Ultrasonics 69 (2016) 74–82

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Ultrasonics
journal homepage: www.elsevier.com/locate/ultras

Measurement of mechanical quality factors of polymers in flexuralvibration for high-power ultrasonic application
Jiang Wu ⇑, Yosuke Mizuno, Marie Tabaru, Kentaro Nakamura
Precision and Intelligence Laboratory, Tokyo Institute of Technology, 4259 Nagatsuta-cho, Midori-ku, Yokohama 226-8503, Japana r t i c l e

i n f oArticle history:
Received 9 September 2015
Received in revised form 1 February 2016
Accepted 24 March 2016
Available online 1 April 2016
Keywords:
Polymer
Q factor
Flexural vibration
High-power ultrasonic applicationa b s t r a c t
A method for measuring the mechanical quality factor (Q factor) of materials in large-amplitude flexuralvibrations was devised on the basis of the original definition of the Q factor. The Q factor, the ratio of thereactive energy to the dissipated energy, was calculated from the vibration velocity distribution. The bar
thickness was selected considering the effect of the thickness on the estimation error. In the experimental
setup, a 1-mm-thick polymer-based bar was used as a sample and fixed on the top of a longitudinal transducer. Using transducers of different lengths, flexural waves in the frequency range of 20–90 kHz weregenerated on the bar. The vibration strain in the experiment reached 0.06%. According to the
Bernoulli–Euler model, the reactive energy and dissipated energy were estimated from the vertical velocity distribution on the bar, and the Q factors were measured as the driving frequency and strain were varied. The experimental results showed that the Q factors decrease as the driving frequencies and strains
increase. At a frequency of 28.30 kHz, the Q factor of poly(phenylene sulfide) (PPS) reached approximately 460 when the strain was smaller than 0.005%. PPS exhibited a much higher Q factor than the other
tested polymers, which implies that it is a potentially applicable material as the elastomer for high-power
ultrasonic devices.
Ó 2016 Elsevier B.V. All rights reserved.

1. Introduction
Metallic compounds, such as stainless steels and aluminumalloys, are commonly used as elastomers in ultrasonic transducers.
To obtain high vibration velocities with lightweight transducers, amaterial with low elastic modulus, low density, and low mechanical loss is required as the elastomer. Among existing functional
materials, polymers usually have low elastic moduli and densities.
It has been believed that, in most cases, polymers are good atabsorbing vibrations because of their high mechanical loss. However, some of the new functional polymers invented in the last
three decades exhibit low mechanical loss. To gain a primary
understanding of the mechanical-loss properties of polymers, it isof significance to evaluate the mechanical loss under ultrasonic frequencies and large amplitudes. As flexural vibrations are widelyused in ultrasonic transducers [1–8], in this study, we focus on
the mechanical-loss properties of polymers in flexural vibrations.
The mechanical quality factor (Q factor) is inversely proportional to the mechanical loss, and it is a commonly-used parameterthat indicates the suitability of a material as a vibrator. Many

⇑ Corresponding author. Tel.: +81 45 924 5052; fax: +81 45 924 5091.
E-mail address: wujiang@sonic.pi.titech.ac.jp (J. Wu).
http://dx.doi.org/10.1016/j.ultras.2016.03.012
0041-624X/Ó 2016 Elsevier B.V. All rights reserved.

methods for Q-factor measurement have been developed: (1)Impact testing method is being widely applied in industry [9].
The Q factor is obtained from the decay in the damped oscillation
curve at the natural frequency of the sample after striking it with ahammer. (2) The resonance curve method is used to evaluate themechanical loss of transducers [4]. The Q factor is calculated fromthe curve of vibration amplitude versus frequency, which provides
an overall result originating from the loss in the elastomer, loss inthe excitor (e.g. piezoelectric ceramic elements), and loss generated on the contacting surface between the elastomer and the excitor. (3) The Q factor can be derived from the attenuation coefficientof a traveling wave. The attenuation coefficient is calculated fromthe amplitude ratio of reflected waves that are generated by a burst
excitation [10]. However, the vibration amplitude is limited
because of the low power of the burst excitation. (4) The attenuation coefficient can be directly calculated from the fitting curve ofthe distance decay of a traveling wave generated by a continuous
excitation. However, it is difficult to separate the traveling wave
from the reflected wave on a vibrating bar, especially for a material
with low mechanical loss.
A method that gives a precise value of the Q factor (or attenuation coefficient) of polymers under ultrasonic frequencies and large
amplitudes is required. In a previous study [11], a method for measuring the Q factors of a cylindrical bar in torsional vibration was

