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An approximated 3-D model of the Langevin transducer
and its experimental validation
Antonio Iula,
a)
Riccardo Carotenuto, and Massimo Pappalardo
Dipartimento di Ingegneria Elettronica, Universita
`
Roma Tre, Via della Vasca Navale, 84, 00146 Roma,
Italy
Nicola Lamberti
Dipartimento d’Ingegneria dell’Informazione ed Ingegneria Elettrica, Universita
`
di Salerno,
Via Ponte Don Melillo, 84084 Fisciano (SA), Italy
共Received 4 September 2001; accepted for publication 16 March 2002兲
In this work, an approximated 3-D analytical model of the Langevin transducer is proposed. The
model, improving the classical 1-D approach describing the thickness extensional mode, allows us
to predict also the radial modes of both the piezoelectric ceramic disk and the loading masses;
furthermore, it is able to describe the coupling between radial and thickness extensional modes. In
order to validate the model, the computed frequency spectrum is compared with that obtained by
measurements carried out on 13 manufactured samples of different thicknesses to diameter ratios.
The comparison shows that the model predicts with quite good accuracy the resonance frequencies
of the two lowest frequency modes, i.e., those of practical interest, all over the explored range.
Finally, the coupling effect between thickness and radial modes on the frontal displacement is
measured and discussed. © 2002 Acoustical Society of America. 关DOI: 10.1121/1.1476684兴
PACS numbers: 43.38.Fx 关SLE兴
I. INTRODUCTION
The Langevin transducer basically consists of a piezo-
electric ceramic disk sandwiched between two cylinder-
shaped loading masses. This structure is usually prestressed
by inserting a bolt along its principal axis in order to increase
the mechanical strength of the piezoelectric ceramic.
This kind of composite transducer is widely used in un-
derwater sonar and communication systems
1–5
as well as in a
large variety of industrial applications
6,7
due to its ability to
vibrate in thickness-extensional mode at low frequency,
avoiding the need for high driving voltages.
The Langevin structure can be analyzed with the classi-
cal one-dimensional theory;
8–10
however, this approach is
able to describe only the thickness-extensional modes and
therefore does not take into account the unavoidable lateral
vibrations of both the piezoelectric ceramic and the loading
masses.
The predictions of the lateral vibrations of the Langevin
transducer is indeed very useful both in power and in broad-
band applications; in fact, in the first case any possible lateral
coupling, which polarizes the motion in some direction other
than axial, must be avoided, while in the second a bandwidth
enlargement can be achieved by exploiting the coupling be-
tween thickness-extensional and lateral modes.
11–13
The 3-D analysis of the Langvein transducer can be per-
formed by using finite element methods 共FEM兲. This ap-
proach is very powerful and is widely used in transducers’
analysis of any geometry.
11–15
However, with respect to ana-
lytical modeling, it gives less physical insight and it is more
time consuming.
Analytical multi-dimensional modeling of piezoelectric
ceramic structures is rather complex, due to the unsolvable
differential coupled equations’ system which describes the
element vibration. Nevertheless, some attempts in this direc-
tion have been made.
16–19
In particular, some of the authors
proposed an approximated 3-D matrix model of cylinder-
shaped piezoelectric ceramics which takes into account the
coupling between thickness and radial modes and which is
also able to describe the interactions with the external
media.
20
In the present work, this 3-D approximated approach is
extended to the classical Langevin configuration, and the im-
provements with respect to the 1-D model are shown. The
proposed model is experimentally validated by comparing
the computed frequency spectrum with that obtained by mea-
surements carried out on 13 manufactured prototypes of dif-
ferent aspect ratios. Finally, measurements of the frontal dis-
placement, carried out by means of an interferometric
technique at the main resonance frequencies, are shown in
order to highlight the effect of the coupling between radial
and thickness modes.
