Frequency response function identification using fused filament fabrication-3D-printed embedded ArUco markers


ACTA IMEKO 
ISSN: 2221-870X 
September 2022, Volume 11, Number 3, 1 - 6 

 

ACTA IMEKO | www.imeko.org September 2022 | Volume 11 | Number 3 | 1 

Frequency response function identification using fused 
filament fabrication-3D-printed embedded ArUco markers 

Lorenzo Capponi1, Tommaso Tocci2, Giulio Tribbiani2, Massimiliano Palmieri2, Gianluca Rossi2 

1 Aerospace Department, University of Illinois at Urbana-Champaign, 61801 Urbana, Illinois 
2 Engineering Department, University of Perugia, 06122 Perugia, Italy  

 

 

Section: RESEARCH PAPER  

Keywords: ArUco; marker detection; non-contact measurement; structural dynamics 

Citation: Lorenzo Capponi, Tommaso Tocci, Giulio Tribbiani, Massimiliano Palmieri, Gianluca Rossi, Frequency response function identification using fused 
filament fabrication-3D-printed embedded ArUco markers, Acta IMEKO, vol. 11, no. 3, article 16, September 2022, identifier: IMEKO-ACTA-11 (2022)-03-16 

Section Editor: Francesco Lamonaca, University of Calabria, Italy 

Received July 24, 2022; In final form September 15, 2022; Published September 2022 

Copyright: This is an open-access article distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, 
distribution, and reproduction in any medium, provided the original author and source are credited. 

Corresponding author: Lorenzo Capponi, e-mail: lcapponi@illinois.edu  

 

1. INTRODUCTION 

When a flexible structure is excited at or close to one of its 
natural frequencies, resonance phenomenon occurs [1], [2]. In 
resonance operating conditions, most of the energy is released 
and the response vibration amplitudes significantly increases. 
This process generally leads to an increasing of the vibration 
fatigue damage [3]-[5]. Due to this, the determination of the 
modal components (i.e., natural frequencies, mode shapes and 
damping) is fundamental in any structural dynamics approach, 
either numerical or experimental, in order to avoid potential 
critical conditions [2]. With this perspective, many experimental 
approaches have been developed in years that allows the 
determination of modal components [6]-[8]. The impact 
excitation using modal hammers and the shaker excitation are the 
two most used full-contact experimental approaches [2], [9]. In 
the last years, many image-based analysis have been introduced 
for the displacement measurements [10], [11] and consequently 
for the structural dynamics due to several operating advantages, 
e.g.  high spatial density, full-field information, no sensors to be 
placed on the structure [12]. Javh et al. [13] proved the hybrid 
modal-parameter identification of full-field mode shapes using a 

DSLR camera for responses far above the camera's frame rate, 
employing the Lucas-Kanade Optical Flow algorithm [14]. 
Gorjup et al. [15] researched on the full-field 3D Operating-
Deflection-Shape (ODS) identification using the frequency 
domain triangulation in the visible spectrum. Capponi et al. [16] 
proposed a methodology based on the thermoelastic principle 
for the visual modal strain determination, that allowed the fatigue 
modal damage identification. 

One of the most promising approaches for deformation, 
displacement and motion detection involves markers, either 
physical or virtual [17]-[19]. Virtual markers are often employed 
as they allow tracking objects in subsequent acquired frames 
without introducing physical targets [20], [21]. When virtual 
markers are not available, physical markers are employed [22], 
[23], and among them, the ArUco marker library (ArUco - 
Augmented Reality University of Cordoba) was found to be one 
of the most effective and robust to detection errors and 
occlusion [24]-[26]. Elangovan et al. [27] used them for decoding 
contact forces exerted by adaptive hands, while Sani and 
Karamian [28] and Lebedev et al. [29] employed them for drone 
quadrotor and UAV autonomous navigation and landing, 
respectively. In relation to the use of fiducial markers for 
vibrations measurement, Abdelbarr et al. [30] researched 