J. Wu et al. / Ultrasonics 69 (2016) 74–82

developed. The Q factors were calculated from the torsional vibration velocity distribution on the side surface based on the original
definition of the Q factor. In the present work, we modify the aforementioned method to measure the Q factor in flexural vibration. In
addition, we discuss the requirements for achieving high measurement precision. When the frequencies and strains are varied, we
measure the Q factors of bars made of polyacetal (POM), acryloni
trile–butadiene–styrene (ABS), poly(ether ether ketone) (PEEK),
and poly(phenylene sulfide) (PPS).
2. Principle of Q-factor measurement
2.1. Original definition of Q factor
Fig. 1(a) shows a vibrating bar. Flexural waves are generated
with a vibration source located at one end. The original definitionof the Q factor is the ratio of the reactive energy Ek to the dissipatedenergy Ep in the sampling part between the cross-sections L1 and
L2:

Q¼

2pEk
:
Ep

ð1Þ

The reactive energy stored in the vibrating bar consists ofkinetic energy and elastic energy. In the time domain, they aretransforming to each other and their sum is constant [12]. Thus,
the reactive energy equals the maximum kinetic energy. The displacement of a small section is a composition of a translationalmotion in the vertical direction and a rotation [13]. Thus, the maximum kinetic energy is given as Z L2 Z L2 
1
1
Ek ¼
d mv 2 þd I u2
2
2
L1
L1
Z L2
Z L2
1
1
v 2 dz þ qbh3
u2 dz;
¼ qbh
2
24
L1
L1

ð2Þ

where v and u represent the zero-to-peak vertical velocity andangular velocity, respectively. m and I are the mass and rotary inertia of the sampling interval, respectively. q is the density, and b and75

h represent the width and thickness of the specimen, respectively.
The rotary inertia I is determined by the width and thickness as
I = bh3/12.
The dissipated energy Ep is the decrease in the active vibrationenergy flowing across L1 and L2:

Ep ¼

2p
 ðP —P Þ;
x0 1 2

ð3Þ

where x0 is the angular frequency, and P1 and P2 represent theactive vibration powers of the wave flowing across L1 and L2,
respectively. If the wave amplitude decreases to zero before reflection because of large energy dissipation, only a traveling wave
exists on the bar, and it is clear that Ep exceeds zero. In most cases,
a wave that consists of a traveling wave and a standing wave is generated. It is necessary to investigate whether Eq. (3) is effective incalculating the dissipated energy of this wave. We assume thatthe wave in Fig. 1(a) is a combination of a traveling wave propagating along the +z axis, W+, and a traveling wave propagating alongthe z axis, W. Further, W is the reflected wave of W+. In Fig. 1(b), P+1 represents the active vibration power of W+ flowing acrossL1, and its positive direction is along the +z axis. On the other hand,P
1 is the active vibration power of W that flows across L1, and its
positive direction is along the z axis. When W+ and W flow across
L1 simultaneously, the active vibration power P1 equals P+1–P
1 . Similarly, P+2 and P
2 represent the active vibration powers of W+ and W
flowing across L2, and P2 = P+2–P
2 . Thus, Eq. (3) can be rewritten as

Ep ¼

2p  þ    þ   2p  þ þ     
 P1 —P 1  P 2 —P2 ¼
 P1 —P2 þ P2 —P1 ;
x0
x0
ð4Þ


where P+1–P+2 and P
2 –P1 represent the dissipated powers of W+ andW, respectively. Because W+ flows across L1 and L2 successively, P+1
is larger than P+2 owing to energy dissipation during propagation.

Similarly, P
2 is larger than P1 . Since W is the reflection wave of
W+, P+2 is larger than P.
Thus,
Ep exceeds zero, which indicates that2

Eq. (3) is effective in calculating the dissipated energy when a traveling wave and a standing wave are generated simultaneously. In
the experiments for Q-factor measurement described later, as
high-amplitude vibrations are generated on long polymer-based
bars, the active vibration power and dissipated power are measureable. Note that it is difficult to estimate Q factors of materials withextremely low mechanical loss, such as sapphire [14], for which P+1
equals P–1 approximately, and P becomes too small to be detected.
In an alternating current circuit, the active power (or real
power) P is calculated from the zero-to-peak voltage u and currenti [12]:

P¼

1
Reðu  i Þ:2

ð5Þ

The voltage u and current i are given as complex numbers; Re
represents the real part and the asterisk indicates the complex conjugate. The active power in a mechanical system is calculated fromthe force (moment) and the translational velocity (angular velocity), which correspond to the voltage and current in an electricalsystem, respectively. The active vibration power in flexural vibration is calculated as [15–17]

P¼

Fig. 1. Q-factor measurement from the vibration velocity distribution. (a) The wave,
consisting of a traveling wave and a standing wave, is generated on the vibratingbar. A composition of two traveling waves, W+ and W, with inverse propagation
directions is assumed. (b) P+1 and P
1 are the active vibration powers of W+ and W+across L1, respectively, and P1 = P+1  P
1 . P2 and P2 are the active vibration powers of
W+ and W across L2, respectively, and P2 = P+2  P
2.

1ReðV  v   M  u Þ;2ð6Þ

where V and M represent the shearing force and bending moment
acting on the cross-section, respectively. The active vibration power
consists of the shearing-force component V  v⁄ and the bendingmoment component M  u⁄. The shearing-force component isV  v⁄, because the positive direction of V on the right lateral surfaceof the bar element (Fig. 2(a)) is +y axis, which is the same as that ofv. The bending-moment component equals M  u⁄ because the

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J. Wu et al. / Ultrasonics 69 (2016) 74–82

Fig. 2. The mechanical models for describing the flexural deformation on a bending bar. (a) The Bernoulli–Euler model and (b) the Timoshenko model.

positive directions of M and u are along the +x and x axis, respectively. The zero-to-peak vertical velocity v is measured directly at
different positions along the z axis. The rotation angle u, shear force
V, and bending moment M need to be theoretically derived from thevertical velocity v. The Bernoulli–Euler model, as shown in Fig. 2(a),is valid for a bar of small thickness. The Timoshenko model, as
shown in Fig. 2(b), is an improved version of the Bernoulli–Euler
model, and it describes the flexural deformation of a thick bar more
precisely [13,18]. Because the deformations generated by the shearing force and rotary inertia are taken into account, the rotation
angle estimated by the Timoshenko model is larger than that estimated by the Bernoulli–Euler model under the same driving force[17]. However, the calculation in the Timoshenko model is more
complicated. To obtain high precision with low calculation cost,choosing a thin sample bar that is valid under the Bernoulli–Euler
model is a reasonable approach. To determine the appropriate
thicknesses of the sample bars, the error in the Bernoulli–Euler
model is estimated quantitatively.
2.2. Comparison of Q factors calculated using Bernoulli–Euler modeland Timoshenko model
According to the Bernoulli–Euler model, the rotation angle u,
shearing force V, and bending moment M for the vertical velocity
v are given as

u¼

dv
;
dz

ð7Þ

dV
d v
¼ EIz  3 ;
dt
dz
3

dM
d v
¼ EIz  2 ;
dt
dz

ð8Þ

2

ð9Þ

where E is the elastic modulus and Iz is the second moment of area.
For a rectangular cross-section, Iz is determined by the width andthickness as Iz = bh3/12. In the Timoshenko model, the rotation
angle u, shearing force V, and bending moment M are given as [17]

dv
V
þ
;
dz jGAdV
dv¼ qA 
;
dz
dt
dM
du
¼ V þ qI z 
;
dz
dt

u¼

ð10Þ

10ð1 þ lÞ:
12 þ 11lv ðt; zÞ ¼ A0 fexp ½jx0 t  ða þ jkÞ  z þ exp ½jx0 t þ ða þ jkÞ  ðz  2lÞg;
ð14Þ
where A0 is the amplitude of the vibration velocity, a is the attenuation coefficient, k is the wavenumber, and l is the total length ofthe vibrating bar. As the length was varied from 0 to 1.5 times
the wavelength, the Q factors were calculated on the basis of the
Bernoulli–Euler model and the Timoshenko model, as QB–E and QT,
respectively. Assuming that the Q factor calculated by the
Timoshenko model gives a precise value, the error in the Bernoulli–Euler model is given as