II. THE MATRIX MODEL
Figure 1 shows the classical Langevin transducer con-
figuration; it is composed of a piezoelectric ceramic disk
with radius a and thickness 2• b, poled along the thickness
direction and electroded on its flat surface. The ceramic disk
is sandwiched between two cylinder-shaped loading masses
having the same radius and thicknesses of 2• b
1
and 2• b
2
,
respectively. When an alternating voltage V is applied to the
electrodes of the piezoelectric ceramic element, all the
modes of the structure can be excited, depending on the fre-
quency of the driving signal.
A 3-D analytical model of cylinder-shaped piezoelectric
a兲
Electronic mail: iula@uniroma3.it
2675J. Acoust. Soc. Am. 111 (6), June 2002 0001-4966/2002/111(6)/2675/6/$19.00 © 2002 Acoustical Society of America
ceramics
20
was recently proposed by some of the authors. It
was derived by assuming that the coordinate axes are pure
mode propagation directions 共i.e., the mechanical displace-
ments u
r
and u
z
depend only on r and z, respectively兲 and by
imposing both electrical and mechanical boundary condi-
tions in an integral way. By means of this model, the piezo-
electric ceramic element has been described as a four-port
system with three mechanical ports 共one for each surface兲
and one electric port. The linear equations which relate the
electrical variables 共current I and voltage V) to the mechani-
cal variables 共forces F
i
and velocities
v
i
, i⫽ 1,...,3) in the
frequency domain are
F
1
⫽
Z
1
j
冋
冉
k
1
aJ
0
共
k
1
a
兲
⫺ J
1
共
k
1
a
兲
k
1
aJ
1
共
k
1
a
兲
冊
⫹
c
12
D
k
1
ac
11
D
册
v
1
⫹
2
ac
13
D
j
共v
2
⫹
v
3
兲
⫹
4bh
31
j
a
I, 共1兲
F
2
⫽
2
ac
13
D
j
v
1
⫹
Z
3
j
冉
v
2
tan
共
2k
3
b
兲
⫹
v
3
sin
共
2k
3
b
兲
冊
⫹
h
33
j
I ,
共2兲
F
3
⫽
2
ac
13
D
j
v
1
⫹
Z
3
j
冉
v
2
sin
共
2k
3
b
兲
⫹
v
3
tan
共
2k
3
b
兲
冊
⫹
h
33
j
I, 共3兲
V⫽
4bh
31
j
a
v
1
⫹
h
33
j
v
2
⫹
h
33
j
v
3
⫹
I
j
C
0
, 共4兲
where c
ij
and h
ij
(i,j⫽ 1,...,3) are the elastic and piezoelec-
tric constants, respectively,
33
S
⫽ 1/

33
S
is the dielectric per-
mittivity,
is the mass density,
v
¯
1
⫽
冑
c
11
D
/
and
v
¯
3
⫽
冑
c
33
D
/
are the wave propagation velocities and k
1
⫽
/
v
¯
1
and k
3
⫽
/
v
¯
3
are the wave numbers in the r and z
directions, respectively, Z
1
⫽
v
¯
1
4
ab and Z
3
⫽
v
¯
3
a
2
are
the piezoelectric ceramic mechanical impedances along the r
and z directions, and C
0
⫽
a
2
/

33
S
2b is the so-called
‘‘clamped capacity’’ of the piezoelectric ceramic.
Following the same approach the two loading masses
are modeled as three-port systems. Each system can be sim-
ply obtained from Eqs. 共1兲–共3兲 by setting to zero the piezo-
electric constants h
31
and h
33
, and, due to the isotropy of the
material, by imposing c
33
⫽ c
11
and c
13
⫽ c
12
, and by sup-
pressing the subscripts for Z and k. With these assumptions
we obtain
F
1
⫽
Z
j
冋
冉
kaJ
0
共
ka
兲
⫺ J
1
共
ka
兲
kaJ
1
共
ka
兲
冊
⫹
c
12
kac
11
册
v
1
⫹
2
ac
12
j
共v
2
⫹
v
3
兲
, 共5兲
F
2
⫽
2
ac
12
j
v
1
⫹
Z
j
冉
v
2
tan
共
2kb
兲
⫹
v
3
sin
共
2kb
兲
冊
, 共6兲
F
3
⫽
2
ac
12
j
v
1
⫹
Z
j
冉
v
2
sin
共
2kb
兲
⫹
v
3
tan
共
2kb
兲
冊
. 共7兲
The full model of the Langevin transducer is easily ob-
tained by connecting the mechanical ports which correspond
to the contacting surfaces 共see Fig. 2兲; in this way, the trans-
ducer is modeled as a six-port system. It should be noted that
the continuity of the velocities at the interfaces between the
piezoelectric ceramic and the masses is imposed only in the
z direction.
