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8/18/2019 A vibrating reed apparatus to measure the natural frequency of multilayer thin films
The natural frequency of an object is its vibrational fingerprint
which depends on its geometry and material properties, and its
determination is of vital importance for predicting resonant dam-
ages, amplifying electrical signals, or optimizing acoustical cav-
ities and resonators [1]. Determination of the natural frequency
plays a very important role in a variety of fields such as mechan-
ical resonators, sensors, instrumentation, and determination of
elastic properties of materials [1–9]. The study of mechanical,
acoustic, and sensing of properties of solid materials has pro-
moted the development of new methods and devices for the pur-
pose of vibration measurement. These devices span from torsion
apparatuses to devices for measuring longitudinal or transverse
vibrations, with a variety of excitation and detection methods [1,5]. One of the applications of vibration measurements consists
in correlating the change in the measured natural frequency to an
elastic property (such as elastic modulus) or existent damage [3–
6]. When the transverse vibrations of a slender cantilever beam
is used to this aim, the method is known as the vibrating reed
method, which has the advantage of being non-invasive, allowing
both frequency and damping analyses. Several vibrating reed
devices have been used in order to determine the material prop-
erties in a wide range of geometries and thicknesses, ranging
from samples in bulk geometry [5, 6] (with length l > 50 mm),
thin films [7–9] ( l15 mm 60< < mm) and nanostructured
films [10, 11] (l < 350 µm). However, given their small thick-
ness, measurements of films at the micrometer thickness and
Measurement Science and Technology
A vibrating reed apparatus to measure the
natural frequency of multilayered thin films
F Gamboa1, A López2,3, F Avilés2, J E Corona1 and A I Oliva1
1 Centro de Investigación y de Estudios Avanzados del IPN, Unidad Mérida, Depto. de Física Aplicada,
km 6 Antigua Carretera a Progreso, 97310 Mérida, Yucatá n, Mexico2 Centro de Investigación Científica de Yucatá n, AC, Unidad de Materiales, Calle 43#130, Col.
Chuburná de Hidalgo, 97200 Mérida, Yucatá n, Mexico3 Facultad de Ingeniería, Universidad Autóma de Yucatá n, Av. Industrias no contaminantes por Perif érico
The first experiment consisted in conducting ten sequential
vibratory measurements, maintaining the beam clamped. The
ten measurements conducted in this way yielded frequen-
cies with an average of 74.6 Hz whose maximum difference
was only 1 mHz, evidencing the high reproducibility of the
apparatus. One of the key factors governing the uncertainty in
this kind of vibration experiments is the boundary conditions
(clamping force). Therefore, a second set of experiments con-
ducted consisted in repeating the vibration measurements ona given sample but removing the sample from the apparatus
(clamp) after each measurement. The straightedge device
shown as #7 in figure 1 assisted in positioning the sample
back at, in principle, the same position, after each test.
Figure 5 shows the results of the ten repetitive experiments
conducted in this way. In this figure, amplitude as a func-
tion of time was directly measured and the FFT was used to
produce the results shown in the frequency domain. As seen
from this figure, a narrow dispersion of the curves with low
experimental uncertainty is achieved in the measurements.
The average frequency measured is 74.6 Hz with maximum
deviations from this value of ±0.2 Hz and a coefficient ofvariation of only 0.18%.
5.2. Measurement of the natural frequency in multilayers
The natural frequency of cantilever beams comprising one
(Kapton), two, (Au/Kapton) and three (Al/Au/Kapton) layers
was measured by means of the constructed apparatus. Figure 6
shows typical vibratory measurements of the three beam
architectures investigated. The left-hand side of figure 6 shows
plots of the directly measured data corresponding to the nor-
malized amplitude of vibration as a function of elapsed time,
indicating the period (T) for the Kapton (a), Au/Kapton (b)
and Al/Au/Kapton (c) beams. The right-hand side of figure 6
(frequency domain) shows the FFT of the correspondingdata in the time domain. Periods of T = 13.4 ms, 12.9 ms and
Figure 6. Representative vibratory measurements conducted on multilayered beams using the constructed apparatus. (a) Kapton, (b) Au/ Kapton, (c) Al/Au/Kapton beams. Left side shows a period (T) in the time domain while right side shows the FFT in the frequency domain.
Table 2. Measured natural frequency of the four layered beamswith 21 mm length and 4.8 mm width.
f n (Hz)
Beam No. Kapton Au/Kapton Al/Au/Kapton
1 ±74.6 0.1 ±7.5 0.2 ±81.1 0.2
2 ±74.6 0.2 ±7.5 0.3 ±80.9 0.2
3 ±74.5 0.2 ±7.6 0.2 ±81.1 0.2
4 ±74.6 0.3 ±7.5 0.3 ±81.2 0.3
Note: The thickness of each layer is indicated in table 1.
Figure 7. Schematic representation of the oscillatory response of a
beam under damped transverse vibrations.
Meas. Sci. Technol. 27 (2016) 045002
8/18/2019 A vibrating reed apparatus to measure the natural frequency of multilayer thin films
12.3 ms corresponding to f 74.6n = Hz, 77.5 Hz and 81.1 Hz
are identified for Kapton (a), Au/Kapton (b) and Al/Au/
Kapton (c) beams, respectively. As seen from this figure an
important frequency shift of at least one order of magnitude
larger than the determined experimental uncertainty of theapparatus (∼0.2 Hz) is detected when additional thin metallic
layers (200–250 nm thick) are added to the Kapton beam.