e¼1

Q BE
:
QT

ð15Þ

Considering the measurement results in the previous study[19], the attenuation coefficient was varied from 0.03 to 60 dB/
cm in our simulation. The other parameters are listed in Table 1.
Fig. 3(a) and (b) shows the results for attenuation coefficients of
0.03 and 30 dB/cm, respectively. In Fig. 3(a), QB–E reached 1040,
which was smaller than QT. This is mainly attributed to the large
reactive energy calculated by the Timoshenko model. As statedabove, under the same driving force, the rotation angle calculatedby the Timoshenko model is larger than that calculated by the Bernoulli–Euler model. Thus, the reactive energy increases as the rotation angle increases. Fig. 3(b) shows that, at an attenuation
coefficient of 30 dB/cm, both QB–E and QT were less than 2. The dissipated energy increases with the attenuation, resulting in adecrease in the Q factor.
Fig. 4 shows the QB–E-to-QT ratios when the attenuation coefficients are 0.03, 0.3, 3, 30, and 60 dB/cm. The QB–E-to-QT ratio wasdetermined by both the attenuation coefficient and thethickness-to-wavelength ratio. When the attenuation coefficientwas varied from 0.03 to 0.3 dB/cm, the QB–E-to-QT ratio did notchange significantly. In contrast, the QB–E-to-QT ratio tended to 1
as the attenuation increased. When the thickness-to-wavelength
ratio was 0.2, the error was smaller than 0.2%. When the

ð11Þ
ð12Þ

where A, the cross-sectional area, is determined by the width andthickness as A = bh. G represents the shear modulus. j is the
Timoshenko coefficient for a rectangular cross-section, and it isdetermined by Poisson’s ratio l [17]:j¼

Considering that the wave is reflected at the free end of the bar,
we assume a harmonic solution for the wave equation asð13Þ

Table 1
Parameters used in simulation.
Parameter

Unit

Value

Elastic modulus, E
Density, q
Poisson’s ratio
Width, b
Length, l
Wavenumber, kGPa103 kg/m3
–
mm
mm
mm1

3.45
1.35
0.36
10
500
0.628

J. Wu et al. / Ultrasonics 69 (2016) 74–82

77

Fig. 3. Q factors calculated by the Bernoulli–Euler model (QB–E), the Timoshenko model (QT), and the QB–E-to-QT ratio as a function of the thickness-to-wavelength ratio, h/k, atattenuations of (a) 0.03 dB/cm and (b) 30 dB/cm.

3.2. Experimental setup

Fig. 4. Error in the Bernoulli–Euler model as a function of the thickness-towavenumber ratio, h/k, at attenuations of 0.03, 0.3, 3, 30.00, and 60.00 dB/cm.

thickness-to-wavelength ratio increased to 0.5, the error increased
to approximately 5% at an attenuation coefficient of 0.03 dB/cm.
In summary, the error in the Bernoulli–Euler model is smaller
for a material with high mechanical loss. The maximum error issmaller than 0.2% if the thickness is less than 0.2 times the wavelength when the attenuation coefficients of the tested polymers are
greater than 0.03 dB/cm. Based on the simulation results, we will
discuss the appropriate thickness of the bar later.
3. Experimental setup

Fig. 5 shows the experimental setup for Q-factor measurementsof the polymer samples. Piezoelectric longitudinal transducers
with resonance frequencies of 30, 42, 65, and 95 kHz were prepared to generate ultrasonic-frequency vibrations. One end of thebar was fixed at the top of the transducer with a bolt. When a driving voltage was applied to the piezoelectric ceramic elements inthe longitudinal transducer, a flexural wave was generated on
the bar. The strain on the vibrating bar was changed by adjusting
the amplitude of the driving voltage. A laser Doppler velocimeter(NLV1232, Polytec, Waldbronn, Germany) mounted on a guide rail
was used to record the vertical velocities on the bar surface at different positions. The amplitude of the vibration velocity and thephase between the velocity and the input signal were detectedwith a lock-in voltmeter (5560, NF electronic instruments, Yokohama, Japan).

3.3. Appropriate thicknesses of polymer-based vibrating bars
To determine the appropriate thickness for the Bernoulli–Euler
model, we calculated the wavelength of the flexural waves using
the expression [17]:

2p
k ¼
f
2

3.1. Sample polymers
In this study, the Q factors of some functional polymers, namely,
POM, ABS, PEEK, and PPS, were experimentally investigated. In
industrial applications, lightweight gears are usually made ofPOM because of its high resistance to abrasion and heat [20]. ABS
has good flowability and high resistance to impact [21]. The current version of PEEK has excellent abrasion resistance and workability [22,23]. PPS is widely employed in the automotive andelectrical industries because PPS has good wear properties [24].
The elastic moduli, densities and Poisson’s ratios of POM, ABS,
PEEK, and PPS tested in the experiments are listed in Table 2.