All the transfer functions of the system can be computed
by loading the five mechanical ports with the mechanical
impedances of the surrounding media, and applying an alter-
nating voltage to the electric port. In the design of the Lange-
vin transducer the most useful relations are the electrical in-
put impedance (Z
i
) and the transmission transfer functions
(TTF
n
), defined, respectively, as
Z
i
⫽
V
I
, 共8兲
TTF
n
⫽
F
n
V
, 共9兲
where the subscript n indicates the mechanical port consid-
ered. The output displacement u
n
at each port is then given
by the relation
FIG. 1. Schematic view of the classical Langevin transducer.
FIG. 2. The six-port system representation.
2676 J. Acoust. Soc. Am., Vol. 111, No. 6, June 2002 Iula
et al.
: 3-D model of Langevin transducer
u
n
⫽
1
j
F
n
Z
n
. 共10兲
III. NUMERICAL RESULTS
Figure 3 shows the frequency spectrum, i.e., the map of
the resonance frequencies of the transducer, computed with
the proposed 3-D model. It is obtained by varying the thick-
ness of the loading masses from 1 to 40 mm. The results
were obtained by assuming PZT-5A by Morgan-Matroc
21
(2a⫽ 20 mm, 2b⫽ 2mm兲 as the piezoelectric ceramic ma-
terial, and steel 共mass density
⫽ 8 kg/m
3
, Young modulus
E⫽ 17.7⫻ 10
10
N/m
2
, Poisson ratio
⫽ 0.3) as the mass ma-
terial.
The diameter of the circles is proportional to the effec-
tive electromechanical coupling factor (k
eff
), which, as it is
well known, is defined as
k
eff
⫽
冑
f
p
2
⫺ f
s
2
f
p
2
, 共11兲
where f
s
and f
p
can be assumed to be the frequencies of
maximum and minimum admittance.
For comparison, Fig. 3 also shows the frequency spec-
trum of the Langevin transducer computed with the classical
1-D thickness extensional model
10
共solid curves T
1
, T
2
, T
3
,
and T
4
), and two straight lines representing the resonance
frequencies of the pure radial modes of the piezoelectric ce-
ramic R
cer
and of the masses R
mass
, which are computed
under the hypothesis of thin disks.
22
By increasing the mass
thickness, the resonance frequency of the pure thickness
mode T
1
decreases, as well as the harmonics corresponding
to T
2
, T
3
, and T
4
, while the R
cer
and R
mass
are constant
because they only depend on the diameter of the structure.
The plot of Fig. 3 shows that the 3-D model is able to
predict both thickness and radial resonance frequencies of
the structure. The plot also shows that there is agreement
between 3-D and 1-D results only in the regions of the spec-
tra where the resonance frequencies of pure modes are suf-
ficiently distant. On the contrary, in regions where pure
modes come closer, the 3-D model is able to predict their
deviation from 1-D trends caused by the coupling existing
between them. It should be noted that the radial resonance
frequency of the masses computed with the 3-D model ap-
proaches R
mass
only when the masses are very thin; else-
where it is higher than R
mass
.
