Engineering Structures xxx (xxxx) xxxx
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Engineering Structures
journal homepage: www.elsevier.com/locate/engstruct
Experimental and numerical vibration analysis of plates with curvilinear
sub-stiffeners
Luca Praticòa, Joel Galosb, , Enrico Cestinoa, Giacomo Frullaa, Pier Marzoccab
⁎
a
b
Department of Mechanical and Aerospace Engineering (DIMEAS), Politecnico di Torino, Torino, Italy
Department of Aerospace Engineering and Aviation, School of Engineering, RMIT University, Melbourne, Australia
ARTICLE INFO
ABSTRACT
Keywords:
Vibration
Finite element
Modal analysis
Laser Doppler vibrometry (LDV)
Curvilinear stiffener
Curvilinear stiffened panels are being developed for aerospace structures and other applications. This paper
presents an experimental and numerical study into the vibration response of three curvilinear stiffened square
plates clamped at the edges. The experimental modal analysis was performed using laser Doppler vibrometry
(LDV) and an impact hammer test. Two of the three plates was constructed with curvilinear stiffener geometry,
while the third was a straight stiffened plate with the stiffeners oriented at an angle to the edge of the plate. Even
though the natural frequencies of the plates are similar, the different stiffening patterns was sufficient in providing unique directional properties and therefore distinct mode shapes. The numerical modal analysis of the
plates was performed using finite element analysis (FEA). A comparison of the experimental and numerical
results was carried out in terms of natural frequencies and mode shapes, using relative differences and modal
assurance criterion (MAC), respectively. The experimental and numerical results were in good agreement for all
the three plates. The difference between experimental and numerical natural frequencies was typically less than
5% and the diagonal MAC values typically ranged from 0.8 to 1. This is in line with previously published results
in the literature. The reasons for differences between the experimental and numerical results, and the practical
significance of the findings, are also discussed.
1. Introduction
Innovative manufacturing technologies now allow for the broadening of the design space to new possibilities in structural tailoring.
Many of these possibilities were not viable from a manufacturing point
of view up to a few years ago. The advent of these technologies facilitated the shift to a cost-oriented monolithic design philosophy of the
flight vehicle. Baron et al. [1,2], demonstrated that the need for exploiting the constitutive mechanical characteristics of composite
structures lead to the avoidance of the traditional design and manufacturing approach, which involves assembling large numbers of mechanically fastened parts. Larger integrated components can be tailored
to maximize both producibility and performance efficiency, while
helping to ensure airframe affordability by reducing machining and
assembly costs. Such kind of integrated components are commonly
referred to as unitized structures. The main advantages of unitized
structures compared to classically assembled ones are described by
Chan et al. [3].
Curvilinear stiffening members, which are an example of unitized
structures, were first hypothesized by Renton et al. [4]. Love [5] have
⁎
shown that for curved beams of arbitrary shape, both axial and torsional deformations are coupled with bending. Often the aim of using
curvilinear stiffening elements is the bending-torsion coupling. This
may be exploited for controlling both natural frequencies and mode
shapes [6].
The study of curvilinear stiffening elements in aeronautical unitized
structures was pioneered by Kapania et al. [7] as an enhancement in the
design space of stiffened panels. In [7] it is observed that, in an optimization framework with buckling constraint, curvilinear-stiffened
plates may lead to valid design solutions. Dang et al. [8] performed an
optimization on damaged unitized structures, demonstrating the effectiveness of curvilinear-stiffened panels in terms of load carrying
capacity. Tamijani and Kapania [9,10] investigated both vibration and
buckling behavior of curvilinear-stiffened plates with mesh-free
methods. In a separate study, Tamijani et al. [11] provided experimental validation of the numerical model.
More recently, coupling effects induced by rectilinear/curvilinear
stiffened structures have been investigated in the framework of aeroelastic optimization in order to postpone critical conditions in high
aspect ratio wings [12–19].
Corresponding author.
E-mail address: joel.galos@rmit.edu.au (J. Galos).
https://doi.org/10.1016/j.engstruct.2019.109956
Received 5 August 2019; Received in revised form 17 November 2019; Accepted 17 November 2019
0141-0296/ © 2019 Elsevier Ltd. All rights reserved.
