Smart Materials and Structures
Smart Mater. Struct. 27 (2018) 095013 (10pp)

https://doi.org/10.1088/1361-665X/aad530

Vibration characteristics of polymer-based
Langevin transducers
Jiang Wu , Yosuke Mizuno

and Kentaro Nakamura

Laboratory for Future Interdisciplinary Research of Science and Technology, Tokyo Institute ofTechnology, Yokohama 226-8503, Japan
E-mail: wujiang@sonic.pi.titech.ac.jp
Received 17 April 2018, revised 17 July 2018
Accepted for publication 23 July 2018
Published 6 August 2018
Abstract

Our previous studies demonstrated that some functional polymers could exhibit satisfactorily
low mechanical loss in high-amplitude bending vibration in the ultrasonic frequency range. In
this study, to explore further applications of these functional polymers in high-power ultrasonictransducers, we focus on Langevin transducers as they are most widely employed in industrial
applications. The developed transducer has a sandwich structure: piezoelectric ceramic elements
are clamped between two polymer cylindrical bodies with a metal screw. The vibration
characteristics of the polymer-based Langevin transducers differ from the metal-based ones
owing to the difference in the material constants. When working in the 1st-order longitudinalmodes, the polymer-based transducers exhibit dumbbell-shaped deformations because of thehigher stiffness of the metal screws. Their mechanical losses reach the maximal values on thevibrating bodies instead of the piezoelectric ceramic elements. Besides, when the strains onpolymer vibrating bodies exceed certain values, there exist sharp reductions in mechanicalquality factors (Q factors), which restrict the maximal vibration velocities of polymer-basedtransducers. The poly phenylene sulﬁde (PPS)-based transducer yields Q factors of 350 and its
vibration velocity linearly increases to 1060 mm s−1, comparable to the maximal values
achievable on metal-based ones. These properties indicate the potential of PPS-based transducers
for ultrasonic applications.
Keywords: Langevin-type transducer, vibration characteristics, poly phenylene sulﬁde, polymer(Some ﬁgures may appear in colour only in the online journal)
under high-amplitude bending vibration [4, 6–10]. In thisstudy, we employ these polymers to substitute the metals as
the vibrating bodies of Langevin longitudinal transducers, andsystematically investigated the vibration characteristics as
they may differ from those of metal-based ones because of thegreat differences in the mechanical constants [7].
First, utilizing functional polymers as well as metals, weprepare several Langevin transducers with identical dimensions. Subsequently, taking poly butylene terehtalate (PBT)and aluminum-based transducers as examples, we compare
their differences in vibration characteristics, including electrical admittances, deformations, dissipated power distributions, mechanical quality factors (Q factors), and achievablemaximum vibration velocities. Finally, we summarize the Q
factors and maximal vibration velocities achievable on thetested transducers.

1. Introduction
Langevin-type ultrasonic transducer is a fundamental component for various kinds of high-power ultrasonic applications,e.g. cleaning, machining, welding, food industry, andenhancement of chemical reaction [1–3]. Conventional Langevin-type ultrasonic transducers consist of metal vibrating
bodies and piezoelectric ceramic elements. Owing to low
densities, functional polymers are attractive candidate materials as vibrating bodies for reducing the weight of transducers [4]. Besides, since polymer components are directly
fabricated via molding rather than machining, the production
efﬁciencies of transducers, particularly those with complex
structures, can be signiﬁcantly improved [5]. A series of ourprevious studies showed that several functional polymers had
the capability of exhibiting relatively low mechanical loss
0964-1726/18/095013+10$33.00

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J Wu et al

Figure 1. Langevin transducer tested in this study: (a) structure and (b) equivalent circuit model.

2. Langevin transducer and its equivalent circuitmodel

The mechanical arm contains three serially connected components, namely the equivalent stiffness k, mass m, anddamper γ. ν denotes the z-axis vibration velocity on the end
surface of the transducer.

