In all mentioned cases, ultrasonic systems for industrial applications are most
often consisted of three elements: excitation generator, which operates in given
frequency range and provides electric supply, electromechanical sandwich transducers,
most often electrostrictive or magnetostrictive, which converts electric energy into
mechanical energy at defined frequency, and medium of propagation, apropos solid or
liquid operating medium, which represents the area of ultrasonic energy radiation.
Power that can be provided by a standard transducer is limited, among other things, by
the volume of used electromechanical transducing material, which is inversely
proportional to the square of frequency. Emitted acoustic power of the transducer is
limited by acoustic impedance of the operating medium and depends on the realized
adaptation. Such ultrasonic system is usually supplied by high voltage, and basically
represents a nonlinear system. Since the interaction between the consisting elements
also strong, only a global model that takes into account all these facts can describe the
entire system in operating conditions. Such model hasn’t been yet realized.
Sandwich transducer, which is most often also called a Langevin’s transducer,
is a half-wave resonant structure that oscillates in thickness (longitudinal,
extension, or axial) direction (Figure 1.1). The very beginning of using of ceramic
materials for narrow-band, low-frequent ultrasonic applications was connected for
transducers made as simple, compact piezoceramic blocks or pipes. However, this
simple construction hasn’t proved very useful, especially for application in power
6
Introduction
ultrasound, due to the low extension resistance of ceramic, as well as due to large
physical dimensions of such transducers at low frequencies. Because of these
difficulties, the original ultrasonic transducer was modified and adapted to new
conditions.
5
7
7
4
6
3
6
9
3
3
4
9
3
8
8
1
1
2
(b)
(a)
Figure 1.1. Variants of half-wave sandwich transducers: (1, 2) duralumin emitter;
(3) PZT piezoceramic rings; (4, 5) steel reflector;
(6, 7) steel bolt; (8) copper contact foils; (9) insulating pipe
In the simplest form sandwich transducer is consisted of active layer or source
of oscillations, made of one or more pairs of piezoceramic rings, mechanically
compressed between metal endings by a bolt. For emitting applications the
piezoceramic should possess great electromechanical coupling factor, high Curie’s
temperature, low dielectric losses, and stable time and temperature characteristics.
1
The most used is lead-zirconium-titanate ceramic (PZT4 and PZT8 ) in shape of
disks or rings. Metal endings are consisted of: reflector, which represents the rear
part of the transducer, and emitter, which transfers oscillations from source to the
operating medium. For the metal endings most often are used materials of different
density, in order to increase the amplitudes of oscillations on the operating surface
of the emitter and decrease amplitudes of oscillations on the reflector surface, as
well as to improve the adjustment with the load. For emitting applications the best
is to use the titanium alloy, which distinguishes with great static and dynamic
strength, low mechanical impedance, and low mechanical factor of loss. However,
most often is for metal endings used cheaper combination with emitter of
duralumin, steel reflector, and steel bolt. Coupling between piezoceramic rings and
metal endings, as well as the increase of extension strength of piezoceramic, are
achieved using mechanical prestressing of the structure in longitudinal direction by
central bolt, which withstands the constant extension pressure. Thus the allowed
amplitude of dynamic mechanical stress, and by that itself the maximum power
intensity too, are considerably increased. Namely, in power ultrasonic transducers,
since a large compression pressure for prestressing of ceramic is applied, at any
1
Vernitron Ltd., now Morgan Matroc Ltd. (http://www.morganmatroc.com)
Introduction
7
level of electric excitation, the ceramic always stays in compressed state. Since the
ceramic has much greater mechanical strength at compression, than at extension,
prestressing increases the reliability of the transducer.
In the simplest prestressed transducers with one piezoceramic ring, one of the
metal endings is on high potential, and the bolt must be placed through an insulating
capsule, in order to be out of contact with metal endings. This brings limits into the
application of the compression force, so the improved versions with one or more pairs
of piezoceramic rings polarized in opposite directions, whereat the excitation voltage is
on the central electrode. At some newer constructions, the simple central bolt is
replaced with a few peripheral bolts. Thus is enabled a more uniformly stressing of the
piezoceramic disks. Piezoceramic rings are mechanically connected serially, and by
the contact foils are electrically connected parallel, in order to achieve greater
ultrasonic power. The rings are with opposite orientation of polarization, and they are
separated by an electrode that is placed in the node of transducer oscillation, and that
electrode is supplied by excitation voltage. Piezoceramic is supplied by an electronic
oscillator set to a fixed frequency, and the transducer and the surrounding medium are
considered as a forced oscillatory system.
Advantages of such design of the ultrasonic sandwich transducers are
following: (1) costs are low, because only thin piezoceramic rings are needed,
in contrast to the case when only piezoceramic material oscillates at given
frequency, and the influence of physical properties variation of the
piezoceramic material on transducer characteristics is decreased; (2) because of
that the capacitance of the piezoceramic ring is high, that is, its electric
impedance is low, which is suitable from aspect of minimal excitation of the
transducer; (3) compression prestressing using metal bolt increases mechanical
strength of the transducer; (4) metal endings are good cooling profiles, well
coupled with ceramic, which is also an advantage at higher levels of excitation;
(5) joining of metal endings using bolt represent a convenient way of fixing of
any further waveguide, that is, a connection metal-metal with endings and
concentrators in the more complex ultrasonic systems.
Main problems connected with design of such transducers are: (1) limits in power,
due to the physical properties of the material from which the transducer is made, (2)
exact determination of the resonant frequencies of the transducer, (3) calculations that
represent three-dimensional problems, (4) problems of transducer assembling.
Members of this family of transducers are important for transfer of ultrasonic
energy from one of the ending metal surfaces of the emitter into the solid or liquid
appliance, while the other metal surface of the reflector emits into the air. If the plane
of the oscillating node divides the ceramic equally, then we have the case of a
symmetric transducer, whereat metal endings may be made of same or different
materials. In the inverse case, when the middle of the ceramic is not in the oscillating
node, we have the case of an unsymmetric transducer. It is usual to name a transducer
with metal endings made of different material unsymmetric, regardless that the neutral
plane is located in the middle of the piezoceramic (Figure 1.1). Operating resonant
frequency of such symmetric and unsymmetric transducers is lower than individual
thickness resonant frequencies of unloaded piezoceramic rings or metal endings. Value
8
Introduction
of the resonant frequency will depend on dimensions and mechanical (acoustic)
characteristics of the metal endings and piezoceramic rings. It is necessary to
emphasize that in case of power transducers their quality depends on several factors,
from which the most important are the quality of the sole piezoceramic plates and
metal endings, sort of material used to make the contact metal foils, processing quality
of the contact surfaces, and the tension force.
1.3. INTRODUCTION INTO MODELING OF ULTRASONIC TRANSDUCERS
At the end of sixties, after appearing of the Rozenberg’s book about power
ultrasonic transducers and fields [51], [52], numerous scientific papers appear in which
are treated different aspects of power ultrasonic technique. The first optimizations of
the sandwich transducer parameters started in the beginning of the seventies, based on
the theory of electromechanical filters. Determination of the resonant frequencies of
the metal endings longitudinal oscillations, which have very complex geometry, was
done using vibrational platforms, and then an adjusting was performed by trimming. In
those first investigations certain conclusions were obtained, whereat the most
important is connected with transducer length, and as an optimum is established that
the half-wave transducer is the most efficient. At the same time there were attempts to
come to the mathematical relations by which could be performed calculations of the
metal endings length. However, such calculations are quite incorrect due to the
complexity of the electromechanical system of the sandwich transducer, complicated
mode of operation and oscillation of the sole piezoceramic plates, and the complexity
of three-dimensional calculation. As the technology of workmanship and quality of the
piezoceramic plates progressed, so transducers with better characteristics were
produced. On the other hand, the procedures of mechanical processing of the metal
endings in order of mechanical coupling improvement were developed, which
increased the efficiency of the transducers itself.
Development of the ultrasonic transducers was accompanied with great scientific
interest for this field, so that many projects were done in the area of perfecting the
materials applied for manufacturing of the transducer component parts, as well as in
the area of electric and mechanical improvements directed to further development of
power ultrasound. Thanks to those technological achievements, nowadays there are
numerous design methods for choosing of optimal form of ultrasonic transducers.
These methods start with mentioned application of initial mathematical formulas and
adjusting (trimming) of the transducers using vibrational platforms (trial and error
method or cut and true method), till today, when using powerful computer systems
attempts are made to perfom a simulation of ultrasonic transducers applying different
numerical procedures (modal analysis, matrix transfer method, finite element method
etc.). Modeling of sandwich transducers was conditioned by development of
piezoceramic disk and ring models and, alongside wit it, by development of metal
component parts of transducer models with same or similar shape. In the following
chapters will be presented the most important existing approaches in modeling of
power ultrasonic transducers, as well as several new approaches, from which the most
significant is based on matrix models of piezoelectric ceramic rings, wherefrom one
Introduction
9
may easily obtain models of piezoceramic disks, as well as metal component parts of
transducers in a shape of ring, disk, or cylinder.
As it was mentioned, the prestressed ultrasonic sandwich transducers, which
mostly oscillate at thickness (longitudinal, extension, or axial) resonant frequency (in
further text: at thickness mode also), are the most often used devices in the power
ultrasound technique. Investigation results here presented are related to an analysis of
sandwich transducers with all consisting parts of circular cross-section. Because of the
transducer prestressing needs, metal emitter, metal reflector and excitation
piezoceramic plates are circular rings with central opening, while the metal bolt is in
shape of solid cylinder. Consequently, transducers here modeled are by its form closest
to the practical ultrasonic sandwich transducers. Design of such ultrasonic systems is a
complex problem, as from the electric, so from the mechanical side. Mathematical
treatment of a model of complete sandwich transducer is difficult, because of its
complexity. Namely, writing of actual equations of such model implies consideration
of five mediums that the transducer components and its boundaries are consisted of, as
well as consideration of piezoelectric characteristics of ceramic, and, finally
assumption that all unknowns are functions of time and of three spatial coordinates.
Even in a less complex model, obtained when the transducer has cylindrical axis
symmetry, such approach in calculation is extremely complicated. For a precise
analysis of such kind of a transducer an ideal model would be three-dimensional, in
which is taken into account coupling of oscillations in radial and thickness direction
(in further text radial and thickness oscillations), whereat one may analyze the effect of
overloading on boundary (contour) surfaces in different directions. In general case the
complete three-dimensional analysis leads to a very complex set of nonlinear
equations, which are practically impossible to solve.
Characteristics of the piezoelectric sandwich transducers in the vicinity of the
fundamental resonant frequency of ultrasonic oscillations may be described by parallel
connection of a serial RLC resonant circuit and a capacitor, whereat this simplest
equivalent scheme is accurate enough for many practical applications. However, such
approach is not appropriate at sandwich transducer construction with several excitation
piezoceramic plates or transducers with metal endings of variable cross-section and
different shape, which are tightened by a metal bolt. This concerns above all to the
impossibility of parameter choice and control of some transducer consisting parts
during its calculation. Because of that, in order to perform the calculation of such
complex oscillatory systems, diverse ways of modeling were developed. Following
chapters are dedicated to the majority of the existing methods and models for analysis
of piezoelectric ceramics and metal component parts of the piezoceramic transducers,
as well as to the existing methods and models for analysis of complete ultrasonic
sandwich transducer.
There are several one-dimensional approaches in modeling of sandwich
transducers, all of them have in common that unknown values are functions of time
and longitudinal coordinate only. At the first group of one-dimensional approaches,
ceramic is represented as a passive, homogenous, isotropic medium, that is, in a way
the metal parts of the transducer are represented. Referent axis coincides with the
polarization axis, which is also the symmetry axis of the transducer, which is the most
often in cylindrical shape. By such an analysis one comes to the general equation of
10
Introduction
the sandwich transducer, which associates the resonant frequency of the transducer
with dimensions of its component parts and their characteristic impedances. Previous
discussion relates to an unloaded transducer. When the transducer excites the acoustic
appliance, resonant frequency will change due to the change of the boundary condition
on the operating surface. If the appliance is known, a new frequency equation must be
derived. Therefore the second group of one-dimensional approaches in modeling of
transducers arose, which imply using of equivalent circuits. Thus, the sandwich
transducer becomes a three-access network also with a three-access equivalent for
piezoceramic rings and line models for metal endings and the prestressing bolt.
Influence of the bolt often hasn’t been taken into account, because its effect has been
considered negligible, due to its low mass regarding the mass of the whole transducer.
These one-dimensional approaches in transducer modeling are mutually similar and
connected. They proved to be especially suitable in modeling of sandwich transducers
with lower operating resonant frequencies, at which the lengths of the metal endings
are great (at the same time much greater regarding the ceramic thickness).
In one-dimensional theory of sandwich transducer design is implied that
transducer oscillates in longitudinal mode, and radial oscillations are neglected. It
means that radial (lateral, cross) dimensions of the transducer should be much smaller
than its length. Generally, when cross dimensions of the transducer are smaller than
one-fourth of the longitudinal oscillations wavelength, then one may use the onedimensional theory and error between designed and measured resonant frequency is
negligible [53]. However, along with development of the ultrasonic technology,
ultrasonic transducers are more and more used in applications like ultrasonic cleaning
and welding, that is in applications that demand great output ultrasonic power. In these
applications, lateral dimensions of the transducer are usually greater than one-fourth of
the longitudinal oscillations wavelength, and then one-dimensional theory of sandwich
transducers design is nor usable. The reverse also stands, that is when resonant
frequency of the transducer increases, longitudinal wavelength and transducer length
proportionally decrease. Based on the assumptions from the one-dimensional theory,
lateral dimensions of the transducers must also be decreased, whereat decrease of the
transducer cross-section decreases its mechanical strength and limits the possibilities of
the transducer. Consequently, lateral dimensions of the transducer cannot be decreased
arbitrarily, so that radial oscillations at such transducer must be taken into account, in
order not to enlarge the error in determination of the resonant frequencies.
So, in case of short metal endings (e.g. at transducers with fundamental
resonant frequency fr around 40kHz, which are used in ultrasonic cleaning systems)
one-dimensional approach is incomplete. Then is necessary some modification of
the one-dimensional theory, by which is enabled modeling and design of such
transducers with short metal endings. Therefore is developed method of seeming
moduli of elasticity, whose basic purpose is oscillating regime analysis of power
metal ultrasonic emitters of cylindrical shape and large dimensions. This method is
also used in design of ultrasonic devices for alteration of oscillation direction. In
this approach one starts from an assumption that mutual coupling of oscillations in
radial and thickness direction generates changes of modulus of elasticity, and
thereby a change of wave propagation velocity in longitudinal direction. In the
continuation of the report, in Chapter 4.2.5, is showed that application of this
Introduction 11
method may be broadened also on piezoelectric ceramics, that is, using this method
one may model a complete ultrasonic sandwich transducer of cylindrical shape.
Comparing experimental results, better results were obtained than in case of
application of classic one-dimensional theory, but only in case of transducers with
longer metal endings, while in case of transducers with short metal endings
deviations were still great. Besides that, application of this method is limited only
to determination of the fundamental resonant frequency of the transducer, while
any other dependence or characteristic parameter of the transducer cannot be
determined, which limits the using of this method regarding the new threedimensional model presented in the last chapter.
In analysis of coupled oscillations of piezoelectric ultrasonic sandwich
transducers are also used different numerical methods for determination of their
frequency characteristics. Numerical models of power ultrasonic systems are
usually based on finite element method - FEM, or boundary element method BEM). Numerical linearized models, based on theory of elasticity, constitutive
piezoelectric equations, and theory of linear acoustics, are the most often used in
analysis of power ultrasonic devices. These models are valid at low levels of
excitation, that is, then they may model with acceptable accuracy characteristics
like: electric impedance, resonant frequencies, coupling coefficients and
characteristics of emitting directionality. Numerical modeling at power excitations
includes diverse nonlinear phenomena in the transfer and oscillation mechanisms in
different mediums, which the analysis of the transducer makes substantially more
complex. These design methods are complicated and demand a lot of computing
time. In the later report, in Chapter 3.2.3, are analyzed axisymmetric oscillations of
metal cylinders using numerical BEM method, and as the most important, metal
cylinders with central opening. Resonant frequencies of the consisting parts of an
ultrasonic sandwich transducer in function of their length are for the first time also
considered in function of relation of their outer and inner diameter, in contrast to
the existing modeling methods relating solid cylinders and disks. The same
procedure is also applied for metal endings and for excitation piezoceramic rings,
whereat for piezoelectric rings were obtained good results with neglecting of
piezoelectric effect and without consideration of ceramic anisotropy. This
procedure is used in order to prove the assumption that mutual coupling of
oscillations in radial and thickness direction of oscillation generates change of
ultrasonic wave propagation velocity in the consisting parts of the transducer.
Since the one-dimensional theory is used for description of oscillations of onedimensional cylindrical structure, it is ordinarily to expect that ultrasonic wave
propagation velocities be functions of frequency. Necessity of correction of
velocities in one-dimensional approach is explained based on the mentioned
analysis of axisymmetric oscillations of finite, homogenous, isotropic, elastic
cylinders with opening (rings). Thereby is also justified application of the method
of seeming modulus of elasticity for modifying of the one-dimensional theory, and
made the first step towards three-dimensional modeling and design of transducers
with short metal endings.
In the next (second) chapter, dedicated to modeling of piezoelectric ceramics,
besides the review of the most important existing models of piezoceramic rings, a
12
Introduction
new approximate three-dimensional matrix model of piezoceramic rings is
presented. Using this model, piezoceramic ring is presented as a 5-access network
with one electric and four mechanical accesses, which correspond to its boundary
surfaces. The proposed model, which describes both thickness and radial modes of
oscillation of piezoceramic ring, as well as their mutual coupling, also enables
analysis of undesired lateral vibrations of the ring, observation of surrounding
medium influence at the boundary surfaces, as well as determination of all transfer
functions. Also is presented the comparison of calculated and experimental results
for piezoceramic rings and disks of various dimensions. Analyses of resonant
frequency spectrums, electromechanical coupling factors and mechanical
displacements depending on various relations of ring dimensions, as well as input
electric impedances of the piezoceramic rings in broad frequency range were done.
Consequently, the proposed model is universal, because for null value of the inner
diameter one also gets the model of a piezoceramic disk.
Similar matrix model is used for modeling of passive metal endings, as 4access networks, whereat four mechanical accesses correspond to their boundary
surfaces, which is presented in the third chapter dedicated to different means of
modeling metal rings and disks (cylinders), in chapter 3.2.4. The model is obtained
by neglecting piezoelectric constants in the proposed model of piezoceramic rings,
as well as taking into account the fact that metal endings are of isotropic material.
Obtained dependences of the resonant frequency from ending dimensions are
compared with analogous results obtained by method of seeming moduli of
elasticity, which up to now represented the best modification of the onedimensional theory, as well as with experimental results obtained using vibrational
platform. Proposed BEM numerical method for analysis of oscillation of
homogenous, isotropic, elastic, axisymmetric metal rings is also discussed in detail
in this chapter, in section 3.2.3, and calculated resonant frequencies are compared
with experimental results. Also are presented comparisons of the proposed threedimensional matrix model with applied BEM numerical approach in modeling of
short metal rings, and advantages and disadvantages of both ways of modeling are
analyzed.
By serial-parallel connecting of the mechanical and electric accesses of even
number of piezoceramic rings, metal endings, and bolt, using their previously
realized models, in the last (fourth) chapter, dedicated to modeling of complete
sandwich transducers, besides different existing models, a global model of an
ultrasonic sandwich transducer is realized. Using this model one may determine
any transfer function, whereat the influence of the surrounding medium is taken
into account, as well as the influence of thickness and radial modes of every
consisting element of the transducer. By dimension varying of some consisting
parts of the transducer, as well as the load value on the mechanical accesses of the
model, one may determine which consisting parts affect mostly the specific
resonant modes of the ultrasonic transducer. The proposed model is suitable for
analysis of transducer resonant modes in broad frequency range. Verification of the
proposed model is performed by comparison of modeled resonant frequency
spectrums, as well as input electric impedances in function of frequency and
specific transducer parameters, with adequate characteristics at concretely realized
Introduction 13
symmetric and unsymmetric ultrasonic sandwich transducers, which are designed
based on the proposed model. Part of the obtained results is published in scientific
journals and reported at symposiums [28], [29], [30], [44], [45], [56], [79], [83],
[94], [97], [98], [99], [100], [102], [103], [104], [107], [111], [125], [128], [131],
[135], [152], while certain part represents results of work in this field unpublished
yet.
In the previous part of the report is remarked that there is number of attempts
to realize a good model of an ultrasonic sandwich transducer. Further will be
treated different approaches in modeling and design of power ultrasonic
transducers, in order to perform a systematization of existing methods, and enable,
in later report, comparison of results obtained by existing methods, and by new,
here proposed methods. However, for better understanding of quoted methods, first
will be presented ways of modeling and determination of resonant frequencies of
specific consisting parts of the sandwich transducers, that is piezoceramic disks and
rings, as well as of solid metal cylinders and metal rings. Due to the assumption
that for solid metal cylinders and piezoceramic disks may be used adequate ring
models, but with null value of inner diameter, most analyses will be done for cited
“perforated” consisting parts. Knowing of resonant frequencies of loaded and
unloaded transducer parts is an initial prerequisite for modeling of a complete
ultrasonic sandwich transducer. In the following sections will be described
methods mentioned in the introduction in more detail.
Realized investigations enabled design and optimization of ultrasonic
sandwich transducers of circular cross-section of any dimensions, using models
that are considerably simpler comparing with numerical methods, whereat the
design is significantly faster. Comparing with traditional one-dimensional models
and methods, results obtained by proposed models have better agreement with
experimental results. Owing to all previously exposed, these investigations enable a
new way of analysis of ultrasonic sandwich transducers from aspect of their
electric and mechanical characteristics, where by alterations and improvements in
modeling of ultrasonic transducers is enabled development of new ultrasonic
systems.
2. MODELING OF PIEZOELECTRIC
CERAMIC RINGS AND DISKS
Modeling of piezoceramic materials represents a complicated problem.
Though the subject of interest here are only piezoceramc rings, their modeling
cannot be completely isolated from the context of modeling of piezoelectric
ceramics with other geometric shapes. Basic approach in modeling and analysis of
piezoelectric ceramics is using of equivalent electromechanical circuits.
Application of these equivalent circuits in the theory of piezoelectric ceramics is
based on the idea that the wave propagation velocity is equivalent to current, and
mechanical force is equivalent to electric voltage. Equivalent circuits are very
powerful tool for analysis and design from several reasons. One of the reasons is
that physical interactions in the material are very accurately copied in the
equivalent circuit, and in a way that include the geometry and dimensions of the
piezoceramic specimen too. This enables easier visualization of the wave
propagation in the structure by simple analysis of the equivalent circuit.
Principally, using of circuits doesn’t provide more information than application of
fundamental piezoelectric equations, but enables a concise and easily readable
form. Equivalent electric circuits may be analyzed using powerful and highly
developed software for network analysis, in order to obtain requested measured
performances.
2.1. ONE-DIMENSIONAL MODELS OF PIEZOCERAMICS
2.1.1. BVD Model of Piezoelectric Ceramics
The oldest model for analysis of oscillation of such electromechanical systems,
which is valid only in the vicinity of isolated resonant mode, represents the BVD
(Butterworth Van-Dyke) model [54]. By this one-access network, input electric
impedance of the piezoceramic element (not only the piezoeramic ring), as a most
often analyzed and used characteristic of the piezoceramic, may be determined in
the vicinity of the isolated resonance using circuit with concentrated parameters,
which represents a parallel connection of serial RLC resonant circuit and static
capacitance of the piezoelectric crystal C0 (Figure 2.1).
16
Modeling of piezoelectric ceramic rings and disks
C1
C0
L1
1
Figure 2.1. Butterworth Van-Dyke equivalent model of piezoceramic
element in the vicinity of its isolated resonant mode
Modeling of piezoelectric ceramics using this circuit is useful only when the
circuit parameters are constant and independent on frequency. In general case
parameters are approximately independent on frequency only in narrow frequency
range around resonant frequency, and only if the observed mode is isolated enough
from other resonant modes. Elements of the equivalent circuit may be calculated
based on known elastic, piezoelectric and dielectric constants, which means that for
this procedure one also must know the type of vibrational modes, which isn’t
possible in most cases occurring in practice. A very descriptive analysis of the
BVD models, as well as of most further quoted traditional one-dimensional models,
altogether with their advantages and disadvantages, was given by Ballato [55].
In piezoceramic disks and rings that are low mechanically loaded, simple ideal
modes (radial or thickness) does not exist, and such modes occur at piezoceramics
that are considerably mechanically loaded in radial or thickness direction. The
situation worsens when the piezoceramic diameter is close to its length. In this case
different modes are not isolated, but coupled.
It was already remarked that in case of isolated modes elements of equivalent
circuit may be calculated based on known elastic, piezoelectric and dielectric
constants, but at coupled modes accuracy of such determined equivalent circuit is
insufficient. Then is better to determine elements of equivalent circuit
experimentally, since primary information about types of vibrational modes is not
needed. In continuation of the exposure is presented original experimental method
for determination of equivalent electric circuit parameters of piezoceramic rings,
and this is the region in which, in current terms of using fast computers, there is
still space for application of this simple model [56].
Piezoceramic ring with coupled resonant modes, which is low mechanically
loaded, may be represented by equivalent electric circuit presented on Figure 2.2.
Parallel combination of serial resonant circuits Rn, Ln, Cn corresponds to different
vibrational modes of piezoceramic. Characterizing of piezoceramic is consisting of
determination of parallel connection elements Rn, Ln, Cn of the serial resonant
circuit and capacitance C0. Further is presented simple way of determination of
these parameters based on the measured curve of input impedance of the
piezoceramic ring in function of frequency. An optimizaton technique that contents
a multidimensional simplex identification algorithm was used.
Modeling of piezoelectric ceramic rings and disks 17
...
C0
...
C1
C2
Cp
Cm
L1
L2
Lp
Lm
1
2
p
m
...
...
Figure 2.2. Equivalent circuit of the piezoceramic ring
with coupled resonant modes
Using common notation, the input electric impedance of the equivalent circuit
from Figure 2.2 amounts:
Z ul =
1
,
(2.1)
m
⎛
Rn
Xn ⎞
⎟
⎜
j
C
ω
+
−
∑
∑
0
2
2
2
2 ⎟
⎜
R
X
+
n =1 Rn + X n
n
=
1
n
n ⎠
⎝
where m is number of resonant circuits and Xn=ωLn-1/(ωCn). If Zul has form of
m
Zul=Rul+jXul, then stands that:
Rul =
X ul = −
m
∑R
n =1
2
2
n
Rn
+ X n2
m
⎛ m
⎞ ⎛
⎞
R
X
⎜⎜ ∑ 2 n 2 ⎟⎟ + ⎜⎜ ω C0 − ∑ 2 n 2 ⎟⎟
n =1 Rn + X n ⎠
⎝ n=1 Rn + X n ⎠ ⎝
m
X
ω C0 − ∑ 2 n 2
n =1 Rn + X n
2
m
⎛ m
⎞ ⎛
⎞
R
X
⎜⎜ ∑ 2 n 2 ⎟⎟ + ⎜⎜ ω C0 − ∑ 2 n 2 ⎟⎟
n =1 Rn + X n ⎠
⎝ n=1 Rn + X n ⎠ ⎝
2
2
,
(2.2)
.
(2.3)
Resonant and antiresonant frequencies of the p-th resonant mode are found from
the condition that at these frequencies Xul=0, that is, this condition may be presented
as:
ω C0 −
where:
Xp
R + X p2
2
p
+ ω ∆C = 0 ,
(2.4)
p −1
m
Xn
X
ω ∆C = −∑ 2
− ∑ 2 n 2.
2
n =1 Rn + X n
n = p +1 Rn + X n
(2.5)
Equation (2.5) represents reactive admittance That originates from all vibrational
modes except the p-th, whereat in the vicinity of the p-th mode the first member has
inductive character, and the second capacitive character. Equivalent capacitance ∆C,
that is influence of other circuit parameters on resonant and antiresonant frequencies of
the p-th mode, will depend on that which member in the expression (2.5) dominates.
By substitution of Xp and solving the equation (2.4) one gets complex expressions
(p)
(p)
for resonant and antiresonant circular frequency of the p-th mode ωr and ωa ,
which depend on C0, Rp, Lp, Cp, but also of ∆C. Besides that, at resonant frequencies is
18
Zul=Rul≠Rn, because Rul depends on all other modes too, that is of Ln, Cn, Rn (n=1÷m).
Namely, existing of other parallel circuits in the equivalent circuit decreases value of
resistance Rul, if one observes the case of one isolated mode Ln, Cn, Rn.
In existing methods for determination of equivalent circuit parameters based
on appropriate measurements, approximate expressions are used for reactive
elements of the equivalent circuit C0, Ln, Cn, which are obtained by essential
(p)
(p)
simplification of previous expressions for ωr and ωa , by neglecting resistance
Rn and treating of resonant modes as isolated (∆C=0) [57]. Besides that, it is taken
that Zul=Rul=Rn at resonant frequencies, which enabled determination of resistance
Rn by direct measuring.
It means that the first assumption is that for the procedure of synthesis of an
equivalent circuit one may use measuring of the spectrum of resonant and
antiresonant frequencies of the real piezoelement, and it is considered that these
frequencies are equal to the serial and parallel frequencies of the isolated p-th
mode:
ω r( p) =
1
L pC p
,
1
ω a( p) =
Lp
C0 C p
.
(2.6)
C0 + C p
The second assumption is that input resistance of the transducer at some
resonant frequency is equal to the correspondent resistance Rn in equivalent circuit.
Capacitance C0 may be found from the condition that sum of all capacitances in the
equivalent circuit must be equal to the capacitance of the transducer at very low
frequencies.
Numerical methods enable obtaining of the mathematical model of impedance
based on the solution of linear equation set, whereat the model contains unknown
elements of the circuit as parameters. Starting from the given model, it is possible,
defining the error function, to determine circuit elements by minimizing the difference
of experimental results and model values with correspondent initial values. The error
function is often defined by sum of squares of mentioned differences. Problems that
exist at such approach (method of the least squares is used), relate to the complexity of
the expression at great number of resonant modes (complex expressions for the first
and second partial derivative), as well as to the ill conditioning of the system due to the
great difference of the coefficients in the system matrix.
Further will be presented one method of fitting of impedance nonlinear
function for defined set of experimental data, which is based on implementation of
the simplex algorithm for minimizing of nonlinear multivariate function, that is, it
will be presented the method used for identification of parameters of the circuit
from Figure 2.2. Fist is treated the mentioned procedure of multidimensional
minimizing of Nelder and Mead [58]. Simplex method is of geometric nature and
it is characteristic by demanding calculation of only the function, but not its
derivatives. Therefore it is not much efficient in sense of number of functional
calculations demanded, however, at such problems it proved to be as a very
efficient method in parameter determination of an equivalent circuit of concrete
piezoceramic ring.
Simplex is a geometric figure that consists, in N dimensions, of N+1 points,
that is, all their mutual links and polygonal surfaces. For two dimensions simplex is
a triangle, and for three dimensions a tetrahedron, not necessarily a regular one.
Generally, only simplexes that are not degenerated are of interest, i.e., that close a
finite internal N-dimensional volume. If any point of a non-degenerated simplex is
taken as an origin, then N remaining points define vector directions that encompass
N- dimensional vector space.
First is necessary to define N+1 point of the originating simplex. The simplex method
implies a series of steps. Most steps start from the simplex point where the function value is
the greatest (“the highest point”), and passes through the opposite side of the simplex to the
“lower point”. These steps are called reflections. Algorithm of Nelder and Mead uses three
possible cases for steps in the simplex algorithm. Beside the mentioned reflection, as a base
step, possible are expansions and contractions, and all these steps are used for calculation of
simplex vertices in the iterative process (Figure 2.3).
xexp
xref
xm
xcont
xG
x'cont
xM
xh
Figure 2.3. Simplex method of Nelder and Mead
Let the vertices of the simplex be denoted as xi, and correspondent values of
function in them as fi (i=0,1,...,N). Vertex xM corresponds to the maximum value of
fM and it is reflected through xG, center of gravity of N remained vertices without
xM, according to expression:
(2.7)
x ref = (1 + α ' )x G − α ' x M ,
+
where α’∈R coefficient of reflection. xref is located on the line xMxG, so that
α’=⎢⎢xrefxG⎢⎢/ ⎢⎢xG xM⎢⎢. Several cases are possible:
a) If fm<fref<fk, where fm is the smallest value of the function related to xm, fref
value for xref and fk the biggest value of the function for all remained simplex
vertices except xM, then xref replaces xM and process is repeating.
b) If reflection causes a new minimum (fref<fm), a new point xexp is calculated
in direction xGxref according to expression:
x exp = γ ' x ref + (1 − γ ' )x G ,
(2.8)
where γ’ is coefficient of expansion (γ ’=⎢⎢xexp xG⎢⎢/ ⎢⎢xref xG⎢⎢ and γ’>1). If
fexp<fm, then xM is replaced by xexp, and xref is kept too.
c) If fref>fk, then a contraction of the simplex is performed. If fref<fM, xref
replaces xM, and on the line xMxG is determined xcont in the following way:
xcont = β ' x M + (1 − β ' )x G ,
(2.9)

where β ’ is coefficient of contraction (β ’=⎢⎢xcont xG⎢⎢/ ⎢⎢xM xG⎢⎢, 0<β ’<1).

20
d) If fcont<fM, xcont replaces xM and the basic procedure is used again (a). If the
contraction is not useful, a new simplex is set, in away that distances xm from all
other vertices decrease by half. Then the basic procedure is used.
Nelder and Mead proposed values α’=1, β ’=1/2, and γ ’=2, which gave in
practice satisfying results. The method converges till the fulfilling of the condition
that in adjacent iterations maximal difference of two values in vector xi is smaller
than fixed value given in advance. Besides that, certain tolerance is also required
for the difference of values fi in adjacent iterations. It is also possible to define the
maximal number of steps.
Based on the previous algorithm is minimized the difference of the frequency
dependence on input impedance zul of the circuit from Figure 2.2 and
correspondent measured impedance function, for piezoceramic ring of PZT8
ceramic with outer diameter 2a=38mm, inner diameter 2b=15mm, and thickness
2h=5mm, which is very often used in power ultrasound technique. Due to the
nature of the impedance function (great range of its changes), recording of the
decreasing characteristic adB=Zul in function of frequency was performed on
automatic network analyzer (HP 3042A Network Impedance Analyzer) for
sinusoidal excitation voltage:
⎛z
⎞
Z ul = 20 log⎜ ul + 1⎟ .
⎝ 50
⎠
(2.10)
First was observed the broad-band characteristic of impedance dependence
on frequency, whereat the analysis encompassed all resonant modes, radial and
thickness. Using this method was obtained that maximal errors in determination of
elements of equivalent connection and impedance are smaller than correspondent
errors obtained by method [57]. Measured and modeled characteristic of impedance
for a concrete unloaded piezoceramic ring are presented simultaneously on Figure
2.4. Characteristic of simulation is obtained by modeling of impedance of the
piezoceramic ring using software Matlab1.
Obtained values of equivalent circuit elements are presented in Table 2.1.
Table 2.1. Values of equivalent circuit elements of piezoceramic ring
n
1
2
3
4
5
6
7
8
1
Rn (Ω)
19,24
1,51
52,99
1,66
7,71
0,79
1,90
11,02
Ln (mH)
29,28
2,03
56,24
6,16
2,56
4,30
0,43
3,34
Cn (pF)
452,98
496,77
6,48
48,39
86,89
40,17
362,05
34,09
Matlab, The MathWorks, Inc. (http://www.mathworks.com)
C0 (nF)
2,05
Zul[dB] 70
60
experiment
model
50
40
30
20
10
0
1
2
3
4
0
1
5 x
5
f [Hz]
Figure 2.4. Experimental and simulated broad-band characteristic of impedance
modulus dependence on frequency for PZT8 piezoceramic ring
Piezoceramic rings are most often used in composite Langevin’s sandwich
transducers for application in powerful systems for ultrasonic cleaning and
welding. Regarding the frequency range of these applications, an identification of
the parameters from Figure 2.2 for frequency range that is of interest (Figure 2.5),
whereat the obtained circuit element values are presented on the very Figure 2.5.
Zul[dB] 70
C0=2,52 nF
C1=424,76 pF
L1=31,20 mH
R1=23,20 W
C2=452,33 pF
L2=2,24 mH
R2=0,53 W
60
50
40
30
20
experiment
model
10
0
0,5
1
1,5
2 x 10
5
f [Hz]
Figure 2.5. Experimental and simulated narrow-band characteristic of impedance
modulus dependence on frequency for PZT8 piezoceramic ring
22
Based on Figures 2.4 and 2.5, justification of the applied procedure is obvious,
as for isolated, so for coupled resonant modes of oscillation of piezoceramic ring.
2.1.2. Mason’s Model of Piezoelectric Ceramics
Use of electric analogies of the one-dimensional acoustic phenomena began
with growing use of piezoceramics, and the most original approach in modeling of
piezoelectric ceramics, and by that itself of the piezoceramic rings and disks, first
was given by Mason [59]. His equivalent circuits still represent base for many
current models. It must be remarked that Mason’s model of piezoelectric ceramics,
which themselves are often called transducers, is the most considered model in
literature. Mason succeeded to show that for one-dimensional analysis even the
greatest problems in deriving of analytical solution of the wave equation in
piezoelectric material may be overcome using the theory of electric networks.
Using the “black box” analogy the piezoelectric disk or ring may be treated as
a device with three accesses, one electric and two mechanical. The first access
corresponds to electric connectors, and the remained two to metalized, circularringed surfaces of the platelet. This case is presented on Figure 2.6. Force direction
and velocity, denoted on Figure 2.6, in case of piezoceramic ring coincides with the
direction of its polarization.
l
F1, v1
v1
F 2 , v2
F1
v2
3-access
network
F2
P
I
V
I
V
(b)
(a)
Figure 2.6. (a) Piezoceramic ring with adopted conventions for senses of forces,
velocities, voltages, and currents; (b) Equivalent electric circuit with three accesses
The following values may be defined: vi is individual velocity of the i-th
surface of piezoceramic (i=1, 2), Fi is force applied or created on the i-th surface, V
is voltage on electrodes of the piezoceramic ring, and I is current that runs through
electrodes. Directions are defined for positive force, velocity, voltage, and current.
One connector line of every pair of the mechanical and electric access is treated
regarding the common zero reference point (mass).
Equations that describe presented network with three accesses are derived
from constitutive equations for piezoelectric ceramic that connect tensors of
relative mechanical strains (dilatations) S, mechanical stresses T, dielectric
displacement D, and electric field E inside the material, with adequate boundary
conditions. These equations follow from fundamental derivations of Mason’s and
Berlincourt’s equations, so that one gets:
Z c ⎡ v1
v 2 ⎤ h33 I
+
,
⎢
⎥+
j ⎣ tg (k l ) sin (k l ) ⎦
jω
Z ⎡ v1
v 2 ⎤ h33 I
+
,
F2 = c ⎢
⎥+
j ⎣ sin (k l ) tg (k l ) ⎦
jω
F1 =
V =
(2.11)
h33
(v1 + v 2 ) + l IS ,
jω
jω ε 33 P
D
/ ρ . Zc is characteristic impedance, k is
whereat is Zc=ρvzP, k=ω/vz, vz = c33
wave number, and vz is velocity of ultrasonic waves in thickness direction. ω=2πf
is circular frequency, ρ is density of piezoceramic, l and P are length and area of
S
the element. h33 is piezoelectric constant, ε33 relative dielectric constant of
D
compressed ceramic, and c33 coefficient of elasticity constants tensor. Thickness
of the piezoelement is considered negligible regarding its lateral dimensions. For
practical use of piezoceramics, determination of input electric impedance zul=V/I is
of greatest interest. If Z1 and Z2 represent mechanical impedances on both
boundary surfaces of the crystal (emitting resistance), whereat stands that F1=-Z1v1
and F2=-Z2v2, input electric impedance in that case will be:
[h / (kv z Z c )] Z c { [(Z1 + Z 2 ) / Z c ] sin(k l ) + 2 j (1 − cos(k l ))}
j
+ 33
. (2.12)
ωC 0
1 + Z 1 Z 2 / Z c2 sin (k l ) − j[(Z 1 + Z 2 ) / Z c ] cos(k l )
2
z ul = −
(
)
Based on equations (2.11), Mason derived equivalent T-networks with
distributed parameters for piezoelectric ceramic, whereat the dimensions of the
piezoceramic are such that it is possible to apply one-dimensional analysis. These
equivalent circuits may be applied for three types of oscillation of piezoelectric
ceramics: (a) platelet with oscillation across thickness; (b) bar with electrodes on
its ends, at which oscillation is performed across length; (c) bar at which oscillation
is performed across length, and electrodes are set laterally. Mason’s equivalent
circuit for the most often occurring case in practice, and which is here of interest in
later considerations, i.e., for the platelet that oscillates across thickness (in
thickness mode, Figure 2.6(a)), is presented in Figure 2.7.
Values of the elements in the circuit from the Figure 2.7 are:
S
P
ε 33
kl
−Zc
, Xb =
.
(2.13)
l
2
sin(k l)
This type of model uses analogy of the transfer line with propagation of the
acoustic wave and it is not limited by condition of resonance vicinity. In order to
realize equivalent circuit of the piezoelectric ceramic, it is necessary to use all
equations (2.11). These equations are in function of three independent (current and
velocity on main surfaces) and three dependent variables (voltage and forces on
main surfaces) and, as it is already mentioned, they define a network with one
C0 =
,
Xa = Zc tg
24
electric and two mechanical accesses. Accordingly, Mason proposed an exact
equivalent circuit that divides piezoceramic material on electric and mechanical
part using ideal electromechanical transformer, capacitance, negative capacitance
and T-network. By defining the transfer ratio of the ideal electromechanical
transformer, one gets equivalent circuit in which electric parameters of the circuit
are determined by values of the coefficients in equations. Model is most often used
for unloaded ceramic or ceramic loaded by one ending mass (impedance), for
analysis of the transient response, determination of the material constants, but for
most of rest applications, too.
v2
v1
jXa
jXa
jXb
-C0
I
F1
F2
C0
V
h33C0:1
Figure 2.7. Mason’s equivalent circuit for piezoceramic in thickness mode of
oscillation
Network with three accesses may be reduced to a network with two accesses if
one considers that a constant acoustic appliance Z2 exists on surface 2, and using
relation F2=-Z2 v2 in the system equations, i.e., if one assumes that the second access is
closed by constant impedance, which is “leaned” to the piezoceramic. Therefore
piezoelement may be treated as a network with two accesses, one electric, and the
other mechanical (surface 1). Further, assuming that the two-access system is supplied
by source at its electric access, or at its remained mechanical access, the network may
be reduced to a Tevenen’s equivalent generator and impedance, that is, to one-access
network. Piezoelement may be conventionally analyzed in two cases, that is, by two
Tevenen’s equivalent circuits. If the generator is connected to an electric port, thereby
is represented the emitting mode of the piezoceramic, i.e., connection of the generator
with mechanical access represents the receiving mode of the piezoceramic.
2.1.3. Redwood’s Version of the Mason’s Model of Piezoelectric Ceramics
Redwood corrected equivalent circuit of Mason and proposed the form in
which elements of the T-network are replaced by transfer line of the characteristic
impedance Zc [60], which, regarding the Mason’s model, did not appreciably
simplified the circuit. However, equivalent circuits like Mason’s original models,
cannot be applied in software for simulation of electric circuits, due to the
frequency dependent elements. Also, artificial line circuits are too complicated and
incorrect at high frequencies. Redwood’s version of the Mason’s model may be
easily modified in order to obtain topology suitable for modeling using computer.
Like the Mason’s, this model is also consisted of capacitance, negative capacitance,
ideal transformer, as well as of transfer line instead of T-network (Figure 2.8).
v2
v1
Zc
-C0
F1
C0
I
V
F2
h33C0:1
Figure 2.8. Redwood’s version of the Mason’s equivalent circuit
for piezoceramic in thickness mode of oscillation
Transfer line in that new circuit is treated as a coaxial line whose sheath is
connected with the secondary of the transformer. Redwood’s equivalent circuit is
particularly useful in determination of time response at excitation of the piezoceramic
by short impulse, especially when the impulse width is smaller than the delay time of
the acoustic wave that propagates through the piezoceramic. Model proposed by
Redwood may be applied in any software for analysis of electric circuits that contains
model of transfer line and correspondent suitable linearly dependent sources.