Please cite this article as: Luca Praticò, et al., Engineering Structures, https://doi.org/10.1016/j.engstruct.2019.109956
Engineering Structures xxx (xxxx) xxxx
L. Praticò, et al.
Structural tailoring of metallic panels through curvilinear stiffeners
was further investigated by Bhatia et al. [20]. They performed an optimization framework for stiffener placement, which avoids the multiminima solution of global gradient-based optimization methods and
saves CPU resources. Kobayashi et al. [21] have shown how curvilinear
internal structures result from a topology optimization using a cellulardivision approach. These types of structures have been shown to be
beneficial for multidisciplinary optimization of air vehicles [22]. Kolonay et al. [23] applied the cellular-division optimization approach to
aircraft lifting surfaces with success. Several types of passive aeroelastic
tailoring, such as curvilinear ribs, spars and stringers, functionally
graded metals and tow steered composites, are examined in Jutte’s
work [24].
Despite the possibilities offered by composite materials, their production cost is often relatively high in early prototyping phases, if
compared to the available manufacturing processes for metal components. Additive manufacturing was considered to produce the test
pieces in the present study, but eventually discarded since it was difficult to ensure material isotropy and homogeneity. This is mainly due
to the directionality inherent to this manufacturing process [25,26]. On
the other hand, CNC machining has proven to be economically feasible
also in the preliminary characterization stages. Low-cost high-performance Aluminum alloy panels, with a non-uniform stiffening pattern,
are considered viable tailored components for future aerospace vehicles
[27,28].
Bushnell and Rankin [29] demonstrated that including small stiffening elements between the conventional primary stiffeners could lead
to an increased buckling resistance. Such small stiffening elements are
referred to as sub-stiffeners when the stiffeners’ height is of the same
order of magnitude of the skin’s thickness. Murphy and Quinn [30]
presented a concept for highly loaded panels with curved sub-stiffeners.
In addition to enhanced buckling resistance performance, substiffened
panels were also proven to have enhanced damage tolerance capabilities, when compared to simply stiffened ones [31–33]. Quinn et al.
[34] inspected the introduction of sub-stiffeners to increase structural
efficiency of integrally machined Aluminum alloy stiffened panels,
obtaining a plate buckling performance gain of 89% over an equivalent
mass at a sub-component level.
Although the dynamics of curvilinear stiffened panel have been
studied experimentally and numerically [10,11], the study of the dynamic behavior of substiffened plates has been greatly neglected, with
no study published to-date on the vibration of curvilinear sub stiffeners
clamped-on-all-edges. The aim of this paper is to investigate the dynamic behavior of curvilinear sub-stiffened panels through experimental and a numerical modal analysis. This yields a complete description of the vibration response and a reliable numerical model for
the test pieces examined here, therefore paving the way for future
studies on the potential for dynamic tailoring of unitized structures.
The rest of the paper is organized as follows. In Section 2, the research methodology is presented with particular attention to the geometry and material of the test pieces here examined. Details about the
carried out experimental and numerical modal analyses are here provided. The results are reported and commented in Section 3. Finally, in
Section 4, the conclusions are drawn and a brief summary of the outcomes is given.
Fig. 1. Plate 1 – geometry and dimensions in mm.
(x ) =
1 +
2
1
b
x
(1)
where b is the plate length along the x-axis, x represents the abscissa
coordinate and (x ) is the local orientation of the stiffener with respect
to the abscissa coordinate. This law was proposed by Gürdal in [35], for
the study of variable stiffness composite panels. Eq. (1) allows the
calculation of the stiffeners path using two parameters, 1 and 2 , the
local orientations of the stiffener at its extremities measured with respect to the x-axis, respectively. The stiffening pattern is achieved by
simple translation of the initial coordinate of the stiffener along the yaxis. It is worth noting that the stiffener curvature and the distance
between the stiffeners varies along the stiffener path and is different for
each plate.
Three stiffening patterns were chosen to produce two different
curvilinear cases (plate 1 and plate 2) and a third linearly stiffened case
(plate 3) as a basis for comparison. The choice of these patterns was
selectively made in order to study how the directional properties can
change for different stiffener curvature values. The mean and maximum
sub-stiffener radius of curvature are reported in Table 1 for each test
piece. Note that the optimization of the sub-stiffeners design is beyond
the scope of the present study.