Figure 1(a) depicts the schematics of tested Langevin transducers. Two pieces of piezoelectric ceramic elements (C213,
Fuji Ceramics, Fujinomiya, Japan) 8 and 20 mm in inner andouter diameters, respectively, and 2.5 mm in thickness areclamped by two cylindrical vibrating bodies with a 30 mm
long screw. The 20 mm diameter vibrating bodies havethroughout holes 7 mm in diameter and 30 mm in length. The
tested transducers have identical dimensions but are made ofdifferent polymers and metals, of which the mechanicalconstants are listed in table 1 [11]. A polar coordinate isestablished on the middle surface of the transducer.
Figure 1(b) represents an equivalent circuit model forevaluating piezoelectric transducers [1, 12]: it comprises anelectrical arm and a mechanical arm coupled by a force factor
A. In the electrical arm, the clamped capacitance Cd and thedielectric loss Rd are parallelly connected. id and im denote thereactive current ﬂowing into Cd and the motional current intoA, respectively. In general, Rd is so large (in the order of MΩ)that the current ﬂowing into Rd is negligibly small [1, 10].

3. Experimental results
3.1. Experimental setup

Figure 2 illustrates the experimental setup. When a sinusoidal
voltage is applied to transducer, the 1st longitudinal vibration
is excited. Here, an in-plane laser Doppler vibrometer(IPV100, Polytec, Waldbronn, Germany) mounted on a linear
guide rail was employed to measure the z-axis vibration
velocity distribution along the axial direction. The amplitude
and phase were recorded by a lock-in voltmeter (5560, NF
Electronic Instruments, Yokohama, Japan). To know the zaxis vibration velocity on the lateral surface, we measured thein-plane vibration velocity on the cylindrical side surface
close to the edge. Using a high frequency power meter (3332,
Hioki E. E. Corp., Nagano, Japan), the voltage u and the input
power p were measured. The motional current im was2

Smart Mater. Struct. 27 (2018) 095013

Table 1. Material constants of tested materials [11].

Material constants

Thermal plastic

Density (×103 kg m–3)

Poisson’s ratio

Longitudinal wave speed in thin rod(m s−1)

PPS
PES
PEEK
PBT
POM
PMMA
PA6

3.45
2.55
3.50
2.90
3.20
3.20
2.60

1.35
1.37
1.32
1.19
1.41
1.18
1.14

0.36
0.35
0.40
0.35
0.35
0.35
0.35

1599
1364
1628
1473
1506
1633
1510

Poly ether imide

PEI

3.00

1.27

0.37

1537

Phenol formaldehyde
resin
Aluminum

PF

11.70

1.46

0.38

2831

—

70.30

2.70

0.35

5103

PolymerAbbreviation

Semi-crystalline

Poly phenylene sulﬁdePoly ether sulfone
Poly ether ether ketone
Poly butylene terehtalate
Poly oxy methylene
Poly methyl methacrylate
Polyamide 6

Amorphous—

3

Thermal setting
plastic
Metal

Young’s modulus (GPa)Crystallinity—

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Figure 2. Testbed for measuring the vibration characteristics of transducers.