2.1.4. KLM Model of Piezoelectric Ceramics
One of disadvantages that may be noticed in the previous models is that it is
required a negative capacitance on the electric access. Although Redwood showed that
this capacitance may be transformed on the acoustic side of the transformer and treated
as part of the acoustic section, that approach was also considered illogical
(unphysical). As an attempt to cut out the circuit elements between the transformer top
and the node of the acoustic transfer line, Krimholtz, Leedom, and Matthae realized
one alternative equivalent circuit, also without losses [61]. Model is most often called
KLM model and offers some advantages in applications in which several
piezoceramics are mechanically connected in a cascade way (multilayer or segment
transducers), and it is used for design of high-frequency transducers for application in
medicine. KLM model for piezoceramic platelet that oscillates across thickness (and
also for a bar that oscillates across length) is presented on Figure 2.9, whereat in case
of piezoceramic platelet, similarly as at Mason’s model, stands that Zc=ρvzP,
S
D
vz = c33
/ ρ , C0=ε33 P/l, and:
26
φ=
v1
F1
ωZc
1
2h33
⎛ lω ⎞
⎟⎟
sin⎜⎜
⎝ 2v z ⎠
,
⎛h
X1 = Z c ⎜⎜ 33
⎝ ωZ c
l/2
l/2
Zc, vz
Zc, vz
2
⎞
⎛ lω ⎞
⎟⎟ sin ⎜⎜ ⎟⎟ ..
⎠
⎝ vz ⎠
(2.14)
v2
F2
I
jX1
C0
V
φ :1
Figure 2.9. KLM equivalent circuit for piezoceramic
in thickness mode of oscillation
This equivalent circuit is also consisted of source and electric network with
frequency dependent components, which is now connected in the middle of
acoustic transfer line from Mason’s model, that is, the line of characteristic
impedance Zc, with velocity vz and length l. Line length is equal to the dimension
of piezoceramic in the direction of ultrasonic wave propagation. In this circuit the
transfer ratio of the transformer became function of frequency, but in that way roles
of electric and mechanical (acoustic) parts are clear separated, so that this circuit is
physically more understandable and enables underlining of clear difference
between electric behavior of elements and wave acoustic behavior of piezoceramic.
Beside the quoted, this circuit eases calculation of electric input impedance for an
arbitrary acoustic appliance, and contrary, when the mechanical impedance is
searched for, in the KLM circuit is easy to see the effect of arbitrary impedance
connected to the electric access onto acoustic transfer line, which is not the case at
correspondent Mason’s circuit.
Similar equivalent circuits may be derived for piezoceramics excited by fields
with nonuniform distributions or for ceramics with nonuniform piezoelectric
properties. Besides that, with certain modifications, one may realize a simple
circuit similar to the circuit from Figure 2.9, which represents a KLM model of a
transducer consisted of several identical mutually connected piezoplatelets. Later
improvements of these models implied use of equivalent circuits with losses in
material, whereat piezoelectric, dielectric, and elastic constants circuit treated as
complex values. In literature about this field there is a more extensive comparison
of the KLM and Mason’s model with losses [62].
2.1.5. PSpice Models of Piezoelectric Ceramics
In this field of application of software for analysis of electric circuits first was
realized Redwood’s version of Mason’s equivalent circuit in the SPICE software, and
the way how it was done is presented in literature [63]. Based on the Redwood’s
model of piezoceramic in thickness mode of oscillation from Figure 2.8, it was
realized a SPICE model of piezoceramic ring of PZT8 ceramic with outer diameter of
2a=38mm, inner diameter 2b=15mm, and thickness 2h=5mm. This SPICE model is on
Figure 2.10 presented in a more actual schematic program editor PSpice2. Thereby is
realized PSpice model of piezoceramic ring with three accesses and without losses,
which is applicable in software packages for simulation of electric circuits.
line
-C0
transformer
Figure 2.10. SPICE (PSpice) version of the Redwood’s model of piezoceramic
Modified circuit uses hybrid representation of the electromechanical
transformer, approximation of negative capacitance, and modified coupling from
the transformer to the acoustic transfer line. Since the SPICE model of the transfer
line did not contain longitudinal inductance in both leads, it could not be used
directly in simulation. Due to the need for an enclosure without inductance,
Redwood’s model is redefined, in a way presented in Figure 2.10. From the Figure
2.10 is obvious in what way are modeled elements of the equivalent circuit. Using
dependent sources was realized model of an ideal transformer, and it was also
approximated negative capacitance -C0 by parallel connection of current source (C0
ICS)/CS and capacitor CS, in the following way:
2
PSpice, OrCad, Inc., Beaverton, OR, USA (http://www.orcad.com)
28
I C = −C 0
dVC
dt
⎛ CS
⎜⎜1 −
⎝ C0
⎞
dV
⎟⎟ ≈ −C 0 C ,
dt
⎠
(2.15)
because C0>>CS. Inserted resistors R1, R2, and Rx, should be such to show
negligible effect in the circuit, and they are used to fulfill demand of PSpice (and
SPICE) that every node has unilateral access to the mass. R1 may be selected in
such way to reflect real dielectric losses in the circuit, if necessary. This model,
when applied in Pspice software, proves as useful supplement during design,
especially for determination of performances of piezoelectric ceramic in function
of emitter or receiver (sensor). Besides that, when the whole sandwich transducer is
being designed, which will be more talked about later, very important is the
simulation of possible configurations of the transducer before the construction
itself. Model enables true simulation of possible material effects of λ/4 adjusting
layers, that is, acoustic adjustment, electric circuits for adjustment, associated
receiving-emitting electronics, and other variables in design, depending on
demanded characteristics of the transducers. Schemes for adjustment of
piezoceramic and the whole transducer are also easily modeled, whereat electric
adjustment is performed by simple addition of adjusting network into the circuit
scheme. Acoustic adjustment between the emitter surface and the appliance is
modeled by addition of transfer line of appropriate characteristic impedance and
delay time. This PSpice model is suitable at impulse-echo simulation, which is
performed by cascade connecting of two circuits of piezotransducers using
depending voltage sources. Also, it is eased determination of transfer
characteristics between different connectors, band-pass width, diagram of electric
and mechanical impedance and transient responses. Here presented model may be
applied in any software for circuit analysis that contains transfer line model and
correspondent linearly dependent sources.
Technique of application of controlled sources instead of transformers,
presented by Leach [64], enabled a more elegant method for implementation of
previously mentioned one-dimensional models of elementary piezoelectric
ceramics in software packages for simulation of electric circuits. Advantage of
using controlled sources over transformer is avoiding of negative capacitance in the
Mason’s model and frequency depending transformer in the KLM model.
However, as it was mentioned, when mechanical and dielectric losses are
significant. e.g., at piezoceramics with low mechanical Q factor, previous models
are incorrect, because they don’t take into account losses in calculations.
Equivalent model for such piezoceramics with losses, which oscillate in thickness
resonant mode, as well as its application in the Pspice software, is presented in
literature [65]. Model is equivalent to the Redwood’s (and by that itself the
Mason’s) model from Figure 2.8, but it is presented in a little bit different form due
to the application in the Pspice software, as given in Figure 2.11.
Equivalent PSpice circuit of the model for piezoceramic with losses from
Figure 2.11 is presented in Figure 2.12(a). Also, in the same figure is presented the
content of the input listing for PSpice simulation of this model. PSpice internal
model of the line with losses is model with distributed parameters L’, R’, C’, G’,
and length l, whereat the characteristic impedance is Zc, function of propagation γp,
and phase velocity vz, respectively:
R '+ jωL'
,
G '+ jωC '
Zc =
(R'+ jωL')(G '+ jωC '),
γ p = α p + jβ p =
(2.16)
vz = ω / β p .
v1
v2
Zc
F1
v1- v2
h33 I
s
(lossy)
C0
+
I
_
+
_
h33 (v1-v2 ) V
s
F2
Figure 2.11. Equivalent circuit of piezoceramic with losses
in thickness mode of oscillation
.SUBCKT PZT E B F
T1 B 1 F 1 LEN=”l” R={“R*”*SQRT(-S*S)}
+ L=”L’” G=0 C=”C’”
V1 1 2
E1 2 0 LAPLACE {I(V2)}={“h33”/S}
V2 E 3
C0 3 0 “C0”
F1 0 3 V1 “h33*C0”
.ENDS
(a)
(b)
Figure 2.12. (a) PSpice equivalent model of piezoceramic with losses from Figure
2.11;
(b) Content of the input listing of the model
Assuming that G’=0 and R’<<ωL’ (at high frequencies), one gets:

L' = Z c / vz ,
C ' = 1 / (v z Z c ),
R ' = R *ω = (L ' / Q ) ω ,
(2.17)
where, like in the previous models, Zc=ρvzP characteristic impedance, ρ density of
piezoceramic, vz velocity of ultrasonic waves in piezoceramic material, and l is
thickness of the piezoceramic platelet. T1 is transfer line with losses, with two
mechanical and one electric access B, F, and E, respectively. C0 is static
30
capacitance and, as hitherto, is defined based on the following expression:
S
C0=ε33 P/l. Independent voltage sources V1 and V2 are of zero value, and they are
used in the circuit as ammeters. Currents of these sources I(V1) and I(V2) control
sources E1 and F1. Controlled voltage source E1 has value E1=h33I(V2)/s, where
s=jω Laplace’s operator. This source is in Pspice applied using Laplace’s function
LAPLACE. Controlled voltage source F1 has value F1=h33 C0 I(V1). Together with
capacitance C0, it replaces correspondent controlled voltage source from Figure
2.11, and realizes the member 1/s.
In contrast to the SPICE model of line with concentrated parameters, PSpice
model with distributed access has some advantages, which above all concern to the
higher accuracy and shorter time of calculation. Besides that, PSpice in contrast of
SPICE allows frequency dependent line parameters R’ and G’, and allows use of
LAPLACE function, as presented in the previously described model.
This model represents powerful tool for simulation of simple and multilayer
piezoceramic sensors and their receiving-emitting electronics, and all that in case
of presence of nonlinear elements and at significant losses. Application of PSpice
enables analysis of piezoceramic as in frequency, so in time domain.
2.1.6. Martin’s Model of Package Piezoceramic Transducers
All previously mentioned one-dimensional models relate to one piezoceramic
element. However, in many applications excitation piezoceramic elements are
composed as multiple identical segments, most often joined in a mechanical
sequence, with different serial-parallel combinations of electric accesses.
At ultrasonic sandwich transducers with only one pair of piezoceramic plates
effective coupling factor, and by that itself the electroacoustic efficiency
coefficient also becomes too low at operation under certain critical frequency. Low
resonant frequency demands great axial thickness of piezoceramic rings, whereby
is decreased capacitance of the transducer, that is, increased its electric impedance.
For excitation of such transducers are needed extremely high voltages, which
makes additional problems in practical realization.
Previous limitations are surpassed using package ultrasonic piezoceramic
excitations, which are consisted of several piezoceramic rings or disks of equal
dimensions and characteristics. Platelets are mechanically connected serially, and
through contact metal foils electrically are connected parallel, in order to achieve
higher ultrasonic power (Figure 2.13(a)).
Total electric capacitance of such piezotransducer is equal to the sum of
individual capacitances of the piezoelements. If such system is directly used as an
ultrasonic transducer, mechanical cascade of n piezoceramic rings would have
approximately n times lower thickness resonant frequency and approximately n
times greater value of effective coupling factor, regarding the correspondent
ultrasonic transducer with only one piezoceramic ring. It enables greater
electroacoustic efficiency in operating at low frequencies.
l
l0
v2
v1
F1 , v 1
F2, v2
jXa
jXa
jXb
P
-C'0
I
F1
F2
C'0
V
N':1
I
V
(b)
(a)
Figure 2.13. (a) Package piezoceramic ultrasonic transducer consisted of n=4
elements; (b) Equivalent circuit of such transducer
Equivalent circuit of segment piezoceramic excitation was realized by Martin
[66]. Determination of the equivalent circuit represents the same procedure used by
Mason, with some similar boundary conditions and assumptions. The circuit is
presented in Figure 2.13(b), whereat is:
C '0 =
S
n 2ε 33
P
− Zc
h C'
kl
, N ' = 33 0 , X a = Zc tg , Xb =
, l=n l0. (2.18)
l
n
2
sin(k l)
2.2. THREE-DIMENSIONAL MODELS OF PIEZOCERAMIC RINGS
One-dimensional models of piezoceramic rings and disks are inconvenient for
determination of the lowest resonant frequencies of radial oscillations, that is, they
are not convenient for modes of oscillation in frequency range interesting, e.g., for
application in ultrasonic cleaning, while they are convenient for determination of
resonant frequencies of thickness oscillations of rings and disks, which are quite
greater than resonant frequencies of the radial oscillations. Also, these models are
applicable for determination of resonant frequencies of very long solid or
punctured piezoceramic cylinders (l>>d).
With development of models of piezoelectric ceramics, occurred a need for
characterizing of piezoelectric materials, that is, for determination of their different
parameters and constants, which wasn’t a simple problem. First occurred studies for
determination of material parameters based on resonant methods. Generally, there
were used approaches in solving wave equations separately for every of the resonant
modes, and solutions were adapted to different geometric shapes of piezoceramic
specimens. This adapting procedure consisted in treating only one uncoupled, onedimensional oscillation of specimens. Results of these analyses were summed in
ANSI/IEEE standards [67]. Accordingly, piezoelectric standards describe analysis of
32
oscillations of piezoelectric materials that have simple geometric shapes. Results are
based on linear piezoelectricity and resonant modes are treated as isolated vibrational
modes. Basic difficulty in use of the previous approach for concrete piezoceramic
material is that, due to the piezoelectric effect and Poisson’s relations, one-dimensional
approach could not be further applied, because in reality in piezoceramic materials
exist coupled modes. For example, when the thickness mode is observed, piezoelectric
D
D
constant h31 and piezoelastic constants c12 and c13 are ignored, although their values
are not small and negligible. On the other hand, when radial modes are observed at
E
piezoceramic rings and disks, piezoelectric constant d33 and elastic constant s33 are
neglected. However, in this last case neglected constant d33 is greater than applied
constant d31. Next difficulty originates from using inconsistent boundary conditions,
that is, at specimen with boundary surfaces without mechanical stresses those stresses
are neglected, as on boundary surfaces, so in the interior of the material. However, this
assumption is wrong, because if components of the mechanical stress vanish on the
external surfaces, they don’t have to be negligible inside the specimen [68].
The next thing that should be mentioned is that, when geometric dimensions
are of same order, frequencies of the resonant modes of radial oscillations occur in
frequency domains close to the frequencies of the thickness mode of oscillation. In
every single case the specimen is excited to oscillation using alternating voltage on
main surfaces perpendicular to the specimen polarization axis z. According to
standard, thickness modes are treated as vibrational modes at constant dielectric
displacement D, and radial modes are implied as vibrational modes at constant
electric field E. However, at same supplying voltage and for a given frequency
range, at the oscillating specimen cannot simultaneously occur vibrational modes,
both at constant dielectric displacement D and constant field E. Using threedimensional approaches these difficulties are surpassed and one comes to coherent
(consistent) results.
In further exposure special attention will be addressed only to modeling of
piezoceramic rings, due to the need for comparison with three-dimensional model
that will here be proposed and analyzed in detail, as well as due to the impossibility
to encompass all published models of piezoceramic elements with rectangular,
cylindrical, or spherical shape by this short review. Piezoelectric ceramics with
such geometry are used in applications that surpass application fields of ultrasound
treated here.
As it was already mentioned, piezoceramic rings are most often used in
Langevin’s sandwich transducers, for application in power ultrasound technique.
At this type of transducer, piezoceramic rings have advantage in use over disks,
which is caused by need for prestressing of sandwich transducer by central bolt.
Knowing of resonant frequencies of piezoceramic rings is an initial condition for
design of different transducers, whose excitation part are these rings. For a precise
analysis of this type of transducer would be ideal a three-dimensional model of
piezoceramic ring, at which is taken into account coupling of radial oscillations
with thickness oscillations, whereat one may analyze the effect of mechanical load
on the ring in different directions. General constitutive piezoelectric equations are
too complicated, so that in general case complete three-dimensional analysis leads
to a very complex set of nonlinear equations, which is the reason why analytical
solution of these equations is not possible to find. However, three-dimensional
model of piezoceramic ring would be very convenient for exact design of
ultrasonic sandwich transducers, due to the strong interaction between the modes of
radial oscillations of the ring and the mode of thickness oscillation of the whole
transducer in real conditions. Besides that, radial vibrations of the ring are quite
different from radial vibrations of the disk with same diameter and thickness,
although disk models were more often used in this field till now, due to the absence
of adequate model of piezoceramic rings.
2.2.1. Berlincourt’s Model of Piezoceramic Rings
Berlincourt proposed a simple model of piezoceramic ring with concentrated
parameters, which is valid only when the inner diameter is close to the outer
diameter, whereat the coupling of radial and thickness modes neglected [69]. He
showed that frequencies of higher harmonics are much greater than frequencies of
the fundamental resonant mode of such ring with close inner and outer diameter,
which is polarized either radially, or by thickness. Because of that the fundamental
resonant mode is isolated and such ring may be easily modeled by simple circuit
with concentrated parameters, which is similar to the circuit presented in Figure
2.1. Such model is obsolete and it is rarely used, because it is impracticable for
analysis of piezoceramic rings in current conditions of their application.
2.2.2. FEM Models of Piezoceramic Rings
Some authors use finite element method (FEM) for analysis of piezoceramic rings
and disks [70], [71], [72]. Algorithms based on the finite element method are capable
to solve steady state problems and in general case they give more options during
analysis of piezoelectric ceramics and ultrasonic sandwich transducers than analytical
models in any segment of analysis. However, nevertheless analytical models are more
often used, because numerical approaches don’t give sufficient insight into the
physical parameters that should be kept under control during characterizing of
piezoceramic material and metal in design of ultrasonic sandwich transducers. Namely,
it is more difficult to notice the influences of the mentioned parameters, because every
new change demands repeated lengthy calculations. FEM methods are very sensitive
to changes of the individual parameters of the piezoceramic material, which demands
exact knowledge of the individual coefficients values. It represents a serious problem,
because small number of piezoceramic manufacturers publishes complete values of
parameters of their piezoceramics, due to their costly and complex measurements in
special conditions defined by ANSI/IEEE standards.
Yet, using of FEM methods for design of piezoelectric ultrasonic sandwich
transducers gives great opportunities for assessment of parameters and transducer
characteristics based on simulation results. The most often during modeling is
repeatedly performed redefining of parameters and optimization of characteristics
before the sole realization of the transducer, with lengthy repeated calculations.
34
However, even such analysis is a great skip in modeling of ultrasonic sandwich
transducers, because in the preceding years was used cut and true method, which
demanded a lot of time and costs, and withal was unreliable.
FEM method represents a powerful tool for modeling, analysis and solving of
complex problems. Concept of solving using this method is based on substituting of
very complicated problems by a simpler problem. Solution of that simpler problem
obviously does not represent exact solution of a more complex problem, but represents
its approximation. However, it is often a method for obtaining of only an approximate
solution, which may be improved after by mesh post processing and by iteration
number increase. One of the first and simplest examples of use of this method is
determination of the circle perimeter, thus that it is approximated by a polygon
perimeter. Using two polygons, one inscribed, and the other circumscribed around the
circle, one may determine the lower and upper limit of the circle perimeter (Figure
2.14(a)). Besides that, by increasing of number of polygon sides, approximate solution
becomes more accurate. This fundamental principle is common for many current
applications of FEM methods, and by that itself for application in the field of power
ultrasound.
Base of this method represents an element. Before performing of any analysis,
one must first form a mesh, which is consisted of elements, mutually connected in
joint points, called nodes or nodal points. Every element is defined by determined
number of nodes, depending on the type (shape) of the chosen element. In Figure
2.14(b) is presented an example of a mesh consisted of elements defined by 8 nodes.
(a)
(b)
Figure 2.14. (a) Application of FEM method for determination of circle perimeter;
(b) Example of mesh consisted of elements that contain 8 nodes
Element nodes are numbered in logical sequence, according to the global
coordinate system. Since changes of the unknown values (mechanical stress,
electric voltage, temperature, etc.) are not known, they are approximated using
interpolation or shape functions. Interpolation functions are defined by values of
the unknowns (variables) in nodes, and they are usually in form of a polynomial,
which must be of specific order, so to generate an approximation of satisfying
precision. When equations of state (constitutive equations) for the whole system
are once defined, new unknowns will be values of variables in nodes. Equations of
state, which are usually in matrix form, are then solved using adequate boundary
conditions, in order to obtain new values of variables in nodes. By using these new
values in nodes, interpolation functions then completely define the set of variables
in the system.
In order to further enable an insight in solving actual problems using FEM
software in the fields that are here of interest, one must first consider a simple
mechanical model, that is oscillatory system consisted of mass, spring, and shock
absorber (damper). This is also useful because the simplest one-dimensional
equivalent model of the whole ultrasonic sandwich transducer, which will be the
subject of analysis in later exposure, is consisted of same components.
Fundamental equation (equation of state) of such system is:
[mg ]
d 2 xr
+ [c]
dxr
+ k' [xr ] = F ,
(2.19)
dt
dt
where mg is mass, c damping, and k’ spring constant. In one-dimensional system all
these values are scalar. If one deals with a much complex system, then mg, c, and k’
become matrices. Interpolation function or shape function in a form of a particular
solution for the previous equation reads:
2
(2.20)
x r = X r e j ωt .
By substitution (2.20) into the fundamental equation (2.19), one gets:
⎛ mg ⎞
⎛ 1 ⎞
⎜
⎟
(2.21)
⎜ [k '] ⎟ Xr = Xr ⎜⎝ ω 2 ⎟⎠ .
⎝
⎠
Equation (2.21) may be expressed using eigenvalues and eigenvectors, as follows:
[ ]
A ν v = ν v λv .
3
4
(2.22)
5
FEM software (Ansys , Algor , Abaqus ) enable solving of this equation,
whereby one gets νv and λv, where νv is shape factor of resonant modes
displacement, and λv determines frequencies at which those modes occur
(since λv = 1 / ω 2 , and thereby f = 1 / λv /(2π ) ). These modes of oscillation are
often called eigenvalue resonant modes, and displacement shapes may be now
easily illustrated using FEM software. For one-dimensional systems these
equations are relatively simple, but when complex three-dimensional models are
used equations become cumbersome and more difficult to solve. It especially
stands for the case of piezoceramics treated here, since here are used piezoceramic
rings and disks, for which stand coupled fundamental piezoelectric equations, and
that connect mechanical, electromechanical, and electric parameters.
3
ANSYS, Swanson Analysis Syst., Houston, PA, USA (http://www.ansys.com)
ALGOR, Algor, Inc., Pittsburgh, PA, USA (http://www.algor.com)
5
ABAQUS, Hibbit, Karlsson & Sorensen, Inc., Pawtucket, RI, USA
(http://www.abaqus.com)
4
36
As already mentioned several times, one of the most essential components of
every sandwich transducer is the piezoceramic ring, although crucial role in
determination of transducer performances have quality, processing and polishing of
all consisting parts of the transducer. Therefore, above all the piezoceramic ring
must be modeled as precise as possible, in order to correct model the whole
ultrasonic sandwich transducer. The first step in this procedure is mesh generation
of the ring. The most optimally is to make a basic plan for the proposed mesh
before writing of the program. This will be later used as a good starting point for
logical node and element numbering and their connecting in the most optimal way.
Once the nodes are defined and generated, elements are then defined between those
nodes. Elements now have characteristics associated with material properties. In
purpose of entering parameters of piezoceramic rings is necessary to know
complete parameters of the piezoceramic, as it is presented in Table 4.3 for
piezoceramics of different manufacturers. FEM software demand three sets of
material constants for simulation of piezoceramic rings and disks in matrix form,
that is, dielectric, piezoelectric, and elastic constants enclosed in tables 4.3 and 4.4
for different types of piezoceramics.
The last input procedure is setting the analysis steps and correspondent
boundary conditions. Usual procedures in analysis of piezoelectric ceramics are
modal analysis of short circuit, modal analysis of open circuit, as well as frequency
analysis. Analyses of short circuit and open circuit give serial and parallel resonant
frequencies, respectively, while frequency analysis gives response of the
piezoceramic element at different frequencies, at input voltage of 1V. In the final
step are also specified demands related to frequency analysis of results.
The simplest case of analysis represents a FEM model of an axisymmetric
piezoceramic ring. Since the ring is symmetric regarding the z-axis, only a model
o
of one small section may be realized, which can then be rotated up to 360 , in order
to obtain a three-dimensional model of the ring. However, since the actual analysis
is related only to this small section, it is logical to expect that all analysis results be
symmetric. Such type of a model requires shorter calculation time than the
complete three-dimensional model. In Figure 2.15 is presented the mentioned
axisymmetric model of a piezoceramic ring [72].
After realization of the input steps, simulation of the ring may be performed
using FEM post-processing software, which enable presentation of the appearance
of the modeled ring, shape of the mechanical displacements of the ring, as well as
different resonant frequencies and ring responses at excitation of 1V. Threedimensional model of a ring generated by Abaqus software using axisymmetric
model from Figure 2.15, is presented in Figure 2.16.
In purpose of presentation of simulation efficiency and reliability in applying
this model, in Figure 2.17 is presented modeled characteristic of input admittance
dependency in function of frequency for piezoceramic ring that is often applied for
realization of sandwich transducers used in systems for ultrasonic cleaning and
welding. It is the case of PZT4 piezoceramic ring with outer diameter of 2a=38mm,
inner diameter 2b=13mm, and thickness 2h=6.35mm.
Figure 2.15. axisymmetric model of a piezoceramic ring
Figure 2.16. Three-dimensional FEM model of a piezoeramic ring
Figure 2.17. Dependence of input admittance of piezoceramic ring on frequency
Obtained by simulation by three-dimensional FEM model at ring excitation of 1V
38
It was obtained a resonant frequency of 42.9kHz, while by measuring the
resonant frequency on the automatic net analyzer for the cited ring, it was obtained
a resonant frequency of 42.37kHz (Figure 2.35(c)), which is very close to the
predicted result. Accordingly, accuracy of the simulated results using FEM
methods is satisfactory, and above required for most applications of piezoceramic
rings.
2.2.3. Brissaud’s 3D Model of Unloaded Piezoceramic Rings
Brissaud developed a three-dimensional model of a ring with zero mechanical
stresses in points on its contour surfaces, in purpose of characterization of
piezoceramic materials [73]. Modeling of a piezoceramic ring by this threedimensional model relates to determination of input electric impedance of the ring,
whereat this model also encompasses radial and thickness modes of oscillation.
This model is general and it is intended, without difference, for modeling as thin,
so the thick piezoceramic rings with different inner diameters. Assuming that
central geometric axes of the ring are the only (pure) directions of oscillation
propagation, general piezoelectric equations are simplified and their analytical
solution is obtained. Assuming that the ring with outer diameter 2a, inner diameter
2b, and thickness 2h is supplied from the voltage source V, an expression is
obtained on whose base one may determine input electric impedance of the ring:
tg(k z h ) ⎫
1 ⎧
S
S
⎬,
⎨1 − [AJ 0 (k r a ) + BY0 (k r a )] k r h31ε 33 − h33ε 33 N 1
jC 0ω ⎩
kzh ⎭
where the values of specific parameters and coefficients are following:
z ul =
S
C 0 = ε 33
A=
N1 =
π (a 2 − b 2 )
(Yb − Ya ) (h33c13D − h31c33D ) ,
2h
B=−
N2
(2.23)
,
(Jb − Ja ) (h33c13D − h31c33D ) ,
N2
D
[(Jb − Ja )Y0 (kr a) − (Yb − Ya ) J0 (kr a)] ,
h33 (JbYa − JaYb ) − h31kr c13
N2
D
(JbYa − J aYb ) − kr c13D
N 2 = c33
D
Ya = kr c11
Y0 (k r a ) −
2
[(Jb − J a )Y0 (kr a ) − (Yb − Ya ) J0 (kr a )] ,
D
J a = kr c11
J 0 (kr a ) −
D
D
c11
− c12
J1 (kr a ) ,
a
D
J b = kr c11
J 0 (kr b ) −
D
D
− c12
c11
J1 (kr b ) ,
b
D
D
c11
− c12
Y1 (k r a ) ,
a
D
Yb = kr c11
Y0 (kr b ) −
(2.24)
D
D
− c12
c11
Y1 (kr b ) .
b
S
Thereat cijD are coefficients of elasticity constants tensor of piezoceramic, ε33 is
dielectric constant in compressed state, hij are tensor elements of piezoelectric
D
D
/ ρ , and vz = c33
/ ρ are
constants (i,j=1,2,3), kr = ω / vr , k z = ω / v z , vr = c11
wave (characteristic, eigenvalue) numbers and phase velocities of two uncoupled
waves in radial and thickness direction, respectively. ω is circular frequency, ρ is
density of piezoceramic, J1 and Y1 are Bessel’s functions of first order, first and
second rank, respectively.
Piezoceramic ring is polarized along axis z, while the lateral surface are
metalized and connected to an alternating current source. This model represents a
model of uncompressed piezoceramic ring without load influence on its boundary
surfaces, that is, circular-ringed surfaces oscillate freely. Therefore, by such
approach cannot be analyzed influence of the external medium, which is modeled
by acoustic impedances that load boundary surfaces. This is a limitation in
application of this model for modeling of complete ultrasonic sandwich
transducers, with certain serious mathematical discrepancies in fulfilling of
mechanical boundary conditions. Model predicts, with certain accuracy, only the
first radial and the first thickness mode of oscillation of piezoceramic ring with
arbitrary ratio of thickness and diameter, while the frequencies of the higher radial
modes of oscillation do not match with experimental results. Based on this model
(and similar models for other geometries of the specimens) Brissaud proposed new
procedures for measuring piezoceramic constants materials. These procedures gave
results that significantly differ from traditional data measured using procedures
from ANSI/IEEE standards. This is because applied boundary conditions in model
are fulfilled only on the circular ring lines, but not on the whole cylindrical and
circular-ringed contour surfaces, so that the solution of fundamental piezoceramic
equations is not valid. As a consequence of this, Brissaud’s model is reduced to
one-dimensional Mason’s model for isolated thickness modes of oscillations.
Disadvantages of this approach in modeling of piezoceramic elements are
presented in literature [74], [75].
2.2.4. Matrix Radial Model of Thin Piezoceramic Rings
In literature [76] is described matrix model of a thin piezoceramic ring in
radial mode of oscillation, which is suitable for predicting of dynamic behavior of
piezoceramic ring when its two plane circular-ringed surfaces are not compressed,
while cylindrical surfaces (inner and outer) are in contact with outer medium. The
ring is modeled as a network with three accesses. Using this model one may easily
calculate the spectrum of radial resonant frequencies and input electric impedance
of a ring with very small thickness. Spectrum of resonant frequencies does not
depend on ring thickness and it is calculated in function of ratio of inner and outer
diameter. Considerable accuracy in using this model is achieved only for the lowest
radial resonant modes.
This model is suitable for determination of interaction of internal and external
cylindrical surface with outer medium. Starting from classical constitutive
equations of piezoceramic materials, as well as from wave differential equation that
40
describes oscillation of a ring in radial direction, one may also determine the
expression for mechanical displacements in radial direction. This matrix model
may describe behavior of the ring as a wave emitter in radial direction for any outer
medium.
Imposing continuity between mechanical stresses and forces on the cylindrical
surfaces of the ring, one gets matrix model of the external behavior of the ring,
which may predict ring behavior as a radial oscillation emitter. For piezoceramic
ring with outer radius a, inner radius b, and thickness 2h, where 2h<<a, with
condition that circular-ringed surfaces are metalized, that polarization is parallel to
z-axis, and that axial symmetry exists, using described mechanical boundary
conditions, one gets the following system of equations:
p
[A1FJ (b ) + B1FY (b )] v1 − 4π hc11p [A2 FJ (b ) + B2 FY (b )] v2 − 2π be31p V ,
F1 = −4π hc11
p
[A1FJ (a) + B1FY (a)] v1 − 4π hc11p [A2 FJ (a) + B2 FY (a)] v2 − 2π ae31p V , (2.25)
F2 = −4π hc11
a2 − b2 p
ε 33V ,
2h
where ∆ J1=aJ1(kYa)-bJ1(kYb) and ∆Y1=aY1(kYa)-bY1(kYb) , kY=ω/vp, (vp)2=c11p/ρ,
and vi and Fi are velocities and forces on cylindrical surfaces (i=1, 2), respectively.
Previous equation system describes external behavior of the ring, which may
be treated as a three-access system (one electric and two mechanical). Thereat, in
previous equations values of specific used parameters and constants are given by
following equations:
⎛
⎞
1
Y1 (kY b ) J1 (k Y a )
⎟,
⎜⎜1 +
A1 =
jω J1 (kY b ) ⎝
J1 (kY b ) Y1 (k Y a ) − J1 (kY a ) Y1 (kY b ) ⎟⎠
⎞
Y1 (kY b )
1 ⎛
⎜⎜
⎟,
A2 = −
jω ⎝ J1 (k Y b ) Y1 (kY a ) − J1 (k Y a ) Y1 (kY b ) ⎟⎠
⎞
J1 (kY a )
1 ⎛
⎜⎜
⎟,
B1 = −
(2.26)
jω ⎝ J1 (k Y b ) Y1 (kY a ) − J1 (kY a ) Y1 (kY b ) ⎟⎠
⎞
J1 (kY b )
1 ⎛
⎜⎜
⎟,
B2 =
jω ⎝ J1 (kY b ) Y1 (kY a ) − J1 (kY a ) Y1 (kY b ) ⎟⎠
p
[A1∆J1 + B1∆Y1 ] v1 − 2π jω e31p [A2 ∆J1 + B2 ∆Y1 ] v2 + jωπ
I = −2π jω e31
( ),
F (r ) = k rY (k r ) − Y (k r ) (1 − σ ) ,
FJ (r ) = kY r J0 (kY r ) − J1 (kY r ) 1 − σ
p
p
Y
Y
0
Y
Y
1
where σ p=c12p/c11p, that is:
2
E
c11p = c11E − c13E / c33
,
2
E
c12p = c12E − c13E / c33
,
ε
p
33
e31p
=ε
S
33
+
2
e33
E
/ c33
,
E
= e31 − e33 c13E / c33
,
where eij are piezoelectric constants.
(2.27)
As mentioned, by this approach is possible to take into account the interaction
of lateral cylindrical surfaces with surrounding, which is realized by loading of
mechanical accesses by acoustic impedances of the medium. Connecting the
electric access to alternating voltage V one may calculate all transfer functions, as
well as the most often required input electric impedance. Finally, using this model,
dielectric and mechanical losses may be taken into account by correspondent
acoustic impedance serially connected with the appliance.
Complete characterizing of radial symmetric oscillation of piezoceramic ring
based on the previous model is presented in literature [77]. Different phenomena in
behavior of resonant frequency spectrum, as well as effective electromechanical
coupling factors (keff) are determined in function of ratio of inner and outer
diameter ranging from 0 to 1. Disadvantage of this model that one cannot simulate
effect of ring loading in direction of polarization axis z, which is necessary during
simulation of complete ultrasonic sandwich transducers.
2.2.5. Three-dimensional Model of Piezoceramic Ring
Loaded on All Contour Surfaces
As mentioned, three-dimensional model of a ring would be convenient for
exact calculation of sandwich transducer, due to the strong interaction between
radial modes of the ring and thickness mode of the whole transducer. The first step
in that direction represents the approximate three-dimensional electromechanical
model [78], but for piezoceramic disks, applicable for any ratio of diameter and
thickness and which enables coupling between thickness and radial modes of
oscillation. Also in boundary cases of thin disk and long thin bar, one gets results
that well agree with corresponding results obtained by classical one-dimensional
models for those cases. However, radial and thickness resonant frequencies of the
ring are very different from radial and thickness oscillations of a disk of same
diameter and thickness. Therefore, in purpose of correct modeling of ultrasonic
sandwich transducers, first is necessary to realize a three-dimensional model of a
piezoceramic ring, which was done in this chapter [79], whereat is used one
approximate approach in fulfilling mechanical and electric boundary conditions
[80].
Basic problem in application of multidimensional models of piezoelectric
ceramics is solving of the system of coupled wave partial differential equations,
which describe element oscillation, so for solving of that problem are usually used
approximate methods [81]. In this chapter is presented new approximate
generalized matrix model of piezoceramic rings, from which one may easily get a
model of piezoceramic disks, and by which one may analyze both the radial and
the thickness oscillations of rings and disks. Model enables predicting of dynamic
behavior of the piezoceramic ring when all contour (boundary) surfaces are in
contact with outer medium, that is, when different mechanical loads are applied on
external surfaces of the ring. The ring is treated as an axisymmetric threedimensional structure in polar-cylindrical coordinate system, whose oscillations in
thickness and radial direction are described by two coupled wave differential
42
equations, with coupled boundary conditions. Solution of this equation system are
two orthogonal wave functions, which depend on time and only one coordinate that
corresponds to the direction of wave propagation, and that fulfill both the
mechanical and electric boundary conditions only in correspondent approximate
integral form. Namely, when one considers boundary surfaces of the ring, applied
integral conditions enable obtaining of approximate model of external behavior of
the ring. Using this model, piezoceramic element is presented in frequency domain
through a 5-access network, with one electric and four mechanical accesses (one
for every contour surface), whereat is possible to determine all relations between
applied input voltage, and forces and velocities on all external surfaces. Thus is
enabled easy determination of input electric impedance of the piezoceramic ring,
its resonant frequency spectrum, effective electromechanical coupling factor, as
well as mechanical displacements in radial and thickness direction. Using this new
model a comparison of obtained results is performed with results obtained using
existing one-dimensional Mason’s model, and existing three-dimensional models
(matrix radial and Brissaud’s). There were noticed changes of fundamental
thickness resonant frequency due to the presence of radial modes, and contrary. In
purpose of verification of obtained model, frequency characteristics of input
impedance dependency are calculated, and determined effective electromechanical
coupling factors, as well as the mechanical displacements in radial and thickness
direction for piezoceramic rings and disks of different dimensions. Also, it is
performed comparison of calculated and experimental results. Comparing to
numerical methods and models, calculations are significantly sped up. The model
may be a simple and useful tool in design optimization of complete ultrasonic
sandwich transducer, whose excitation part in practical applications are just
described piezoceramic rings.
2.2.5.1. Analytical Model
Piezoceramic elements that will be subject of analysis in this chapter are, as in
hitherto analysis, piezoceramic rings polarized across thickness (that is with
polarization parallel to z-axis), with outer radius a, inner radius b, and thickness 2h,
and with completely metalized circular-ringed (plane) surfaces on which is
supplied alternating excitation voltage. Dimensions of the ring and polarcylindrical coordinate system with origin in the ring centre, are defined in Figure
2.18(a). Every ring surface is loaded by acoustic impedance Zi, where vi and Fi are
velocities and forces on those contour surfaces Pi (i=1, 2, 3, 4) (Figure 2.18(b)).
Piezoceramic materials, and by that itself the piezoceramic rings too, are
characterized by tensors of their elastic, piezoelectric, and dielectric constants. The
most often used set of constitutive piezoelectric equations (of four existing)
presents tensors of mechanical stresses T and electric field E inside the material in
function of tensor of relative mechanical deformations (dilatations) S and dielectric
displacement D.
z
I
V
r
b F v1
1
a
(a)
v1
F3
v3
v2
F1
5-access
network
v3
2h
F3
F2
F 4 v2
v4
V
F2
v4
F4
I
(b)
Figure 2.18. Loaded piezoceramic ring: (a) geometry and dimensions;
(b) ring as a 5-access network
In case of piezoceramic ring from Figure 2.18(a), which will be analyzed
furthermore, components of electric field Er and Eθ are equal to zero on two plane
surfaces, because they are metalized, and it is also assumed that they are negligible
(equal to zero) everywhere inside the material. Besides that, due to the axial symmetry,
only symmetric (radial and thickness) modes of oscillation are excited. Accordingly,
all values are independent on angle θ, so the displacement uθ is equal to zero.
Proposed model of a piezoceramic ring is obtained with assumption that coordinate
axes r and z are directions of pure (uncoupled) modes of wave propagation, with
mechanical displacements in radial and thickness direction ur=ur(r, t) and uz=uz(z, t).
With these assumptions, the most often used set of constitutive equations, which
describe oscillation of a piezoceramic ring, is reduced to the following system of
equations in polar-cylindrical coordinate system [67]:
Trr = c11D S rr + c12D S θθ + c13D S zz − h31 D z ,
Tθθ = c12D S rr + c11D Sθθ + c13D S zz − h31 D z ,
D
T zz = c13D S rr + c13D S θθ + c33
S zz − h33 D z ,
(2.28)
S
E z = − h31 S rr − h31 S θθ − h33 S zz + D z / ε 33
,
S
where cijD are coefficients of elasticity constants tensor; ε33 is dielectric constant
of the ring in compressed state; hij are elements of piezoelectric constants tensor
(i,j=1, 2, 3).
Relations between components of relative deformations tensor S and
mechanical displacement vector u are following (Spq=0, if p≠q):
(2.29)
S rr = ∂ u r / ∂ r , Sθθ =u r / r , S zz = ∂ u z / ∂ z .
Differential equations that describe oscillation of elastic (deformable) body in
radial and thickness direction may be also used for approximate description of
piezoceramic ring oscillation, whereat are not taken into account influences of
forces due to the acting of electric values. Quoted equations are obtained from the
condition of dynamic balance in the following form [73], [78]:
∂Trr Trr − Tθθ
∂ 2 ur
∂Tzz
∂ 2uz
(2.30)
,
ρ
.
+
=ρ
=
∂r
r
∂z
∂t 2
∂t 2
44
By substitution of (2.29) into (2.28) and (2.28) into (2.30), one gets differential
equations of oscillation in radial and thickness direction in the following form:
⎛ 2
∂ 2 ur
1 ∂ur ur ⎞⎟
D ⎜ ∂ ur
ρ
,
c11
=
+
−
⎜ ∂r 2 r ∂r r 2 ⎟
∂t 2
⎠
⎝
(2.31)
2
2
∂
u
∂
u
D
z
c33
= ρ 2z .
2
∂z
∂t
Assuming that the waves are harmonic ( Dz = D0 e jωt ), components of the
mechanical displacement in radial and thickness direction are solutions of previous
equations and they are presented through two orthogonal wave functions:
ur (r, t ) = [A J1 (kr r ) + B Y1 (kr r )] e jωt ,
(2.32)
u z (z, t ) = [C sin (k z z ) + D cos(k z z )] e jωt ,
where
previously
was
defined
D
that k r = ω / v r , k z = ω / v z , vr = c11
/ρ ,
and
D
v z = c33
/ ρ wave (characteristic, eigenvalue) numbers and phase velocities of two
uncoupled waves in radial and thickness direction, respectively. Also, in the previous
exposure was defined that ω is circular frequency, ρ density of piezoceramic, and that J1
and Y1 are Bessel’s functions of first order, first and second rank, respectively. Consequence
of choosing such orthogonal functions for displacements is that boundary conditions cannot
be fulfilled in every point on external surfaces exactly, but only approximately.
2.2.5.2. Model of External Behavior of Piezoceramic Ring
Constants A, B, C, and D in the assumed solution are calculated using
mechanical boundary conditions, which imply that all external surfaces are in
contact with surrounding medium (same or different, infinite or definite) and with
assumption that there is continuity of velocities on those surfaces, because the
purpose of this model is to describe behavior of piezoceramic ring as a transducer:
∂ ur
∂ ur
= − v 2 e j ωt ,
= v1 e jωt ,
∂ t r =b
∂ t r =a
(2.33)
∂ uz
∂ uz
j ωt
j ωt
= − v3 e ,
= v4 e .
∂ t z =h
∂ t z =− h
Using these boundary conditions, that is, by substitution of (2.32) into (2.33),
unknown constants are calculated, so that one gets:
A v + A2 v2
B v + B2 v2
,
,
A= 1 1
B= 1 1
jω
jω
(2.34)
v3 + v 4
v 4 − v3
,
,
C=−
D=
2 jω sin(kz h )
2 jω cos(k z h )
where newly introduced constants A1, A2, B1, and B2 are defined in the following way:
A1 =
A2 =
Y1 (kr a )
,
J1 (kr b )Y1 (kr a ) − J1 (kr a )Y1 (kr b )
Y1 (kr b )
,
J1 (kr b )Y1 (kr a ) − J1 (kr a )Y1 (kr b )
(2.35)
J1 (kr a )
,
B1 =
J1 (kr a )Y1 (kr b ) − J1 (kr b )Y1 (kr a )
J1 (kr b )
.
J1 (kr a )Y1 (kr b ) − J1 (kr b )Y1 (kr a )
External behavior of the ring id determined from the condition of continuity
of mechanical stresses and forces on its external surfaces. However, two orthogonal
functions (2.32) do not fulfill these conditions. Therefore is applied a compromise
in modeling of external behavior of the ring, that is, it is considered that force on
every external surface is determined (equilibrated) by integral of mechanical stress
on that surface. Accepting these equations it is assumed that equilibrium is yet
fulfilled, and in an integral form [80]:
B2 =
∫P Trr (b)dP = −F1 ,
∫P Trr (a)dP = −F2 ,
∫P Tzz (h)dP = −F3 ,
∫P Tzz (− h)dP = − F4 ,
1
2
(2.36)
3
4
whereat P1 and P2 are cylindrical lateral surfaces for r=b and r=a, and P3 and P4 are
plane circular-ringed surfaces for z=h and z=-h, respectively, so that one may write:
h
2π b Trr (b ) dz = − F1 ,
∫
−h
h
2π a Trr (a ) dz = − F2 ,
∫
−h
a
(2.37)
2π Tzz (h )r dr = − F3 ,
∫
b
a
2π Tzz (− h )r dr = − F4 .