The global geometry of the specimens was selected to be representative of a real-scale aeronautical component. All three test plates
were manufactured from 5083-O Aluminum (Table 2). This is a generalpurpose Aluminum-magnesium alloy. The measured weight of the test
pieces is reported in Table 3.
The design of the frame arises from the need of providing a reliable
fixed boundary condition for the plate/stiffeners edges and constraining
the plates during the manufacturing process to meet the precision requirements.
2. Research methodology
2.1. Materials and test plates
This study relates to square symmetrically substiffened plates
clamped on all four edges. The test pieces include sub-stiffened plates
surrounded by thick frames fabricated from a single-piece Aluminum
plate through CNC machining. The geometrical features of the three
plates are shown in Figs. 1–3. The stiffener path follows the law:
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Table 1
Sub-stiffeners’ radii of curvature.
Plate no.
Mean Radius of Curvature
[mm]
Maximum Radius of Curvature
[mm]
1
2
3
1451.5
867.4
–
1613.5
889.5
–
Table 2
Al 5083-O mechanical and physical properties.
Property
Value
Young’s Modulus, E (GPa)
Poisson’s ratio,
Mass density, (kg/m3)
72.0
0.33
2,660
Table 3
Plate mass.
Plate no.
Mass [Kg]
1
2
3
2.681
2.617
2.653
support allows bolting the plates on top of it and its tabs permit the
support-plate assembly to be fixed on the test table. Steel was chosen as
a material for the piece to be significantly stiffer compared to the plates.
The volume was adjusted in the design phase to provide a weight of at
least three times the one of the plates. The side tabs were welded to the
main body. Fig. 4 shows the steel frames geometry and dimensions. The
total weight of the frame is measured to be 6.8 kg. The plates were
bolted on top of the support with sixteen M10 bolts. To establish equal
constraints on each edge of the plate, a torque wrench was used for the
measurement of the fastening torque.
Fig. 2. Plate 2 – geometry and dimensions in mm.
2.2. Experimental modal analysis
Two kind of tests were performed. The first was a hammer test using
an impact hammer and accelerometers. The second test was performed
using laser Doppler vibrometry for vibration measurement.
2.2.1. Hammer test
The experimental setup used for the hammer test is shown in Fig. 5.
The experimental parameters used for the acquisitions with the hammer
test technique are listed in Table 4. A steel hammer tip was used to
obtain the widest frequency span possible, ensuring the excitation of
the largest number of modes for each test piece. The acquisition time
was chosen to capture the full response of the plates. Due to the stochastic nature of this kind of test, a high number of averages-permeasurement is necessary. Averaging allows for random signal components canceling and overall noise reduction. The hammer test was
performed using LMS TestLab software and a LMS SCADAS frontend
[36].
Fig. 5 shows the hammer and the sensors are connected to the
SCADAS frontend. This permits the optimal signal conditioning and the
storage of data in the form of frequency response functions (FRFs). The
SCADAS is then connected to a laptop with the LMS TestLab software,
for data collection and analysis. The modal model and mode shape
derivation was carried out in LMS TestLab using the PolyMax and
Maximum Likelihood Modal Model (MLMM) algorithms.
Twenty-five equally spaced measurement points were distributed on
the plate’s skin as shown in Fig. 6. Identical measurement point locations were used for all three plates. Each plate was constrained by
bolting it on top of the steel support presented in the previous section.
Fig. 3. Plate 3 – geometry and dimensions in mm.
To ensure the fixing of the plate itself during the modal testing
phase, while allowing for the transverse vibration, a steel support was
designed and manufactured to match the plate geometry. The steel
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Fig. 4. Steel support - geometry and dimensions in mm.
Subsequently the steel frame was fixed to the test table. Each test was
carried out roving the twelve available PCB Piezotronics accelerometers
[37] to cover all the measurement’s locations. The excitation of each
test piece was provided by means of a hammer strike towards the
bottom left corner of each plate, as shown in Fig. 6. This provides a nonsymmetrical excitation that triggers the largest number of modes possible, avoiding structural nodes. Since the energy at the spectrum’s
highest frequencies are too low for a proper excitation delivery, the
analyzed bandwidth was limited from 0 to 1.7 kHz. Hence, the following analysis pertains to the first four bending modes of each plate.