calculated as
im =

2p
.
u

were 14.6 and 39.5 kHz, respectively, slightly higher thanthose obtained from the admittance characteristics. It often
occurred because of the mesh size and the modeling of thecontacting area. Figure 4(a) illustrates that the PBT-basedtransducer exhibits a dumbbell-shaped deformation, different
from that of the aluminum one shown in ﬁgure 4(b).
Figures 4(c) and (d) show the z-axis vibration velocity distributions of the PBT- and aluminum-based transducers,
respectively. It can be observed that the z-axis vibration
velocity distribution of the PBT-based transducer deviatesfrom the cosine waveform. The strain distribution shown inﬁgure 5(a) was derived from the FEA results. The strainreached the maximal values at z=−12 and 12 mm, close to
the lateral surfaces of the screw (at z=−15 and 15 mm),probably because of higher stiffness of the aluminum screw.
On the basis of the vibration velocity distribution, the local
dissipated power was estimated by the method given inappendix. As ﬁgure 5(b) shows, since there exist measurement errors caused by the non-uniformities in materials[14–17], we draw the envelopes to clearly illustrate the dissipated power distribution. The dissipated powers reach theirmaximal values at z=−15 and 19 mm, where the transducers exhibit higher strains. The temperature on the surface of
the PBT-based transducer was measured with an infrared
camera (i7, FLIR Systems, Wilsonville, USA) after it wascontinuously excited for approximately 5 min. Figure 5(c)demonstrates that the polymer vibrating bodies exhibit relatively high temperature compared to piezoelectric ceramicelements. The local dissipated powers near z=−10 and
10 mm are relatively low compared to the values nearz=−15 and 19 mm, but these regions have almost the same
temperatures. In addition to the mechanical loss, there exists
frictional loss on the thread ridges between vibrating bodies
and screws; that leads to the temperature rises near z=−10
and 10 mm [18]. The distributions of the vibration velocity
and strain, dissipated power, and temperature on the surfaceof the aluminum-based transducer are illustrated inﬁgures 5(d)–(f), respectively. Note that the temperatureshown in ﬁgure 5(f) were measured after a 10 min continuous(1 )Here, u, im, and v are indicated as the zero-to-peak values.
3.2. Electrical admittanceFirst, the electrical admittances of the PBT- and aluminumbased transducer were measured with an impedance analyzer(4294A, Agilent, Santa Clara, USA). In the admittance of the
PBT-based transducer shown in ﬁgure 3(a), there was aresonance response at 14.1 kHz. The admittance loop shownin ﬁgure 3(b) shows that the susceptance is much higher thanthe conductance; this indicates that the mechanical vibrationenergy is far lower than the electrically stored energy.
Figure 3(c) shows the admittance of the aluminum-basedtransducer. At 38.5 kHz, there existed an observable response.
Figure 3(d) demonstrates that the motional admittance of thealuminum-based transducer was 12.5 mS, over 1000 times thevalue of the PBT-based one. In this case, the current dominantly ﬂows into the mechanical arm. Table 2 gives theequivalent circuit parameters of these two transducers. The
force factor is lower for the polymer-than for the aluminumbased transducer because of lower Young’s moduli and lower
densities of polymers [7]. Though the polymer-based transducer exhibits relatively low equivalent stiffness and mass, its
damper is relatively high owing to the relatively low Q factor(see section 3.4). The low force factor and the high damper of
the PBT-based transducer lead to the weak mechanical resonance [13].
3.3. Deformation and dissipated power

Subsequently, the deformations of the PBT- and aluminumbased transducers operated in the 1st-order longitudinalmodes were investigated via ﬁnite element analysis (FEA).
The model was meshed into several thousands of cuboid
elements with sizes of 0.5 mm. The simulated resonancefrequencies of the PBT- and aluminum-based transducers
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Figure 3. (a) Admittance characteristics of PBT-based transducer and (b) its admittance loop near 14.4 kHz. (c) Admittance characteristics ofaluminum-based transducer and (d) its admittance loop near 38.5 kHz.

Table 2. Measured values of equivalent circuit parameters of PBT- and aluminum-based transducers.

Material

Clamped capacitance Cd(nF)

Force factor A(N V−1)

Equivalent stiffness k(106 N m−1)

Equivalent mass m(10−3 kg)

Damper γ(N s m−1)

PBT
Aluminum

2.11
2.05

0.012
0.133

76.4
647.2

9.3
11.4

12.2
1.4

surfaces of PBT- and aluminum-based transducers wererespectively kept at approximately 60 and 100 mm s−1; andthe measured power consumptions were 26.9 and 7.0 mW,
respectively. The sums of local dissipated powers were estimated to be 18.4 and 4.4 mW for the PBT- and aluminumbased transducers, respectively. The other powers are
assumed to frictionally dissipate on the thread ridges between
vibrating bodies and screws [18].

excitation as the temperature rise was not observable in shorttime. Figure 5(e) demonstrates that higher power is dissipatedin the piezoelectric ceramic elements (the part betweenz=−2.5 and 2.5 mm) than in the aluminum vibrating bodies.
Since the piezoelectric ceramic elements have lower heat
capacities and lower heat conductivity [19], they exhibit
higher temperature rises than the aluminum vibrating bodies.
During measurements, the vibration velocities on the end
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Figure 4. Vibration modes of (a) PBT-based and (b) aluminum-based transducers. In (c) and (d), the solid curves are their z-axis vibration

velocity distributions obtained through FEA, and they agree with the experimental results (dots). Cosine waves (dashed curves) are given forcomparison.