Table 2Mechanical parameters of the tested polymers.
Polymer

Elastic modulus,
E (GPa)Density, q(103 kg/m3)

pﬃﬃﬃﬃﬃﬃﬃﬃﬃ
E=q(103 m/s)

Poisson’s
ratio, l

POM
ABS
PEEK
PPS

2.80
2.40
3.50
3.45

1.41
1.05
1.28
1.35

1.41
1.51
1.65
1.59

0.35
0.36
0.40
0.36

sﬃﬃﬃﬃ
sﬃﬃﬃﬃﬃﬃﬃ
EIz
hE;¼ 1:814  
f
qA
q

ð16Þ

where f is the driving frequency. Substituting the elastic modulus
and density of PPS into Eq. (16), the wavelengths of the PPS-based
bars were calculated as a function of thickness at 30, 42, 65, and95 kHz. Fig. 6 shows the results. When the thickness was 1 mm
and the frequency was 95 kHz, the wavelength was approximately
5.5 mm and the thickness-to-wavelength ratio was approximately
0.18. When the frequency decreased to 30 kHz, the thickness-towavelength ratio decreased to 0.1 monotonically. According to thesimulation results in Fig. 4, the error in the Bernoulli–Euler model
was smaller than 0.2% if the thickness-to-wavelength ratio was varied from 0.1 to 0.18, which can be tolerated in this measurement.
When the thickness increased to 3 mm, the thickness-towavelength ratio reached 0.31 at 95 kHz and the error increased
to 2% for a material with low mechanical loss. Fig. 6 is also effective
in determining the appropriate thicknesses of the POM-, ABS-, andPEEK-based vibrating bars, because the tested polymers do not differ significantly in terms of their elastic moduli and densities. Considering the workability, polymer samples of 1 mm in thickness,
10 mm in width, and 300 mm in length were used in this study.
The theoretical error in the Bernoulli–Euler model is 0.2%.

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J. Wu et al. / Ultrasonics 69 (2016) 74–82

Fig. 5. Experimental setup for Q factor measurement by the devised method.

"
#
3
2

EIz
d v ðzÞ 
d v ðzÞ dv ðzÞ:PðzÞ ¼
 Im
 v ðzÞ 

3
2
dz
2x0
dz
dz

ð19Þ

The 1st-, 2nd-, and 3rd-order spatial derivatives of thevibration velocities are calculated in the k space using theexpression [14]:d v
nn ¼ ðjkÞ  CðkÞ;dznCn ðkÞ ¼ C

Fig. 6. Wavelength, k, of the flexural wave for the PPS-based vibrating bar as afunction of the thickness, h.

3.4. Measurement procedure
To calculate the Q factor from the discrete measurement resultsof the vibration velocity distribution, Eqs. (2) and (6)–(9) need tobe discretized. In previous studies [14–16], the data was processedin the wavenumber space (k space) to suppress the error generated
in high-order finite difference calculations. The procedures are as
follows:(1) Implementing the spatial Fourier transform, the k-space
spectrum Craw(k) of the vibration velocity distribution v(z)is obtained.
(2) A low-pass filter is employed in the k space to reduce thenoise of the high-wavenumber components. The functionof the low-pass filter L(k) is given as [15]LðkÞ ¼8>
1>>< 1  2 exp S  1 >>>: 12 exp S 

k
kc

2

1

k
kc

2


;


; k 6 kc ;where n represents the order of the derivative and Cn(k) is
defined as the k-space spectrum of the nth-order derivativeof the filtered vibration velocity. Using the inverse spatial
Fourier transform, the filtered vibration velocity distributionvf, and the 1st-, 2nd-, and 3rd-order derivatives of the vibration velocity (dv/dz, d2v/dz2, and d3v/dz3) are calculated fromC(k), C1(k), C2(k), and C3(k), respectively.
(4) The reactive-energy distribution DEk(z) and the localdissipated-power distribution DP(z) are given as

1
1
dv ðzÞ
3
DEk ðzÞ ¼ qbh  v 2 ðzÞ  Dz þ qbh 224dzDPðzÞ ¼ PðzÞ  Pðz þ DzÞ;

2

 Dz;

ð21Þ
ð22Þ

where Dz is the sampling interval along the z axis, which is
set to 0.1 times the wavelength in this study. The reactiveenergy Ek and dissipated energy Ep between L1 and L2 aregiven as

Ek ¼

L2
X
DEk ðzÞ;

ð23Þ

L1
2
2p X
DPðzÞ:

L

Ep ¼

x0 L1

ð24Þ

(5) Finally, the Q factors are calculated with Eq. (1).