The accelerometers were attached to each test piece with Petro
Mounting Wax. To avoid measurement distortions from the unequal
mass distribution due to the accelerometer mass, small iron weights
with mass approximately equal to a single accelerometer, were attached
in all the free measurement locations.
The excitation is enforced through a periodic chirp signal. This type of
signal permits the setting of a continuous sweep from a minimum to a
maximum frequency, providing the excitation of every frequency inbetween. The experimental parameters used for the acquisitions with
the LDV test technique are listed in Table 4. The span was set to 2 kHz
to be sufficiently wide to capture the first four bending modes of vibration. Being the excitation signal periodic and controlled in amplitude, a low number of averages was considered sufficient. The acquisition time matches the sweep duration of the chirp signal. All the test
pieces were placed at a distance of 0.7 m from the scanning laser head.
The sensitivity was set to 2 mm/s/V and a 20 kHz low pass filter was
used to remove high frequency noise from the signal. The experimental
setup for LDV test is shown in Fig. 7.
The input signal is controlled from the PSV Software and generated
inside the control box. Once amplified, the signal is sent to the electrodynamic shaker. The connection between the shaker and the test
piece is rigid. The shaker head is attached with hot glue in the same
location where the excitation was applied during the hammer test.
Through the load cell, the force applied to the test piece is measured
2.2.2. Laser Doppler vibrometry (LDV)
The LDV test was performed using an electrodynamic shaker and a
Polytec PSV-400 scanning laser head with OFV-5000 control box [38].
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Fig. 5. Schematic of hammer test experimental setup.
2.3. Numerical modal analysis
Table 4
Experimental parameters used in experimental testing.
Property
Hammer test Parameters
LDV test Parameters
Span (Hz)
Spectral lines
Frequency resolution (Hz)
Acquisition time (s)
Averages per measurement
3200
8192
0.39
2.56
10
2000
6400
0.3125
3.2
3
The numerical modal analysis was performed using the Finite
Element (FE) method. All simulations were preformed using MSC
Patran/Nastran® software package. Nastran® SOL103 was used for the
natural frequencies and mode shapes retrieval.
2.3.1. Finite element models description
The FE models mirror the geometry features of the three test pieces.
The same CAD files used for CNC manufacturing were imported in MSC
Patran® and meshed through its auto-mesh algorithm. Standard
CTETRA solid elements with four vertex nodes and six midsize nodes
were used [39]. The material properties assigned to each element are
the ones reported in Table 2. To validate the numerical model, a
comparison with results available from the literature was performed. In
particular, the plate with curvilinear stiffeners presented in [11] was
taken as a benchmark. The plate was meshed with the previously described method while retaining the original geometry, material and
boundary conditions. A convergence test was carried out refining the
mesh size. The results in terms of natural frequencies are reported in
Table 5. Following the same mesh-refinement criteria, a convergence
test was performed using both free-free and fixed boundary conditions
and the results are reported in Table 6. The comparison with the results
reported in [11] demonstrates that the present formulation is in good
agreement with both Meshfree and Ritz Method and it can therefore be
considered validated. Furthermore, it was shown that an acceptable
convergence has been achieved through mesh refinement. An average
and the measurement signal is sent back to the control box. The vibration of the test piece is measured from the scanning laser head and
sent to the control box as well. Both input and output signals are
measured in the time domain. Once they reach the control box, the FFT
of both is computed and the resulting FRF sent to the PC for data visualization.
LDV measurements are difficult on polished Aluminum surfaces due
to poor return light. Black tape was attached to each plate’s top surface
to provide the maximum amount of light to be scattered back into the
instrument. About 400 measurement points, equally distributed on the
surface of the test piece were used in LDV testing. LDV provides a
contactless measurement of vibrations, avoiding mass loading problems. The boundary conditions implemented in the present test were
identical to those used in the hammer test, with the plate and steel
support frame assembly fixed vertically. This allowed the positioning of
the shaker behind the plate, thus leaving the front surface of the plate
free for the laser measurement.
