Dynacomp2012
CONSIDERING HONEYCOMB SANDWICH BEAM DESIGN
PARAMETER VARIABILITY FROM VIBRATION MEASUREMENT
DATA: A STOCHASTIC APPROACH
Stijn Debruyne†, Dirk Vandepitte†, Loujaine Mehrez†, Eric Debrabandere*
† Department of Mechanical Engineering
Katholieke Universiteit Leuven (KUL)
Celestijnenlaan 300b, 3001 Heverlee, Belgium
e-mail: stijn.debruyne@khbo.be,
dirk.vandepitte@mech.kuleuven.be,loujaine.mehrez@mech.kuleuven.be
*
Department of Industrial Science
Katholieke Hogeschool Brugge Oostende
Zeedijk 101, Oostende, Belgium
e-mail: eric.debrabandere@khbo.be
Key words: Thermoplastic honeycomb beams, experimental modal analysis, finite element
modelling, design parameter variability, Random Fields.
Abstract:
Honeycomb panels are sandwich structures with a high specific strength and stiffness,
together with a low areal mass. They are complex but regular structures with a high number
of design parameters that govern their dynamic behaviour.
Firstly this paper gives some insight into the physics of the glass fibre weave reinforced
thermoplastic honeycomb beams used in this study. The resonance behaviour of these freely
suspended beams is governed by a whole set of design parameters, each having some kind
of physical randomness. In this study two independent elastic design parameters are
considered as random fields, the Young’s modulus of the homogenized skin material and the
out-of-plane shear modulus of the homogenized core material. The goal of this study is to
estimate the probability density functions of these parameters from the variability on the
experimentally determined resonance behaviour of a set of 22 virtually identical honeycomb
beams. A suitable FE model is constructed and updated by means of the results of the
experimental modal analysis.This process is thoroughly outlined. The obtained values for the
considered design variables from the FEM updating process are then considered as
stochastic random fields. The process of estimating both aleatory and epistemic uncertainty
of these random fields is discussed in depth.
1. Introduction
Honeycomb panels are geometric complex but regular structures. Such panels consist of a
honeycomb core that is bonded to thin face sheets. The structure of a typical panel is shown
in fig. 1. Their dynamic behavior, e.g. resonance frequencies and mode shapes, is
influenced by a high number of parameters related to the beam geometry and the elastic
properties of the materials used. Variability on the parameters that govern a beam’s dynamic
Stijn Debruyne, Dirk Vandepitte, Loujaine Mehrez and Eric Debrabandere.
behavior results in a variability in its dynamic structural behavior. In case a set of samples is
used as in this study, one can consider inter-sample variability and spatial variability within
one sample.
The elastic mechanical properties of a typical honeycomb core are described and
analytically calculated by Gibson & Ashby [1]. They propose formulas for calculation of the
in-plane and out-of-plane elastic moduli and Poisson ratios of the core. The main work on the
dynamics of sandwich panels is related to conventional foam-core structures. Little work has
been carried out on honeycomb panels. Nilsson & Nilsson [2] tried to analytically predict
natural frequencies of a honeycomb sandwich plate with free boundary conditions using
Blevins [3] formula in which areal mass and equivalent bending stiffness are frequency
dependent. Another, more practical way to predict natural frequencies and mode shapes of a
honeycomb panel is by means of Finite Element analysis. In the past years, different new
approaches have been developed which incorporate high order shear deformation of the
core. Work in this area has been carried out by Topdar [4] and Qunli Liu [5][6][7]. The latter
stated that the shear moduli of the core are important factors in the determination of the
values of the natural frequencies and the sequence of mode shapes, especially at high
frequencies. At low frequencies natural frequencies are mostly determined by the bending
stiffness of the panel. For the Monopan beams the governing parameters are treated in
section 1.
skin
honeycomb core
Figure 1: a typical honeycomb sandwich panel
The use of vibration measurement data for the identification of elastic material properties is
treated by Louwagie [8]. This work discusses in detail how Young’s moduli, shear moduli and
Poisson ratio’s of laminated materials can be obtained from modal data as resonance
frequencies and mode shapes.
