0037549717738068

Simulation
Applications
Harmonic stress analysis on Coffea
arábica L. var. Colombia fruits in order
to stimulate the selective detachment:
A finite element analysis
Simulation: Transactions of the Society for
Modeling and Simulation International
1–12
Ó The Author(s) 2017
DOI: 10.1177/0037549717738068
journals.sagepub.com/home/sim
Hector A Tinoco and Fabio M Peña
Abstract
This study shows a finite element harmonic stress analysis to evaluate the stress performance at the pedicel interface of
a fruit–peduncle system of Coffea arábica L. var. Colombia plants. The aim was to study detachment of fruits subjected to
mechanical vibrations. A model of the coffee fruit–peduncle system is designed computationally to reproduce its topology in all ripening stages using a proposed numerical procedure. Young’s modulus, Poisson ratio, and density were
adjusted using analytical models for all ripening stages. The glomerulus of three fruits in different ripening stages was proposed and these were combined in four groups. Based on a detachment model, it was verified which fruit was detached
first when a chirp force signal was applied on each glomerulus. Results indicated that dynamic excitations applied in
between 130 to 150 Hz detached only ripe fruits, since fruits that were in other ripening stages were not stimulated until
detachment in that bandwidth.
Keywords
Harmonic analysis, coffee fruits, selective harvesting, detachment of fruits, finite element analysis, Coffea arábica, finite
element method
1. Introduction
Colombian coffee production impacts the economy of
approximately 563,000 families located in 12 states of
Colombia.1 Colombian coffee is recognized internationally
for the quality associated with the aroma and taste.2 To
achieve these levels of quality, ripe coffee fruits are
selected and harvested manually; the process only allows
2.5% of unripe fruits. The ideal conditions of the coffee
crops in the Colombian mountain are given between 1200
and 1800 m.a.s.l. (meters above sea level) to 19 and
21.5°C grades of temperature. In Colombia, coffee harvesting is one of the most complex and expensive stages
of the coffee production since topographical conditions are
not ideal, and the selectivity of ripe fruits and the spatial
distribution of the crop present a challenge to use machinery in-situ. Therefore, there is a great interest to study
diverse ways of harvesting ripe coffee fruits in a selective
way without using robust machinery.
In recent decades, mechanical harvesting of coffee
fruits has played an important role 3–4 in the production
chain of coffee. Especially, countries like Brazil have
taken advantage with respect to harvesting since this process has been essential to augment operational capacity.5
Several technologies that use vibrations of low frequency
(2–30 Hz) for harvesting fruits have been implemented
with success in apples, peaches, cherries and olives.6–8
These have overtaken limitations as slope and topography
conditions, spatial distribution of the crop and relation
between weight and detachment force of the fruits.
However, the use of these technologies depends on exposure factors. In Colombia, coffee crops are found on
mountainous terrains and therefore, this aspect presents a
difficulty to apply the mechanization by vibrations with
the success seen in Brazil, where the terrain is flat.5
To detach a fruit from a tree, the detachment force at
the interface between the fruit and the tree should exceed
a limit value. This value is achieved by means of two techniques, direct forces in the fruit and inertial motions done
Experimental and Computational Mechanics Laboratoy, Universidad
Autónoma de Manizales (UAM), Manizales-Caldas, Colombia.
Corresponding author:
Hector A Tinoco, Experimental and Computational Mechanics Laboratoy,
Universidad Autónoma de Manizales (UAM), Antigua Estación del
Ferrocarril, Edificio Sacatin C.P. 170001, Manizales-Caldas, Colombia.
Email: [email protected]
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Simulation: Transactions of the Society for Modeling and Simulation International 00(0)
by the fruit.9 There are different ways to detach a fruit; we
can impose bending, torsion, axial, and shear forces.10 In
this study, we focused on axial forces since experimental
measurements are given for coffee fruits Coffea arábica L.
var. Colombia.