ð17Þk > kc ;4. Experimental results

where kc is the cutoff wavenumber and S is the filter-shapecoefficient. The optimal values of kc and S have been investigated in a previous study [14]: the ratio of the cutoff
wavenumber kc to the propagating wavenumber k0 was set
to 1.8 and the filter-shape coefficient S was set to 10. The filtered k-space spectrum C(k) is given asCðkÞ ¼ LðkÞ  Craw ðkÞ:

ð20Þ

ð18Þ

(3) Substituting Eqs. (7)–(9) into Eq. (6), and implementing discretization, the active vibration power flow is given as

4.1. Dissipated power
Fig. 7(a) shows the vibration velocity distribution on the PPSbased bar at a frequency of 28.30 kHz and driving voltage of
100 V. The maximum vibration velocity reached 280 mm/s. The
spectrum of the vibration velocity is shown Fig. 7(b). The propagating wavenumber k0 was approximately 0.616 mm1 (k = 10.2 mm)
at 28.30 kHz. The cut-off wavenumber of the low-pass filter was
set to 0.9 mm1. Fig. 7(c) and (d) shows the reactive-energy distribution DEk(z) and the local-dissipated-energy distribution DP(z)along the z axis. Clearly, the reactive energy and the local

J. Wu et al. / Ultrasonics 69 (2016) 74–82

79

Fig. 7. Example of Q-factor calculation. (a) Vibration velocity distribution at a driving frequency of 28.30 kHz and zero-to-peak voltage of 100 V. (b) Spectrum of the vibration
velocity and the low-pass filter. (c) Reactive-energy distribution, DEk(z), and (d) local-dissipated-power distribution, DP(z), along the z axis.

dissipated energy decreased as the propagating distance increased.
In this study, the integral interval (L1–L2) for calculating Ek and Ep
were 125–175 mm.
Based on the Bernoulli–Euler model [13,18], the strain c on thebending bar is given as

M hc¼  :EIz 2

ð25Þ

Substituting Eq. (9) into Eq. (25), we obtain

c¼

h

d v
2



2x0 dz2:

ð26Þ

The 2nd-order derivative of the vibration velocity was obtainedusing Eq. (20). The maximum strain of the 1-mm-thick PPS-based
bar in the range of 125–175 mm is 0.037%. Using Eqs. (23) and(24), we calculated that the reactive energy and dissipated energy

in the range of 125–175 mm are 4.900  106 J and 0.085  106 J,
respectively. Thus, the Q factor of PPS is 362 at a driving frequencyof 28.30 kHz and strain of 0.037%.
4.2. Q factors of the tested polymers
Fig. 8 shows the Q factors of the PPS-, PEEK-, POM-, and ABSbased bars as the strain is varied from 0.005% to 0.060% at
40.99 kHz. The Q factor of the PPS-based bar was approximately
350 at 0.005%. As the strain increased, the Q factors of the PPS-,PEEK-, POM-, and ABS-based bars decreased. The Q factor of the
PPS-based vibrating bar decreased to approximate 150 at 0.058%,
which was 0.4 times that at 0.005%. The Q factor of the PEEKbased bar was approximately 90 at 0.006%. The Q factors of the
POM- and ABS-based bars were approximately 20, which were
only 0.06 times the maximum Q factor of the PPS-based bar.

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J. Wu et al. / Ultrasonics 69 (2016) 74–82

Fig. 8. Q factors of the PPS-, PEEK-, POM-, and ABS-based vibrating bars as afunction of the strain at a frequency of 40.99 kHz.