factors of several polymer materials measured by our developed method [17]. Clearly, when the strain on the polymer
vibrating body exceeds 0.004%, the Q factor of PBTdecreases by approximately 70. Whereas the strain is 0.001%
on the piezoelectric ceramic element; this value is far lower
than the value corresponding to the rapid Q factor reduction(0.027%). Clearly, the large increase in mechanical loss of thepolymer vibrating body causes the sharp reduction in the Q
factor of the entire transducer. Figure 8 demonstrates how the
Q factors of transducers made of other polymer materials
depend on their vibration velocities. The maximal Q factors of
POM-, PBT-, and PMMA-based transducers were respectively 42, 71, and 47, limited by the Q factors of POM, PBT,
and PMMA (respectively 60, 90, and 70; see ﬁgure 7). Notethat since the Q factor of a material Qmat excludes themechanical loss on contact surfaces between vibrating bodies
and piezoelectric ceramic elements, as well as those on threadridges, it is higher than the Q factor of the transducer made of
the material. The maximal vibration velocities of POM- andPMMA-based transducers can be linearly increased to 60 and
100 mm s−1, respectively, corresponding to the strains where
the Q factors of POM and PMMA sharply decrease (respectively 0.003% and 0.006%; see ﬁgure 7). These results
validate our aforementioned conclusion. Besides, the transducers made of semi-crystalline polymers, e.g. PPS andPEEK, exhibited relatively high Q factors compared to that of
the amorphous polymer except the PA6-based transducer witha Q factor of lower than 10. The PPS-based Langevin-typetransducer yielded Q factors of approximately 350; this highvalue originates from low mechanical loss of PPS [6].
Another transducer capable of exhibiting Q factors of higher
than 100 was made of PES. Interestingly, both PPS and PES
are sulfur compounds [11].

3.4. Q factor

Then, the resonance frequency and the Q factor were measured as functions of the vibration velocity. Here, the Q
factors of the PBT-based transducer were obtained from theresonance frequency fr and the bandwidth Δf corresponding
to 0.707 times of the maximal vibration velocity [1, 4, 12]:
Q=

fr
.
Df

(2 )

As shown in ﬁgure 6(a), the Q factor had almost no variation
when the vibration velocity increased from 25 to 64 mm s−1.
However, it yielded a sharp reduction when the vibration
velocity exceeded 64 mm s−1. In the meantime, the resonancefrequency decreased from 14.39 to 14.36 kHz. As shown inﬁgure 6(b), the Q factors of the aluminum-based transducer in
low-amplitude region (<800 mm s−1) were also calculated
from the resonance curves. On the other hand, the Q factors inhigh-amplitude region were estimated from the transient
response to avoid the effect caused by the temperature rise
[20–22]. The vibration velocity dependence of Q factor of thealuminum-based transducer provided the similar tendencywith that of the PBT-based one. At the vibration velocity of1500 mm s−1, the maximal strains on both the aluminum
vibrating body and the piezoelectric ceramic element reached
0.029%. In general, Q factors of transducers reﬂect themechanical losses of component materials. According to theprevious reports [16, 20], the Q factors of aluminum andpiezoelectric ceramics sharply decrease at the strains of
0.060% and 0.027%, respectively. Thus, the signiﬁcant Q
factor reduction of the aluminum-based transducer at over1500 mm s−1 originates from the sharp increase in themechanical loss of the piezoelectric ceramic elements[16, 22]. In the case of the PBT-based transducer, the strain
on the vibrating body was approximately 0.004% at
64 mm s−1. Figure 7 shows the strain dependences of Q
6

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Figure 5. Strains (red curves) of (a) PBT- and (d) aluminum-based transducers derived from the simulated vibration velocities (black curves).

Local dissipated powers of (b) PBT- and (e) aluminum-based transducers. The gray curves are the estimated local dissipated powers, and theblack curves are their envelopes. (c) and (f) respectively show the temperature distributions on the surfaces of the PBT- and aluminum-basedtransducers.

vibration velocity probably originates from the nonlinearity in
Young’s modulus in high-amplitude region [23]. For practicalapplications of polymers in Langevin transducers in the
future, two important indicators for practical applications, i.e.
maximal vibration velocities in linear regions vl and thesaturated vibration velocities vs of the tested transducers, aresummarized in table 3. In contrast to most of the polymerbased transducers incapable of providing the performancecomparable to metal-based ones dominantly owing to their
low Q factors, the PPS-based transducer yields a maximal
vibration velocity of 1060 mm s−1 in linear region. Considering its satisfactorily high vibration velocity and chemical

3.5. Achievable maximum vibration velocity

Finally, we investigate how linearly the vibration velocity v
increased as the motional current im becomes higher. As
shown in ﬁgure 9(a), initially, the vibration velocity of the
PBT-based transducer linearly increased to 64 mm s−1. Then,in the im range from 0.9 to 1.7 mA, the vibration velocity had
a nonlinear increase. At 1.7 mA, the vibration velocity
approached its saturated value, 104 mm s−1. The vibration
velocity of the aluminum-based transducer shown inﬁgure 9(b) has the similar tendency. As mentioned above, thevibration velocity can be linearly increased until the Q factor
of material provides a marked reduction. The saturation in7

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J Wu et al

Figure 6. Resonance frequencies (black curves) and Q factors (red curves) as functions of the vibration velocities of (a) PBT- and (b)aluminum-based transducers. The open and closed bullets represent the results measured from the frequency response and the transient curve,
respectively.