Fig. 6. Accelerometer measurement grids
for (a) plate 1, (b) plate 2 and (c) plate 3,
used in the hammer test. The hammer impact location is indicated in blue (point 21)
and the measurement points are indicated in
red. Note that the impact location corresponds with a measurement location. (For
interpretation of the references to color in
this figure legend, the reader is referred to
the web version of this article.)
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Fig. 7. Schematic of LDV experimental setup.
element size of 4 mm was chosen for all three models. The number of
nodes and degrees of freedom for the three models is reported in
Table 7.
Table 6
Convergence study – Plate 1′s first natural frequency [Hz]. * denotes rigid body
modes excluded.
Plate 1 – free-free*
2.3.2. Boundary conditions
Given the experimental boundary conditions, the test plate frame
areas between the bolted joints are technically free to move in the
transverse direction, but they are not free to move towards the steel
support. Modelling limitations make the definition of a directional
boundary condition difficult, especially because of the linearity inherent to Nastran® SOL103. Moreover, it is unclear to which extent the
bolts fix the plate to the steel support or vibrate with it. Finally, linearity constraints inherent to the numerical modal analysis algorithm
allow just for the exploitation of glued contacts (no friction modelled),
hence limiting the modelling possibilities. Because of the above listed
modelling limitations, the modelled boundary conditions were chosen
as follows. No steel support and bolts were modelled, reducing the
analysis to a single component i.e. the test plate. Since the test plate
holes are not threaded, the only ensured contact area between the
washer/bolt and the plates is the one on the circular edges of the
support frame holes. Hence, the boundary conditions of the numerical
model are enforced by constraining all six degrees of freedom on the
circular edges of the frame’s holes.
Average element size [mm]
Natural Frequency [Hz]
8
204.07
6
203.56
4
202.53
Plate 1 – fixed
Average element size [mm]
Natural Frequency [Hz]
8
572.20
6
570.73
4
566.66
Table 7
Characteristics of the FE models of the curvilinear stiffened plates.
Plate no.
No. of nodes
No. of DOFs
1
2
3
212,599
204,999
298,753
1,275,594
1,229,994
1,792,518
rel. diff . % =
| num
(
exp |
num + exp
2
)
·100
(2)
where
indicates the natural circular frequency, the subscript num
indicates the FE results and the subscript exp indicates the experimental
results.
MAC is an indicator ranging from zero to one, used to determine the
similarity of two mode shapes. It is the normalized dot product of two
complex modal vectors at each common node:
2.3.3. Correlation method
The correlation between the experimental and numerical results is
enforced on a double basis: the comparison of natural frequencies and
mode shapes is made using relative differences and Modal Assurance
Criterion (MAC), respectively.
The relative difference for the natural frequencies yields to:
Table 5
Numerical procedure validation – Natural frequencies [Hz] for a curvilinear-stiffened plate as reported in [11].
Mode
1
2
3
4
5
6
Meshfree (Tamijiani et al. [11])
13.34
29.80
79.96
108.85
111.05
194.43
Ritz Method (Tamijiani et al. [11])
Experimental (Tamijiani et al. [11])
13.18
29.68
80.34
109.81
111.17
192.50
8.98
26.91
70.81
103.34
–
178.15
6
FEA
Average Element size [mm].
8
6
4
13.06
29.52
79.88
108.35
109.73
194.73
13.06
29.51
79.88
108.35
109.69
194.67
12.97
29.41
79.50
108.04
109.32
194.13
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Table 8
Natural frequencies and mode shapes of the first four bending modes of the curvilinear
stiffened plates.
MACij =
shapes retrieved in experimental and numerical tests found using Eq.
(3) are reported in Figs. 10–12.
Table 8 shows that the natural frequencies of the three plates are
very similar. This is reasonable considering that, as indicated in
Figs. 1–3, the test pieces have the same dimensions, are made of the
same material and have similar weights. However, it can be noticed
that the obtained mode shapes are significantly different. This is attributed to the different stiffening patterns used in each plate. The
waves of each mode shape are oriented accordingly to the stiffener local
orientation along the width of the plate. This is especially evident in
higher order mode shapes.