Different kinds of equipment based on vibrations have
been designed and investigated for coffee harvesting
applying motions on the trunk and branches.11–13
However, the results showed a low performance in the
selectivity of ripe fruits due to the fact that frequency
ranges were established experimentally. From a mechanical point of view, it is necessary to know the natural frequencies that characterize the system in order to excite
only ripe fruits in the frequency spectrum and posteriorly
to know the amount of energy necessary for the detachment. Therefore, harvesting coffee fruits by mechanical
vibrations requires a knowledge of the physical parameters
involved in the detachment, including the variation in the
ripening stages. We can mention those parameters related
to natural frequencies (fruits, branches, trunk or whole
tree). The knowledge of mechanical parameters of any
subsystem of the coffee tree will permit us obtain a higher
performance in the detachment of fruits as well as to
diminish the impact of the motions over the tree (e.g.,
defoliation by vibrations and detachment of unripe fruits).
In order to describe the dynamic behavior of any subsystem in the coffee the mechanical properties of these should
be known (i.e., fruit–peduncle). With this focus, it is more
feasible to describe the fruit–peduncle system since this
presents small topological differences compared with the
geometry of the tree.11,14,15 Recently, Tinoco et al.15 carried out a modal analysis on the fruit–peduncle system to
classify the natural frequencies in different ripening stages.
The study determined the vibration modes and natural frequencies, called selective frequencies. These showed significant differences between the ripening stages evidencing
the frequency intervals of ripe fruits that can be isolated
from unripe fruits.
The aim of this study is to evaluate the detachment of
coffee fruits from the harmonic stresses generated at the
interfaces of the pedicel–fruit and pedicel–peduncle by
means of a harmonic finite element analysis. For this purpose, the fruit topology (computer-aided design models) is
generated by using the methodology reported by Tinoco
et al.15 Different values for the Young’s moduli are proposed using semi-experimental data of elasticity obtained
for the different substructures (fruit–pedicel–peduncle).15
For the finite element analysis, the glomerulus of three
fruits in different ripening stages are constructed and combined in four groups. A detachment model is used to verify
which fruit could be detached firstly when a chirp force
signal is applied to each group in the frequency interval 0–
400 Hz.
Figure 1. Simplified geometry of coffee fruit.
2. Geometric model for the fruit Coffea
arábica L. var. Colombia
In order to represent the fruit geometry of Coffea arábica
L. var. Colombia during its growth from anthesis until
ripening, let us assume that the fruit is oriented as shown
in Figure 1. The fruit presents three directions that define
its topology, which are represented by the unitary vectors
e1 , e2 , and e3 . Such that e1 and e2 are the equatorial directions, knowing that f1 and f2 are the equatorial diameters,
such that f1 5 f2 and f3 is the polar diameter. Here, the
diameters of the fruit will be called orthogonal diameters.
The directions e1 , e2 , and e3 coincide with coordinate system x, y, and z which are orthogonal by definition. It is
important to mention that the geometric parameters vary in
function of days after the anthesis (DAAs) or days of
ripening. Since, the orthogonal diameters change with the
time of ripening as mentioned by Carvajal-Herrera et al.16
The hyperplanes established by e1 e2 , e3 e1 and
e2 e3 are symmetry in the fruit, as evidenced in Figure 1.
Let us consider that one-eighth of the fruit is limited by
the set of functions fi (t), 8i = 1, 2, 3 at the planes defined
by ek ej , 8kj = 12, 23, 31, as observed in Figure 2(a).
The setfi (t) is obtained from Bézier cubic functions constructed from the points of control P1 ek ej , P2 ek ej , P3 ek ej ,
and P4 ek ej established in each plane ek ej (see
Figure 2(b)). Functions that describe the form of each
eighth are delimited within the following set of parametric
equations:
fi (t) = ðXi (t), Yi (t)Þ 2 R2 jXi (t)
ð1Þ
= ai T t, Yi (t) = bi T t, 8t 2 (0, 1)g
where ai T = ½ a1i a2i a3i a4i , bi T = ½ b1i b2i b3i b4i T
and t= t3 t2 t 1 . Each eighth of the fruit is
Tinoco and Peña
3
Figure 2. (a) Boundaries of one-eighth of coffee fruit. (b) Plane e1 e2 of the eighth with dimensions and points of control.