As far as the values of k
eff
are concerned, Fig. 3 shows
that in the regions where two modes are strongly coupled,
these modes present very similar values of k
eff
, even if in
‘‘undisturbed’’ regions these values are quite different. This
behavior indicates that, in coupling regions, the two modes
cannot be considered pure modes, because their vibrational
characteristics are somehow mixed. On the other hand, when
the transducer has an aspect ratio for which coupling be-
tween modes does not occur, the k
eff
value permits us to
establish for each resonance frequency the nature of the cor-
responding mode 共radial or thickness兲, without resorting to
comparisons with 1-D models.
We also computed, for each sample, the electrical input
impedance, the transmission transfer function, and the frontal
displacement at one end surface as a function of frequency.
The transducer was assumed to work in air; however, in or-
der to take into account the internal losses, the mechanical
ports were loaded with specific acoustic impedances of 0.1
MPa•s/m. As an example, Fig. 4 shows the results obtained
for a transducer with mass thicknesses of 14 mm. Figure 4共a兲
shows the magnitude of the input impedance versus fre-
quency. Comparing this plot with Fig. 3, we can recognize
that the resonance frequencies correspond to the first thick-
ness mode (T
1
3D
), and the radial modes of the piezoelectric
ceramic (R
cer
3D
) and of the masses (R
mass
3D
), respectively. From
the plot of the transmission transfer function 关Fig. 4共b兲兴,it
can be seen that the maximum value, as expected, is obtained
at the resonance frequency of the thickness mode; however, a
quite good response is also observed at the resonance fre-
quency of the radial mode R
cer
3D
. The response at the reso-
nance frequency of the radial mode R
mass
3D
is negligible. Simi-
lar considerations can be made for the frontal displacements
关see Fig. 4共c兲兴. Finally, it should be noted that, as the model
was derived assuming a pistonlike motion, the force and the
displacement of Figs. 4共b兲 and 4共c兲 are the mean values on
the terminal surfaces.
IV. EXPERIMENTAL VALIDATION
In order to experimentally validate the proposed model,
we manufactured 13 Langevin transducers of different
lengths ranging from 2.5 to 32.5 mm. Piezoelectric ceramic
and mass materials and dimensions are those described in the
previous section.
Each prototype was prestressed with the jig shown in
Fig. 5. The Langevin structure, whose masses are provided
with flanges, is placed into the jig, composed of two threaded
elements. By means of a jig adapter, a torque wrench is used
to tighten the ceramic disk between the two masses, obtain-
ing a good control of the prestress, which is applied only to
the piezoelectric ceramic element. In order to avoid shear
stresses during wrenching operations, a thin layer of grease
was laid at the interfaces between the piezoelectric ceramic
and the masses. Measurements were conducted with the
transducer mounted in the jig; this experimental solution
avoids the need for a hole in the structure to insert a pre-
FIG. 3. The frequency spectrum versus the thickness of the loading masses.
2677J. Acoust. Soc. Am., Vol. 111, No. 6, June 2002 Iula
et al.
: 3-D model of Langevin transducer
stress bolt and, therefore, allows a more realistic comparison
with the results of the model, which is derived for a cylinder
and not for ring-shaped elements. In fact, as shown in recent
works,
23,24
the frequency behavior of the radial modes of
disks and rings is substantially different.
Figure 6 shows the experimental frequency spectrum;
also in this case, the diameter of the circles is proportional to
the k
eff
. In this plot, the frequency spectrum computed with
the proposed model is reported in solid curves.
The experimental results reported in Fig. 6 were ob-
tained by applying to the Langevin transducer a light pre-
stress 共corresponding to an applied torque of about 2 N•m兲,
just sufficient to ensure the mechanical contact between the
piezoelectric ceramic and the masses. This is the best experi-
mental approximation of the hypotheses imposed deriving
the model, which does not take into consideration any pre-
stress, and describes the contact between the piezoelectric
ceramic and the masses by imposing the continuity of dis-
placements only in the thickness direction. In fact, both the
light prestress and the use of grease should allow radial slid-
ing.