ABSTRACT 
The assessment of modal components is a fundamental step in structural dynamics. While experimental investigations are generally 
performed through full-contact techniques, using accelerometers or modal hammers, this research proposes a non-contact Frequency 
Response Function identification measurement technique based on ArUco square fiducial markers displacement detection. A video of 
the phenomenon to be analyzed is acquired, and the displacement is measured through markers, using a dedicated tracking algorithm.  
The proposed method is presented using a harmonically excited fused filament fabrication-3D-printed flexible structure, equipped with 
multiple embedded-printed markers, whose displacement is measured with an industrial camera. Comparison with numerical simulation 
and an established experimental approach is finally provided for the results validation. 

mailto:paul@regtien.net


 

ACTA IMEKO | www.imeko.org September 2022 | Volume 11 | Number 3 | 2 

structural 3D displacement using ArUco markers, while the study 
of Kalybek et al. [31] provides one of the first evidence of the 
capability of optical vibration monitoring systems in modal 
identifications. Recently, Tocci et al. [32] presented an ArUco 
marker-based vibration displacement technique, provided with 
an uncertainty analysis, based on acquisition parameters 
influence investigation.  

In this research, the ArUco markers are employed for the 
determination of the Frequency Response Function (FRF) of a 
flexible structure using image-based analysis. In this study, the 
growing potential of 3D printing is exploited: the tested structure 
is realised in Polylactic Acid using Fused Filament Fabrication 
3D printing methodology. The employed markers are still 
realised using 3D printing methodology and they are generated 
as embedded in the structure during a unique printing job. An 
established experimental FRF assessment technique and a 
numerical model are also provided for the results validation. 

The manuscript is organized as follows. In Sec. 2 the 
theoretical background of structural dynamics and marker 
detection is given. In Sec. 3, the proposed approach is presented 
and in Sec. 4 the experimental campaign and the numerical model 
are described. Sec. 5 gives the results while Sec. 6 draws the 
conclusions. 

2. THEORETICAL BACKGROUND 

2.1. Structural dynamics 

Flexible structures can be represented by N-degrees of 
freedom (DOFs) systems using [8]: 

𝑀 �̈�(𝑡) + 𝐷 �̇�(𝑡) + 𝐾 𝑥(𝑡) = 𝐷 �̇�(𝑡) + 𝐾 𝑦(𝑡), (1) 

where 𝑴, 𝑫 and 𝑲 are the mass, damping and stiffness 
matrices, while 𝒚(𝑡) and 𝒙(𝑡) are the excitation and the response 
displacements of the DOFs, respectively. Assuming a harmonic 

excitation 𝒚(𝑡)  =  𝑌 𝑒 𝑖𝜔𝑡  and a response 𝒙(𝑡)  =  𝑋 𝑒 𝑖𝜔𝑡 , Eq. 
(1) can be written as[8]: 

(−ω2 + i ω 𝐷 𝑀−1 + 𝐾 𝑀−1) 𝑋(ω)
= (i ω 𝐷 𝑀−1 + 𝐾 𝑀−1) 𝑌(ω).  

(2) 

From Eq. (2), the displacement-response amplitude is 
obtained as [8]: 

α(ω) =
𝑋(ω)

𝑌(ω)
=

i ω 𝐷 𝑀−1 + 𝐾 𝑀−1

−ω2 + i ω 𝐷 𝑀−1 + 𝐾 𝑀−1
 (3) 

where 𝜶(𝜔) defines the receptance matrix, that is also know 
as the Frequency-Response Function (FRF) from displacement 

to displacement [2], [8]. Using the eigenvalues notation, 𝜶(𝜔), 
which relates the j-th response to the k-th excitation, can be 
written as [2], [8]: 

α𝑗𝑘 (ω) = ∑ (
𝑅𝑗𝑘𝑟

i 𝜔 − 𝜆𝑟
+

𝑅 ∗𝑗𝑘𝑟

i 𝜔 − 𝜆 ∗𝑟
)

𝑁

𝑟=1

, (4) 

where r is the eigenvalue index (i.e., the mode index), * stands 

for the complex conjugate notation, 𝑅𝑗𝑘𝑟  is the modal constant 

and 𝜆𝑟  is r-th eigenvalue [2]. 
In the experimental modal analysis, several approaches for the 

excitation and the measurement of the structural dynamics can 
be employed [2]. The Frequency Response Function of a system 
can be experimentally determined using FRF estimators [8]. 
When there is no input noise and the output noise is 

uncorrelated, the α̂(ω) is used: 