∫
b
Equations (2.28), (2.29), (2.32), (2.34), and (2.37), and classical relation
between current I and dielectric displacement Dz:
a
∂Dz
I = 2π
r dr = jωπ a 2 − b 2 D0 e jω t = jωπ a 2 − b 2 Dz , (2.38)
∂t
b
that is Dz=I/[jωπ(a2-b2)], lead to the linear system of equations, which, using the
matrix of dimensions 5x5 describes external behavior of the ring, connecting
∫
(
)
(
)
46
electric (voltage V and current I) with mechanical values (forces Fi and velocities
vi) in frequency domain (Figure 2.18(b)):
⎡ F1 ⎤ ⎡ z11
⎢ F ⎥ ⎢z
⎢ 2 ⎥ ⎢ 21
⎢ F3 ⎥ = ⎢ z13
⎢ ⎥ ⎢
⎢ F4 ⎥ ⎢ z13
⎢⎣ V ⎥⎦ ⎢⎣ z15
z12
z13
z13
z22
z23
z23
z33
z23
z34
z23
z25
z34
z35
z33
z35
z15 ⎤ ⎡ v1 ⎤
⎥
z25 ⎥ ⎢⎢v2 ⎥⎥
z35 ⎥ ⎢ v3 ⎥ .
⎥⎢ ⎥
z35 ⎥ ⎢v 4 ⎥
z55 ⎥⎦ ⎢⎣ I ⎥⎦
(2.39)
Matrix elements are following:
− 4π h D
D
[1 − kr b(A1 J0 (kr b ) + B1Y0 (kr b ))] ,
z11 =
c12 − c11
jω
4π h D
D
[1 + kr a(A2 J0 (kr a ) + B2 Y0 (kr a ))] ,
c12 − c11
z 22 =
jω
{
}
{
}
z12 =
D
− 4π kr bhc11
[A2 J0 (kr b ) + B2 Y0 (kr b )],
jω
z 21 =
D
− 4π kr ahc11
[A1 J0 (kr a ) + B1Y0 (kr a )],
jω
z13 =
z23 =
z33 =
D
2π bc13
,
z15 =
4π bhh31
,
j ωP
D
2π ac13
,
jω
z25 =
4π ahh31
,
jω P
jω
(2.40)
D
D
c33
kz P
c33
kz P
, z34 =
,
jω tg(2 kz h )
jω sin (2 kz h )
z35 =
1
h33
, z55 =
,
jω
jωC0
S
whereat is P=π (a2-b2) ring area, and C0=(ε33 P)/(2h) is so-called capacitance of
the compressed ceramic. Impedance matrix does not contain zero elements, so that
all surface forces Fi and voltage V depend on all velocities vi and current I. It refers
to the conclusion that proposed model is capable to describe coupling of the
thickness and radial resonant modes.
Concerning the electric field in z direction, substituting (2.29) and (2.32) into
the last equation in (2.28), one gets the field Ez. The electric boundary condition is
fulfilled through integral condition for the electric field, that is, voltage V is
h
determined by integration Ez along z-axis (V =
∫E d
z
z ).
−h
However, thus obtained stress is function of r coordinate, which is in contrast with
assumption that surfaces normal (orthogonal) to z-axis (P3 and P4) are metalized and,
according to that, equipotential too, and this is direct consequence of the choice of
orthogonal functions (2.32) as solutions of wave equations. This is surpassed using one
alternative approach, that is, repeated integration of the voltage V and along r-axis
[80], in order to make V independent on r, and thus is obtained the last equation in
(2.39). Consequence of this approximate integral condition is that on external surfaces
of the ring may be observed only mean values of the force and velocity, and one may
not notice their values in every point of the surface.
Equation system (2.39) describes external behavior of the ring, whereat it
represents 5-access network with one electric and four mechanical accesses. By such
analysis it is possible to take into account interaction of all external surfaces of the ring
with surrounding by connecting acoustic impedances of the medium to the mechanical
accesses. Acoustic impedances will be real if the observed surrounding medium is
infinite, or complex, if the surrounding medium is bounded in observed direction.
ω
Lastly, connecting the alternating voltage V=V0ej t to the electric access, one may
present (analyze) external behavior of the piezoceramic ring as a transducer. Thereat
for arbitrary acoustic appliances, for different ratios of inner and outer ring diameter
(b/a), as well as for different ratios of thickness and outer diameter (h/a), are most
often determined input electric impedance (V/I), emitting transfer function (Fi /V) and
receiving transfer function (V/Fi) (where index i denotes observed mechanical access).
In the model is considered piezoelectric material without losses. Using this
model, mechanical and dielectric losses may be taken into account by introducing
of complex elastic and dielectric constants or connecting correspondent acoustic
impedances serially with the appliance.
2.2.5.3. Numerical Results
2.2.5.3.1. Input Electric Impedance of the Piezoceramic Ring
Connecting the forces and velocities on external surfaces through acoustic
impedances (Fi=-Zi vi, i=1, 2, 3, 4), one may modify the z matrix in the equation
system (2.39), wherewith one gets linear system that connects input voltage and
current, so that one may easily determine the input electric impedance zul=V/I. In
purpose of comparison of obtained results with existing one-dimensional thickness
Mason’s model, three-dimensional matrix radial model of thin ring, as well as the
three-dimensional Brissaud’s model, it was determined the modulus of the input
electric impedance of the piezoceramic ring of PZT8 piezoceramic material
(Zul=20log( zul [Ω ] /50+1) [dB]), whereat the parameters of the piezoceramic
material used in analysis are presented in Table 4.3. Figure 2.19 presents obtained
results if it is implied that the surrounding medium is air, that is, external acoustic
impedances are Zi=400 Rayl 6 [63].
In order to present the possibility of analysis of concrete piezoceramic rings
using the proposed model, it is determined the input electric impedance for the
already mentioned PZT8 piezoceramic ring with dimensions 2a=38mm, 2b=15mm,
2h=5mm, because exactly such rings are used in the fourth chapter in specific
practically realized ultrasonic sandwich transducers.
6
1 Rayl=10 Pa⋅s⋅m-1=10 N⋅s⋅m-3
70
1D thickness model
realized 3D model
60
R2
R1
(a)
T1
Zul [dB]
50
40
R3
30
R4
20
10
0
0
1
2
3
f [Hz]
4
5
x 10
5
6
(a)
70
matrix radial model
realized 3D model
60
R2
R1
(b)
T1
50
Zul [dB]
48
40
30
R4
20
10
0
0
1
2
3
f [Hz]
(b)
4
5
x 10
5
6
70
60 R1
3D Brissaud’s
model
realized
3D model
R2
(c)
T1
Zul [dB]
50
40
30
R4
20
10
0
0
1
2
3
f [Hz]
4
5
x 10
5
6
(c)
Figure 2.19. Comparison of input impedances of unloaded PZT8 piezoceramic ring
with dimensions 2a=38mm, 2b=15mm, 2h=5mm in case of proposed threedimensional model and: (a) one-dimensional thickness model [59],
(b) matrix radial model [76], and
(c) Brissaud’s three-dimensional model [73]
Quoted dependence is compared with analogous characteristic obtained by
standard Mason’s thickness one-dimensional model [59] (Figure 2.19(a)). One may
notice that for the first thickness mode (T1) exists great agreement for both models,
whereat are interesting small deviations that exist due to the strong coupling of
adjacent radial modes with thickness mode in the proposed three-dimensional
model. As expected, one-dimensional model in contrast of the three-dimensional
model neglects radial resonant modes of oscillation.
Characteristic of the impedance obtained by model proposed in this chapter, is
compared in Figure 2.19(b) with input impedance obtained by the matrix radial
three-dimensional model of very thin rings [76]. There is great agreement of the
first two radial modes between presented characteristics, especially for the first
radial mode (R1), while for the other radial modes there is big disagreement due to
the great influence of the thickness mode in the new three-dimensional model.
From Figure 2.19(b) is obvious that matrix radial model for thin rings cannot
enable predicting of the thickness mode T1.
50
Beside these mentioned models, in Chapter 2.2.3 is remarked that exists a
Brissaud’s three-dimensional model of the unloaded ceramic [73], which
contains serious limitations and faults. In Figure 2.19(c), which represents
comparison of impedances obtained by this model and proposed matrix threedimensional model, one may see great differences in predicting the resonant
modes, as well as illogical radial mode R4 (region marked by arrow), where
impedance characteristic first reaches a maximum, and then minimum, which
isn’t real. Besides that, this Brissaud’s three-dimensional model does not enable
analysis of influences of mechanical loads on external surfaces of the ring,
while by the proposed three-dimensional model one may analyze those
influences, which will be presented later.
In order to further present possibilities of the proposed model, in Figure
2.20 is presented impedance dependence of the PZT8 ring with dimensions
2a=38mm, 2b=15mm, in function of the frequency and ring thickness 2h
ranging 0÷120mm. Characteristic parameters of the PZT8 piezoceramic
material, used in ring modeling, are presented in Table 4.3. As it is logical to
expect, increase of the ring thickness has the greatest influence on the thickness
resonant mode, which shifts to the region of the lower frequencies, and smaller
influence on radial resonant modes, although they too are shifting due to the
coupling with the thickness mode. Also, change of the ring thickness also
affects the value of the ring capacitance, so that change of the impedance level
occurs, too.
Besides this case, in Figure 2.21 is presented impedance of the same ring
with dimensions 2a=38mm, 2h=5mm in function of the ratio of the inner and
outer radius b/a, whereat that ratio is ranging 0÷1. It is obvious that change of
the radius ratio has the greatest influence on behavior of the radial resonant
modes, whereat an interesting phenomenon occurs, that the first radial resonant
mode shifts towards lower frequencies, while other radial resonant modes tend
towards high frequencies, whereat they substantially affect the thickness
resonant mode. This influence of the inner radius on thickness resonant
frequency hasn’t been analyzed yet, neither in the field of piezoceramic ring
modeling, and by that itself nor in the analysis of the complete ultrasonic
sandwich transducer.
Like in the previous case, change of the value of the metalized ring surfaces
due to the inner radius change, generates change of its capacitance, and by that
itself the change of the impedance level.
In Figure 2.22(a) is presented characteristic of impedance dependency on
frequency for PZT8 piezoceramic ring from the previous analysis (Figure 2.19),
whereat the ring surfaces are loaded by different acoustic impedances: first
piezoceramic ring oscillates freely in the air (solid line), then the ring is loaded
on one metalized surface while the rest surfaces are free (dashed line), and in
the end the ring is loaded by same great impedances on both metalized surfaces
(dot line).
Zul
[dB]
Z [dB]
60
40
20
0
0.012
0.01
0.008
2h
2b [m]
[m]
0.006
0.004
0.002
0
0
3
2
1
6
5
4
x 10
f [Hz]
5
f [Hz]
Figure 2.20. Input impedance change of the PZT8 ring of dimensions 2a=38mm,
2b=15mm depending on frequency and ring thickness
1
[dB]
ZZul
[dB]
0.8
60
0.6
40
a1/a2
20
0.4
b/a
0
0
1
x 10
5
0.2
2
f [Hz]
3
f [Hz]
4
5
6
0
Figure 2.21. Input impedance change of the PZT8 ring of dimensions 2a=38mm,
2h=5mm, depending on frequency and inner and outer radius ratio
70
60
Z1=Z2=Z3=Z4=400 Rayl
R2
R1
Z1=Z2=Z3=400 Rayl
Z4=5 MRayl
Zul [dB]
50
(a)(a)
T1
Z1=Z2=400 Rayl
Z3=Z4=5 MRayl
40
R3
30
R4
20
10
0
0
1
2
70
60
3
f [Hz]
4
5
x 10
Z1=Z2=Z3=Z4=400 Rayl
5
6
(b)(b)
T1
R2
R1
Z1=3 MRayl
Z2=Z3=Z4=400 Rayl
50
Zul [dB]
52
Z1=Z2=3 MRayl
Z3=Z4=400 Rayl
40
R3
30
R4
20
10
0
0
1
2
3
f [Hz]
4
5
x 10
Figure 2.22. Input impedance of the PZT8 ring of dimensions
2a=38mm, 2b=15mm, 2h=5mm, for different acoustic loads:
(a) in thickness direction and (b) in radial direction
5
6
It is obvious that acoustic load in thickness direction affects mostly the
thickness oscillation mode of the ring, and its influence on radial modes is
negligible, for the adopted ring dimensions. On the other hand, as expected,
increase of the acoustic load in radial direction affects significantly the radial
resonant modes (Figure 2.22(b)), and has almost none influence on thickness
oscillation mode of the ring in this concrete case. Here is also observed unloaded
ring that oscillates in air (solid line), ring loaded on inner cylindrical surface
(dashed line), and ring loaded on both cylindrical lateral surfaces (dot line).
Previous conclusions may be presented even more clearly through threedimensional graphs of input impedance of this PZT8 piezoceramic ring in function
of frequency and applied acoustic impedance. Input impedance from Figure 2.22(a)
is corresponded by graph from Figure 2.23, while input impedance from Figure
2.22(b) is corresponded by graph presented in Figure 2.24. In these Figures are
clearly noticed values of acoustic loads at which specific resonant modes
disappear: thickness at load in direction of polarization axis (Figure 2.23) or radial
at lateral load of the ring (Figure 2.24). Naturally, since the modes are coupled,
these influences are not isolated, as one may see in Figure 2.23 where changes of
the thickness mode also affect changes of the third and fourth radial mode, which
are closest to the thickness mode, and at high loads also the changes of the quite
distant second radial mode.
ZulZ [[dB]
dB]
40
35
30
25
20
0
15
2
4
10
0
6
1
2
3
x 10
5
f [Hz
]
f [Hz]
x 106
8
4
5
10
Z33=Z44
Z3=Z4 [[Rayl]
Rayl]
6
Figure 2.23. Change of the input impedance of the PZT8 ring of dimensions
2a=38mm, 2b=15mm, 2h=5mm, with frequency and acoustic load
on metalized circular-ringed surfaces
x 10
6
54
[dB]
ZulZ[dB
]
60
50
40
30
20
0
0.5
10
1
0
0
1.5
1
2
2
3
x 10
5
f [Hz]
f [Hz]
2.5
4
5
3
x 106
x 10
Z11=Z22
[Rayl]
Z1=Z2 [Rayl
]
6
Figure 2.24. Change of the input impedance of the PZT8 ring of dimensions
2a=38mm, 2b=15mm, 2h=5mm, with frequency and acoustic
load on lateral cylindrical surfaces
Mutual position of the radial and thickness resonant modes depend on the ratio of
inner and outer radius, as well as of thickness of the piezoceramic ring, which may be
seen in Figures 2.20 and 2.21, as well as later in experimentally recorded graphs. For
specific values of these dimensions resonant modes may be very close, and previously
analyzed loading of the ring by acoustic impedances may generate not only decrease
(Figure 2.22), but also the frequency shift of the resonant modes. For example, at ring
with small outer diameter and great length, cylindrical lateral ring surfaces are also
large, so the influence of lateral acoustic impedances on radial modes is also great. In
those cases disappearing of the radial modes due to the lateral loads may generate
frequency shifts of the thickness mode, because of their strong coupling. Even such
detailed analyses are possible using the proposed model, but they will not be
considered further because of massiveness.
2.2.5.3.2. Frequency Spectrum of Piezoceramic Ring
In purpose of completion of frequency behavior of the proposed model of
piezoceramic rings and disks, as well as to present possibilities of the model in
describing fundamental resonant modes and their harmonics, it is determined the
frequency spectrum of unloaded piezoceramic ring in function of ratio of its thickness
6
and outer diameter. Thereat are taken into account resonant frequencies at which
occurs minimum of the input electric impedance. In Figure 2.25 is presented the
obtained resonant frequency spectrum for ratio of thickness and outer ring diameter
ranging 0÷1, for the case of PZT8 ring of dimensions 2a=38mm, 2b=15mm.
x 10
6
5
IV3D
5
4
f [Hz]
III3D
3
IV3D
2
II3D
III3D
II1D
I1D
1
II3D
I3D
I3D
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
h/a
Figure 2.25. Frequency spectrum of the PZT8 piezoceramic ring
With ratio b/a=15/38 in function of its thickness:
proposed model (
) and Mason’s model (
)
For observed piezoceramic material and for a given ring thickness, in the
spectrum is easy to notice that frequencies of specific modes do not depend on
thickness, and that these are radial resonant modes, while the frequencies of the
thickness modes are decreasing functions of thickness. Coupling between thickness
and radial modes is clearly noticed in the regions of lower frequencies, where
significant shifting of the resonant frequencies exists, that is, curves are not neither
constant lines, nor they have pure hyperbolic form. Results obtained by the
proposed model are compared with the two lowest resonant modes obtained using
traditional one-dimensional thickness Mason’s model for the same piezoceramic
ring [59]. Mason’s model is in the most of the literature about modeling of
ultrasonic sandwich transducers considered as a good representation of the physical
behavior of such piezoceramic ring. Frequencies of the thickness modes obtained
by Mason’s model coincide with frequencies of the presented modes only in case
of piezoceramic rings with great length (piezoceramic pipes with ratio h/a>>1).
56
In order to further analyze frequency characteristics of the unloaded
piezoceramic ring, again is determined frequency spectrum of the ring based
on the minimum of the input impedance, but now in function of the ratio of
outer and inner radius b/a ranging 0÷1. Figure 2.26 presents such frequency
spectrum of the PZT8 ring with thickness 2h=5mm and outer diameter
2a=38mm.
x 10
6
5
5
f [Hz]
4
IV
3
III
2
II
1
I
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
b/a
Figure 2.26. Frequency spectrum of the PZT8 piezoceramic ring in function
of opening dimensions obtained by the proposed model, for cases:
h/a=5/38 (
) and h/a= 0 (
) and using traditional onedimensional theory of radial oscillations of piezoceramic rings (
)
In contrast of the previous dependency from the Figure 2.25, where some parts of
the spectrum are independent on ring thickness, one needs to notice that here don’t
exist spectrum regions completely independent on dimensions in radial direction.
Some interesting conclusions, noticed in Figure 2.21, may now be analyzed more
detailed based on the shape of the frequency curves in the spectrum: when ratio b/a
grows from 0 to 1, resonant frequency of the first mode declines, while resonant
frequencies of the higher modes have other trend: they first reach minimum, and then
abruptly grow to higher frequencies, with obvious influence of the mode coupling on
its shape. Therefore is logical to associate these modes with different physical
dimensions in radial direction: first mode, which has not higher harmonics, should be
associated with the mean diameter of the ring, which grows with increase of the ratio
b/a, while the frequencies of the other modes (the second and its harmonics) are
determined by ring thickness in radial direction (a-b), which declines with increase of
b/a. When the ratio b/a→0, resonant frequencies of different modes approximate
resonant frequencies of the piezoceramic disk of same diameter and thickness. It is
obvious that, depending on the ring opening size, its resonant frequencies may be
either greater or smaller than resonant frequencies of correspondent modes at disk.
Influence of the ring thickness, that is, thickness resonant modes that depend on the
ring thickness, on the shape of the higher resonant modes may be also seen in Figure
2.26, where is, besides the cited case, also presented radial frequency spectrum
obtained for a ring of negligible thickness (2h≈0). Results obtained for the case of a
thin ring are analogous with results that may be obtained using matrix threedimensional radial model of a thin piezoceramic ring [76], so that the proposed model
is more general and extends the mentioned matrix model. It is obvious that, if one
decreases the ring thickness 2h, frequencies of the higher resonant modes grow much
faster to infinite frequencies. In that case, when b/a→1, one may easily determine the
fundamental resonant frequency, while other resonant frequencies are very high and
may be considered infinite, so the first resonant mode is in that case isolated from
other modes, and may be easily modeled by circuit with concentrated parameters [69].
Results of modeling from Figure 2.26, coincide with results obtained by this
Berlincourt’s model. Beside the cited dependencies on frequencies from ring
dimensions, in Figure 2.26 is also presented analogous frequency spectrum obtained
using traditional one-dimensional theory of radial oscillation of piezoceramic rings
[59], with visible deviations of resonant mode frequencies regarding the proposed
model, even in case of the lowest (I) resonant mode. Dependencies obtained by this
model have the same trend as the dependencies obtained by matrix three-dimensional
model of a thin piezoceramic ring [76], and like it, they don’t predict coupling of the
resonant modes.
Previous conclusions about possibilities of the proposed model for determination
of the resonant frequencies may be simultaneously confirmed on three-dimensional
dependencies for frequency spectrums of the first (Figure 2.27) and second (Figure
2.28) resonant mode in function of thickness and inner opening ring size, for the case
of PZT8 piezoceramic ring. By thicker lines in Figure 2.27 are presented the lowest (I)
resonant modes from Figures 2.25 and 2.26, and in Figure 2.28 second in turn (II)
resonant modes from the same graphs. The most important conclusion in the previous
analysis represents the fact that inner diameter ring size significantly and nonsinglevalued (complexly) affects all frequencies of the resonant ring modes. Therefore in
analysis and design of ultrasonic sandwich transducers cannot be all the same if for
excitation are applied piezoceramic disks or rings. This fact was neglected in almost all
existing models of ultrasonic sandwich transducers.
Beside the previous considerations, it is possible to perform an analysis of
the influence of specific piezoelectric coefficients on frequency spectrum shape
of the piezoceramic ring. Thereat, the more significant influence of the
observed coefficient on frequencies of the thickness or radial mode of
oscillation occurs depending on its definition, that is, crucial is influence on
that resonant mode for which the parameter is associated by its definition.
58
f [Hz]
44
xx10
10
f [Hz]
7
6
5
4
3
2
0
0.2
0
0.4
h/a b/a2
0.2
0.4
0.6
0.6
0.8
0.8
1
b/a
a1/a2
1
Figure 2.27. Frequencies of the first resonant mode of the PZT8 piezoceramic ring
in function of normalized thickness and normalized inner radius
f [Hz]
[Hz]
x 105 5
x 10
4
3
2
1
0
0
0.2
0
0.4
h/a
b/a2
0.2
0.4
0.6
0.6
0.8
0.8
1
a1/a2
b/a
1
Figure 2.28. Frequencies of the second resonant mode of the PZT8 piezoceramic
ring in function of normalized thickness and normalized inner radius
Due to massiveness this analysis is omitted too, and it must be remarked again that
all previous analyses are very sensitive to the changes of the used piezoceramic
material parameters. Influence of these parameters, especially about validity and
coefficient values available from different piezoceramic manufacturers, will be
talked about during later exposure.
2.2.5.3.3. Effective Electromechanical Coupling Factor
Effective electromechanical coupling factor keff is usually the most significant
and the most used parameter, which characterizes the efficiency of the piezoelectric
material as a transducer. This parameter may be determined the most easily by
experiment. It is used to assess the ability of the piezoelectric element to, for a
given resonant mode and given ceramic geometry, convert mechanical energy into
electric energy, and vice versa. Effective electromechanical coupling factor keff is
in range from 0 to 1, and for a good transducer is always greater than 0.5. This
factor may be, as it is known, calculated for every resonant mode using classical
relation [67]:
2
=
keff
fa2 − fr2
fa2
,
(2.41)
where for observed oscillation mode of the piezoceramic element without losses, fa
and fr represent frequencies at which input ring impedance reaches maximum, i.e.,
minimum, respectively. Using the model proposed in this chapter, applying the
equation (2.41) one may determine keff of the piezoceramic ring for every resonant
mode. Figure 2.29 presents dependence of the effective electromechanical coupling
factor in function of ratio b/a and h/a, for fundamental (lowest) resonant mode of
the ring of PZT8 material. Solid lines present effective electromechanical coupling
factors of the PZT8 ring observed in previous analyses, and all that in case of ring
with ratio b/a=15/38 and with variable thickness (h/a), as well as the same ring
with ratio h/a=5/38 and with variable ratio of inner and outer radius (b/a). As
expected, greater keff is obtained in regions in which the first resonant mode is
predominantly the thickness resonant mode, and it is the region of great lengths of
the specimen (h/a>>1), as well as in cases when the inner radius of the ring is small
(b/a≈0), that is, when ring changes into a disk. As seen from the Figure 2.29, when
b/a grows to 1, keff declines. This is because radial oscillations of the rings with
growing inner radius are substantially different from radial oscillations of disks.
Decreasing h/a, keff also decreases, because radial and thickness dimensions of the
specimen become close and significant coupling of radial and thickness oscillation
arises, which at the worst decreases effective coupling factor of the observed
thickness mode. Here is essential to emphasize that on the presented graph are
marked calculated constant values of the dynamic coupling factor keff for boundary
cases, which correspond to the values of the static coupling factors obtained by
one-dimensional analyses (kp, k31, and k33). In static conditions, spatial distribution
60
of the mechanical stress and electric field is uniform and it is possible to find
expressions for coupling factors (most often called material coupling factors) in
closed form, for any one-dimensional configuration of mechanical stress and
electric field. In the case of ring here observed, geometry of the element allows
one-dimensional observations in case of zero and infinite b/a and h/a. Dynamic
coupling factor of the oscillating ring is variable in the scope of these limits. Good
agreement of these values with calculated dynamic coupling factor is also
confirmation of the validity of the piezoceramic ring model and correctness of the
approach used in its realization. From Figure 2.29 one may see that for the case of
a very thin disk (h/a→0, b/a→0), effective dynamic electromechanical coupling
factor keff has value of 0.469. Thus is achieved good agreement with the value
obtained by division of static planar coupling factor k p =0.51 for PZT8 material
[82], which corresponds to the radial oscillation of a thin disk, with correctional
coefficient 1.13 for a thin disk [69]: k p /1.13=0.451. This correction takes into
account certain dynamic conditions in calculations. If the ratio h/a for the previous
case increases, keff grows, reaches maximum for h/a≈1 (region marked by arrow),
and then declines and inclines to a constant value keff=0.616. This value also agrees
well with one-dimensional approximation for the case of a long solid cylinder, that
is, with the ratio between static coupling factor k33=0.64 for PZT8 material [82],
and correctional factor 1.05 [69]: k33/1.05=0.610. If in case of a thin ring (h/a≈0)
b/a increases from 0 to 1, keff decreases. For b/a≈1 one gets the approximation
already explained through the model with concentrated parameters, whereat one
may see from the Figure 2.29 that this approximation is applicable already for
b/a>0.6 (keff is from that value on practically independent of b/a). In that case,
energy in the thin-walled ring is uniformly distributed and the obtained value
keff=0.302 coincides with correspondent static coupling factor defined for that
dimension ratio: k31 =0.3, for PZT8 material [82].
In Figure 2.30 is presented dependency of the effective electromechanical
coupling factor for second resonant mode in function of ring dimensions. It is
presented the region that is the most interesting from aspect of practical
applications (h/a<1). From the Figure 2.30 is obvious that keff of the second
resonant mode has the reverse trend regarding the coupling factor of the first mode,
that is, it begins to grow in those regions in which keff of the first mode begins to
decline due to the coupling with this mode. keff of the second resonant mode grows
in the regions in which the second mode is firmly coupled with the first mode and
has resonant frequencies in its vicinity. keff has the greatest values for those
dimension values, for which this second mode itself represents a thickness resonant
mode (maximal keff). It means that in some regions the second resonant mode has
greater coupling factor, although the first mode is excited at lower frequencies.
Here too is by thicker lines set aside the case of PZT8 ring with ratio b/a=15/38
and with variable thickness (h/a), as well as the case of the same ring with ratio
h/a=5/38 and with variable ratio of inner and outer radius (b/a), which is used later
in practical realization of the ultrasonic sandwich transducers, modeled in last
chapter.
0.616
kkeff
eff
0.7
0.6
0.469
0.5
0.4
0
0.3
3
0 .5
a1/a2
b/a
2
1
h/a
b/a2
0.302
0 1
Figure 2.29. Effective electromechanical coupling factor keff for the first resonant
mode PZT8 of the piezoceramic ring in function of ratios b/a and h/a
keff
keff
0.6
0.5
0.4
0.3
0.2
0.1
0
1
0
0.8
0.6
b/a2
h/a
0.4
0.2
0.5
a1/a2
b/a
0 1
Figure 2.30. Effective electromechanical coupling factor keff for the second
resonant mode of the PZT8 piezoceramic ring in function of ratios b/a and h/a
62
2.2.5.3.4. Components of the Mechanical Displacement of the Piezoceramic Ring Points
In order to further analyze the nature of the resonant modes, it is necessary to
determine the components of the mechanical displacements of the piezoceramic
ring points in radial and thickness direction.
Displacements ur and uz in r and z direction are presented through two
orthogonal wave functions (2.32), whereat, if it is assumed that piezoceramic ring
is mechanically isolated, one should apply boundary conditions on the external
surfaces that reflect this strain state without mechanical stress. Since equations
(2.32) do not fulfill these boundary conditions, it is applied the condition that
integral of the mechanical stress on every external surface is equal to zero, which
corresponds to the condition that resulting force on those contour surfaces is equal
to zero, on which base one may determine unknown constants A, B, C, and D:
∫P Trr (b)dP = 0 ,
∫P Tzz (h)dP = 0 ,
1
3
∫P Trr (a)dP = 0 ,
∫P Tzz (− h)dP = 0 .
2
(2.42)
4
By substitution of constitutive equations (2.28) into the boundary conditions
defined by equations (2.42) and using known relations (2.29), one gets an equation
system by which one may determine unknown constants A, B, C , and D. Solving
this system, with condition of sinusoidal excitation ( Dz = D0 e jωt ), one gets:
A=
k2 − k5
B,
k4 − k1
B=
k 9 − k3
C,
kb
C=
k9 kb − ka k10
,
k3 kb − ka k8
D = 0 , (2.43)
where:
J1 (kr b ) D D
c12 −c11 ,
b
Y (k b ) D D
D
k2 = 2h c11
kr Y0 (kr b ) + 2 h 1 r c12
−c11 ,
b
(
(
D
k1 = 2h c11
kr J0 (kr b ) + 2 h
D
k3 = 2 c13
sin (kz h ) ,
)
)
J1 (kr a ) D D
c12 −c11 ,
a
Y (k a ) D D
D
k5 = 2 h c11
kr Y0 (kr a ) + 2 h 1 r c12
−c11 ,
a
D
k4 = 2 h c11
kr J0 (kr a ) + 2 h
[ ( )
[ ( )
(
(
( )] ,
( )] ,
D
k6 =c13
aJ1 kr a − bJ1 kr b
D
k7 =c13 aY1 kr a − bY1 kr b
a2 − b2 D
k8 =
c33 kz cos kz h
2
( ),
k9 = 2 h h31 D0 ,
k10 =
a2 − b2
h33 D0 ,
2
)
)
(2.44)
as well as:
ka =
k2 − k5
k1 + k2 ,
k4 − k1
kb =
k2 − k5
k6 + k 7 .
k4 − k1
(2.45)
First are determined radial displacements of the ring points function of r/a for
five rings with same outer radius and different inner radius (b/a=0; 0.2; 0.4; 0.6;
0.8), and all that in case of two different ring thickness (2h=5mm and 2h=20mm).
In Figures 2.31 and 2.32 are presented these radial displacements of points between
inner and outer cylindrical contour ring surface in function of r/a, whereat by
comparison of corresponding graphs one may see influence of increase of the inner
ring opening and its thickness on radial displacements shape for the first four
resonant modes.
Presented mechanical displacements are normalized with radial displacement
of the points on internal cylindrical surface (r=b) for the first mode of the PZT8
ring with dimensions 2a=38mm, 2b=15mm, 2h=5mm, while the distances r in
radial direction are normalized with outer radius a. If, for example, one observes
the first few lowest modes of this ring (b/a=0.4), one may notice that for the first
and the third radial mode internal and external cylindrical surface move in same
directions, while at the second mode, which is also radial, observed surfaces move
in opposite directions (which means that for this resonance case at value r/a=0.7
exists circular line that represents wave node, for which is displacement ur=0), etc.
Directions of movement of quoted surfaces are presented on adequate graphs. With
increasing of the inner radius (that is, increasing b/a), are linearized displacements
of the first and the second radial mode of ceramic with 5mm thickness and all four
modes of ceramic with 20mm thickness, while in other cases displacements mostly
keep classical theoretical form of Bessel’s functions.
In Figure 2.33 are 2.34 presented thickness displacements of the points
between circular-ringed boundary surfaces for the first four resonant modes, for the
rings with same outer radius and different inner radius (b/a=0; 0.4; 0.6; 0.8), and
with same length values as in previous analysis of radial displacements (2h=5mm
and 2h=20mm). Here too are mechanical displacements of the points in thickness
direction normalized with thickness displacement of the points on circular-ringed
surface (z=h) for the first mode of the PZT8 ring with dimensions 2a=38mm,
2b=15mm, 2h=5mm, while the distances z in thickness direction are normalized
with half of thickness h. It is obvious at all presented cases, that plane, metalized
surfaces always move in opposite directions. As expected, the greatest influence on
the thickness displacement appearance has ceramic thickness, while influence of
the inner opening exists, and it is greater at thicker ceramic, while at thinner
ceramic it exists only for larger inner diameters. At thicker ceramics are also
increased cylindrical lateral boundary surfaces, so the influence of the radial
oscillations on thickness displacements due to the presence of greater lateral load is
also greater.
64
Figure 2.31. Normalized radial displacements of the points between cylindrical
surfaces for a ring with thickness 2h=5mm
Figure 2.32. Normalized radial displacements of the points between cylindrical
surfaces for a ring with thickness 2h=20mm
66
Figure 2.33. Normalized thickness displacements of the points between circularringed surfaces with thickness 2h=5mm
Figure 2.34. Normalized thickness displacements of the points between circularringed surfaces with thickness 2h=20mm
68
2.2.5.4. Experimental results
In purpose of obtaining experimental verification of the proposed model, the input
electric impedance of different piezoceramics with cylindric shape in function of
frequency is measured. These results are compared with correspondent values obtained
using the described model. Calculated and experimental results are obtained using
PZT4 and PZT8 piezoelectric rings and disks [82], with several different characteristic
dimensions. As in the case of PZT8 piezoceramic, used parameters of the PZT4
piezoceramic materials are also presented in Table 4.3. In every example piezoceramic
element is excited to oscillate supplying alternating voltage on electrodes that are
placed on the main surfaces normal to the polarization axis z. Experimental impedance
dependences of the piezoceramic specimens are measured by automatic network
analyzer (HP 3042A Network Impedance Analyzer). Figure 2.35 presents
experimental and simulated moduli of input electric impedance in dB, in function of
frequency, for three different PZT4 and PZT8 rings that oscillate in air, with following
dimensions: (a) 2a=10mm, 2b=4mm, 2h=2mm, (b) 2a=38mm, 2b=13mm, 2h=4mm,
(c) 2a=38mm, 2b=13mm, 2h=6.35mm. Form and calculated values of impedances, as
well as of the calculated resonant and antiresonant frequencies, are very close to the
correspondent experimentally obtained results, and all that as for the first radial mode
R1, so for the first thickness mode T1.
The first radial mode R1 and thickness mode T1 are the most often used modes in
practical applications. The proposed model predicts these modes with sufficient
accuracy. Thickness oscillation mode T1 is the most often used in high-frequency
applications. However, in applications essential in this analysis, one must use lower
frequencies. In that sense two solutions for obtaining of lower operating resonant
frequencies are possible: application of Langevin's sandwich tansducer, but also
application of the simple piezoceramic ring or disk, which oscillates at its first radial
mode R1. In fact, at this radial resonant mode, a significant stress exists in thickness
direction due to the elastic coupling. Analysis of such thickness oscillation of the ring
or disk is enabled by this three-dimensional model, which considers coupling of the
thickness and radial oscillations, whereby is possible to optimize the ring or disk
geometry in purpose of increase of the thickness displacement during oscillation at the
first radial mode.
In Figure 2.36 same dependences are presented for the case of PZT4 and PZT8
disks, which also oscillate in air, with dimensions: (a) 2a=50mm, 2h=3mm, (b)
2a=20mm, 2h=5mm, (c) 2a=38mm, 2h=6.35mm (case from Figure 2.35(c) without
inner opening). In case of disks, experimental and modelled dependences in the region
of the first thickness mode even better coincide than in case of rings, and results in the
region of the first radial mode are also satisfying. Also in case of piezoceramic rings,
as well as in case of piezoceramic disks, when higher radial resonant modes are
observed, modelled resonant frequencies of the radial modes are mostly greater than
measured, and rarely smaller than measured resonant frequencies. Possible cause for
that is presence of other types of vibrational modes in rings and disks, which are not
encompassed by the model: for example, edge and flexion (thickness-shear) modes
occurring between the first radial and first thickness mode are not encompassed by the
model, and their presence is noticeable especially at specimens in shape of a disk. This
may be best seen from the dependence presented in Figure 2.36(b), where some
resonant modes in the experimental characteristic in the vicinity of the modelled radial
mode R2, do not represent radial resonances. Due to the absence of these modes in the
model, modeled impedance dependence is mainly above the measured characteristic
and it does not “go down” at greater frequencies. Exception is the case of the disk from
Figure 2.36(a), where missing modes are weakly coupled with modes present in the
model and don't have great influence on the impedance characteristic. Besides that,
due to the fact that model does not consider mechanical and dielectric losses, minimal
and maximal impedance values at radial and thickness resonant frequencies are more
noticable at calculated dependences, than those at measured dependences, in all
analyzed cases. However, besides all these facts, general trend of impedance behaviour
may be noticed in all observed cases.
Results presented in Figures 2.35 and 2.36 nevertheless cannot be used for very
precise comparison of measured and theoretical results, less due to the limited
measuring accuracy, and more because only typical values for constants PZT4 and
PZT8 material are used [82]. Namely, differences obtained between modelled and
experimental dependences, even in regions around the first radial and first thickness
mode, where it was not expected, arise also due to the fact that simulated dependences
are obtained using coefficients that ceramics manufacturers give based on standard
one-dimensional
measurings. By such measurings one obtains piezoelectric
parameters that somewhat differ from their current values. It is possible, using fitting
procedure of experimental and modelled dependences to correct slightly initial
coefficients, whereby for all resonant modes encompassed by model one gets better
agreement with experiment. This optimization technique, which contains
multidimensional simplex algorithm of minimization for determination of unknown
piezoceramic coefficients, is already presented in Chapter 2.1.1. Besides that, some
ceramic manufacturers (Table 4.3) publish special piezoceramic parameters for the
finite element method, and special for analytical methods. If one applies parameters of
the Pz26 piezoceramic (Ferroperm7) for the finite element method, for rings and disks
from Figures 2.35 and 2.36, almost in all cases one gets excellent coincidence of
antiresonant frequencies, and for all resonant modes encompassed by the model, while
in the region of the resonant frequencies exists great deviation.
In literature [83] is presented Matlab software for determination of input
impedance of the piezoceramic ring from Figure 2.35(b) using the proposed threedimensional model. Other softwares, used in this chapter for three-dimensional
calculations, are similar and they are not presented in literature [83], because with
certain simple modifications presented software may easily enable modelling of all
analyzed dependences. Also, due to its massiveness, here are not presented software
used to reproduce existing one-dimensional (thickness and radial) and threedimensional models, and which are used for the purpose of comparison with the
proposed model.
7
Ferroperm Piezoceramics A/S
(http://www.ferroperm-piezo.com)
70
80
eksperiment
model
70
R1
T1
R2
60
40
30
R3
20
10
0
0
2
4
6
8
10
f [Hz]
(a)
12
14
16
18
x 10 5
80
eksperiment
model
70
R1
60
R2
T1
R4
50
Zul [dB]
Zul [dB]
50
40
R3
30
20
R5
10
0
0
1
2
3
4
f [Hz]
(b)
5
6
7
x 10 5
8
80
70
eksperiment
model
R2
R1
T1
60
Zul [dB]
50
R3
40
30
20
10
0
0
0.5
1
1.5
2
2.5
3
f [Hz]
3.5
4
4.5
5
x 10 5
(c)
Figure 2.35. Modulus of input electric impedance in function of frequency for
PZT rings: Zul=20log( z ul [Ω ] /50+1). Ri (i=1, 2, 3, 4, 5) are radial modes,
and T1 is thickness mode: (a) 2a=10mm, 2b=4mm, 2h=2mm (PZT8);
(b) 2a=38mm, 2b=13mm, 2h=4mm (PZT4);
(c) 2a=38mm, 2b=13mm, 2h=6.35mm (PZT4)
By proposed model, using matrix of dimensions 5x5, one may describe
external behavior of the piezoceramic ring in frequency domain, whereby one may
easily determine all transfer functions of the element. Model takes into account as
coupling of the radial and thickness modes, so the mechanical interactions of the
ring with surrounding, as on plane circular-ringed surfaces, as well as on
cylindrical lateral surfaces. Also, using such analysis one may determine:
frequency spectrum, mechanical displacements, and effective electromechanical
coupling factor, valid for any ratio of b/a and h/a. When the inner diameter of the
ring decreases (b→0), electric impedance of the ring coincides with the impedance
of the corresponding disk, because ring degenerates into a disk of same outer
diameter. Therefore, proposed model is general and extends the model [78]. This
new three-dimensional approach will be extended, in order to enable modeling of
more complex structures, such are classical Langevin’s sandwich transducers
(Chapter 4.3.2).
60
(a)
eksperiment
model
R1
50
T1
R2
Zul [dB]
40
30
20
10
0
0
1
2
3
4
5
f [Hz]
(a)
6
7
8
9
10
x 10 5
80
(b)
70
R1
T1
eksperiment
model
R2
60
50
Zul [dB]
72
40
30
20
10
0
0
0.5
1
1.5
2
2.5
3
f [Hz]
(b)
3.5
4
4.5
5
x 10 5
80
eksperiment
model
R1
70
60
R2
T1
R3
Zul [dB]
50
40
30
R4
20
10
0
0
0.5
1
1.5
2
2.5
f [Hz]
(c)
3
3.5
4
4.5
5
x 10 5
Figure 2.36. Modulus of input electric impedance in function of
frequency for PZT disks: Zul=20log( z ul [Ω ] /50+1).
i (i=1, 2, 3, 4) are radial modes, and T1 is thickness mode:
(a) 2a=50mm, 2b=3mm (PZT8); (b) 2a=20mm, 2b=5mm (PZT4);
(c) 2a=38mm, 2b=6.35mm (PZT4)
3. MODELING OF METAL CYLINDRICAL
RESONATORS
Analysis of oscillation of different metal elements has been usually connected
with solving some practical problem and it has been performed because of two
essential, but mutually confronted reasons. At some applications, presence of the
oscillations is undesirable, and adequate analysis should enable elimination or
decrease of undesired oscillation. On the other hand, in some devices like the
Langevin’s ultrasonic transducers, one tends to obtain as great amplitudes of
mechanical oscillations as possible at its operating, emitting end, so in those cases
an analysis of oscillation is also necessary for their proper design and function. In
numerous applications of power ultrasound different metal waveguides are
designed to oscillate at resonant conditions, because great output oscillation
amplitudes are necessary (even several tens of µm). In most number of
applications, metal waveguides are solid cylinders, which oscillate at fundamental
thickness resonant mode, as half-wave resonators. On one end of the metal
waveguide is connected an ultrasonic transducer, which excites waveguide
oscillation, and on the other, operating end of the metal waveguide is emitted
ultrasonic power. In order to obtain as great efficiency as possible, that is,
increasing of emitted ultrasonic power, it is necessary that excitation transducer
and metal operating tool oscillate at same frequency. This is the first reason
because determination of the resonant frequencies of metal waveguides became
essential, especially in the design procedure of ultrasonic devices for welding.
Beside such “independent” oscillation of metal operating tools, metal endings are,
as mentioned several times, also consisting parts of complex, excitation ultrasonic
sandwich transducers. In such cases, resonant frequency of the complete ultrasonic
transducer is different from resonant frequency of a single metal ending. Because
of that, determination of resonant frequencies (modeling) of single metal endings is
an initial condition for design of more complex ultrasonic sandwich transducer.
There are a few rules during design, about which one needs to take care. First,
demanded operating resonant frequency, and material of which the ending is made
(metal solid cylinder or ring), determine its overall dimensions, of which is crucial
the ending length. In addition, distribution of mechanical stresses in points along
the ending must be such to guarantee that its operating lifetime is as long as
possible. In many applications is also demanded an amplification of oscillation
76
Modeling of Metal Cylindrical Oscillators
amplitudes on the operating end of the ending. The most significant parameters that
characterize metal parts in ultrasonic resonators are resonant frequency and
corresponding resonant mode.
3.1. ONE-DIMENSIONAL MODELS OF METAL RESONATORS
Analysis of oscillation of metal cylindrical resonators is treated in detail in
literature. As noticed, studies of oscillation of cylinders of finite length were most
often related to solving of some practical problem. Application of metal cylinders
in ultrasonic sandwich transducers demands knowing of their characteristics in the
frequency range that is of interest, that is, in range in which cylinders have resonant
modes of oscillation, and where they cannot be represented by concentrated masses
anymore. Besides that, second basic demand is need for knowing the types of
resonant modes that are excited (radial, thickness or longitudinal, flexural or
transversal, edge, torsional, etc.). Recognizing of useful resonant modes in overall
frequency range is a basic condition for accessing to design of efficient ultrasonic
transducers.
