This apparatus measures the natural frequency of multilayer thin films with high precision and low uncertainty. It uses a cantilever beam sample holder that clamps one end of the film. An air pulse excites vibrations at the free end, which are detected by a laser and photosensor. Repeated tests of a Kapton sample yielded a measurement uncertainty of 1 mHz. The apparatus was used to measure the natural frequency of Kapton beams alone and with additional thin layers of gold and aluminum deposited on top, detecting small frequency shifts. Finite element modeling supported the accuracy of the experimental measurements.

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A vibrating reed apparatus to measure the natural frequency of multilayered thin films
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1. Introduction
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,whichhastheadvantageofbeingnon-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, Yucatán, Mexico
2
Centro de Investigación Científica de Yucatán, AC, Unidad de Materiales, Calle 43 #130, Col.
Chuburná de Hidalgo, 97200 Mérida, Yucatán, Mexico
3
Facultad de Ingeniería, Universidad Autóma de Yucatán, Av. Industrias no contaminantes por Periférico
Norte, 97310 Mérida, Yucatán, Mexico
E-mail: ffgamboa@mda.cinvestav.mx
Received 9 November 2015, revised 15 December 2015
Accepted for publication 11 January 2016
Published 15 February 2016
Abstract
An apparatus for measuring the natural frequency of sub-micrometric layered films in
cantilever beam configuration is presented. The instrument comprises a specially designed test
rig with a sample holder, an electronic excitation source, a vibration sensor and an automated
software for the excitation and data recollection. The beam is excited by means of an air pulse
and the oscillation amplitude of its free end is measured through a laser diode-photosensor
arrangement. The instrument provides a very low uncertainty (∼1 mHz, for frequencies of the
order of tens Hz) for repeated sequential tests and the major source of uncertainty (∼0.2 Hz,
corresponding to a coefficient of variation of 0.18%) arises from the difficulty of placing
the sample in an exactly identical location upon clamping. This high sensitivity renders the
capability of measuring very small frequency shifts upon deposition of sub-micrometric films
over thicker substrates. In order to assess the reliability of the apparatus, cantilever beams of
125 μm thick neat Kapton (substrate) and thin layered films of Au/Kapton and Al/Au/Kapton
of 200–250 nm film thickness were fabricated and their natural frequency and damping factor
were measured. Calculations of the natural frequency of such beams by finite element analysis
further support the accuracy of the experimental measurements.
Keywords: vibration measurements, natural frequency, thin films
(Some figures may appear in colour only in the online journal)
0957-0233/16/045002+8$33.00
doi:10.1088/0957-0233/27/4/045002Meas. Sci. Technol. 27 (2016) 045002 (8pp)

3. F Gamboa et al
2
below can be very challenging. Thin films are frequently part
of micro- and nano-electromechanical systems, and knowledge
of their fundamental vibratory frequency is useful for a correct
design, operation and reliability [12–14]. Applications of thin
films as electronic conductors and in nanosensors and actuators
are extensive. For example, curved beams used in atomic force
microcopy are used for measuring frequency shifts and damping
factors [11]. Thin films can also be used as bilayers cantilevers
forremotetemperaturesensors[15].Adoublysupportedmetallic
beam configuration can be used for studying the influence of
overlapping length and adhesive joints, to obtain the variation of
the resonance frequency, peak amplitudes and loss factor which
is useful for vibration control applications [16]. In particular,
accurate measurement of mechanical properties of materials in
thin film geometry is known to be a challenging task [8, 17, 18].
The instrument designed for such an aim must provide high acc­
uracy and low experimental uncertainty, and a vibratory device
seems to be excellent candidate for such a task. Although a few
existing vibratory instruments may satisfy these requirements
[19–22], they either do not provide the precision demanded for
thin film architectures, measure the resonant frequency (instead
of the natural one) and/or need cumbersome instrumentation.
In a previous work [22], the authors developed a simpler vibra-
tory device for measuring the resonant frequency of thin canti­
lever beams, based on frequency sweeps using piezo­electric
excitation. However, in many applications, the natural frequency
is needed and its measurement may be influenced by the amount
of damping. Therefore, a precise apparatus using a cantilever
beam configuration, a dedicated system of air-pulse excitation
and a photosensor is reported here to produce and measure free
vibrations. Although the instrument could be used for materials
or structures with thickness ranging from nm to a few mm, given
its high resolution, the major application in mind focuses on
determining the small frequency shifts produced by adding thin
(sub-micrometer) layers to multi-layered architectures, which
can in turn be correlated to their material properties.
In order to evaluate the operation of the apparatus for the
intended application, thin films comprising one, two and
three layers were fabricated and their natural frequency and
damping factor were measured. To further support the reli-
ability of the apparatus, finite element analysis was used to
predict the natural frequency of such layered systems and
their results were compared to the measurements conducted
with the constructed apparatus.
2. Apparatus design and automation
2.1. Mechanical design
The mechanical components of the vibrational apparatus are
identified in figure 1. The overall dimensions of the apparatus
are 100 mm by 100 mm by 220 mm high. The apparatus com-
prises a 13 mm thick, 100 mm by 100 mm aluminum base-
plate (#1) for the mechanical support of all components.
It is made of solid aluminum contributing to the mechanical
stability. This base-plate rests on a rubber pad (#2) which
acts as a vibration isolator for reducing external perturbations.
Four vertical screws (#3) join the base plate with an upper
auxiliary plate (#4), which supports the main elements of
the apparatus. A C-shaped arm (#5) is fixed at its center. The
ends of the C-shaped arm contain both the sample holder (#6)
and the straightedge mechanism (#7) for adjusting the sample
to similar conditions at each test for reproducibility. Since an
imperfect clamping is one of the most significant error sources
[4, 23], the designed straightedge device plays an important
role on the measurements, especially when the cantilever
beam needs to be taken out of the rig and measured several
times. A close-up of one end of the C-shaped arm is detailed
at the right side of figure 1. The sample is clamped between
a thin rubber strip (#8) and the top surface of an end of the
C-shaped arm (#5) by means of a flexible steel sheet (#9).
Two screws (#10) fasten the steel sheet at one end while a
Figure 1. Mechanical components of the vibrating apparatus. 1- Base metallic plate; 2- Rubber pad; 3- Join screws; 4- Auxiliary plate;
5- C-shape arm; 6- Sample holder; 7- Straightedge mechanism; 8- Thin rubber strip; 9- Flexible steel sheet; 10- Fixing screws for the steel
sheet; 11- Adjustment screw; 12- Vibration sensor; 13- Air valve; 14- Adjustable height crossbar.
Meas. Sci. Technol. 27 (2016) 045002