The comparison between computed and experimental re-
sults shows that the model predicts with quite good accuracy
the two lowest resonance frequencies all over the explored
range. In particular, it is noteworthy the agreement observed
in the region where these two modes are strongly coupled,
i.e., where the diameter and the whole length of the structure
are comparable. A further agreement can be observed by
comparing the behavior of the experimental k
eff
in Fig. 6
with that computed with the 3-D model in Fig. 3.
As far as the higher frequencies modes are concerned,
measurements only partly confirm the results of the model;
this is probably due to the presence of other modes of differ-
ent nature which are not predicted by the model.
In order to better investigate the effect of the coupling
FIG. 4. Modulus of the input impedance 共a兲, modulus of the transmission
transfer function 共b兲, and modulus of the frontal displacement 共c兲 for a
transducer with masses of 14-mm length.
FIG. 5. 共a兲 Exploded view of the transducer and of the jig. 共b兲 The asembled
structure, the accessory torque wrench, and its jig adapter.
FIG. 6. Comparison between measured and computed frequency spectra.
2678 J. Acoust. Soc. Am., Vol. 111, No. 6, June 2002 Iula
et al.
: 3-D model of Langevin transducer
between the thickness and the radial modes, we measured the
frontal displacement of the terminal surface by means of an
interferometric technique, for three different transducers at
the main resonance frequencies.
Figure 7 shows the shapes of the frontal displacement
measured on the sample with masses of 32.5-mm length at
the three lowest resonance frequencies. As can be seen from
the plot of Fig. 6, for this aspect ratio, the resonance frequen-
cies are quite distant; therefore the modes can be considered
uncoupled. The lowest frequency mode presents a clear pis-
tonlike motion, and can be considered as a pure thickness
extensional mode. Also, the second mode has an almost flat
axial displacement and can be recognized as the first har-
monic of the fundamental thickness mode. The third mode
has the typical displacement shape of radial modes.
25
The displacements of the transducer with masses of
25-mm length are shown in Fig. 8. For this aspect ratio, the
low-frequency mode can still be recognized as a pure thick-
ness extensional mode, due to the flatness of its displace-
ment. However, it is not possible to distinguish which of the
two other modes is the thickness extensional and which is
the radial, because the shapes of the displacement are very
similar and the amplitudes comparable. By the observation
of Fig. 6, it is evident that these two modes are strongly
coupled; in the figure, we indicated the lower frequency
mode as R
cer
M
only because its k
eff
has a greater value than the
other.
The displacements at the first two resonance frequencies
for the transducer with masses of 5-mm length are shown in
Fig. 9. The low-frequency mode can be recognized as a ra-
dial mode both for its displacement shape and for the high
k
eff
value shown in Fig. 6. It should be noted that, at this
frequency, the mean displacement of the transducer is com-
parable with those observed for the fundamental thickness
mode of transducers of Figs. 7 and 8. The second mode, for
this sample, is R
mass
M
.
V. CONCLUSION
In this work, an approximated 3-D model of the Lange-
vin transducer has been proposed. The model represents an
FIG. 8. Frontal displacement of the transducer with masses of 25-mm
length.
FIG. 7. Frontal displacement of the transducer with massses of 32.5-mm
length.
FIG. 9. Frontal displacement of the transducer with masses of 5-mm length.
2679J. Acoust. Soc. Am., Vol. 111, No. 6, June 2002 Iula
et al.
: 3-D model of Langevin transducer
improvement with respect to the 1-D approach because it is
able to describe the radial modes of the structure and their
coupling with thickness extensional modes. The comparison
with experimental results shows that, for any aspect ratio of
the transducer, the model is able to predict with quite good
accuracy the two lowest resonance frequencies, which are
those mainly used in practical applications.
The proposed model therefore seems to be a useful ana-
lyical tool which permits us to extend the design of Langevin
transducers also to structure with the diameter comparable or
greater than the total length. It can be used in broadband
applications, where the coupling between radial and thick-
ness modes can be exploited to enlarge the bandwidth, and in
power applications due to its capability to predict the funda-
mental resonance frequency of the transducer for any aspect
ratio.
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