Fig. 9(a) shows the Q factors of the PPS-based bar versus thestrain at 28.30, 40.99, 63.84, and 90.23 kHz. The maximum Q factor
reached approximately 460 at a driving frequency of 28.30 kHz andstrain of 0.050%. The attenuation coefficient corresponding to the
maximum Q factor was approximately 0.05 dB/cm. When thestrain was smaller than 0.020%, the Q factors did not change significantly. In contrast, the Q factors decreased sharply when the strain
exceeded 0.020%, especially at 28.30 and 40.99 kHz. As shown in
Fig. 9(b), the maximum Q factor of the PEEK-based vibrating bar
was 115 and decreased as the strain increased.
Fig. 10 shows the Q factors of the PPS- and PEEK-based bars as afunction of the driving frequency when the strains are 0.005–
0.010% and 0.035–0.040%. The Q factors of the PPS- and PEEKbased bars decreased as the driving frequency increased. The fitting curves show that the Q factor is inversely proportional tothe driving frequency in the range of 30–100 kHz. The decreasein the Q factors with increasing vibration velocity and driving frequency may be attributed to the hysteretic strain–stress behaviorof polymers [19,25].
We draw the following conclusions about the Q factors of thetested polymers: (1) The Q factors of the polymer-based vibratingbars decrease as the vibration velocity and driving frequency
increase. (2) When the driving frequency is 28.30 kHz and he strain
is smaller than 0.005%, the Q factor of the PPS-based vibrating bar
reaches 460, which is larger than those of other tested polymers
under the same strain and driving frequency.
4.3. Comparison with experimental results obtained via impact testing
For comparison, the Q factors of the PPS- and PEEK-based cantilevers were also measured using the impact testing method [9].
The experimental setup is shown in Fig. 11. Cantilevers withlengths of 100, 150, and 200 mm were sandwiched between two

Fig. 10. Q factors of the PPS- and PEEK-based bars as a function of the driving
frequency in the strain ranges of 0.005–0.01% and 0.035–0.04%. The Q factors of the
PPS- and PEEK-based vibrating bars at a natural frequency of 0.08 kHz are measured
by the impact testing method.

Fig. 11. Experimental setup for Q-factor measurement by the impact testing
method.

stainless-steel plates with a vise. The width and thickness of thecantilevers were 10 and 1 mm, respectively. A 20-mm-diameterball made of stainless steel was released from the position above
the free end of the cantilever to give a transient mechanical input.
The strain on the cantilever was changed by adjusting the initialheight of the ball. The laser Doppler velocimeter (see Section 3.2)
was used to measure the vibration velocity at 50 mm from thefixed end. In theory, the vibration velocity on the vibrating bar isa damped oscillation with an exponential envelope, which is givenasv ðtÞ ¼ V 0  efx t  sinðxn t þ hÞ;nFig. 9. Q factors of (a) the PPS-based and (b) PEEK-based vibrating bars as a function of the strain.

ð27Þ

J. Wu et al. / Ultrasonics 69 (2016) 74–82

81

definition, the Q factor was precisely measured for a bar with sufficiently small thickness. The Q factor decreased as the driving frequencies and strains increased. At 28.30 kHz, the Q factor of PPS
reached approximately 460 when the strain on the bar was smaller
than 0.005%, which indicated that PPS can be potentially employed
as an elastomer in ultrasonic transducers. In the future, using thismethod, we will explore the mechanical loss of other functional
polymers.
Acknowledgement

Fig. 12. Q factors of the PPS- and PEEK-based bars measured by the impact testing
method.

where V0, 2f, xn, and h represent the initial vibration velocity,
damping coefficient, natural frequency of the bar, and phase,respectively. The Q factor is given as [12]

Q¼

1
:
2f

ð28Þ

The Q factor was calculated by fitting the measured data withthe envelope of Eq. (27). The measurements were carried out 20
times and the averaged value was selected as the Q factor of thepolymer at a certain strain.
Fig. 12 shows the Q factors measured by the impact testing
method. The natural frequencies of the 100-, 150-, and 200-mmlong polymer-based bars were approximately 0.18, 0.08, and0.05 kHz, respectively. In the range of 0.005–0.010%, the Q factorof the PPS-based cantilever was approximately 580, which was
twice as high as that in the range of 0.035–0.040%. The Q factorof the PEEK-based cantilever was 150 in the range of 0.005–
0.010% and decreased to 90 as the strain increased to 0.040%. The
experimental results also demonstrate that the Q factor is independent of frequency in the range of 0.050–0.180 kHz.
The Q factors of the PPS- and PEEK-based bars at 0.08 kHz measured using the impact tested method are shown in Fig. 10. They
are in accordance with the frequency- and strain-dependences of
the Q factor measured by the devised method described above.
However, these Q factors are lower than the values extrapolated
from the results obtained by the devised method. This may be
attributed to the mechanical loss generated on the contacting surface between the cantilever and the plates. In the devised method(Fig. 5), the energy loss is distributed in the transducer, contacting
surface, and polymer-based bar. Since the active vibration power
flowing across a cross-section is calculated, the dissipated energy,
which reflects the energy loss generated on a certain part, can be
estimated. In contrast, using the impact testing method (Fig. 11),
it is difficult to separate the energy loss generated on the bar fromthat on the contacting surface between the plate and the polymerbased bar. In the impact testing method, the energy loss in thevibration system is used as the dissipated energy in calculations,which yields a smaller Q factor than that obtained by the devised
method.
5. Conclusions
As the first step in investigating the possibility of employing
polymers as elastomers in high-power ultrasonic applications, we
devised a method for evaluating the mechanical loss in ultrasonicfrequencies and high vibration amplitudes, and we measured the
Q factors of some functional polymers. Based on its original