Describing variability can be done in various ways. However, in any statistical analysis the
issue of gathering and obtaining enough data is essential. In this study a sample population
of 22 is used as a basis for an extensive statistical analysis. A survey of uncertainty
treatment in Finite Element Analysis is given by Moens [9]. The focus of the work presented
in this article is to make optimal use of the statistical information available from the limitedsize experimental data. Processes, vibration behavior in specific in this study, can be
regarded as stochastic processes [10] of which the outcome is governed by a set of
stochastic random variables. In this area a recently developed approach in analyzing
variability is describing the quantity or process of interest as a stochastic Random Field (RF).
The random field theory is extensively studied by Ghanem [11][12]. Soize [13], Desceliers
[14], Arnst [15] and Perrin [16] implemented and adapted this theory for inverse problems
and for cases where only limited experimental data is available. Mehrez [17,18] adopted this
method to describe the variability of the Young’s modulus of composite beams from
experimentally determined frequency response functions, thus by also solving an inverse
problem.
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Stijn Debruyne, Dirk Vandepitte, Loujaine Mehrez and Eric Debrabandere.
A first part in this paper describes how resonance frequencies and mode shapes of
honeycomb sandwich beams with free boundary conditions are determined experimentally.
These honeycomb beams are made from Monopan panels. Panels of this type have a
cylindrical honeycomb core from Polypropylene (PP). The core is welded to the
Polypropylene skin by means of a welding foil. The skin consists of a symmetric glass fibre
woven fabric (Twintex) with a Polypropylene matrix and a theoretical thickness of
approximately 0.7 mm. To smoothen the outer surfaces of the panel, a polypropylene
finishing foil is welded there. A set of 22 virtually identical Monopan beams with dimensions:
850 x 50 x 25 mm are used for this study. The problem of measurement uncertainty and
modal parameter estimation uncertainty is also addressed.
In the second part it is outlined how the measured resonance frequencies and mode shapes
are processed through FEM updating to create a database of values for two important design
parameters of the considered honeycomb beams, the Young’s modulus of the skin in length
direction (Es) and the out-of-plane shear modulus of the honeycomb core (Gc). These
parameters determine the bending stiffness of a beam section.
The third part explains the general principles of the Random Field methodology and its
adoption in this study. Using this method leads to an estimation of the real probability density
functions for the two studied parameters at the measurement locations, along with
confidence intervals (CI) of the estimations. The obtained results are given and extensively
discussed. The confidence intervals represent the epistemic uncertainty which arises from
the lack of sufficient statistical data.
The fourth section explains how the uncertainty from the EMA and FEM processes are
eliminated from the total variability estimated by the RF representation of the chosen design
parameters. This results in estimated PDF’s that describe the purely physically related
design parameter variability, which is the issue of interest in this research.
The fifth part discusses the physical relevance of the obtained results. The relation between
the estimated probability density functions and the plausible real physical variability of the
two design variables is discussed in detail. The so called aleatory uncertainty includes all
physically related variability; this is discussed in detail.
2. Experimental Modal Analysis
Each beam of a set of 22 is meshed to obtain 17 evenly spaced measurement points. The
beams are elastically suspended to attain free boundary conditions. The excitation is done
with an impulse hammer at all 17 locations; the beam’s vibrations are captured with a light
weight accelerometer of 0.4 grams. The experimental set up is shown in figure 2. From the
measured Frequency Response Functions (FRF) [19] modal parameters such as resonance
frequencies, damping ratios and mode shapes are estimated using Test.lab from LMS. As
mentioned earlier natural frequencies and mode shapes are the quantities of interest here. In
the range of 100 to 1800 Hz 8 modes are captured and used for further analysis. Table 1
gives an overview of the obtained resonance frequencies for all beams.
Experimentally determined Frequency Response Functions are subject to different kinds of
variability. First of all there is measurement uncertainty inherent to the measurement method
and the hardware used (transducers etc.). Secondly the process of modal parameter
estimation is subject to some variability as the estimation algorithms used have a finite
precision. The third variability is related to the real physical variability of the different beam
design parameters that govern a beam’s mass and bending stiffness.
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Stijn Debruyne, Dirk Vandepitte, Loujaine Mehrez and Eric Debrabandere.