α̂(ω) =
𝑆𝑦�̂� (ω)

𝑆𝑦�̂� (ω)
 (5) 

where 𝑆𝑦�̂� (ω) is the cross-spectrum between the input 

excitation y and the output response x and 𝑆𝑦�̂� (ω) is the auto-
spectrum of the input excitation y (the ^ symbol denotes the 
estimation from measurements), defined as: 

𝑆𝑦�̂� (𝜔) =
1

𝑇
[𝑌∗̂(𝜔) �̂�(𝜔)], (6) 

𝑆𝑦�̂� (𝜔) =
1

𝑇
[𝑌∗̂(𝜔) �̂�(𝜔)] (7) 

where T is the measurement time length, �̂�(ω) and �̂�(ω) are 
the spectra of the input excitation and of the response, 
respectively. 

For the experimental FRF reconstruction, the modal 
parameters identification is required, and, for this purpose, 
different methods can be used [33], [34]. The preferred 
procedure in experimental modal analysis consists of using the 
Least Square Frequency Domain (LSFD) approach for the modal 

constants identification from the eigenvalues  λ�̂�  obtained 
through the Least Square Complex Frequency (LSCF) and the 
stabilisation chart [35]. 

2.2. ArUco marker detection 

An ArUco marker is a square-marker composed by a wide 
black border, that facilitates its detection in the image, and an 
inner binary-matrix, which determines its identification number 
[24], [25]. An example of ArUco marker is presented in Figure 1. 

The identification of an ArUco marker in a captured frame 
requires several computational steps [32], that, as well as for the 
generation of the marker, are provided from the OpenCV 
Python dedicated library. However, properly developed image-
processing (e.g., filters and thresholds) can facilitate the pattern 
recognition. The marker detection is based on its 4 corners 
identification in each captured frame (see Figure 1). From the 
corners, the spatial coordinates of the centre of the marker 

(𝑥𝑐 , 𝑦𝑐 ) are evaluated frame-by-frame during the acquisition [32]: 

𝐶 = (𝑥c, 𝑦c) = �⃗� ⋅ (
1

4
∑ |𝑥r|

4

r=1

,
1

4
∑ |𝑦r|

4

r=1

) (8) 

where (𝑥𝑟 , 𝑦𝑟 ) are the coordinates of the r-th vertex and �⃗� is the 
calibration factor from pixel units to SI units, defined as the ratio 

between the side length of the physical marker in SI units 𝑑𝑆𝐼  and 

 

Figure 1. Example of an ArUco marker from Original dictionary: corners and 
reference system.  



 

ACTA IMEKO | www.imeko.org September 2022 | Volume 11 | Number 3 | 3 

the average of the four side lengths (in pixels) of the captured 

marker in the FOV 𝑑𝑝𝑥⃗⃗ ⃗⃗ ⃗⃗ ⃗ ∶ 

�⃗� = 𝑑𝑆𝐼  /  𝑑𝑝𝑥⃗⃗ ⃗⃗ ⃗⃗ ⃗ [m/pixels]. (9) 

The calibration factor is evaluated at each new acquired 
frame. In this way, if the marker is subjected to non-planar 
displacements or deformations during the measurement, the 
calibration factor is again estimated. The time-history of the 

centre the marker 𝐶(𝑡) is obtained by tracking it during the 
acquisition. 

3. ARUCO MARKER-BASED FREQUENCY-RESPONSE 
FUNCTION IDENTIFICATION 

With this study, a method for the experimental Frequency 
Response function identification is proposed. As discussed in 

Sec. 2.1, the receptance matrix α̂(ω), estimated from 
experiments, can be determined using Eq. (5). In Eq. (6), the 

spectrum of the excitation input �̂�(ω) and of the structure 

response �̂�(ω) are determined from the ArUco marker centre 
displacement time histories 𝐶𝑦 (𝑡) and 𝐶𝑥 (𝑡) (see Eq. 8): 