(c) Digital image to approximate the topology of the coffee fruit (case: unripe 220).
symmetric with respect to symmetry planes. The vectors
ai and bi are defined from the formulation of Bézier functions by the following expressions:
ai = Mpxi
ð2Þ
bi = Mpyi
ð3Þ
where px = f P1xi P2xi P3xi P4xi gT and pyi =
T
P1yi P2yi P3yi P4yi
are the coordinates of the
(see
points of control, and Pn ek ej = Pnxi Pnyi
Figure 2b). For Equations (2) and (3), the matrix M is constant and it is determined from cubic Bézier curves17 as:
2
1
6 3
M=6
4 1
1
3 3
6 3
3
0
0
0
3
1
07
7
05
0
The coordinates of the control pointsP1 ek ej and P4 ek ej in
a principal plane correspond to fj 2 and fk =2, e.g. in
Figure 2(b) the principal plane is e1 e2 . The value of
coordinates for the points P2 ek ej and P3 ek ej in directions
ej and ek is known only partially due to that the projections of coordinates (on ej and ek ) that segment the fruit
are unknown (see Figure 2(c)).
In the study carried out by Tinoco et al.,15 six parameters were established lk , 8k = 1, 2, 3, 4, 5, 6 that define
the shape of the coffee fruit in the three principal planes.
Each coordinate lk represents the control variables that
describe the parametric functions of the geometry. The
parameters lk are projections on radii fi =2, 8i =
1, 2, 3, which should be approximated in the hyperplanes
ek ej , 8kj = 12, 23, 31 for different stages of ripening
(e.g., unripe, semi-ripe and ripe). The topology of the fruit
is completely represented byfi (t) and each function is
applied in each principal plane ek ej , . The approximation
process of the parametric functions fi (t) was carried out by
means of a technique of digital photography in which the
pictures were scaled into a grid as illustrated Figure 2(c).
This procedure was mentioned by Tinoco et al.15 For each
ripening stage, the values of lk were determined from the
pictures of the fruits. The parameters li , 8i = 3, 4, 5, 6 were
obtained as fix with the aim to optimize the value of l1
(plane e1 e2 ) from the experimental data of volume.
Lineal relations between parameters li and orthogonal diameters were established for nine ripening states. These estimations of li , 8i = 3, 4, 5, 6 are the same for all ripeness
stages and these are described in the following form:
f1
f
l2 = 0:60
, l3 = 0:65 1 ,
2
2
f3
f3
f
, l5 = 0:75
and l6 = 0:75 2
l4 = 0:65
2
2
2
ð4Þ
For the determination of parameters, the fruits were selected
by means of a random sampling in the nine ripening stages.
As a final procedure, the parameter l1 was determined from
minimization of the fruit volume. As an initial value, an
inequality was established as 0:5(f1 =2) 4 l1 4 (f1 =2) and
the variable l1 is converted in a minimization parameter.
The optimal value of l1 was computed from a computeraided design (CAD) model done in SolidWorksÒ from
Dassault Systems. Comparisons were performed with experimental volumes determined from measurements of mass
obtained by Carvajal-Herrera et al.16 which were converted
to volume data with density reported by Ciro et al.18 The
values of l1 are called ideal values for the volume and these
are shown in Table 1. These values can be interpolated with
the aim to find any parametric value of l1 for any ripening
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Simulation: Transactions of the Society for Modeling and Simulation International 00(0)
Figure 3. Geometric models of the fruit–peduncle structure designed in SolidWorksÒ. (a) Unripe 196. (b) Unripe 210.