The frequency variability due to the measurement itself can be caused by the transducers
and the data acquisition system. Performing a series of 30 calibrations yield a normalized
random error on the amplitude of less than 0.007 % for both acceleration and force
transducers and no traceable frequency error. For all measurements the FFT analyzer was
set at a frequency resolution of 0.25 Hz giving a maximum frequency error of 0.22 % for the
first mode at approximately 115 Hz.
The measurement amplitude error on experimentally determined FRF’s can be estimated
from analyzing the coherence function [20]. A coherence value less than 1 indicates a non
perfect correlation between the measured acceleration and the input force signal. A poor
coherence can be caused by uncorrelated noise in the signal measurement, system nonlinearities, leakage in the signal analysis, … Expression 1 calculates the normalized random
error  r ( H )   / H on the amplitude of the transfer function H as a function of the coherence
value  2 and the number of averages Na for one FRF measurement.
 r (H ) 
1  2
2 N a 2
(1)
In this study the number of averages is set at 20. For the first resonance frequency the worst
coherence value is 0.96, resulting in a random normalized error of 0.07 %. This error
maximizes to 0.12 % for mode 6. A Monte Carlo (MC) routine is used to give this error
estimation its random character.
The modal parameters of interest are estimated from the measured Frequency Response
Functions (FRF) by means of the Poly-reference Least Squares Frequency-domain algorithm
(p-LSCF) or better known as the PolyMAX algorithm [21]. This algorithm fits a right matrix
fraction model on the measured FRFs in a least squares way. The algorithm produces a
stabilization chart from which modal parameters such as resonance frequency values and
damping ratios can be read. The measured Frequency Response Functions are then
reconstructed using the estimated resonance frequencies and damping ratios. For each
measured FRF a global amplitude error is calculated between the measured and synthesized
function that can easily be translated to an estimated mode shape variability. Resonance
frequency variability is estimated from a sensitivity analysis were the sensitivity of each
estimation algorithm parameter is studied.
The two significant sources of variability in the EMA process are studied for the first 6
resonance frequencies and mode shapes. Thus for each beam, resonance frequency and
mode shape variability is quantified. These results are further on related to the variability of
the two design parameters considered in this study.
Figure 2: Measurement set up
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Stijn Debruyne, Dirk Vandepitte, Loujaine Mehrez and Eric Debrabandere.
Beam \ mode number
1
2
3
4
5
6
7
8
1
113.5 287.5 502.5 730.5
975
1225 1474 1715
2
119.5
520.5 757.5
1009
1260 1519 1758
3
118
296.5 517.5 750.5
1000
1248 1504 1747
299
4
116.5 293.5
988.5
1237 1484 1720
5
115.5
291
510.5 740.5
512
744
984
1235 1488 1719
6
119
299
523.5
757
1002
1252 1508 1744
7
117.5
297
522
760
1005
1260 1522 1769
8
117
295
519.5
761
1009
1263 1522 1768
9
120
10
115.5
292
511
744
987.5
1239 1488 1728
11
118.5
298
522.5
766
1015
1268 1524 1763
12
117
295.5
518
759.5
1008
1265 1520 1767
13
117.5
297
522.5
763
1013
1265 1524 1771
14
118
297
522
765
1011
1268 1524 1765
15
118
298
1024
1278 1539 1788
16
118.5
298
524
767.5
1013
1269 1534 1765
17
118
298
525.5
771
1024
1277 1535 1778
119.5 300.5 524.5
767
1014
1271 1531 1783
18
301.5 528.5 768.5 1018.5 1270 1528 1776
524.5 770.5
19
118
297
762.5
1012
1264 1521 1763
20
119
299
523.5 763.5
1013
1260 1522 1761
21
119
299
523.5
763
1012
1260 1519 1761
524
761
1009
1264 1523 1766
22
119.5 300.5
522
Table 1: Resonance frequencies from experimental modal analysis (EMA), expressed in
Hz.