3.1.1. Analysis of Oscillation of Long Half-wave Resonators
If the lateral dimensions of the resonator small comparing the wavelength, its
oscillation is determined by one-dimensional wave equation of longitudinal wave
propagation in a long waveguide (bar), and solutions of this equation are available
for several shapes (profiles) of resonators. Analytical solutions may be derived for
exponential, conical, or catenoidal resonators, while for some shapes it is not
possible to determine analytical solutions, so for their analysis are used numerical
procedures. In most applications, metal resonators are solid cylinders with constant
circular cross-section, which oscillate at fundamental longitudinal mode (half-wave
resonators). This case of a metal resonator, which will be analyzed further, is
presented in Figure 3.1.
uz(z)
Tzz(z)
z
l
Figure 3.1. Half-wave resonator of length l, with constant cross-section
Assuming that the resonator is made of isotropic material, that wave propagation
is without losses, as well as in condition of linear elasticity and uniform propagation
on the cross-section of the resonator, wave equation for propagation of longitudinal
oscillations in direction of z-axis of the resonator is [53]:
∂ 2 uz (z, t )
77
∂ 2 uz (z, t ) ,
(3.1)
∂t 2
∂z 2
where uz (z, t ) is displacement in z direction, and it is both function of time t and
coordinate z. v0 is velocity of longitudinal waves propagation in a thin cylinder (bar):
v0 = EY / ρ . Solution of equation (3.1) in case of harmonic oscillation amounts to:
(
= v02
)
u z (z, t ) = A1e − jk0z + A2 e jk0z e jωt = u z (z ) e jωt ,
(3.2)
where A1 and A2 are constants, ω is circular frequency, and k0=ω/v0 is axial wave number.
In further consideration is of interest only the time-independent part of the
equation (3.2), that is, displacement uz(z). For the half-wave resonator from the
Figure 3.1, boundary conditions are obtained from the assumption that ends of the
resonator have zero mechanical stresses, that is:
duz ( z)
duz ( z )
= 0 and
=0.
(3.3)
dz z =0
dz z =l
According to that, displacement function uz(z) may be presented in the following form:
uz (z ) = uzm cos(k0 z ) ,
(3.4)
where uzm is maximal amplitude of displacement at the ends of the cylinder. This
resonant oscillatory mode is called fundamental longitudinal mode of oscillation.
Based on expressions (3.3) and (3.4) one may obtain the frequency equation k0l=π,
which links the resonator length l and resonant frequency f in a following way:
π π v 0 v0
,
(3.5)
l=
=
=
k0
ω
2f
where ω=2πf. Length l is often presented as λ/2, where λ is wavelength.
Mechanical stress in z direction Tzz(z) is associated with longitudinal strain in z
direction Szz, that is, with displacement uz(z) in a following way:
du z (z )
Tzz (z ) = EY S zz ( z ) = EY
.
(3.6)
dz
By substitution of v0 = EY / ρ , where ρ is material density of the resonator
and EY its Young’s modulus of elasticity, equation (3.6) becomes:
Tzz (z ) = −ωρv0 uzm sin(k0 z ) .
(3.7)
In contrast to the cosine function of mechanical displacement, which has maximums at
the cylinder ends, maximal value of the sine mechanical stress is placed in the middle of the
cylinder (z=l/2), where the zero mechanical displacement is, that is, the nodal plane of the
cylinder oscillation. Maximal value of the mechanical stress in the cylinder depends of
frequency, material properties, and maximal amplitude of mechanical displacement. At
distance z along the resonator, velocity of points (particles) vz(z,t) in z direction is defined in
the following way:
∂u (z, t )
= vz (z )e jωt = jωuzm cos(k0 z )e jωt .
v z (z, t ) = z
(3.8)
∂t
At distance z, axial force of extension F(z) is defined as:
F (z ) = P Tzz (z ) ,
(3.9)
where P is the area of the cylinder cross-section.
78
The characteristic that is the most significant in analysis of wave propagation
in solid materials is mechanical impedance Zm(z), and it is defined as a relation of
force F(z) and point velocity vz(z) for a given cross-section:
F (z )
Z m (z ) =
.
(3.10)
v z (z )
Mechanical impedance has zero value on the free ends of the cylinder (F(z)=0
for z=0 and z=l), and becomes infinite in the nodal plane (for z=l/2), where vz(z)=0.
Using expressions (3.8) and (3.9), Zm(z) becomes:
Zm (z ) = jρPv0 tg(k0 z ) ,
(3.11)
which is identical to the input impedance of the short-circuit (unloaded) transfer
line with characteristic impedance Zc=ρPv0 and velocity v0. Such way of modeling
of thin metal resonators using transfer line is often used for one-dimensional
modeling of metal endings in complete ultrasonic sandwich transducers. Further
will be shown that such way of modeling of metal cylindrical resonators, at which
dimensions in transversal (radial) direction are greater than its length (thickness), is
neither suitable for analysis of such metal resonators, nor for analysis of complete
ultrasonic sandwich transducers with such short endings.
Analysis presented here is valid only in ideal case when mechanical
displacements are uniform on the cross-section of the resonator (cylinder). In real
cases, if wavelength is not much greater comparing with dimensions in radial
direction of the cylinder, wave propagation is not uniform, but distorted due to the
displacements in radial direction generated by Poisson’s effect, that is, in direction
normal to the sense of wave propagation. It leads to non-uniform output amplitudes
of oscillation, which will be more discussed later.
3.1.2. Rayleigh’s Correction of the Wave Propagation Velocity
As already mentioned, considerations presented in the previous section are of
approximate nature, where the basic assumption is that after the passing of the
wave cross-sections remain planar, changing only their dimensions and positions,
but with still present parallelism of those sections. Increase of calculation accuracy
of obtained solutions lead to major mathematical limitations and difficulties due to
the nonnegligible resonator diameter, so that further more accurate solutions
reduced to relatively simple calculations for metal bars with simple cross-sections,
most often with circular shape. From the mentioned analyses the most significant is
analysis of propagation velocity decrease of longitudinal waves in cylinders with
nonnegligible diameter. Considering the propagation velocity decrease of
longitudinal waves with increase of cylinder diameter, Rayleigh proposed a
formula that enables certain corrections of that velocity, due to the mentioned
effects of mechanical displacements in radial direction [84]. Therefore, due to the
Poisson’s effect and radial movement that is annexed to the longitudinal wave
propagation, a decrease of longitudinal wave propagation velocity occurs. This
79
formula at finite diameters increases the accuracy of velocity calculation, only in
cases when the ratio of diameter and wavelength is d/λ < 0.4.
Namely, Rayleigh showed that displacement in radial direction generates decrease
of resonant frequency of the oscillating cylinder. For a given cylinder of length l, based
on the relation between circular frequency, wave propagation velocity, and length
(ω=πvz/l), decrease of the resonant frequency is physically identical to the wave
propagation velocity decrease vz. Corrected value of this velocity for cylindrical
resonator with present displacements in radial direction amounts to:
v z = v0
1
,
(3.12)
1 2 2 2
1 + υ k0 d
8
where υ is Poisson’s ratio of the cylinder material, and d its diameter.
Equation (3.12) may be linearized due to the fact that the member 1 υ 2 k02 d 2 is
8
much smaller than the half-wave resonator length (d<<l). Thus, expression (3.12)
may be written in form of:
1
⎛
⎞
(3.13)
v z = v0 ⎜1 − υ 2 k02 d 2 ⎟ ,
⎝ 16
⎠
which represents Rayleigh’s correction of longitudinal wave propagation velocity in a
cylinder of finite diameter, for any ratio of cylinder length and diameter. Similar
correctional formula also exists for propagation velocity of radial oscillations of a disk.
In purpose comparison with other theories, expression (3.13) is often presented in
literature in dimensionless notation, linking length l and diameter d of the cylinder.
Based on the expression for wave number k0=ω/v0 and expression for cylinder length
that oscillates at longitudinal mode l=vz/(2f), dimensionless wave number k0l may now
be linked with dimensionless wave number k0d in a following way:
⎡ ⎛ υ k d ⎞2 ⎤
(3.14)
k0 l = π ⎢1 − ⎜ 0 ⎟ ⎥ .
⎢⎣ ⎝ 4 ⎠ ⎥⎦
In later exposure will be performed comparison of the equation (3.14) with
experimental results, as well as with results obtained based on more accurate,
three-dimensional analyses of longitudinal wave propagation velocity in cylinders.
3.1.3. Experimental Studies of Metal Cylinder Oscillation
At the very beginning of appearing of theoretical analyses from this field, the
only experimental paper in literature about oscillation of cylinders of finite length
was published by McMahon [85]. By these measurements were encompassed
twenty lowest resonant modes of oscillation of aluminum and steel cylinders (that
are most often used in power ultrasound technique) with ratio of length and
diameter ranging 0<l/d<1.7, that is, for normalized wave numbers ranging
1.2<k0d<6.2. Namely, as mentioned, besides that characteristic dimensions of the
cylinder may be presented in dimensionless notation (l/d), by using the wave
(characteristic, eigenvalue) number k0=ω/vz one may also define dimensionless
80
(normalized) wave numbers k0d and k0l. Quoted fields and used materials almost
completely cover fields and materials that are of interest in design of ultrasonic
cylindrical resonators (and transducers).
In Figure 3.2 is presented part of the recorded experimental frequency
spectrum for an aluminum cylinder with following parameters: υ=0.344 and
v0=5150m/s. Presented longitudinal resonant modes are symmetric (uz(z)=uz(-z)) or
antisymmetric (uz(z)=-uz(-z)) regarding the middle (central) plane of the cylinder.
McMahon denoted symmetric modes by even numbers, and antisymmetric modes
by odd numbers.
k0d
l
Figure 3.2. Lowest branches of the experimental
resonant
d
frequency spectrum of McMahon, for an aluminum cylinder
These results may be further analyzed from the aspect of application of solid
cylinders of finite length in the field of power ultrasound. Longitudinal mode of the
cylinder, presented in Figure 3.2 by the spectrum branch 2, is the only one that
fulfills the condition that oscillation amplitude of the output surface is as more
uniform as possible, because McMahon affirmed, analyzing the output surface
displacement, that only in that case on the output surface do not exist nodal lines.
For small values of ratio l/d (l/d<0.2), this mode corresponds to the radial mode of
oscillation of a thin disk (k0d≈4.4). For great values l/d (l/d>1.5) this resonant
mode represents a half-wave longitudinal mode of a thin bar. In the range that is of
interest in this consideration (0.6<l/d<2), resonant frequency of the longitudinal
mode of the cylinder with dimensions from that range may be graphically
determined based on the Figure 3.2.
Based on the presented spectrum, one may notice some dimensions of the
cylinder for which the branch of the resonant mode 2 intersects other spectrum
81
branches, that is, dimensions for which occurs interference of these modes in the
cylinder. Based on the Figure 3.2, it is obvious that for l/d≈0.77 mode 2 is strongly
coupled (it interferes) with mode 1. It means that, if one realizes a metal resonator with
this dimension ratio, these two resonant frequencies will be very close, and in practice
can be hardly distinguished. One may conclude that there is no value l/d for which the
difference between resonant frequency of the longitudinal mode and any other
resonant frequency of other modes is greater than 10%. If the resonant frequency of
any undesired mode for a concrete resonator is too close to the longitudinal resonance
(especially if the difference is smaller than 5%), such experimental frequency spectrum
must be used in order to determine those dimensions of the resonator that will enable
avoiding of this situation.
Until then there hasn’t been existed an adequate theoretical analysis that would
predict the resonant (eigenvalue) frequencies of finite length metal cylinders with
satisfying accuracy, with ratio of diameter and length (d/l) less than 1. The only
exception is the previously analyzed Rayleigh’s correction (3.14), which is derived for
a part of the spectrum branch 2 from Figure 3.2, and which is on the same figure
presented by dashed curve A (curve B represents Rayleigh’s correction of the radial
oscillations of a thick disk). Namely, The greatest practical interest have cylindrical
resonators at which is the ratio of diameter and length in range from 0 to 1. Potrebno je
bilo razmatrati i prisustvo ostalih rezonantnih modova koji bi mogli da budu spregnuti
sa debljinskim modom. Primary goal was to find a formula that would serve for
calculation of resonant conditions for thickness mode of oscillation of such cylinders.
It was also necessary to consider the presence of other resonant modes that could be
coupled with thickness mode. That was the reason to approach to more complex
theoretical analyses of oscillation of metal cylinders, which are presented shortly in the
following chapters.
3.1.4. Method of Seeming Elasticity Moduli
In design of ultrasonic sandwich transducers, which are here subject of later
considerations, the greatest interest is resonant mode of longitudinal oscillations in
metal cylinders. Simple, approximate theory for determination of this mode of
oscillation for a given cylinder, which is based on assumptions introduced by Mori
[86], enabled analysis of the observed mode through the coupling of solution of wave
equation for longitudinal oscillations in a long cylinder with negligible cross
dimensions and equation for radial oscillations in a thin disk. Thus originated method
of seeming elasticity moduli. This method substantially improved one-dimensional
model of a line, which was mostly used for modeling of metal cylinders. In this
approach, one starts from the assumption that mutual coupling of oscillations generates
changes of the elasticity modulus. It is used in analysis of oscillation of thick and thin
metal disks and rings, with any outer or inner diameter. Besides that, this method is
used in design of ultrasonic devices for changing of oscillation direction [87].
Oscillations of the points of contour circular-ringed (plane) surfaces in longitudinal
direction and oscillations of the points on contour cylindrical lateral surfaces in radial
direction are in mutual antiphase. In that case, seeming elasticity modulus (EYz) in
82
longitudinal (z) direction is smaller than Young’s elasticity modulus (EY). Therefore
longitudinal strain in z direction (Szz) contains not only the strain induced by
longitudinal mechanical stress (Tzz), but also the strain generated by Poisson’s effect
generated due to the radial oscillation. Accordingly, longitudinal strain in presence of
radial oscillation is greater than in case without radial oscillation. As a result of this,
seeming elasticity modulus (EYz=Tzz/Szz) is smaller than Young’s elasticity modulus
(EY). Basically, this decrease of the elasticity modulus is analogous to the decrease of
wave propagation velocity. In the same way one may define the seeming elasticity
modulus in radial direction (EYr). It must be remarked that seeming elasticity moduli
are now functions of geometric dimensions of the oscillator. Associating the frequency
equations in longitudinal and radial direction one gets resonant frequencies in function
of oscillator dimensions, which agree much better with experimental results for the
fundamental resonant mode than results obtained using classical one-dimensional
theory. As already remarked in the introduction, here is the mentioned method
extended on piezoceramic rings too, so that in the last chapter, dedicated to modeling
of sandwich transducers using this method, is realized the model of a complete
ultrasonic sandwich transducer (Chapter 4.2.5).
In extension will be presented Mori’s approximate theory for determination of
resonant conditions for longitudinal oscillation mode of a solid cylinder, in broad
range of cylinder length and diameter ratios, and there will be derived a simple
formula, which proved to be of great practical significance. Application of this method
at metal rings will be presented in Chapter 4.2.5, altogether with analysis of complete
ultrasonic sandwich transducers using this method. As remarked, Mori’s theory is
based on the assumption that current resonant mode of the longitudinal waves, in the
cylinder with diameter and length ratio close to one, may be observed through
interaction of two orthogonal waves. One is longitudinal wave in the thin bar, and the
other is radial extensional wave in the thin disk. Interaction of the two waves is
realized by introducing of wave coupling factor nS, which is based on certain
assumptions for mechanical stresses in the cylinder, as it is explained in extension.
It has been already shown that resonant length of the cylinder with small diameter
is given by expression k0l=π. Wave equation for harmonic oscillation of a thin disk is:
d 2 ur 1 dur ⎛ 2 1 ⎞
(3.15)
+
− ⎜ kr + 2 ⎟ur = 0 ,
dr 2 r dr ⎝
r ⎠
where ur is displacement in radial direction. Wave number for radial oscillations kr
is defined as:
kr = ω
(
ρ 1 −υ 2
).
(3.16)
EY
Analysis of the fundamental extensional mode of axisymmetric radial
oscillations of a thin disk leads to the following equation:
kr d ⎛ kr d ⎞
⎛k d⎞
(3.17)
J0 ⎜
⎟ = (1 − υ ) J1 ⎜ r ⎟ ,
2
⎝ 2 ⎠
⎝ 2 ⎠
where J0 and J1 are Bessel’s functions of first order, zero and first rank. First root
of the equation (3.17) is:
83
kr d
(3.18)
= α'' ,
2
whereat solution for α’’ depends of Poisson’s ratio, and it may be approximated by
expression:
α ' ' = 1.84 + 0.68υ .
(3.19)
Now is necessary to link somehow the solution of the equation k0l=p with the
solution of the equation (3.18). It is known that for derivation of equation k0l=π is
assumed that in thin cylinder both the radial and the tangential mechanical stresses are
equal to zero. Mori assumed here that with increase of diameter also grow these
mechanical stresses, and that they may be then approximated by following
expressions:
1
1
T zz and Tθθ =
Tzz .
nS
nS
Applying the Hooke’s law, mechanical strain Szz in z direction is then:
⎛
⎞
1
[Tzz − υ (Trr + Tθθ )] or S zz = 1 Tzz ⎜⎜1 − 2υ ⎟⎟ .
S zz =
EY
EY
⎝ nS ⎠
Trr =
(3.20)
(3.21)
Thus one may define equivalent elasticity modulus for oscillations in direction
of axis z, which Mori named seeming elasticity modulus:
−1
⎛ 2υ ⎞
,
(3.22)
⎜
⎟
EYz = EY ⎜1 −
⎟
⎝ nS ⎠
which is equivalent to the defining of wave propagation velocity decrease. Corrected length
of the resonator, due to the seeming elasticity modulus may be obtained from equation:
−1 / 2
⎛ 2υ ⎞
.
(3.23)
⎟⎟
k 0 l = π ⎜⎜1 −
⎝ nS ⎠
On the other hand, for pure radial oscillations of a thin disk, in onedimensional theory is implied that mechanical stress is Tzz=0. Here also Mori
adopted an approximation for mechanical stress Tzz in disk at which thickness is
increasing, using the same wave coupling factor nS:
(3.24)
Tzz = n S Trr .
Using the Hooke’s law, mechanical strains in radial and tangential direction are:
1
[Trr (1 − n S υ ) − υ Tθθ ] , Sθθ = 1 [Tθθ − υ Trr (1 + n S )] . (3.25)
S rr =
EY
EY
Mechanical stress in radial direction Trr is in function of Srr and Sθθ, and is mount to:
[
]
Trr = EY (S rr + υ S θθ ) 1 − υ 2 − n S υ (1 + υ ) .
(3.26)
Seeming elasticity modulus for radial oscillations is defined by comparison of
the previous expression for strain Trr with equation for case in which is Tzz=0. For
the case of a thin disk (Tzz=0 and nS=0) radial mechanical stress becomes:
(
)
Trr = EY (S rr + υ S θθ ) 1 − υ 2 .
(3.27)
Combining equations (3.26) and (3.27) one gets seeming elasticity modulus
for radial oscillations:
84
)[
(
]
EYr = EY 1 − υ 2 1 − υ 2 − n S υ (1 + υ ) .
(3.28)
The corrected value of the cylinder diameter may now be obtained by
substitution of expression (3.28) into (3.18). It is often more convenient to express
the dimensionless wave number using wave number k0 instead of kr. Based on
equations (3.18), (3.27), and (3.28) one gets:
k0 d
−1 / 2
.
(3.29)
= α ' ' 1 − υ 2 − n S υ (1 + υ )
2
By elimination of the coupling factor nS from equations (3.23) and (3.29), one
finally gets the relation between length l and diameter d of the cylinder, for a
distinct resonant frequency and known Poisson’s ratio:
2
2 ⎛ k0 d ⎞
1− 1−υ ⎜
⎟
2
2α ' ' ⎠
⎛ k0l ⎞
⎝
.
(3.30)
⎜
⎟ =
2
⎝ π ⎠
k
d
⎛
⎞
1 − ⎜ 0 ⎟ 1 − 3υ 2 − 2υ 3
2
'
'
α
⎝
⎠
Results obtained based on the method of seeming elasticity moduli using the
last equation show deviations ranging 3÷6% regarding the McMahon’s results from
Figure 3.2. Solution of the equation (3.30) has two asymptotic values (k0d/π→0
represents oscillation of a thin bar, while k0d/π→1.41 produces radial oscillations
of a disk). Applying the Mori’s theory, one gets too short cylinder when one
designs the cylindrical ultrasonic resonator directly, which may lead to too high
resonant frequencies. However, this method approximates experimental results
pretty well, and, which is the most important, it is easy for application in practical
realizations. Based on the obtained dependency (3.30), one may also analyze the
influence of the Poisson’s ratio on cylinder dimensions, whereat is essential to
emphasize that it significantly affects the solutions of the equation (3.30), as well
as the effect of the wave propagation velocity change v0 at defined frequency.
−1
[
]
(
)
(
)
3.2. THREE-DIMENSIONAL MODELS OF METAL RESONATORS
In former ultrasonic practice have been used metal cylindrical endings with or
without openings, whereat, at present internal opening, in calculations have been
neglected influences of its dimensions, especially metal endings that oscillate at
thickness resonant frequency. In this chapter will be shown that this influence is
not negligible. If the radial dimensions of the ending are smaller than one-fourth of
the wavelength of thickness oscillations, radial oscillations are negligible. Then one
may, based on the one-dimensional wave equation of longitudinal oscillations
propagation in long cylinder (bar), find simple analytical solutions for
determination of resonant frequencies. As already mentioned, in this approach are
point displacements in thickness direction uniform on the cross-section of the
oscillator. If the wavelength may be compared with dimensions in radial direction,
wave propagation is distorted (deformed) due to the influence of the radial
displacements generated by Poisson’s effect, which leads to nonuniform
amplitudes of thickness oscillation. Analysis of oscillation of metal cylinders must
then be performed using three-dimensional models.
85
3.2.1. Hutchinson’s Theory of Oscillation of Metal Cylinders
In general case, if one wants a complete analysis of oscillation of elastic cylinders,
it is necessary to solve partial differential equations of linear elasticity theory, in
function of three spatial coordinates and time, with correspondent boundary (contour)
and initial conditions. Treated material is isotropic, and wave propagation is without
losses and uniformly on the cross-section of the oscillator. Such analysis doesn’t
represent a problem if the cylinder is of infinite length. Solutions of the mentioned
equations were first formulated Pochhamer (1876) and Chree (1884), and these
equations were first applied by Bancroft [88]. At most applications, metal cylinders
oscillate at fundamental thickness resonant mode. When cross (radial) dimensions are
not small enough comparing with wavelength, wave propagation is not uniform on the
cross-section normal to the sense of wave propagation. During design of oscillator this
leads to nonuniform output amplitudes of oscillation. Besides that, wave propagation
velocity (v) decreases due to this dispersion effect. For some values of Poisson’s ratio
(υ) Bancroft presented the wave propagation velocity decrease as a function of
diameter (d=2a) and wavelength (λ) for an infinite cylinder.
Detailed analysis of solving the Pochhamer-Chree equations of movement of solid
cylinder with infinite length in cylindrical coordinate system, is presented in literature
[53]. With assumption of zero mechanical stresses on cylindrical (circumferential)
surface, these equations are reduced to simpler expressions, which may be presented
through one dependence, that is, through the following characteristic equation:
4k 2 µ J 0 ' (α a ) J1 ' (β a ) −
(3.31)
⎤ ,
⎡
⎛
λ ω 2ρ
ρω 2 ⎞⎟
− ⎜ 2k 2 −
J1 (β a ) ⎢ 2 µ J 0 ' ' (α a ) − m
J 0 (α a )⎥
⎜
µ ⎟⎠
λm + 2 µ
⎝
⎦⎥
⎣⎢
where λm and µ are Lame’s coefficients, J0 and J1 are Bessel’s functions, and J1’
and J0’’ their derivatives with respect to time t, and :
α2 =
ρω 2
λm + 2 µ
− k2,
β2 =
ρω 2
− k2 .
µ
(3.32)
Final result of such analysis is dependence of longitudinal wave propagation
velocity decrease with increase of ratio a/λ. Approximation is insofar closer as smaller
is the ratio of element radius and wavelength a/λ and it may be seen in Figure 3.3. In
Figure 3.3 are presented longitudinal velocities for the first three resonant modes, as
solutions of the Pochhammer-Chree equations for a solid steel bar of infinite length
(υ=0.29). Solutions (v=ω/k) are normalized with the solution for the longitudinal
velocity of the wave in a thin cylinder v0 = EY / ρ . In purpose of comparison, in the
same figure is presented by dashed line longitudinal velocity for the first resonant
mode of the same steel bar, which is obtained using Rayleigh’s correction of that
velocity presented by equation (3.13). As mentioned earlier, this correction is valid in
the limited region a/λ, that is, for small values of cylinder diameter.
86
1.4
1.2
v /(ρ/EY)-0.5
1
0.8
0.6
0.4
Rayleighova
aproksimacija (3.13)
Rayleigh’s
approximation
0.2
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
a/λ
Figure 3.3. Longitudinal wave velocity for the first three resonant modes
in infinite steel bar
Accurately solving of the equation (3.31) leads to very complicated
dependences, with minor practical application. Besides that, if the radius a is small
comparing the wavelength (αa<<1 and βa<<1), then Bessel’s functions may be
expanded into series. If in that expanding members αa and βa with power greater
than one are not taken into account, equation (3.31) reduces to the following
relatively simple form:
⎧⎪
⎫
⎡
λm ρω 2 ⎤
2⎪
2 2
µ ⎨ 2k 2 − µ ⎢α 2 +
(3.33)
⎥ − 2k ⎬ βa − 2k α = 0 ,
(
)
+
µ
λ
2
µ
⎪⎩
⎪⎭
m
⎣⎢
⎦⎥
whereat is then v = v0 = EY / ρ , that is, thus one obtains the expression for
velocity v identical to the expression for longitudinal wave velocity in infinite bar
v0, using previously described one-dimensional approximate method. If in
expanding of the Bessel’s functions are also taken into account members αa and βa
of second order, one gets that:
2
E ⎛ 1
⎞
v 2 = Y ⎜1 − k 2 a 2υ 2 ⎟ ,
(3.34)
ρ ⎝ 4
⎠
which is identical to the expression (3.13) that represents Rayleigh’s correction of
longitudinal wave propagation velocity in cylinder of finite diameter.
Introduction into serious and detailed analysis of axisymmetric oscillations of
finite cylinders is represented by the theory of Hutchinson [89]. His approach is based
on the choice of mechanical displacements in a form of function series with unknown
coefficients, which fulfill fundamental wave equations and boundary conditions.
(
)
87
However, even for the simplest cases of solid cylinders, calculations using this method
are complicated and lasting. One good method for obtaining of a similar approximate
solution is presented in the paper of Rumerman [90], and by it one may determine
resonant frequencies, as for solid cylinders, so for the cylinders with central openings.
The method is also based on expanding of mechanical displacements into a function
series, whereat displacements correspond to the resonant modes that may be expected
in cylinders. The first paper related to the analysis of unsymmetrical (antisymmetric)
oscillations of finite cylinders was published by Rasband [91], whereat numerical
results were not available in it. Finally, the complete description of symmetric and
unsymmetrical oscillation of metal cylinders was again published by Hutchinson [92].
Numerical results showed complete agreement with experimental results of McMahon
[85]. This approach is completely reproduced in extension and it will be used for
comparison with the method for analysis of oscillation of metal (and even
piezoceramic) cylinders proposed in Chapter 3.2.4, as well as for verification of
performed experimental results for solid metal cylinders.
As already explained, presented Pochhammer-Chree solution of general
equations of linear elasticity (3.31) stands only for an infinite long circular bar,
with zero mechanical stresses on the circumferential surface. This solution does not
include existing of plane circular-ringed ends of the bar, which are also without
mechanical stresses. If the expressions (3.32) are written in following form:
1 − 2υ ,
(3.35)
α r2 + β r2 = Ω 2 and α r2 + δ r2 = Ω 2
2(1 − υ )
where dimensionless wave numbers αr=ka, βr=βa, δr=αa are associated with
dimensionless circular frequency Ω = (ω a ) / µ / ρ , and if one applies generally
known expressions for Bessel’s function derivatives, characteristic equation (3.31)
may be also written in the form (3.36), which also often occurs in literature from this
field:
(
)
4αr2 βr δ r J0 (βr )J1 (δ r ) − 2δ r Ω2 J1 (βr )J1 (δ r ) + α r2 − βr2 J1 (βr )J0 (δ r ) = 0 . (3.36)
Hutchinson’s basic idea was that, nevertheless, the solution of the
characteristic equation for a cylinder of infinite length may be also extended to the
case of finite length cylinder if one introduces appropriate boundary conditions on
the plane (circular-ringed) ends of cylinder, that is, if one considers that axial
(longitudinal) mechanical stress and radial displacement on cylinder ends are equal
to zero. Therefore, starting from the Pochhamer-Chree equations, Hutchinson
realized a method for analysis of oscillation for a finite cylinder without existing of
mechanical stresses on any boundary (contour) surface.
Determination of the resonant frequency spectrum is performed using the
exact solutions of general equations of linear elasticity, obtained for the case of
infinite cylinder with zero mechanical stresses on the cylindrical (circumferential)
surface, and those solutions represent radial and axial displacement, as well as the
axial, radial, and tangential mechanical stress. These variables are further chosen in
form of series, which part by part fulfill mentioned boundary conditions that axial
(longitudinal) mechanical stress and radial displacement on the cylinder ends are
equal to zero. This leads to the eigenvalue matrix, whose dimensions are
determined by number of members of every series:
2
88
Bmn ⎤
⎡ Am
⎢
⎥
A=⎢
O
(3.37)
⎥.
⎢⎣ Amn
Bm ⎥⎦
Coefficients of the eigenvalue matrix are transcendent functions of frequency and
they are given by following expressions (Ji represent Bessel’s functions of first
rank, order i):
⎡
J (β ) ⎤
⎛ d ⎞ ⎧⎪
Am = ⎜ ⎟ ⎨4α r2δ r J1 (δ r ) ⎢ J0 (β r ) − 1 r ⎥ +
βr ⎦
⎝ 2 ⎠ ⎪⎩
⎣
(
)
+ α r2 − β r2 J1 (β r )
(
)(
[(α
2
r
)
⎡ 4α 2 β 2
β 2 − α r2 2δ r2 − Ω 2
Bmn = 2⎢ 2 r r2 + r
α 'r2 −δ r2
⎣⎢α 'r − β r
(
[
]
− β r2 J0 (δ r ) + 2δ r J1 (δ r ) ⎫⎪
⎬,
βr
⎪⎭
)(
)⎤⎥δ
⎦⎥
r sin
⎡ 4α 2 β 2
β 2 − α r2 2δ r2 − Ω 2
Amn =⎢ 2 r r2 + r
α 'r2 −δ r2
⎣⎢ α 'r − β r
(
)
(β r d / 2 )sin(δ r d / 2 ) / β r ,
)⎤⎥ δ J (δ )J (β ) / β
⎦⎥
r 1
r
1
r
r
(3.38)
,
]
Bm = 4α r2δ r sin (δ r d / 2 ) cos(β r d / 2 ) + β r2 − α r2 2sin (β r d / 2 ) cos(δ r d / 2 ) / β r / 2 ,
with an exception that coefficient A0 has doubled value than the general expression
for Am, given in (3.38). In the previous expressions are omitted some indexes for
simplicity. In equations (3.38a) and (3.38d) indexes for αr, βr and δr are m. In
equations (3.38b) and (3.38c) indexes for αr, βr and δr are n, and index for αr’ is m.
αr’ are defined as follows:
- in (3.38a) are αrm=2mπ/d, m=0, 1, 2, … ,
- in (3.38b) αrn are roots of the function J1(αrn), αrm’=2mπ/d ,
- in (3.38c) αrn=2nπ/d, αrm’ are roots of the function J1(αrm’) ,
- in (3.38d) αrm are roots of the function J1(αrm), including αrm=0 .
In equation (3.37) determinant of the matrix A must be equal to zero, wherefrom
one determines eigenvalue frequencies of a finite cylinder. Coefficients of the matrix
for known δr and υ are functions only of dimensionless circular frequency Ω, which is
obvious based on expressions (3.38) and (3.35). Eigenvalue frequencies are
determined by finding the zero values of the determinant, using high-speed computers.
Satisfying accuracy of final results is obtained using matrix with dimensions 20x20. In
purpose of completion of the frequency behavior of the finite length cylinder, lastly is
determined the resonant frequency spectrum of a steel cylinder in function of its
length. It must be remarked that here expressions for mechanical displacements
include two types of motion, that is, symmetric and antisymmetric movement of the
cylinder surfaces, which correspond to even and odd resonant modes of the cylinder.
Here are further determined both the even and the odd resonant modes.
In Figure 3.4 are presented previously quoted approaches in modeling of metal
cylinders in case of a steel cylinder (υ=0.29), where the dimensionless circular
frequency Ω is given in function of ratio of ending length l and its diameter d. In
89
this figure are first presented solutions of general equations of linear elasticity for a
free cylinder of finite length, obtained by the previously described Hutchinson’s
method of assuming solutions in form of series. Accuracy of the presented
solutions depends of the number of series members. Here are presented
fundamental (first) even and odd resonant mode, obtained by Hutchinson’s
method, and those resonant modes correspond to the spectrum branches 1 and 2
from the McMahon’s experimental Figure 3.2, respectively. Besides that, here are
presented analogous results obtained using the solutions of the Pochhammer-Chree
equations (3.36) for a steel cylinder of infinite length, as well as the resonant
modes obtained using one-dimensional line model, by finding the solution of the
equation for mechanical impedance (3.11), in case of Zm=0 and Zm→∞.
4
3.5
osnovni
fundamental
parni
mod
even mode
3
2.5
Ω 2
osnovni
fundamental
neparni
mod
1.5
odd mode
1
one-dimensional
theory teorija
jednodimenzionalna
Pochhammer-Chree
solution
Pochhammer-Chree
re{enje
Hutchinson’s
method
Hutchinsonov
metod
0.5
0.5
1
1.5
l/d
2
2.5
3
Figure 3.4. Comparison of different methods of resonant frequency spectrum
determination of a steel cylinder, in function of its normalized length
From Figure 3.4 is obvious that, for cylinders of great length, different presented
solutions asymptotically approach, that is, at such cylinders determination of resonant
frequencies of some modes is not critical in its design, in contrast of short cylinders
(resonators) with ratio of length and diameter less than 1.
3.2.2. Finite Element Method
Al previously mentioned theoretical analyses are not easy for analytical
formulation, and generating of numerical results demands al lot of computer time.
For practical application such numerical solutions are not too useful. In this field of
numerical modeling today are also available software packages for application of
finite element method (Ansys, Algor, Abaqus), which provide satisfying precision
at comparison of calculated and experimental frequency characteristics, but which
also demand a lot of computer time and additional adjustments and fitting of
parameters and characteristics. Basic remarks related to the application of this
method in the field of piezoelectric ceramics modeling, and which are exposed in
90
Chapter 2.2.2, stand also here for the case of isotropic metal cylinders, disks, and
rings, and they will not be further considered in particular.
3.2.3. Numerical Analysis of Axisymmetric Oscillations of Metal Rings
Analysis of ultrasonic oscillation of metal rings was not given enough space in
literature, yet mostly are analyzed solid metal cylinders. Reason for this is that
there have been analyzed ultrasonic transducers with metal endings of great length,
at which, during analysis of longitudinal oscillation modes, inner diameter of the
cylinder with opening doesn’t play crucial role in determination of resonant
frequency. However, at analysis of oscillation of ultrasonic transducers for greater
operating resonant frequencies with short metal rings, and which are here subject
of further analysis, ending shape has great influence on fundamental resonant
frequency. Differences in frequency spectrum for metal disks and rings with same
outer diameter and thickness, that is, influence of the inner diameter at rings, will
be illustrated using approximate theory of axisymmetric oscillations of metal rings
with finite length. For determination of resonant frequencies of metal rings is used
BEM numerical method (boundary element method), similar to the method
presented in literature [93], which represents a new approach in analysis of short
metal endings in the field of power ultrasound [94], and that will be described in
detail in this chapter.
Numerical methods found extensive application in analysis and calculation of
oscillatory systems with coupled oscillations, and enabled analysis of distribution
(dispersion) of resonant modes and determination of resonant frequencies. Since
large amount of data has to be processed, numerical methods demand a lot of time
for obtaining satisfying accuracy of results. Yet, in this chapter is nevertheless first
proposed one numerical approach for determination of resonant frequencies of
metal rings, due to the significance of influence analysis of inner opening size of
the ring to its resonant frequency characteristic. Theoretical consideration of
oscillation of metal rings are based on different approximate theories, as well as on
the exact three-dimensional theory. However, most of these analyses relates to the
thin-walled rings (with close inner and outer diameter) or rings of infinite length.
Besides that, approximate theories are accurate only in the framework of limited
range of frequencies and wavelengths, as well as for certain dimension ratios, due
to different approximations included into the formulation of such theories.
For hollow metal cylinders of infinite length, without mechanical stresses on
cylindrical surfaces, the solution of problem of their elastic oscillation is contented
in the correspondent frequency equation, obtained by solving the general
equations of linear elasticity, just as in case of solid metal cylinders of infinite
length. This equation is obtained by satisfying the relevant boundary conditions on
cylindrical surfaces of the element (ending), without considering the influence of
circular-ringed ends of a finite ringed ending. However, in ultrasonic devices are
most often used elements of finite length, and in them ending effects cannot be
neglected, especially for short specimens in a shape of ring or disk. For such
endings in a shape of ring or disk of finite length, the problem is furthermore
complicated due to serious mathematical limitations. Namely, boundary conditions
91
with zero mechanical stresses cannot be simultaneously exactly fulfilled both on
the cylindrical and circular-ringed surfaces of the ending.
In the extension of exposure is performed numerical analysis of dispersion of
axisymmetric waves in unloaded metal finite cylindrical endings with opening
(metal rings). Special emphasis will be put on the dispersion of the frequency
spectrum. Using the exact equations of three-dimensional problem of linear
elasticity, there were analyzed frequency spectrums of rings of finite thickness
(length), with different ratios of inner and outer diameter, which are valid for any
range of frequencies and wavelength, as well as for any dimension ratio. It is
assumed an axisymmetric ring movement of the ring that oscillates at different
resonant modes, as well as symmetric movement regarding its central plane (z=0).
Boundary conditions with zero mechanical stresses will be satisfied exactly on the
cylindrical surfaces of the ring. Frequencies of vibrational modes thus obtained
from the frequency equation, determine the modes that will be superimposed so to
approximately satisfy boundary conditions with zero mechanical stresses on the
planar circular-ringed surfaces of the ring. Therefore, real, imaginary, and complex
branches of the correspondent dispersion spectrum of an infinitely long ring are
superimposed so to satisfy boundary conditions on the planar surfaces of the finite
ring with great accuracy level. It means that, beside the propagating modes (real
branch of the spectrum), analysis also encompasses the damped modes (imaginary
and complex branch of the spectrum), because of which are boundary conditions
on the planar surfaces satisfied with great accuracy. As a final result of this
analysis, it is possible to determine dependences of resonant frequencies of finite
metal rings (even of the piezoceramic rings), with arbitrary ratio of inner and outer
diameter, from its thickness (length). Purpose of such analysis of resonant
frequencies of ultrasonic transducer consisting parts is to present limitations of the
one-dimensional models of metal endings, and point out the necessity of
application of the three-dimensional matrix model of metal rings proposed in
extension of this exposure (Chapter 3.2.4). Since this analysis is based on the exact
equations of the three-dimensional problem of linear elasticity, it represents a
measure for comparison of validity of different methods that use approximate
equations of the three-dimensional problem of linear elasticity for obtaining the
frequency spectrum of metal rings, among which is also the method proposed in
Chapter 3.2.4.
3.2.3.1. Determination of the Frequency Equation
Subject of observation are axisymmetric oscillations of the homogenous,
isotropic, elastic metal cylinder with opening (ring), which is in its shape identical
with the piezoceramic ring presented in Figure 2.18. Thereby is enabled that in
later exposure within this chapter one gets the three-dimensional model of a metal
ring, based on the three-dimensional model of the piezoceramic ring proposed in
Chapter 2.2.5. Therefore, let there are the outer and inner radius of the metal ring a
and b, respectively, and let there is the ring thickness 2h (Figure 3.15(a)). It is
assumed that the central plane of the ring is located at z=0, so that its end surfaces
lie at z=±h.
92
Movement of the ring points is presented by the Lame’s partial differential
equation in vector form [53]:
(λm + µ ) graddivu + µ ∇ 2 u = ρ ∂
2
∂t
u
2
,
(3.39)
which gives relation between the displacement vector of the ring points u, and
Lame’s constants of ring material λm and µ in dynamic conditions, and in which ρ
is density, t is time, and ∇ 2 is Laplace’s differential operator. For axisymmetric
movement, solutions of the equation (3.39) are radial, tangential, and axial
(thickness) component of the displacement vector u:
∂ Y (α r )
∂ J (β r )
∂ Y (β r ) ⎤ j (ω t −k z )
⎡ ∂ J (α r )
−B 0
ur = ⎢ − A 0
,
+ Ck 0
+ Dk 0
⎥e
∂r
∂r
∂r
∂r
⎣
⎦
uθ = 0 ,
(3.40)
j (ω t −k z )
,
uz = j Ak J 0 (α r ) + BkY0 (α r ) + C β 2 J 0 (β r ) + Dβ 2 Y0 (β r )] e
where:
[
(
)
(
)
α 2 = ω 2 / v d2 − k 2 , β 2 = ω 2 / v s2 − k 2 ,
vd =
λm + 2µ
=
ρ
E Y (1 − υ )
vs =
ρ (1 + υ )(1 − 2υ ) ,
(3.41)
EY
µ
=
ρ
2 ρ (1 + υ ) .
ω is circular frequency, υ is Poisson’s ratio, k=ω/v=2π/λ is axis wave number (v is
phase velocity of the wave, λ is wavelength), vd and vs are phase velocities of the
compressional (longitudinal) and equivolume (transversal) waves in an infinite
medium, respectively, and A, B, C, and D are constants. Equations (3.41) include two
types of motion, from which one is symmetric, regarding the central plane, while the
other is antisymmetric movement regarding the central plane [92]. In this part of the
exposure, which relates to metal rings, a numerical analysis of both types of movement
is performed, although antisymmetric movement (determination of the so-called even
resonant modes) is not of interest in design of the ultrasonic sandwich transducers. For
the symmetric displacement (odd resonant modes), members e − jkz in expressions
(3.40) are replaced by coskz and sinkz, respectively, while for the antisymmetric
movement they are replaced by sinkz and coskz, respectively.
If Trr and Trz are normal and shear mechanical stress, then boundary conditions
with zero mechanical stresses in the points on the cylindrical surfaces of the ring
give following boundary conditions on those surfaces:
(Trr)r=a=0, for every z and t,
(Trz)r=a=0, for every z and t,
(3.42)
(Trr)r=b=0, for every z and t,
(Trz)r=b=0, for every z and t,
whereat the expressions for mechanical stress tensor components in function of the
displacement vector components of the ring points following:
93
∂u ⎞
∂u
∂u ⎞
⎛ ∂u u
⎛ ∂u
Trr = λm ⎜ r + r + z ⎟ + 2 µ r , Trz = µ ⎜ r + z ⎟ ,
∂r ⎠
r
∂z ⎠
∂r
⎝ ∂z
⎝ ∂r
∂u ⎞
u
∂u
⎛ ∂u
(3.43)
Tzz = λm ⎜ r + r + z ⎟ + 2 µ z .
∂z ⎠
r
∂z
⎝ ∂r
By substitution of the expression for displacements (3.40) into the boundary
conditions (3.43), one gets the system of four homogenous algebraic equations in
function of the unknown constants A, B, C, and D. For obtaining of the nontrivial
solution of these algebraic equations, determinant of the system must be equal to zero:
A11 A12 A13 A14
where:
(
= (ξ
A21
A31
A22
A32
A23
A33
A24
=0,
A34
A41
A42
A43
A44
(3.44)
)
/ 2 ) Y (α a ) + (α a )Y (α a ),
A11 = ξ 2 − Ω 2 / 2 J0 (α a ) + (α a ) J1 (α a ),
A12
2
−Ω
2
0
1
A13 = ξ [(β a ) J0 (β a ) − J1 (β a )],
A14 = ξ [(β a ) Y0 (β a ) − Y1 (β a )],
A21 = −2ξ (α a ) J1 (α a ),
A22 = −2ξ (α a ) Y1 (α a ),
(
= (2ξ
)
A
− Ω ) Y (β a ),
= (b / a ) (ξ − Ω / 2 ) J (α b ) + (α b ) J (α b ),
= (b / a ) (ξ − Ω / 2 ) Y (α b ) + (α b )Y (α b ),
A23 = 2ξ 2 − Ω 2 J1 (β a ),
24
A31
A32
2
2
2
2
2
1
2
2
0
1
0
1
(3.45)
2
A33 = ξ (b / a ) [(β b ) J0 (β b ) − J1 (β b )],
A34 = ξ (b / a ) [(β b ) Y0 (β b ) − Y1 (β b )],
A41 = −2ξ (b / a )(α b ) J1 (α b ),
A42 = −2ξ (b / a )(α b ) Y1 (α b ),
(
= (b / a ) ( 2ξ
)
− Ω ) Y (β b ).
