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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, 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

Norte, 97310 Mérida, Yucatá n, Mexico

E-mail: ffgamboa@mda.cinvestav.mx

Received 9 November 2015, revised 15 December 2015

Accepted for publication 11 January 2016Published 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)

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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

for remote temperature sensors [15]. A doubly supported metallic

beam configuration can be used for studying the influence of

overlapping length and adhesive joints, to obtain the variation ofthe 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 ofthe 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 steelsheet; 11- Adjustment screw; 12- Vibration sensor; 13- Air valve; 14- Adjustable height crossbar.

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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 alaser 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 

Figure 2 shows a schematic of the excitation and sensing arrange-

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 adequateoperation 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 frommeasurement 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.

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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 beamsfabricated.

Table 1.  Thickness, elastic modulus ( E  ) and density (ρ ) of thebeam 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 sameKapton beam, removed from the grip and placed back.

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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.

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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.

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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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Table 3.  Logarithmic decrement (δ  ) and damping factor (ζ  )corresponding to vibration experiments of Kapton, Au/Kapton andAl/Au/Kapton beams.

Beam   δ ζ 

Kapton 0.0121 0.0019

Au/Kapton 0.0192 0.0030

Al/Au/Kapton 0.0219 0.0034

Table 4.  Measured natural frequency and FEA predictions of thelayered beams.

 f n (Hz)

Beam Measured FEA

Kapton   ±4.6 0.3 74.6

Au/Kapton   ±7.5 0.3 77.7

Al/Au/Kapton   ±81.2 0.3 80.8

Meas. Sci. Technol. 27 (2016) 045002

8/18/2019 A vibrating reed apparatus to measure the natural frequency of multilayer thin films

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