3. Finite Element Modeling
The honeycomb beams are modeled using the shell-volume-shell (SVS) principle. The
honeycomb core is modeled as an homogeneous orthotropic material using volume
elements; the face sheets are also modeled as orthotropic materials but using shell
elements. Figure 3 shows the FE model used. After a convergence study the mesh size was
set at 100 x 6 x 3 elements. The bending stiffness and beam mass are the main factors that
determine the dynamic behavior of the beams. The bending stiffness of the beams is mainly
governed by the Young’s modulus of the skin Es (in length direction) and the out-of-plane
core shear modulus Gc . With the model updating process in mind the FE model is divided
into 17 zones, each of them centered around the corresponding measurement point. Each
zone in the FE model has a constant value for Es and Gc. Initial values for these parameters
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Stijn Debruyne, Dirk Vandepitte, Loujaine Mehrez and Eric Debrabandere.
were determined experimentally by performing three point bending tests on each beam
sample according to Fan [22]. Ten modes are calculated using free boundary conditions.
Measured Frequency Response Functions are first processed to obtain resonance
frequencies and mode shapes.
Figure 3: FE model of the honeycomb beams.
These are used for updating the FE model. For this process three parameters, Es , Gc and ρc
(core density), are used. At this stage of the research these parameters are assumed to be
independent which is true for the FE model but may not exactly hold for the physical beam.
Each FE model is thus updated using 51 updating parameters. Figure 4 shows the
histograms of the obtained Es and Gc values after model updating for measurement point 5.
For the updating process the first 6 resonance frequencies and corresponding mode shapes
are used. The target for the model updating process is a minimization of the differences
between EMA and FE resonance frequencies and an optimization of the MAC – values of
corresponding EMA and FE mode shapes. This is done in a least-squares sense. Table 2
gives the differences between EMA and FE resonance frequencies and mode shapes after
FE model updating. Frequency error is very low but mode shape error is as high as 10 % for
the sixth mode. this indicates that the FE model is good for frequency calculation but less
adequate for mode shape calculation. Even if the sensitivity of Es and Gc to mode shapes is
rather little this modeling error should be taken into account for further analysis. Standard
Monte Carlo method [23] is used to give the FE modeling error its random character.
frequency error (%) mode shape error (%)
0.13
1.02
0.29
2.23
0.22
2.76
0.09
4.70
0.31
9.67
0.25
8.06
Table 2: Comparison between EMA and FE resonance frequencies and mode shapes
after FE model updating
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Stijn Debruyne, Dirk Vandepitte, Loujaine Mehrez and Eric Debrabandere.
4. Describing design parameters as random fields
The quantities of interest in this study exert a certain variability, shown by the histograms in
figure 6. The dynamic behavior of the considered honeycomb beams can be regarded as a
stochastic process [24] governed by a set of stochastic variables, Es and Gc in this case.
These stochastic variables cannot be determined exactly, they can only be estimated
statistically.
Due to the limited size of the available experimental data (only 22 beams) it is difficult to
perform an accurate estimate of the distribution type of Es and Gc and of the epistemic
uncertainty involved. Advanced techniques should thus be applied to adequately estimate
the true variance of the two obtained mixed experimental/numerical databases. The purpose
of using the method described in this section is twofold. The first concern is to estimate the
true probability density distributions of the two considered databases and to exclude all
variability that is not directly related to the real Es and Gc variability. The second purpose is to
estimate the epistemic uncertainty caused by the limited statistical data available in this
study. Recently special numerical techniques are developed to deal with these kind of
problems. The two parameters of interest are modeled as independent Random Fields. This
section outlines in depth how this is done and what the results are.
The random field that describes a physical parameter or process is denoted by F. In this
research measurements are very time consuming and samples are expensive, hence the
limited amount of statistical data. In such a case the true covariance matrix can be
estimated from the available experimental database, with dimensions M  N exp , of
discrete values of the random field . The covariance matrix of a set of discrete values,
which is the experimental data, can be spectrally decomposed using the Karhunen –
Loève (KL) series expansion.