(c) Semi-ripe 224. (d) Ripe 231. (e) Overripe 238. Image taken from Tinoco et al, Elsevier, copyright 2014.15
Table 1. Minimized values of l1 ,15 dimensions mm. Table taken
from Tinoco et al., Elsevier, copyright 2014.15
Days
Minimized (l1 )
182
189
196
203
210
217
224
231
238
3.2803
3.7054
3.4806
3.9570
3.8466
4.0999
4.7572
4.7807
4.2757
3. Physical-mechanical properties of the
fruit–peduncle system of Coffea arábica
L. var. Colombia
In this section, we are going to show different approximations for mechanical properties of the fruit and the
pedicel–peduncle system. For this purpose, an analytical
model is proposed to represent the behavior of all properties. The mathematical model that will be applied to all
experimental values obtained for the coffee subsystems in
other researches is called Boltzmann Sigmoidal Model
(BSE) represented by the following equation:
time td . The polynomial l1 describes with the other parameters l1 + i topology of fruit in its ripening process and it
is written as follows:
y=
A1 A2
+ A2
1 + e(xx0 )=dx
ð6Þ
where A1 , A2 , x0 , and dxare constants to be determined.
ð5Þ
3.1. Density, Poisson ratio, and Young’s modulus of a
Coffea arábica L. var. Colombia fruit
Equations established for lk are used to describe the
complete geometry of fruit Coffea arábica L. var.
Colombia, if the values of fi , 8i = 1, 2, 3 and td are
known. When the parameters of lk are determined, each
function fi (t) is described in continuous form in each main
plane and then CAD software can be used for modeling
the geometry of the coffee fruit. In our study, the coffee
fruits were modeled in SolidWorksÒ and these are
depicted in Figure 3. The fruit models are the same
obtained in the study carried out by Tinoco et al.15 The
presented models will be used in the finite element analysis proposed in this study, posteriorly.
It is important to mention that the procedures shown in
this section were replicated from the study carried out by
Tinoco et al.15
Previous studies have shown an interest for characterizing
the coffee fruit and its subsystems. In the study done by
Ciro et al.18 the average density of coffee fruit Coffea arábica L. var. Colombia was measured by calculating the
buoyancy force when the fruit was immersed in distilled
water. In conclusion, it was determined that the fruit density was constant
in all ripening stages and was approxi
mately 1.07g cm3 .
Table 2 presents BSE model parameters for Young’s
modulus of coffee fruit Coffea arábica L. var. Colombia as
a function of ripening time. A semi-empirical Young’s modulus was determined by Tinoco et al.15 with a numerical
iterative procedure using the firmness indices. These values
found for Young’s moduli satisfy the firmness indices
(K1 ,K2 and K3 ) of the fruit at the orthogonal directions as
demonstrated by Tinoco et al.15 The estimations were
l1 = 3e6 td 4 + 0:0021td 3 0:664td 2
+ 91:224td 4684:9, 8td 2 (182, 238) days
Tinoco and Peña
5
Table 2. Young’s modulus fitted for the semi-empirical data
approximated by FEM taken with permission from Tinoco
(2017), Taylor and Francis, copyright 2017.19
Table 3. BSE calculated parameters for density, Poisson’s ratio
and Young’s modulus for the pedicel–peduncle Coffea arábica L.
var. Colombia structure.
Ef
A1
A2
x0
dx
Chi-square
R2
Prob. > F
Analysis of variance
18.844
5.654
220.423
1.834
0.758
0.98
A1
A2
x0
dx
Densityrf
Poissonnf
Young’s
modulus Ef
1.01
1.5
22.05
0.28
0.34542
0.3181
21.95411
0.28
3.939
2.26
22.05
0.28
BSE: Boltzmann Sigmoidal Model
5:22 × 107
FEM: Finite Element Method.
ripe
Ef [Mpa]
semi-ripe
Approx. BSE
unripe
Experimental-FEM approx.
td [days]
Figure 4. Approximated elastic modulus from FEM and BSE
model taken with permission from Tinoco (2017), Taylor and
Francis, copyright 2017.19
BSE: Boltzman Sigmoidal Model; FEM: Finite Element Method.
obtained with convergence errors of the firmness indices less
than 5%, this requirement was included in the proposed
algorithm by them. The determined values are depicted in
Figure 4 with red markers. Values of coefficients for analytical model adjusted for Young’s modulus described in
Equation (5) are listed in Table 2. A graphical representation
of the analytical model can be seen in Figure 4 as a black
line. The statistical approximation indicates that R2 is 0.98
with a probability F far below 5%, therefore the estimated
BSE model of Young’s modulus present great correlation
with the experimental and semi-empirical (Ef ) data.