Plates 1 and 3 have similar sub-stiffeners orientation and their mode
shapes follow the stiffening pattern orientation. This is valid for the first
four modes but becomes more evident in higher order modes. As shown
in Table 2, additional finite element modelling was performed to assess
how the curvature of the sub-stiffeners affects the free vibration characteristics of the plates. The results show that mode shapes are similar
(i) T (j ) |2
| num
exp
(i) T (i ) )( (j) T
( num
num
exp
(j )
exp )
(3)
where (i) and (j) are i-th and j-th mode eigenvectors respectively, the
subscript num indicates the FE results and the subscript exp indicates
the experimental results.
3. Results and discussion
Experimental and numerical modal analysis results are shown in
Tables 8 and 9. The FRFs measured during the experimental tests, the
ones retrieved through numerical finite element modelling and the ones
synthetized in PolyMax as modal models for the impact test are shown
in Fig. 8. The relative differences of the natural frequencies of the plates
for the experimental setups and the numerical model, found using Eq.
(2), are reported in Fig. 9. The MAC matrices comparing the mode
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Table 9
Higher order mode shapes of plates 1 and 3 determined numerically.
Fig. 9. Comparison of the relative differences in natural frequencies found via
experimental and numerical techniques for (a) plate 1, (b) plate 2 and (c) plate
3.
Additional tests were carried out at a later stage to measure the
plate frame vibrations. Some accelerometers were placed on the fixed
frame and an additional hammer test was performed hitting the plates
in the same location as in the previous tests. It was observed that some
of the measured modes are induced by the experimental boundary
conditions. The frame-induced modes were identified by superimposing
the FRFs measured on the middle plate and on the plate frame for each
test piece. The overlapping peaks are considered being due to the
contribution of the frame only. Considering that the experimental
boundary conditions effect is not modelled in the FE model, the modes
of the support frame are excluded in the modal model retrieval without
Fig. 8. Experimental and numerical FRFs for (a) plate 1, (b) plate 2 and (c)
plate 3.
up to Mode 8. After Mode 8, the local effect of the sub-stiffener curvature becomes dominant, differentiating the shapes of Plate 1 and
Plate 3. An exception lies in symmetrical modes (e.g. Mode 11 through
13), involving larger contributions from the middle plate rather than
from the sub-stiffeners.
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Fig. 10. MAC matrices for the comparison of mode shapes for plate 1. (a) Comparison of hammer and LDV tests, (c) hammer test and FE, and (c) LDV test and FE.
loss of accuracy in the analysis. Fig. 8 shows how the synthetized modal
model were retrieved without considering the frame-induced modes.
In Fig. 9 the natural frequencies retrieved from the two experimental setups present a relative difference less than 10% and typically
around the 5%. This trend agrees with previously published results in
the literature [11]. Mode one and four are generally well approximated
by the FE model. Slight discrepancy in modes two and three is due to
the choice of the excitation point location during the experiments. This
choice is the result of a trade-off analysis carried out in order to best
excite the largest number of modes, specifically avoiding structural
nodes. Moreover, modes that are non-symmetrical with respect to the
horizontal and vertical axes of the test plate, are generally more affected by the boundary conditions. This is reasonable because the fixing
on all four sides of the test plate is exactly what permits the exploitation
of the directional properties conferred by the stiffening patterns and
differentiates the currently examined plates from common non-stiffened isotropic ones.
As reported from Figs. 10–12, understandable differences appear in
mode shapes retrieved from the two experimental setups. This is mainly
due to some physical differences in between the two, including: different sensors used, different excitation methods, and possible mass
loading effect. Nevertheless, the MAC matrices indicate good agreement
in between the retrieved mode shapes. The diagonal elements of MAC
matrices mostly range from 0.8 to 1 when comparing both the experimental setups with the numerical FE model. Out-of-diagonal elements
appear to be mostly lower than 0.05 with few exceptions, never exceeding 0.1.
