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] 2 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 4 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 6 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 8 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 10 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. References 1. FNC (Federación Nacional de Cafeteros de Colombia). Cafés de origen, http://www.federaciondecafeteros (2017; accessed 16 October 2017). 2. Lopez-Fisco HA, Ramı́rez CA, Oliveros CE, et al. 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Eng Agrı́c 2007; 27: 714–721. 13. Santos FL, De Queiroz DM, Pinto FDADC, et al. Efeito da frequência e amplitude de vibracxão sobre a derric xa de frutos de café. Rev Bras Eng Agrı́c 2010; 14: 425–431. 14. Gaskin E, Martı́nez A and Llanes O. Búsqueda de modos de vibración apropiados para la cosecha selectiva del café. In XII Congreso de Maquinaria y Mecanización Agrı́cola AGROMEC 2007; 16: 1–6. 15. Tinoco HA, Ocampo DA, Peña FM, et al. Finite element modal analysis of the fruit-peduncle of Coffea arábica L. var. Colombia estimating its geometrical and mechanical properties. Comput Electron Agric 2014; 108: 17–27. 16. Carvajal-Herrera JJ, Aristizabal-Torres ID and OliverosTascón CEO. Evaluación de propiedades fı́sicas y mecánicas del fruto de café (Coffea arábica L. var. Colombia) durante su desarrollo y maduración. Dyna 2012; 79(173): 116–124. 17. Riškus A. Approximation of a cubic Bézier curve by circular arcs and vice versa. Inf Technol Control 2006; 35: 1–8. 18. Ciro HJ, Álvarez F and Oliveros CE. Estudio experimental de la dinámica de las vibraciones longitudinales y transversales aplicadas a las ramas de café. Rev Fac Nac Agron 1998; 51: 245–275. 19. Tinoco HA. Modeling elastic and geometric properties of Coffea arábica L. var. Colombia fruits by an experimentalnumerical approach. Int J Fruit Sci 2017; 17: 159–174. 20. Hall AJ, Minchin PE, Clearwater MJ, et al. A biophysical model of kiwifruit (Actinidia deliciosa) berry development. J Exp Botany 2013; 64: 5473–5483. 21. He J and Fu ZF. Modal analysis. Oxford: ButterworthHeinemann, 2001. 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.

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