During the ripening process, Young’s modulus is a
property which varies during the fruit growth, as evidenced in Figure 5. Starting from the day 200, small
changes in the Young’s modulus are manifested until the
day 210. From this day, the rate of change of the elastic
property is evident such that the fruit loses 62.80% of its
initial Young’s modulus to day 224. Subsequently, in the
overripe stage, day 238, the fruit loses approximately an
additional 8% of its elastic capacity. In mechanical terms,
the loss of elastic capacity means that the fruit has higher
possibility of changing its shape if a low pressure is
applied internally. As a result, the strains in the fruit
increase easily. However, inside the fruit, the biophysical
phenomena are more complex as explained by Hall et al.20
However, these can be seen from a macro scale as consequences of biological complexities.
For Coffea arábica L. var. Colombia fruit, the absolute
value of Poisson’s ratio was not determined. However,
Poisson’s ratio value is irrelevant for this study since the
fruit–peduncle system acts under pendular dynamics. This
means that fruit is not subjected to considerable deformation amplitudes when these are compared with deformations of the structure pedicel–peduncle.
3.2. Density, Poisson ratio and Young’s modulus of
the pedicel–peduncle structure
In similar way as described above, the BSE was used in
order to approximate the experimental data from Coelho
et al.5 for density, Poisson ratio and Young’s modulus of
pedicel–peduncle of coffee Coffea arábica L. var.
Colombia structures, as shown in Figure 5. Table 3 shows
the calculated parameters of BSE for each property.
4. Finite element model harmonic
analysis of different groups of fruits
In this section, we will study the stresses generated at the
pedicel–fruit and pedicel–peduncle interfaces at different
stages of coffee fruit ripening. The aim of the study is to
analyze those stresses generated by a dynamic stimulus on
a three- fruit glomerulus by means of a harmonic force
(see Figure 7). A total of four glomeruli of three fruits each
were proposed and seven ripening fruit stages were
defined. Placing of the fruits was randomly assigned, as
can be observed in Figure 7. For the simulations, a computer SONY VAIO PC (M350 2.27 GHz i3 CPU, 8 GB
RAM) with Windows 7 operating system was used to perform the finite element analysis in ANSYSÒ Workbench
16 software. There were necessary CAD models of each
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Simulation: Transactions of the Society for Modeling and Simulation International 00(0)
Poisson ratio
Experimental data (Coehlo et al. 2015)
Fitted Boltzman Model
Experimental data (Coehlo et al. 2015)
Fitted Boltzman Model
190
200
210
[Days]
220
240
230
190
180
200
210
220
230
240
[Days]
MPa
180
Experimental data (Coehlo et al. 2015)
Fitted Boltzman Model
180
190
200
210
[Days]
220
230
240
Figure 5. Experimental and BSE approximate models for density, Poisson ratio and Young’s modulus of pedicel–peduncle of Coffea
arábica L. var. Colombia structures as a function of ripening days.