A43 = (b / a ) 2ξ 2 − Ω 2 J1 (β b ),
2
A44
2
2
2
1
Complex transcendent equation, obtained from expression (3.44), is called the
frequency (characteristic) equation, and it links the normalized circular frequency
(Ω=ω a/vs), normalized axis wave number (ξ=αr=k a), and different physical and
geometric parameters of the metal ring. Physical parameters are wave propagation
velocity (v), Poisson’s ratio of the ring material (υ), and density (ρ), while the
normalized thickness of the ring wall (δ=1-b/a) is geometric parameter. For
individual values of υ and δ, frequency equation gives dispersion curves, that is,
94
dependences between Ω andξ. Thus, one gets infinitely many curves, which are
usually called branches of the dispersion spectrum. Since the boundary conditions
(3.43) are independent of z, it means that the characteristic equation is the same
both for symmetric and for antisymmetric displacement. For real normalized
frequencies (Ω), solutions of the characteristic equation give real, imaginary, and
complex values of the normalized wave number (ξ), which are associated to the
correspondent wave. Thereat are real wave numbers associated to the propagating
waves (modes), imaginary wave numbers are associated to the spatially damped
waves (modes), while the complex wave numbers are associated to the both, so that
imaginary and complex wave numbers are essential in the vicinity of the finite ring
ends (z=±h). As already mentioned, depending of the nature of the ξ, branches of
the dispersion spectrum are called real, imaginary, or complex, whereat for a
specific frequency exists finite number of real and imaginary branches, and infinite
number of complex branches. Solutions (eigenvalues) for ξ in conjugated-complex
plane are neglected in case of rings with infinite length, because those values of ξ
then lead to infinite values for displacements. For finite rings this problem does not
exist and, accordingly, conjugated-complex values of ξ are included at satisfying
the boundary conditions on the plane surfaces of the ring with finite length. Since
the distribution of mechanical stress cannot be satisfied in function of finite number
of real values ofξ, it is necessary to include the contribution of imaginary and
complex eigenvalues ofξ, in order to satisfy the boundary conditions on the plane
surfaces of the ring with acceptable accuracy.
In this chapter are determined real, imaginary, and complex branches of the
dispersion spectrum in ranges 0 ≤ Ω ≤ 10 and 0 ≤⏐ξ⏐≤ 14 for metal rings of
duralumin (Figure 3.5) and steel (Figure 3.6), and ranges 0 ≤ Ω ≤ 30 and 0 ≤⏐ξ⏐≤
15 for PZT8 piezoceramic ring (Figure 3.7). Thereat is the piezoceramic ring
considered (analogously to the metal endings) as a passive medium, physically
represented by its modulus of elasticity EY, Poisson’s ratio υ and density ρ (not
taken into account the piezoelectric properties and anisotropy of the
piezoceramic). In the Table 3.1 are quoted geometric dimensions and values of the
physical constants for all three elements, which are used in this analysis [82], [95].
Including the conjugated-complex branches in the considered ranges too, for every
Ω <10 and Ω <30 characteristic equation (3.44) gives 9 branches of the dispersion
spectrum, that is, 9 solutions for ξ, both for metal rings and for PZT8 piezoceramic
ring. The way these dependences are obtained is presented in literature [83], using
the software Mathematica1, in case of duralumin ring from Figure 3.5.
The first real branch (Im(ξ)=0) corresponds to the fundamental resonant mode
and exists at all frequencies, whereat Ω grows with increase of Re(ξ). The second,
third, fourth, and fifth branch start from the first, second, third, and fourth
boundary frequency, respectively, etc. Boundary frequencies are obtained as
solutions of the equation (3.44), if ξ=0. Real branches have minimums, or they are
also only ascending functions Re(ξ).
1
Mathematica, Wolfram, Inc. Research (http://www.wolfram.com)
Ω
10
8
6
4
2
0
10
5
Re(ξ)
0
-10
-5
10
5
0
Im(ξ)
Figure 3.5. Normalized frequency in function of normalized wave number,
for a duralumin ring with radius ratio b/a=8/40:
real branches of spectrum
imaginary branches of spectrum
complex branches of spectrum
Ω
10
8
6
4
2
0
10
5
Re(ξ)
-5
0
5
10
Im(ξ)
for a steel ring with radius ratio b/a=8/40:
0
-10
95
96
Ω
30
25
20
15
10
5
0
30
20
Re(ξ)
10
0
-10
0
Im(ξ)
10
for a ring of PZT8 ceramic with radius ratio b/a =15/38:
Table 3.1. Geometric dimensions and parameters of rings used in analysis
dural D5
20
4
0.34
tool steel
20
4
0.29
E
E
− s12
/ s11
= 0 .3
EY (Pa)
ρ (kg/m3)
vd (m/s)
0.74⋅1011
2790
2.18⋅1011
7850
1/s33E=0.74⋅1011
7600
6389
6033
3622
vs(m/s)
3146
3281
1936
a (mm)
b (mm)
υ
19
7.5
Imaginary branches of spectrum form loops in regions of the first and second
boundary frequency, that is, in regions of existing the third and fourth boundary
frequency. Thereat, imaginary loops join specific real branches of the dispersion
spectrum, and also have minimums and do not touch the zero frequency plane. In
case of PZT8 piezoceramic, the greatest frequency range is observed, and it is
possible to notice four imaginary loops. Complex branches of the spectrum “well
up” from the extreme points (relative maximums and minimums) of the real or
imaginary spectrum branches, and intersect the zero frequency plane as complex
spectrum branches. For a specific complex branch, Ω decreases with increase ofξ.
Within the specific range of the normalized frequency and normalized wave
number, number of real, imaginary, and complex branches decreases with increase
of the inner diameter of the ring of same material.
97
3.2.3.2. Analysis of longitudinal wave propagation velocity of metal rings
In this part of exposure numerical analysis will be interrupted for a moment,
because it is useful to perform an analysis of wave propagation velocity based on the
expressions hither derived. For design and modeling of ultrasonic transducers,
knowing of wave propagation velocities of different resonant modes is very essential,
because with known velocities and defined dimensions, resonant frequencies of
different modes may be easily determined for metal rings of practical significance. In
general case, as already remarked, decrease of resonant frequency is physically
identical with decrease of wave propagation velocity. In the proposed numerical
approach, equation (3.44) may be also used for association of wave propagation
velocity v with wavelength λ, whereat the constants in calculation are Poisson’s ratio,
as well as the inner and outer radius. Namely, applying this equation, wave
propagation for specific modes may be analyzed based on dependence of longitudinal
wave propagation velocity in infinite hollow cylinder from the ratio of outer radius and
wavelength, for a given ratio of inner and outer radius. Typical dispersion curves
(vI÷vVI), which correspond to the roots of equation (3.44) for real ξ, are presented in
Figure 3.8, in case of duralumin cylinder with ratio b/a=8/40. Thereat, except the first
several resonant modes, remained higher modes of oscillation have no practical
significance, and the most important is the lowest resonant mode with wave
propagation velocity vI. Similar analysis of the influence of cross dimensions on
longitudinal wave propagation velocity in an axisymmetric elastic body with central
opening, in case of steel specimens, is presented in paper [96].
At small values a/λ (a/λ<0.2, Figure 3.8), oscillation across the whole crosssection of the cylinder is uniform, and only the fundamental longitudinal mode of
oscillation exists, whereat the longitudinal wave velocity for the fundamental mode (vI)
approximates velocity in a very thin infinite metal bar: v0 = EY / ρ =5150m/s.
Dispersion of the fundamental mode is especially small for values a/λ<0.1. With
increase of the ratio a/λ, due to the numerous reflections of the waves inside the
cylinder, whose radial dimensions are not any more negligible, a spatial system of
waves that propagate in different directions arises, so velocity decrease occurs. At
great ratios of a/λ, oscillation intensity decreases inside the cylinder, and grows near
the cylinder surface. Then the wave velocity of the fundamental mode vI inclines to
constant value equal to the velocity of Rayleigh’s surface waves v =2937m/s, which
represents the solution of the following equation [53]:
6
4
2
2
⎛ ⎛ v ⎞2 ⎞
⎛v ⎞ ⎞
⎛v ⎞ ⎛
⎞
⎛v ⎞
⎟⎟ − 8 ⎜⎜ ⎟⎟ + 8 ⎜⎜ ⎟⎟ ⎜ 3 − 2 ⎜⎜ s ⎟⎟ ⎟ − 16 ⎜1 − ⎜⎜ s ⎟⎟ ⎟ = 0 . (3.46)
⎜
⎟
⎜
⎟
v
⎝ vd ⎠ ⎠
⎝ vs ⎠ ⎝
⎠
⎝ vs ⎠
⎝ ⎝ d⎠ ⎠
Situation at cylinders with finite length is additionally complicated, because in
that case there also arises the wave reflection from the boundary surfaces in the
sense of propagation of longitudinal waves, so that velocity dispersion at finite
cylinders is more complex regarding the described velocity dispersion of infinite
cylinders. Therefore, when the ratio of length and diameter decreases, velocity
dispersion in one-dimensional modeling is the most important problem, since the
⎛v
⎜⎜
⎝ vs
98
adopted line velocity is fixed and it is not a function of frequency, which will be
talked about more in the Chapter 3.2.3.3.
v 8000
7000
vV
vVI
6000
v0
5000
vII
4000
vI
v
3000
vIV
vIII
2000
vz
1000
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
a/λ
Figure 3.8. Longitudinal wave propagation velocity for a duralumin cylinder
of infinite length with opening (b/a=8/40)
Because of all quoted, if one uses the line model for description of oscillation
of metal rings of finite thickness and nonnegligible diameter, longitudinal wave
velocity for the fundamental mode must be decreased regarding the velocity v0.
Velocity decrease should be performed based on the first (lowest) dependence vI
from the Figure 3.8 for fundamental resonant mode (if it is still matter of a cylinder
long enough). However, at short metal rings, at which the thickness is smaller than
diameter, the situation is the most complex. Occurrence of the method of seeming
elasticity modulus enabled improvement in determination of the longitudinal wave
velocity for the fundamental resonant mode in such cases. The velocity vz, obtained
using the mentioned method, but at metal cylinders with opening (rings), is also
presented in Figure 3.8. Thereat is [86]:
vz =
EYz
ρ
, EYz =
EY
, nS =
1 − 2υ / n S
⎛ R' v 0
1 − υ 2 − ⎜⎜
⎝ ωa
υ (1 + υ )
2
⎞
⎟⎟
⎠ ,
(3.47)
where nS determines the coupling degree of oscillations, and constant R’ duralumin
with ratio b/a=8/40 amounts to R’=1.8 [97]. This approximation is valid at smaller
99
values of a/λ, because at greater values of a/λ the first resonant mode, to which this
method relates, is no longer longitudinal, but passes into a radial mode. Therefore,
at greater values of a/λ, that is, smaller ratios of thickness and outer diameter of the
ring, and this method makes great error in design. Reason for this is that this
procedure does not treat simultaneously several coupled resonant modes, which is
necessary in design of ultrasonic sandwich transducers, but it is modeled only the
fundamental resonant mode, with influence of radial dimensions and remained
modes taken into calculation, through the Poisson’s ratio. Nevertheless, method of
seeming elasticity modulus represents a very useful approximation in the field of
design of short ultrasonic emitters with great power, and it represented the best
modification of the one-dimensional theory hither.
3.2.3.3. Determination of Resonant Frequencies of Metal Rings
In order to determine further theoretically the resonant frequencies of metal rings
by the proposed numerical approach, one should satisfy boundary conditions without
mechanical stresses on all surfaces. However, this is a serious theoretical limitation,
because if the boundary conditions are exactly satisfied on the dominant ring surfaces,
it is not possible to satisfy exactly the boundary conditions with zero mechanical
stresses on the remained ring surfaces. Therefore is applied an appropriate approximate
approach, that is, the method of superimposing of finite number of resonant modes, so
that conditions on the remained surfaces are satisfied with as small error as possible. It
is assumed that, for defined δ, characteristic equation gives 2m+1 values ξ on defined
frequency Ω. Individual displacements that correspond to those wave numbers, are
linearly superimposed introducing 2m+1 unknown constants, whereby one gets
resulting displacement. Amplitudes of those displacements will be chosen so that
2m+1 independent boundary conditions are satisfied as accurately as possible on the
planar ring surfaces. If for an individual δ=1-b/a exists 2m+1 spectrum branches, one
should satisfy following 2m+1 conditions on the planar ring surfaces in 2m+1 chosen
points:
Tzz =
2 m +1
∑ Ai Tzz i, rm1 = 0,
z = ±h,
(3.48)
z = ± h,
(3.49)
i =1
where rm1=b + m1 (a-b) / m ,
m1=0,1,2,...m, and
Trz =
2 m +1
∑ Ai Trz i, rm 2 = 0,
i =1
where rm2=b + m2 (a-b) / (m+1), m2=1,2,...m.
Thereat are Tzz i, rm1, and Trz i, rm2 mechanical stresses (normal and shear), which
correspond to the i-th branch of the dispersion spectrum, and which are calculated for
r=rm1, that is, r=rm2. Ai is amplitude constant, which corresponds to the i-th branch,
and Tzz and Trz are resulting normal and shear mechanical stress, respectively.
Equations (3.48) and (3.49) represent a system of 2m+1 independent
homogenous equations, which content 2m+1 unknown constants Ai. For obtaining
of nontrivial solution, determinant of the system should be equal to zero:
100 Modeling of Metal Cylindrical Oscillators
∆=
Tzz 1, r 0
Tzz 2, r0
K K Tzz 2 m +1, r 0
Tzz 1, r 1
Tzz 2, r 1
K K Tzz 2 m +1, r 1
K
K
K K
K
K
K
K K
K
Tzz 1, rm
Tzz 2, rm
K K Tzz 2 m +1, r m
Trz 1, r 1
Trz 1, r 2
Trz 2, r 1
Trz 2, r2
K K Trz 2 m +1, r 1
K K Trz 2 m +1, r 2
K
K
K K
K
K
K
K K
K
Trz 1, rm
Trz 2, r m
= 0.
(3.50)
K K Trz 2 m +1, r m
For a given frequency f, characteristic equation (3.50) contains ring thickness
l=2h as the only unknown parameter, and gives infinitely many solutions for l. Taking
different values for frequency, one may determine dependence of the resonant
frequency and ring thickness. Transcendent equation (3.50) is different for symmetric
and antisymmetric motion, and here it will be performed an analysis for both types of
displacement. There are analyzed characteristics of duralumin and steel metal rings
with dimensions 2a=40mm, 2b=8mm, and PZT8 ring with dimensions 2a=38mm,
2b=15mm, because just those rings are later used in realization of ultrasonic sandwich
transducers. Frequency spectrums of the resonant modes (dependences f of l) for
symmetric and antisymmetric displacement, for values of l from 0 to 120mm, for
metal endings and PZT8 piezoceramic, are presented in Figures 3.9, 3.10, and 3.11.
The way previously described procedure is concretely applied, using the software
Mathematica, is presented in literature [83], for the case of determination of the first
resonant mode of duralumin ring from Figure 3.9.
Dependences of the resonant frequencies from duralumin and steel ring thickness,
presented in Figures 3.9 and 3.10, show that rings made of the cited metals, and with
same dimensions, have very similar resonant frequency spectrums, which may be also
seen from Figures 3.5 and 3.6. In both cases are presented the three lowest odd
resonant modes for symmetric motion of the ring (solid line) and three lowest even
resonant modes for antisymmetric motion of the ring (dashed line), in order to
illustrate great complexity of the spectrum in the region of small ring thicknesses.
For the demand of this analysis are significant only the symmetric modes, and
further considerations are related just to those resonant modes. It is obvious that
frequency of these modes does not decline uniformly with thickness in the whole
observed range of f and l, because in some regions spectrums have gradual shape due
to the mode coupling, whereat in those spectrums small changes of thickness generate
great frequency changes, and vice versa. It is first part of the spectrum around 75kHz
in Figures 3.9 and 3.10, for both metal rings, when all spectrum branches have gradual
structure, and also the part of the spectrum where the first complex branch of the
dispersion spectrum occurs (Figures 3.5 and 3.6). In addition, around 90kHz the third
mode has gradual structure, as well as the higher modes that are not presented here,
and this region of the frequency spectrum corresponds to the resonant frequency of the
edge (end) mode, for which dominant displacements are around the planar ring
surfaces.
x 10
f[Hz] 18
101
4
IIIV
IIV
IVV
VV
VIV
16
14
IINa
12
IIINa
IINs
10
IIINs
INs
8
6
IV
INa
4
line theory (index V)
numerical method (indexes Ns and Na)
2
0
0
0.02
0.04
0.06
0.08
0.1
l [m]
0.12
Figure 3.9. Dependence of the resonant frequency from length for a duralumin
ring (b/a=8/40): s – symmetric modes, a – antisymmetric modes
f[Hz] 18
x 10
4
IIIV
IIV
IVV
VV
VIV
16
14
IINa
IIINa
12
IINs
10
IIINs
INs
8
6
IV
INa
4
2
0
0
0.02
0.04
0.06
0.08
0.1
l [m]
0.12
Figure 3.10. Dependence of the resonant frequency from length for a steel ring
(b/a=8/40): s – symmetric modes, a – antisymmetric modes
5
f[Hz]
[Hz]
3
x 10
IIV
2.5
IIIV IVV VV VIV
2
IINs
1.5
1
IVNs
IV
0.5
IIINs
INs
VIIINs
VNs
VINs
VIINs
INa
0
0
0.02
0.04
0.06
0.08
0.1
l [m]
0.12
Figure 3.11. Dependence of the resonant frequency from length for a PZT8
ceramic ring (b/a=8/40): s – symmetric modes, a – antisymmetric modes
At PZT8 piezoceramic ring, the gradual parts of the spectrum are in other regions,
because it is the matter of a ring with substantially different dimensions and
characteristics, that is, in regions around 45kHz, 140kHz, and 170kHz. Thereat are
presented the first eight modes for symmetric (odd) resonant modes (solid line)
and, for clearness, only the first even resonant mode (dashed line). In all cases, on
the given graphs there are also presented dependence characteristics of the resonant
frequency from thickness, in case of one-dimensional modeling, in case when the
metal rings may be presented by line model. In case of unloaded ring, dependences
of the resonant frequency from thickness (Zc is characteristic line impedance) [94]
are calculated by finding the zero values of the input impedance of the shortcircuited line without losses (Zul=j Zc tgkl):
f =N
v0
,
2l
N = 1, 2, 3,...;
v0 =
EY
ρ
.
(3.51)
It is obvious that at those ring thicknesses, especially in the region l/(2a)<2, line
model cannot even nearly represent the real frequency spectrum at axisymmetric
oscillation of rings. This may be even better noticed if the frequency spectrum is
presented in somewhat different form, that is, if the product fxl is presented in function
of normalized length l/(2a). In Figure 3.12 are presented quoted dependences for the
103
first three resonant modes of a duralumin ring with ratio b/a=8/40, which correspond
to odd modes INs, IINs, IIINs from Figure 3.9, obtained by numerical method. On the
same graph is presented constant value fxl = v0 / 2 , obtained based on expression
(3.51) for fundamental resonant mode (N=1) in case of a line. Based on presented
dependences, it is obvious that the line model, with the velocity equal to the wave
propagation velocity v0 in an infinitely long cylinder, may be applied for great ring
lengths, when the solutions obtained by different approaches get closer asymptotically.
At small ring thicknesses, product fxl, which is obtained using numerical method, is
approximately constant and it has substantially lower value than in case of an infinite
cylinder. It means that the line model may be used in that region of l/(2a), but with
substantially smaller wave propagation velocity. In purpose of comparison, on the
same graph are also presented the first three resonant modes obtained by the method of
seeming elasticity moduli, also based on equation (3.51), where v0 should be
substituted by vz from expression (3.47), for three successive values ’=1.80; 4.71;
8.23 [97]. Resonant modes obtained by this method follow general trend of the real
resonant modes, although differences between the compared dependences are not
negligible.
Resulting values for displacements ur and uz may now be easily determined and
drawn by substitution of values of calculated constants Ai into the corresponding
expressions. For every value of frequency f and length l, one may determine the
displacement uz on the endmost, planar, emitting surface. In the frequency range in the
vicinity of the fundamental resonant mode, one may in the previous figure link the
frequencies that correspond to those pairs of f and l for which the displacements uz are
maximal. Thus, one gets the main mode in the observed region, which is in Figure
3.13 presented by thicker line. Similar graphs may be also obtained for rings made of
other materials, which are of practical interest. By dashed line it is again presented
wave propagation velocity for a pure one-dimensional thickness mode, obtained based
on the line model. Differences between this velocity and the real velocity of the
thickness mode show that one-dimensional model is not adequate for modeling of such
metal rings. Product of frequency and length (expressed in Hz⋅m), which is for the
main mode presented in Figure 3.13, is equal to the half of the searched line velocity v0
(in m/s) (equation 3.51).
Based on the previously exposed, one may conclude that improvement of onedimensional modeling of metal rings is possible to perform by line velocity fitting,
based on the dependence for the main mode presented in Figure 3.13. It is clear that
this approach has no practical significance, because it is too complicated to perform
complete previous numerical procedure for determination of the dispersion spectrum
of finite rings and determine maximal displacements uz, in order to obtain the
dependence on which base would be improved only the line model. Thereat one
should not forget that overall previous numerical procedure must be repeated at every
new radius ratio b/a. Because of that in extension of the paper is proposed an
approximate three-dimensional matrix model of metal rings, whose application is
substantially simpler than application of numerical approach, and which gives resonant
frequency spectrums that agree well with experimental results.
fxl 4000
[Hz m]
3500
3000
v0 /2
2500
III
2000
I
II
1500
line model
method of seeming moduli EY
proposed numerical method
1000
500
0
0
1
0.5
1.5
2
2.5
l/(2a)
l/a
3
Figure 3.12. Dispersion of resonant modes of duralumin ring with b/a=8/40
fxl 4000
[Hz m]
3500
main mode of oscillation
3000
v0 /2
2500
III
2000
II
I
1500
1000
500
0
0
0.5
1
1.5
2
2.5l/(2a)
Figure 3.13. Main mode of oscillation for case from Figure 3.12
l/a2
3
105
If δ=1, previous numerical procedure gives solutions for oscillation of a solid
cylinder (or disk). Because of massiveness, this procedure is not presented again,
but in Figure 3.14 there are presented by solid line four lowest resonant modes of a
solid duralumin cylinder, obtained in that way. In the same figure, this normalized
frequency spectrum of a disk is compared with frequency spectrum of a ring made
of same material and same outer diameter (dashed line), with ratio b/a=8/40 (part
of the spectrum from Figure 3.9). It is obvious that values of the resonant
frequencies of some modes change because of the presence of inner opening, and
these changes are greater if the ring opening diameter is greater. Because of that, as
already mentioned, it is not all the same in modeling of power ultrasonic
transducers if the consisting elements of the transducers are metal and
piezoceramic rings or disks. It must be remarked that the spectrum of a solid
cylinder, obtained by this method, is identical with spectrum obtained by
Hutchinson’s method [89] (equation (3.37)), which is in that purpose reproduced
for the case of duralumin cylinder, and whose mere results are presented in Figure
3.14. In further analysis, resonant frequencies of the solid cylinders are of interest
due to the modeling of the bolt and bolt head in ultrasonic sandwich transducers,
which by its shape represent solid cylinders.
W 10
9
disk
ring
8
7
6
III
5
IV
4
II
3
I
2
1
0
0
0.5
1
1.5
2
2.5
3
l/(2a)
Figure 3.14. Dependence of the normalized resonant frequency on normalized
length for: (a) duralumin disk (
); (b) duralumin ring with b/a=8/40 (
)
Nevertheless, the most important contribution in determination of resonant
frequencies in this part of the analysis is connected to the metal rings, because the
bolt length is quite greater than its diameter, so for the bolt one may also apply the
line model, and determination of resonant frequencies dependence from length is
not critical for it in design. Numerical analysis for such case is not presented here,
although it may be easily reproduced by denormalization of values for the case of
solid cylinder from Figure 3.14. Thickness of the piezoceramic platelets during
realization of ultrasonic transducers is fixed, and usually are adopted piezoceramic
rings with thickness of 5mm or 6.35mm. Because of that, it is possible in practice
to adjust the resonant frequency of the transducer mostly by choice of metal rings.
Therefore, the most attention is paid to determination of their characteristics.
Presented numerical approach for determination of resonant frequency spectrums
of metal rings may be useful for analysis of advantages and disadvantages of analyzed
one-dimensional ring models, as well as for verification of the three-dimensional
matrix model proposed in the next chapter. However, because of its voluminosity and
complexity, this approach has no significance in modeling of complete ultrasonic
sandwich transducers, because by this approach one doesn’t get ring model by which
could be easily analyzed influences of different parameters and loads, as in case of the
proposed (even in case of one-dimensional) model.
3.2.4. Three-dimensional Matrix Model of Metal Rings
3.2.4.1. Analytical Model
Based on the three-dimensional model of piezoceramic rings presented in
Chapter 2.2.5, in this chapter is obtained three-dimensional model of metal rings
(and disks). Proposed model relates to the metal rings whose appearance and
dimensions are presented in Figure 3.15(a).
z
F3
v3
v1
F2
v2
F1
F1
v1
r
2h
v3
v2
4 access
network
F3
F2
v4
F4
b
a
(a)
F4
v4
(b)
Figure 3.15. Loaded metal ring: (a) geometry and dimensions;
(b) metal ring as a 4-access network
Using this model, metal ring is modeled in electromechanical circuit by 4access network (Figure 3.15(b)), whereat are Fi and vi (i=1, 2, 3, 4) forces and
velocities on outer surfaces of the ring, as in the case of piezoceramic rings. The
model is obtained simply, based on the derived model of piezoceramic rings, by
107
simple neglecting the piezoelectric constants h31 and h33 in expressions (2.40).
Besides that, expressions (2.40) are simplified in a way that, due to the material
isotropy, following relations are adopted for material constants [53]:
(3.52)
c11 = c33 = λm + 2 µ ,
c12 = c13 = λm ,
thereat are Lame’s coefficients:
υ EY
EY
.
(3.53)
λm =
,
µ=
(1 + υ )(1 − 2υ )
2(1 + υ )
With these assumptions, linear equations that link mechanical values on the
external ring surfaces are simpler, that is, system (2.39) is reduced to the following
equation system:
⎡ F1 ⎤ ⎡ z11 z12 z13 z13 ⎤ ⎡ v1 ⎤
⎥⎢ ⎥
⎢ ⎥ ⎢
⎢ F2 ⎥ = ⎢z21 z22 z23 z23 ⎥ ⎢v2 ⎥
(3.54)
⎢ F3 ⎥ ⎢ z13 z23 z33 z34 ⎥ ⎢ v3 ⎥ ,
⎥⎢ ⎥
⎢ ⎥ ⎢
⎣ F4 ⎦ ⎣ z13 z23 z34 z33 ⎦ ⎣v4 ⎦
whereat the impedance matrix elements are defined by the following expressions:
{
{
z 22
}
}
− 4πh
c12 − c11 [ 1 − kb (A1 J 0 (kb ) + B1Y0 (kb ))] ,
jω
4πh
=
c − c [ 1 + ka (A2 J 0 (ka ) + B2 Y0 (ka ))] ,
jω 12 11
z11 =
z12 =
− 4π k b h c11
[A2 J0 (kb) + B2 Y0 (kb)],
jω
z 21 =
− 4π k a h c11
[A1 J0 (ka) + B1Y0 (ka)],
jω
2π bc12
,
jω
z23 =
2π a c12
,
jω
c11k π a 2 − b 2
,
jω tg(2 kh )
z34 =
c11k π a 2 − b 2
,
jω sin (2 kh )
z13 =
z33 =
(3.55)
(
)
(
)
in which are the values of introduced integration constants A1, A2, B1, and B2,
determined from boundary conditions (2.33) in form (2.35):
Y1 (ka )
,
A1 =
J1 (kb )Y1 (ka ) − J1 (ka )Y1 (kb )
A2 =
B1 =
B2 =
Y1 (kb )
,
J1 (kb )Y1 (ka ) − J1 (ka )Y1 (kb )
J1 (ka )
,
J1 (ka )Y1 (kb ) − J1 (kb )Y1 (ka )
J1 (kb )
.
J1 (ka )Y1 (kb ) − J1 (kb )Y1 (ka )
(3.56)
3.2.4.2. Numerical Results
In ultrasonic transducers, such metal ring is connected (joined) with the
piezoceramic ring via one its circular-ringed (planar) surface, and because of that
the input mechanical impedance of the metal ring, by which the piezoceramic ring
is loaded, is essential. By introducing the relations between forces and velocities on
external surfaces through acoustic impedances (Fi=-Zivi, i=1, 2, 3, 4), and their
substituting into the equation system (3.54), one may determine any mechanical
impedance. In Figure 3.16 are presented input mechanical impedances Zm=F4/v4
for a duralumin ring with dimensions: 2a=40mm, 2b=8mm, 2h=18mm, as well as
for a steel ring of same outer and inner diameter, with thickness 2h=11mm. Metal
rings with these dimensions are consisting parts of an ultrasonic transducer with
operating resonant frequency of 40kHz, which will be later modelled. Matlab
software for determination of mechanical impedance, given in Figure 3.16 for a
duralumin ring, is presented in literature [83], and it is similar to the software for
determination of input electric impedance of the piezoceramic ring, also presented
in literature [83].
For design and optimization in the field of power ultrasound, more significant
than the presented dependence, is the dependence of the resonant frequency from
the metal rings dimensions, which may be also determined applying the proposed
model. In Figures 3.17 and 3.18 are presented the three lowest branches of the
resonant frequency spectrum in function of length, for the case of duralumin and
steel rings, with radius ratio b/a=8/40. Presented characteristics are very similar,
although the physical parameters of the quoted metals differ a lot.
In order to present the improvements obtained by quoted modeling approach,
in Figure 3.17 is also presented the frequency spectrum obtained applying the
method of seeming elasticity moduli. It is obvious that the first resonant modes for
both models almost identical, which was logical to expect, because the method of
seeming elasticity moduli reduces to determination of the first resonant mode.
However, higher resonant modes obtained by the proposed matrix model predict
the mode coupling, so the resonant curves are not uniform as in case of application
of the seeming elasticity moduli method, which is closer to the real case. As
already mentioned, the method of seeming elasticity moduli represents
modification of the one-dimensional line theory, so that its application in
equivalent circuits would reduce to a two-access network, with artificially modified
line parameters. Thus, one may not realize analyses of oscillation in radial direction
that is enabled by the proposed three-dimensional matrix model, especially in
complex devices with several serial-parallel connections of the mechanical
accesses that exist in the field of power ultrasound.
In Figure 3.18 are simultaneously presented frequency spectrums for a steel
ring and steel disk with same outer diameter, obtained by the proposed threedimensional matrix model. Thereby is confirmed the statement obtained applying
the numerical method that differences in frequency spectrums increase with
increase of the inner opening diameter of the ring, which in endmost case generates
changes of the resonant frequencies of the ultrasonic sandwich transducers modes.
109
120
duralumin
steel
100
Zm [dB]
80
60
40
20
0
0.5
0
1.5
1
2
f [Hz] x 10 5
2.5
Figure 3.16. Mechanical impedance of free metal rings:
a) duralumin (
) 2a=40mm, 2b=8mm, 2h=18mm;
(b) steel (
) 2a=40mm, 2b=8mm, 2h=11mm
5
2.5 x 10
III
three-dimensional model
method of seeming moduli EY
2
II
f [Hz]
1.5
1
I
0.5
0
l [m]
0
0.02
0.04
0.06
0.08
0.1
0.12
Figure 3.17. Frequency spectrum of a duralumin ring with ratio b/a=8/40 in
function of its length: proposed model (
) and Mori’s model (
)
2.5
x 10 5
ring
disk
2
1.5
III
f [Hz]
II
1
I
0.5
0
0
0.02
0.04
0.06
l [m]
0.08
0.1
0.12
Figure 3.18. Frequency spectrum of a steel ring in function of its length and
opening dimensions, obtained by proposed model for the case
of a ring b/a=8/40 (
) and disk b/a=0 (
)
This conclusion may be even better confirmed based on the Figure 3.19.
Namely, beside the mentioned dependence of the resonant frequency from the ring
length, it is also essential the dependence of the resonant frequency from the ratio
of inner and outer ring radius. In Figure 3.19 is presented quoted dependence for
the case of duralumin ring with thickness of 18mm. In the presented figure,
frequencies of the lowest resonant mode decrease with increase of b/a, frequencies
of the second resonant mode do not depend much from the inner opening
dimension of the ring, while the frequencies of the third and higher resonant modes
grow with increase of ratio b/a.
As in case of piezoceramic ring, one may here also observe the influence of
specific mechanical loads on different surfaces of the metal ring onto the input
mechanical impedance and resonant frequency spectrum; however, such analysis for
an isolated metal ring is not too interesting. This analysis will be performed in the last
chapter dedicated to the modeling of sandwich transducers, when much more
interesting is the influence of mechanical load of metal rings, because then they will be
consisting parts of the complete ultrasonic sandwich transducers. In addition, as at
piezoceramic rings, here too one may determine displacements ur and uz for different
resonant modes, for rings of different dimensions and made of different materials.
x 10
4
111
5
3.5
3
III
f [Hz]
2.5
2
1.5
II
1
0.5
I
0
0
0.1
0.2
0.3
0.4
0.5
b/a
0.6
0.7
0.8
0.9
1
Figure 3.19. Frequency spectrum of a duralumin ring with thickness
2h=18mm in function of ratio b/a
3.2.4.3. Comparison of Numerical and experimental Results
In order to test the validity of the proposed matrix model of metal rings,
like in the case of piezoceramic rings, there are experimentally determined resonant
frequencies of metal rings and solid cylinders with different dimensions and made
of different materials. Measurements are performed using vibrational platform of
the company Herfurth2, whose measuring range is from 15kHz to 50kHz. Obtained
experimental results are compared with analogous characteristics obtained applying
the described numerical method, as well as applying the proposed threedimensional matrix model. First are presented dependences of resonant frequencies
from the length of metal rings and cylinders made of duralumin (Figure 3.20) and
steel (Figure 3.21). There are presented fundamental even and odd resonant modes
obtained applying the presented numerical method. Thereat are presented in same
figures by circlets experimental resonant frequencies measured on the vibrational
platform, for unloaded specimens with geometry presented in Figure 3.15(a). As
one may see in Figures 3.20 and 3.21, frequencies obtained by numerical analysis
agree very well with measured frequencies in observed range, although for the
2
Herfurth, Hamburg-Altona, Sonotroden-Meßgerät type USM4
given materials are adopted typical values for the Young’s modulus EY and
Poisson’s coefficient υ [95]. At concrete specimens, those values may deviate from
adopted values, which leads to certain differences in the compared results. This
may be particularly noticed at the steel endings from Figure 3.21, where the
theoretical resonant frequencies are in all cases above the measured resonant
frequencies. Minor decrease of the value EY contributes to the coincidence of the
presented results. It has to be noticed that all performed theoretical analyses are
very sensitive to the values of coefficients EY and υ, so that small changes of these
constants generate great changes of resonant frequencies.
It is especially interesting to compare the experimental frequency
spectrums of piezoceramic rings and disks with analogous theoretical spectrums
obtained applying numerical analysis of axisymmetric extension oscillations of
isotropic (metal) rings, whereat are not taken into account piezoelectric properties
and anisotropy of the ceramic [98], [99], [100]. In that case, frequency
spectrum depends from Poisson’s ratio of the ceramic material (for piezoceramic is
E
E
E
, µ = 1 / s44
).
υ = −s12
/ s11
8
x 10 4
Is
Ia
IIs
7
IIIs
IIa
IIIa
f [Hz]
6
5
4
3
duralumin:
I) 2a=40mm, 2b=0
II) 2a=40mm, 2b=8mm
III) 2a=51mm, 2b=0
2
0
0.01
0.02
0.03
0.04
0.05
l [m]
0.06
0.07
0.08
0.09
Figure 3.20. Frequencies of the first symmetric (s) and antisymmetric (a)
modes for different duralumin specimens:
comparison of calculated and experimental results
0.1
113
Theoretically and experimentally is determined the frequency spectrum of
piezoceramic rings and disks with Poisson’s ratio 0.3, for different ratios of thickness
and outer diameter. Dimensions of the analyzed PZT8 rings are presented in Table 3.2.
Obtained normalized frequency spectrums for the five lowest resonant modes are
presented in Figure 3.22 [98]. Thereat are, in purpose of evaluation of such way of
modeling, in the same figures presented experimental values of resonant frequencies
for the lowest modes of the PZT8 rings from Table 3.2. Experimental results are
obtained using automatic network analyzer HP 3042A, by measuring the frequencies
at which the modulus of the input electric impedance is minimal. These experimental
values are normalized in the earlier mentioned way (Ω=ω a/vs), for comparison with
theoretical spectrum. It is obvious that by such analysis one may almost completely
predict also the frequency spectrum of the piezoelectric ceramics, with no need for
additional correction of the applied constants by experimental results fitting.
9
x 10 4
Is
8
Ia
IIs
7
IIa
f [Hz]
6
5
4
steel:
I) 2a=40mm, 2b=0
II) 2a=51mm, 2b=0
3
2
0
0.01
0.02
0.03
0.04
0.05
l [m]
0.06
0.07
0.08
0.09
Figure 3.21. Frequencies of the first symmetric (s) and antisymmetric (a)
modes for different steel specimens:
comparison of calculated and experimental results
1. ring
2. ring
Table 3.2. PZT8 rings used in analysis
2a (mm)
2b (mm)
2h (mm)
38
15
5
50
20
6.35
b/a
0.39
0.40
0.1
Ω
20
2a=38mm, 2b=15mm, 2h=5mm
18
16
14
12
10
8
6
4
2
0
0
Ω
0.5
1
1.5
l/(2a)
2
20
2a=50mm, 2b=20mm, 2h=6.35mm
18
16
14
12
10
8
6
4
2
0
0
0.5
1
1.5
l/(2a)
2
Figure 3.22. Calculated frequency spectrums and measured results
for the first and the second PZT8 ring from Table 3.2
In purpose of completion the description of the frequency behavior of
cylindrical elements, an analogous analysis for piezoceramic disks is performed
[99]. In order to test the validity of such disk modeling, as in previous case,
experimentally determined certain resonant frequencies of the piezoceramic disks
are compared with analogous characteristics obtained by computer applying the
proposed numerical approach. Dimensions of the PZT8 disks used in this analysis
are presented in Table 3.3, and the correspondent frequency spectrum, compared
with experimental results for the five lowest resonant modes, is given in Figure
3.23.
Ω
115
25
PZT8
20
15
10
5
I II III
0
Ω
0
IV
0.1
0.2
0.3
0.4 l/(2a) 0.5
25
BaTiO3
20
15
10
5
I II III
0
0
IV
0.5
1
l/(2a)
1.5
Figure 3.23. Comparison between calculated and measured
frequency spectrum for PZT8 and BaTiO3 disks
Since the Poisson’s ratio for BaTiO3 disks is the same as for the PZT8 ceramic, in
the same figure are presented calculated frequency spectrums and experimental results
for BaTiO3 disks from Table 3.4 [100]. Comparing the frequency spectrum obtained
applying the described numerical method, with experimental measurements of
resonant frequencies of disks from Tables 3.3 and 3.4, one may notice that it is realized
great precision in predicting the resonant modes in all cases. Thereat is presented the
frequency range that is most often used in practical applications.
Table 3.3. PZT8 disks used in analysis and experimental measurements
2a (mm)
2h (mm)
h/a
50
3
0.060
30.9
2.1
0.068
24.9
2.5
0.100
38.4
6.35
0.165
Table 3.4. BaTiO3 disks used in analysis and experimental measurements
2a (mm)
2h (mm)
h/a
10.1
0.5
0.049
18.7
1.5
0.080
10.2
1.2
0.118
9.8
10
1.020
Finally, in Figures 3.24 and 3.25 are presented frequency spectrums of the
lowest (first) resonant modes of different rings and solid cylinders of duralumin
and steel, respectively, applying the proposed three-dimensional matrix model.
These frequency spectrums of resonant modes are compared with experimental
resonant frequencies, obtained by vibrational platform. Material parameters used in
modeling (simulation) of metal rings are presented in Table 3.1.
x 10 4
5
II I
I
III
II
IV
III
4.5
IV
f [Hz]
4
3.5
3
2.5
0.04
duralumin:
I) 2a=40mm, 2b=0
II) 2a=40mm, 2b=8mm
III) 2a=51mm, 2b=0
IV) 2a=51mm, 2b=16mm
0.05
0.06
0.07
0.08
0.09
l [m]
Figure 3.24. Frequency spectrums of the first resonant mode
of different duralumin specimens
0.1
5
117
x 10 4
I
II
I
II
4.5
f [Hz]
4
3.5
3
2.5
0.04
steel:
I) 2a=40mm, 2b=0
II) 2a=51mm, 2b=0
0.05
0.06
0.07
l [m]
0.08
0.09
0.1
Figure 3.25. Frequency spectrums of the first resonant mode
of different steel specimens
Obviously, the proposed three-dimensional matrix model has ability to predict
frequency spectrums of metal rings with any dimensions, and with precision not
less than in case of application of numerical method. Therefore, great precision in
determination of resonant frequencies is preserved, and the more essential from
that is given possibility of determination different transfer functions of a loaded
metal ring, as well as its componential displacements. Obtaining good results in
case of analysis of unloaded metal rings is justification for application of this
model in case of loaded rings, during modeling the complete ultrasonic sandwich
transducers presented in the last chapter.
4. MODELING OF POWER ULTRASONIC SANDWICH
TRANSDUCERS
As it is mentioned several times, the most important narrow-band piezoelectric
transducer for applications in power ultrasound is the generally known prestressed
sandwich transducer, which represents a modification of the primary Langevin’s
transducer. Basic requirements that must fulfill power ultrasonic transducer for
applications as welding of metal and plastics, drilling or cleaning, are consisted in
following: (1) it is necessary to convert certain amount of electric power into
mechanical vibration power at defined frequency; (2) internal power losses must be
low, since the dissipation is a limitation factor in determination of power that
should be transferred; (3) electric impedance must be low, so that minimal
excitation voltage is required, and the adjustment with supplying source is eased;
(4) it is necessary to enable coupling with waveguides without losses in a more
complex oscillatory system.
Transducer is the most important part of a high power ultrasonic system. As it
is mentioned in the introduction, it is made as a prestressed sandwich transducer
with two or more platelets made of PZT ceramic. Sandwich transducer has a
structure metal-piezoceramic-metal, whereat it is most often made with two
piezoceramic platelets, one reflecting and one emitting metal ending, central bolt
that realizes firm connection of the structure, and electric pins made of soft metals.
For metal endings, most often are used: aluminum, steel, titanium, and magnesium,
which will be more talked about later. Nowadays are about 80% of applied
ultrasonic transducers just of this type. The described structure of the prestressed
sandwich transducer is nowadays irreplaceable in many low-frequency ultrasonic
applications, especially in use of sonars, and in the field of power ultrasound
(ultrasonic cleaning and welding). Such type of transducer is designed for
operation at frequencies between 15÷150kHz (mostly 20÷60kHz), with power
intensities greater than 40W/cm2 [101]. Power ultrasonic transducers that have the
cited shape and characteristics are presented in Figure 1.1 and in extension their
appearance, design, and modelling will be analyzed more detailed.
In this chapter are first analyzed most of the existing models of sandwich
transducers, and then is performed an original modeling of ultrasonic sandwich
transducers with different dimensions and with different combinations of applied
120 Modeling of Power Ultrasonic Sandwich Transducers
materials, using the general one-dimensional model, method of seeming elasticity
moduli, as well as applying the new three-dimensional matrix model.
As it is mentioned, in industrial applications of high power ultrasound (ultrasonic
cleaning, welding, etc.) are necessary half-wave transducers with resonant frequency
ranging 20÷60kHz. At such demands, if a monolithic piezoceramic transducer would
be made of, e.g., only PZT4 piezoceramic, at which the sound speed is about 3200m/s,
it would have length of 3.5÷9cm. It means that for achieving high power in such a
structure would be needed large lateral dimensions of the piezoceramic transducer.
Besides that, such transducer is inefficient because of great oscillation energy losses,
which are inversely proportional to the relatively small (regarding, e.g., metals)
mechanical goodness factor.