𝑌�̂� (𝜔) = ∫ 𝐶𝑦(𝑡)
∞

−∞

  𝑒 −i𝜔𝑡  𝑑𝑡 (10) 

𝑋�̂� (𝜔) = ∫ 𝐶𝑥 (𝑡)
∞

−∞

  𝑒 −i𝜔𝑡  𝑑𝑡 (11) 

Then, the receptance α̂(ω) is estimated using: 

α̂(ω) =
𝑌𝐶

∗̂(ω) 𝑋�̂� (ω)

𝑌𝐶
∗̂(ω) 𝑌�̂� (ω)

 (12) 

Finally, FRFs reconstruction using LSFD and LSCF 
approaches is performed. 

4. EXPERIMENTAL RESEARCH 

4.1. Setup 

In this research, a Y-shaped specimen, shown in Figure 2, was 
used [4], [12]. In particular, this geometry was chosen due to its 
structural dynamic properties. In fact, by using two steel weights 
(each of 360 g), fixed to each of the arms, the structural dynamics 
were adjusted to the research needs. The Y-shaped sample was 
realised in White PLA, using an Ultimaker3 3D printer (100% 

infill and 0.1 mm of layer height). Default values for other 
printing parameters were used. In the printing process, three 8x8 
mm2 ArUco markers were embedded in the last four layers of 
the Y-sample geometry and printed using Black PLA material in 
one printing process (see Figure 2). The sample was mounted on 
an electro-dynamical shaker (Sentek L1024 with PA115 Power 
Amplifier), as shown in Figure 3. On the shaker fixation, a fourth 
ArUco marker (printed in b&w on a standard 80 g/m2 paper in 
8x8 mm2) was rigidly glued for the input excitation measurement. 
The marker detection was performed using a FLIR Backfly S 5 
MP monochrome camera with Sony IMX250 sensor and Fujinon 
12 mm optic mounted. The resolution of the camera was settled 
at 1000 × 850 pixels and the frame rate at 160 fps. The setup 
consists also of a PCB-352C34 accelerometer, bonded on the 
shaker fixation for controlling and measuring the input 
excitation, and of a PCB-352C23/NC, fixed on one Y-sample 
arm for the response measurement. The main specifications of 
the accelerometers used are shown in Table 1.  

For the excitation, a sine-sweep of 0.5 g of constant amplitude 
from 5 Hz to 80 Hz was given to the shaker (close-loop control), 
with a sweep-rate of 16 Oct/min (i.e., approximately 4 sweeps in 
68 seconds). The sweep rate was carefully chosen in order to 
excite the natural frequencies of the sample. However, the 
measurement with the camera was limited at approximately 45 
seconds due to hardware and memory limitations. 

4.2. Data acquisition 

The marker used in this research are from the ArUco Original 
library, identified as shown in Figure 4. In particular, the markers 
with ID1 and ID7 are considered as input reference while the 
markers on the two arms (i.e., ID2 and ID5) as output 
displacement. The displacement of the four markers centre 
point, captured during the experiment and evaluated using Eq. 8 
and Eq. 9, is shown in Figure 5. 

 

Figure 2. Y-shaped specimen with installed sensors.  

Table 1. Technical specifications of accelerometers used. 

Specifications PCB-352C34 PCB-352C23/NC 

Sensitivity 100 mV/g (±10 %) 5 mV/g (±20 %) 

Measurement 
Range 

±490 m/s² pk ±9810 m/s² pk 

Frequency 
Range (±5 %) 

0.5 to 10000 Hz 2 to 10000 Hz 

Resonant 
Frequency 

≥50 kHz ≥70 kHz 

Broadband 
Resolution 

0.0015 m/s2 rms 0.03 m/s2 rms 

Non-Linearity ≥ 1 % ≥ 1 % 

Transverse 
Sensitivity 

≥ 5 % ≥ 5 % 

 

Figure 3. Experimental setup. 



 

ACTA IMEKO | www.imeko.org September 2022 | Volume 11 | Number 3 | 4 

Once the displacement time-histories are obtained, the FRFs 
can be evaluated through Eq. 12, and, finally, reconstructed using 
LSFD and LSCF approaches. Similarly, the reference and the 
response accelerations time-histories are measured during the 
excitation (see Figure 6) and the accelerance FRF is evaluated [2]. 