BSE: Boltzmann Sigmoidal Model
fruit in the following ripening stages: unripe 182, unripe
189, unripe 196 (Figure 3(a)), unripe 210 (Figure 3(b)),
semi-ripe 224 (Figure 3(c)), ripe 231 (Figure 3(d)), and
overripe 238 (Figure 3(e)). For each CAD model, the geometry of the pedicel was assumed as being cylindrical (see
Figure 3) with dimensions reported by Coelho et al.5 for
Coffea arábica. Geometric models of the glomerulus were
generated with SolidWorksÒ software with the methodology explained in Section 2, which was proposed by
Tinoco et al.15 Elastic properties of the fruit and peduncle–
pedicel were assigned using BSE approximations for each
ripening stage, as described in Section 3. Kinematical considerations for the fruit–peduncle model are prescribed by
the following boundary conditions: (a) clamped at the origin O (Figure 6(a)), and (b) peduncle–pedicel and pedicel–
fruit are considered bonded interfaces. Fruit–pedicel and
pedicel–peduncle interfaces are also bonded, without relative displacement among them. Unstructured meshes are
used for simulation of each structure and tetrahedral finite
elements are used, as shown in Figure 6(a).
It is important to denote that in reality, fruits can be in
different configurations as bonded and grouped. But, it
does not mean that the results can change since when we
analyze mechanical systems in frequency, there are
dependencies on three parameters (mass, stiffness, and
damping) and not by boundary conditions (linear system). We can have dependency of external sources if an
external force like vibro-impact and dry friction perturb
the system. In reality, coffee fruits are independent systems which are coupled with other subsystems (pedicel,
peduncle, branches, and so on). It indicates that their
mechanical impedances (natural property of the system
in frequency) are too. In modal analysis theory, if the
fruit is shared with others, it does not affect the natural
frequency values, but it will affect the vibration mode of
the fruit, constrained it, as explained by He et al.21 in the
modal analysis theory.
Applied
force on
the glomerulus is defined
asP = 0:5^i + 0^j + 0^k sin (vt)N and the application is
done in a frequency range 0 400Hz. For the harmonic
stress analysis, the stresses will be determined at the interfaces of the pedicel (inferior and superior, see Figure 6(b))
with the following stress intensity determined by the following expression:
Tinoco and Peña
7
Figure 6. (a) Finite element mesh model for the fruit–peduncle system. (b) Pedicel interfaces.
Figure 7. Groups of three fruits in different ripening stages, scaled fruits: (a) 224, 182, 231. (b) 182, 231, 238. (c) 196, 231, 217.
(d) 189, 231, 224.
si = sp(Max) sp(Min)
ð7Þ
Where sp(Max) the maximum principal stress and sp(Min)
is the minimum principal stress.
It is important to point out that the stresses will be
obtained on the area of the inferior and superior interfaces
in the pedicel as described in Figure 6(b). For the finite
element analysis, the following groups of fruits are considered: group 1 (Figure 7(a)); group 2 (Figure 7(b)); group 3
(Figure 7(c)) and group 4 (Figure 7(d)) in which are
demarked the ripening days on each fruit. The location
and ripening were defined randomly.
For the analysis, a detachment model was considered
with the aim of understanding which fruit will be detached
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Simulation: Transactions of the Society for Modeling and Simulation International 00(0)
Figure 8. Normalized detachment model: Force ratio
determined with data obtained by Carvajal-Herrera et al.16
in the glomerulus in the simulation. For this purpose, data
reported by Carvajal-Herrera et al.16 of detachment forces
in axial tension are considered to establish a detachment
model which is plotted in Figure 8. The model is normalized with respect to the necessary force to detach a ripe
fruit of 231 DAAs. We can see that variations in the
experimental data present a great dispersion, therefore it is
observed that if a glomerulus is excited mechanically, the
detachment forces of non-ripe fruits should not reach 0.7
(70%) of the detachment force of a ripe fruit. This idea is
the main hypothesis used for the analysis in the four
groups of fruits (see Figure 7). A Boltzmann model was
approximated to represent the force ratio, but it is not
important in the analysis.