From the covariance matrix the eigenvalues with their corresponding eigenvectors can be
calculated, leading to a corresponding eigenvalue system. When F denotes the
~
~
discretised random field and F is the vector containing the N exp mean values of F , thus
for every physical measurement point. The discretised random field F can be written as a
KL series expansion in terms of the eigenvalues and eigenvectors of the covariance
matrix C F~  and a finite set of random variables  (k ) with zero mean, generally a non
Gaussian distribution and most often statistically uncorrelated. The discretised random
field F can be expressed following a KL series expansion as,
F  F~ 
N exp

k 1
(k )
k  k
(2)
µ
Modeling the random field F comes down to determining the statistics of the random vector
 µ   (k ) which has a zero mean and the µ x µ identity matrix as covariance matrix. The
 
joint density of  µ however is not known and has to estimated from the available random field
realizations (experimental data). This estimation process uses a Hermite PC expansion with
Bayesian Inference [17,18]. The random variables  (k ) are expressed as multi-dimensional
Hermite polynomials. Methodologies based on the Maximum Likelihood principle and the use
of Bayesian Inference algorithm have been proposed to estimate the PC coefficients. A
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Stijn Debruyne, Dirk Vandepitte, Loujaine Mehrez and Eric Debrabandere.
Random Walk Metropolis Hastings Markov Chain Monte Carlo algorithm has been adopted
here for the purpose.
In this study the two design parameters of interest are considered as two independent
Random Fields. A third order Polynomial Chaos expansion is used and after a convergence
study the number of terms in the KL – series expansion was set at 9. In case of the Es
database the largest 9 eigenvalues of the covariance matrix cover over 96 % of the available
statistical information. As mentioned before it is important to know the influence of having
only limited statistical data. Bayesian Inference is used to estimate the 95 % confidence
intervals (CI)of the estimated PDF’s; this is a representation of the epistemic uncertainty. For
both design parameters figure 4 shows the estimated PDF’s for measurement point 5
together with its 95 % confidence interval. The latter are narrow indicating that the use of 22
beam samples is adequate for statistical analysis. However the obtained probability density
functions of Es and Gc include all types of variability from EMA, FEM and the real variability of
these two parameters. The next section focuses on estimating the real variability of these
parameters.
Figure 4: Estimated PDF’s and epistemic uncertainty: 95 % confidence intervals for
Es and Gc for measurement point 5.
5. Estimating real variability of the considered parameters Es and Gc
In previous section the probability density functions of the two parameters Es and Gc ware
estimated using the Random Field methodology. In this section the variability part from FEM
and EMA is excluded from this in order to end up with the real physical variability of the
considered parameters.
A first issue is to translate the observed FEM and EMA variability to a extra variability on Es
and Gc. There for the sensitivity matrices of the FE model are used. For all 17 measurement
points and for all 22 test beams the resulting variability on both considered model parameters
is calculated. As the standard Monte Carlo method was used to give the FEM and EMA
variability a random character the resulting Es and Gc variability is normally distributed. Due
to the overall low variability on experimental and numerical results the translated Es and Gc
variability is very small and its influence on the eventual estimated PDF’s of these
parameters will be negligible.
The results from the elimination process are the estimated variability of Es and Gc that is
related to physical variability of the honeycomb design variables. Since the variability due to
FEM and EMA uncertainty is not correlated to the physical variability of any honeycomb
beam design parameter the PDF related to FEM and EMA can simply be subtracted from its
respective PDF. The finally obtained probability density functions representing the physical
design parameter variability for measurement point 5 are shown in figure 5. Results for all
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Stijn Debruyne, Dirk Vandepitte, Loujaine Mehrez and Eric Debrabandere.
measurement points are similar. The next section studies the relation between the
considered parameter variability of the numerical beam model on the one hand and the
variability of the underlying physical parameters that determine Es and Gc on the other hand.
Figure 5: Physically related probability density functions for Es and Gc compared to the total
variability, measurement point 5.
6. Relating the obtained variability of Es and Gc to variability of real honeycomb
skin and core parameters
In this section the obtained variability for Es and Gc is related to the variability of
corresponding physical parameters of the skin and core of the honeycomb sandwich beams.