Alternating excitation force at compact (monolithic) piezoelement is really
efficient only in the central plane of such half-wave transducer, where the maximal
mechanical stress amplitude is Tmax, while the endings of the compact
piezotransducer mostly act as inert masses. Therefore, the parts of the piezoceramic
towards the ends may be replaced by appropriate cheaper nonpiezoelectric metal
endings with far higher mechanical goodness factor. Such structure is known as
complex or Langevin’s half-wave sandwich transducer and it is presented in Figure
4.1.
r
lr
ure
e
lp
le
lp
radna
operating
sredina
medium
T
Tmax
u
uem
Figure 4.1. Symmetric half-wave cylindrical sandwich transducer consisted of same
metal endings, and with equal cross-section areas of elements, with given distribution
characteristics of displacement amplitudes (u) and mechanical stress (T)
Comparing the compact piezoceramic transducers, the sandwich transducers
have greater total mechanical goodness factor in unloaded state, whereat is also
lower the internal temperature of the piezoceramic. Besides that, total
electroacoustic coefficient of efficiency is greater than the one at the more
expensive complete piezoceramic transducer.
Modeling of Power Ultrasonic Sandwich Transducers 121
Construction of the sandwich transducer, presented on the symmetric
transducer example from Figure 4.1 altogether with the principle of operation, at
the same time represents the simplest case of a sandwich transducer. Endings r and
e are made of same material (by that itself they have same acoustic properties),
they are of same length and same cross-sections, which are equal to the crosssection areas of the central ceramic platelets (rings). Thereat the displacement
amplitudes on the end surfaces of endings r and e are equal (ure=uem), as well as the
amplitudes of their velocities (vre=vem).
If the surface at the end of ending e is loaded (e.g., by liquid with density ρW and
with sound speed vW), one may determine the maximal intensity of ultrasound in the
liquid, assuming that into the liquid are emitted ideal planar waves, which will be
talked about more in later exposure [95]. In most applications are used transducers at
which are needed higher values of ultrasound intensity and greater width of application
frequency range, than those which may be realized using symmetric sandwich
transducers from Figure 4.1. Because of that is performed modification of the previous
construction, in order to obtain necessary properties of transducers.
If the endings are made of different materials, whereat the plane between the
piezoceramic platelets, where the mechanical stress amplitude is maximal, onwards
represents the nodal plane of oscillation, emitting intensities on output surfaces of
the metal endings become different (Figure 4.2).
r
lr
ure
e
lp
lp
Tmax
u
operating
radna
medium
sredina
le
T
uem
Figure 4.2. Symmetric half-wave transducer consisted of different metal endings, and
with different cross-section areas of elements, with given
distribution characteristics of displacement amplitudes (u) and mechanical stress (T)
The most interesting case from aspect of ultrasonic sandwich transducers
design is when the emitting intensity of one ending is minimal, and of the other one
is maximal. Because of different roles, the ending r is called reflector ending
(reflector), and ending e is called emitting ending (emitter), so the basic structure
of such sandwich transducer, which is onwards too called symmetric, may be
defined as a reflector-piezoelement-emitter structure. The sandwich transducer
from Figure 4.2 is in literature often called an unsymmetrical transducer, although
is more correct to call unsymmetrical the transducer at which the nodal plane of
oscillation is not between the piezoceramic rings. Concerning the choice of
material for endings, in practice is frequent use of steel-PZT-magnesium
combination, or quite cheaper steel-PZT-aluminum combination. Table 4.1
contains characteristic values essential for material properties for both types of
endings, which are mostly used for high power transducers manufacturing [95].
Table 4.1. Physical properties of materials for high power transducers:
(1) Langley-Hidurax Special; (2) brass (Naval brass) BS 251;
(3) titanium alloy ICI 318A with composition 90% Ti-6% Al-4% V
VALUE
MATERIALS FOR ENDING r
MATERIALS FOR ENDING e
tool
steel
7.85
5250
4.12
aluminum
bronze (1)
8.50
4070
3.46
Naval
brass (2)
8.30÷8.45
3400÷3240
2.82÷2.74
titanium
alloy (3)
4.42
4900
2.17
duralumin
2.79
5130
1.43
magnesium
1.74
4800
0.835
0.58
0.69
0.85÷0.88
1.11
1.68
2.88
EY (10 N/m )
2.18
1.43
0.74
0.42
∆l/l ∆Tp(10 / C)
-
0.95÷0.89
0.35
18÷20
1.06
0.29
11÷16
0.36
9
0.34
23
0.28
26
Qm
≥1400
≥17000
≥3000
≥24000
≥50000
-
ρ (103 kg/m3)
v0 (m/s)
ρ vz (107 kg/m2s)
(ρpvp)/(ρr,evre,em)
11
2
υ
-6 o
Values of the maximal mechanical stress Tmax at nonprestressed transducer
may be drastically reduced (5÷10 times regarding the nominal value), because of
static thermal stressing that issue from the substantially different dilatation
coefficients of glued parts due to heat. Stresses of the disk surface layers become
nonhomogenous because of that, which, along with substantial dissipation at great
excitation intensities, creates indefinite conditions that must be taken into account
in transducer design. By that itself, the value of the maximally possible intensity of
ultrasonic emitting for the given transducer also drastically declines.
In order to enable emitting of higher intensity ultrasound, the prestressing of
the sandwich transducer is performed. All elements of the sandwich transducer are
tightened by central metal bolt, whereby is realized initial mechanical stress in
longitudinal direction. Such great prestress more or less compensates the value
Tmax, which enables emitting of ultrasound of substantially higher intensity. When
it is the question of high power, nowadays are almost as a rule used such
Langevin’s half-wave ultrasonic transducers, whose appearance is presented in
Figure 4.2.
Torsion moment for central bolt tightening is consisted of two components:
one needed for winding the bolt into the emitter threads, and the other needed for
sliding of the bolt head across the external surface of the reflectors ending r.
Material of which the bolt is made should be able to withstand high static stressing,
as well as high dynamic mechanical stresses (that is, high dynamic tensions). The
highest tension strength have bolts made of titanium alloy, with very small losses at
high oscillation amplitudes, and also are used special steel bolts, whereat the bolt
tightening is done by moment-wrench with 70÷80 Nm, whereat mutual rotation of
elements must not occur.
Combination of the bolt and bolt head is characterized by mechanical resonant
frequency that often exceeds the resonant frequency of the main combination of
piezoceramic central rings and metal endings. It leads to the negligible increase of
total resonant frequency and results in additional thread wearing, which, along with
given static prestressing, now also exchange dynamic mechanical energy. Resonant
frequency of the bolt and harmful wearing may be adequately decreased by
negligible decrease of the bolt diameter on short part of its length, in the middle of
the bolt.
At sandwich transducers with only one pair of piezoceramic platelets, the
effective coupling factor, and by that itself the electroacoustic efficiency
coefficient, become too low during operation bellow specific critical frequency.
Low resonant frequency demands great axial thickness of piezoceramic rings lp,
whereby is decreased the transducer capacitance, that is, increases its electric
impedance. For excitation of such transducers, there is a need for extremely high
voltages, which creates additional problems in practical realizations.
Previous limitations are surpassed using package ultrasonic piezotransducers
that consist of several piezoceramic rings with equal dimensions and
characteristics. Platelets are mechanically connected serially, and through the
contact metal foils they are electrically connected parallel, for achieving of higher
ultrasonic power (Figure 4.3).
Figure 4.3. Package piezoceramic ultrasonic transducer
consisted of 6 elements
Total electric capacitance of such piezotransducer is equal to the sum of
individual capacitances of piezoelements. If such system were applied at sandwich
transducers, mechanical cascade of n piezoceramic pairs would have approximately
n times lower thickness resonant frequency and approximately n times greater
value of effective coupling factor regarding the adequate transducer with only one
pair of piezoplatelets. It enables greater electroacoustic efficiency during operation
at low frequencies.
Since ultrasonic sandwich transducer represents a set of elements made of
different materials, design and optimization of multilayer (complex) sandwich
transducers is complicated problem, which implies knowledge of electronics,
mechanics, acoustics, as well as the acoustic properties of the applied materials.
Optimal combination of appropriate materials may be found by method of
adjustment (trial and error method), but it demands a lot of time and cost, so
because of that it is approached to modeling and simulation of transducers in the
procedures of design and optimization of ultrasonic transducers. Modeling is based
on application of the simplest mechanical and electric analogies, and all the way to
the use of standard software for electric circuit simulation, as Pspice, or application
of different mathematical software (Matlab, Mathematica, or FEM software, as
Ansys, Algor, or Abaqus). Nevertheless, analysis of complete sandwich transducers
here started applying the trial and error method, and it was given an example of
design using this method in case of specific sandwich transducer with a reflector
made of heavy metal, in order to perceive the possibilities of such way of
transducer design.
4.1. APPLICATION OF EXPERIMENTAL METHODS IN
ULTRASONIC TRANSDUCER DESIGN
4.1.1. Design of Sandwich Transducers Using the Trial and Error Method
Determination of metal ending length for piezoelectric sandwich transducers is
based on knowing of longitudinal oscillation resonant lengths of homogenous
metal bars for λ/2-like way of oscillation [102]. In that case length of metal bar is:
λ v ( f ,l / d ) ,
(4.1)
l= = z
2
2 fr
whereat is vz(f, l/d) sound speed in metal bar of diameter d and length l, fr is
resonant frequency of the transducer, and λ=vz/fr. Length of the symmetric
sandwich transducer with piezoceramic in the middle of the metal bar, which has
structure presented in Figure 4.1, assuming that lr=le, amounts to:
(4.2)
l' = 2le + 2l p ,
wherefrom is the length of one metal ending:
le =
1
(l '−2l p ) .
2
(4.3)
As mentioned, in practice are used sandwich transducers with endings made of
different metals, where one serves as an emitter, and the other as an ultrasound
reflector in purpose of performing the amplification of direct emission of
ultrasound. In that case, the ending lengths differ, too. Real value of length le
depends on the kind of metal and piezoceramic used as a piezotransducer.
Therefore, for le one may give a semiempirical formula:
1
(4.4)
le = l'− A ⋅ l p ,
2
where constant A depends on the kind of the metal and of piezoceramic, as well as
from the operating frequency. If in equation (4.4) l’ is substituted according to
expression (4.1), one gets:
v
(4.5)
le = z − A ⋅ l p ,
4 fr
where vz is the sound speed in long metal bar (l>3d).
For design of the endings and determination of the constant A it is logical to
use diagrams similar to the experimental McMahon’s diagram, presented in Figure
3.2, or experimental curves presented earlier in Chapter 3.2.4.3. These diagrams
contain experimental values measured on the vibrational platform for duralumin
and steel bars, and represent dependence of the longitudinal oscillation resonant
frequency on length, for λ/2-like way of oscillation. Since during workmanship of
transducer elements there must be drilled holes due to mechanical joining, given
diagrams have to be recorded separately for endings without openings, for elements
with bolt opening through the whole ending, as well as for elements both with
opening and with saddle, for different cross-sections and diameters of the
mentioned cylindrical endings. From given dependences one may notice that in
ultrasonic transducer design, beside lateral dimensions, great influence have both
the shape and the construction of the endings. However, these diagrams stand for
individual, free metal endings, and they are not of very much importance for
defining metal ending models that would be used in design and modeling of
complete ultrasonic sandwich transducers, which as a unique system have quite
different resonant frequency characteristics. Because of that, for determination of
constant A in the expression (4.5) it is crucial the experience of the designer of such
system, which represents the essence of trial and error method application.
4.1.1.1. Example of Design of a Transducer with Heavy Metal Reflector
An example of application of the trial and error method is presented for the case
of design of a transducer with heavy metal reflector, whose application here makes
sense because of specific, and in literature insufficiently available, characteristics of
such reflector, as well as the transducer with such reflector itself. In the electric sense,
the piezoelectric sandwich transducer, analyzed in a broader frequency range,
represents predominantly capacitive impedance, except in finite number of resonant
frequency regions, in which impedance has complex character [102]. Properly
designed transducer possesses frequency interval of natural resonant mechanical
oscillation, which in case of mechanically unloaded transducer coincides with the
interval of electric resonant oscillation of the transducer. Electronic ultrasonic
generators the transducer is connected to, change the region of electric resonant
oscillation, and since the transducer is in contact with operating medium, which loads
it additionally, the change of the resonant mechanical oscillation interval occurs too. It
leads to insufficient coincidence of these intervals, which affects negatively the
transducer efficiency. The aim of design of the transducer and its accompanying
electronic generator is that these intervals in real operating conditions mutually
coincide and follow themselves synchronically, if the load regime is variable. Then
one achieves maximal stability and efficiency of transducer oscillation.
Coincidence of the electric and mechanical resonant frequencies implies that
those frequencies are in mutually close frequency vicinity of the impedance
extremum. Which one of the two resonant regimes should be chosen as operating,
depends on the characteristic of the medium that receives acoustic energy, whereat
in applications in which the transducer is used as a source of maximal oscillation
amplitude, always is used operation in the region of transducer impedance
minimum. It means that problem of power transfer during conversion of electric
energy into acoustic energy is reduced to it that total input electric power is
delivered to the transducer in form of active power, e.g., to eliminate the reactive
electric power that would return from transducer to generator, and thereat as large
part of that input power as possible be transfomed into mechanical work, that is, to
consume as less input energy as possible on transducer heating. A simplified
scheme of the measuring circuit by which one may simply determine input electric
impedance of an ultrasonic sandwich transducer, that is, of any individual
piezoceramic element, is presented in Figure 4.4.
zul
g
50 Ω
V1
V2
p
50 Ω
Vg
Figure 4.4. Scheme of the measuring device for determination of ultrasonic
transducer input electric impedance
Thereat the alternating excitation voltage Vg, the internal resistance of the
voltage source Rg, and resistance Rp are known, while zul is the unknown
transducer impedance. Measuring the voltage V1 and V2, and using the voltagedistributing frame, one comes to the expression for input electric impedance of the
transducer in function of their relation, that is, in function of frequency:
⎛V
⎞
(4.6)
zul ( f ) = p ⎜⎜ 1 − 1⎟⎟ .
⎝ V2
⎠
During graphic interpretations of the characteristics of ultrasonic transducers
and piezoceramics, further is, as hither, instead of measuring of the voltage relation
V1/V2, performed measuring of the voltage attenuation in dB (expression (2.10)):
V
⎛z
⎞
adB = Zul = 20 log 1 = 20 log⎜ ul + 1⎟ ,
(4.7)
50
V2
⎝
⎠
whereat is, based on attenuation characteristic, possible to determine impedance
dependence characteristic on frequency for sandwich transducers and
piezoceramics using the expression:
⎛ a dB
⎞
zul ( f ) = p ⎜⎜10 20 − 1⎟⎟ .
(4.8)
⎜
⎟
⎝
⎠
Using heavy metal, which is characterized by specific density even three times
greater than the steel density, it is possible to construct an ultrasonic transducer based
on PZT ceramic. Such transducer, besides that it is characterized by smaller
dimensions regarding the classical steel-PZT-aluminum transducer, uses as a reflector
material with substantially higher melting temperature than the melting temperature of
steel. In this chapter is presented the construction, and considered electric
characteristics of this type of transducer applying the trial and error method [103].
In Figure 4.5(a) is presented the appearance of designed ultrasonic sandwich
transducer that contains two PZT piezoceramic rings, with two-sided applied silver
coatings, which serve as excitation electric contacts of the whole ultrasonic transducer.
In purpose of the transducer excitation voltage decrease a parallel connection of two
piezoceramic rings is used, whereat the polarization is such that external platelet sides
are at mass potential, while the middle connection is at higher potential. Metal endings
are connected with piezoceramic platelets by steel bolt, where the bolt is so tightened
that operating point of the variable mechanical stress located in the region of pure
compression. Namely, as already mentioned, in this region the piezoelement has
maximal strength and enables obtaining of maximal ultrasonic power. Central bolt has
such construction that enables the prestressing change. Tight mechanical connection
between the piezoceramic rings, electrodes, and metal endings is also provided by
gluing and tightening the contact surfaces.
In purpose of studying the possibilities of application of heavy metal as a
reflector, two types of transducers that are designed in that purpose, are analyzed
parallel. The first, already mentioned transducer contains heavy metal reflector
(Figure 4.5(a)), and the second, earlier designed transducer is with classical steel
reflector (Figure 4.5(b)), which has fundamental resonant frequency of 26.06kHz.
The last one transducer will be used in chapters 4.2.2, 4.2.4, and 4.3.3 for
comparison with characteristics of transducers designed by different, here proposed
sandwich transducer models. The used heavy metal represents an industrial alloy
based on wolfram (92% W, 4% Ni, 4% Fe), with specific density 17.4⋅103kg/m3. In
both cases, the emitter was made of duralumin.
Ratio of wave strengths of piezoceramic rings and metal endings is set such
that energy is emitted predominantly in one direction (in direction of the operating
emitter). Thus, the ultrasonic transducer acts as a half-wave resonator. During the
transducer operation, in it occur mechanical stresses that are contributed by all its
constitutive elements. In this case, maximal stress of transducer prestressing is
30÷35MPa. In order to achieve that stress, besides the mechanical prestressing, it is
also necessary precise planparalell workmanship of all joints and fine processing of
joining surfaces of the transducer elements.
4.1.1.2. Experimental results
Comparison of these transducers may be performed through the frequency
characteristics of their electric input impedances, which are presented in Figure 4.6,
and which are recorded by network analyzer HP 3042A (Hewlett Packard). For
final setting of transducer dimensions by the trial and error method is used
40
5
40.2
15.4
43
42
17.4
65.6
3.7
84.1
5
5
12
3.7
8
17.3
31.5
13.1
38
38
40
40
(a)
(b)
te{ki metal
heavy
metal
~elik
steel
aluminijum
aluminum
Figure 4.5. Designed ultrasonic sandwich transducers with
heavy metal reflector (a) and with steel reflector (b)
vibrational platform of the Herfurth Company. Final dimensions of both
transducers are given in the very Figure 4.5, where the heavy metal reflector is
12mm, and the steel reflector is 31.5mm long. Using the measuring circuit from
Figure 4.4, at resonant frequencies denoted in Figure 4.6, one gets the following
impedance values: for the transducer with heavy metal reflector zul,min=17.45Ω, and
for transducer with steel reflector zul,min=11.51Ω. Comparing the impedance
frequency narrow-band characteristics from Figure 4.6, one may notice that
transducers have practically equivalent characteristics. Beside those characteristics,
in Figure 4.7 are presented broadband characteristics of impedance dependence on
frequency for both transducers, measured in frequency range from 0.5÷49.5kHz.
Thereat the most outstanding resonant mode determines the fundamental resonant
frequency of the transducer, and thereat is isolated enough, that is, distant by
frequency from the remained resonant modes of the impedance-frequency
characteristic of the transducer that may exist. In this case, there occurs only one
higher additional resonant frequency mode at transducers with heavy metal
reflector. For practical application the best are transducers that have only one
solitary enough and very outstanding resonant region, which is, beside good
electric characteristics one of the basic demands in design of an optimal transducer.
u 1 (dB)
Zullog
(dB)
20
u2
50
40
(a)
30
(a)
(b)
(b)
20
10
(a) fr=25.97 kHz
(b) fr=26.06 kHz
2.6
1.8
f (kHz)
20
25
30
Figure 4.6. Narrow-band characteristics of impedance dependence on transducer
frequency with heavy metal reflector (a) and with steel reflector (b)
Based on all cited, one may conclude that electric characteristics of the transducer
with reflector based on heavy metal are equivalent to the characteristics of the
transducer with steel reflector. Thereat the dimensions of the newly constructed
transducer are substantially smaller. Thereby is, using the trial and error method,
enabled application of heavy (and high-temperature) metals for workmanship of
reflectors in all applications where smaller transducer dimensions are demanded.
Results exposed in this chapter relate to the construction and application of ultrasonic
transducers used in sonotrodes for application in metallurgy [104].
Development and application of different models for design and optimization of
power ultrasonic transducers are improved with occurrence of possibility to combine
the design with series of experimental measurements, which are performed by laser
interferometers [105], [106], [107]. Based on such approach it is possible to assess
very quickly the quality of every applied model or concretely realized ultrasonic
transducer. It is possible to determine distribution of oscillation amplitudes along any
direction in the transducer that oscillates in air or liquid, whereby is determined the
position of the wave nodes and bellies and enables optimal choice of location at which
the transducer is fixed. Such analyses also serve for determination of appropriate
sensor position on a transducer, whereby one gets returning signals in real time for
automatic control of the transducer and ultrasonic process.
Zul
20(dB)
log u 1 (dB)
u2
(a)
(a)
(b)
(b)
f (kHz)
Figure 4.7. Broadband characteristics of impedance dependence on frequency of
the transducer with heavy metal reflector (a) and with steel reflector (b)
4.1.2. BVD Model of Ultrasonic Sandwich Transducers
Characteristics of the ultrasonic piezoelectric sandwich transducers in the
vicinity of the fundamental resonant frequency may be, as well as of the
piezoceramic itself, described by electromechanical analogies represented by a
serial LC resonant circuit that is in parallel connection with static capacitance of the
transducer, whereat is such equivalent scheme also accurate enough for many
practical applications. Several authors dealt with determination of the circuit
elements by procedures based on solving the wave equation, and all that in case of
simple composite piezotransducers with one ending added, as well as in case of
more complex transducers with two metal endings [101], [108]. Resistance in the
equivalent electric circuit depends partly on the losses in the transducer itself and
partly on the medium conditions on the operating end of the transducer. Such
simple model was useful for defining the most significant parameters of the
piezoelectric sandwich transducers, which determine the criteria for making a good
transducer, and these are electric and mechanical impedance, electric and
mechanical Q factor, electromechanical coupling factor, electric losses,
electromechanical and mechanoacoustic efficiency [109]. However, such approach
is not appropriate in sandwich transducer constructions with several excitation
piezoceramic platelets, or transducers with metal endings of variable cross-section
and different form, tightened by metal bolt.
Such way of sandwich transducer modeling kept its application until now in
design of adaptation circuits, but for already concretely realized sandwich
transducers [110]. All values of equivalent circuit elements may be evaluated based
on data about geometric dimensions, properties of applied materials, and using
certain empiric rules. If the transducer is designed properly, so that desired
resonant frequency is at enough distance from other resonant frequencies,
mentioned data and experimental measurements will provide enough accurate
values of the equivalent circuit elements. Based on such equivalent circuits, it is
possible using the filter theory to design adaptation circuits that, when they are
inserted between the supplying source and the transducer, enable control of the
transducer characteristics [111]. However, if the resonant modes are insufficiently
isolated, accuracy of such equivalent circuit is insufficient and ultrasonic
transducer then must be represented by an equivalent scheme that contains parallel
combination of serial
resonant circuits, which correspond to different resonant
modes of the transducer. Then is better to determine the elements of the equivalent
circuit elements experimentally, and one original experimental method for
determination of the parameters of such equivalent circuit is already presented in
Chapter 2.1.1 [56]. Values of the static capacitance and elements of the serial
resonant circuits are determined based on measured frequency dependence of the
input electric impedance of the transducer, using an optimization technique that
contains multidimensional simplex identification algorithm.
4.1.2.1. Example of BVD Model Application in Design of the Circuit for
Ultrasonic Transducer Adjustment
In this chapter there is described design and realization of an original
adaptation circuit of ultrasonic transducers using the BVD model. Using the filter
theory, it is possible to design adaptation circuits which, when inserted between the
source and transducer, enable band-pass control. The usual method that leads to a
modest increase of the band-pass for such transducers is the use of serial
connection of inductance and transducer. However, much more complex adaptation
networks may be designed, whereat is enabled greater range in adjusting the
transfer characteristics of the transducer. The design method of such adaptation
circuits is discussed in this chapter, and a concrete example of a band-passer that
illustrates performance improvement for a practical ultrasonic transducer is
presented. Results of simulation and measuring on the realized circuit are in good
agreement and they confirm correctness of such theoretical approach.
The transducer operates in resonant regime, whereat is the Q-factor of the
transducer (electric and mechanical) very high, which means that the impedance
change in the vicinity of the resonant frequency is quite great. As an illustration of
the transducer impedance character may serve measured characteristics of the
impedance dependence on frequency in Figures 4.6 and 4.7. Else, the sole design
of the transducer used in this chapter is not considered in this part of exposure.
As already mentioned, equivalent model of the ultrasonic transducer, as well
as in case of piezoceramic, represents a serial resonant circuit consisted of
electromechanical analogies of the resonant structure represented by L1, C1, and R1,
in parallel connection with static capacitance of the piezoelectric crystal C0 (Figure
4.8). Here too the resistance 1 depends partly on losses in the transducer itself and
partly on the medium conditions on the operating end of the transducer.
C1
C0
L1
1
Figure 4.8. Equivalent BVD model of an ultrasonic sandwich transducer
All values of the equivalent circuit may be evaluated applying the design rules
that demand as input mechanical dimensions, properties of the applied materials
and certain empiric rules. If the transducer is designed properly, so that desired
resonant frequency is at enough distance from other resonant modes, experimental
measurements will provide enough accurate values of the equivalent circuit
components. It is necessary for later correct design of the adaptation circuits.
Measuring the electric frequency characteristics of a concrete transducer in
operating conditions (Figure 4.13) the following values are obtained:
fr=41.263kHz, R1=47.492Ω, C1=1.072nF, L1=13.878mH, and C0=5.886nF.
Different dimensions and choice of the piezoeramic type and choice of
adjusting mechanical elements, whereat is such approach in design limited to
transducers operating at high frequencies, may achieve certain control of the
transfer characteristic, whereat it is primarily meant the width of the band-pass. At
conventional transducers, which operate at frequencies of several tens of kHz, such
method of band-pass adjusting is not acceptable. Usual method that leads to a
modest increase of band-pass for such transducers is the use of serial connection of
inductance and transducer.
At such adjustment, design of the compensational inductance represents specific
problem [45]. Namely, when an ultrasonic transducer is excited, due to the
piezoelectric effect on the transducer is generated alternating voltage of several kV.
Practically, the whole generated voltage is located on the serial inductance, which
means that its design must be performed as at high-voltage inductances with ferritic
core. Although this adjusting procedure has some credits in efficient coupling of the
source and transducer, properties of the band-passer filter are limited. If the resistance
1 is considered as an output of the circuit, in the position of one resonant response
occurs double resonance. Therefore this method in the vicinity of the fundamental
resonant frequency creates two discrete resonant frequencies that depend on the
inductance value, whereat one doesn’t get frequency characteristic of the band-passer
filter. If one considers as an output the complete transducer, that is, the impedance that
loads the generator, increase of the serial inductance decreases the fundamental
resonant frequency. It may be noticed in Figure 4.9, which represents dependence of
the transducer input impedance on frequency, obtained by PSpice software for electric
circuit simulation for different values of compensational inductance.
50
i
m
p
e
d
a
40
n
s
a
30
20
10
L=5mH
bez kalema
without coil
L=2mH
0
10Kh
20Kh
30Kh
40Kh
50Kh
60Kh
A
Frequency
Figure 4.9. Influence of the compensational inductance on input impedance
of the transducer
Certain improvements may be obtained by adding the parallel capacitance to
the transducer, whereby is effectively increased C0. Hereby is obtained mutual
approaching of the previously mentioned resonant minimums. Waviness of the
response is quite great, and it may be decreased only by adding a resistor serially
with compensational coil, which is in such applications unacceptable because of
substantial power losses on the serial resistor.
Technique of adaptation circuit design is based on assumption that the serial
resonant circuit in the equivalent model may be treated as a serial LC branch of the
ladder filter, whose terminate element is resistance R1 (Figure 4.10). Then the static
capacitance of the piezoelectric ceramic C0 may be observed as a part of the
capacitance C2 in the arresting LC branch.
L3
C3
C1
L2
C0
L1
1
C2
Figure 4.10. Scheme of the adjustment circuit with the transducer circuit
Propagating this idea, one may realize a ladder filter, whereat limitations exist
since the components in the ladder connection with equivalent transducer circuit
may be modified within the range acceptable for practical realization.
Components of the equivalent circuit L1, C1, C0, and 1 are input data in the
synthesis procedure and after that, one should choose the desired filter family. Here
is further designed Chebyshev’s adaptation filter, which is of first order, and at
which one should choose definite value of the waviness Apmax [110]. Thereat,
principal limitations that the transducer imposes in that case are central frequency
of the band-passer filter, which must coincide with serial resonant frequency
determined by C1 and L1, as well as the fact that waviness of the band-passer filter
Apmax and band-pass width are mutually linked (the wider the range, the greater the
waviness, and vice versa).
Based on the chosen value Apmax, one may come to the normalized lowpassing prototype of the third order filter presented in Figure 4.11.
L3n
L1n
C2n
R1n=1Ω
Figure 4.11. Low-passing equivalent circuit of an adjusted transducer
Transfer function of the circuit from Figure 4.11 is:
1
H (s ) =
,
(4.9)
3
a1 s + a2 s 2 + a3 s + 1
whereat is: a1=(L1nC2nL3n)/ 1n, a2=L3nC2n, and a3=(L1n+L3n)/ 1n. Comparing the
modulus of the transfer function (4.9) with corresponding transfer function of the
Chebyshev’s third order filter one gets expressions that link elements of the
prototype from Figure 4.11 (and on which base are easily obtained L1n, C2n, and
L3n in closed form):
ε
3
L33n − ε L23n − = 0 ,
(4.10)
2
2
2ε
(4.11)
L1n = 3ε − L3n + 2 ,
L3n
4ε
C2 n =
,
(4.12)
L1n L3n
where ε = 10 0.1 Ap max − 1 .
Transformation of the low-passing element into a band-passer is further step in
purpose of denormalization of ladder filter data. Every coil of the low-passing filter
is transformed into a serial LC circuit, while every capacitor of the low-passing
filter is transformed into a parallel LC circuit. Resonant frequency of all LC circuits
(serial and parallel) is the same and coincides with central frequency of the bandpasser filter fr. Expressions on which base are determined values of denormalized
elements are [112]:
Li =
Ci =
fr
Lin , i = 1, 3 ,
ωr
∆f
∆f 1
fr Lin
C2 =
L2 =
1
1ω r
fr
C2n
∆f
, i = 1, 3 ,
(4.13)
(4.14)
1
1ω r
∆f 1 1
fr C2n ω r ,
,
(4.15)
(4.16)
where ωr=2π fr, and ∆f is band-pass.
Verification that such procedure gave correct results may be obtained by
modeling the adaptation circuit, that is, by its electric simulation.
4.1.2.2. Numerical and Experimental Results
Concrete Langevin’s transducer, which is here subject of analysis, is designed for
application in the system for ultrasonic cleaning, although the adaptation procedure
may be generally applied for any transducer used in the power ultrasound technique.
Based on the measured dependence characteristic of the input transducer impedance
on frequency, it is decided to design an adaptation circuit with Chebyshev’s
characteristic in the band-pass, so that response transducer impedance is presented by
the attenuation characteristic given in Figure 4.13(a).
Design procedure for adaptation circuit encompasses application of adequate
Matlab software in the part for normalization, transformation of low-passing
element into a band-passer, and graphic presentation of results. Thus are for value
Apmax=0.5dB obtained following values of the normalized elements: L1n=0.79814,
C2n=1.300145, and L3n=1.346486; that is, previously described synthesis procedure
gave following values of the adaptation circuit: L3=23.4mH, C3=635.4pF,
L2=1.48µH, C2=10µF, ∆f=434.7 Hz.
Verification of the synthesis is performed applying the PSpice software for
electric circuit simulation, whereat are the input impedance characteristics of the
circuit in case of output on the resistor R1 (a) and on the whole transducer (b), with
and without adaptation, given in Figure 4.12.
In Figure 4.13 is presented experimental broadband curve of input impedance
dependence on frequency for a concrete transducer without adaptation circuit (a)
and with adaptation circuit (b), which is recorded by automatic network analyzer.
Same narrow-band characteristic is presented in Figure 4.14.
Previous calculation is derived at Apmax=0.5dB. Using equations (4.13) and
(4.14) one may find dependence of the obtained band-pass on the maximal
attenuation in the band-pass for concrete transducer, which is presented in Figure
4.15 for values Apmax up to 3 dB.
100
Zul[dB]
output on the resistor R1
with
adaptation
sa prilag.
80
60
(a)
40
bez prilag.
without
adaptation
20
0
36Kh
38Kh
40Kh
42Kh
44Kh
46Kh
Frequency
50
Zul[dB]
output on the whole transducer
40
with
adaptation
sa prilag.
30
(b)
20
bez prilag.
without adaptation
10
0
36Kh
38Kh
40Kh
42Kh
44Kh
46Kh
a
Frequency
Figure 4.12. Characteristic of the input impedance of a transducer with and
without adaptation, obtained by circuit simulation
Therefore, in this chapter is presented an example in which is still
irreplaceable application of the BVD model in design of the adaptation circuit of
concretely realized power ultrasonic sandwich transducers. An adaptation circuit
for concrete transducer used in ultrasonic system for cleaning and degreasing is
designed. Also, there are presented comparisons of some solutions with adopted
solution of the band-passer filter obtained applying the filter theory. There are
given simulated and experimental diagrams of correspondent waveforms measured
on the transducer. Obtained results confirm correctness of the applied original
theoretical approach and advantages of the proposed solution.
Zul (dB)u1
20log
(dB)
u2
80
60
(b)
(b)
40
(a)
20
(a)
f (kHz)
fr=41.263 kHz
0
0.5
20
40
60
79.5
Figure 4.13. Input impedance of the transducer with and without adaptation,
recorded by network analyzer
Zul (dB)u1
20log
(dB)
u2
40
30
(a)
(b)
20
(b)
10
(a)
f (kHz)
0
36
38
40
42
44
46
Figure 4.14. Narrow-band characteristic of the input impedance of the transducer
with and without adaptation, recorded by network analyzer
∆f (Hz)
1000
900
800
700
600
500
400
300
200
100
0
0
0.5
1
1.5
2
2.5
3
Apmax(dB)
Figure 4.15. Dependence of the band-pass on maximal
attenuation in the band-pass for a concrete transducer
4.2. ONE-DIMENSIONAL MODELS OF PIEZOCERAMIC
ULTRASONIC SANDWICH TRANSDUCERS
As mentioned in the introduction, in literature there exist several one-dimensional
approaches in modeling the sandwich transducers and all have in common that
unknown values are functions of time and only of the longitudinal coordinate. In this
analysis, the first attempt of realization of an acceptable model of ultrasonic sandwich
transducer represents the improvement of the one-dimensional equivalent circuit. At
first are in concrete examples illustrated possibilities of several existing onedimensional approaches to the design of ultrasonic sandwich transducers, that is, to
determination of dimensions and resonant frequencies of different transducers.
Thereat, these models do not take into account piezoelectric and anisotropic properties
of excitation piezoceramic rings. Then is realized general one-dimensional model of
ultrasonic transducers, which includes all its consisting parts and piezoproperties of the
excitation ceramic. As mentioned in the introduction, most of the one-dimensional
models do not include into the model the influence of the prestressing bolt, or includes
just some of its parts. In the transducers with small length of the metal endings,
presence of the bolt becomes significant in determination of the resonant frequency,
while in the transducers with longer metal endings, influence of the bolt on the
resonant frequency may be neglected.
4.2.1. Langevin’s Equation
For design of the simple sandwich transducers, the most often is applied a
simple frequency equation, which is called the Langevin’s equation [113]. As
simple transducers here are implied the symmetric transducers, whereat the
endmost endings are cylindrical nonconical elements, made of same material and
with same cross-section. Basic sandwich transducer is designed as a symmetric
half-wave (λ/2) resonant structure. It means that every λ/4 section may be observed
separately. Langevin’s equation links the resonant frequency with characteristic
impedances, sound speeds, and dimensions of transducer elements. It may be the
most easily derived based on the simplified equivalent circuit for piezoceramic
loaded at one end by metal ending, and all that using the transfer line analogy, both
for the piezoceramic and for the metal ending. When the properties of applied
materials (ceramic thickness, ceramic area, as well as the demanded operating
resonant frequency), are known, one may determine the needed length of the metal
ending. This is, also, a very flexible method of transducer design, because many
different forms and material combinations give same resonant frequency. This
design method is very popular because of its simplicity, and besides its
disadvantages, it still remains the base for design of many transducers.
Langevin’s equation for both λ/4 sections of the λ/2 sandwich transducer
from figure 4.1 reads:
Zep tg(kele ) tg k pl p = 1 ,
(4.17)
Zrp tg(kr lr ) tg k pl p = 1 ,
( )
( )
where indexes e, p, and r relate to the emitter, piezoceramic, and reflector, so that,
e.g., for the transducer from Figure 1.1(a) stands: le=l1+l2, lp=2⋅l3, lr=l4+l5. Besides
that is Zrp=Zcr /Zcp, Zep=Zce /Zcp, and Zcn=ρn vn Pn are characteristic impedances for
n= e, p, and r. ρn, Pn and ln are densities, areas, and lengths (thicknesses) of the
corresponding transducer elements, respectively, and kn is the wave number:
kn =
2π
λn
=
ω
vn
= 2π f
ρn
EYn
.
(4.18)
Equation (4.17), formed from the resonance condition, may be written in form
of (n=e,r), for n-th ending,:
⎛ ω l ⎞ ⎛ ω l p ⎞⎟
tg⎜⎜ n ⎟⎟ tg⎜
= qn .
(4.19)
⎝ vn ⎠ ⎜⎝ v p ⎟⎠
Angles ωl/v have values from 0÷π/2 rad and may be presented on a diagram where
qn is used as parameter (Figure 4.16), and for clearness, the axes are in form of:
lp
l
2 ωln
2 ωl p
and
.
(4.20)
= n
=
1
1
π vn
π
vp
λn
λp
4
4
Family of curves ln/(λn/4)=f(lp/(λp/4)) is given in Figure 4.16 for practical
values qn ranging 0.4÷4.
It is often defined the amplification of the ultrasonic oscillations amplitude Gn
on the ends of the endings, at constant mechanical stress amplitude Tmax [95]:
⎛ ω lp ⎞
⎟ ,
(4.21)
Gn = qn2 − qn2 − 1 sin 2 ⎜
⎜ vp ⎟
⎝
⎠
(
)
4ln/λn
1.0
qn
0.8
4
2.5
1.5
0.6
1
0.6
0.4
0.4
0.2
4lp/λp
0
0.2
0.4
0.6
0.8
1.0
Figure 4.16. Demanded ending length ln for given piezoceramic
ring with thickness lp, in function of the operating frequency
whereat the coefficients qn and Gn are equal to one for both λ/4 sections of the
symmetric sandwich transducer from Figure 4.1, while at the symmetric transducer
from Figure 4.2, for individual λ/4 section, they are greater or smaller than one,
depending on the ending construction. Gn has value ranging between qn2 and 1, and
it rises if ρnvn declines. Corresponding dependences Gn=f(lp/(λp/4)) are presented
in Figure 4.17.
Gn
qn
4
10
5
2.5
1.5
2
1
1
0.5
0.6
0.2
0.4
4lp/λp
0.1
0
0.2
0.4
0.6
0.8
1.0
Figure 4.17. Realized amplification of the ultrasonic oscillations amplitude as a
function of operating frequency for a given piezoceramic ring of thickness lp
Based on the previous diagrams one gets impression that for detailed calculations in
transducer design it is enough to choose adequate materials with known parameters (ve, vr,
vp, that is, λe, λr, and λp). However, practical transducer calculations, beside the quoted, are
also caused by shapes, that is, by the construction of piezoelements and endings.
It is very important what the ratio of mechanical carrier of ultrasound and onefourth of wavelength is. For example, if that ratio rises from 1 to 2, in that case
effective phase velocity of the longitudinal waves vz (first resonant mode)
automatically declines substantially bellow the value known as “thin bar velocity”,
which is a general name for the phase velocity v0 = EY / ρ in a thin bar. Besides
that, there occurs one more oscillatory regime, at which is very great phase velocity
(second resonant mode). Necessary corrections of the velocity vz depend on the value
of the Poisson’s coefficient υ, which will be talked about later.
Table 4.1, beside other important properties for high power transducers, contains
numerical data for v0, as for ending e, whose emission is of great intensity (Ge>>1), so
for the ending r, whose emission intensity is as small as possible (Gr<1).
Concerning the piezoceramic material, the piezoceramic has mostly small sound
speed, and at frequencies of 40÷50kHz the ratio of piezoceramic disk diameter and
one-fourth of wavelength must be greater than 1. Based on the diagram from Figure
4.17, one may notice that the greatest amplification of oscillation amplitude Ge, as well
as the greatest ratio Ge/Gr, which is essential in this case, is obtained when the
wavelength at the piezoceramic material is 20 or 30 times greater than the
piezoceramic disk (ring) thickness. It leaves large space for the choice of the lowest
possible resonant frequency. Nevertheless, one cannot choose extremely low applied
frequencies for the given pair of piezoceramic rings because of two factors:
1) Effective piezoelectric coupling factor becomes too low at extremely low
frequencies. Although the reflector ending with qr<1 insignificantly improves the
coupling, the emitter with qe>1 reduces it, and thereupon the total coupling does not
differ much from the one that would have sandwich transducer with qn=1. qn=ρpvpPp /
ρnvnPn is the ratio of the characteristic acoustic impedances of the central piezoceramic
part and the n-th ending, whose cross-section areas are Pp and Pn (Pn≥ Pp, Figure 4.2).
Based on diagram presented in Figure 4.18, for piezoceramic platelet made of
PZT ceramic with diameter dp and length lp=6.35mm stands: keff≤0.5k33 for λp≥25lp,
i.e., fr≤20kHz [95].
2) At lower frequencies emitting waves are not planar any more, so now the
emitting intensity of the transducer that emits ultrasonic waves into, e.g., liquid
medium is modified multiplying by emitting coefficient ζ’ in water, whose
dependence regarding the de,r/(λW/4) is presented in Figure 4.19. One may notice from
the diagram that at low frequencies ultrasound intensity abruptly declines with
frequency (but not linearly).
Besides that, a notable effective mass is added to the transducer, that is, a
liquid load, which also affects the ultrasound intensity decrease in that liquid
medium. This attenuation is determined by coefficient ζ’’, whose dependence is
also presented in Figure 4.19.
1.0
8/π2
<1
(keff/k33)2
1 >1 qn
0.5
0.25
0
0.15
0.5
4lp/λp
1.0
Figure 4.18. Diagram of minimal thickness lp of the ring or disk made of PZT
piezoceramic: shaded area represents the region where
keff ≤ 0.5k33, fr ≤ 20kHz at thickness lp=6.35mm
Based on minimal allowed coefficient ζ’, which is approximately equal to
0.75, one gets minimal value of the area diameter that emits ultrasound, and which
is approximately equal to the half of the ultrasound wavelength in operating, liquid
medium.
Applying these conclusions to the case of PZT piezoceramic disks of diameter
dp=38.1mm with aluminum or magnesium ending, with insignificantly greater
diameter de, one may find that minimal applied frequency amounts to around
20kHz. Same value of the minimal frequency is obtained if the coupling factor is
taken as a criterion (Figure 4.18). For frequencies bellow 20kHz one pair of
piezoceramic platelets in insufficient.
1.2
ζ''
ζ'
ζ'
1.0
0.8
0.6
ζ''
ρ w, v w
de
0.4
0.2
0
2
4
6
8
10
12
4de,r/λW
Figure 4.19. Diagram of minimal diameters de and dr of the endmost ending areas
that emit ultrasound: shaded area represents the region of unacceptably small
value of the real part of the emitting coefficient
4.2.2. Equation of a Half-wave (λ/2) Sandwich Transducer
It is already mentioned that it is often desirable to make the endmost endings
of material with different mechanical impedances, whereby the transducer becomes
a velocity transformer (Figure 4.2). The best combination is steel-titanium for
metal endings, where titanium, which has the role of an emitter, has greater
ultrasound velocity than steel because of its small acoustic impedance. Titanium
also has significant advantage because of its mechanical strength, but usually is for
transducer making used cheaper combination with emitter of duralumin. Limit
mechanical stresses of the titanium are about 10 times greater than the
correspondent stresses of duralumin, while maximal pressures which the titanium
withstands without danger of cracking is around 7 times greater than the
correspondent pressures of duralumin [114].
It is possible that nodal plane of oscillations divides the piezoceramic equally,
and in that case, transducers are called symmetric, too [113], [115], [116]. As
already mentioned, the same Langevin’s equations, as in the previous case of the
transducers with identical endings, may be used in design of such transducers with
different endings, if the cited symmetry exists. In the opposite case, when the
middle of the piezoceramic is not in the oscillation node, it is the matter of an
unsymmetrical transducer [113], [117], [118]. Frequency equation for this case is
derived in the same way as at symmetric transducer. This type of the transducer is
useful if the transducer is fixed on the external cylindrical surface part of the one of
endmost endings, whereat is demanded minimal oscillation damping, so that fixing
is performed on the metal ending in the region of oscillation node. It is believed for
these sandwich transducers too, that through emitter, piezoceramic, and reflector
are propagating planar longitudinal waves with velocity v0 = EY / ρ (bolt
influence is still being neglected). Boundary (contour) surfaces of the transducer
are unloaded. Emitter, piezoceramic, and reflector are passive mediums physically
represented by their elasticity moduli EY and densities ρ, and geometrically
represented by their length l and cross-section P. Reference axis coincides with the
polarization axis, which is at the same time symmetry axis of the transducer with
cylindrical shape. By such analysis, one comes to the general equation of the
sandwich transducer, which brings into relation the resonant frequency of the
transducer with dimensions of the emitter ending, piezoceramic, and reflector
ending, and their characteristic impedances. Therefore is clear that here too many
different forms and material combinations will give same resonant frequency.