5. Results and discussion
In this section, we investigate the effects of a harmonic
force applied on each glomerulus. The main objective of
the analysis is to compute the stresses generated at the pedicel interfaces (see Figure 6) in the bandwidth 0 to 400 Hz
to stimulate all fruits. The evaluation of the stresses is an
important characteristic for the selective detachment of
fruits that are completely ripened, fruits of 231 DAAs for
Coffea arábica L. var. Colombia in this case. Figure 9
shows the stress intensities determined in the bonding
areas of the pedicel for the four groups. Stress intensities
correspond to the maximum differences in the principal
stresses and these are computed by ANSYS. Stresses were
determined at the inferior (Figure 9(a)) and superior
(Figure 9(b)) interfaces of the pedicel. We observed that
for each ripening stage, several resonance peaks can be
identified. However, our aim was to choose the higher values of stress for the ripe fruits (231 DAAs) with respect to
other fruits in order to activate the detachment. A
resonance peak is distinguished in 138 Hz for the first
group of fruits, this peak is marked with a black bullet.
The peaks were marked in all cases of identification (all
groups) of resonances for ripe fruits in its higher values of
stresses as seen in Figure 9. In the second group, the resonance peak is identified in 140 Hz; in the third group, the
resonance is 148 Hz and in the fourth is 150 Hz, respectively. We can see that the stresses are higher at the inferior interfaces, where the fruits are bonded to the pedicel
and this happens in all groups of fruits. Observing Figure
9, we can see that in all bandwidths (0–400 Hz) different
peaks appear that correspond to each fruit as well as to the
structure where the pedicel–peduncle is coupled. However,
our interest is focused on selecting resonances of ripe fruits
only. We see that the variation in the identified natural frequencies of ripe fruits for the four groups is 13 Hz (138–
150 Hz). To carry out a stress analysis, an interval of 20
Hz (130–150) will be taken into account in which the maximum values of stresses are chosen numerically, with the
aim to demonstrate that the maximum values are produced
on the ripe fruits (231 DAAs) as shown in Figure 10. In
this figure, the maximum values of stress intensities are
reported in the frequency interval 130–150 Hz.
In Figure 10(a), it is shown that the mean of stress
intensities at the pedicel present higher values in the ripe
fruits than the other fruits. To compare the stress values
with respect to the detachment model (see Figure 8), the
stress values are normalized with respect to those computed for ripe fruits, such as that done with the detachment
model. In Figure 10(b) the comparisons are shown. We
can see that in the group 1, if the ripe fruit is detached
(force ratio 1), the other fruits reach detachment force
ratios between 0.2–0.65 which are not enough to detach
those fruits according to the prescribed model in Figure 8.
In the second group of fruits, force ratio reaches 0.2 for
the fruit 182 DAAs and 0.78 for an overripe fruit 238
DAAs. Results in group 2 mean that only two fruits of the
group are detached (231 and 238), and it indicates that the
unripe fruit does not reach detachment. Force ratios for
the third group show that unripe fruits (196 and 217
DAAs) reach ratios of 0.07–0.15, it means that when a ripe
fruit is detached the forces at the interfaces reach a 15% of
the detachment force of the ripe fruit. For the fourth group,
force ratios achieve a range between 0.15–0.3 in the
unripe fruits of 189 and 224 DAAs. In all cases is demonstrated that the ripe fruits reach detachment forces in the
interval 130–150 Hz and the other unripe fruits achieve a
65% which are insufficient forces to detach the fruits
according to the detachment model obtained experimentally. Therefore, we can say that the ripe fruits can be harvested with vibrations without affecting fruits in an unripe
stage, since fruits in complete ripening stages are dynamically stimulated in their natural frequencies.
In dynamics, for each natural frequency, kinematics is
associated with the whole structure, called a vibration
Tinoco and Peña
Figure 9. Stress intensity in the frequency spectrum. (a) Inferior interface. (b) Superior interface.
9
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Simulation: Transactions of the Society for Modeling and Simulation International 00(0)
Figure 10. (a) Maximum normalized stresses in the frequency spectrum (130–150 Hz). (b) Normalized force ratios with respect to
the determined force in a ripe fruit for each group of fruits.