Figure 7. Variations in Q factors of PBT, POM, and PMMA against

Figure 8. Q factors of transducers as functions of the vibration
velocities on the end surfaces.

strains.

1. The mechanical loss of polymer-based Langevintransducers reached their maximal values on vibrating
bodies rather than piezoelectric ceramic elements.
Whereas conventional metal-based ones exhibited theirmaximal mechanical loss on the piezoelectric ceramicelements.
2. When the strain exceeded a certain value, the polymerbased transducer exhibited a sharp reduction in Q
factor, which originated from the sharp increase in themechanical loss of the polymer vibrating body. The
attainable maximum vibration velocity of the transducer
depended on the characteristics of each polymer
material.

resistance, it is possibly applicable in special circumstances.
For instance, PPS-based transducers can generate ultrasounds
into acid or alkaline solutions to enhance chemical reactions,or be utilized as sensors for robots used in chemicalplants [24].

4. Conclusions
After carrying out a systematical investigation on the vibration characteristics of polymer-based Langevin transducers,
we have drawn the following conclusions:8

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Figure 9. Vibration velocities as functions of the motional currents of (a) PBT- and (b) aluminum-based transducers.
Table 3. Maximal vibration velocities in linear region and saturated

vibration velocities of transducers made of different materials.
Material

Maximal vibration velocity
in linear region, vl (mm s−1)

Saturated vibration
velocity, vs (mm s−1)

PPS
PES
PEEK
PBT
POM
PMMA
PA6
PEI
PF
Aluminum

1060
108
116
64
60
100
9
32
132
1400

1120
148
179
104
94
128
13
58
150
2000
Figure A1. Conceptual view of calculation method of local

dissipated power from longitudinal vibration velocity distribution ontransducer surface.

3. Most of the tested polymer-based transducers yielded Q
factors of lower than 100, except that the PPS- andPES-based transducers yielded Q factors of approximately 350 and 115, respectively.
4. The vibration velocity of the PPS-based Langevintransducer linearly increased to 1060 mm s−1.

JSPS KAKENHI Grant Number 17J05057 and JST A-step
program Grant Number AS2711904S.

We anticipate that these results will provide adequate
information for further design of polymer-based transducers.
Towards practical applications, other properties of polymerbased transducers, e.g. long-term durability, should beassessed. Usage of stiff polymers, e.g. carbon- or glass-ﬁberenhanced PPS/PEEK [4] and thermal annealing on polymers[17] are also worth exploring in the future.

Appendix
As shown in ﬁgure A1, we assume three adjacent sampling
points I, II, and III on the cylindrical vibrating body, and thecross-sections LL and LR in the middle of points I and II, andII and III, respectively. When the points I, II, and III
respectively have the vibration velocities vI=VIexp(−jθI),
vII=VIIexp(−jθII), and, vIII=VIIIexp(−jθIII), where Vi
represents the amplitude and θi denotes the phase of thevibration velocity at the ith point (i=I, II, and III), the active
powers ﬂowing into the small region between LL and LR are

Acknowledgments
The authors thank Daicel Corporation for providing various
kinds of transducer components. This work was supported by9Smart Mater. Struct. 27 (2018) 095013

J Wu et al

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where E denotes the Young’s modulus, ω represents theangular frequency, Δz is the interval between the sampling
points (=0.5 mm in this study), and S is the cross-sectional
area. Similarly, the active power ﬂowing out of the region isexpressed as [14]PR =ES
· VII VIII sin (q II - q III ).
2wDz

(A-2)

The power dissipated in this region is calculated as [6, 16]
Pdis = P L - P R.

(A-3)

ORCID iDs
Jiang Wu https://orcid.org/0000-0002-6899-5601
Yosuke Mizuno https://orcid.org/0000-0002-3362-4720
Kentaro Nakamura https://orcid.org/0000-00032899-4484

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