Since the material, resonant frequency, and any two of the three lengths are chosen,
one may, based on the frequency equation of the sandwich transducer, determine
the unknown dimension.
Therefore, workmanship of the metal-piezoceramic-metal multilayer
transducers, which oscillate at given resonant frequency fr in direction of the
consisting elements thickness, is based on the general equation that represents the
sandwich transducer. Assuming that planar longitudinal waves are propagating
through the emitter, ceramic, and reflector with velocity vi = EYi / ρ i , one comes
to the mentioned general equation of the sandwich transducer [115]:
(
)
( )
,
+ Zrp sin(kr lr )cos(k pl p )cos(ke le ) + Zep cos(kr lr )cos(k pl p )sin(ke le ) = 0
cos(kr lr )sin k pl p cos(ke le ) − Zrp Zep sin(kr lr )sin k pl p sin(ke le ) +
(4.22)
where the meanings of the used indexes are explained in the analysis of the
equation (4.17). By substituting lr=λr/4, equation (4.22) passes into the well-known
Langevin’s equation (4.17). It means that when the reflector length is known, or
the reflector material and frequency are fixed, transducer dimensions depend only
on the emitter-piezoceramic joint, in which the layer thicknesses are determined by
equation (4.17).
Equation (4.22) may be presented graphically through the dependence le=f(lr),
if the ceramic thickness lp considers as a known parameter. As an illustration of the
previously cited, in Figure 4.20 is presented the dependence of the emitter length
on the reflector length, for a sandwich transducer with total thickness of the PZT8
ceramic lp=10mm, for resonant frequency of the transducer fr=26.06kHz.
Transducer with this resonant frequency has the form presented in Figure 1.1(a)
(that is, in Figure 4.5(b)) and, as already mentioned, it is constructed in purpose of
experimental verification of the three-dimensional matrix model proposed in this
chapter (Chapter 4.3.2.). Thereat the transducer dimensions were the following, for
the symbols from Figure 1.1(a): l1=18.7mm, l2=23.3mm, 2a1=2a2=40mm,
l3=5mm, 2a3=38mm, 2b3=15mm, l4=16.3mm, l5=15.2mm, 2a4=2a5=40mm,
l6=l1+2l3+l4, 2a6=8mm, l7=8mm, 2a7=13.15mm, where ai and bi are external and
internal radii of the correspondent elements, respectively.
9
8
fr =26.06 kHz
7
le (cm)
6
5
4
3
2
1
2
4
6
8
10
lr (cm)
Figure 4.20. Graphical presentation of the sandwich transducer equation (4.22)
Based on the Figure 4.20, for real emitter length le=42mm, the calculated reflector
length is lr=56.5mm, that is, for real reflector length lr=31.5mm is determined the
emitter length le=51.3mm. Accordingly, disadvantage of the quoted approach in
transducer design faces in great deviations for calculated ending lengths regarding
their real dimensions. That is because the equation (4.22) stands in case of great length
of the transducer consisting parts, that is, in case when the wave propagation velocity
in the transducer elements is approximately constant. Considerations in the previous
part of the paper have approximate nature, although generally they yet provide useful
results (data), because obtained values for ending lengths are always greater than the
final dimensions, so by subsequent shortening and adjusting by vibrational platform or
network analyzers one may achieve the demanded resonant frequency. That is why
this method is still often used in spite of the cited disadvantages. Approximation is as
closer as the ratio of the element diameter and longitudinal oscillations wavelength is
smaller.
Since the unknown value cannot be expressed explicitly, usually are used curves
for universal design [115]. Since there are infinitely many solutions of the frequency
equation of the sandwich transducer, every solution may be assigned by an
amplification index of the oscillation amplitude [116], which determines the transducer
quality. Maximal amplification index is obtained in function of the thicknesses of the
three consisting transducer components. Introducing this parameter, by analysis of the
sandwich equation, it is proved that, when the reflector and the emitter are made of
different materials, optimal transducer performances are obtained if: (1) transducer
with the emitter has length of λ/4, which was known till then too, whereat as a special
case of the general frequency equation one gets the Langevin’s equation, and if (2) the
piezoelectric layer has zero thickness. The last fact shows that one cannot realize an
optimal transducer design, because in practice one must use piezoceramic layer with
finite thickness [117].
Frequency equation of the sandwich transducer oscillation may be extended to the
case of multielement sandwich transducer that is further developed and used, and
which may be designed by summing of the equations of the simple λ/4 sandwiches
[113]. In this transducer type one may use a specific even number of piezoceramic
rings, whereby is enabled transducer capacitance increase and obtaining of the desired
electric impedance range. Such construction implies increased losses, and thereby
greater heating, because of the increased number of contact surfaces. In this analysis
are significant transducers with one and two pairs of piezoceramic rings (solution
adopted in purpose of greater power achievement).
In case of shorter metal endings, that is, in transducers with resonant frequency
around fr=40kHz, which have the form presented in Figure 1.1(b), application of
equation (4.22) gives even more unfavorable results. Then is necessary modification of
the one-dimensional theory, in order to enable modeling and design of transducers
with short metal endings. To illustrate this, for concrete realized ultrasonic sandwich
transducer from Figure 1.1(b), whose measured fundamental resonant frequency is
fr=41.6kHz, are determined values of the resonant frequency by different onedimensional ways of modeling, for known transducer dimensions: l1=18mm,
2a1=40mm, l3=5mm, 2a3=38mm, 2b3=15mm, l4=11.2mm, 2a4=40mm, l6=l1+2l3+l4,
2a6=8mm, l7=8mm, 2a7=12.8mm, where also, ai and bi are corresponding external and
internal radii, and li are lengths of the specific transducer elements. Obtained values are
presented in table 4.2 (transducer II), whereat are, in purpose of comparison, also
presented the resonant frequencies obtained in the same way for the transducer from
Figure 1.1(a), with fundamental resonant frequency fr=26.06kHz (transducer I).
Parameters and constants of the materials used are presented in the Table 3.1 for metal
parts of the transducer, while the parameters of the PZT8 ceramic, which is equivalent
to the piezoceramic the transducers are realized by, are presented in Table 4.3.
Table 4.2. Values of the resonant frequencies obtained applying existing ways of
sandwich transducers modeling
Langevin’s equation [113] for both (λ/4)
transducer halves:
Zep tg(ke le ) tg k pl p = 1
( )
Zrp tg(kr lr ) tg(k pl p ) = 1
fr [kHz]
for
transd. I
Deviation
from
measured fr
fr [kHz]
for
transd. II
Deviation
from
measured fr
26.63
2.19%
52.46
26.11%
27.14
4.14%
50.93
22.43%
26.78
2.76%
51.89
24.74%
27.70
6.29%
53.05
27.52%
28.07
7.71%
49.78
19.66%
Equation of (λ/2) sandwich [115], [123]:
(
)
( )
+ Z rp sin(kr lr )cos(k p l p )cos(ke le ) +
+ Z ep cos(kr lr )cos(k p l p )sin(ke le ) = 0
cos(kr lr )sin k p l p cos(ke le ) −
− Z rp Z ep sin(kr lr )sin k p l p sin(ke le ) +
Ueha’s equation [124] for both (λ/4) transducer
halves:
Zc 2 tg(k2 l2 ) − Zc6 ctg(k6 (l1 + l3 )) −
I)
⎛
⎛Z
⎞⎞
− Zc1 ctg⎜⎜ k1l1 + arcctg⎜⎜ c 3 ctg(k3l3 )⎟⎟ ⎟⎟ = 0
⎝ Zc1
⎠⎠
⎝
Z c 5 tg (k 4 l 5 ) + Z c 7 tg (k 6 l 7 )
− Z c 6 ctg(k 6 (l 3 + l 4 )) −
II)
− Z c4
⎛
⎛Z
⎞⎞
ctg⎜⎜ k 4 l 4 + arcctg⎜⎜ c 3 ctg(k 3 l 3 )⎟⎟ ⎟⎟ = 0
Z
⎝ c4
⎠⎠
⎝
There is an obvious disagreement between the calculated and experimental
resonant frequencies, especially for the transducer with short metal endings
(fr=41.6kHz), while much better results are obtained in case of a transducer with
longer metal endings (fr=26.06kHz). This is a consequence of application of
constant velocity values of longitudinal oscillations of metal endings and ceramic,
which do not depend on frequency or length, and which stand for long metal
E
ρ3 ) , where
cylinders (bars) and piezoceramic ( vi = EYi / ρi , i=1,4,6; v3 = 1 /(s33
s33E is piezoelectric constant): v1=5150 m/s (duralumin), v3=3122 m/s (PZT8),
v4=v6=5270 m/s (steel).
Great differences in literature between the calculated and experimental results,
presented in Table 4.2, are explained by the fact that resonant frequencies of such
structure depend on the prestressing value in the piezoceramic material, i.e., that static
prestressing will modify the values of the piezoceramic parameters (constants), and
that these are values which must be used in any electric calculation in design. Yet, the
most recent studies showed that those reasons were not crucial, although there were
changes of the quoted parameters due to the prestressing, especially for the PZT4
ceramic [119], i.e., the greatest influence to the resonant frequency yet had the
geometric and physical characteristics of the metal endings. Accordingly, quoted
deviations of the calculated and experimental results are rather consequence of the
disadvantages of the cited models, than they are consequence of the material
characteristics changes due to prestressing. Namely, theoretical analyses [120], and
experimental results within this chapter, showed that resonant and antiresonant
frequencies of the considered transducers vary quite a little in the prestress range from
30MPa to 50MPa. It is showed that physical properties of the ceramic do not change
significantly for the prestress values up to 50MPa, so that possible reason for the
frequency shift is change of the effective contact surface between the transducer parts
due to the mechanical prestressing [121]. Also, the Young’s elasticity modulus for
duralumin, and thereby the wave propagation velocity in it, stay practically constant
when the material is exposed to a static compression up to 50MPa [122]. Because of
all cited, in this chapter are not considered further the prestressing influences during
the realization of an ultrasonic transducer model.
4.2.3. Ueha’s Equation
In purpose of design of the prestressed Langevin’s sandwich transducers, in
paper [124] Ueha proposed a simple one-dimensional model for analysis of the
resonant conditions of the transducers with endings made of different metals, with
two excitation piezoceramic rings and a flat head bolt in the reflector region.
Thereby is obtained structure with two completely different halves of the
transducer, considered regarding the middle of the piezoceramic. It is used an
assumption that in unloaded transducer that oscillates in λ/2-like thickness mode
the nodal oscillation plane divides the piezoceramic in two equal parts, so that two
different, independent equations are proposed, each for a single separately treated
λ/4-like section (Figure 1.1(a)).
Characteristic frequency equations for the bottom and top half of the
transducer from Figure 1.1(a) are given by the following expressions, respectively:
⎛
⎛Z
⎞⎞
Zc 2 tg(k2 l2 ) − Zc6 ctg(k6 (l1 + l3 )) − Zc1 ctg⎜⎜ k1l1 + arcctg⎜⎜ c3 ctg(k3l3 )⎟⎟ ⎟⎟ = 0 ,
⎝ Zc1
⎠⎠
⎝
(4.23)
⎛
⎞⎞
⎛Z
Zc 5 tg(k4l5 ) + Zc 7 tg(k6l7 ) − Zc6 ctg(k6 (l3 + l4 )) − Zc 4 ctg⎜ k4l4 + arcctg⎜⎜ c3 ctg(k3l3 )⎟⎟ ⎟ = 0.
⎟
⎜
⎠⎠
⎝ Zc 4
⎝
Because of the model simplicity it is possible, therefore, to take into account
the influence of the prestressing bolt too, whereby is broadened the basic
Langevin’s equation from the previous models. The method of prestressing bolt
choice is presented, as well as the distribution of the static mechanical stress in the
points of the prestressed piezoceramic element. Optimal values of the bolt diameter
are also obtained in function of metal ending dimensions and transducer diameter,
by analysis of the mechanical stress state (distribution). Great accuracy in
determination of resonant frequencies is achieved, using both equations of this
model, which uses the transfer line analogy. It must be remarked that this model
gives good results, but only in case of low-frequency ultrasonic transducers with
metal endings of great length [94].
4.2.4. General One-dimensional Model of Sandwich Transducer
Previous discussion relates to an unloaded transducer. A good analysis of a sandwich
transducer and the majority of the previously mentioned design approaches, which are
reduced to the solution of frequency equations without consideration of piezoelectric
characteristics of the applied ceramics, are given in paper [95]. If the transducer emits
towards the complex acoustic appliance, the resonant frequency will change because of the
boundary condition change on the operating surface [125]. If the appliance is known, there
must be derived a new frequency equation of the longitudinal oscillations. Therefore is
using of equivalent circuits in transducer modeling a better approach. One-dimensional
equivalent circuits for sandwich transducers use Mason’s theory for piezoelectric ceramics,
but along with passive elements represented by the generally known symmetric T
quadripole [59]. Thus the sandwich transducer becomes a network with one electric and
two mechanical accesses, with also three-access equivalent for piezoceramic rings and twoaccess T-networks (transfer line models) for the endmost endings and the prestressing bolt.
In some papers bolt influence often was not taken into account [126], [127], [128], because
its effect was considered negligible, or it was taken only the parallel connection of the
ceramic and the bolt part that passes through it [113], [123], [129]. Complete onedimensional model of the piezoelectric sandwich transducer along with all concomitant
parts (adapting layers and shanks), which is used for underwater applications, is presented in
paper [130], while the complete model of a similar transducer for ultrasonic cleaning
application is presented in paper [131]. In both cases are also taken into account dimensions
of the bolt and bolt head.
êrpak derived expressions for calculations of the dynamic impedances of the
sandwich transducers with different construction via concentrated parameters, for
several excitation piezoceramic platelets [123]. A one-dimensional theory without
losses was used, when the load is connected to one mechanical output of the
transducer. The solution method is based on using of the equivalent circuit with
distributed parameters, which is transformed into a multiple Tevenen’s
network.
Active element is represented by a set of identical, thickness polarized piezoelements,
which are mechanically connected serially, and electrically connected parallel, and
which satisfy the conditions for which the Martin’s equivalent scheme is applicable
[66].
Beside the mentioned analysis of the thickness influence of the applied
piezoceramic [117], it is analyzed the influence of the piezoceramic location along the
transducer on the oscillatory characteristics of the power ultrasonic transducers, but
now by an equivalent circuit, whereat the Mason’s equivalent circuit is reduced to an
equivalent circuit with concentrated parameters [132]. Based on numerous
experimental results, it is showed the significant influence of the piezoceramic location
on input electric impedance and on the Q factor of the transducer, as well as on the
mechanical displacements. The possibility of prediction of the transducer resonant
frequency is achieved, but it is not achieved prediction of the equivalent circuit
elements characteristics based on such analysis, which is justified by the piezoceramic
characteristics change at high mechanical stresses and powerful electric excitations.
Based on the already published PSpice models for simulation of individual
piezoelectric elements [63], [65], PSpice models of ultrasonic transducers are realized,
which could include electric and acoustic (mechanical) adjustment [133], temperature
and frequency dependences of the transducer characteristics [134], as well as the
influence of the prestressing bolt [135]. Naturally, limitations of these PSpice models
are same as the well-known limitations of the one-dimensional models [59]. Using
these models with associated excitation electronic generator, one may realize the
simulation of the complete ultrasonic system. Based on electromechanical analogies,
the transient emitted power at transducer emission may be easily obtained as a product
of force and velocity on the operating surface. In standard models, excitation generator
is considered as an ideal voltage source, which gives an ideal sinusoidal or impulse
excitation at fixed frequency. Nevertheless of the available software for energetic
electronics circuits modeling, a more detailed analysis of an excitation generator has
been usually omitted in design procedures of power ultrasonic systems, but attention
has been dedicated mostly to the ultrasonic transducers themselves.
In this chapter certain improvement in transducer modeling is obtained first
by realization of an one-dimensional transducer model by equivalent circuit, which
includes all parts of the transducer, and which takes into account piezoelectric
properties of applied excitation rings [131], [135]. Yet, this model enables to
predict only the thickness resonant modes, and accordingly it does not take into
account inevitable radial oscillations of the piezoceramic and metal parts of the
transducer. The idea that in calculation of such complex oscillatory system, excited
by piezoceramic rings, are used matrices of equivalent z, y, or a parameters, and
particularly the network with three accesses for active elements and adequate
network with two accesses for passive elements, isn’t a new one, but here is
presented the complete scheme that includes all consisting transducer elements.
Thereby is created a base for later application of the seeming elasticity moduli
method in the field of modeling the complete ultrasonic transducers. Besides that,
in contrast to the application of the equations from Table 4.2, by which one may
determine only the resonant frequency, and any other analysis isn’t possible, by
such transducer modeling is possible to determine electric and mechanical
impedances, electroacoustic efficiency coefficient, oscillation amplitudes, and also
is possible the analysis of losses and sensitivity for a specific sandwich transducer.
Elements of the transducer from Figure 1.1, which are the consisting part of
the model, are: frontal emitter ending (1, 2), piezoceramic rings (3), reflector
ending (4, 5), and prestressing bolt (6, 7). Active element in the circuit is a pair of
identical, thickness polarized, piezoceramic rings, which are mechanically
connected serially, and electrically connected parallel, and for which is applied the
known Mason’s model. Influence of the contact foils in further transducer
modeling will be omitted, because of its small thickness.
In Figure 4.21 are again presented geometry and definitions of the connectors
for excitation piezoceramic ring, which oscillates in thickness mode, as already
presented in Figure 2.7 [59]. Model is consisted of capacitance C0=ε33S⋅P/l,
negative capacitance -C0, ideal transformer, and T-network (line). F1, F2, and v1, v2
are forces and individual velocities on the ring surfaces, h33 is piezoelectric
constant, and ε33S relative dielectric constant of the compressed ceramic. Further is
mechanically serial and electrically parallel connection of such piezoceramic rings
in the model of a complete sandwich transducer presented in blocks.
l
v2
v1
Z31
F1 , v1
Z31
Z32
F2 , v2
-C0
I
F1
F2
P
I
C0
V
h33C0:1
V
(a)
(b)
Figure 4.21. (a) Piezoceramic ring in thickness mode;
(b) Mason’s equivalent circuit
Using the rules for obtaining the resulting quadripoles at mechanical serial or
parallel connecting of elements, for the transducer from Figure 1.1(a) is obtained a
complete circuit of the electromechanical model of the sandwich transducer with
distributed parameters, which is presented in Figure 4.22 [131]. These rules are very
simple. On every joining surface is observed the distribution of forces. If only two
elements are mutually directly joined, the force is continual and those elements
quadripoles) are connected in a cascade way (directly). In case when force, transferred
from the surface of one element, is shared among two or three elements on the other
side of the joint, a serial connection of those elements is necessary. In other words, it
means that mechanical accesses of the elements that are in contact, that is, their contact
surfaces, have same velocities. Accordingly, in the proposed electric model these
accesses must be connected by the same current (velocity) loop.
Z 61
Z21
Ze
Z61
Z62
Z71
Z72
Z21
Z51
Z22
Z52
Z11
Z11
keramika
1
keramika
2
Z41
Z71
Zr
Z51
Zr
Z41
ceramic 2
ceramic 1
Z12
Z42
~
Figure 4.22. General one-dimensional electromechanical
model of a sandwich transducer
Passive elements (emitter, reflector, and bolt) are in the equivalent scheme of
the whole transducer represented by the generally known symmetric T quadripoles
(lines). Elements of the scheme that correspondent to isotropic, symmetric endings
of different lengths and materials, piezoceramic rings, as well as the elements of
the metal bolt scheme, are determined based on the expression:
− j Z ci
k l
Zi1 = j Z ci tg i i ,
Z i2 =
,
(4.24)
2
sin(k i l i )
whereat is Zci=ρi⋅vi⋅Pi and ki=ω /vi, for i=1, 3, 4, and 6 (Zc1=Zc2, Zc4=Zc5, and
Zc6=Zc7). Zci are characteristic impedances, ρi are densities, li and Pi are lengths
and surfaces of the elements, and vi are velocities of the longitudinal ultrasonic
waves (whereat is ρ1=ρ2, ρ4=ρ5, ρ6=ρ7, and v1=v2, v4=v5, v6=v7).
Reflector and emitter ending are connected mechanically serially with rings,
whereat they are closed by acoustic impedances Ze and Zr, which are in the
considered case negligible, because the experimental measurements on the realized
transducers are too performed for the free, unloaded transducers, which oscillate in
air. Metal bolt extends along the whole structure, and because of that, it is
connected mechanically parallel with other elements in the scheme. The expression
for sandwich transducer input impedance observed on the electric accesses is
complicated, and because of massiveness omitted.
Presented complete model may be used for modeling of sandwich transducers
with different forms. For example, input impedance of the transducer from Figure
1.1(b) is obtained if one puts P2=0 and P5=0 into the model from Figure 4.22.
Proposed model of the sandwich transducer assumes that circuit elements are
without losses. Losses may be included by treating the piezoconstants and elasticity
constants of the metal parts of the transducer in form of complex numbers.
4.2.4.1. Comparison of Numerical and Experimental Results
Verification of the proposed model is performed by modeling the dependence
characteristics of the input electric impedance on frequency for unloaded ultrasonic
sandwich transducers from Figure 1.1, whose operating resonant frequencies are
26.06kHz and 41.6kHz, respectively. Values of specific dimensions for both
transducers are already presented earlier, while the parameters and constants of the
used materials are presented in Table 3.1 and in Table 4.3. Experimental characteristics
of the impedance dependence on frequency for both transducers are recorded by a
network analyzer, and compared with analogous characteristics obtained by the
proposed transducer model from Figure 4.22. Cited characteristics, which encompass
the fundamental and several higher resonant modes, are presented in Figures 4.23 and
4.24, whereat is, as in case of piezoceramic rings, because of great range of impedance
change analyzed the attenuation function in dB. Besides that, in the same figures are
presented dependence characteristics of the input impedance on frequency obtained by
the existing one-dimensional model from literature [113], [123], which contains only
the parallel connection of the piezoceramic and the bolt part that passes through it.
In Figure 4.23 are presented the experimental and both modeled impedance
dependences for the transducer with resonant frequency fr=26.06kHz, whereat one
may notice great similarity of all quoted characteristics. Better results for the
fundamental resonant mode are obtained using the proposed general onedimensional model, because obtained resonant frequency amounts to 25.4kHz, that
is, the error in determination of the resonant frequency is 2.53%. Resonant
frequency obtained by the existing model that takes into account the bolt part
amounts to 27.5kHz, with error in calculation of 5.53%. Since this is the case of a
transducer with longer metal endings, such one-dimensional model is quite
acceptable for its analysis, too. Regarding the higher resonant modes, their general
form may be predicted by the proposed general model, but the deviations of the
resonant frequency values are great. By the existing model that takes into account
the bolt part, one may not simultaneously model both the second and the third
resonant mode, which means that in the complete one-dimensional transducer
model, as well as in the real transducer, one of those resonant modes results from
the prestressing bolt. If prediction of almost all resonant modes exists, like in this
case, with assumption that ultrasonic wave velocities in some transducer elements
are not constant any more, one may perform fitting of the modeled and
experimental dependence in the vicinity of every resonant mode. Thus one may,
based on experimental measurements in the vicinity of every resonant mode,
correct afterwards the initial values of correspondent wave velocities in
longitudinal direction, as well as the initial values of the piezoconstants, for every
realized transducer, if the sandwich transducer model is considered satisfying in
that frequency range [135].
In Figure 4.24 are presented analogous characteristics for a transducer with
resonant frequency fr=41.6kHz. By the proposed general one-dimensional model is
obtained fundamental resonant frequency of 45.3kHz, so that the error regarding
the measured resonant frequency is 8.89%. In case of a model that takes into
account the bolt part, resonant frequency is 54.2kHz, with error of 30.29%. Based
on the comparison of two modeled characteristics, one may assume that the second
resonant mode, which is obtained by the proposed general model, results from the
bolt influence. Higher resonant modes obtained by modeled characteristics show
great deviation from the experimental results, which represents a limitation in
application of this model by which, therefore, one may analyze only the
fundamental resonant mode.
Based on the previous analysis and values from Table 4.2, it is obvious that
accuracy of the existing model that takes into account the bolt part [113], [123] for
determination of the resonant frequencies is close to the accuracies of the onedimensional expressions from Table 4.2, while by the proposed one-dimensional
general model one gets slightly better results in case of the transducer with short
metal endings. Yet, even those results are quite imprecise, because one gets
substantial differences between the proposed one-dimensional theory and
experiments, so that further modification of this one-dimensional model is
necessary, too.
70
60
Zul [dB]
50
40
30
20
10
0
0
1
2
3
4
5
6
7
x 10
f [Hz]
4
Figure 4.23. Input impedance in function of frequency for ultrasonic transducer
with fundamental resonant frequency of 26.06kHz: proposed general onedimensional model (
), model [113], [123] with one bolt part
(
) and experimental results (
)
70
60
Zul [dB]
50
40
30
20
10
0
0
2
4
6
8
f [Hz]
10
12
14
x 10
4
Figure 4.24. Input impedance in function of frequency for ultrasonic transducer
with fundamental resonant frequency of 41.6kHz: proposed general onedimensional model (
), model [113], [123] with one bolt part
(
)
4.2.5. Design of Ultrasonic Sandwich Transducers
by Seeming Elasticity Moduli
All previously described accesses in transducer modeling are mutually similar
and related. They proved especially appropriate in modeling the sandwich
transducers with lower operating resonant frequencies, in which the lengths of the
reflector and the emitter are great, that is, quite greater regarding the piezoceramic
thickness. In case of short endings (e.g., in transducers with fundamental resonant
frequency fr=40kHz, which are used in ultrasonic cleaning systems) such approach
is incomplete. Modification of the one-dimensional theory is then necessary, by
which is enabled modeling and design of transducers with short metal endings.
The first attempt of modification of the one-dimensional theory in ultrasonic
sandwich transducers design, as well as at metal rings and disks, represented the
introduction of the seeming elasticity moduli method. Application of this method
in ultrasonic sandwich transducers design is presented in paper [136]. Subject of
analysis is a transducer with identical metal endings of large rectangular crosssection, and with circular excitation piezoceramic rings, but without consideration
of prestressing bolt influence. Resonant frequencies obtained from the frequency
oscillation equation for that case, show better agreement with measured results
regarding the results obtained by classical one-dimensional analysis. However, all
these conclusions are derived based only on one realized transducer with metal
endings of great length. In this chapter is for the first time applied the seeming
elasticity moduli method in design of several different sandwich transducers, with
metal rings of different dimensions and made of different materials, and with
present bolt and excitation piezoceramic rings in the model [97].
The idea of modification of the one-dimensional theory in the field of
Langevin’s sandwich transducer design originates even from Kikuchi [108], for
cases when oscillating elements by its form do not represent neither enough long
bars, nor enough thin disks, comparing their diameters and length (thickness), and
when character of their oscillations becomes extremely complex. Kikuchi’s
analyses showed that it was needed to introduce correctional factors during design
of Langevin’s transducers which consisted of arbitrary combination of endings
made of different materials, and which were neither too thin, nor too long, and to
which the transducers treated in this analysis belonged.
The first factor that has to be introduced is the form factor, that is, the factor of
the transducer shape. Besides that, one may also introduce a factor determined by
different mechanical impedances of applied materials. There is also a factor
conditioned by mutual contact of elements, whose influence may be decreased
improving the acoustic contact between the transducer parts. In addition, one may
introduce the factor that corrects the assumption that the piezoelectric medium is
isotropic. Individual determination of all these factors is very difficult, so that for a
specific group of transducers one may determine experimentally the correctional
factor that represents a product of all mentioned factors, which represents a basic
limitation in application of this approach.
Because of that is in this part of exposure is applied different approach in
modification of the one-dimensional theory, that is, coupled oscillations of sandwich
transducers of circular cross-section are considered, based on seeming elasticity
moduli. As explained in the previous chapters, in oscillators of large dimensions, like
thick circular disks and cylinders with thick walls, two types of oscillations,
longitudinal and radial, are mutually orthogonally coupled. Accordingly, the design of
such oscillators is very complicated. This problem is simplified introducing the
seeming elasticity moduli method. Because of the need for transducer prestressing,
metal emitter, metal reflector, and excitation piezoceramic platelets are circular rings
with central opening, while the metal bolt is in shape of a solid cylinder. Accordingly,
transducers treated in this analysis are by its shape the closest to the practical ultrasonic
transducers. In the extension of the exposure are explained seeming elasticity
constants, the frequency equation is determined, and lastly the theoretical and
experimental results are presented.
In Figure 4.25 is presented the simplest version of a symmetric piezoceramic
sandwich transducer with central bolt, which is the most suitable for explanation of
this method. This transducer form is adopted because for it is the most easily to
apply the Langevin’s and Ueha’s equation for one-half of this transducer (Table
4.2), because then the plane of oscillation node is for sure between the
piezoceramic rings. Because of that, the conclusions about possibilities for
application of this method through the cited expressions, derived based on the
comparison of theoretical and experimental results for such concretely realized
symmetric transducers will be certainly valid. Theory of transducer design is based
on assumption that the neutral plane is located between the piezoceramics,
although similar procedure may be also performed in case of unsymmetrical
transducers. Method of seeming elasticity moduli is in this analysis for the first
time applied on symmetric transducers with form presented in Figure 4.25, and
based on obtained results it is also extended on unsymmetrical transducers with
short endings from Figure 1.1, applying the complete one-dimensional model from
Figure 4.22. As mentioned, hither is in literature presented application of this
method for a symmetric transducer with great ending lengths of rectangular crosssection, and without consideration of prestressing bolt influence. Thereat the
conclusions are derived based on only one realized transducer [136].
Lengths of specific transducer elements from Figure 4.25 are li (i=1, 2, 3),
outer diameters are ai (i=1, 2, 3), and bi are inner diameters (i=1, 2). Thereat the
radial dimensions of endings 2a1 are close to the longitudinal dimensions l1, which
means that one-dimensional theory cannot be used directly in transducer design.
The seeming elasticity moduli now differ from the Young’s modulus, and they
depend not only on the material parameters, but also on geometric dimensions, as
well as on the vibrational mode of the oscillator that is in question.
At piezoceramic circular platelets with central opening, with piezoelectric
effect neglected and anisotropy taken into account, the expressions for seeming
elasticity moduli are the following [136]:
[ (
)]
EYr 2 = s11E 1 − υ122 − υ13 n S 2 (1 + υ12 )
−1
,
(4.25)
−1
E
(1 − 2υ 31 / nS 2 ) ,
EYz 2 = s33
where: EYr2=Trr2/Srr2, EYz2=Tzz2/Szz2, υ12=-s12E/s11E, υ13=-s13E/s11E, υ31=-s13E/s33E.
[
]
2
2
3
1
1
l1''
l1'
l1
l2
l2
l1''
l1'
l1
Figure 4.25. Half-wave symmetric sandwich transducer
Trr2, Tzz2 and Srr2, Szz2 are componential mechanical stresses and relative strains
along polar coordinates, sijE are piezoelectric elasticity constants, nS2=Tzz2/Trr2 is
the coupling coefficient between the longitudinal and radial oscillations of the ring.
The seeming elasticity moduli of the metal endings and the bolt are given by
the following expressions (i=1, 3) [86]:
[
]
EYri = EYi 1 − υ i2 − υ i n Si (1 + υ i ) ,
−1
EYzi = EYi (1 − 2υ i / n Si ) ,
−1
(4.26)
where υi is Poisson’s material ratio, EYi Young’s elasticity modulus of the material,
and nSi=Tzz i /Trr i, as well as in case of ceramic, determines the coupling degree of
the oscillations.
Based on the previous analysis and on expressions for seeming elasticity
moduli for specific consisting transducer parts, coupled oscillations of the
transducer are separated into two types: longitudinal and radial. These oscillations
are not independent yet, but mutually connected by the virtual coupling
coefficients. Accordingly, assuming that zero boundary conditions for mechanical
stresses are fulfilled on external surfaces, the characteristic frequency equation may
be obtained by combination of the characteristic frequency equation for the radial
and longitudinal oscillations of the sandwich transducer. For simplicity, only the
first fundamental resonant mode of the transducer is observed.
Characteristic frequency equation for the radial oscillations of the
piezoceramic ring is [136]:
kr 2 a2 Y0 (kr 2 a2 ) − (1 − υ12 )Y1 (kr 2 a2 )
=
kr 2 a2 J 0 (kr 2 a2 ) − (1 − υ12 )J1 (kr 2 a2 )
,
(4.27)
kr 2 b2 Y0 (kr 2 b2 ) − (1 − υ12 )Y1 (kr 2 b2 )
kr 2 b2 J 0 (kr 2 b2 ) − (1 − υ12 )J1 (kr 2 b2 )
where a2 and b2 are outer and inner radius of the piezoceramic ring. J0, J1, and Y0,
Y1 are Bessel’s functions of the first and second rank, zero and first order,
kr2=ω /vr2, vr 2 = EYr 2 / ρ 2 and ρ2 is the piezoceramic. For given values υ12, a2,
and b2, the solution of the expression (4.27) is:
k r 2 a2 = R2 '.
(4.28)
R2’ is the solution of the characteristic frequency equation for radial oscillations,
and it is a function only of υ12 and of ratio a2/b2. Now one may define:
2
1 − υ122 − (v 02
R 2 ' 2 ) / (ω 2 a 22 ) ,
(4.29)
nS 2 =
υ13 (1 + υ12 )
where v022=1/(s11Eρ2).
Analogous frequency equation for radial oscillations stands also for the metal
endings, where υ12 and υ13 should be substituted by υ1, R2’ with R1’, and a1 and b1
will be outer and inner radii of the metal endings. In this case is:
2
1 − υ12 − (v 01
R1 ' 2 ) / (ω 2 a12 )
(4.30)
,
n S1 =
υ1 (1 + υ1 )
where v012=E Y1/ρ1, E Y1 is Young’s modulus, and ρ1 is the metal.
Regarding the bolt, since the bolt is solid cylinder, the frequency equation for
radial oscillations is [136]:
(4.31)
kr 3 a3 J 0 (kr 3 a3 ) = (1 − υ 3 ) J1 (kr 3 a3 ),
whose solution is kr3a3=R3’, and a3 is the bolt radius.
In order to derive the characteristic frequency equation for longitudinal
oscillations, one starts from the assumption that the oscillation node plane is
located between the piezoceramic rings. Based on the frequency equations
of Langevin and Ueha from Table 4.2, obtained for the same case in
classical one-dimensional design, one may determine frequency equations
in function of the seeming elasticity moduli for the transducer parts in front
of and behind the nodal plane, which are in this case equal (Figure 4.25).
Thus, one gets the following expressions, respectively:
ρ1v z1 P1 tg(k z1l1 ) tg(k z 2 l2 ) = ρ 2 v z 2 P2 ,
ρ 2 v z 2 P2
ctg k z 2 l2 ) +
ρ1v z1 P1
+ ρ 3 v z 3 P3 ctg k z 3 (l1 '+l2 ) − ρ1v z1 P1 tg k z1l1 ' ' = 0,
ρ1v z1 P1 ctg( k z1l1 '+ arcctg
(4.32)
(4.33)
where kzi=ω /vzi, v zi = EYzi / ρ i and Pi are cross-section surfaces of specific
consisting parts of the transducer (i=1, 2, 3). In this method, ultrasound wave
velocities in specific elements are functions only of frequency and outer and inner
radius, while dependence on lengths and again on frequency occurs in equations
(4.32) and (4.33) (in the model itself).
Accordingly, frequency equations (4.32) and (4.33) for coupled oscillations, since
in expressions (4.25) are included expressions (4.29) and (4.30) too, do not depend
only on material parameters, longitudinal dimensions, and frequency, but on cross
dimensions too, which is different regarding the traditional one-dimensional theory.
Presented method of transducer design is based on assumption that neutral plane is
located between the piezoceramics, although a similar procedure may be also
performed in case of unsymmetrical transducers. Based on the obtained results one
could now determine the displacement distribution on the transducer surfaces, which is
not a simple problem because of the three-dimensional character of oscillations.
4.2.5.1. Comparison of Numerical and Experimental Results
In order to analyze the possibilities of this method for design of complete
sandwich transducers, there are analyzed symmetric and unsymmetrical sandwich
transducers with long metal endings, which are designed by three-dimensional
matrix model (Chapter 4.3.2), and realized because of its experimental verification.
Resonant frequencies of these transducers may be determined by vibrational
platform or network analyzer. In the symmetric transducer, the emitter and the
reflector are made of duralumin of same length. In the unsymmetrical transducer,
the emitter is made of duralumin, and the reflector is made of steel. In both
transducers is used steel bolt. In both cases, the piezoceramic rings are made of
material equivalent to the PZT4 piezoceramic. Parameters of all used materials are
presented in Table 3.1 and Table 4.3. Geometric dimensions of the symmetric
transducer are following (for symbols from Figure 4.25): l1’=16.2mm, l2=6.35mm,
2a1=40mm, 2b1=10mm, 2a2=38mm, 2b2=13mm, and 2a3=10mm. Dimensions of
the unsymmetrical transducer, according to the symbols from Figure 1.1, are:
l1=11.3mm, 2a1=2a2=40mm, 2b2=10mm, l3=6.35mm, 2a3=38mm 2b3=13mm,
l4=6mm, l5=4.88mm, 2a4=2a5=40mm, l6=l1+2l3+l4, 2a6=10mm, l7=8mm,
2a7=14mm, where ai and bi are outer and inner radius, and li is the length of the
corresponding element.
In symmetric transducer is performed simultaneous shortening of the both metal
endings length (and starting from some value of l1 simultaneously is shortened the
steel bolt, too), and experimentally are determined the dependences of the resonant
frequencies on length of duralumin l1. The thickness of the piezoceramic rings was
constant. These results are compared with analogous dependences of the resonant
frequency on length, which are obtained based on the frequency equations (4.32) and
(4.33), as well as with same dependence obtained based on the proposed complete
one-dimensional model from Figure 4.22. In Figures 4.26, 4.27, and 4.28 are presented
those dependences for this transducer, respectively, altogether with same onedimensional dependences obtained without application of seeming elasticity moduli.
Thereat, in models by which the graphs presented in Figures 4.27 and 4.28 are
obtained, is also taken into account the bolt shortening.
In Figure 4.26 is presented dependence of the resonant frequency of the first
mode on the duralumin ending length for a symmetric transducer, which is
obtained applying the Langevin’s equation. As one may notice, the resonant
frequencies obtained after the described correction of the elasticity modulus of the
piezoceramic rings, as well as of the metal endings for every their individual
length, in range of greater ending lengths agree better with measured frequencies,
than the results obtained by one-dimensional theory. In case of small metal ending
lengths, there are deviations between the experimental results and the results
obtained by this method. Accordingly, the application of the seeming elasticity
moduli method on the Langevin’s equation is somewhat justified in the region of
greater ending lengths.
In case of application of Ueha’s equation, whose solution is presented in Figure
4.27, also for the first resonant mode of the same transducer, the error in determination
the resonant frequencies exists both in the region of the short and in the region of the
long metal endings, whereat is for the short endings the error smaller regarding the
case from the Figure 4.26. In addition, there exists a value of the ending length (around
30mm) for which the experimental and calculated results coincide.
In Figure 4.28 are presented dependences of the resonant and antiresonant
frequency of the first mode on the ending length for the same symmetric
transducer, obtained applying the proposed general one-dimensional model. In
contrast to the previous two cases, here experimental results, beside the values of
the resonant frequencies (fr), contain also the values of the antiresonant frequencies
(fa) at which the input impedance of the transducer is maximal, because it is
possible to perform an analysis of both dependences by the proposed model (beside
some other characteristics mentioned earlier that may be analyzed).
It is obvious that one may perform prediction of the quoted resonant frequencies
in wide range of ending lengths by proposed general one-dimensional model, without
application of the seeming elasticity moduli method, whereat only for very small
lengths one gets deviations between the calculated and experimental results. Using the
seeming elasticity moduli method one gets results that deviate from measured results
in the whole observed length range, although the error obtained applying this method
for short endings is smaller than the error obtained applying the one-dimensional
model, especially for values of antiresonant frequencies.
Figure 4.26. Dependence of the resonant frequency of the first mode on ending
length for a symmetric transducer, obtained applying the Langevin’s equation
Figure 4.27. Dependence of the resonant frequency of the first mode on ending
length for a symmetric transducer, obtained applying the Ueha’s equation
fr
fa
fr
fa
fa
fr
Figure 4.28. Dependences of the resonant and antiresonant frequency of the first
mode on ending length for a symmetric transducer, obtained applying the proposed
one-dimensional model
The last conclusion stands even more in case of unsymmetrical sandwich
transducers from Figure 1.1(a), where the elasticity moduli of all transducer
consisting parts are corrected. In unsymmetrical transducer, only the duralumin
shortening (l2) is performed, and dependences analogous to the graphs from the
previous figure are presented in Figure 4.29. Here too, applying the proposed
general one-dimensional model one gets satisfying results in the region of greater
emitter lengths, while the results obtained by the seeming elasticity moduli method
are practically useless, and all that for the values of resonant and antiresonant
frequencies.
7
fr
fa
6
fa
5
fr
4
fa
fr
3
2
1
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
Figure 4.29. Dependences of the resonant and antiresonant frequency of the first
mode on ending length for an unsymmetrical transducer, obtained applying the
proposed one-dimensional model
Accordingly, if the applied one-dimensional model is more complete, that is, if
its accuracy is greater, there is no need for correction of the elasticity modulus of
the transducer consisting parts, because one obtains results that show greater
deviations from the experimental results, except in the region of small ending
lengths, when the one-dimensional models are useless. Such great deviations in
case of application of the seeming elasticity moduli method are consequence of the
consideration only the first resonant mode in all transducer consisting parts (only
R’imin is taken in the solutions of the expressions (4.28) and (4.31)).
4.3. THREE-DIMENSIONAL MODELS OF ULTRASONIC
SANDWICH TRANSDUCERS
4.3.1. Numerical FEM and BEM Methods
One-dimensional models are unsuitable for characterization of the transducers
with great band-pass, as the transducers for application in sonars are, because their
characteristics are valid only around the fundamental longitudinal resonant frequency.
In broadband transducers, the transducer form becomes more complex and other
resonant modes essentially affect the fundamental resonant mode. Because of that, in
this field became the application of the finite element method, which uses the threedimensional approach in transducer analysis and surpasses the limitations of the onedimensional models [137], [138], [139]. This method is used for any transducer
configuration and for materials with different characteristics. It enabled full
characterization of the resonant modes of the transducers, that is, determination of the
mechanical displacements, resonant and antiresonant frequencies, coupling
coefficients, impedance, etc. Flexibility of this method brought to its almost
irreplaceable application in sonar design.
Beside the sonars, in the literature about power ultrasound are available several
examples from other ultrasound application fields, which relate to the analysis of the
classical Langevin’s transducers applying the finite element method [140], [141],
[142]. In those papers are presented models of power ultrasound transducers, and
methods that enable analysis of the mechanical prestressing influence in piezoceramic,
determination of resonant and antiresonant frequencies, coupling coefficients,
transducer impedance, oscillation amplitudes, and ultrasonic fields form. Losses may
be included into the model using complex physical constants. Differences between the
simulated and experimental amounts of the mentioned values usually have been
explained by dispersion of the physical constants of the used PZT ceramics or by
composite structure of the Langevin’s transducer, because the connection between the
transducer’s parts was considered as an ideal joint in the model, which was not the
case in reality [142]. The speed of the computer simulation of the operating regime and
transducer oscillation is a limitation factor in choosing this method for sandwich
transducer modeling, so no much attention is further dedicated to the application of
this method.
Ttransducer, the liquid as an operating medium through which the ultrasonic waves
propagate, and their coupling, are often modeled by the finite element method
combined with the boundary element method (BEM) [143], [144]. These models are
linear, and they are based on the theory of elasticity and constitutive piezoelectric
equations. Accordingly, the transducers are designed for lower excitation levels and
their performances in operating conditions cannot be predicted. Basic postulates of the
pure and the combined finite element method are the same. The boundary element
method does not demand mesh forming in the domain of liquid, and because of that,
its use is justified in cases where the calculation time should be significantly reduced
by such analysis approach. Such approach is used at high frequencies, when the
dimensions of the liquid appliance are much greater than the wavelength of the
ultrasonic waves. Combination of the mentioned methods may be also used at graded
circular transducers that emit in air, for numerical prediction of the oscillation
amplitude distribution and the diagram of emission directionality [145].
Basic limitation of the previously mentioned models is their linearity. Power
ultrasonic devices always contain nonlinear mechanisms, which modify not only
the device performances, but also may cause significant physical phenomena, like
the ultrasonic cavitation, acoustic levitation, etc. Modeling of nonlinearities in the
transducer materials is nowadays a very current topic in the field of numerical
modeling [146].