4.3. Finite Element model 

A finite element model of the Y-shaped specimen is prepared 
using a commercial software. Figure 7 shows the realized 
numerical model. 

The structure has been meshed using solid element with ten 
degree of freedom for node for a total of 89962 elements and 
131580 nodes. To model the external masses attached to the free 
end of the structures, two-point mass, with a mass equal to 0.36 
kg each one, are rigidly connected to the holes on the arms of 
the structure. Moreover, to accurately replicate the experimental 
test, an additional mass of 1000 kg (namely large mass in Figure 7) 
was connected to the constrain zone of the structure. The 
displacement along the Y axis was not constrained, while all the 
other degree of freedom were fixed. In such a way it was possible 
to use the Large Mass Method to evaluate the frequency response 
both in terms of displacement and in terms of acceleration. 

To calculate the numerical frequency response function 
(Shown in Sec. 5 the modal approach was used. For this reason, 
a modal analysis was necessary to obtain the natural frequencies 
of the system and the modal shapes in the point where the 
responses should be addressed. All the frequency response 

function shown in Sec. 5 were obtained considering a percentage 
damping equal to 1% constant for each vibrating mode.   

5. RESULTS 

Four different FRFs are obtained from the combination of 
the two markers as input and the two as output. However, as 
expected, the displacement measured with ID1 and ID7 markers 
is totally comparable and the same consideration can be 
performed for ID2 and ID5 markers, due to geometrical 
considerations. Due to this, for the sake of clarity, only ID1-ID2 
markers FRF will be shown and considered in the further 
discussion.  

From the results in Figure 8, the goodness of the numerical 
model is verified, comparing the FRF obtained performing the 
accelerometer-base experiments. 

 

Figure 4. ArUco markers employed.  

 

Figure 5. Measured displacement of the detected markers. 

 

Figure 6. Reference and response accelerations.  

 

Figure 7. FE model of the Y-shaped specimen. 

 

Figure 8. Experimental and numerical acceleration Frequency Response 
Functions comparison.  



 

ACTA IMEKO | www.imeko.org September 2022 | Volume 11 | Number 3 | 5 

In the considered frequency range, three predominant natural 
frequencies are clearly identified at approximately 17 Hz, 48 Hz 
and 69 Hz from both experiments and numerical model. 

Finally, the comparison between the verified numerical model 
and the proposed approach is performed in terms of 
displacement FRF (see Figure 9). 

In the same considered frequency range, the same natural 
frequencies are identified using the ArUco markers with high 
accuracy. The obtained natural frequencies are presented in 
Table 2. 

A slightly decreasing of the third natural frequency is detected 
with respect to the numerical model. However, the experimental 
approaches give similar results, and this deviation can be 
attributed to the numerical model setting. 

6. CONCLUSIONS 

This study researches the modal components identification 
using a non-contact measurement approach based on ArUco 
marker displacement detection. Even though the established full-
contact methods are widely used for several research 
applications, the required instrumentation is expensive and 
delicate, and the experimental procedures are time consuming 
for a full-field comprehensiveness of the dynamics of a structure. 
On the other hand, the proposed method proved high accuracy 
on the assessment of natural frequencies of a structure with a 
relatively low computational effort and extremely lower budget 
sensors instrumentation: each ArUco marker can be considered 
as a sensor, and if multiple markers are placed and detected in 
the field of view of the camera, more information on the 
dynamics of the structure can be easily provided. Moreover, 
using the 3D printing technology, embedded sensors are 
demonstrated to be effective and reliable. Further employment 
of ArUco markers in structural dynamics will be investigated. 

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Figure 9. Experimental and numerical displacement Frequency Response 
Functions comparison.  

Table 2. Natural frequencies obtained for each technique used. The standard 
deviation on each value is ±1.28 Hz. 

Technique 
1st Mode 

Frequency / Hz 
2nd Mode 

Frequency / Hz 
3rd Mode 

Frequency / Hz 

ArUco markers 17.2 48.9 64.00 

Accelerometers 17.2 49.0 63.9 

Numerical model 17.2 48.9 69.5 

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