Table 2 lists a summary of the fundamental frequencies
measured (average value and standard deviation), considering
the four tested replicates for each layered system. An increase
in f n is observed when each layer is added, which corresponds
to the added mass and stiffness upon film deposition. An
important feature to point out is that such changes in f n are
due to the deposit of very thin (200 and 250 nm thick) metallic
films and the vibratory apparatus constructed has enough
resolution to detect such small changes in natural frequency.
These changes in frequency can be associated to the change in
the effective stiffness of the beam, and, if a proper data reduc-
tion model is used, the elastic modulus of each layer can be
obtained by this technique, see e.g. [26].
5.3. Damping analysis
In actual free vibration experiments, the magnitude and fre-
quency of oscillations are affected by damping. Vibration
theory recognizes a difference between the frequency of
damped vibration ( f d) and the natural frequency ( f
n) by intro-
ducing a damping factor (ζ ) such as [1],
f f 1 .d2
nζ = − (1)
Several vibratory instruments base their performance on
conducting frequency sweeps and detecting the maximum
amplitude of vibration, thus determining a resonant frequency.
However, in many applications (such as those involving mat-
erial property determination or in structural design) the actualnatural frequency is needed. Measurement of f n demand
free vibration experiments, such as those conducted herein.
Therefore, damping is an integral part of a free vibration
experiment/instrument and its quantification allows esti-
mating differences between f d and f
n, which are of particular
importance close to resonance.
In free vibration experiments, the amplitude of oscilla-
tion decreases with the elapsed time because of friction with
the air and test rig. This damping can be characterized by the
damping factor (ζ ), which is a function of the logarithmic dec-
rement (δ ). This decrement δ is defined as the ratio of two
consecutive amplitudes W 1 and W 2 (see figure 7), i.e.
W
W ln .
1
2
δ = (2)
The damping factor ζ can be determined from δ by means of
the relationship [1],
2
.2 2( )
ζ δ
π δ =
+ (3)
For the case of the investigated beams, figure 8 shows two
consecutive amplitudes (normalized) considering that W 1 = 1,
which facilities the calculations of the damping factor. For
the cases presented in figure 8, W 2 = 0.9880, 0.9815 and0.9800 for the Kapton, Au/Kapton and Al/Au/Kapton layered
Figure 8. Close-up of the first oscillation amplitude used to determine the damping factor of the layered beams. (a) Kapton, (b) Au/Kapton,(c) Al/Au/Kapton. Insets show the full oscillatory signal for 1 s.
Meas. Sci. Technol. 27 (2016) 045002
8/18/2019 A vibrating reed apparatus to measure the natural frequency of multilayer thin films
systems, respectively. The inset in figure 8 shows the com-
plete vibratory oscillation for 1 s, indicating a slow decay in
the vibrating amplitude. The vibratory parameters (ζ and δ )
extracted from the vibratory curves measured are listed in
table 3. Very small damping factors ranging between 0.0019
and 0.0034 were obtained for all the investigated multilayer
system, given their low mass. Therefore, using equation (1)
the ratio f f / d n
is very close to 1 for all cases, indicating that the
instrument rightfully measures the natural frequency.
5.4. Comparison with finite element analysis
FEA was used to predict the fundamental frequency of the
tested beams in order to further support the reliability of ourapparatus. Table 4 shows the FEA predictions of the natural
frequency along with the average and standard deviation of
the measured frequency. An excellent agreement is observed
between the measured data and the FEA predictions. The
slight differences observed are practically within the exper-
imental scattering, which provides further reliability to the
constructed apparatus for measuring natural frequencies of
thin multilayer beams.
6. Conclusions
A vibratory apparatus was introduced for measuring the naturalfrequency of thin (micrometric or sub-micrometric) layered
beams. The apparatus consists of an aluminum frame with a
C-shaped arm holding the sample in cantilever configuration.
The excitation-sensing arrangement uses a controlled air-pulse
applied at the free-end of the cantilever beam and an optical
system for sensing the vibratory amplitude. A commercial data
acquisition board and an in-house software were used for the
control and data acquisition. High reproducibility was found
in the constructed apparatus with a maximum uncertainty of
1 mHz (for frequencies of the order of tens Hz) if the sample
is not removed from the clamp. When the sample is removed
from the apparatus and placed back, the coefficient of variationof ten measurements is only ∼0.2%. The amount of damping
was small enough to not affect the determination of natural
frequencies. Kapton, Au/Kapton and Al/Au/Kapton layered
beams were fabricated and their natural frequency was mea-
sured using this apparatus. The average measured frequency for
the three layered system was 74.6 Hz (Kapton), 77.5 Hz (Au/
Kapton) and 81.2 Hz (Al/Au/Kapton) and the shifting upon
thin film deposition is at least an order of magnitude larger
than the detected experimental uncertainty of the apparatus.
The measured frequencies for the multilayered beams agree
well with finite element analysis computations, which pro-
vide further confidence to the apparatus. With an appropriate
data reduction model, this shift could used, for example, for
determination of elastic modulus or assessing delamination or
damage in multilayered beams and others thin film structures.
Acknowledgments
The authors wish to thank O Gómez (CINVESTAV), Alejandro
May (CICY) and Cesar Villanueva (FI-UADY) for their tech-
nical support.
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