The authors thank the Daicel Corporation for providing somepolymer samples in this study.
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J. Wu et al. / Ultrasonics 69 (2016) 74–82
Jiang Wu was born in Liaoning, China, on January 29,
1988. He received the B.E. degree in mechanical engineering from Dalian University of Technology (DUT),
China, in 2010. From 2010 to 2012, he studied in theState Key Laboratory of Robotics and System, Harbin
Institute of Technology (HIT), China, and received the M.
E degree in mechatronic engineering from HIT in 2012.
He is a presently a doctoral candidate in Precision andIntelligence Laboratory, Tokyo Institute of Technology(TITECH), Japan. His research interest includes piezoresistive and piezoelectric materials, permanent-magnetbased sensing technology, ultrasonic transducers andactuators, and ferroelectric film.

Yosuke Mizuno was born in Hyogo, Japan, on October
13, 1982. He received the B.E., M.E., and Dr.Eng. degrees
in electronic engineering from the University of Tokyo,
Japan, in 2005, 2007, and 2010, respectively.
From 2007 to 2010, he was engaged in Brillouin opticalcorrelation-domain reflectometry for his Dr.Eng. degree
at the University of Tokyo. From 2007 to 2010, he was a
Research Fellow (DC1) of the Japan Society for the Promotion of Science (JSPS). From 2010 to 2012, as a
Research Fellow (PD) of JSPS, he worked on polymer
optics at Tokyo Institute of Technology, Japan. In 2011,
he stayed at BAM Federal Institute for Materials
Research and Testing, Germany, as a Visiting Research Associate. Since 2012, he has
been an Assistant Professor at the Precision and Intelligence Laboratory, Tokyo
Institute of Technology, where he is active in fiber-optic sensing, polymer optics,
and ultrasonics.
Dr. Mizuno is the winner of the Funai Research Award 2010, the Ando Incentive
Prize for the Study of Electronics 2011, the NF Foundation R&D Encouragement
Award 2012, and the Challenging Research Award 2013. He is a member of the IEEE,
the Japanese Society of Applied Physics (JSAP), and the Institute of Electronics,
Information, and Communication Engineers (IEICE) of Japan.

Marie Tabaru was born in Tokyo, Japan. She received
her B.Eng. degree in Electrical and Electronic Engineering from Tokyo Institute of Technology in 2002 and her
M.E. degree in Human System Science in 2005. She
studied image signal processing of medical ultrasoundat the University of Washington, Seattle, WA from 2002
to 2003. She received the Best Student Paper Award inEngineering Acoustics from the Acoustical Society ofAmerica in 2006. She received her D.E. degree from
Tokyo Institute of Technology in 2007. She has been an
associate professor of the Precision and Intelligence
Laboratory, Tokyo Institute of Technology since 2013.
She is a member of the Acoustical Society of Japan, the Japan Society of Ultrasonics
in Medicine, and the Institute of Electronics, Information and Communication
Engineers.

Kentaro Nakamura was born in Tokyo, Japan, on July 3,
1963. He received the B.E., M.E., and Dr.Eng. degrees
from Tokyo Institute of Technology, Japan, in 1987,
1989, and 1992, respectively.
Since 2010, he has been a Professor at the Precision andIntelligence Laboratory, Tokyo Institute of Technology.
His research field is the applications of ultrasonic waves,measurement of vibration and sound using opticalmethods, and fiber-optic sensing.
Prof. Nakamura is the winner of the Awaya Kiyoshi
Award for encouragement of research from the Acoustical Society of Japan (ASJ) in 1996, and the Best Paper
Awards from the Institute of Electronics, Information and Communication Engineers (IEICE) in 1998 and from the Symposium on Ultrasonic Electronics (USE) in2007 and 2011. He also received the Japanese Journal of Applied Physics (JJAP)
Editorial Contribution Award from the Japan Society of Applied Physics (JSAP) in2007. He is a member of the IEEE, the ASJ, the JSAP, the IEICE, and the Institute ofElectrical Engineers of Japan (IEEJ).