Since the considered honeycomb beams are 1 – dimensional structures the skin is modeled
as an isotropic material. The real skin of the tested honeycomb beams is made of glass fibre
weave reinforced Polypropylene. Its Young’s modulus is there for governed by a number of
design parameters of this composite layer such as the glass fibre volume fraction (vf), the
glass fibre orientation angle and the crimp of the reinforcing weave. The latter was
determined experimentally for measurement point 5. For each beam the orientation angle
was measured at 10 spots around measurement point 5. From these measurements it is
clear that these small fibre angle variations do not account for the experienced variability of
Es. The fibre volume fraction vf is unlikely to vary large among the different measurement
intervals as its correlation length is much smaller than this interval length by approximately a
factor 7. Another parameter that governs the skin’s elasticity modulus is the crimp of the
glass fibre weave. A higher crimp causes a decrease of the elasticity modulus.
To validate the physical relevance of the obtained Es variability different series of static
tensile tests were performed each on 25 skin samples with dimensions: 200 x 30 mm. The
experienced normalized standard deviation from this tests varies between 8.5 and 11 %.
Assuming that Es variability decreases linearly with increasing sample width the equivalent Es
variability from the tensile tests should lie in the 4.25 – 5.5 % range. This is of the same order
as the estimated variability of the Es database.
There are several physical design variables of the core that influence its out-of-plane shear
stiffness. The estimated variability of the out-of-plane shear modulus Gc can thus be traced
back to the underlying dimensional parameters that govern the shear stiffness. These are the
cell wall thickness, cell diameter and the core height, the stacking sequence of the cylinders
and the adhesion between them. FE simulated shear tests were carried out to study the
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Stijn Debruyne, Dirk Vandepitte, Loujaine Mehrez and Eric Debrabandere.
sensitivity of Gc with respect to the related dimensional core parameters mentioned. Table 2
relates the normalized relative sensitivity coefficients from the FE model to the expected
relative variability of Gc by using experimental values for core height, cell wall thickness and
cell diameter determined through measurements on 330 core cells.
During the production process of the honeycomb beams the Polypropylene core cylinders
are somewhat welded together. Local changes in core cylinder adhesion affect the out-ofplane core shear stiffness. To have an estimation of the real Gc variability dynamic shear
tests were carried out on 38 samples with dimensions 50 x 50 mm, the same size as a
measurement interval. A mean value of 30.02 MPa with a normalized random error of 17.8 %
was obtained. FE calculations were carried out to study the effect of core cylinder adhesion.
In case of a perfect adhesion of all core cylinders in the unit cell a shear modulus of 43.4
MPa is obtained while in case of zero adhesion of the cylinder walls the shear modulus
becomes 20.2 MPa. Taking the mean value from experiments into account one can conclude
that physically there is a partial adhesion between the cylinder walls as can be expected.
Although it is impossible to determine Es and Gc values of the considered honeycomb beams
related experiments on other samples and the use of simulated tests give significant insight
in the relation between the experienced variability from the stochastic analysis and the
physical variability of the real skin and core design parameters.
7. General conclusions
In this study two important elastic parameters of honeycomb sandwich beams are treated as
stochastic variables governing the resonant behavior of the beams, representing the process
of interest. Experimental resonant vibration data of a limited number of test beams is used as
an input database for Finite Element model updating. The database of Es and Gc values
resulting from this process is further analyzed using the Random Field methodology to obtain
optimal estimations of the variability of the two design parameters in the form of Probability
Density Functions. Applying this method yields an estimation of a confidence interval for
each estimated PDF, representing the epistemic uncertainty.
From a study of the contribution of EMA and FEM variability to the Es and Gc variability one
concludes that this contribution is minimal as long as the quality of the EMA process is very
good. If the measurement quality is not excellent this reflects in high errors on measured
mode shapes which in that case makes their use for Model Updating somewhat doubtful.
A study was made to relate the estimated Probability Density Functions to plausible
variability of physical honeycomb design parameters of core and skin. By performing related
tests and by simulating shear tests these relations were studied successfully.
Although it is impossible to determine the variability of the two considered elastic parameters
in a direct deterministic way it is shown that it is possible to perform adequate estimates
using stochastic methods.
Further research will focus on the application on honeycomb sandwich panels rather than on
beams.
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Stijn Debruyne, Dirk Vandepitte, Loujaine Mehrez and Eric Debrabandere.
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