4. F Gamboa et al
3
screw at the mid-­section (#11) is used to adjust the applied
pressure to the sample. The block with the vibration sensor
(#12) is located under the sample edge. The block contains a
laser diode and a photosensor to detect the oscillation ampl­
itudes of the sample. The block height can be changed in order
to adjust for the zone of the best sample’s optical reflection.
An air valve (#13) is positioned above the free end of the
sample which is fixed through an adjustable support system
(#14). For the sample’s oscillation, a controlled short pulse
of air of a few milliseconds of duration is applied at the free
end of the sample.
2.2. Excitation and sensing arrangement
Figure2showsaschematicoftheexcitationandsensingarrange-
ment that comprises the sample holder, an air valve and a vibra-
tion sensor arrangement. The sample is clamped by the holder
at one of its ends producing a cantilever beam ­configuration. A
miniature air valve (MB332-VB33-L203, Gems Sensors and
Controls, CT, USA) is gated during 20 ms to produce a short air-
pulse which is concentrated on a small area (∼1 mm2
) at the free
end of the sample. The air supplied to the valve is previously fil-
tered and its pressure is controlled to 1 psi by means of a pressure
regulator. The oscillation ampl­itude at the free end of the sample
is measured through a laser diode-photosensor arrangement
(see bottom part of figure 2). This design was chosen because
it permits an easy sample handling with high accuracy. A light
beam emitted by an OPV332 laser diode (OPTEK Technology
Inc., TX, USA) is reflected by the bottom surface of the sample
and detected by an UDT-455 photosensor (OSI Electronics, CA,
USA). For better pick up, both sensor components are posi-
tioned at an angle of 30° with respect to the perpendicular axis
of the incidence plane. The laser beam spot is 0.4 mm2
while
the photosensor has an active area of 5.1 mm2
. The photovoltage
registered is a function of the displacement of the reflection sur-
face with respect to the position of the photosensor [24]. This
arrangement requires a sample surface with enough reflectance
to detect the vibration frequency. The reflectance of the Kapton
foil used in this work as substrate was enough for an adequate
operation of the apparatus. However if the sample surface is not
reflective, a small area (∼6 mm2
) can be covered with a thin
reflective coating to fulfill this purpose.
2.3. Control and data acquisition
A schematic of the control and data acquisition system of
the vibration apparatus is shown in figure 3. A data acquisi-
tion board NI USB-6216 (National Instruments Corp., TX,
USA) carries out the tasks of controlling and acquiring the
different signals used. A dedicated program developed in
LabVIEW (National Instruments Corp., TX, USA) controls
and acquires the measurements. The air-pulse is executed
by means of a digital output of the board and a conditioning
circuit connected to the air valve, by sending a 20 ms pro-
grammed voltage pulse. A conditioning circuit increases
the voltage in order to control the air valve. The laser
diode is powered by means of the board and a constant cur­
rent circuit is applied in order to maintain a constant light
intensity. The oscillatory signal produced by the air-pulse
is sensed by the photosensor and registered by an analog
input of the board. The program developed in LabVIEW
synchronizes the air-pulse and the data acquisition of the
vibration signal, generating in this way data of intensity
(voltage or vibration amplitude) as a function of elapsed
time. The natural frequency could be simply obtained from
measurement of the time period of the sinusoidal wave, but
Figure 2. Excitation and sensing arrangement.
Figure 3. Schematic of the control system and data acquisition.
Meas. Sci. Technol. 27 (2016) 045002