The partial lack of accuracy in the correlation can be justified making
the following observations. The clamped boundary conditions modelled
are just an approximation of the ones employed in the experimental
phase. The difference in stiffness in between the experimental fixture and
the modelled one may be largely accountable for the discrepancies observed. FE analyses carried out in the present work linearize the behavior
of the examined components. Non-linearities due to friction and contacts
with the experimental boundary conditions are not modelled. No
damping is modelled in the present FEA. Different mode shapes may
affect differently the boundary conditions’ influence on the dynamic
behavior of the component, for example providing enhanced strain in
highly constrained areas. The interaction of the experimental boundary
conditions with the test object may lead to the appearance of modes that
are not due to the dynamics of the components. This may be particularly
true for components/boundary conditions combinations that develop
compounded directional properties. The trade-off in choosing the excitation point is key for the triggering of bending modes. Some modes
Fig. 11. MAC matrices for the comparison of mode shapes for plate 2. (a) Comparison of hammer and LDV tests, (c) hammer test and FE, and (c) LDV test and FE.
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Fig. 12. MAC matrices for the comparison of mode shapes for plate 3. (a) Comparison of hammer and LDV tests, (c) hammer test and FE, and (c) LDV test and FE.
may always be less excited than others, thus leading to a less accurate
measurement. Different type of test induces not only different boundary
conditions, but also different structural modifications inherent in the
measurement technique (i.e. stinger or accelerometers attachment
against impulse excitation and contactless measurement).
The components examined in the present study might be used in the
aeronautical field in order to tailor the dynamic, buckling and damage
tolerance properties of highly stressed areas of the wing box. Moreover,
they might be employed as core layer for newly designed sandwich
panels. Particularly, in the case of structures with relatively complex
geometries such as the present ones, non-linearities due to damping and
geometrical properties may arise, making the outcome of experimental
tests different from FE predictions at times. Enhancing the reliability of
the numerical model through correlation with experimental results,
opens the possibility for use in more complex load cases and paves the
way for future enquiries.
CRediT authorship contribution statement
4. Conclusions
Declaration of Conflicting Interests
The present study concerns an experimental and numerical study of
the vibration response of two curvilinear stiffened and one straight
stiffened square plates clamped at the edges. The three plates are made
of Aluminum 5083-O and manufactured by CNC machining in the spirit
of unitized components. Since the height of the stiffeners is within the
same order of magnitude of the skin’s thickness, the present plates can
be labelled as sub stiffened. Experimental modal analysis was performed using laser Doppler vibrometry (LDV) and an impact hammer
test. The different stiffening patterns adopted are demonstrated to be
sufficient in providing unique directional properties and distinct mode
shapes for each of the plates here analyzed. Such components allow for
easy tailoring of directional properties to meet design specifications in
highly loaded panels and might find application in particularly stressed
areas of the wing box. Moreover, they might be employed as core layer
in purposely-designed sandwich panels. The development of a reliable
numerical model allows for further studies of more complex load cases.
The comparison of experimental results with the FE outcomes was
carried out in terms of natural frequencies and mode shapes, using
relative differences and a modal assurance criterion (MAC). The difference margin between experimental and numerical natural frequencies is typically underneath the 5% threshold with few exceptions.
An isolated case of maximum relative difference amounting to 9% is
retrieved for plate 1 third mode. Diagonal MAC values are mostly
ranging from 0.8 to 1, with few exceptions. Hence, the results here
presented validate the examined numerical model for the forecast of the
dynamic behavior of the three test pieces.
The author(s) declared no potential conflicts of interest with respect
to the research, authorship, and/or publication of this article.
Luca Praticò: Data curation, Visualization, Methodology, Writing original draft. Joel Galos: Methodology, Visualization. Enrico Cestino:
Supervision, Conceptualization, Writing - review & editing. Giacomo
Frulla: Supervision, Conceptualization, Writing - review & editing. Pier
Marzocca: Supervision, Conceptualization, Writing - review & editing.
Acknowledgements
The authors acknowledge the support of Julian Bradler and Huw
James at RMIT University Bundoora Campus in the experimental setup
preparation. The authors are also grateful for the assistance of Albert
Maberley and Paul Spithill at RMIT Advanced Manufacturing Precinct
in specimen manufacturing. Special thanks go to Dr. Francesco Danzi
for his support during the configuration’s choice and the preliminary
stages of the present study.
Appendix A. Supplementary material
Supplementary data to this article can be found online at https://
doi.org/10.1016/j.engstruct.2019.109956.
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