Tinoco and Peña
11
Figure 11. Vibration mode: (a) Group 1. (b) Group 2. (c) Group 3. (d) Group 4.
mode, which is the vibration shape taken by the structure
in the natural frequency. It is important to point out that
those dynamic characteristics are important to understand
the detachment of the fruits. From the simulations in
ANSYS Workbench 16, the kinematics of the vibration
modes was calculated and this is shown in Figure 11 for
the identified natural frequencies; 138, 140, 148 and 150
Hz of each group of fruits. The motion refers to the displacements that the fruits experiment in the associated vibration mode with each identified natural frequency. In group
1, we see that the ripe fruit presents a bending mode and
the displacement ratio shows that the ripe fruit has higher
amplitude than the other fruits. For group 2, it is seen that
the principal motion is developed in the overripe fruit (238
DAAs). However, the stresses developed at the pedicel are
bigger in the ripe fruit than in the overripe as evidenced in
Figure 10. The most interesting thing is that the unripe
fruit presents small motions and only ripe and overripe
fruits are stimulated dynamically. In group 3, it is seen that
the ripe fruit is excited axially showing that is displaced in
bigger amplitudes with respect to the other two. The
motions of the unripe fruits developed forces much lower
in the pedicel than those of the ripe fruit, as discussed
before. Finally, in group 4 the results are the same as the
other groups, but the displacement of the ripe fruit is different. We observed that the ripe fruit performs a rotational motion with respect to the pedicel increasing the
stresses there, which are bigger that those computed for
the other two fruits. In conclusion, it is analyzed that the
studied frequency interval will contribute to the selective
harvesting which could be advantageous in the moment of
detaching ripe fruits since these are stimulated dynamically and selectively.
6. Conclusions
A finite element harmonic analysis was performed to evaluate the performance of the stresses at the interfaces of the
pedicel–fruit and pedicel–peduncle of Coffea arábica
L. var. Colombia fruits coupled to a glomerulus of three
fruits in the frequency spectrum of 0–400 Hz. We observed
that natural frequencies found between 130–150 Hz
increased the stresses at the interfaces of the ripe fruits in
all groups. Results demonstrated that in all cases, the ripe
fruits can be harvested selectively according to a detachment model applied. In a real context, the fruit systems
could be excited in the desired frequency ranges to only
harvest coffee-ripe fruits.
Acknowledgments
The authors wish to recognize the work carried out by the
mechanical engineering student Mateo Vargas for their help in
this study. Further, to extend a recognition to the department of
Mechanics and Production of Universidad Autónoma de
Manizales (www.autonoma.edu.co) for the time assigned to the
research
12
Simulation: Transactions of the Society for Modeling and Simulation International 00(0)
Declaration of conflicting interest
The authors declare that there is no conflict of interest.
Funding
This research received no specific grant from any funding agency
in the public, commercial, or not-for-profit sectors.
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Author biographies
Hector Andres Tinoco received his BSc in Mechanical
Engineering degree from the Universidad Autonóma de
Manizales, Manizales-Caldas, Colombia, in 2005 and
MSc degree in Mechanical Engineering with specialization in Computational Mechanics from University of
Campinas, Campinas, Brazil, in 2011, respectively. Since
2012, He has been working as an Associate Professor in
the Department of Mechanics and Production at
Universidad Autónoma de Manizales. Currently, he is the
head of the Experimental and Computational Mechanics
Laboratory at the same university. His research interests
include modeling and simulation of multiphysical systems,
optimization applied to structural analysis, structural
health monitoring, mechanical vibrations and energy harvesting applications.
Fabio Marcelo Peña received his BSc in Mechanical
Engineering degree from Universidad Nacional de
Colombia, Bogota, Colombia, in 1990 and his MSc degree
in Materials and Manufacturing Process from Universidad
Nacional de Colombia, Bogota, Colombia, in 2005.
Currently, he is associate professor from 1994 at
Universidad Autónoma de Manizales in which he has
gained a wide experience in the application of the
mechanical design in industrial projects. His research
interests include finite element analysis, mechanical
design and maintenance.