Finite element method further found its application in simulation of
nonlinearities and losses in element joints, in power ultrasonic transducers at high
mechanical stresses [147], [148], [149]. Namely, the design and simulation of
power transducers is complicated if in design one takes into account that at high
excitation levels piezoelectric, dielectric, and elastic properties of the piezoelectric
material change. Bias, which compensates small extension strength of the used
piezoceramic, must be greater than the dynamic mechanical stress on the
component joints during oscillation at high excitations. Determination of the
optimal prestressing is essential in transducer design, because on one side it
determines mechanical losses in the contacts, and by that itself the transducer
efficiency, and on the other side, together with thermal stressing due to the heating,
it may generate the piezoceramic damage. Accordingly, nonlinearities and losses in
the contacts between the metal endings and the piezoceramic demand a transducer
model based on nonlinear constitutive piezoelectric equations. By such model,
based on the three-dimensional finite element method, quoted losses and
nonlinearities may be analyzed in function of applied electric voltages or
mechanical prestressing.
Until now only several special nonlinear models are developed. Next essential
thing in this field of transducer design is connecting the mechanical and electric
elements of the model, especially when nonlinearities are included in both models.
Some papers dealing with that are published in the field of microelectromechanical
systems, and research in this field represents future trend in power ultrasonic
device modeling [150]. Future studies will be also focused on the interface between
the energetic electronics and ultrasonic transducers.
4.3.2. Three-dimensional Matrix Model of Piezoceramic
Ultrasonic Sandwich Transducers
In the paper [151] the ultrasonic sandwich transducer is modeled for the first
time by an analytical matrix model, using the approximate three-dimensional
model of piezoceramic elements with cylindrical shape, presented in literature [78].
Using this model, which describes both the thickness and the radial modes of
oscillation, as well as their mutual coupling, piezoelectric element is presented as a
4-access network with one electric and three mechanical accesses, which
correspond to its main contour surfaces. By cascade connecting the piezoceramic
disk model with metal endings model, which are derived based on the
piezoceramic model, a complete model of ultrasonic Langevin’s transducer is
obtained. Using this model one may determine any transfer function, whereat is
taken into account the external medium influence, as well as the influence of the
thickness and radial modes of every consisting transducer part. However, this
model is appropriate only for analysis of the sandwich transducers that have been
joined by gluing in manufacturing process, since in such model it is not possible to
take into account influence of the prestressing bolt and of internal openings of the
transducer consisting elements. Because of the significant differences between the
frequency characteristics of the piezoceramic and metal disks and rings, modeling
the sandwich transducers tightened by metal bolt using this model gives significant
deviations regarding the experimental results [152].
In this chapter is presented new three-dimensional matrix model of a classical
ultrasonic sandwich transducer. Ultrasonic sandwich transducer is modeled using
approximate three-dimensional matrix model of piezoceramic rings, presented in
Chapter 2.2.5. Using this model, which describes both the thickness and the radial
modes, as well as their mutual coupling, the piezoelectric element is presented as a
network with five accesses (one electric and four mechanical accesses), which
correspond to its main (contour) surfaces. In Figure 2.18(b) is presented in blocks
the mentioned model with input electric (voltage and current) and output
mechanical values (forces and velocities on the cylindrical surfaces, and forces and
velocities on the circular-ringed surfaces). Similar approach is also applied for
modeling the passive metal consisting parts, based on three-dimensional matrix
model of metal rings presented in Chapter 3.2.4 (Figure 3.15(b)), and that as a
network with three or four accesses, with corresponding mechanical values on the
accesses (forces and velocities on the cylindrical surfaces, and forces and velocities
on the circular-ringed surfaces). Here will be presented a transducer model with
one pair of piezoceramic rings, which are mechanically connected serially, and
electrically connected parallel. By mutual connection of piezoceramic rings, metal
endings, and the bolt, one gets the three-dimensional model of a complete
ultrasonic sandwich transducer. By this model too, one may determine any transfer
function, whereat here too is taken into account the influence of the external
medium, as well as the influence of both the thickness and the radial oscillation
modes of every consisting transducer part.
Linear equations that connect electric and mechanical values in the frequency
domain, in case of a piezoceramic ring are presented by equations (2.39), that is,
(2.40), so that they will not be treated here again. Same approach is used for
modeling the metal endings, whereat is taken that the values of the piezoelectric
constants are equal to zero, and the material isotropy is taken into account, so that
linear equations that connect mechanical values on the external surfaces of the
metal ring in frequency domain are given through expression (3.54), that is, (3.55).
Thus, metal rings are presented by four-access networks, while the bolt parts are
presented by three-access networks. The model of an ultrasonic sandwich
transducer is obtained by connecting the mechanical accesses, based on the really
existing surface contacts that those accesses represent.
Using this procedure for obtaining the resulting equivalent circuits at
mechanical serially-parallel and electric parallel elements connecting, a threedimensional electromechanical model of a sandwich transducer, whereat the access
number depends on that if it is the matter of symmetric or unsymmetrical
transducer. Transducer models for both cases are presented in Figures 4.30 and
4.31. If one connects corresponding acoustic impedances of the surrounding
medium Zi, which are small for an unloaded transducer, to the mechanical accesses
that correspond to the free surfaces, and if one applies an alternate excitation
voltage on the electric accesses, it is possible to determine any transfer function of
such system.
Z6b
Z6a
6
Z5
5
Z6c
Z7
7
6
~
PZT
ring 1
V
Z1a
5
Z1b
2
1
8
PZT
ring 2
Z3
3
Z4a
4
7
Z8
Z2a
3
8
9
4
Z2b
9
Z9
Z4c
Z4b
Figure 4.30. Model of a symmetric ultrasonic sandwich transducer
4.3.3. Comparison of Numerical and Experimental Results
In extension, the model is used for determination of input electric impedance of a
sandwich transducer, as well as for determination of frequency dependence of the first
(thickness) resonant mode on dimensions of the metal endings, for different transducer
construction. Verification of the proposed transducer model is first performed by
modeling the dependence characteristic of input impedance on frequency, for
previously analyzed ultrasonic sandwich transducers with excitation PZT8
piezoceramic rings, whose operating resonant frequencies are 26.06kHz and 41.6kHz.
Z6b
Z8b
Z9
9
Z6c Z8a
Z6a
6
8
Z5
5
7
~
PZT
ring 1
V
Z1a
Z7
Z10
10
Z1b
6
8 9
5
7
1
10
2
3
11
4
PZT
ring 2
Z3
3
Z4a
4
Z2a
Z2b
11
Z11
Z4c
Z4b
Figure 4.31. Model of an unsymmetrical ultrasonic sandwich transducer
Besides that, it is determined the input impedance of the transducer with
excitation PZT8 piezoceramic rings, duralumin emitter and steel reflector and bolt,
which has fundamental resonant frequency of 41.28kHz, and which is similar in
shape and dimensions to the transducer with frequency of 41.6kHz. Thereat its
dimensions, according to the symbols from Figures 1.1(b), are following:
l1=16.5mm, 2a1=39.8mm, l3=6.35mm, 2a3=38mm 2b3=13mm, l4=10mm,
2a4=40mm, l6=l1+2l3+l4, 2a6=8mm, l7=8mm, 2a7=13.15mm. Characteristics of
impedance dependence on frequency for cited transducers are obtained applying
the model of an unsymmetrical transducer from Figure 4.31 and they are presented
in Figures 4.32, 4.33, and 4.34, respectively. In purpose of comparison, in all
figures is also presented the impedance characteristic obtained by measuring by
automatic network analyzer on realized transducers with quoted dimensions and
material combinations. When alternating voltage is connected to the piezoceramic
ring electrodes, all presented resonant modes of such structure may be excited,
depending on frequency of the excitation generator.
As one may notice from Figures 4.32, 4.33, and 4.34, realized three-dimensional
transducer model might predict the fundamental thickness resonant frequency with great
accuracy, which is not the case with the one-dimensional models. Besides that, the higher
resonant modes may be also predicted by this model with satisfying accuracy, in broad
frequency range, which also is not the case with previously analyzed transducer models.
Yet, there are some resonant modes that are not encompassed by this method, which is not
unexpected, and about what was already discussed during modeling the piezoceramic and
metal rings, because models of these elements did not encompass all their existing resonant
modes in the observed frequency range. Differences that one may notice in Figures 4.32,
4.33, and 4.34 for minimal and maximal impedance values at resonant frequencies are
typical for the analysis without losses. Absolute values of impedance at resonant
frequencies are completely determined by losses. Dielectric and mechanical loss angles for
PZT ceramic, in which the losses are dominant, are not known precisely, because the values
of the excitation levels vary, and usually are obtained by subsequent fitting of
characteristics.
Like in the case of piezoceramic rings, here too, in complete sandwich transducers, the
possibilities of the model may be presented by frequency spectrum calculation, that is, by
determination of dependences of the fundamental (thickness) resonant frequencies on
dimensions of specific endings. Experimental verification of this model is performed by
comparison of the calculated frequency spectrum with measurements on concrete
specimens, whereat is noticed great agreement of calculated and measured resonant
frequencies. These dependences, obtained applying the proposed model of symmetric and
unsymmetrical transducer from Figures 4.30 and 4.31, are presented in Figures 4.35, 4.36,
4.37, and 4.38, for the case of symmetric transducers, and Figures 4.39 and 4.40, for
unsymmetrical transducers. In all these transducers are used ceramics of corresponding
diameter, with thickness l1=l2=6.35mm, made of material that is equivalent to the PZT4
ceramic. Dimensions of both transducer types, which are used in analyses, are presented on
the very figures, and they correspond to the symbols from Figure 4.30 for symmetric
transducers, and to the symbols from Figure 4.31 for unsymmetrical transducers.
Experimental dependences of the frequency spectrum on the ending lengths for a
symmetric transducer from Figure 4.35, and unsymmetrical transducer from Figure 4.39,
are used earlier too, in analysis of the seeming elasticity moduli method 4.28 and 4.29). In
this case are presented theoretical and experimental dependences of the resonant frequencies
of the first and the second resonant mode on the ending length, for the symmetric transducer
from Figure 4.35 whereat for both modes one may notice great agreements of the compared
dependences in the observed range. In Figure 4.36 is presented on the same graph
dependence of the resonant frequency on length, obtained by the proposed threedimensional matrix model for the observed transducer, if the piezoceramic platelets and
metal endings would be disks without openings, which corresponds to the application of the
model from literature [151]. Transducer parts should be joined by gluing in that case, that is,
in the observed case there is no prestressing bolt. This analysis shows too, that during the
sandwich transducer modeling it is not real to approximate the piezoceramic and metal rings
by disks, because then one gets quite greater values of the resonant and antiresonant
frequencies. Same results are obtained in all considered symmetric and unsymmetrical
sandwich transducers.
70
60
Zul [dB]
50
40
30
20
10
0
0
1
2
3
4
5
f [Hz]
6
4
x 10
Figure 4.32. Dependence characteristic of the input impedance on frequency of the
ultrasonic transducer with resonant frequency of 26.06kHz: proposed threedimensional model (
)
70
60
Zul [dB]
50
40
30
20
10
0
0
5
10
f [Hz]
15
x 10
4
Figure 4.33. Dependence characteristic of the input impedance on frequency of
the ultrasonic transducer with resonant frequency of 41.6kHz: proposed threedimensional model (
)
70
60
Zul [dB]
50
40
30
20
10
0
0
2
4
6
10
8
f [Hz]
12
4
x 10
Figure 4.34. Dependence characteristic of the input impedance on frequency of
the ultrasonic transducer with resonant frequency of 41.28kHz: proposed threedimensional model (
)
7
fa
fr
fr
fa
6
measuring results; threedimensional model of the
complete transducer
fa
5
fr
4
3
2
1
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
Figure 4.35. Frequency spectrum in function of duralumin ending lengths l3+4=l5+6
for a symmetric transducer with dimensions: l1= l2=6.35mm, 2a1=2a2=38mm,
2b1=2b2=13mm, l8=l1+ l2, l3= l5= l7= l9=16.2mm, 2a3=2a4=2a5=2a6=40mm,
2b3=2b4=2b5=2b6=10mm, 2a7=2a8=2a9=10mm
6
5.5
fr
fa
fa
5
4.5
fr
4
fa
3.5
fr
3
2.5
2
1.5
1
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
Figure 4.36. Frequency spectrum in function of duralumin ending lengths l3+4=l5+6
for a symmetric transducer with dimensions: l1= l2=6.35mm, 2a1=2a2=50mm,
2b3=2b4=2b5=2b6=16mm, 2a7=2a8=2a9=16mm
6
5.5
fr
fa
fa
5
4.5
fr
4
3.5
3
2.5
2
1.5
1
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
Figure 4.37. Frequency spectrum in function of steel ending lengths l3+4=l5+6 for a
symmetric transducer with dimensions: l1= l2=6.35mm, 2a1=2a2=38mm,
2b3=2b4=2b5=2b6=17mm, 2a7=2a9=17mm, 2a8=10mm
5
fr
4.5
fa
4
fa
3.5
fr
3
2.5
2
1.5
1
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
Figure4.38. Frequency spectrum in function of steel ending lengths l3+4=l5+6 for a
symmetric transducer with dimensions: l1= l2=6.35mm, 2a1=2a2=50mm,
2b3=2b4=2b5=2b6=24.5mm, 2a7=2a9=24.5mm, 2a8=16mm
7
fr
fa
6
5
fa
4
fr
3
2
1
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
Figure 4.39. Frequency spectrum in function of duralumin ending lengths l3+4 for
an unsymmetrical transducer with dimensions: l1= l2=6.35mm, 2a1=2a2=38mm,
2b1=2b2=13mm, l3= l11=11.3mm, l8=l9=8mm, l5=l7=6mm, l6=4.88mm,
2b6=2b5=2a8=2a7=14mm, 2a3=2a4=2a5=2a6=40mm, 2b3=2b4=2b7=10mm,
2a9=2a10=2a11=10mm, l10=l1+ l2+ l7
5
fr
4.5
fa
4
3.5
fa
3
2.5
fr
2
1.5
1
0.025
0.03
0.035
0.04
0.045
0.05
Figure 4.40. Frequency spectrum in function of duralumin ending lengths l3+4 for
an unsymmetrical transducer with dimensions: l1= l2=6.35mm, 2a1=2a2=50mm,
2b1=2b2=20mm, l3= l11=28mm, l8=l9=10mm, l5=l7=10mm, l6=21mm,
2b6=2b5=2a8=2a7=22mm, 2a3=2a4=2a5=2a6=51mm, 2b3=2b7=14mm, 2b4=0,
2a9=2a10=2a11=14mm, l10=l1+ l2+ l7
Based on the presented dependences one may conclude that there exists a great
agreement of the resonant and antiresonant frequencies obtained by the proposed threedimensional transducer model with experimental results, as in the region of great ending
lengths, that is, in the area of the low frequencies, so in the region of short endings, at higher
frequencies. Thereby is decreased the error in predicting the resonant frequencies, which is
present in one-dimensional models. Concerning the one-dimensional theory, for same
ending lengths it gives better results in case of transducers from Figures 4.37 and 4.39,
regarding the transducers from Figures 4.38 and 4.40, because they have smaller crosssection, and then the transducer is closer to the ideal one-dimensional model. In threedimensional model in both cases are obtained satisfying results, independently of the
specific relations of the transducer dimensions, and thereby of the relation of the crosssection and length. In case of the proposed three-dimensional model, there also exist
nonnegligible deviations of calculated and measured results, but only in the region of small
ending lengths. This is a consequence of the fact that the transducer parts are joined by a
metal bolt, whereat is during ending length decrease, starting from the specific ending
length, the bolt shortened too, whereby is changed the piezoceramic prestressing force, and
by that itself changed the piezoceramic parameters and the connection between the elements
is weakened. In the transducer model is taken into account the bolt shortening, but it is not
considered the piezoceramic characteristics change due to the changed prestressing
conditions. In order to illustrate how the changes of the specific parameters of the sandwich
transducer look like in different prestressing conditions, In Figures 4.41, 4.42, and 4.43 are
presented the prestressing influences on the changes of resonant and antiresonant
frequencies, on the changes of the minimal transducer impedance at fundamental resonant
frequency, and on the Q-factor changes, obtained experimentally. Thereat are analyzed two
similar transducers, whose dimensions are presented in Figure 4.40, and concrete lengths of
their emitters were l3+l4=46mm and l3+l4=48mm.
Based on the presented graphs, it is obvious that prestressing improves the mechanical
contact between the parts, which leads to significant decrease of mechanical losses in
contact zones. Number of contact surfaces is increased by presence of soft copper foils
between the transducer parts. At low pressures, there is no existence of a sufficient
mechanical contact between the parts, so that minimal demanded pressure value is about
30MPa, when resonant and antiresonant frequency, minimal impedance, and Q-factor cease
to depend significantly on the pressure value. The obtained optimal prestressing value,
which is most often determined experimentally, confirms theoretical assumptions from this
field [119], [120]. In addition, the equivalent resistance (minimal impedance) is substantially
reduced at high-pressure values. In the treated three-dimensional model, one starts from the
assumption that the connection of the transducer consisting parts is such that there exists
good mechanical contact between them, i.e., the prestressings are greater than 30MPa.
2.6
fa
2.5
2.4
fr
2.3
2.2
2.1
2
0
5
10
15
20
25
30
Figure 4.41. Experimental dependences of the resonant (bottom graphs) and
antiresonant (top graphs) frequency in function of mechanical
prestressing of the transducer from Figure 4.40
50
45
40
35
30
25
20
15
10
0
5
10
15
20
25
30
Figure 4.42. Experimental dependences of the minimal impedance (resistance) of
the transducer at resonant frequency in function of mechanical
prestressing of the transducer from Figure 4.40
500
450
400
350
300
250
200
150
100
0
5
10
15
20
25
30
Figure 4.43. Experimental dependences of the Q- factor in function of
mechanical prestressing of the transducer from Figure 4.40
It should be remarked that all theoretical analyses presented in this chapter are
very sensitive even to small changes in parameter values of the used materials, which
especially relates to the parameters of the used piezoceramic rings. Values of the
piezoceramic parameters are measured at low levels of excitation signals, that is, at
same conditions that existed during the measuring by network analyzer in
experimental parts of this analysis, so it is considered that during modeling one may
use catalogue values of the piezocoefficients. Like in the case of the one-dimensional
model, here too may be subsequently fitted the initial values of the piezoconstants and
parameters for metal endings based on the experimental measurements, in the vicinity
of every resonant mode, for every realized transducer. By such approach, in the case of
treated transducers, one gets small changes of the coefficients of piezoceramic and
metal endings regarding the initial data.
In Table 4.3 are presented values of the piezoceramic parameters of different
manufacturers, which are close to the used PZT4 and PZT8 ceramics by its properties
and which are also used in the field of power ultrasound. PZT4 and PZT8
piezoceramics are modified lead-zirconium-titanate ceramics (Cr doped in PZT4, Fe
and Mn in PZT8), so that small changes in composition generate great changes of the
characteristics of these piezoceramics. It may be noticed based on the comparative
Table 4.3, for similar piezoceramics from different manufacturers. PZT4 and PZT8 are
ceramics that have relatively small piezoelectric, dielectric, and elastic constants. PZT4
piezoceramics is very suitable for applications in power ultrasound because it may be
undergone to great stressing (greater than the PZT8 ceramic), since it has small
dielectric losses and Curie’s temperature above 300oC. PZT8 piezoceramics is
specially developed for transducers that demand great excitation voltage. At powerful
excitations it has much lower dielectric losses and greater resistance to depolarization,
two times greater mechanical Qm-factor, but also smaller coupling factors than the
PZT4 piezoceramic, while its Curie’s temperature is 300oC.
Presented Table 4.3 contains piezoceramic material parameters, obtained
from different piezoceramic manufacturers, which are most often used in the power
ultrasound technique, and which are by its characteristics similar to the generally
known and most often used PZT4 and PZT8 piezoceramics. Standard symbols for
specific parameters are used [67]: e33S is dielectric constant in compressed state
(e0=8.85x10-12 F/m); cijE, cijD, and sijE are piezoelastic constants; eij and hij are
piezoelectric constants (i, j=1, 2, 3, 4), kp, k31, and k33 are coupling factors of the
material, and r is density. Only three piezoceramic manufacturers have complete
data necessary for modeling and simulation of their ceramics by models that are
more complex. Here are presented values of only those parameters that are in this
analysis used for calculation in some one-dimensional and three-dimensional
models. Beside the characteristics of the PZT4 and PZT8 ceramics (Vernitron1), the
characteristics of the following ceramics are presented: EC64 and EC69 (Edo
Western2), Pz24, Pz26, and Pz26 for FEM users (Ferroperm3).
1
Vernitron Ltd., now Morgan Matroc Ltd.: http://www.morganmatroc.com
Edo Corporation/Western Division: http://www.edocorp.com
3
Ferroperm Piezoceramics A/S: http://www.ferroperm-piezo.com
2
Other piezoceramic manufacturers have available data for their piezoceramic
materials that are sufficient only for their one-dimensional modeling. In Table 4.4
are presented parameters of such piezoceramics, which by their properties also
correspond to the characteristics of the PZT4 and PZT8 piezoceramics, the most
often used in the power ultrasound technique. Characteristics of the following
ceramics are presented: SONOX P4 and SONOX P8 (CeramTec4), APC 840 and
APC 880 (American Piezo Ceramics5), PXE 41 and PXE 42 (Philips6) and HYP42
(Hwang Sun 7).
Table 4.3. Piezoceramic Parameters of Different Manufacturers
Symbol
Unit
PZT4
PZT8
EC64
EC69
PZ24
PZ26
PZ26 for
FEM
users
ε33S/ε0
1010 N/m2
1010 N/m2
1010 N/m2
1010 N/m2
1010 N/m2
1010 N/m2
1010 N/m2
1010 N/m2
10-12 m2/N
10-12 m2/N
10-12 m2/N
10-12 m2/N
10-12 m2/N
C/m2
C/m2
108 V/m
108 V/m
kg/m3
635
13.9
7.78
7.43
11.5
14.5
8.39
6.09
15.9
12.3
-4.05
-5.31
15.5
39.0
-5.2
15.1
-9.2
26.8
-0.58
-0.33
0.70
7500
582
13.7
6.97
7.16
12.4
14.0
7.28
6.08
16.1
11.5
-3.38
-4.69
13.5
31.9
-4.0
13.8
-7.8
26.9
-0.51
-0.30
0.64
7600
645
13.9
7.78
7.43
11.5
14.5
8.39
6.09
15.9
12.8
-4.2
15.0
-5.2
15.1
-9.2
26.8
-0.60
-0.35
0.71
7500
646
14.9
8.11
8.11
13.2
15.2
8.41
7.03
16.9
10.1
-3.4
13.5
-4.1
14.0
-7.7
26.4
-0.52
-0.31
0.62
7500
239
16.2
8.84
8.75
13.4
16.3
8.94
8.07
18.1
10.5
-3.13
-4.77
13.6
23.0
-1.45
9.9
-6.88
47.0
-0.494
-0.292
0.659
7700
700
16.8
11.0
9.99
12.3
16.9
11.2
9.33
15.8
13.0
-4.35
-7.05
19.6
33.2
-2.8
14.7
-4.52
23.7
-0.568
-0.327
0.684
7700
518
15.5
9.41
7.99
11.0
16.2
10.1
8.64
15.1
12.2
-5.13
-5.13
16.0
36.7
-5.48
13.6
-12.0
29.7
-0.575
-0.310
0.707
7650
c11E
c12E
c13E
c33E
c11D
c12D
c13D
c33D
s11E
s12E
s13E
s33E
s44E
e31
e33
h31
h33
kp
k31
k33
ρ
In all theoretical analyses presented in this chapter are obtained similar
dependences, if one applies parameters of the piezoceramics equivalent to the
PZT4 or PZT8 piezoceramics from other manufacturers. The only exception
represents the case presented in Figure 4.44.
4
CeramTec N.A.E.A.: http://www.ceramtec.com
APC International Ltd.: http://www.americanpiezo.com
6
Philips: http://www.philips.com
7
Hwang Sun Enterprice Co., Ltd.: http://www.hes.com.tw
5
Table 4.4. Piezoceramic Parameters of Different Manufacturers
Symbol
ε33S/ε0
E
s33
h33
kp
k31
k33
ρ
Unit
SONOX4
SONOX8
APC
840
APC
880
PXE 41
PXE 42
HYP 42
-
660
18.1
29.3
-0.57
-0.31
0.68
7650
540
13.7
36.6
-0.55
-0.30
0.68
7700
602
17.4
31.2
-0.59
-0.35
0.72
7600
616
15.0
26.3
-0.50
-0.30
0.62
7600
554
14.6
37.4
-0.64
-0.38
0.74
7900
676
15.0
31.3
-0.61
-0.34
0.70
7800
698
19
27.3
-0.66
-0.36
0.72
7700
10-12 m2/N
108 V/m
kg/m3
In Figure 4.44 is given dependence characteristic of the input impedance on
frequency for the transducer from Figure 4.34, obtained applying the threedimensional model with data for Pz26 piezoceramic (Ferroperm), which is equivalent
to the PZT4 ceramic used in this analysis, but here are used parameters recommended
from the manufacturer for finite element method users. These parameters are presented
in the last column of the Table 4.3, and they are obtained by parameter modification
based on experimental measurements, with rationale that, beside the fundamental
resonant mode, with these parameters is also satisfying the prediction of higher
resonant modes of the piezoceramic. In application of this method in power ultrasonic
transducers design most often are determined zero values of the input admittance of
the piezoceramic and the whole transducer, so that, if the transducer impedance is
observed, for given data one should expect satisfying results in the region of
antiresonant frequencies. It may be noticed in Figure 4.44, where antiresonant
frequencies, calculated by the model, are close to the real antiresonant frequencies.
However, calculated resonant frequencies much differ from experimental resonant
frequencies, which means that using different data for same piezoceramic coefficients,
depending if it is about analytical and FEM methods, is not a good approach in
transducer modeling. Our further investigations in this field will relate to the
realization of an acceptable model of a sandwich transducer applying the finite element
method, which will not contain previous discrepancies.
In order to further present the possibilities of the proposed model for ultrasonic
sandwich transducer analysis, as in the case of the piezoceramic rings here are
analyzed changes of the input transducer impedance in function of frequency and
specific transducer parameters, that is, in function of different dimensions of the
consisting elements and different acoustic impedances that load the external
surfaces. Thereat is analyzed a concrete sandwich transducer with 5mm thick PZT8
rings and with fundamental resonant frequency of 41.6kHz, whose impedance
characteristic is presented in Figure 4.33, and whose dimensions, according to the
symbols from Figure 4.31, are following: l1=l2=5mm, 2a1=2a2=38mm,
2b1=2b2=15mm, l3=l11=18mm, l8=l9=8mm, l5=l7=11.2mm, l4=l6=0, l10=l1+l2+l7,
2a9=2a10=2a11=8mm,
2b6=2b5=2a8=2a7=12.8mm,
2b3=2b4=2b7=8mm,
2a3=2a4=2a5=2a6=40mm. Because of the mutual coupling of the oscillations, one
cannot claim for a specific resonant mode of a transducer that it originates from
radial or thickness mode of only one transducer consisting element, i.e., that it is
determined only by one its dimension. Resonant modes of the transducer depend
on the coupling of several specific modes, that is, they simultaneously depend on
several dimensions, but one may notice which dimensions of the consisting
transducer parts affect mostly on the observed resonant mode.
70
60
Zul [dB]
50
40
30
20
10
0
0
2
4
6
f [Hz]
8
10
12
4
x 10
Figure 4.44. Dependence characteristic of the input impedance on frequency
of the ultrasonic transducer from Figure 4.34, if one uses FEM data for Pz26
piezoceramic ring: proposed three-dimensional model (
)
and experimental results (
)
First is in Figure 4.45 presented dependence of the input impedance of this
transducer in function of frequency and emitter length, if the emitter length increases
for 30mm regarding the primary length of 18mm. Impedance characteristic of this
transducer from Figure 4.33 contains six main resonant modes in the observed
frequency range, whereat is in all these analyses the most important behavior of the
lowest resonant modes, especially the first mode, which is the operating resonant mode
of the transducer. From Figure 4.45 is obvious that at small increase of the emitter
length, a frequency decrease of the first, and especially the second and the sixth
resonant mode occurs, while the frequencies of the third, fourth, and fifth mode do not
depend on this change. At great emitter length, resonant frequencies of the second,
fourth, and sixth mode do not depend on the emitter length, while the frequencies of
the first, third, and fifth mode are functions of the emitter length. At great emitter
lengths, between the third and the fourth mode occurs another resonant mode.
Accordingly, only the first mode is the true thickness resonant mode, because it
depends on the emitter length in the whole observed range. Analyzing by the proposed
model one may, therefore, assume that if one would increase the emitter length, the
fundamental resonant frequency would decline, but approaching of the first and the
second mode would perform, which is unfavorable, because the second mode is
undesirable in the vicinity of the fundamental resonant mode. Presence of the
undesirable (parasite) modes in the vicinity of the operating frequency may
significantly affect the quality of ultrasonic cleaning or welding. According to the form
of presented modes, if the emitter length would be smaller than the initial length, the
first and the second mode would also approach, so that adopted emitter length of
18mm is optimal and then the distance between the first and the second resonant mode
is the greatest. The second resonant mode is, accordingly, highly determined by the
emitter length, but it does not depend on that dimension.
In literature [83] is presented Matlab software for obtaining the input impedance
dependence on frequency and on emitter length from Figure 4.45, for the previously
analyzed unsymmetrical transducer. Modifying this software, that is, changing the
dimensions and material parameters of the transducer consisting parts, one may
determine all dependences of the impedance or frequency spectrum analyzed in this
chapter, as for the unsymmetrical, so for the symmetric transducers.
There is an inverse situation in case of change of the emitter cross-section, and
that case is presented in Figure 4.46, where it is given the input impedance change in
function of frequency and outer diameter of the emitter, which changes in range from
14mm to 40mm. In that case, the resonant frequencies of the second, fourth, and sixth
mode decrease with increase of the outer emitter diameter, while the frequencies of the
first, third, and fifth mode do not change under effect of this change. The thickness
mode almost none depends on this change, which was logical to expect, and on the
other hand, behavior of the fourth and the sixth mode shows that these modes depend
mostly on radial dimensions of the emitter. Decreasing the outer diameter of the
emitter the first and the second resonant mode are separated, which is desirable from
the excitation aspect, but thereby is decreased the cross-section of the emitter regarding
the ceramic, which limits the transducer possibilities.
Similar analysis may be performed in case of the observed transducer with
constant dimensions, with emitter of 18mm, if different acoustic loads are connected to
the operating emitter end and to the external cylindrical emitter surface. Characteristics
of the input transducer impedance in that case are presented in Figures 4.47 and 4.48.
Based on application of the proposed model, it is obvious that the second undesirable
resonant mode will decrease during emitter loading in operating conditions (Figure
4.47), and it will not affect the fundamental resonant mode, so this is a confirmation of
the proper transducer design, too. This dependence on acoustic load on the operating
surface shows too that the first, second, and sixth mode depend on the emitter
characteristics in thickness direction. In Figure 4.45, the second and the third resonant
mode are in mutually sufficient distance at great emitter lengths, while they are close at
small emitter lengths. The dependence presented in Figure 4.47 also confirms strong
mutual coupling of these modes, for the initial emitter length of 18mm, because at high
loading of the emitter the second mode disappears and approaches to the third resonant
mode, whereat it affects its form. High mechanical load on the cylindrical surface of
the emitter shows, based on the Figure 4.48, that the greatest change is in the fourth
resonant mode, while there is significant change in the second and the sixth resonant
mode, which is analogous to the conclusions obtained based on Figure 4.46.
0.05
Zul [dB]
0.045
50
40
0.04
30
0.035
20
l3+l4 [m]
0.03
10
0.025
0
0
0.02
5
x 10
f [Hz]
4
10
15
Figure 4.45. Change of the input impedance of a transducer
in function of frequency and emitter length l3+l4
0.04
Zul [dB]
50
0.035
40
0.03
2a3=2a4 [m]
30
0.025
20
10
0.02
0
0
5
x 10
0.015
f [Hz]
10
15
4
Figure 4.46. Change of the input impedance of a transducer
in function of frequency and outer diameter of the emitter 2a3=2a4
Zul [dB]
50
40
30
0
20
Z4b [Rayl]
10
5
x 106x
10
6
0
0
5
x 10
f [Hz]
10
10
15
4
Figure 4.47. Change of the input impedance of a transducer in function of
frequency and acoustic load Z4b on the plane surface of the emitter
Zul [dB]
50
40
30
0
20
10
5
6
Z3=Z4ax [Rayl
]
10
0
0
5
x 10
4
f [Hz]
10
15
10
frequency and acoustic load Z3=Z4a on the external cylindrical emitter surface
Same dependences for a steel reflector are presented in Figures 4.49, 4.50, 4.51, and
4.52. In Figure 4.49 is presented dependence of the input impedance of the transducer in
function of frequency and reflector length, if the reflector length increases for 30mm
regarding the primary length of 11.2mm. This is much greater range of length change,
regarding the case from Figure 4.45, respecting that the reflector is shorter and that its
characteristics are different. Small increase of the reflector length generates frequency
decrease of the first, second, fourth, and the sixth resonant mode, although those changes
are not so outstanding like in the case of emitter length change, while the frequencies of the
third and the fifth mode do not depend on this change. At great reflector length the second
resonant mode approaches to the first resonant mode, frequencies of the third and the fourth
mode decrease, and the sixth mode becomes the fifth. Respecting the drastic changes of the
sixth mode, one may conclude that the reflector dimensions in thickness direction primarily
determine this mode. Fundamental thickness resonant mode, as in case of emitter length
change, depends on the reflector dimensions in the whole observed range. Since the relative
changes of the reflector length are greater regarding the emitter length change from Figure
4.45, one may conclude based on Figure 4.49 that changes of the fundamental resonant
frequency are much smaller regarding the case of emitter length change. It means that
thickness resonant mode is far less sensitive to the reflector length changes, regarding the
same emitter length changes, which means that for a given material combination one may
adjust the desired resonant frequency much faster by emitter length change, which is
favorable from the aspect of metal processing, too. Thereby is realized one of the basic
demands in transducer design that transducer characteristics depend as less as possible on
the reflector characteristics.
In Figure 4.50 is given the change of the input impedance in function of frequency and
outer diameter of the reflector, which changes in range from 14mm to 40mm. Then the
resonant frequencies of the third, and especially the fourth mode, decrease with increase of
the outer diameter of the reflector. In addition, the frequencies of the first (thickness) mode
here also do not change under the influence of this change, which also stands for the
frequencies of the second, fifth, and the sixth resonant mode. If it is the matter of a
transducer with reflector with constant length of 11.2mm, which is loaded by high acoustic
impedance on its plane surface, one obtains conclusions analogous to the conclusions
derived based on Figure 4.49. In that case too, which is presented in Figure 4.51, one may
notice that with load increase, the second, fourth, and the sixth resonant mode decrease. The
same characteristic in case of load change on external cylindrical surface of the reflector is
presented in Figure 4.52, whereat from the presented dependence one may notice the
changes of only the third resonant mode. The last two cases are presented in purpose of
completion the behavior of the transducer characteristics in function of reflector
characteristics, and they have no practical meaning, because in all ultrasonic applications the
reflector is unloaded. Therefore, in this case too was proved that transducer characteristics
did not depend much on the reflector characteristics, i.e., in this case on the way of its
mechanical loading.
In design of ultrasonic sandwich transducers one may influence on the transducer
characteristics mostly by choosing the characteristics and dimensions of the metal endings,
because it is implied that there will be used piezoceramic rings with available thickness,
which cannot be influenced on, and which represents a constant in design.
0.045
0.04
Zul [dB]
0.035
0.03
40
l5+l6 [m]
0.025
20
0.02
0
0.015
0
5
x 10
f [Hz]
4
10
15
0.01
frequency and reflector length l5+l6
Zul [dB]
0.04
50
0.035
40
0.03
30
2a5=2a6 [m]
20
0.025
10
0.02
0
0
5
x 10
4
f [Hz]
10
0.015
15
frequency and outer diameter of the reflector 2a5=2a6
Zul [dB]
50
40
30
20
0
10
6
Z6b [Rayl
x 1 0]
5
0
0
5
x 10
f [Hz]
4
10
15
10
frequency and acoustic load Z6b on the plane reflector surface
Zul [dB]
50
40
30
0
20
10
5
x 10
6
Z5=Z6a [Rayl]
0
x 106
0
5
x 10
4
f [Hz]
10
10
15
frequency and acoustic load Z5=Z6a, on the external cylindrical reflector surface
0.015
Zul [dB]
40
0.01
l1=l2 [m]
20
0
0
5
x 10
0.005
10
f [Hz]
15
4
frequency and piezoceramic length l1= l2
Zul [dB]
50
40
30
20
10
0
0
2
4
6
x 10
Z1a=Z2a [Rayl]
6
8
10
0
5
f [Hz]
15
10
x 10
4
frequency and acoustic load Z1a=Z2a on the external cylindrical surfaces on both
piezoceramic rings
In Figure 4.53 is presented the way that the choice of the thicker piezoceramic
rings would reflect on the transducer characteristics. It is obvious that increase of
the piezoceramic thickness generates decrease of the frequencies of all resonant
modes, whereat the relative position of the first two resonant modes does not alter
with this change. When one talks about the piezoceramic rings in a transducer,
beside the presented dependence, it makes sense to analyze only the case from
Figure 4.54. Based on the impedance characteristic from Figure 4.54, in case of
piezoceramic rings load on external cylindrical surfaces, one may conclude that
eventually fixing of the transducer in the region of piezoceramic rings would
generate drastic shifting of the frequencies of the fundamental resonant mode
towards the higher resonant frequencies, and even disappearing of this mode at
high loads. It means that high piezoceramic load in radial direction generates
decrease of the radial modes of the piezoceramic rings, which is even presented in
the second chapter (Figure 2.22(b)), so thereby it generates disappearing of the
fundamental radial mode of the piezoceramic rings, which is located just at the
thickness resonant frequency of the whole transducer. Therefore, behavior of the
impedance characteristic from Figure 4.54 is a consequence of the high coupling of
the fundamental radial mode of the piezoceramic rings and the thickness resonance
of the whole transducer, especially for the transducers with fundamental resonant
frequency around 40kHz, so that knowing of the lowest radial modes of the rings is
essential in analysis of the complete sandwich transducer. This analysis was not
possible in case of the one-dimensional modeling of the sandwich transducer.
Attaching (fixing) of the symmetric transducers in operating conditions is
usually performed such that in the region of oscillation node, between the
piezoceramic rings, is inserted a thin duralumin ring, whereat the fixing is
performed on its external cylindrical surface. This case is easily modeled by
including this duralumin ring into the model Figure 4.30. Analyzing the load of
this ring in radial direction, on the external cylindrical surface, one may show that,
in contrast to the previous case, in this case changes of the fundamental resonant
frequency of the transducer are negligible.
By this model it is possible to determine any transfer function of the
transducer, whereat besides the input electric impedance the most often is
determined the transfer function Fi/V, where by Fi is denoted the surface force on
the observed mechanical access. Besides that, it is possible to perform an analysis
of the sensitivity of the specific transducer characteristics to the values of the
piezoelectric and elastic constants of the piezoceramic rings, as well as to the
values of the metal rings parameters. In addition, as in the case of the piezoceramic
rings in the second chapter, here too is possible to determine the effective
electromechanical coupling factor keff in function of frequency and specific
dimensions of the consisting parts for any resonant mode of the sandwich
transducer. This possibility is very important, because based on the previous
analysis it is obvious that by the proposed model one may obtain a transducer with
particular resonant frequency using several length combinations of transducer
consisting parts. However, since it is important to obtain a transducer with as great
as possible keff, only some dimension combinations will enable as great as possible
ability of converting the electric into the mechanical energy. Based on the previous
analysis, it is possible to perform design of an ultrasonic sandwich transducer by
the proposed model for any demanded resonant frequency, but not for any
dimension ratio of transducer consisting parts, yet only for those dimension
combinations that enable as great as possible emitted ultrasonic power. Therefore,
using this model one may easily predict what those dimensions are.
Based on all mentioned considerations it is possible to make easily selection of
materials and dimensions of the sandwich transducer consisting parts. Thus, using
this model it is possible to evaluate very quickly the quality of every ultrasonic
transducer that should be realized.
In contrast to the former approaches in transducer design, here is in input
impedance analysis great attention dedicated to the form and the position of the
higher resonant modes, and to the influence of the specific parameters on their
behavior. The first reason for such approach is the need that higher, parasite modes,
have as small as possible influence on the resonant frequency, that is, to
be
on as great distance from it as possible. This model enabled to determine what
length or diameter, and of which consisting part one should change, in order to
realize the desired separation of modes. The second reason for analysis of the
higher resonant modes is that during long practice of the transducer production,
based on the form of the higher resonant modes, was noticed that inadequate choice
of some transducer elements lead to an irregular form of some of the higher
resonant modes. Analysis using proposed model may show which dimension, and
which element it is, based on the influence of that element on the observed
resonant mode. If some higher resonant mode is damped because of a design error,
and if this may be noticed in the static measuring conditions applied in this
analysis, regardless if that resonant mode is isolated from other modes, it will
certainly reflect on the characteristics of the fundamental resonant mode in
dynamic operating conditions too, regardless that the impedance characteristic in
the vicinity of the thickness mode is satisfying in static measuring conditions.
Applying this model one may determine the cause of such irregularities in realized
transducer. During the realization of numerous transducers it was noticed that
influence of the reflector on the transducer characteristics was small, so by
changing of the emitter characteristics the most often was performed adjusting of
the resonant frequency. This model confirmed that assumption, based on the
previously performed analyses. In ultrasonic technique, it was seldom approached
to the design of the transducers with entirely new frequencies and characteristics,
due to the impossibility of analysis of different parameters influence, and that is
now enabled by the proposed three-dimensional model. Usually designs finished
with copying those transducers that showed satisfying characteristics in practice.
This model unfolds great possibilities in solving this problem during designing the
new transducers.
Previous analysis of the impedance characteristics on dimensions in thickness
and radial direction shows that it is realized a model by which one may very
quickly perform a synthesis, and evaluate the quality of every sandwich transducer
that could be realized, which was the goal of this analysis. An ideal model, which
would take into account all parameters, boundary conditions, all existing resonant
modes, and states which the sandwich transducer characteristics depend on, cannot
be realized at all, so in this analysis one tended to the obtaining of an as complete
as possible, but simple model, which, although approximate, takes into
consideration as many initial parameters as possible. It was showed that, although
the transducer characteristics depended on many parameters, nevertheless from all
parameters the greatest influence had the length (thickness) of the transducer
consisting parts. In contrast to the finite element method, which is complicated for
such parameter analysis, the proposed three-dimensional model is simple.
However, although the model is simple, calculations performed to determine
specific transducer characteristics are very complicated, as in case of determination
of the transducer characteristics in this chapter, so in determination of the
characteristics of the piezoceramic and metal rings from previous chapters, and this
stands either for the three-dimensional model, or for the numerical method
(Chapter 3.2.3.). These calculations are not presented in detail, but the procedures
how certain characteristics are obtained are described, and endmost results of these
calculations are contained in the computer programs enclosed in literature [83]. By
this model, one obtains very quickly returning information about the change
directions of different characteristics at varying the specific parameters, which was
the basic purpose of modeling in this analysis, aimed to assist the designers of new
systems.
Therefore, comparing with the one-dimensional theory, obtained results better
agree with experimental results. Based on the presented model, it is possible to
perform a synthesis of the sandwich transducers with characteristics defined in
advance and with given resonant frequency, which was the purpose of this analysis.
Based on the designing results obtained using the proposed model of ultrasonic
sandwich transducers, it is realized a great number of transducers not presented
here, and which are tested in many applications. Measuring results of electric
characteristics of such obtained electromechanical systems confirm correctness of
such approach in design.
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Prof. Milan Đ. Radmanović, Ph.D. was born on the 15th of October 1948,
in Poljane, Republic of Croatia. Graduated in 1972 on Faculty of Electronics
in Niš, where he also acquired the title of master of electrical engineering in
1977, and the title of doctor of technical sciences in 1988. Currently on
position of Professor Associate in the Department of Electronics of the Faculty
of Electronics in Niš. At undergraduate and postgraduate studies teaches the
subjects of Energetic electronics, Electroenergetic transducers, and Control of
electroenergetic transducers. Supervised and took part in realization of 14
projects financed by Ministry for Science and Technology of Republic of
Serbia. Author or coauthor over one hundred papers, reported or printed in
journals and paper collections, and coauthor of one university textbook. Scope
of his scientific interest is energetic electronics, and modeling and design of
power ultrasonic transducers and generators.
Dragan D. Mančić, Ph.D. was born on the 9th of April 1967 in Visočka
Ržana near Pirot. Graduated on Faculty of Electronics in Niš in 1991, and
became master of science on the same faculty in 1995. In 2002 acquired the
doctor’s degree, also on Faculty of Electronics in Niš. Dragan Mančić, Ph.D. is
a reader on the Faculty of Electronics for section of electronics, where he
teaches in courses for Electronics, Automation, and Industrial energetics in
subjects Energetic electronics and Control of electroenergetic transducers.
Coauthor of one university textbook. He published over sixty scientific and
expert papers in international and national journals and collections of papers
from international and national conferences. He deals with energetic
electronics, and modeling and design of power ultrasonic transducers and
generators.
Edition: Monographies
⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯
ISBN 86-80135-87-9

Univerzitet u Ni{u

Transcription

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