5. F Gamboa et al
4
for a more accurate definition of the natural frequency a
fast Fourier transform (FFT) was applied to the measured
amplitude data in the time domain, producing corresp­
onding frequency versus amplitude plots. The natural
frequency was then identified from the frequency corresp­
onding to the peak amplitude in the frequency domain.
All experiments were conducted at lab temperature, which
was ∼23 °C. Temperature variations of a couple of Celsius
did not affect our measurements.
3. Samples and material properties
In order to evaluate the performance of the vibration appa-
ratus, several rectangular beams of 24.0 mm as total length
and 4.8 mm width were obtained from a 125 μm thick 500 HN
Kapton®
foil (DuPont). The free beam length (effective
length) of the cantilever is l = 21.0 mm, and 3 mm overhang
was allowed for edge clamping, see figure 4. First, ten repeti-
tive vibration measurements of a baseline Kapton beam were
conducted in order to evaluate the uncertainty of the appa-
ratus. Afterwards, a group of four rectangular Kapton beams
of identical dimensions were cut as substrates for subsequent
metallic film deposit using a thermal evaporation technique.
Before the film deposition, the Kapton substrates were ultra-
sonically cleaned with isopropyl alcohol and distilled water.
During the film growth the thickness of the films were mea-
sured in situ with a quartz crystal sensor and monitored with
a Maxtek 400 controller with ±0.1 nm accuracy. Initially,
four of the Kapton substrates were placed closely inside the
thermal deposition chamber in order to deposit a 250 nm thick
gold (Au) layer to produce four Au/Kapton identical speci-
mens. After vibration measurements of the Au/Kapton beams,
a new 200 nm thick layer of aluminum (Al) was deposited
over them, forming in this way four Al/Au/Kapton three-lay-
ered samples, see figure 4. New vibration measurements of
those three-layered samples were performed. Given that the
vibratory measurement technique is not destructive, it allows
the use of the same specimens for sequential film deposition.
Table 1 shows a summary of the thicknesses, elastic modulus
(E ) and density (ρ) of the Kapton substrate and the deposited
metallic layers. The values E and of ρ were taken from refer-
ences [10] and [25].
4. Finite element analysis
The vibratory measurements conducted on single-­layered
and multilayered thin films were further supported by
calcul­ations of the natural frequency based on finite
­element analysis (FEA). The model was constructed with
dimensions consistent with the experimental conditions
and the layer properties given in table 1. Tridimensional
FEA was conducted by using the commercial software
ANSYS®
employing a solid layered element (‘SOLID46’)
with translational degrees of freedom at each node. This
layered element allows the definition of layer-by-layer
properties which is suitable for modeling multi-layered
materials. A typical layered beam was constructed with
3930 solid elements using a mesh of 30 elements in the
width direction and 131 elements in the length direction
with one element through the thickness. Zero deflec-
tion at the clamped edge was considered and the natural
frequency of the beam under transverse vibrations was
numerically found by solving the resulting modal eigen-
value problem.
5. Results and discussions
5.1. Reproducibility and uncertainty
The uncertainty and reproducibility of the apparatus were first
evaluated. To this aim, natural frequency measurements of
the Kapton beams were conducted as indicated in section 2.3.
Figure 4. Schematic of the Al/Au/Kapton multilayered beams
fabricated.
Table 1. Thickness, elastic modulus (E ) and density (ρ) of the
beam constituents.
Material Thickness (μm) E (GPa) ρ (kg m−3
)
Substrate (Kapton) 125 3.64 1420
Film #1 (Au) 0.25 69.1 19 320
Film #2 (Al) 0.20 78.0 2699 Figure 5. Ten measurements of natural frequency of the same
Kapton beam, removed from the grip and placed back.
Meas. Sci. Technol. 27 (2016) 045002

6. F Gamboa et al
5
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 on
a 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 of
variation 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 corresponding
data 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 beams
with 21 mm length and 4.8 mm width.
fn (Hz)
Beam No. Kapton Au/Kapton Al/Au/Kapton
1 ±74.6 0.1 ±77.5 0.2 ±81.1 0.2
2 ±74.6 0.2 ±77.5 0.3 ±80.9 0.2
3 ±74.5 0.2 ±77.6 0.2 ±81.1 0.2
4 ±74.6 0.3 ±77.5 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

7. F Gamboa et al
6
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 the
apparatus (∼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 fn 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 fn 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 ( fd) and the natural frequency (fn) by intro-
ducing a damping factor (ζ) such as [1],
f f1 .d
2
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 actual
natural frequency is needed. Measurement of fn 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 fd and fn, 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 W1 and W2 (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 W1 = 1,
which facilities the calculations of the damping factor. For
the cases presented in figure 8, W2 = 0.9880, 0.9815 and
0.9800 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. F Gamboa et al
7
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 our
apparatus. 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 ­natural
frequency 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 variation
of 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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Au/Kapton ±77.5 0.3 77.7
Al/Au/Kapton ±81.2 0.3 80.8
Meas. Sci. Technol. 27 (2016) 045002

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